(Advances in Design and Control) Hassan K Khalil - High-Gain Observers in Nonlinear Feedback Control-Society for Industrial and Applied Mathematics (SIAM) (2017)

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<p>High-Gain Observers</p><p>in Nonlinear</p><p>Feedback Control</p><p>DC31_Khalil_FM_04-06-17.indd 1 4/20/2017 9:37:48 AM</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>Advances in Design and Control</p><p>SIAM’s Advances in Design and Control series consists of texts and monographs dealing with all areas of design</p><p>and control and their applications. Topics of interest include shape optimization, multidisciplinary design, trajectory</p><p>optimization, feedback, and optimal control. The series focuses on the mathematical and computational aspects of</p><p>engineering design and control that are usable in a wide variety of scientific and engineering disciplines.</p><p>Editor-in-Chief</p><p>Ralph C. Smith, North Carolina State University</p><p>Editorial Board</p><p>Series Volumes</p><p>Khalil, Hassan K., High-Gain Observers in Nonlinear Feedback Control</p><p>Bauso, Dario, Game Theory with Engineering Applications</p><p>Corless, M., King, C., Shorten, R., and Wirth, F., AIMD Dynamics and Distributed Resource Allocation</p><p>Walker, Shawn W., The Shapes of Things: A Practical Guide to Differential Geometry and the Shape Derivative</p><p>Michiels, Wim and Niculescu, Silviu-Iulian, Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-</p><p>Based Approach, Second Edition</p><p>Narang-Siddarth, Anshu and Valasek, John, Nonlinear Time Scale Systems in Standard and Nonstandard Forms:</p><p>Analysis and Control</p><p>Bekiaris-Liberis, Nikolaos and Krstic, Miroslav, Nonlinear Control Under Nonconstant Delays</p><p>Osmolovskii, Nikolai P. and Maurer, Helmut, Applications to Regular and Bang-Bang Control: Second-Order Necessary</p><p>and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control</p><p>Biegler, Lorenz T., Campbell, Stephen L., and Mehrmann, Volker, eds., Control and Optimization with Differential-Algebraic</p><p>Constraints</p><p>Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization,</p><p>Second Edition</p><p>Hovakimyan, Naira and Cao, Chengyu, L1 Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation</p><p>Speyer, Jason L. and Jacobson, David H., Primer on Optimal Control Theory</p><p>Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second Edition</p><p>Shima, Tal and Rasmussen, Steven, eds., UAV Cooperative Decision and Control: Challenges and Practical Approaches</p><p>Speyer, Jason L. and Chung, Walter H., Stochastic Processes, Estimation, and Control</p><p>Krstic, Miroslav and Smyshlyaev, Andrey, Boundary Control of PDEs: A Course on Backstepping Designs</p><p>Ito, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and Applications</p><p>Xue, Dingyü, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design with MATLAB</p><p>Hanson, Floyd B., Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation</p><p>Michiels, Wim and Niculescu, Silviu-Iulian, Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based</p><p>Approach</p><p>Ioannou, Petros and Fidan, Barıs, Adaptive Control Tutorial</p><p>Bhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix Problems</p><p>Robinett III, Rush D., Wilson, David G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic Programming for</p><p>Optimization of Dynamical Systems</p><p>Huang, J., Nonlinear Output Regulation: Theory and Applications</p><p>Haslinger, J. and Mäkinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation, and Computation</p><p>Antoulas, Athanasios C., Approximation of Large-Scale Dynamical Systems</p><p>Gunzburger, Max D., Perspectives in Flow Control and Optimization</p><p>Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and Optimization</p><p>Betts, John T., Practical Methods for Optimal Control Using Nonlinear Programming</p><p>El Ghaoui, Laurent and Niculescu, Silviu-Iulian, eds., Advances in Linear Matrix Inequality Methods in Control</p><p>Helton, J. William and James, Matthew R., Extending H∞ Control to Nonlinear Systems: Control of Nonlinear Systems</p><p>to Achieve Performance Objectives</p><p>¸</p><p>Siva Banda, Air Force Research Laboratory</p><p>Stephen L. Campbell, North Carolina State University</p><p>Michel C. Delfour, University of Montreal</p><p>Fariba Fahroo, Air Force Office of Scientific Research</p><p>J. William Helton, University of California, San Diego</p><p>Arthur J. Krener, University of California, Davis</p><p>Kirsten Morris, University of Waterloo</p><p>John Singler, Missouri University of Science and</p><p>Technology</p><p>Stefan Volkwein, Universität Konstanz</p><p>DC31_Khalil_FM_04-06-17.indd 2 4/20/2017 9:37:48 AM</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>Society for Industrial and Applied Mathematics</p><p>Philadelphia</p><p>High-Gain Observers</p><p>in Nonlinear</p><p>Feedback Control</p><p>Hassan K. Khalil</p><p>Michigan State University</p><p>East Lansing, Michigan</p><p>DC31_Khalil_FM_04-06-17.indd 3 4/20/2017 9:37:48 AM</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>is a registered trademark.</p><p>Copyright © 2017 by the Society for Industrial and Applied Mathematics</p><p>10 9 8 7 6 5 4 3 2 1</p><p>All rights reserved. Printed in the United States of America. No part of this book may be</p><p>reproduced, stored, or transmitted in any manner without the written permission of the publisher.</p><p>For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street,</p><p>6th Floor, Philadelphia, PA 19104-2688 USA.</p><p>Trademarked names may be used in this book without the inclusion of a trademark symbol. These</p><p>names are used in an editorial context only; no infringement of trademark is intended.</p><p>MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information,</p><p>please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA,</p><p>508-647-7000, Fax: 508-647-7001, info@mathworks.com, www.mathworks.com.</p><p>Simulink is a registered trademark of The MathWorks, Inc.</p><p>Publisher David Marshall</p><p>Acquisitions Editor Elizabeth Greenspan</p><p>Developmental Editor Gina Rinelli Harris</p><p>Managing Editor Kelly Thomas</p><p>Production Editor Louis R. Primus</p><p>Copy Editor Bruce Owens</p><p>Production Manager Donna Witzleben</p><p>Production Coordinator Cally Shrader</p><p>Compositor Lumina Datamatics</p><p>Graphic Designer Lois Sellers</p><p>Library of Congress Cataloging-in-Publication Data</p><p>Names: Khalil, Hassan K., 1950-</p><p>Title: High-gain observers in nonlinear feedback control / Hassan K. Khalil,</p><p>Michigan State University, East Lansing, Michigan.</p><p>Description: Philadelphia : Society for Industrial and Applied Mathematics,</p><p>2017. | Series: Advances in design and control ; 31 | Includes</p><p>bibliographical references and index.</p><p>Identifiers: LCCN 2017012787 (print) | LCCN 2017016390 (ebook) | ISBN</p><p>9781611974867 (ebook) | ISBN 9781611974850 (print)</p><p>Subjects: LCSH: Observers (Control theory) | Feedback control systems. |</p><p>Stochastic control theory.</p><p>Classification: LCC QA402.3 (ebook) | LCC QA402.3 .K4287 2017 (print) | DDC</p><p>629.8/314--dc23</p><p>LC record available at https://lccn.loc.gov/2017012787</p><p>DC31_Khalil_FM_04-06-17.indd 4 4/20/2017 9:37:48 AM</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>Contents</p><p>Preface vii</p><p>1 Introduction 1</p><p>1.1 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1</p><p>1.2 Motivating Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2</p><p>1.3 Challenges</p><p>with real eigenvalues outperform the one with complex eigenvalues.</p><p>Between these two designs, the one with distinct eigenvalues is slightly better than</p><p>the one with multiple eigenvalues. These observations are typical for observers with</p><p>dimension other than three. 4</p><p>2.4 Reduced-Order Observer</p><p>Reconsider the system (2.12)–(2.15):</p><p>ẇ = f0(w, x, u), (2.37)</p><p>ẋi = xi+1+ψi (x1, . . . , xi , u) for 1≤ i ≤ ρ− 1, (2.38)</p><p>ẋρ =φ(w, x, u), (2.39)</p><p>y = x1. (2.40)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>24 CHAPTER 2. HIGH-GAIN OBSERVERS</p><p>0 0.02 0.04 0.06 0.08 0.1</p><p>−0.4</p><p>−0.2</p><p>0</p><p>0.2</p><p>0.4</p><p>0.6</p><p>0.8</p><p>1</p><p>Time</p><p>x̃</p><p>1</p><p>A</p><p>B</p><p>C</p><p>0 0.02 0.04 0.06 0.08 0.1</p><p>−80</p><p>−60</p><p>−40</p><p>−20</p><p>0</p><p>20</p><p>Time</p><p>x̃</p><p>2</p><p>0 0.02 0.04 0.06 0.08 0.1</p><p>−3000</p><p>−2000</p><p>−1000</p><p>0</p><p>1000</p><p>2000</p><p>Time</p><p>x̃</p><p>3</p><p>Figure 2.1. Simulation of Example 2.1. Design A has three multiple eigenvalues at −1;</p><p>design B has complex eigenvalues on the unit circle in a Butterworth pattern; design C has real</p><p>eigenvalues at −0.5, −1, −1.5.</p><p>The dimension ρ of the standard high-gain observer (2.17)–(2.18) is the same as the</p><p>dimension of the vector x. Since the measured output is x1, it is possible to design</p><p>an observer of dimension ρ− 1 to estimate x2 to xρ. In this section we derive such</p><p>reduced-order observer and calculate an upper bound on the estimation error. The</p><p>derivation procedure is similar to the one used in linear systems.9 We first assume that</p><p>ẏ −ψ1(y, u) = x2 is the measured output and design an observer driven by ẏ, then we</p><p>apply a change of variables to eliminate the dependence on ẏ.</p><p>Let</p><p>ξi = xi+1 for 1≤ i ≤ ρ− 1.</p><p>Then</p><p>ξ̇i = ξi+1+ψi+1(y,ξ1, . . . ,ξi , u) for 1≤ i ≤ ρ− 2, (2.41)</p><p>ξ̇ρ−1 =φ(w, y,ξ1, . . . ,ξρ−1, u), (2.42)</p><p>ẏ −ψ1(y, u) = ξ1, (2.43)</p><p>which is similar to (2.38)–(2.40) except that the dimension of ξ is ρ− 1 instead of ρ.</p><p>A high-gain observer of dimension ρ− 1 is given by</p><p>˙̂</p><p>ξi = ξ̂i+1+ψ</p><p>s</p><p>i+1(y, ξ̂1, . . . , ξ̂i , u)</p><p>+</p><p>βi</p><p>εi</p><p>[ẏ −ψ1(y, u)− ξ̂1] for 1≤ i ≤ ρ− 2, (2.44)</p><p>˙̂</p><p>ξρ−1 =φ0(y, ξ̂1, . . . , ξ̂ρ−1, u)+</p><p>βρ−1</p><p>ερ−1</p><p>[ẏ −ψ1(y, u)− ξ̂1], (2.45)</p><p>9 See, for example, [9, Section 9.3.2] or [126, Chapter 15].</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>2.4. REDUCED-ORDER OBSERVER 25</p><p>where ε is a sufficiently small positive constant, β1 to βρ−1 are chosen such that the</p><p>polynomial</p><p>sρ−1+β1 sρ−2+ · · ·+βρ−2 s +βρ−1 (2.46)</p><p>is Hurwitz, and the functions ψs</p><p>2 to ψs</p><p>ρ−1 and φ0 are defined in Section 2.2. In partic-</p><p>ular, they satisfy (2.20), (2.21), and (2.23). The only difference is that the y argument</p><p>of ψs</p><p>i is not saturated. With the change of variables</p><p>zi = ξ̂i −</p><p>βi</p><p>εi</p><p>y, 1≤ i ≤ ρ− 1, (2.47)</p><p>we arrive at the reduced-order observer</p><p>żi = zi+1+</p><p>βi+1</p><p>εi+1</p><p>y</p><p>+ψs</p><p>i+1</p><p>�</p><p>y, z1+</p><p>β1</p><p>ε</p><p>y, . . . , zi +</p><p>βi</p><p>εi</p><p>y, u</p><p>�</p><p>−</p><p>βi</p><p>εi</p><p>�</p><p>z1+</p><p>β1</p><p>ε</p><p>y +ψ1(y, u)</p><p>�</p><p>for 1≤ i ≤ ρ− 2, (2.48)</p><p>żρ−1 =φ0</p><p>�</p><p>y, z1+</p><p>β1</p><p>ε</p><p>y, . . . , zρ−1+</p><p>βρ−1</p><p>ερ−1</p><p>y, u</p><p>�</p><p>−</p><p>βρ−1</p><p>ερ−1</p><p>�</p><p>z1+</p><p>β1</p><p>ε</p><p>y +ψ1(y, u)</p><p>�</p><p>, (2.49)</p><p>x̂i = zi−1+</p><p>βi−1</p><p>εi−1</p><p>y for 2≤ i ≤ ρ. (2.50)</p><p>The bound on the estimation error is given in the following theorem.</p><p>Theorem 2.2. Under the stated assumptions, there is ε∗ ∈ (0,1] such that for 0 0 is the solution of the Lyapunov equation P E +</p><p>ET P =−I . The derivative V̇ satisfies the inequality</p><p>εV̇ ≤−‖η‖2+ 2k1ε‖P‖ ‖η‖</p><p>2+ 2ε‖P‖M‖η‖.</p><p>Proceeding as in the proof Theorem 2.1, it can be shown that</p><p>‖η(t )‖ ≤max</p><p>¦</p><p>b e−at/ε‖η(0)‖,εcM</p><p>©</p><p>for some positive constants a, b , c . From (2.52) we see that ‖η(0)‖ ≤ k2/ε</p><p>ρ−1, where</p><p>k2 > 0 depends on the initial conditions x(0) and z(0). From (2.47) and (2.52) we have</p><p>|ξi − ξ̂i |=</p><p>�</p><p>�</p><p>�ερ−1−iηi</p><p>�</p><p>�</p><p>�≤max</p><p>�</p><p>b k2</p><p>εi</p><p>e−at/ε,ερ−i cM</p><p>�</p><p>,</p><p>which yields (2.51) since xi − x̂i = ξi−1− ξ̂i−1. 2</p><p>While the bound on the estimation error for the reduced-order observer in Theo-</p><p>rem 2.2 takes the same form as the one for the standard observer in Theorem 2.1, the</p><p>standard observer has better performance, especially in the presence of measurement</p><p>noise. When both observers are linear, the frequency response of the standard ob-</p><p>server rolls off at high frequency, unlike the frequency response of the reduced-order</p><p>observer, which stays flat at high frequency. This point is illustrated in Example 1.4</p><p>of Section 1.2.</p><p>2.5 Multi-output Systems</p><p>The high-gain observer of Section 2.2 can be extended to classes of multi-output sys-</p><p>tems. In the simplest such class, the system is represented by</p><p>ẇ = f0(w, x, u), (2.54)</p><p>ẋ i</p><p>j = x i</p><p>j+1 for 1≤ j ≤ ρi − 1, 1≤ i ≤ q , (2.55)</p><p>ẋ i</p><p>ρi</p><p>=φi (w, x, u) for 1≤ i ≤ q , (2.56)</p><p>yi = x i</p><p>1 for 1≤ i ≤ q , (2.57)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>2.5. MULTI-OUTPUT SYSTEMS 27</p><p>-</p><p>-</p><p>x</p><p>u</p><p>- w</p><p>-</p><p>-</p><p>-</p><p>x</p><p>w</p><p>u</p><p>-φ1 ∫</p><p>-</p><p>x1ρ1 ∫</p><p>-</p><p>x11 = y1</p><p>-</p><p>-</p><p>-</p><p>-</p><p>x</p><p>w</p><p>u</p><p>-φ2 ∫</p><p>-</p><p>x2ρ2 ∫</p><p>-</p><p>x21 = y2</p><p>-</p><p>-</p><p>-</p><p>-</p><p>x</p><p>w</p><p>u</p><p>-</p><p>φq ∫</p><p>-</p><p>xqρq ∫</p><p>-</p><p>xq1 = yq</p><p>-</p><p>1</p><p>Figure 2.2. Block diagram representation of the multi-output system (2.54)–(2.57).</p><p>where</p><p>x i = col(x i</p><p>1 , x i</p><p>2 , . . . , x i</p><p>ρi</p><p>),</p><p>x = col(x1, x2, . . . , xq ) ∈ Rρ,</p><p>ρ= ρ1+ · · ·+ρq .</p><p>The functions f0,φ1,. . . ,φq are locally Lipschitz in their arguments, and w(t ) ∈W ⊂</p><p>R`, x(t ) ∈X ⊂ Rρ, and u(t ) ∈U ⊂ Rm for all t ≥ 0, for some compact sets W , X , and</p><p>U . A block diagram of the system is shown in Figure 2.2. It has q chains of integrators</p><p>with ρi integrators in each chain whose states are x i</p><p>1 , x i</p><p>2 , . . . x i</p><p>ρi</p><p>. The measured outputs</p><p>are the states of the first integrators of these chains. The q chains are coupled by the</p><p>input to the last integrator in each chain, the function φi , which could depend on all</p><p>integrator states as well as the state w and the input u. Without loss of generality, we</p><p>assume that each chain has at least two integrators, that is, ρi ≥ 2; a chain of a single</p><p>integrator can be included in the equation ẇ = f0.</p><p>The model (2.54)–(2.57) arises naturally in mechanical and electromechanical sys-</p><p>tems, where displacement variables are measured while their derivatives (velocities,</p><p>accelerations, etc.) are not measured. For example, the m-link robot manipulator is</p><p>modeled by [146].</p><p>M (q)q̈ +C (q , q̇)q̇ +Dq̇ + g (q) = u,</p><p>where q is an m-dimensional</p><p>vector of generalized coordinates representing joint posi-</p><p>tions, u an m-dimensional control (torque) input, and M (q) a symmetric positive defi-</p><p>nite inertia matrix. The terms C (q , q̇)q̇ , Dq̇ , and g (q) account for centrifugal/Coriolis</p><p>forces, viscous damping, and gravity, respectively. If the measured variables are the</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>28 CHAPTER 2. HIGH-GAIN OBSERVERS</p><p>joint positions, this model takes the foregoing form with x i</p><p>1 = qi , x i</p><p>2 = q̇i , ρi = 2, for</p><p>i = 1, . . . , m, and the functions φi are given by</p><p>col(φ1, . . . ,φm) =M−1[u −C q̇ −Dq̇ − g ].</p><p>The multivariable normal form of a nonlinear system having a well-defined (vec-</p><p>tor) relative degree is a special case of (2.54)–(2.57) [64, Section 5.1]. An m-input–</p><p>m-output nonlinear system of the form</p><p>χ̇ = f (χ )+</p><p>m</p><p>∑</p><p>i=1</p><p>gi (χ )ui , yi = hi (χ ) for 1≤ i ≤ m</p><p>has vector relative degree {ρ1,ρ2, . . . ,ρm} in an open setR if, for all χ ∈R ,</p><p>Lg j</p><p>Lk−1</p><p>f hi (χ ) = 0 for 1≤ k ≤ ρi − 1</p><p>for all 1≤ i ≤ m and 1≤ j ≤ m, and the m×m matrix</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>Lg1</p><p>Lρ1−1</p><p>f h1 · · · · · · Lgm</p><p>Lρ1−1</p><p>f h1</p><p>Lg1</p><p>Lρ2−1</p><p>f h2 · · · · · · Lgm</p><p>Lρ2−1</p><p>f h2</p><p>...</p><p>...</p><p>...</p><p>...</p><p>Lg1</p><p>Lρm−1</p><p>f hm · · · · · · Lgm</p><p>Lρm−1</p><p>f hm</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>is nonsingular.</p><p>A high-gain observer for the system (2.54)–(2.57) is taken as</p><p>˙̂x i</p><p>j = x̂ i</p><p>j+1+</p><p>αi</p><p>j</p><p>ε j</p><p>(yi − x̂ i</p><p>1) for 1≤ j ≤ ρi − 1, (2.58)</p><p>˙̂x i</p><p>ρi</p><p>=φi0(x̂, u)+</p><p>αi</p><p>ρi</p><p>ερi</p><p>(yi − x̂ i</p><p>1) (2.59)</p><p>for 1≤ i ≤ q , where ε is a sufficiently small positive constant and the positive constants</p><p>αi</p><p>j are chosen such that the polynomial</p><p>sρi +αi</p><p>1 sρi−1+ · · ·+αi</p><p>ρi−1 s +αi</p><p>ρi</p><p>is Hurwitz for each i = 1, . . . , q . The functionφi0(x, u) is a nominal model ofφi (w, x,</p><p>u), which is required to satisfy the inequality</p><p>‖φi (w, x, u)−φi0(z, u)‖ ≤ Li‖x − z‖+Mi (2.60)</p><p>for all w ∈W , x ∈X , z ∈ Rρ, and u ∈U .</p><p>Theorem 2.3. Under the stated assumptions, there is ε∗ ∈ (0,1] such that for 0</p><p>of Theorem 2.2, a similar</p><p>separation result holds for the reduced-order observer of Section 2.4. The proofs of</p><p>such results are almost the same as the proof of Theorem 3.1 of Section 3.1.</p><p>10Throughout the book, sliding mode control is implemented with the signum function sgn(s) replaced</p><p>by sat(s/µ). By abuse of notation, we refer to such control as sliding mode control.</p><p>31</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>32 CHAPTER 3. STABILIZATION AND TRACKING</p><p>For minimum-phase nonlinear systems in the normal form, the tracking problem</p><p>reduces to a stabilization problem when the state variables are chosen as the tracking</p><p>error and its derivatives. Section 3.3 presents the design of output feedback tracking</p><p>control using the same separation approach that is delineated in Sections 3.1 and 3.2.</p><p>3.1 Separation Principle</p><p>3.1.1 Problem Statement</p><p>Consider the multi-input–multi-output nonlinear system</p><p>ẇ = f0(w, x, u), (3.1)</p><p>ẋ =Ax +Bφ(w, x, u), (3.2)</p><p>y =C x, (3.3)</p><p>z =ψ(w, x), (3.4)</p><p>where u ∈ Rm is the control input, y ∈ Rq and z ∈ Rs are measured outputs, and</p><p>w ∈ R` and x ∈ Rρ constitute the state vector. The ρ×ρ matrix A, the ρ× q matrix</p><p>B , and the q ×ρmatrix C , given by</p><p>A= block diag[A1, . . . ,Aq], Ai =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 · · · · · · 0</p><p>0 0 1 · · · 0</p><p>...</p><p>...</p><p>0 · · · · · · 0 1</p><p>0 · · · · · · · · · 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>ρi×ρi</p><p>,</p><p>B = block diag[B1, . . . ,Bq], Bi =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>...</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>ρi×1</p><p>,</p><p>C = block diag[C1, . . . ,Cq], Ci =</p><p>�</p><p>1 0 · · · · · · 0</p><p>�</p><p>1×ρi</p><p>,</p><p>where ρ = ρ1 + · · ·+ ρq , represent q chains of integrators. Equations (3.2) and (3.3)</p><p>are the same as (2.55) to (2.57) of Section 2.5 but are written in a more compact form.</p><p>The functions f0, φ, and ψ are locally Lipschitz in their arguments for (w, x, u) ∈</p><p>Dw×Dx×Rm , where Dw ⊂ R` and Dx ⊂ Rρ are domains that contain their respective</p><p>origins. Moreover, f0(0,0,0) = 0, φ(0,0,0) = 0, and ψ(0,0) = 0. Our goal is to design</p><p>an output feedback controller to stabilize the origin.</p><p>The model (3.1)–(3.4) arises naturally in mechanical and electromechanical sys-</p><p>tems, where displacement variables are measured, while their derivatives (velocities,</p><p>accelerations, etc.) are not measured. The multivariable normal form of a nonlinear</p><p>system having a well-defined (vector) relative degree is a special case of (3.1)–(3.3) [64,</p><p>Section 5.1]. If y is the only measured variable, (3.4) is dropped. However, in many</p><p>problems, we can measure some state variables in addition to those at the end of the</p><p>chains of integrators. For example, a magnetic-levitation system is modeled by [80,</p><p>Appendix A.8].</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.1. SEPARATION PRINCIPLE 33</p><p>ẋ1 = x2,</p><p>ẋ2 =−b x2+ 1−</p><p>4c x2</p><p>3</p><p>(1+ x1)2</p><p>,</p><p>ẋ3 =</p><p>1</p><p>T (x1)</p><p>�</p><p>−x3+ u +</p><p>βx2x3</p><p>(1+ x1)2</p><p>�</p><p>,</p><p>where the dimensionless variables x1, x2, and x3, are the position of a magnetic ball,</p><p>its velocity, and the electric current in an electromagnet, respectively. Typically, we</p><p>measure the ball position x1 and the current x3. The model fits the form (3.1)–(3.4)</p><p>with (x1, x2) as the x component and x3 as the w component. The measured outputs</p><p>are y = x1 and z = x3.</p><p>Another source for the model (3.1)–(3.4) where (3.4) is significant arises in systems</p><p>in which the dynamics are extended by adding integrators. Consider a single-input–</p><p>single-output nonlinear system represented by the nth-order differential equation</p><p>y (n) = f</p><p>�</p><p>y, y (1), . . . , y (n−1), v, v (1), . . . , v (m−1), v (m)</p><p>�</p><p>,</p><p>where v is the input, y is the output, and f (·) is a sufficiently smooth function in a do-</p><p>main of interest. This nonlinear input–output model reduces to the transfer function</p><p>H (s) =</p><p>bm s m + bm−1 s m−1+ · · ·+ b0</p><p>s n + an−1 s n−1+ · · ·+ a0</p><p>, bm 6= 0,</p><p>for linear systems. We extend the dynamics of the system by adding a series of m</p><p>integrators at the input side and define u = v (m) as the control input of the extended</p><p>system. The dimension of the extended system is n+m. Taking the state variables as</p><p>w =</p><p></p><p></p><p></p><p></p><p></p><p>v</p><p>v (1)</p><p>...</p><p>v (m−1)</p><p></p><p></p><p></p><p></p><p></p><p>, x =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>y</p><p>y (1)</p><p>...</p><p>...</p><p>y (n−1)</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>,</p><p>the state model of the extended system takes the form (3.1)–(3.4) with</p><p>ẇ =Au w +Bu u,</p><p>ẋ =Ax +Bφ(w, x, u),</p><p>y =C x,</p><p>z = w,</p><p>where</p><p>Au =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 · · · · · · 0</p><p>0 0 1 · · · 0</p><p>...</p><p>...</p><p>0 · · · · · · 0 1</p><p>0 · · · · · · · · · 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>m×m</p><p>, Bu =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>...</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>m×1</p><p>,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>34 CHAPTER 3. STABILIZATION AND TRACKING</p><p>A=</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 · · · · · · 0</p><p>0 0 1 · · · 0</p><p>...</p><p>...</p><p>0 · · · · · · 0 1</p><p>0 · · · · · · · · · 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>n×n</p><p>, B =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>...</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>n×1</p><p>, C =</p><p>�</p><p>1 0 · · · · · · 0</p><p>�</p><p>1×n ,</p><p>φ(w, x, u) = f (x1, x2, . . . , xn , w1, w2, . . . , wm , u) .</p><p>The measured variables are y and the whole vector w.</p><p>It is seen in Section 2.2 that a high-gain observer can be designed for the system</p><p>ẇ = f0(w, x, u),</p><p>ẋi = xi+1+ψi (x1, . . . , xi , u) for 1≤ i ≤ ρ− 1,</p><p>ẋρ =φ(w, x, u),</p><p>y = x1</p><p>if the functions ψ1 to ψρ−1 are known. In feedback control we can allow uncertainty</p><p>in these functions by applying a change of variables that transforms the system into</p><p>the single-output case of (3.1)–(3.4). Suppose ψ1 to ψρ−1 are sufficiently smooth. The</p><p>change of variables</p><p>z1 = x1, z2 = ż1, . . . , zρ = żρ−1</p><p>transforms the system into the form</p><p>ẇ = f̄0(w, z, u),</p><p>żi = zi+1 for 1≤ i ≤ ρ− 1,</p><p>żρ = φ̄(w, z, u),</p><p>y = z1.</p><p>It can be verified that</p><p>zi = xi + ψ̄i (x1, . . . , xi−1) for 2≤ i ≤ ρ</p><p>for some functions ψ̄i . The change of variables is a global diffeomorphism. By design-</p><p>ing the output feedback controller in the z-coordinates, there is no need to know the</p><p>change of variables. The functions f̄0 and φ̄ depend on the uncertain functions ψ1 to</p><p>ψρ−1, but, as seen in Section 2.2, the high-gain observer does not require knowledge</p><p>of f̄0 and is robust with respect to φ̄. Therefore, as long as a robust state feedback</p><p>controller can be designed to tolerate the uncertainty in f̄0 and φ̄, the output feedback</p><p>controller will be robust with respect to these two functions.</p><p>3.1.2 Main Result</p><p>We use a two-step approach to design the output feedback controller. First, a partial</p><p>state feedback controller that uses measurements of x and z is designed to asymptoti-</p><p>cally stabilize the origin. Then a high-gain observer is used to estimate x from y. The</p><p>state feedback controller is allowed to be a dynamical system of the form</p><p>ϑ̇ = Γ (ϑ, x, z), u = γ (ϑ, x, z), (3.5)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.1. SEPARATION PRINCIPLE 35</p><p>where γ and Γ are locally Lipschitz functions in their arguments over the domain of</p><p>interest and globally bounded functions of x. Moreover, γ (0,0,0) = 0 and Γ (0,0,0) =</p><p>0. A static state feedback controller u = γ (x, z) will be viewed as a special case of</p><p>the foregoing equation by dropping the ϑ̇-equation. For convenience, we write the</p><p>closed-loop system under state feedback as</p><p>χ̇ = f (χ ), (3.6)</p><p>where χ = col(w, x,ϑ). The output feedback controller is taken as</p><p>ϑ̇ = Γ (ϑ, x̂, z), u = γ (ϑ, x̂, z), (3.7)</p><p>where x̂ is generated by the high-gain observer</p><p>˙̂x =Ax̂ +Bφ0(z, x̂, u)+H (y −C x̂). (3.8)</p><p>The observer</p><p>gain H is chosen as</p><p>H = block diag[H1, . . . , Hq], Hi =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>αi</p><p>1/ε</p><p>αi</p><p>2/ε</p><p>2</p><p>...</p><p>αi</p><p>ρi−1/ε</p><p>ρi−1</p><p>αi</p><p>ρi</p><p>/ερi</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>ρi×1</p><p>, (3.9)</p><p>where ε is a small positive constant and the positive constants αi</p><p>j are chosen such that</p><p>the polynomial</p><p>sρi +αi</p><p>1 sρi−1+ · · ·+αi</p><p>ρi−1 s +αi</p><p>ρi</p><p>is Hurwitz for i = 1, . . . , q . The functionφ0(z, x, u) is a nominal model ofφ(w, x, u),</p><p>which is required to be locally Lipschitz in its arguments over the domain of interest</p><p>and globally bounded in x. Moreover, φ0(0,0,0) = 0.</p><p>Theorem 3.1. Consider the closed-loop system of the plant (3.1)–(3.4) and the output</p><p>feedback controller (3.7)–(3.8). Suppose the origin of (3.6) is asymptotically stable andR</p><p>is its region of attraction. Let S be any compact set in the interior of R and Q be any</p><p>compact subset of Rρ. Suppose (χ (0), x̂(0)) ∈S ×Q. Then</p><p>• there exists ε∗1 > 0 such that for every 0 0, there exist ε∗2 > 0 and T2 > 0, both dependent on µ, such that for</p><p>every 0 0, there exists ε∗3 > 0, dependent onµ, such that for every 0 0 such that for every 0 0 such that for every 0 0, {V (χ )≤ c} is a compact subset ofR .</p><p>Boundedness: Let S be any compact set in the interior ofR . Choose positive con-</p><p>stants b and c such that c > b >maxχ∈S V (χ ). Then</p><p>S ⊂Ωb = {V (χ )≤ b} ⊂Ωc = {V (χ )≤ c} ⊂R .</p><p>For the boundary-layer system, the Lyapunov function W (η) = ηT P0η, where P0 is</p><p>the positive definite solution of the Lyapunov equation P0A0+AT</p><p>0 P0 =−I , satisfies</p><p>λmin(P0)‖η‖</p><p>2 ≤W (η)≤ λmax(P0)‖η‖</p><p>2,</p><p>∂W</p><p>∂ η</p><p>A0η=−‖η‖</p><p>2.</p><p>Let Σ = {W (η) ≤ %ε2} and Λ = Ωc ×Σ. Due to the global boundedness of F and ∆</p><p>in x̂, there are positive constants k1 and k2 independent of ε such that</p><p>‖F (χ , D(ε)η)‖ ≤ k1, ‖∆(χ , D(ε)η)‖ ≤ k2</p><p>for all χ ∈Ωc and η ∈ Rρ. Moreover, for any 0</p><p>Ẇ ≤− 1</p><p>2λmax(P0)ε</p><p>W for W (η)≥ %ε2.</p><p>By the comparison lemma [78, Lemma 3.4],</p><p>W (η(t ))≤W (η(0))exp(−σ1 t/ε)≤</p><p>σ2</p><p>ε2(ρmax−1)</p><p>exp (−σ1 t/ε), (3.16)</p><p>where σ1 = 1/(2λmax(P0)) and σ2 = k2λmax(P0). The right-hand side of (3.16) equals</p><p>%ε2 when</p><p>σ2</p><p>ε2(ρmax−1)</p><p>exp (−σ1 t/ε) = %ε2 ⇔ t =</p><p>ε</p><p>σ1</p><p>ln</p><p>�</p><p>σ2</p><p>% ε2ρmax</p><p>�</p><p>.</p><p>Choose ε2 > 0 small enough that</p><p>T (ε) def=</p><p>ε</p><p>σ1</p><p>ln</p><p>�</p><p>σ2</p><p>% ε2ρmax</p><p>�</p><p>≤ 1</p><p>2 T0</p><p>for all 0 0 such that for all ‖χ ‖ ≤ r and ‖η‖ ≤ r , ‖∆(χ , D(ε)η)‖</p><p>≤ L3‖η‖ for some L3 > 0. On the same set, there are positive constants L4 and L5 such</p><p>that ‖∂ V /∂ χ ‖ ≤ L4 and ‖F (χ , D(ε)η)− F (χ , 0)‖ ≤ L5‖η‖. Consider the composite</p><p>Lyapunov function V1(χ ,η) =V (χ )+</p><p>p</p><p>W (η) :</p><p>V̇1 =</p><p>∂ V1</p><p>∂ χ</p><p>F (χ , D(ε)η)+</p><p>1</p><p>2</p><p>p</p><p>W</p><p>�</p><p>−1</p><p>ε</p><p>‖η‖2+ 2ηT P0B∆</p><p>�</p><p>≤</p><p>∂ V1</p><p>∂ χ</p><p>F (χ , 0)+</p><p>∂ V1</p><p>∂ χ</p><p>[F (χ , D(ε)η)− F (χ , 0)]</p><p>− 1</p><p>2ε</p><p>p</p><p>λmax(P0)</p><p>‖η‖+</p><p>‖P0B‖L3</p><p>p</p><p>λmin(P0)</p><p>‖η‖</p><p>≤−U (χ )− 1</p><p>4ε</p><p>p</p><p>λmax(P0)</p><p>‖η‖</p><p>−</p><p>�</p><p>1</p><p>4ε</p><p>p</p><p>λmax(P0)</p><p>− L4L5−</p><p>‖P0B‖L3</p><p>p</p><p>λmin(P0)</p><p>�</p><p>‖η‖.</p><p>Choose ε7 > 0 such that</p><p>1</p><p>4ε7</p><p>p</p><p>λmax(P0)</p><p>− L4L5−</p><p>‖P0B‖L3</p><p>p</p><p>λmin(P0)</p><p>> 0.</p><p>Then, for 0 0 such that for every</p><p>0 0 such that for every 0 0 such</p><p>that for every 0</p><p>AND TRACKING</p><p>1680 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 1999</p><p>Fig. 1. Recovery of region of attraction:�� = 0 (solid), �</p><p>� = 0:007 (dashed),�� = 0:057 (dash-dotted), and�� = 0:082 (dotted).</p><p>as in our previous work [1], [14]–[17], [20], [21], [23], but to</p><p>any kind of system that fits the model (1)–(4).</p><p>A. Example 1</p><p>We consider a second-order system having an exponentially</p><p>unstable mode, together with a bounded linear controller that</p><p>achieves a finite region of attraction. The system is</p><p>(78)</p><p>(79)</p><p>where the control is .</p><p>We consider a full-order high-gain linear observer (i.e.,</p><p>) with . In this example we show how</p><p>the output feedback controller recovers the region of attraction</p><p>achieved under state feedback.</p><p>Fig. 1 shows the region of attraction under state feedback</p><p>control, in addition to three compact subsets that are recovered</p><p>using the high-gain observer. In each case the compact subset</p><p>is specified, then a design parameter is found through</p><p>multiple simulations at different points of the subset such that</p><p>for every the output feedback controller is able to</p><p>recover the given subset, i.e., it is a part of the region of</p><p>attraction of the new closed-loop system. The boundis tight</p><p>in the sense that for there is a part of the given set that</p><p>is not included in the region of attraction. The bounds’s</p><p>for these subsets are and , respectively,</p><p>starting from the smallest subset. Notice that the bigger the</p><p>subset, the smaller the bound. In all cases we take .</p><p>B. Example 2—Inverted Pendulum</p><p>We consider the inverted pendulum-on-a-cart problem given</p><p>in [3]. The system consists of an inverted pendulum mounted</p><p>on a cart free to move on a horizontal plane. The equations</p><p>of motion are given by</p><p>(80)</p><p>(81)</p><p>(82)</p><p>(83)</p><p>where is the mass of the cart, is the mass of the ball</p><p>attached to the free end of the pendulum,is the length of</p><p>the pendulum, is the gravitational acceleration, is the</p><p>coefficient of viscous friction opposing the cart’s motion,</p><p>is the cart’s displacement, is the cart’s velocity, is the</p><p>angle that the pendulum makes with the vertical, andis</p><p>the pendulum’s angular velocity. The values of the different</p><p>parameters of the model are , Kg,</p><p>m/s , m, and Kg/s. The nominal</p><p>value of is . It is shown in [3] that the state feedback control</p><p>(84)</p><p>Figure 3.1. Recovery of the region of attraction in Example 3.1: ε? = 0.007 (dashed),</p><p>ε? = 0.057 (dash-dotted), and ε? = 0.082 (dotted).</p><p>where the estimates x̂1 and x̂2 are provided by the high-gain observer</p><p>˙̂x1 = x̂2+(1/ε)(y − x̂1),</p><p>˙̂x2 = (1/ε</p><p>2)(y − x̂1).</p><p>We show how the output feedback controller recovers the region of attraction achieved</p><p>under state feedback. Figure 3.1 shows the region of attraction under state feedback</p><p>control (solid), in addition to three compact subsets that are recovered using the high-</p><p>gain observer. In each case the compact subset is specified, then a design parameter ε?</p><p>is found through multiple simulations at different points of the subset such that for</p><p>every ε ≤ ε? the output feedback controller is able to recover the given subset; i.e., it</p><p>is a subset of the region of attraction of the closed-loop system under output feedback.</p><p>The bound ε? is tight in the sense that for ε > ε? there is a part of the given set that is</p><p>not included in the region of attraction. The bounds ε?’s for these subsets are 0.082,</p><p>0.057, and 0.007, respectively, starting from the smallest subset. Notice that the bigger</p><p>the subset, the smaller the bound ε?. In all cases we take x̂(0) = 0.</p><p>Example 3.2. Consider the inverted pendulum-on-a-cart system, modeled by [80,</p><p>Appendix A.11].</p><p>ẋ1 = x2,</p><p>ẋ2 =</p><p>1</p><p>∆(x1)</p><p>�</p><p>(m+M )m g L sin x1−mLcos x1(u +mLx2</p><p>2 sin x1− k x4)</p><p>�</p><p>,</p><p>ẋ3 = x4,</p><p>ẋ4 =</p><p>1</p><p>∆(x1)</p><p>�</p><p>−m2L2 g sin x1 cos x1+(J +mL2)(u +mLx2</p><p>2 sin x1− k x4)</p><p>�</p><p>,</p><p>where m is the mass of the pendulum, M the mass of the cart, L the distance from the</p><p>center of gravity to the pivot, J the moment of inertia of the pendulum with respect to</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.1. SEPARATION PRINCIPLE 43</p><p>the center of gravity, k a friction coefficient, x1 the angular rotation of the pendulum</p><p>from the upward vertical position (measured clockwise), x3 the displacement of the</p><p>pivot, g the acceleration due to gravity, and</p><p>∆(x1) = (J +mL2)(m+M )−m2L2 cos2 x1 ≥ (J +mL2)M +mJ > 0.</p><p>The model parameters are</p><p>m = 0.1, M = 1, k = 0.1, J = 0.008, L= 0.5, g = 9.81.</p><p>The measured outputs are x1 and x3, and the goal is to stabilize the origin x = 0.</p><p>We start by the design of state feedback control. Assuming the domain of interest</p><p>is limited to |x1|</p><p>the initial</p><p>conditions between the position and its estimate. It also shows the control saturation</p><p>during the peaking interval. This is done for a linear observer with ε = 0.001 and</p><p>initial conditions x(0) = col(1,0,1,0) and x̂(0) = 0. Figure 3.3 shows how the output</p><p>feedback controller recovers the trajectories achieved under state feedback. We use a</p><p>linear high-gain observer with two values of ε: 0.03 and 0.005. The initial conditions</p><p>13See [92] for the details of the singular perturbation analysis.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.1. SEPARATION PRINCIPLE 45</p><p>0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01</p><p>0</p><p>200</p><p>400</p><p>600</p><p>x</p><p>2</p><p>a</p><p>n</p><p>d</p><p>x̂</p><p>2</p><p>Time</p><p>0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01</p><p>0</p><p>200</p><p>400</p><p>600</p><p>x</p><p>4</p><p>a</p><p>n</p><p>d</p><p>x̂</p><p>4</p><p>Time</p><p>0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2</p><p>−100</p><p>−50</p><p>0</p><p>50</p><p>100</p><p>u</p><p>Time</p><p>Figure 3.2. Simulation of Example 3.2. Peaking of the velocity estimates and saturation</p><p>of the control: the states x2 and x4 (solid) and their estimates x̂2 and x̂4 (dashed).</p><p>0 1 2 3</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>x</p><p>1</p><p>Time</p><p>0 0.5 1 1.5 2</p><p>−8</p><p>−6</p><p>−4</p><p>−2</p><p>0</p><p>2</p><p>4</p><p>x</p><p>2</p><p>Time</p><p>0 1 2 3 4 5</p><p>−1</p><p>0</p><p>1</p><p>2</p><p>3</p><p>4</p><p>x</p><p>3</p><p>Time</p><p>0 1 2 3 4 5</p><p>−2</p><p>0</p><p>2</p><p>4</p><p>6</p><p>8</p><p>10</p><p>x</p><p>4</p><p>Time</p><p>Figure 3.3. Simulation of Example 3.2. Trajectory convergence: state feedback (solid);</p><p>output feedback with ε= 0.03 (dashed) and ε= 0.005 (dotted).</p><p>are x(0) = col(1,0,1,0) and x̂(0) = 0. Intuitively we expect that a nonlinear observer</p><p>that includes a model of the system’s nonlinearities would outperform a linear one</p><p>when the model is accurate and ε is not too small. Figure 3.4 shows that this intu-</p><p>ition is justified. In Figures 3.4(a) and (b) we compare between a linear observer and a</p><p>nonlinear one with ε= 0.03. The advantage of the nonlinear observer over the linear</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>46 CHAPTER 3. STABILIZATION AND TRACKING</p><p>0 1 2 3</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>x</p><p>1</p><p>Time</p><p>(a)</p><p>0 1 2 3</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>x</p><p>1</p><p>Time</p><p>(c)</p><p>0 1 2 3 4 5</p><p>−1</p><p>0</p><p>1</p><p>2</p><p>3</p><p>4</p><p>x</p><p>3</p><p>Time</p><p>(b)</p><p>0 1 2 3 4 5</p><p>−1</p><p>0</p><p>1</p><p>2</p><p>3</p><p>4</p><p>x</p><p>3</p><p>Time</p><p>(d)</p><p>Figure 3.4. Simulation of Example 3.2. Effect of nonlinearity in the observer: state</p><p>feedback (solid); with linear observer (dashed); with nonlinear observer (dotted). ε= 0.03 in (a) and</p><p>(b) and ε= 0.001 in (c) and (d).</p><p>one is clear. In Figures 3.4(c) and (d) we repeat the comparison with ε = 0.001. This</p><p>time we cannot differentiate between the performance of the two observers. As ε de-</p><p>creases, the effect of model uncertainty diminishes. In these two simulations the initial</p><p>conditions are x(0) = col(1,0,1,0) and x̂(0) = 0.</p><p>Example 3.3. Consider the simplified PVTOL (Planar Vertical Take-off and Landing)</p><p>aircraft, modeled by 14</p><p>ẋ1 = x2, ẋ2 = −u1 sin x5+µu2 cos x5,</p><p>ẋ3 = x4, ẋ4 = u1 cos x5+µu2 sin x5− g ,</p><p>ẋ5 = x6, ẋ6 = λu2,</p><p>where x1, x3, and x5 are the horizontal coordinate, the vertical coordinate, and the in-</p><p>clination of the aircraft, respectively. A linearizing dynamic state feedback controller</p><p>is given by</p><p>ẋ7 = x8, ẋ8 = −ν1 sin x5+ ν2 cos x5+ x2</p><p>6 x7,</p><p>u1 = x7+</p><p>µ</p><p>λ</p><p>x2</p><p>6 ,</p><p>u2 =</p><p>1</p><p>λx7</p><p>(−ν1 cos x5− ν2 sin x5− 2x6x8),</p><p>where ν1 and ν2 are the control inputs to the linearized system. The closed-loop system</p><p>under the linearizing feedback has an equilibrium point at xeq = col(x̄1, 0, x̄3, 0, 0, 0,</p><p>14The model and the state feedback design are taken from [107].</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.1. SEPARATION PRINCIPLE 47</p><p>g , 0), ū = col(g , 0), and ν̄ = col(0,0). The linearizing effect of this dynamic controller</p><p>can be seen by applying the change of variables</p><p>χ1 = x1− x̄1−</p><p>µ</p><p>λ sin x5, χ2 = x2−</p><p>µ</p><p>λ x6 cos x5,</p><p>χ3 = −x7 sin x5, χ4 = −x8 sin x5− x6x7 cos x5,</p><p>χ5 = x3− x̄3−µ/λ+</p><p>µ</p><p>λ cos x5, χ6 = x4−</p><p>µ</p><p>λ x6 sin x5,</p><p>χ7 = x7 cos x5− g , χ8 = x8 cos x5− x6x7 sin x5</p><p>to obtain</p><p>χ̇1 = χ2, χ̇2 = χ3, χ̇3 = χ4, χ̇4 = ν1,</p><p>χ̇5 = χ6, χ̇6 = χ7, χ̇7 = χ8, χ̇8 = ν2.</p><p>In the new coordinates, the equilibrium point is at the originχ = 0. It can be stabilized</p><p>by a linear state feedback control of the form</p><p>ν1 =−(k1χ1+ k2χ2+ k3χ3+ k4χ4), (3.24)</p><p>ν2 =−(k1χ5+ k2χ6+ k3χ7+ k4χ8). (3.25)</p><p>For the purpose of simulation, we take x̄1 = x̄3 = 2, g = λ = 1, and µ = 0.5. The</p><p>feedback gains k1 = 24, k2 = 50, k3 = 35, and k4 = 10 assign the closed-loop eigenvalues</p><p>at −1, −2, −3, and −4.</p><p>Now suppose we only measure the position variables x1, x3, and x5 and set y1 = x1,</p><p>y2 = x3, and y3 = x5. We want to design an observer to estimate the velocity variables</p><p>x2, x4, and x6. Noting that the nonlinear functions sin x5 and cos x5 depend only on</p><p>the measured output y3, the system takes the form</p><p>ẋ =Ax +Bφ(u, y), y =C x,</p><p>where</p><p>A= block diag[A1,A2,A3], Ai =</p><p>�</p><p>0 1</p><p>0 0</p><p>�</p><p>for i = 1,2,3,</p><p>B = block diag[B1,B2,B3], Bi =</p><p>�</p><p>0</p><p>1</p><p>�</p><p>for i = 1,2,3,</p><p>C = block diag[C1,C2,C3], Ci =</p><p>�</p><p>1 0</p><p>�</p><p>for i = 1,2,3,</p><p>φ(u, y) = col(−u1 sin y3+µu2 cos y3, u1 cos y3+µu2 sin y3− g , λu2).</p><p>If the nonlinear function φ(u, y) is exactly known, we can design an observer that</p><p>yields linear error dynamics [80, Section 11.3]. Assuming perfect knowledge of the</p><p>parameters λ and µ, we design the reduced-order observer</p><p>ż1 = −(z1+ y1)− u1 sin y3+µu2 cos y3, x̂2 = z1+ y1,</p><p>ż2 = −(z2+ y2)+ u1 sin y3+µu2 sin y3− g , x̂4 = z2+ y2,</p><p>ż3 = −(z3+ y3)+λu2, x̂6 = z3+ y3,</p><p>whose eigenvalues are assigned at−1. The estimation errors x̃2 = x2− x̂2, x̃4 = x4− x̂4,</p><p>and x̃6 = x6 − x̂6 satisfy the equations ˙̃xi = −x̃i , for i = 2,4,6. The output feedback</p><p>controller uses the estimates x̂2, x̂4, and x̂6 to replace x2, x4, and x6, respectively. In</p><p>view of the linear error dynamics, the output feedback control stabilizes the system</p><p>at the desired equilibrium point. We refer to this control as the linear-error-observer</p><p>controller.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>48 CHAPTER 3. STABILIZATION AND TRACKING</p><p>On the other hand, if there is uncertainty in modeling the nonlinearity φ(u, y)</p><p>or if we want the trajectories under output feedback to come arbitrarily close to the</p><p>trajectories under state feedback, we use a high-gain observer. We design the reduced-</p><p>order high-gain observer</p><p>ż1 = −(z1+ y1)/ε− u1 sin y3+ µ̂u2 cos y3, x̂2 = (z1+ y1)/ε,</p><p>ż2 = −(z2+ y2)/ε+ u1 sin y3+ µ̂u2 sin y3, x̂4 = (z2+ y2)/ε,</p><p>ż3 = −(z3+ y3)/ε+ λ̂u2, x̂6 = (z3+ y3)/ε,</p><p>where λ̂ and µ̂ are nominal values of λ andµ. The observer eigenvalues are assigned at</p><p>−1/ε. In the simulation, we take λ̂= λ= 1 and µ̂=µ= 0.5. We saturate the dynamic</p><p>feedback controller as follows: ẋ7 = 20 sat(·/20), ẋ8 = 200 sat(·/200), u1 = 40 sat(·/40),</p><p>and u2 = 200 sat(·/200). These saturation levels have been determined from extensive</p><p>simulations done to see the maximal values that the state trajectories would take when</p><p>the initial state is in a region of interest around xeq . The model nonlinearities are</p><p>naturally globally bounded, so we do not need to saturate the nonlinearities in the</p><p>observer. We refer to this control as the high-gain-observer controller. The difference</p><p>between the two controllers is in the choice of the observer eigenvalues, −1/ε versus</p><p>−1, and the use of saturation in the high-gain-observer controller.</p><p>To compare the performance of the two controllers, we perform simulations with</p><p>x(0) = col(1,0,1,0,0,0,1,0) and z(0) = 0. Figure 3.5 shows that as ε decreases, the</p><p>trajectories under output feedback approach</p><p>the ones under state feedback. The figure</p><p>shows only x1 to x4, but the recovery holds for the entire state x1 to x8. Figure 3.6</p><p>illustrates that the trajectory recovery property does not hold for the linear-error ob-</p><p>server controller. It might appear that if the observer eigenvalues of the linear-error</p><p>observer are chosen as large as in the high-gain observer, the trajectory recovery prop-</p><p>0 1 2 3 4 5</p><p>0.8</p><p>1</p><p>1.2</p><p>1.4</p><p>1.6</p><p>1.8</p><p>2</p><p>Time</p><p>x</p><p>1</p><p>0 1 2 3</p><p>−2</p><p>−1</p><p>0</p><p>1</p><p>2</p><p>3</p><p>4</p><p>5</p><p>Time</p><p>x</p><p>2</p><p>0 1 2 3 4 5</p><p>0.8</p><p>1</p><p>1.2</p><p>1.4</p><p>1.6</p><p>1.8</p><p>2</p><p>Time</p><p>x</p><p>3</p><p>0 1 2 3</p><p>−1.5</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>Time</p><p>x</p><p>4</p><p>Figure 3.5. Simulation of Example 3.3. Trajectory convergence: state feedback (solid);</p><p>high-gain-observer output feedback with ε= 0.01 (dash-dotted) and ε= 0.002 (dashed).</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.1. SEPARATION PRINCIPLE 49</p><p>0 2 4 6 8</p><p>0.6</p><p>0.8</p><p>1</p><p>1.2</p><p>1.4</p><p>1.6</p><p>1.8</p><p>2</p><p>Time</p><p>x</p><p>1</p><p>0 2 4 6 8</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>Time</p><p>x</p><p>2</p><p>0 2 4 6 8</p><p>0.8</p><p>1</p><p>1.2</p><p>1.4</p><p>1.6</p><p>1.8</p><p>2</p><p>Time</p><p>x</p><p>3</p><p>0 2 4 6 8 10</p><p>−0.4</p><p>−0.2</p><p>0</p><p>0.2</p><p>0.4</p><p>0.6</p><p>Time</p><p>x</p><p>4</p><p>Figure 3.6. Simulation of Example 3.3. High-gain versus linear-error observer: state feed-</p><p>back (solid); high-gain-observer controller with ε = 0.002 (dashed); linear-error-observer controller</p><p>(dash-dotted).</p><p>0 2 4 6</p><p>0</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>Time</p><p>x</p><p>1</p><p>0 1 2 3</p><p>−10</p><p>−8</p><p>−6</p><p>−4</p><p>−2</p><p>0</p><p>2</p><p>Time</p><p>x</p><p>2</p><p>0 2 4 6</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>Time</p><p>x</p><p>3</p><p>0 1 2 3</p><p>−10</p><p>−8</p><p>−6</p><p>−4</p><p>−2</p><p>0</p><p>2</p><p>Time</p><p>x</p><p>4</p><p>Figure 3.7. Simulation of Example 3.3. High-gain versus linear-error observer: state feed-</p><p>back (solid), high-gain-observer controller with ε = 0.002 (dashed); linear-error-observer controller</p><p>with eigenvalues at −500 (dash-dotted).</p><p>erty will be realized. Figure 3.7 shows that this is not the case. In the simulation, the</p><p>eigenvalues of the linear-error observer are assigned at −500, which coincide with the</p><p>eigenvalues of the high-gain observer with ε= 0.002. This simulation emphasizes the</p><p>fact that trajectory recovery is due to the combined effect of fast observer dynamics</p><p>and global boundedness of the control input and the controller nonlinearities.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>50 CHAPTER 3. STABILIZATION AND TRACKING</p><p>3.2 Robust Stabilization of Minimum-Phase Systems</p><p>In this section we consider the normal form of a single-input–single-output minimum-</p><p>phase nonlinear system that has relative degree ρ≥ 1. Our goal is to design an output</p><p>feedback controller to stabilize the origin in the presence of matched model uncer-</p><p>tainty and time-varying disturbance. We start by considering the relative-degree-one</p><p>case where a sliding mode controller can be designed without observers. Then, we</p><p>show how to use a high-gain observer to reduce a relative-degree-higher-than-one sys-</p><p>tem to a relative-degree-one system.</p><p>3.2.1 Relative-Degree-One System</p><p>When ρ= 1, the normal form is given by</p><p>η̇= f0(η, y), ẏ = a(η, y)+ b (η, y)u +δ(t ,η, y, u), (3.26)</p><p>where η ∈ Rn−1, u ∈ R, and y ∈ R. We assume that the functions f0, a, and b are locally</p><p>Lipschitz, f0(0,0) = 0, a(0,0) = 0, and the origin of η̇ = f0(η, 0) is asymptotically</p><p>stable.15 Equation (3.26) includes a matched time-varying disturbance δ, which is</p><p>assumed to be piecewise continuous and bounded in t and locally Lipschitz in (η, y, u)</p><p>uniformly in t . Let D ⊂ Rn be a domain that contains the origin and assume that for</p><p>all (η, y) ∈ D , b (η, y) ≥ b0 > 0. Because the origin of η̇ = f0(η, 0) is asymptotically</p><p>stable, the sliding surface is taken as y = 0. We assume that there is a (continuously</p><p>differentiable) Lyapunov function V (η) that satisfies the inequalities</p><p>φ1(‖η‖)≤V (η)≤φ2(‖η‖) (3.27)</p><p>∂ V</p><p>∂ η</p><p>f0(η, y)≤−φ3(‖η‖) ∀ ‖η‖ ≥φ4(|y|) (3.28)</p><p>for all (η, y) ∈ D , where φ1 to φ4 are class K functions. Inequality (3.28) implies</p><p>regional input-to-state stability of the system η̇= f0(η, y) with input y.16 We take the</p><p>control as</p><p>u =ψ(y)+ v,</p><p>where ψ is locally Lipschitz with ψ(0) = 0. The function ψ could be zero or could be</p><p>chosen to cancel known terms on the right-hand side of ẏ, which depend only on y.</p><p>Thus,</p><p>y ẏ = y [a(η, y)+ b (η, y)ψ(y)+δ(t ,η, y,ψ(y)+ v)+ b (η, y)v]</p><p>≤ b (η, y)</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>a(η, y)+ b (η, y)ψ(y)+δ(t ,η, y,ψ(y)+ v)</p><p>b (η, y)</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>|y|+ yv</p><p>�</p><p>.</p><p>We assume that we know a locally Lipschitz function%(y)≥ 0 and a constantκ0∈ [0,1)</p><p>such that the inequalities</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>a(η, y)+ b (η, y)ψ(y)+δ(t ,η, y,ψ(y))</p><p>b (η, y)</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤ %(y) (3.29)</p><p>15By definition, (3.26) is minimum phase when the origin of the zero dynamics η̇= f0(η, 0) is asymptoti-</p><p>cally stable [64].</p><p>16See [80, Chapter 4] for the definition of regional input-to-state stability.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.2. ROBUST STABILIZATION OF MINIMUM-PHASE SYSTEMS 51</p><p>and</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>δ(t ,η, y,ψ(y)+ v)−δ(t ,η, y,ψ(y))</p><p>b (η, y)</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤κ0 |v | (3.30)</p><p>hold for all t ≥ 0, (η, y) ∈D , and v ∈ R. Taking</p><p>v =−β(y) sat</p><p>�</p><p>y</p><p>µ</p><p>�</p><p>, (3.31)</p><p>where β(y) is a locally Lipschitz function that satisfies</p><p>β(y)≥</p><p>%(y)</p><p>1− κ0</p><p>+β0, (3.32)</p><p>for some β0 > 0, ensures that</p><p>y ẏ ≤−β0(1− κ0)b (η, y)|y| ≤ −β0b0(1− κ0)|y| for |y| ≥µ. (3.33)</p><p>Define the classK function φ by φ(r ) = φ2(φ4(r )) and suppose µ, c > µ, and c0 ≥</p><p>φ(c) are chosen such that the set</p><p>Ω= {V (η)≤ c0}× {|y| ≤ c} (3.34)</p><p>is compact and contained in D . Using (3.28) and (3.33) it can be shown that V̇ ≤ 0</p><p>on the boundary V = c0 and y ẏ ≤ 0 on the boundary |y|= c . Therefore, the set Ω is</p><p>positively invariant. By the same inequalities, the set</p><p>Ωµ = {V (η)≤φ(µ)}× {|y| ≤µ} (3.35)</p><p>is positively invariant. Inside Ω, (3.33) shows that for |y| ≥µ,</p><p>d</p><p>d t</p><p>|y|= d</p><p>d t</p><p>p</p><p>y2 =</p><p>y ẏ</p><p>|y|</p><p>≤ −β0b0(1− κ0).</p><p>By integration,</p><p>|y(t )| ≤ |y(t0)| −β0b0(1− κ0)(t − t0)</p><p>for t ≥ t0. Since the right-hand side of the preceding inequality reaches zero in finite</p><p>time, we conclude that the trajectory enters the set {|y| ≤ µ} in finite time. Once it</p><p>is in, it cannot leave because y ẏ 1 is given by</p><p>η̇= f0(η,ξ ), (3.36)</p><p>ξ̇i = ξi+1</p><p>for 1≤ i ≤ ρ− 1, (3.37)</p><p>ξ̇ρ = a(η,ξ )+ b (η,ξ )u +δ(t ,η,ξ , u), (3.38)</p><p>y = ξ1, (3.39)</p><p>where η ∈ Rn−ρ, ξ = col(ξ1,ξ2, . . . ,ξρ), u ∈ R, and y ∈ R. Let D ⊂ Rn be a domain</p><p>that contains the origin and assume that for all (η,ξ ) ∈ D , u ∈ R, and t ≥ 0, the</p><p>functions f0, a, and b are locally Lipschitz, f (0,0) = 0, a(0,0) = 0, the matched dis-</p><p>turbance δ is piecewise continuous and bounded in t and locally Lipschitz in (η,ξ , u)</p><p>uniformly in t , and the origin of η̇= f0(η, 0) is asymptotically stable.</p><p>From the properties of high-gain observers, presented in Section 2.2, and the sepa-</p><p>ration principle of Section 3.1, we expect that if we can design a partial state feedback</p><p>controller, in terms of ξ , which stabilizes or practically stabilizes the origin, we will be</p><p>able to recover its performance by using a high-gain observer that estimates ξ . Based</p><p>on this expectation we proceed to design a controller assuming that ξ is measured.</p><p>With measurement of ξ , we can convert the relative-degree-ρ system into a relative-</p><p>degree-one system with respect to the output</p><p>s = k1ξ1+ k2ξ2+ · · ·+ kρ−1ξρ−1+ ξρ. (3.40)</p><p>The normal form of this system is given by</p><p>ż = f̄0(z, s), ṡ = ā(z, s)+ b̄ (z, s)u + δ̄(t , z, s , u), (3.41)</p><p>where</p><p>z =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>η</p><p>ξ1</p><p>...</p><p>ξρ−2</p><p>ξρ−1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, f̄0(z, s) =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>f0(η,ξ )</p><p>ξ2</p><p>...</p><p>ξρ−1</p><p>ξρ</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, ā(z, s) =</p><p>ρ−1</p><p>∑</p><p>i=1</p><p>kiξi+1+ a(η,ξ ),</p><p>b̄ (z, s) = b (η,ξ ), δ̄(t , z, s , u) = δ(t ,η,ξ , u),</p><p>with ξρ = s −</p><p>∑ρ−1</p><p>i=1 kiξi . The functions f̄0, ā, b̄ , and δ̄ inherit the properties of</p><p>f0, a, b , and δ, respectively. In particular, b̄ (z, s) ≥ b0 > 0 for all (η,ξ ) ∈ D . The</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.2. ROBUST STABILIZATION OF MINIMUM-PHASE SYSTEMS 53</p><p>zero dynamics of (3.41) are ż = f̄0(z, 0). This system can be written as the cascade</p><p>connection</p><p>η̇= f0(η,ξ )</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>ξρ=−</p><p>∑ρ−1</p><p>i=1 kiξi</p><p>, ζ̇ = F ζ , (3.42)</p><p>where</p><p>ζ =</p><p></p><p></p><p></p><p></p><p></p><p>ξ1</p><p>ξ2</p><p>...</p><p>ξρ−1</p><p></p><p></p><p></p><p></p><p></p><p>, F =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 0 · · · 0</p><p>0 0 1 · · · 0</p><p>...</p><p>...</p><p>0 · · · 0 1</p><p>−k1 −k2 · · · −kρ−2 −kρ−1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>.</p><p>Because the origin η= 0 of η̇= f0(η, 0) is asymptotically stable, the origin z = 0 of the</p><p>cascade connection (3.42) will be asymptotically stable if the matrix F is Hurwitz [80,</p><p>Appendix C.1]. Therefore, k1 to kρ−1 in (3.40) are chosen such that the polynomial</p><p>λρ−1+ kρ−1λ</p><p>ρ−2+ · · ·+ k2λ+ k1 (3.43)</p><p>is Hurwitz.</p><p>In view of the converse Lyapunov theorem for asymptotic stability [78, Theorem</p><p>4.17] and the local Lipschitz property of f̄0(z, s) with respect to s , it is reasonable</p><p>to assume that there is a (continuously differentiable) Lyapunov function V (z) that</p><p>satisfies the inequalities</p><p>φ1(‖z‖)≤V (z)≤φ2(‖z‖), (3.44)</p><p>∂ V</p><p>∂ η</p><p>f̄0(z, s)≤−φ3(‖z‖) ∀ ‖z‖ ≥φ4(|s |) (3.45)</p><p>for all (η,ξ ) ∈ D , where φ1 to φ4 are class K functions. We note that if ρ = n, the</p><p>zero dynamics of (3.41) will be ζ̇ = F ζ .</p><p>Thus, we have converted a relative-degree-ρ system into a relative-degree-one sys-</p><p>tem that satisfies the assumptions of Section 3.2.1. We proceed to design a sliding mode</p><p>controller as in Section 3.2.1, with the exception that the functions ψ, %, and β will</p><p>be allowed to depend on the whole vector ξ and not only the output s since we are</p><p>working under the assumption that ξ is measured. Let</p><p>u =ψ(ξ )+ v</p><p>for some locally Lipschitz function ψ, with ψ(0) = 0, and suppose we know a locally</p><p>Lipschitz function %(ξ )≥ 0 and a constant κ0∈ [0,1) such that the inequalities</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>∑ρ−1</p><p>i=1 kiξi+1+ a(η,ξ )+ b (η,ξ )ψ(ξ )+δ(t ,η,ξ ,ψ(ξ ))</p><p>b (η,ξ )</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤ %(ξ ) (3.46)</p><p>and</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>δ(t ,η,ξ ,ψ(ξ )+ v)−δ(t ,η,ξ ,ψ(ξ ))</p><p>b (η,ξ )</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤κ0 |v | (3.47)</p><p>hold for all t ≥ 0, (η,ξ ) ∈D , and v ∈ R. Similar to (3.31), take v as</p><p>v =−β(ξ ) sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>54 CHAPTER 3. STABILIZATION AND TRACKING</p><p>where β(ξ ) is a locally Lipschitz function that satisfies</p><p>β(ξ )≥</p><p>%(ξ )</p><p>1− κ0</p><p>+β0 (3.48)</p><p>for some β0 > 0. The overall control is</p><p>u =ψ(ξ )−β(ξ ) sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>def= γ (ξ ). (3.49)</p><p>Define the classK function φ by φ(r ) = φ2(φ4(r )) and suppose µ, c > µ, and c0 ≥</p><p>φ(c) are chosen such that the set</p><p>Ω= {V (z)≤ c0}× {|s | ≤ c} with c0 ≥φ(c) (3.50)</p><p>is compact and contained in D . As in Section 3.2.1, we conclude that Ω is positively</p><p>invariant, and for all initial states in Ω, the trajectories enter the positively invariant</p><p>set</p><p>Ωµ = {V (z)≤φ(µ)}× {|s | ≤µ} (3.51)</p><p>in finite time.</p><p>If δ(t , 0, 0, 0) = 0, the origin of η̇ = f0(η, 0) is exponentially stable, and f0(η, 0)</p><p>is continuously differentiable in some neighborhood of η = 0, then for sufficiently</p><p>small µ the origin (z = 0, s = 0) of the closed-loop system is exponentially stable</p><p>and Ω is a subset of its region of attraction. This can be shown as follows. By the</p><p>converse Lyapunov theorem [78, Theorem 4.14], there is a Lyapunov function V0(η)</p><p>that satisfies the inequalities</p><p>d1‖η‖</p><p>2 ≤V0(η)≤ d2‖η‖</p><p>2,</p><p>∂ V0</p><p>∂ η</p><p>f0(η, 0)≤−d3‖η‖</p><p>2,</p><p>∂ V0</p><p>∂ η</p><p>≤ d4‖η‖</p><p>in some neighborhood Nη of η = 0, where d1 to d4 are positive constants. By the</p><p>local Lipschitz property of f0, there is a neighborhood N of (η,ξ ) = (0,0) such that</p><p>‖ f0(η,ξ )− f0(η, 0)‖ ≤ L‖ξ ‖. Chooseµ small enough thatΩµ ⊂Nη×N . Consider the</p><p>composite Lyapunov function</p><p>V = αV0(η)+ ζ</p><p>T Pζ + 1</p><p>2 s2, (3.52)</p><p>where P = P T > 0 is the solution of the Lyapunov equation P F + F T P =−I and α is</p><p>a positive constant to be determined. Inside Ωµ, it can be shown that V̇ ≤−Y T QY ,</p><p>where</p><p>Y =</p><p></p><p></p><p>‖η‖</p><p>‖ζ ‖</p><p>|s |</p><p></p><p> , Q =</p><p></p><p></p><p>α`1 −α`2 −(α`3+ `4)</p><p>−α`2 1 −`5</p><p>−(α`3+ `4) −`5 (`6/µ− `7)</p><p></p><p></p><p>for some positive constants `1 to `7. Choose α max</p><p>Ω</p><p>{|ξi |}, 1≤ i ≤ ρ,</p><p>and take ψs andβs as the functions ψ andβwith ξi replaced Mi sat(ξi/Mi ). Alterna-</p><p>tively, let</p><p>Mψ >max</p><p>Ω</p><p>{|ψ(ξ )|}, Mβ >max</p><p>Ω</p><p>{|β(ξ )|},</p><p>and take ψs (ξ ) = Mψ sat(ψ(ξ )/Mψ) and βs (ξ ) = Mβ sat(β(ξ )/Mβ). In either case,</p><p>the control is taken as</p><p>u =ψs (ξ )−βs (ξ ) sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>.</p><p>We may also saturate the control signal itself. Let</p><p>Mu >max</p><p>Ω</p><p>{|ψ(ξ )−β(ξ ) sat(s/µ)|}</p><p>and take the control as</p><p>u =Mu sat</p><p>�</p><p>ψ(ξ )−β(ξ ) sat(s/µ)</p><p>Mu</p><p>�</p><p>.</p><p>The high-gain observer</p><p>˙̂</p><p>ξi = ξ̂i+1+</p><p>αi</p><p>εi</p><p>(y − ξ̂1), 1≤ i ≤ ρ− 1, (3.53)</p><p>˙̂</p><p>ξρ = a0(ξ̂ )+ b0(ξ̂ )u +</p><p>αρ</p><p>ερ</p><p>(y − ξ̂1) (3.54)</p><p>is used to estimate ξ by ξ̂ , where ε is a small positive constant; α1 to αρ are chosen</p><p>such that the polynomial</p><p>sρ+α1 sρ−1+ · · ·+αρ−1 s +αρ (3.55)</p><p>is Hurwitz; a0(ξ ) and b0(ξ ) are locally Lipschitz, globally bounded functions of ξ ,</p><p>which serve as nominal models of a(η,ξ ) and b (η,ξ ), respectively; and a0(0) = 0. The</p><p>functions a0 and b0 are not allowed</p><p>to depend on η because it is not estimated by the</p><p>observer (3.53)–(3.54). The choice a0 = b0 = 0 results in a linear observer. This choice</p><p>is typically used when no good models of a and b are available or when it is desired to</p><p>use a linear observer. The output feedback controller is taken as</p><p>u = γs (ξ̂ ), (3.56)</p><p>where γs (ξ̂ ) is given by</p><p>ψs (ξ̂ )−βs (ξ̂ ) sat( ŝ/µ) or Mu sat</p><p>ψ(ξ̂ )−β(ξ̂ ) sat( ŝ/µ)</p><p>Mu</p><p>!</p><p>and</p><p>ŝ =</p><p>ρ−1</p><p>∑</p><p>i=1</p><p>ki ξ̂i + ξ̂ρ.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>56 CHAPTER 3. STABILIZATION AND TRACKING</p><p>The properties of the output feedback controller are stated in the following</p><p>theorems.</p><p>Theorem 3.2. Consider the system (3.36)–(3.39) and let k1 to kρ−1 be chosen such that</p><p>the polynomial (3.43) is Hurwitz. Suppose there exist V (z), %(ξ ), and κ0, which satisfy</p><p>(3.44) to (3.47), and β is chosen to satisfy (3.48). Let Ω and Ωµ be defined by (3.50) and</p><p>(3.51), respectively. Consider the high-gain observer (3.53)–(3.54) and the output feedback</p><p>controller (3.56). LetΩ0 be a compact set in the interior ofΩ andQ be a compact subset of</p><p>Rρ. Suppose the initial states satisfy (η(0),ξ (0)) ∈Ω0 and ξ̂ (0) ∈Q. Then there exists ε∗,</p><p>dependent on µ, such that for all ε ∈ (0,ε∗), the states (η(t ),ξ (t ), ξ̂ (t )) of the closed-loop</p><p>system are bounded for all t ≥ 0 and there is a finite time T , dependent on µ, such that</p><p>(η(t ),ξ (t )) ∈ Ωµ for all t ≥ T . Moreover, if (ηr (t ),ξr (t )) is the state of the closed-loop</p><p>system under the state feedback controller (3.49) with ηr (0) = η(0) and ξr (0) = ξ (0), then</p><p>given any λ > 0, there exists ε∗∗ > 0, dependent on µ and λ, such that for all ε ∈ (0,ε∗∗),</p><p>‖η(t )−ηr (t )‖ ≤ λ and ‖ξ (t )− ξr (t )‖ ≤ λ ∀ t ∈ [0,T ]. (3.57)</p><p>3</p><p>Theorem 3.3. Suppose all the assumptions of Theorems 3.2 are satisfied, the origin of η̇=</p><p>f0(η, 0) is exponentially stable, f0(η, 0) is continuously differentiable in some neighborhood</p><p>η= 0, and δ(t , 0, 0, 0) = 0. Then, there exists µ∗ > 0, and for each µ ∈ (0,µ∗) there exists</p><p>ε∗ > 0, dependent on µ, such that for all ε ∈ (0,ε∗), the origin of the closed-loop system</p><p>under the output feedback controller (3.56) is exponentially stable and Ω0×Q is a subset</p><p>of its region of attraction. 3</p><p>Theorems 3.2 and 3.3 show that the output feedback controller (3.56) recovers the</p><p>stabilization or practical stabilization properties of the state feedback controller (3.49)</p><p>for sufficiently small ε. It also recovers its transient behavior, as shown by (3.57).</p><p>We note that the inequalities of (3.57) will hold for all t ≥ 0 when µ is chosen small</p><p>enough. This is so because both (η(t ),ξ (t )) and (ηr (t ),ξr (t )) enter Ωµ in finite time.</p><p>By choosing µ small enough, we can ensure that ‖η(t )‖ ≤ λ/2 and ‖ηr (t )‖ ≤ λ/2 for</p><p>all t ≥ T for some T > 0. Hence, ‖η(t )−ηr (t )‖ ≤ λ for all t ≥ T . A similar argument</p><p>holds for ξ (t )− ξr (t ).</p><p>Proof of Theorem 3.2: Write the trajectory under state feedback as χr = col(ηr ,ξr )</p><p>and the one under output feedback as χ = col(η,ξ ). As in the proof of Theorem 3.1,</p><p>represent the dynamics of the observer using the scaled estimation errors</p><p>ϕi =</p><p>ξi − ξ̂i</p><p>ερ−i</p><p>, 1≤ i ≤ ρ, (3.58)</p><p>to obtain</p><p>εϕ̇ =A0ϕ+ εB0∆(t ,χ , ξ̂ ),</p><p>where</p><p>A0 =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>−α1 1 0 · · · 0</p><p>−α2 0 1 · · · 0</p><p>...</p><p>. . .</p><p>...</p><p>−αρ−1 0 1</p><p>−αρ 0 · · · · · · 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, B0 =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>...</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.2. ROBUST STABILIZATION OF MINIMUM-PHASE SYSTEMS 57</p><p>and</p><p>∆(t ,χ , ξ̂ ) = a(η,ξ )− a0(ξ̂ )+ [b (η,ξ )− b0(ξ̂ )]γs (ξ̂ )+δ(t ,η,ξ ,γs (ξ̂ )).</p><p>The matrix A0 is Hurwitz by design. Because a0(ξ̂ ), b0(ξ̂ ), and γs (ξ̂ ) are globally</p><p>bounded in ξ̂ ,</p><p>|∆(t ,χ , ξ̂ )| ≤ L1 ∀ χ ∈Ω, ξ̂ ∈ Rρ, t ≥ 0, (3.59)</p><p>where throughout the proof Li , for i = 1,2, . . ., denote positive constants independent</p><p>of ε. Let P be the positive definite solution of the Lyapunov equation PA0+AT</p><p>0 P =−I ,</p><p>W (ϕ) = ϕT Pϕ, and Σ = {W (ϕ) ≤ kε2}. The first step of the proof is to show that</p><p>the constant k > 0 in the definition of Σ can be chosen such that, for sufficiently small</p><p>ε, the set Ω×Σ is positively invariant; that is, χ (t0) ∈ Ω and ϕ(t0) ∈ Σ imply that</p><p>χ (t ) ∈Ω and ϕ(t ) ∈Σ for all t ≥ t0. Using (3.59), it can be shown that, for all χ ∈Ω,</p><p>εẆ =−ϕTϕ+ 2εϕT PB0∆ ≤ −‖ϕ‖</p><p>2+ 2εL1‖PB0‖ ‖ϕ‖</p><p>≤− 1</p><p>2‖ϕ‖</p><p>2 ∀ ‖ϕ‖ ≥ 4εL1‖PB0‖.</p><p>Using the inequalities</p><p>λmin(P )‖ϕ‖</p><p>2 ≤ ϕT Pϕ ≤ λmax(P )‖ϕ‖</p><p>2,</p><p>we arrive at</p><p>εẆ ≤−σW ∀W ≥ ε2W0, (3.60)</p><p>where σ = 1/(2λmax(P )) and W0 = λmax(P )(4L1‖PB0‖)2. Taking k =W0 shows that</p><p>ϕ(t ) cannot leave Σ because Ẇ is negative on its boundary. On the other hand, for</p><p>ϕ ∈Σ, ‖ξ−ξ̂ ‖ ≤ L2ε. Using the Lipschitz property ofβ,ψ, and sat(·), it can be shown</p><p>that ‖γs (ξ )−γs (ξ̂ )‖ ≤ L3ε/µ, whereµ appears because of the function sat(s/µ). Since</p><p>γs (ξ ) = γ (ξ ) in Ω, we have</p><p>‖γ (ξ )− γs (ξ̂ )‖ ≤</p><p>εL3</p><p>µ</p><p>. (3.61)</p><p>Using (3.41), (3.46), (3.47), and (3.61), it can be shown that, for |s | ≥µ,</p><p>s ṡ = s{ā(z, s)+ b̄ (z, s)γ (ξ )+ δ̄(t , z, s ,γ (ξ ))+ b̄ (z, s)[γs (ξ̂ )− γ (ξ )]</p><p>+ δ̄(t , z, s ,γs (ξ̂ ))− δ̄(t , z, s ,γs (ξ ))}</p><p>≤ b (η,ξ ) [%(ξ )+ κ0 β(ξ )−β(ξ )+ L4ε/µ] |s |</p><p>≤ b (η,ξ ) [−β0(1− κ0)+ L4ε/µ] |s |.</p><p>Taking ε≤µβ0(1− κ0)/(2L4) yields</p><p>s ṡ ≤− 1</p><p>2 b0β0(1− κ0)|s |.</p><p>With this inequality, the analysis of Section 3.2.1 carries over to show that the trajec-</p><p>tory (η(t ),ξ (t )) cannot leave Ω and enters Ωµ in finite time.</p><p>The second step of the proof is to show that for all χ (0) ∈ Ω0 and ξ̂ (0) ∈ Q, the</p><p>trajectory (χ (t ),ϕ(t )) enters Ω×Σ within a time interval [0,τ(ε)], where τ(ε)→ 0</p><p>as ε→ 0. Due to the scaling (3.58), the initial condition ϕ(0) could be of the order of</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>58 CHAPTER 3. STABILIZATION AND TRACKING</p><p>1/ερ−1. Because Ω0 is in the interior of Ω and the control γs (ξ̂ ) is globally bounded,</p><p>there is time T1 > 0, independent of ε, such that χ (t ) ∈Ω for t ∈ [0,T1]. During this</p><p>time interval, (3.59) and consequently (3.60) hold. It follows that [80, Theorem 4.5]</p><p>W (t )≤max</p><p>¦</p><p>e−σ t/εW (0),ε2W0</p><p>©</p><p>≤max</p><p>�</p><p>e−σ t/εL5</p><p>ε2(ρ−1)</p><p>,ε2W0</p><p>�</p><p>,</p><p>which shows that ϕ(t ) enters Σ within the time interval [0,τ(ε)], where</p><p>τ(ε) =</p><p>ε</p><p>σ</p><p>ln</p><p>�</p><p>L5</p><p>W0ε</p><p>2ρ</p><p>�</p><p>.</p><p>By l’Hôpital’s rule it can be verified that limε→0 τ(ε) = 0. By choosing ε small enough,</p><p>we can ensure that τ(ε)</p><p>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12</p><p>1.4 Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14</p><p>2 High-Gain Observers 17</p><p>2.1 Class of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 17</p><p>2.2 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18</p><p>2.3 Lyapunov and Riccati Equation Designs . . . . . . . . . . . . . . . . . 21</p><p>2.4 Reduced-Order Observer . . . . . . . . . . . . . . . . . . . . . . . . . . 23</p><p>2.5 Multi-output Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26</p><p>2.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30</p><p>3 Stabilization and Tracking 31</p><p>3.1 Separation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32</p><p>3.2 Robust Stabilization of Minimum-Phase Systems . . . . . . . . . . . 50</p><p>3.3 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60</p><p>3.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70</p><p>4 Adaptive Control 73</p><p>4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73</p><p>4.2 State Feedback Adaptive Control . . . . . . . . . . . . . . . . . . . . . 76</p><p>4.3 Output Feedback Adaptive Control . . . . . . . . . . . . . . . . . . . 80</p><p>4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85</p><p>4.5 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97</p><p>4.6 Approximation-Based Control . . . . . . . . . . . . . . . . . . . . . . . 101</p><p>4.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106</p><p>5 Regulation 107</p><p>5.1 Internal Model Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 108</p><p>5.2 Integral Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109</p><p>5.3 Conditional Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118</p><p>5.4 Conditional Servocompensator . . . . . . . . . . . . . . . . . . . . . . 126</p><p>5.5 Internal Model Perturbation . . . . . . . . . . . . . . . . . . . . . . . . 139</p><p>5.6 Adaptive Internal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 145</p><p>5.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158</p><p>v</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>vi CONTENTS</p><p>6 Extended Observer 159</p><p>6.1 Motivating Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160</p><p>6.2 Feedback Control via Disturbance Compensation . . . . . . . . . . 164</p><p>6.3 Nonminimum Phase Systems . . . . . . . . . . . . . . . . . . . . . . . 194</p><p>6.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210</p><p>7 Low-Power Observers 211</p><p>7.1 Cascade Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212</p><p>7.2 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219</p><p>7.3 Cascade Observer with Feedback Injection . . . . . . . . . . . . . . . 224</p><p>7.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236</p><p>8 Measurement Noise 237</p><p>8.1 Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237</p><p>8.2 Closed-Loop Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245</p><p>8.3 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253</p><p>8.4 Reducing the Effect of Measurement Noise . . . . . . . . . . . . . . . 260</p><p>8.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276</p><p>9 Digital Implementation 279</p><p>9.1 Observer Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 280</p><p>9.2 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284</p><p>9.3 Measurement Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298</p><p>9.4 Multirate Digital Control . . . . . . . . . . . . . . . . . . . . . . . . . . 301</p><p>9.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309</p><p>Bibliography 313</p><p>Index 323</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>Preface</p><p>This book is the result of over twenty-five years of research on high-gain observers</p><p>in nonlinear feedback control. Most of the results of the book were generated by</p><p>my research team at Michigan State University. I am indebted to my students and</p><p>collaborators who worked with me on these problems. I am also indebted to many</p><p>colleagues whose work on high-gain observers motivated many of our results. I am</p><p>grateful to Michigan State University for an environment that allowed me to write the</p><p>book and to the National Science Foundation for supporting my research.</p><p>The book was typeset using LATEX. Computations were done using MATLAB and</p><p>Simulink. The figures were generated using MATLAB or the graphics tool of LATEX.</p><p>Hassan Khalil</p><p>vii</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>Chapter 1</p><p>Introduction</p><p>1.1 Brief History</p><p>The use of high-gain observers in feedback control appeared first in the context of</p><p>linear feedback as a tool for robust observer design. In their celebrated work on loop</p><p>transfer recovery [36], Doyle and Stein used high-gain observers to recover, with ob-</p><p>servers, frequency-domain loop properties achieved by state feedback. The investiga-</p><p>tion of high-gain observers in the context of robust linear control continued in the</p><p>1980s, as seen in the work of Petersen and Hollot [116] on H∞ control. The use of</p><p>high-gain observers in nonlinear feedback control started to appear in the late 1980s</p><p>in the work of Saberi [86, 129], Tornambe [151], and Khalil [37]. Two key papers,</p><p>published in 1992, represent the beginning of two schools of research on high-gain ob-</p><p>servers. The work by Gauthier, Hammouri, and Othman [50] started a line of work</p><p>that is exemplified by [21, 25, 35, 49, 51, 57, 154]. This line of research covered a wide</p><p>class of nonlinear systems and obtained global results under global growth conditions.</p><p>The work by Esfandiari and Khalil [39] brought attention to the peaking phenomenon</p><p>as an important feature of high-gain observers. While this phenomenon was observed</p><p>earlier in the literature [109, 117], the paper [39] showed that the interaction of peak-</p><p>ing with nonlinearities could induce finite escape time. In particular, it showed that, in</p><p>the lack of global growth conditions, high-gain observers could destabilize the closed-</p><p>loop system as the observer gain is driven sufficiently high. It proposed a seemingly</p><p>simple solution for the problem. It suggested that the control should be designed as a</p><p>globally bounded function of the state estimates so that it saturates during the peaking</p><p>period. Because the observer is much faster than the closed-loop dynamics under state</p><p>feedback, the peaking period is very short relative to the time scale of the plant vari-</p><p>ables, which remain very close to their initial values. Teel and Praly [149, 150] built</p><p>on the ideas of [39] and earlier work by Tornambe [152] to prove the first nonlinear</p><p>separation principle and develop a set of tools for semiglobal stabilization of nonlinear</p><p>systems. Their work drew attention to [39], and soon afterward many leading nonlin-</p><p>ear control researchers started using high-gain observers; cf. [5, 8, 26, 54, 62, 65, 67, 71,</p><p>72, 89, 94, 98, 99, 102, 103, 118, 119, 123, 128, 133, 138, 139, 141, 158]. These papers</p><p>have studied a wide range of nonlinear control problems, including stabilization, regu-</p><p>lation, tracking, and adaptive control. They</p><p>‖</p><p>2+ c5‖χ ‖ ‖ϕ‖−</p><p>1</p><p>ε</p><p>‖ϕ‖2+ c6‖ϕ‖</p><p>2</p><p>for some positive constant c5 and c6. The foregoing inequality can be written as</p><p>V̇c ≤−</p><p>1</p><p>2 c3(‖χ ‖</p><p>2+ ‖ϕ‖2)− 1</p><p>2</p><p>�</p><p>‖χ ‖</p><p>‖ϕ‖</p><p>�T � c3 −c5</p><p>−c5 (2/ε− c3− 2c6)</p><p>��</p><p>‖χ ‖</p><p>‖ϕ‖</p><p>�</p><p>.</p><p>The matrix of the quadratic form can be made positive definite by choosing ε small</p><p>enough. Then, V̇c ≤ −</p><p>1</p><p>2 c3(‖χ ‖2 + ‖ϕ‖2), which shows that the origin of (3.63) is</p><p>17See (3.52).</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.2. ROBUST STABILIZATION OF MINIMUM-PHASE SYSTEMS 59</p><p>exponentially stable and all trajectories in Ωµ × Σ converge to the origin. Since all</p><p>trajectories with χ (0) ∈Ω0 and ξ̂ (0) ∈Q enter Ωµ×Σ, we conclude that Ω0×Q is a</p><p>subset of the region of attraction. 2</p><p>Example 3.4. Consider the pendulum equation</p><p>θ̈+ sinθ+ c1θ̇= c2u,</p><p>where 0≤ c1 ≤ 0.2 and 0.5≤ c2 ≤ 2. To stabilize the pendulum at (θ = π, θ̇ = 0), we</p><p>take x1 = θ−π and x2 = θ̇, which yield the state equation</p><p>ẋ1 = x2, ẋ2 = sin x1− c1x2+ c2u.</p><p>With s = x1+ x2, we obtain</p><p>ṡ = x2+ sin x1− c1x2+ c2u.</p><p>It can be shown that</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>(1− c1)x2+ sin x1</p><p>c2</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤ 2(|x1|+ |x2|).</p><p>Therefore, the control is taken as</p><p>u =−2(|x1|+ |x2|+ 1) sat(s/µ).</p><p>Inside the boundary layer {|s | ≤µ}, the system is given by</p><p>ẋ1 =−x1+ s , ṡ = (1− c1)x2+ sin x1− 2c2(|x1|+ |x2|+ 1)s/µ</p><p>and has an equilibrium point at the origin. The derivative of V = 1</p><p>2 x2</p><p>1 +</p><p>1</p><p>2 s2 satisfies</p><p>V̇ =−x2</p><p>1 + x1 s +(1− c1)(s − x1)s + s sin x1− 2c2(|x1|+ |x2|+ 1)s2/µ</p><p>≤−x2</p><p>1 + 1.2|x1| |s | − (1/µ − 1)s2 = −</p><p>�</p><p>|x1|</p><p>|s |</p><p>�T � 1 −0.6</p><p>−0.6 (1/µ − 1)</p><p>��</p><p>|x1|</p><p>|s |</p><p>�</p><p>.</p><p>For µ 0.1 and 0 0 over Dη ×Dξ .</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.3. TRACKING 61</p><p>The term δ is a matched time-varying disturbance that is assumed to be piecewise</p><p>continuous and bounded in t and locally Lipschitz in (η,ξ , u) uniformly in t . When</p><p>the relative degree ρ = n, η and its equation are dropped. The goal of the tracking</p><p>problem is to design a feedback controller such that the output y asymptotically tracks</p><p>a reference signal r , that is, limt→∞[y(t )−r (t )] = 0, while ensuring boundedness of all</p><p>state variables. We assume that the reference signal satisfies the following assumption.</p><p>Assumption 3.1. r (t ) and its derivatives up to r (ρ)(t ) are bounded for all t ≥ 0,</p><p>and the ρth derivative r (ρ)(t ) is a piecewise continuous function of t . Moreover, R =</p><p>col(r, ṙ , . . . , r (ρ−1)) ∈Dξ for all t ≥ 0.</p><p>The reference signal r (t ) could be specified, together with its derivatives, as given</p><p>functions of time, or it could be the output of a reference model driven by a command</p><p>input uc (t ). In the latter case, the assumptions on r can be met by appropriately</p><p>choosing the reference model. For example, for a relative-degree-two system, a refer-</p><p>ence model could be a second-order linear time-invariant system represented by the</p><p>transfer function</p><p>ω2</p><p>n</p><p>s2+ 2ζ ωn s +ω2</p><p>n</p><p>,</p><p>where the positive constants ζ and ωn are chosen to shape the reference signal r (t )</p><p>for a given input uc (t ). The signal r (t ) is generated by the state model</p><p>ẏ1 = y2, ẏ2 =−ω</p><p>2</p><p>n y1− 2ζ ωn y2+ω</p><p>2</p><p>n uc , r = y1.</p><p>If uc (t ) is piecewise continuous and bounded, then r (t ), ṙ (t ), and r̈ (t ) will satisfy</p><p>Assumption 3.1.</p><p>In the rest of this section we present two designs of the tracking controller. The</p><p>first design is based on feedback linearization and applies to a special case of (3.64)–</p><p>(3.67) where ρ= n, the functions a, and b are known, and δ = 0. The second design</p><p>uses sliding mode control and applies to the full model (3.64)–(3.67) even when f0,</p><p>a, and b are uncertain and δ 6= 0. The latter design will require a bounded-input–</p><p>bounded-state property of the system η̇= f0(η,ξ ).</p><p>3.3.1 Feedback Linearization</p><p>Consider a special case of (3.64)–(3.67) with no η and no disturbance, that is,</p><p>ξ̇i = ξi+1 for 1≤ i ≤ n− 1, (3.68)</p><p>ξ̇n = a(ξ )+ b (ξ )u, (3.69)</p><p>y = ξ1, (3.70)</p><p>and suppose a and b are known. The change of variables</p><p>e1 = ξ1− r, e2 = ξ2− r (1), . . . , en = ξn − r (n−1) (3.71)</p><p>transforms the system (3.68)–(3.69) into the form</p><p>ėi = ei+1 for 1≤ i ≤ n− 1, (3.72)</p><p>ėn = a(ξ )+ b (ξ )u − r (n). (3.73)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>62 CHAPTER 3. STABILIZATION AND TRACKING</p><p>In these coordinates, the goal of the design is to ensure that the vector e=col(e1, . . . , en)=</p><p>ξ−R is bounded for all t ≥ 0 and converges to zero as t tends to infinity. Boundedness</p><p>of e implies boundedness of ξ sinceR is bounded. The limit limt→∞ e(t ) = 0 ensures</p><p>that the tracking error e1 = y − r converges to zero. A state feedback controller that</p><p>achieves this goal will use e . A high-gain observer driven by e1 will generate an esti-</p><p>mate</p><p>ê of e , which replaces e in the control law after saturating the control to suppress</p><p>the effect of the peaking phenomenon. Rewrite the ė -equation as</p><p>ė =Ae +B</p><p>�</p><p>a(ξ )+ b (ξ )u − r (n)</p><p>�</p><p>,</p><p>where the pair (A,B) represents a chain of n integrators. By feedback linearization,</p><p>the state feedback control</p><p>u =</p><p>1</p><p>b (ξ )</p><p>�</p><p>−a(ξ )+ r (n)−Ke</p><p>�</p><p>results in the linear system</p><p>ė = (A−BK)e , (3.74)</p><p>in which K =</p><p>�</p><p>K1 K2 · · · Kn</p><p>�</p><p>is designed such that A−BK is Hurwitz. Therefore,</p><p>e(t ) is bounded and limt→∞ e(t ) = 0. Consequently, ξ = e +R is bounded. Let P1</p><p>be the symmetric positive definite solution of the Lyapunov equation</p><p>P1(A−BK)+ (A−BK)T P1 =−Q1,</p><p>where Q1 =QT</p><p>1 > 0. Then, Ω = {eT P1e ≤ c} is a positively invariant compact set of</p><p>the closed-loop system (3.74) for any c > 0. Choose c > 0 such that for every e ∈ Ω,</p><p>ξ = e +R ∈ Dξ . Suppose the initial error e(0) belongs to S , a compact set in the</p><p>interior of Ω. In preparation for output feedback, saturate a(ξ ), b (ξ ), and Ke outside</p><p>the compact set Ω. Take Mi >maxe∈Ω |ei | for i = 1, . . . , n. Define as (e ,R) by</p><p>as (e ,R) = a(e +R)|ei→Mi sat(ei/Mi )</p><p>and bs (e ,R) by</p><p>bs (e ,R) =max</p><p>¨</p><p>b0, b (e +R)</p><p>�</p><p>�</p><p>�</p><p>�</p><p>ei→Mi sat(ei/Mi )</p><p>«</p><p>.</p><p>The functions as and bs are globally bounded in e , as = a and bs = b for e ∈ Ω,</p><p>and bs (e ,R) ≥ b0 for all e ∈ Rn . The term Ke =</p><p>∑n</p><p>i=1 Ki ei is saturated as</p><p>∑n</p><p>i=1 Ki Mi sat(ei/Mi ), which is equal to Ke for e ∈Ω.</p><p>Because the functions a and b are known, the high-gain observer and output feed-</p><p>back control can be taken as</p><p>˙̂e =Aê +B</p><p>�</p><p>as (ê ,R)+ bs (ê ,R)u − r (n)</p><p>�</p><p>+H (ε)(y − r − ê1), (3.75)</p><p>u =</p><p>−as (ê ,R)+ r (n)−</p><p>∑n</p><p>i=1 Ki Mi sat(êi/Mi )</p><p>bs (ê ,R)</p><p>, (3.76)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.3. TRACKING 63</p><p>where</p><p>H (ε) = col(α1/ε,α2/ε2, . . . ,αn/ε</p><p>n)</p><p>with α1 to αn chosen such that the polynomial</p><p>s n +α1 s n−1+ · · ·+αn−1 s +αn</p><p>is Hurwitz and ε is a small positive constant. Substitution of u from (3.76) into (3.75)</p><p>simplifies the observer equation to</p><p>˙̂e =Aê −B</p><p>n</p><p>∑</p><p>i=1</p><p>Ki Mi sat(êi/Mi )+H (ε)(e1− ê1). (3.77)</p><p>Theorem 3.4. Suppose Assumption 3.1 is satisfied and consider the closed-loop system</p><p>of the plant (3.68)–(3.70) and the output feedback controller (3.76)–(3.77). LetQ be any</p><p>compact subset of Rn . Then, there exists ε∗ > 0 such that for every 0 0, there exists ε∗∗ > 0, dependent on λ, such</p><p>that for every 0 0, ε∗1 > 0, and T (ε) > 0, with</p><p>limε→0 T (ε) = 0, such that for every 0 0 such that (3.78) is satisfied for all t ≥ T2. Showing (3.78) on the</p><p>compact interval [0,T2] is done as in the performance recovery part of the proof of</p><p>Theorem 3.1. 2</p><p>Example 3.5. Consider the pendulum equation</p><p>ẋ1 = x2, ẋ2 =− sin x1− c1x2+ c2u, y = x1,</p><p>where c1 = 0.015 and c2 = 0.5. We want the output y to track r (t ) = cos t . Taking</p><p>e1 = x1− r = x1− cos t , e2 = x2− ṙ = x2+ sin t</p><p>yields</p><p>ė1 = e2, ė2 =− sin x1− c1x2+ c2u + cos t .</p><p>The state feedback control</p><p>u = 2(sin x1+ 0.015x2− cos t − 2e1− 3e2)</p><p>assigns the eigenvalues of A−BK at−1 and−2. The solution of the Lyapunov equation</p><p>P1(A− BK) + (A− BK)T P1 = −I is P1 = [ 1.25 0.25</p><p>0.25 0.25 ]. Set Ω = {eT P1e ≤ 3}. It can be</p><p>shown that maxe∈Ω |e1|= 1.7321 and maxe∈Ω |e2|= 3.873. We saturate e1 at ±2 and e2</p><p>at ±5. The observer gains α1 and α2 are chosen as 2 and 1, respectively, to assign the</p><p>eigenvalues of A−HoC at−1 and−1. Thus, the observer and output feedback control</p><p>are taken as</p><p>˙̂e1 = ê2+(2/ε)(e1− ê1),</p><p>˙̂e2 =−4 sat(ê1/2)− 15 sat(ê2/5)+ (1/ε</p><p>2)(e1− ê1),</p><p>u = 2 [sin(ê1+ cos t )+ 0.015(5 sat(ê2/5)− sin t )− cos t</p><p>−4 sat(ê1/2)− 15 sat(ê2/5)] .</p><p>The term ê1 in sin(ê1+ cos t ) is not saturated because the sine function is bounded.</p><p>Figure 3.9 shows the response of the system under state and output feedback for</p><p>two different values of ε with initial conditions x(0) = 0 and ê(0) = 0. Panel (a)</p><p>compares the output under output feedback with ε = 0.01 to the reference signal.</p><p>Panels (b) and (c) illustrate that the trajectories under output feedback approach the</p><p>ones under state feedback as ε decreases. 4</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.3. TRACKING 65</p><p>0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>y</p><p>a</p><p>n</p><p>d</p><p>r</p><p>(a)</p><p>0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5</p><p>−1</p><p>−0.5</p><p>0</p><p>e</p><p>1</p><p>(b)</p><p>0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5</p><p>0</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>Time</p><p>e</p><p>2</p><p>(c)</p><p>Figure 3.9. Simulation of Example 3.5. (a) shows the reference r (dashed) and the output</p><p>y under output feedback with ε = 0.01. (b) and (c) show the tracking errors e1 and e2 under state</p><p>feedback (solid), output feedback with ε = 0.1 (dash-dotted), and output feedback with ε = 0.01</p><p>(dashed).</p><p>3.3.2 Sliding Mode Control</p><p>Consider the system (3.64)–(3.67), where f0, a and b could be uncertain. The change</p><p>of variables</p><p>e1 = ξ1− r, e2 = ξ2− r (1), . . . , eρ = ξρ− r (ρ−1)</p><p>transforms the system into the form</p><p>η̇= f0(η,ξ ), (3.79)</p><p>ėi = ei+1, 1≤ i ≤ ρ− 1, (3.80)</p><p>ėρ = a(η,ξ )+ b (η,ξ )u +δ(t ,η,ξ , u)− r (ρ)(t ). (3.81)</p><p>We start by designing a state feedback controller using sliding mode control. The</p><p>sliding manifold is taken as s = 0, where</p><p>s = k1e1+ · · ·+ kρ−1eρ−1+ eρ,</p><p>in which k1 to kρ−1 are chosen such that the polynomial</p><p>λρ−1+ kρ−1λ</p><p>ρ−2+ · · ·+ k1</p><p>is Hurwitz. The derivative of s is given by</p><p>ṡ =</p><p>ρ−1</p><p>∑</p><p>i=1</p><p>ki ei+1+ a(η,ξ )+ b (η,ξ )u +δ(t ,η,ξ , u)− r (ρ)(t )</p><p>def= b (η,ξ )u +∆(t ,η, e ,R , r (ρ), u).</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>66 CHAPTER</p><p>3. STABILIZATION AND TRACKING</p><p>Suppose</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>∆(t ,η, e ,R , r (ρ), u)</p><p>b (η,ξ )</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤ %</p><p>�</p><p>e ,R , r (ρ)</p><p>�</p><p>+ κ0 |u|, 0≤κ0 0.</p><p>Then</p><p>u =−β</p><p>�</p><p>e ,R , r (ρ)</p><p>�</p><p>sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>(3.82)</p><p>ensures that s ṡ ≤ −β0b0(1− κ0)|s | for |s | ≥ µ. Setting ζ = col(e1, . . . , eρ−1), it can be</p><p>seen that ζ satisfies the equation</p><p>ζ̇ = (Ac −Bc K)ζ +Bc s ,</p><p>where K =</p><p>�</p><p>k1 k2 . . . kρ−1</p><p>�</p><p>and the pair (Ac ,Bc ) represents a chain of (ρ− 1)</p><p>integrators. The matrix Ac −Bc K is Hurwitz by design. Let P be the solution of the</p><p>Lyapunov equation P (Ac −Bc K)+(Ac −Bc K)T P =−I . The derivative of V = ζ T Pζ</p><p>satisfies the inequality</p><p>V̇ =−ζ T ζ + 2ζ T PBc s ≤−(1−θ)‖ζ ‖2 ∀ ‖ζ ‖ ≥ 2‖PBc‖ |s |/θ,</p><p>where 0 µ.</p><p>Because e = col(ζ , s −Kζ ), there is ρ2 > 0 such that ‖e‖ ≤ ρ2c for all e ∈ Ωc . Since</p><p>ξ = e+R andR ∈Dξ , by choosing c small enough we can ensure that for all e ∈Ωc ,</p><p>ξ ∈Dξ and ‖ξ ‖ ≤ ρ2c+ρ3 for some ρ3 > 0. The setΩc is positively invariant because</p><p>s ṡ 0. We turn now to the equation η̇ = f0(η,ξ ) and impose the following</p><p>assumption to ensure boundedness of η.</p><p>Assumption 3.2. There is a continuously differentiable function V0(η), classK functions</p><p>φ1 and φ2, and a nonnegative continuous nondecreasing function χ such that</p><p>φ1(‖η‖)≤V0(η)≤φ2(‖η‖),</p><p>∂ V0</p><p>∂ η</p><p>f0(η,ξ )≤ 0 ∀ ‖η‖ ≥ χ (‖ξ ‖)</p><p>for all (η,ξ ) ∈Dη×Dξ . Moreover, there is c0 ≥φ2(χ (ρ2c+ρ3)) such that {V0(η)≤ c0}</p><p>is compact and contained in Dη.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>3.3. TRACKING 67</p><p>Assumption 3.2 ensures that the set {V0(η)≤ c0}×Ωc is positively invariant because</p><p>on the boundary {V0 = c0},</p><p>φ2(‖η‖)≥ c0 ≥φ2(χ (ρ2c +ρ3))⇒‖η‖ ≥ χ (ρ2c +ρ3)⇒‖η‖ ≥ χ (‖ξ ‖)⇒ V̇0 ≤ 0.</p><p>If all the assumptions hold globally, that is, Dη ×Dξ = Rn , and φ1 is class K∞, the</p><p>constants c and c0 can be chosen arbitrarily large, and any compact set of Rn can be</p><p>put in the interior of {V0(η)≤ c0}×Ωc .</p><p>For minimum-phase systems, where the origin of η̇= f0(η, 0) is asymptotically sta-</p><p>ble, Assumption 3.2 holds locally [80, Lemma 4.7]. It will hold globally or regionally</p><p>if η̇ = f0(η,ξ ) is input-to-state stable or regionally input-to-state stable, respectively.</p><p>However, the assumption might hold while the origin of η̇= f0(η, 0) is not asymptot-</p><p>ically stable, as in the system</p><p>η̇=−</p><p>|ξ |</p><p>|ξ |+ 1</p><p>η+ ξ ,</p><p>where, with ξ = 0, the origin of η̇= 0 is stable but not asymptotically stable. Assump-</p><p>tion 3.2 is satisfied because the derivative of V0 =</p><p>1</p><p>2η</p><p>2 satisfies V̇0 ≤ 0 for |η| ≥ |ξ |+ 1.</p><p>In preparation for output feedback, we saturate the control outside the compact set</p><p>Ωc . Because the saturation function is bounded, we only need to saturateβ</p><p>�</p><p>e ,R , r (ρ)</p><p>�</p><p>.</p><p>Take Mi ≥maxe∈Ωc</p><p>|ei | for i = 1, . . . ,ρ and define βs</p><p>�</p><p>e ,R , r (ρ)</p><p>�</p><p>by</p><p>βs</p><p>�</p><p>e ,R , r (ρ)</p><p>�</p><p>= β</p><p>�</p><p>e ,R , r (ρ)</p><p>�</p><p>�</p><p>�</p><p>�</p><p>ei→Mi sat(ei/Mi )</p><p>.</p><p>The function βs is globally bounded in e and βs =β for all e ∈ Ωc . The linear high-</p><p>gain observer</p><p>˙̂ei = êi+1+</p><p>αi</p><p>εi</p><p>(y − r − ê1), 1≤ i ≤ ρ− 1, (3.83)</p><p>˙̂eρ =</p><p>αρ</p><p>ερ</p><p>(y − r − ê1) (3.84)</p><p>estimates e by ê , where ε is a sufficiently small positive constant and α1 to αρ are</p><p>chosen such that the polynomial</p><p>λρ+α1λ</p><p>ρ−1+ · · ·+αρ−1λ+αρ</p><p>is Hurwitz. The output feedback tracking controller is given by</p><p>u =−βs</p><p>�</p><p>ê ,R , r (ρ)</p><p>�</p><p>sat</p><p>�</p><p>ŝ</p><p>µ</p><p>�</p><p>, (3.85)</p><p>where ŝ = k1 ê1+ · · ·+ kρ−1 êρ−1+ êρ.</p><p>Theorem 3.5. Suppose Assumptions 3.1 and 3.2 hold and consider the closed-loop system</p><p>of the plant (3.79)–(3.81) and the output feedback controller (3.83)–(3.85). Let S ∈ Rn</p><p>be a compact set in the interior of {V0(η)≤ c0}×Ωc , and letQ be any compact subset of</p><p>Rρ. Then, there exists ε∗ > 0 (dependent on µ) and for every 0 0 and T2 > 0, there exists ε∗1 > 0, dependent on µ, λ, and T2,</p><p>such that for every 0 0. Hence,</p><p>‖e(t )− er (t )‖ ≤ λ for all t ≥ T3.</p><p>Proof of Theorem 3.5: With ϕi = (ei − êi )/ε</p><p>ρ−i for i = 1, . . . ,ρ, the closed-loop</p><p>system is represented by</p><p>η̇= f0(η,ξ ),</p><p>ζ̇ = (Ac −Bc K)ζ +Bc s ,</p><p>ṡ = b (η,ξ )ū +∆(t ,η, e ,R , r (ρ), ū)+δ1(t ,η, e , ê ,R , r (ρ)),</p><p>εϕ̇ =A0ϕ+ εBδ2(t ,η, e , ê ,R , r (ρ)),</p><p>where ϕ = col(ϕ1, . . . ,ϕρ), A0 =A−HoC , Ho = col(α1, . . .αρ), B = col(0, . . . , 0, 1),</p><p>ū =−β</p><p>�</p><p>e ,R , r (ρ)</p><p>�</p><p>sat(s/µ),</p><p>δ1 = b (η,ξ )(u − ū)+∆(t ,η, e ,R , r (ρ), u)−∆(t ,η, e ,R , r (ρ), ū),</p><p>δ2 = a(η,ξ )+ b (η,ξ )u +δ(t ,η,ξ , u)− r (ρ).</p><p>The matrix A0 is Hurwitz by design. Similar to the boundedness part of the proof of</p><p>Theorem 3.1, it can be shown that there exist % > 0, ε∗1 > 0, and T (ε) > 0, with</p><p>limε→0 T (ε) = 0, such that for every 0</p><p>Since T (ε)→ 0 as ε→ 0, we can ensure that (3.86) is satisfied over [0,T (ε)] by choos-</p><p>ing ε small enough. On the compact time interval [T (ε),T2], (3.86) follows from</p><p>continuous dependence of the solution of differential equations on initial conditions</p><p>and parameters [78, Theorem 3.4]. 2</p><p>Example 3.6. Reconsider the pendulum tracking problem</p><p>ė1 = e2, ė2 =− sin x1− c1x2+ c2u + cos t</p><p>from Example 3.5 and suppose that c1 and c2 are uncertain parameters that satisfy</p><p>0≤ c1 ≤ 0.1 and 0.5≤ c2 ≤ 2. Taking s = e1+ e2, we have</p><p>ṡ = e2− sin x1− c1x2+ c2u + cos t = (1− c1)e2− sin x1+ c1 sin t + cos t + c2u.</p><p>It can be verified that</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>(1− c1)e2− sin x1+ c1 sin t + cos t</p><p>c2</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤</p><p>|e2|+ 2.1</p><p>0.5</p><p>≤ 2|e2|+ 4.2.</p><p>We take β= 2|e2|+ 5, which yields the control</p><p>u =−(2|e2|+ 5) sat</p><p>�</p><p>e1+ e2</p><p>µ</p><p>�</p><p>.</p><p>From the equation ė1 =−e1+ s , we see that the derivative of 1</p><p>2 e2</p><p>1 satisfies</p><p>e1 ė1 =−e2</p><p>1 + e1 s ≤−(1−θ)e2</p><p>1 ∀ |e1| ≥</p><p>|s |</p><p>θ</p><p>,</p><p>where 0</p><p>and represent the extended system by a state-space model. The states of</p><p>these integrators are z1 = u, z2 = u (1), up to zm = u (m−1). The control input of the</p><p>extended system is taken as v = u (m). With x1 = y, x2 = y (1), up to xn = y (n−1), the</p><p>extended state model is given by</p><p>ẋi = xi+1, 1≤ i ≤ n− 1,</p><p>ẋn = f0(x, z)+θT f (x, z)+ [g0(x, z)+θT g (x, z)]v,</p><p>żi = zi+1, 1≤ i ≤ m− 1,</p><p>żm = v,</p><p>y = x1,</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>(4.2)</p><p>where</p><p>x = col(x1, . . . , xn), z = col(z1, . . . , zm).</p><p>We note that the state of this system comprises z1 to zm , which are available for feed-</p><p>back because they are the states of the augmented integrators, and x1 to xn , which can</p><p>be estimated from y using a high-gain observer. Based on the properties of high-gain</p><p>observers, we anticipate that the performance of a state feedback controller that uses</p><p>the full state (x, z) can be recovered by an output feedback controller in which x is</p><p>substituted by its estimate x̂. Once v has been determined as a function of (x̂, z), it</p><p>will be integrated m times to obtain the actual control u. Thus, the m integrators are</p><p>actually part of the feedback controller.</p><p>The system is required to satisfy the following two assumptions.</p><p>Assumption 4.1.</p><p>g0(x, z)+θT g (x, z)≥ ḡ > 0 ∀ x ∈ Rn , z ∈ Rm , θ ∈ Ω̂.</p><p>Assumption 4.2. There exists a global diffeomorphism</p><p>�</p><p>η</p><p>x</p><p>�</p><p>=</p><p>�</p><p>T1(x, z,θ)</p><p>x</p><p>�</p><p>def= T (x, z,θ) (4.3)</p><p>with T1(0,0,θ) = 0, which transforms (4.2) into the global normal form</p><p>η̇= p(η, x,θ),</p><p>ẋi = xi+1, 1≤ i ≤ n− 1,</p><p>ẋn = f0(x, z)+θT f (x, z)+ [g0(x, z)+θT g (x, z)]v,</p><p>y = x1.</p><p></p><p></p><p></p><p></p><p></p><p>(4.4)</p><p>Assumption 4.1 implies that the system (4.2) has relative degree n. Under this</p><p>assumption, the system is locally transformable into the normal form [64]. Assump-</p><p>tion 4.2 requires the change of variables to hold globally. In the special case when g0</p><p>and g are independent of zm , the mapping T1 can be taken as</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>4.1. PROBLEM STATEMENT 75</p><p>ηi = zi , 1≤ i ≤ m− 1,</p><p>ηm = zm −</p><p>∫ xn</p><p>0</p><p>1</p><p>G(x1, . . . , xn−1,σ , z1, . . . , zm−1,θ)</p><p>dσ ,</p><p>where G = g0+θ</p><p>T g .</p><p>Let</p><p>ei = xi − r (i−1), 1≤ i ≤ n,</p><p>e = col(e1, . . . , en),</p><p>R = col(r, r (1), . . . , r (n−1)).</p><p>The error vector e satisfies the equation</p><p>ė =Ae +B</p><p>¦</p><p>f0(x, z)+θT f (x, z)+ [g0(x, z)+θT g (x, z)]v − r (n)</p><p>©</p><p>, (4.5)</p><p>where</p><p>A=</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 0 · · · · · · 0</p><p>0 0 1 · · · · · · 0</p><p>...</p><p>...</p><p>0 0 · · · · · · 1 0</p><p>0 0 · · · · · · 0 1</p><p>0 0 · · · · · · 0 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, B =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>...</p><p>0</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>.</p><p>If θ was known, a state feedback controller could have been taken as</p><p>v =</p><p>−Ke + r (n)− f0(x, z)−θT f (x, z)</p><p>g0(x, z)+θT g (x, z)</p><p>,</p><p>which results in the system</p><p>ė = (A−BK)e .</p><p>Choosing K such that Am = A− BK is Hurwitz ensures that e(t ) converges to zero</p><p>as t tends to infinity. The boundedness of all variables will follow if the solution η(t )</p><p>of η̇ = p(η, x,θ) is bounded for every bounded x(t ), which is a stronger version of</p><p>the minimum-phase assumption.21 For later use with adaptive control, we make the</p><p>following assumption.</p><p>Assumption 4.3. The system ˙̄η = p(η̄,R ,θ) has a unique bounded steady-state solu-</p><p>tion ηs s (t ).</p><p>22 Moreover, with η̃ = η− ηs s , there is a continuously differentiable function</p><p>V1(t , η̃), possibly dependent on θ, that satisfies the inequalities</p><p>ρ1‖η̃‖</p><p>2 ≤V1(t , η̃)≤ ρ2‖η̃‖</p><p>2, (4.6)</p><p>∂ V1</p><p>∂ t</p><p>+</p><p>∂ V1</p><p>∂ η̃</p><p>[p(ηs s + η̃, e +R ,θ)− p(ηs s ,R ,θ)]≤−ρ3‖η̃‖</p><p>2+ρ4‖η̃‖ ‖e‖ (4.7)</p><p>globally, where ρ1 to ρ4 are positive constants independent ofR and θ.</p><p>21The minimum-phase assumption requires the equation η̇ = p(η, 0,θ) to have an asymptotically stable</p><p>equilibrium point [64].</p><p>22See [64, Section 8.1] for the definition of the steady-state solution of a nonlinear system.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>76 CHAPTER 4. ADAPTIVE CONTROL</p><p>Inequalities (4.6) and (4.7) imply that the solution of η̇= p(η, x,θ) is bounded for</p><p>every bounded x(t ).</p><p>A special case of (4.1) arises when the functions f0, f , g0, and g depend only on</p><p>y, y (1), . . . , y (n−1). In this case no integrators are augmented with the system, and (4.2)</p><p>takes the form</p><p>ẋi = xi+1, 1≤ i ≤ n− 1,</p><p>ẋn = f0(x)+θ</p><p>T f (x)+ [g0(x)+θ</p><p>T g (x)]u,</p><p>y = x1,</p><p></p><p></p><p></p><p>(4.8)</p><p>and Assumptions 4.2 and 4.3 are not needed.</p><p>The assumption that the model (4.1) depends linearly on the unknown parameters</p><p>may require redefinition of physical parameters, as shown in the next example.</p><p>Example 4.1. A single-link manipulator with flexible joints and negligible damping</p><p>can be modeled by [146].</p><p>I q̈1+M g L sin q1+ k(q1− q2) = 0,</p><p>J q̈2− k(q1− q2) = u,</p><p>where q1 and q2 are angular positions and u is a torque input. The physical parameters</p><p>g , I , J , k, L, and M are all positive. Taking y = q1 as the output, it can be verified that</p><p>y satisfies the fourth-order differential equation</p><p>y (4) =</p><p>g LM</p><p>I</p><p>(ẏ2 sin y − ÿ cos y)−</p><p>�k</p><p>I</p><p>+</p><p>k</p><p>J</p><p>�</p><p>ÿ −</p><p>g kLM</p><p>I J</p><p>sin y +</p><p>k</p><p>I J</p><p>u.</p><p>Taking</p><p>θ1 =</p><p>g LM</p><p>I</p><p>, θ2 =</p><p>k</p><p>I</p><p>+</p><p>k</p><p>J</p><p>, θ3 =</p><p>g kLM</p><p>I J</p><p>, θ4 =</p><p>k</p><p>I J</p><p>we obtain</p><p>y (4) = θ1(ẏ</p><p>2 sin y − ÿ cos y)−θ2 ÿ −θ3 sin y +θ4u,</p><p>which takes the form (4.1) with f0 = 0, g0 = 0,</p><p>f =</p><p></p><p></p><p></p><p></p><p>ẏ2 sin y − ÿ cos y</p><p>−ÿ</p><p>− sin y</p><p>0</p><p></p><p></p><p></p><p></p><p>, g =</p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p>, and θ=</p><p></p><p></p><p></p><p></p><p>θ1</p><p>θ2</p><p>θ3</p><p>θ4</p><p></p><p></p><p></p><p></p><p>.</p><p>Knowing bounds on the physical parameters, we can define a compact convex set Ω</p><p>such that θ ∈ Ω. If Ω is defined such that θ4 is bounded from below by a positive</p><p>constant on a compact convex set Ω̂ that contains Ω in its interior, Assumption 4.1</p><p>will be satisfied. 4</p><p>4.2 State Feedback Adaptive Control</p><p>We start by deriving a state feedback adaptive controller. Let P = P T > 0 be the</p><p>solution of the Lyapunov equation</p><p>PAm +AT</p><p>m P =−Q, Q =QT > 0</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>4.2. STATE FEEDBACK ADAPTIVE CONTROL 77</p><p>and consider the Lyapunov function candidate</p><p>V = eT Pe + 1</p><p>2 θ̃</p><p>T Γ−1θ̃,</p><p>where Γ = Γ T > 0, θ̃ = θ̂ − θ, and θ̂ is an estimate of θ to be determined by the</p><p>adaptive law. The derivative of V is given by</p><p>V̇ =−eT Qe + 2eT PB</p><p>�</p><p>f0+θ</p><p>T f +(g0+θ</p><p>T g )v +Ke − r (n)</p><p>�</p><p>+ θ̃T Γ−1 ˙̂</p><p>θ.</p><p>By adding and subtracting θ̂T ( f + g v), V̇ is rewritten as</p><p>V̇ =−eT Qe + 2eT PB</p><p>h</p><p>f0+ θ̂</p><p>T f +(g0+ θ̂</p><p>T g )v +Ke − r (n)</p><p>i</p><p>+ θ̃T Γ−1</p><p>� ˙̂</p><p>θ− Γ (2eT PB)( f + g v)</p><p>�</p><p>.</p><p>Taking</p><p>v =</p><p>−Ke + r (n)− f0(e +R , z)− θ̂T f (e +R , z)</p><p>g0(e +R , z)+ θ̂T g (e +R , z)</p><p>def= ψ(e , z,R , r (n), θ̂) (4.9)</p><p>and setting</p><p>ϕ(e , z,R , r (n), θ̂) = 2eT PB</p><p>h</p><p>f (e +R , z)+ g (e +R , z)ψ(e , z,R , r (n), θ̂)</p><p>i</p><p>, (4.10)</p><p>we end up with</p><p>V̇ =−eT Qe + θ̃T Γ−1[</p><p>˙̂</p><p>θ− Γϕ].</p><p>Taking</p><p>˙̂</p><p>θ= Γϕ yields V̇ =−eT Qe , which can be used to show that e(t ) converges to</p><p>zero as t tends to infinity. However, we need to keep θ̂ in Ω̂. This can be achieved by</p><p>adaptive laws with parameter projection,23 which can be chosen such that</p><p>θ̃T Γ−1(</p><p>˙̂</p><p>θ− Γϕ)≤ 0 (4.11)</p><p>and θ̂ ∈ Ω̂ for all t ≥ 0. We give two choices of the adaptive law for the cases when Ω</p><p>is a closed ball and a convex hypercube, respectively. In the first case, Ω= {θT θ≤ k}.</p><p>Let Ωδ = {θT θ≤ k +δ}, where δ > 0 is chosen such that Ωδ ⊂ Ω̂. The adaptive law</p><p>is taken as</p><p>˙̂</p><p>θ=Π Γϕ, where the `× `matrix Π is defined by</p><p>Π(θ̂,ϕ) =</p><p></p><p></p><p></p><p></p><p></p><p>I − (θ̂</p><p>T θ̂−k)</p><p>δθ̂T Γ θ̂</p><p>Γ θ̂θ̂T if θ̂T θ̂ > k and θ̂T Γϕ > 0,</p><p>I otherwise.</p><p>(4.12)</p><p>With</p><p>˙̂</p><p>θ=Π Γϕ, θ̂ cannot leaveΩδ because, at the boundary θ̂T θ̂= k+δ, if θ̂T Γϕ > 0,</p><p>θ̂T ˙̂</p><p>θ= θ̂T</p><p>�</p><p>I − δ</p><p>δθ̂T Γ θ̂</p><p>Γ θ̂θ̂T</p><p>�</p><p>Γϕ =</p><p></p><p>1− δθ̂</p><p>T Γ θ̂</p><p>δθ̂T Γ θ̂</p><p></p><p> θ̂T</p><p>Γϕ = 0,</p><p>23 See, for example, [63].</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>78 CHAPTER 4. ADAPTIVE CONTROL</p><p>and if θ̂T Γϕ ≤ 0,</p><p>θ̂T ˙̂</p><p>θ= θ̂T Γϕ ≤ 0.</p><p>It can also be seen that with</p><p>˙̂</p><p>θ =Π Γϕ, (4.11) is satisfied. This is obvious when Π= I</p><p>since</p><p>˙̂</p><p>θ= Γϕ. Otherwise,</p><p>θ̃T Γ−1(</p><p>˙̂</p><p>θ− Γϕ) = θ̃T Γ−1(Π− I )Γϕ</p><p>= θ̃T Γ−1</p><p></p><p>−</p><p>(θ̂T θ̂− k)</p><p>δθ̂T Γ θ̂</p><p>Γ θ̂θ̂T</p><p></p><p>Γϕ</p><p>=−</p><p>θ̂T θ̂− k</p><p>δθ̂T Γ θ̂</p><p>!</p><p>θ̃T θ̂θ̂T Γϕ k, θ̂T Γϕ > 0, and</p><p>θ̃T θ̂= θ̂T θ̂−θT θ̂≥ ‖θ̂‖2−‖θ‖ ‖θ̂‖= ‖θ̂‖(‖θ̂‖−‖θ‖)> 0</p><p>since ‖θ‖2 ≤ k and ‖θ̂‖2 > k.</p><p>In the case of a convex hypercube,</p><p>Ω= {θ | ai ≤ θi ≤ bi , 1≤ i ≤ p},</p><p>and Γ is taken as a positive diagonal matrix. Let</p><p>Ωδ = {θ | ai −δ ≤ θi ≤ bi +δ, 1≤ i ≤ p},</p><p>where δ > 0 is chosen such that Ωδ ⊂ Ω̂. The adaptive law is taken as</p><p>˙̂</p><p>θi = πiγiϕi ,</p><p>where γi > 0 and πi is defined by</p><p>πi (θ̂,ϕ) =</p><p></p><p></p><p></p><p>1+(bi − θ̂i )/δ if θ̂i > bi and ϕi > 0,</p><p>1+(θ̂i − ai )/δ if θ̂i 0,</p><p>˙̂</p><p>θi =</p><p>1+</p><p>bi − θ̂i</p><p>δ</p><p>!</p><p>γiϕi =</p><p>�</p><p>1+</p><p>bi − bi −δ</p><p>δ</p><p>�</p><p>γiϕi = 0,</p><p>and if ϕi ≤ 0,</p><p>˙̂</p><p>θ= γiϕi ≤ 0. Similarly, at the boundary θ̂i = ai −δi , if ϕi bi , ϕi > 0, and θ̃i = θ̂i −θi > 0. If πi = 1+(θ̂i − ai )/δ,</p><p>(πi − 1)θ̃iϕi = (θ̂i − ai )θ̃iϕi/δ c1 + c2. It follows from (4.11) that V̇ ≤ −eT Qe . Hence, the set {V ≤ c3}</p><p>is positively invariant. Consequently, e(t ) ∈ E def= {eT Pe ≤ c3} for all t ≥ 0. Let</p><p>ρ=maxe∈E ‖e‖. From Assumption 4.3, we see that</p><p>V̇1 ≤−ρ3‖η̃‖</p><p>2+ρ4ρ‖η̃‖.</p><p>Thus, we can choose c4 > 0 large enough so that the set {V1(t , η̃)≤ c4} is positively in-</p><p>variant and that for all e(0) ∈ E0 and z(0) ∈ Z0, η̃(0) is in the interior of {V1(t , η̃)≤ c4}.</p><p>Since {V1(t , η̃)≤ c4} ⊂ {ρ1‖η̃‖2 ≤ c4}, mapping {ρ1‖η̃‖2 ≤ c4} into the z-coordinates,</p><p>we can find a compact set Z ∈ Rm such that z(t ) ∈ Z for all t ≥ 0. Take S ></p><p>max |ψ(e , z,R , z, r (n), θ̂)|, where the maximization is taken over all e ∈ E = {eT Pe ≤</p><p>c3}, z ∈ Z , R ∈ Yr , |r (n)| ≤ Mr , and θ̂ ∈ Ωδ . Define the saturated control function</p><p>ψs by</p><p>ψs (e , z,R , r (n), θ̂) = S sat</p><p>ψ(e , z,R , r (n), θ̂)</p><p>S</p><p>!</p><p>, (4.15)</p><p>where sat(y) = sign(y)min{|y|, 1}.</p><p>24A parameter projection that confines θ̂ toΩwill be discontinuous. The parameter projection used here</p><p>is locally Lipschitz and confines θ̂ to Ωδ ⊃Ω.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>80 CHAPTER 4. ADAPTIVE CONTROL</p><p>4.3 Output Feedback Adaptive Control</p><p>The output feedback controller is given by</p><p>˙̂</p><p>θ=Π(θ̂,ϕ(ê , z,R , r (n), θ̂))Γϕ(ê , z,R , r (n), θ̂), (4.16)</p><p>v =ψs (ê , z,R , r (n), θ̂), (4.17)</p><p>where ê is provided by the high-gain observer</p><p>˙̂ei = êi+1+(αi/ε</p><p>i )(e1− ê1) for 1≤ i ≤ n− 1,</p><p>˙̂en = (αn/ε</p><p>n)(e1− ê1),</p><p>«</p><p>(4.18)</p><p>in which ε is a small positive constant and α1 to αn are chosen such that the polynomial</p><p>s n +α1 s n−1+ · · ·+αn−1 s +αn (4.19)</p><p>is Hurwitz. The closed-loop system under output feedback control is given by (4.2),</p><p>(4.16), (4.17), and (4.18).</p><p>To show convergence of e and θ̂ under output feedback control, we need the fol-</p><p>lowing assumption. Let</p><p>wr (t ) = f (R , zs s )+ g (R , zs s )ψ(0, zs s ,R , r (n),θ), (4.20)</p><p>where zs s is the unique solution of ηs s = T1(R , zs s ,θ) in which ηs s is defined in</p><p>Assumption 4.3.</p><p>Assumption 4.4. wr satisfies one of the following three conditions:</p><p>(a) wr is persistently exciting and ẇr is bounded.</p><p>(b) wr = 0.</p><p>(c) There is a constant nonsingular matrix M , possibly dependent on θ, such that</p><p>M wr (t ) =</p><p>�</p><p>wa(t )</p><p>0</p><p>�</p><p>,</p><p>where wa is a persistently exciting and ẇa is bounded.</p><p>Recall that w is persistently exciting if there are positive constants β, β1, and β2</p><p>such that [63].</p><p>β1I ≤</p><p>∫ t+β</p><p>t</p><p>w(τ)wT (τ) dτ ≤β2I ∀ t ≥ 0.</p><p>In case (c) of Assumption 4.4, define vectors θa and θ̂a , of the same dimension as wa , by</p><p>(M−1)T θ=</p><p>�</p><p>θa</p><p>θb</p><p>�</p><p>, (M−1)T θ̂=</p><p>�</p><p>θ̂a</p><p>θ̂b</p><p>�</p><p>.</p><p>Theorem 4.1. Consider the closed-loop system formed of (4.2), (4.16), (4.17), and (4.18).</p><p>Suppose Assumptions 4.1 to 4.4 are satisfied, θ̂(0) ∈ Ω, e(0) ∈ E0, z(0) ∈ Z0, and ê(0) is</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>4.3. OUTPUT FEEDBACK ADAPTIVE CONTROL 81</p><p>bounded. Then there is ε∗ > 0 such that for all 0 0 such</p><p>that the set</p><p>Σ= {V ≤ c3} ∩ {θ̂ ∈Ωδ}× {V1 ≤ c4}× {V2 ≤ c5ε</p><p>2}</p><p>is positively invariant for sufficiently small ε. InsideΣ, ‖ξ ‖=O(ε), and the saturation</p><p>of ψ is not effective, that is, ψs =ψ. Therefore, e satisfies the equation</p><p>ė =Am e −Bθ̃T ( f + gψ)+ ε∆1(·),</p><p>where ‖∆1‖ ≤ k1 for some k1 independent of ε. Thus,</p><p>V̇ ≤−eT Qe + kε ≤ −c0eT Pe + kε</p><p>=−c0V + 1</p><p>2 c0θ̃</p><p>T Γ−1θ̃+ kε≤−c0V + c0c2+ kε</p><p>for some constant k > 0, where c0 = λmi n(Q)/λmax (P ). When ε</p><p>As shown earlier, the choice of c4 ensures that {V1 ≤</p><p>c4} is positively invariant. The derivative of V2 satisfies</p><p>εV̇2 ≤−k2V2+ k3ε</p><p>p</p><p>V 2 ≤−</p><p>1</p><p>2 k2V2 ∀V2 ≥ (2k3ε/k2)</p><p>2. (4.24)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>82 CHAPTER 4. ADAPTIVE CONTROL</p><p>Taking c5 > (2k3/k2)</p><p>2 ensures that {V2 ≤ c5ε</p><p>2} is positively invariant, which completes</p><p>the proof that Σ is positively invariant. Next we show that there is time T (ε), with</p><p>l i mε→0T (ε) = 0, such that the trajectories of the closed-loop system are confined to</p><p>Σ for all t ≥ T (ε). This follows from two observations. First, because the initial con-</p><p>ditions (e(0), θ̂(0), η̃(0)) are in the interior of the set {V ≤ c3}∩ {θ̂ ∈Ωδ}×{V1 ≤ c4},</p><p>v is bounded uniformly in ε due to saturation, and θ̂ is confined to Ωδ by parameter</p><p>projection, there is time T0 > 0, independent of ε, such that (e(t ), θ̂(t ), η̃(t )) ∈ {V ≤</p><p>c3} ∩ {θ̂ ∈ Ωδ} × {V1 ≤ c4} for all t ∈ [0,T0]. On the other hand, (4.24) shows that</p><p>there time T (ε), with limε→0T (ε) = 0, such that V2 ≤ c5ε</p><p>2 for all t ≥ T (ε).25 For</p><p>sufficiently small ε, T (ε) 0 and k5 > 0, together with Assumption 4.3, (4.31), and (4.32), it can be</p><p>shown that the derivative of W with respect to (4.25) satisfies</p><p>Ẇ ≤−χ T Nχ , (4.33)</p><p>26The boundedness of żs s follows from the boundedness of η̇s s , which is the case because ηs s satisfies the</p><p>equation η̇s s = p(ηs s ,R ,θ).</p><p>27See, for example, [75, Section 13.4].</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>4.4. EXAMPLES 85</p><p>where χ = col(‖e‖,‖θ̃a‖,‖η̃‖,‖ξ ‖) and</p><p>N = 1</p><p>2</p><p></p><p></p><p></p><p></p><p>2(αk4+δ6−δ10) −δ11 −(βρ4+δ12) −(αk5+δ8+σ2)</p><p>−δ11 2δ7 −δ13 −(δ9+σ1)</p><p>−(βρ4+δ12) −δ13 2βρ3 −σ3</p><p>−(αk5+δ8+σ2 −(δ9+σ1) −σ3 2(1− εσ4)/ε</p><p></p><p></p><p></p><p></p><p>.</p><p>Choose β large enough to make</p><p>�</p><p>2δ7 −δ13</p><p>−δ13 2βρ3</p><p>�</p><p>positive definite; then choose α large enough so that</p><p></p><p></p><p>2(αk4+δ6−δ10) −δ11 −(βρ4+δ12)</p><p>−δ11 2δ7 −δ13</p><p>−(βρ4+δ12) −δ13 2βρ3</p><p></p><p></p><p>is positive definite. Finally, choosing ε small enough makes N positive definite. Hence,</p><p>limt→∞χ (t ) = 0, [78, Theorem 8.4], which proves (4.21) and (4.23). In case (a) of</p><p>Assumption 4.4, θ̃= θ̃a , which proves (4.22). �</p><p>It should be noted that in cases (b) and (c) of Assumption 4.4, the proof does</p><p>not imply exponential convergence because the right-hand side of (4.33) is only neg-</p><p>ative semidefinite. In particular, the right-hand side of (4.33) is a quadratic form of</p><p>(e , θ̃a , η̃,ξ ), while the Lyapunov function W is a quadratic form of (e , θ̃, η̃,ξ )The con-</p><p>struction of W is a key point in the proof. While the perturbation termsΛe andΛθ on</p><p>the right-hand side of (4.28) satisfy the growth conditions (4.29), the constantsδ1 toδ5</p><p>are not necessarily small. Consequently, we see in (4.31) that the right-hand side con-</p><p>tains the positive termδ10‖e‖2, which could dominate the negative term−δ6‖e‖2. We</p><p>overcome this difficulty by including αV in the composite Lyapunov function W and</p><p>choose α large enough</p><p>to ensure that the negative term −αk4‖e‖2 dominates δ10‖e‖2</p><p>and other cross-product terms. In case (a) of Assumption 4.4, θ̃ = θ̃a and the proof</p><p>shows that the closed-loop system has an exponentially stable equilibrium point at</p><p>(e , θ̃, η̃,ξ ) = 0.</p><p>4.4 Examples</p><p>We use three examples to illustrate the adaptive controller of the previous section. The</p><p>first example is a linear system that shows how the algorithm that is developed for non-</p><p>linear systems applies to linear systems as a special case. The example also elaborates</p><p>on the persistence of excitation conditions of Assumption 4.4. The second example re-</p><p>visits the tracking problem of the pendulum that is treated in Example 3.5 by feedback</p><p>linearization when the systems parameters are known. The same feedback strategy is</p><p>used here but with unknown parameters. The example allows us to compare the use</p><p>of feedback linearization with and without adaptation. The last example considers a</p><p>nonlinear system with zero dynamics and illustrates the extension of the system by</p><p>adding integrators at the input side.28</p><p>28The zero dynamics are the dynamics of the system when the output is identically zero.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>86 CHAPTER 4. ADAPTIVE CONTROL</p><p>Example 4.2. A linear system whose transfer function is</p><p>G(s) =</p><p>c</p><p>(s + a)(s + b )</p><p>,</p><p>with c > 0, can be represented by the second-order differential equation</p><p>ÿ =−θ1y −θ2 ẏ +θ3u,</p><p>where θ1 = ab , θ2 = a + b , and θ3 = c . We assume that the parameters a, b , and c</p><p>are unknown but with known bounds. The differential equation can be represented</p><p>by the state model (4.2) with no z variable and with f0 = 0, g0 = 0,</p><p>θ=</p><p></p><p></p><p>θ1</p><p>θ2</p><p>θ3</p><p></p><p> , f =</p><p></p><p></p><p>−x1</p><p>−x2</p><p>0</p><p></p><p>=</p><p></p><p></p><p>−y</p><p>−ẏ</p><p>0</p><p></p><p> , and g =</p><p></p><p></p><p>0</p><p>0</p><p>1</p><p></p><p> .</p><p>With</p><p>A=</p><p>�</p><p>0 1</p><p>0 0</p><p>�</p><p>and B =</p><p>�</p><p>0</p><p>1</p><p>�</p><p>,</p><p>the matrix K =</p><p>�</p><p>k1 k2</p><p>�</p><p>is chosen such that A− BK is Hurwitz. The state feedback</p><p>adaptive controller is given by</p><p>ψ(e ,R , r̈ , θ̂) =</p><p>−Ke + r̈ − θ̂T f</p><p>θ̂T g</p><p>=</p><p>−k1e1− k2e2+ r̈ + θ̂1(e1+ r )+ θ̂2(e2+ ṙ )</p><p>θ̂3</p><p>.</p><p>To examine Assumption 4.4, consider</p><p>ψ(0,R , r̈ ,θ) =</p><p>r̈ +θ1 r +θ2 ṙ</p><p>θ3</p><p>,</p><p>which, by (4.20), yields</p><p>wr =</p><p></p><p></p><p>−r</p><p>− ṙ</p><p>( r̈ +θ1 r +θ2 ṙ )/θ3</p><p></p><p> .</p><p>We consider three cases of the reference signal: r = 1, r = sinωt , and r = 1+ sinωt ,</p><p>whereω> 0. In the first case,</p><p>wr =</p><p></p><p></p><p>−1</p><p>0</p><p>θ1/θ3</p><p></p><p> and</p><p></p><p></p><p>−1 0 0</p><p>0 1 0</p><p>θ1/θ3 0 1</p><p></p><p></p><p>︸ ︷︷ ︸</p><p>M</p><p>wr =</p><p></p><p></p><p>1</p><p>0</p><p>0</p><p></p><p> .</p><p>Hence, case (c) of Assumption 4.4 is satisfied with wa = 1, and</p><p>(M−1)T θ=</p><p></p><p></p><p>0</p><p>θ2</p><p>θ3</p><p></p><p> , (M−1)T θ̂=</p><p></p><p></p><p></p><p>−(θ̂1−θ1θ̂3/θ3)</p><p>θ̂2</p><p>θ̂3</p><p></p><p></p><p></p><p>.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>4.4. EXAMPLES 87</p><p>Thus, θa = 0 and θ̂a = −(θ̂1 − θ1θ̂3/θ3). Theorem 4.1 shows that limt→∞[θ̂a(t )−</p><p>θa(t )] = 0, that is, limt→∞[θ̂1(t )−θ1θ̂3(t )/θ3] = 0. When r = sinωt ,</p><p>wr =</p><p></p><p></p><p>− sinωt</p><p>−ω cosωt</p><p>(−ω2 sinωt +θ1 sinωt +θ2ω cosωt )/θ3</p><p></p><p></p><p>and</p><p></p><p></p><p>−1 0 0</p><p>0 −1/ω 0</p><p>(θ1−ω2)/θ3 θ2/θ3 1</p><p></p><p></p><p>︸ ︷︷ ︸</p><p>M</p><p>wr =</p><p></p><p></p><p>sinωt</p><p>cosωt</p><p>0</p><p></p><p> .</p><p>Therefore, case (c) of Assumption 4.4 is satisfied with wa = col(sinωt , cosωt ), which</p><p>is persistently exciting because</p><p>∫ t+π/ω</p><p>t</p><p>�</p><p>sinωτ</p><p>cosωτ</p><p>��</p><p>sinωτ</p><p>cosωτ</p><p>�T</p><p>dτ = 1</p><p>2 (π/ω)I</p><p>and</p><p>(M−1)T θ=</p><p></p><p></p><p>−ω2</p><p>0</p><p>θ3</p><p></p><p> , (M−1)T θ̂=</p><p></p><p></p><p></p><p>−θ̂1+(θ1−ω2)θ̂3/θ3</p><p>−ωθ̂2+ωθ2θ̂3/θ3</p><p>θ̂3</p><p></p><p></p><p></p><p>.</p><p>Thus,</p><p>θa =</p><p>�</p><p>−ω2</p><p>0</p><p>�</p><p>and θ̂a =</p><p>�</p><p>−θ̂1+(θ1−ω2)θ̂3/θ3</p><p>−ωθ̂2+ωθ2θ̂3/θ3</p><p>�</p><p>.</p><p>Theorem 4.1 shows that limt→∞[θ̂a(t )−θa(t )] = 0, that is,</p><p>lim</p><p>t→∞</p><p>[θ̂1(t )− (θ1−ω</p><p>2)θ̂3(t )/θ3−ω</p><p>2] = 0 and lim</p><p>t→∞</p><p>[θ̂2(t )−θ2θ̂3(t )/θ3] = 0.</p><p>Finally, when r = 1+ sinωt ,</p><p>wr =</p><p></p><p></p><p>−1− sinωt</p><p>−ω cosωt</p><p>(−ω2 sinωt +θ1+θ1 sinωt +θ2ω cosωt )/θ3</p><p></p><p></p><p>=</p><p></p><p></p><p>−1 −1 0</p><p>0 0 −ω</p><p>θ1/θ3 (θ1−ω2)/θ3 θ2ω/θ3</p><p></p><p></p><p>︸ ︷︷ ︸</p><p>Q</p><p></p><p></p><p>1</p><p>sinωt</p><p>cosωt</p><p></p><p> .</p><p>The vector w̄r = col(1, sinωt , cosωt ) is persistently exciting because</p><p>∫ t+π/ω</p><p>t</p><p>w̄r (τ)w̄</p><p>T</p><p>r (τ) dτ = (π/ω) diag(1, 1</p><p>2 , 1</p><p>2 ),</p><p>and Q is nonsingular since det(Q) =ω3/θ3 > 0. Thus, wr is persistently exciting as in</p><p>case (a) of Assumption 4.4. It follows from Theorem 4.1 that limt→∞[θ̂i (t )−θi ] = 0</p><p>for i = 1,2,3.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>88 CHAPTER 4. ADAPTIVE CONTROL</p><p>To complete the design, suppose a ∈ [−1,1], b ∈ [−1,1], and c ∈ [1,2]. Then</p><p>Ω= {θ | θ1 ∈ [−1,1], θ2 ∈ [−2,2], θ3 ∈ [1,2]}.</p><p>With δ = 0.1,</p><p>Ωδ = {θ | θ1 ∈ [−1.1,1.1], θ2 ∈ [−2.1,2.1], θ3 ∈ [0.9,2.1]}.</p><p>The matrix K is taken as K =</p><p>�</p><p>2 3</p><p>�</p><p>to assign the eigenvalues of Am = A− BK at</p><p>−1, −2. The solution of the Lyapunov equation PAm +AT</p><p>m P =−I is</p><p>P = 1</p><p>4</p><p>�</p><p>5 1</p><p>1 1</p><p>�</p><p>and PB = 1</p><p>4</p><p>�</p><p>1</p><p>1</p><p>�</p><p>.</p><p>Thus,</p><p>ψ=</p><p>−2e1− 3e2+ r̈ + θ̂1(e1+ r )+ θ̂2(e2+ ṙ )</p><p>θ̂3</p><p>and</p><p>ϕ = 2eT PB( f + gψ) = 1</p><p>2 (e1+ e2)</p><p></p><p></p><p>−(e1+ r )</p><p>−(e2+ ṙ )</p><p>ψ</p><p></p><p> .</p><p>The adaptive law is</p><p>˙̂</p><p>θi =πiγiϕi for i = 1,2,3, where</p><p>π1(θ̂1,ϕ1) =</p><p></p><p></p><p></p><p>1+(1− θ̂1)/0.1 if θ̂1 > 1 and ϕ1 > 0,</p><p>1+(θ̂1+ 1)/0.1 if θ̂1 2 and ϕ2 > 0,</p><p>1+(θ̂2+ 2)/0.1 if θ̂2 2 and ϕ3 > 0,</p><p>1+(θ̂3− 1)/0.1 if θ̂3 c1+ c2. Now we compute the maximum of |ψ| for</p><p>eT Pe ≤ c3 and θ̂ ∈Ωδ :</p><p>|ψ|=</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>(−2+ θ̂1)e1+(−3+ θ̂2)e2+ r̈ + θ̂1 r + θ̂2 ṙ</p><p>θ̂3</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤ 1</p><p>0.9</p><p>[|(−2+ θ̂1)e1+(−3+ θ̂2)e2|+ | r̈ + θ̂1 r + θ̂2 ṙ |].</p><p>It can be verified that for all three cases of r ,</p><p>| r̈ + θ̂1 r + θ̂2 ṙ | ≤ 7.71.</p><p>On the other hand, the maximum of |(−2+θ̂1)e1+(−3+θ̂2)e2| over eT Pe ≤ c3 is given</p><p>bypc3</p><p>�</p><p>(−2+ θ̂1) (−3+ θ̂2)</p><p>�</p><p>P−1/2</p><p>, where P−1/2 is the inverse of the square root</p><p>matrix of P . Maximizing this norm over θ̂ ∈ Ωδ , we arrive at the bound |ψ| ≤ 30.57.</p><p>We take S = 32. The output feedback controller is given by</p><p>u = 32 sat</p><p>−2ê1− 3ê2+ r̈ + θ̂1(ê1+ r )+ θ̂2(ê2+ ṙ )</p><p>32 θ̂3</p><p>!</p><p>,</p><p>˙̂e1 = ê2+(2/ε)(y − r − ê1),</p><p>˙̂e2 = (1/ε</p><p>2)(y − r − ê1),</p><p>˙̂</p><p>θ1 = γ1π1(θ̂1,ϕ1)ϕ1, ϕ1 =−</p><p>1</p><p>2 (ê1+ ê2)(ê1+ r ),</p><p>˙̂</p><p>θ2 = γ2π2(θ̂2,ϕ2)ϕ2, ϕ2 =−</p><p>1</p><p>2 (ê1+ ê2)(ê2+ ṙ ),</p><p>˙̂</p><p>θ3 = γ3π3(θ̂3,ϕ3)ϕ3, ϕ3 =</p><p>1</p><p>2 (ê1+ ê2)u,</p><p>where the observer gains assign the observer eigenvalues at −1/ε and −1/ε.</p><p>Figures 4.1 to 4.4 show simulation results. The simulation is carried out with a =</p><p>0.5, b = −1, c = 1.5, ω = 2, and ε = 0.01. Consequently, θ1 = −0.5, θ2 = −0.5, and</p><p>θ3 = 1.5. The initial conditions of x, ê , and θ̂ are taken to be zero, except θ̂3(0) = 1.</p><p>Figures 4.1 and 4.2 are for the case r = 1. Figure 4.1 shows the tracking errors e1 and</p><p>e2 and the control u in two different time scales. Zooming on the initial period shows</p><p>the control saturating at 32 during the peaking period of the observer. Figure 4.2</p><p>shows the parameter estimates θ̂1 to θ̂3. As expected, they do not converge to</p><p>the</p><p>true values of θ1 to θ3, which are shown by the dashed lines. As we saw earlier, in</p><p>this case θ̂1 − θ1θ̂3/θ3 = θ̂1 +</p><p>1</p><p>3 θ̂3 converges to zero as t tends to infinity, which is</p><p>demonstrated in Figure 4.2(d). Figure 4.3 shows simulation results when r = sin2t .</p><p>Figures 4.2(a) and (b) demonstrate that the tracking errors e1 and e2 converge to zero.</p><p>Figures 4.3(c) and (d) show the parameter estimates θ̂1 to θ̂3, which do not converge to</p><p>the true values of θ1 to θ3. We saw earlier that θ̂1−(θ1−ω2)θ̂3/θ3−ω2 = θ̂1+3θ̂3−4</p><p>and θ̂2 − θ2θ̂3/θ3 = θ̂2 +</p><p>1</p><p>3 θ̂3 converge to zero, which is shown in Figures 4.3(e) and</p><p>(f). Finally, Figure 4.4 shows simulation results when r = 1+ sin2t . As expected, the</p><p>tracking errors e1 and e2 converge to zero, and the parameter estimates converge to the</p><p>true parameters. 4</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>90 CHAPTER 4. ADAPTIVE CONTROL</p><p>0 1 2 3 4 5</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>Time</p><p>T</p><p>ra</p><p>ck</p><p>in</p><p>g</p><p>er</p><p>ro</p><p>rs</p><p>(a)</p><p>e</p><p>1</p><p>e</p><p>2</p><p>0 0.02 0.04 0.06 0.08 0.1</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>Time</p><p>T</p><p>ra</p><p>ck</p><p>in</p><p>g</p><p>er</p><p>ro</p><p>rs</p><p>(b)</p><p>e</p><p>1</p><p>e</p><p>2</p><p>0 1 2 3 4 5</p><p>0</p><p>5</p><p>10</p><p>15</p><p>20</p><p>25</p><p>30</p><p>35</p><p>Time</p><p>C</p><p>on</p><p>tr</p><p>ol</p><p>u</p><p>(c)</p><p>0 0.02 0.04 0.06 0.08 0.1</p><p>0</p><p>5</p><p>10</p><p>15</p><p>20</p><p>25</p><p>30</p><p>35</p><p>Time</p><p>C</p><p>on</p><p>tr</p><p>ol</p><p>u</p><p>(d)</p><p>Figure 4.1. Simulation of Example 4.2 with r = 1. (a) and (b) show the tracking error</p><p>e1 and e2 on two different time scales. (c) and (d) do the same for the control signal u.</p><p>0 2 4 6</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>0.2</p><p>Time</p><p>θ̂</p><p>1</p><p>(a)</p><p>0 2 4 6</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>2.5</p><p>Time</p><p>θ̂</p><p>2</p><p>(b)</p><p>0 2 4 6</p><p>0.8</p><p>1</p><p>1.2</p><p>1.4</p><p>1.6</p><p>Time</p><p>θ̂</p><p>3</p><p>(c)</p><p>0 2 4 6</p><p>−0.5</p><p>0</p><p>0.5</p><p>Time</p><p>θ̂</p><p>1</p><p>+</p><p>1 3</p><p>θ̂</p><p>3</p><p>(d)</p><p>Figure 4.2. Simulation of Example 4.2 with r = 1. The parameter estimates do not</p><p>converge to the true values of the parameters, shown by the dashed lines, but θ̂1+</p><p>1</p><p>3 θ̂3 converges to</p><p>zero.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>4.4. EXAMPLES 91</p><p>0 5 10 15 20</p><p>−0.4</p><p>−0.3</p><p>−0.2</p><p>−0.1</p><p>0</p><p>Time</p><p>e 1</p><p>(a)</p><p>0 5 10 15 20</p><p>−2</p><p>−1</p><p>0</p><p>1</p><p>Time</p><p>e 2</p><p>(b)</p><p>0 5 10 15 20</p><p>−1</p><p>0</p><p>1</p><p>2</p><p>θ̂1</p><p>θ̂2</p><p>Time</p><p>θ̂</p><p>1</p><p>a</p><p>n</p><p>d</p><p>θ̂</p><p>2</p><p>(c)</p><p>0 5 10 15 20</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>2.5</p><p>Time</p><p>θ̂</p><p>3</p><p>(d)</p><p>0 5 10 15 20</p><p>−2</p><p>−1</p><p>0</p><p>1</p><p>2</p><p>3</p><p>Time</p><p>θ̂</p><p>1</p><p>+</p><p>3</p><p>θ̂</p><p>3</p><p>−</p><p>4</p><p>(e)</p><p>0 5 10 15 20</p><p>−1</p><p>0</p><p>1</p><p>2</p><p>Time</p><p>θ̂</p><p>2</p><p>+</p><p>1 3</p><p>θ̂</p><p>3</p><p>(f)</p><p>Figure 4.3. Simulation of Example 4.2 with r = sin2t . The tracking errors e1 and e2</p><p>converge to zero. The parameter estimates do not converge to the true values of the parameters, but</p><p>θ̂1+ 3θ̂3− 4 and θ̂2+</p><p>1</p><p>3 θ̂3 converge to zero.</p><p>0 5 10 15</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>e</p><p>1</p><p>e</p><p>2</p><p>Time</p><p>T</p><p>ra</p><p>ck</p><p>in</p><p>g</p><p>er</p><p>ro</p><p>rs</p><p>(a)</p><p>0 5 10 15 20</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>0.2</p><p>Time</p><p>θ̂</p><p>1</p><p>(b)</p><p>0 5 10 15 20</p><p>−1.5</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>2.5</p><p>Time</p><p>θ̂</p><p>2</p><p>(c)</p><p>0 5 10 15 20</p><p>0.8</p><p>1</p><p>1.2</p><p>1.4</p><p>1.6</p><p>1.8</p><p>2</p><p>2.2</p><p>Time</p><p>θ̂</p><p>3</p><p>(d)</p><p>Figure 4.4. Simulation of Example 4.2 with r = 1+ sin2t . The tracking errors e1 and</p><p>e2 converge to zero, and the parameter estimates converge to the true values of the parameters.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>92 CHAPTER 4. ADAPTIVE CONTROL</p><p>Example 4.3. Reconsider the tracking problem for the pendulum equation</p><p>ẋ1 = x2, ẋ2 =− sin x1−θ1x2+θ2u, y = x1</p><p>from Example 3.5, where the goal is to have the output y track the reference r (t ) =</p><p>cos t . In Example 3.5 feedback linearization is used when the parameters θ1 = 0.015</p><p>and θ2 = 0.5 are known. Suppose these parameters are unknown but we know that</p><p>θ1 ∈ [0,0.1] and θ2 ∈ [0.5,2]. With the change of variables</p><p>e1 = x1− r = x1− cos t , e2 = x2− ṙ = x2+ sin t ,</p><p>the system is represented by</p><p>ė =</p><p>�</p><p>0 1</p><p>0 0</p><p>�</p><p>︸ ︷︷ ︸</p><p>A</p><p>e +</p><p>�</p><p>0</p><p>1</p><p>�</p><p>︸︷︷︸</p><p>B</p><p>[− sin(e1+ cos t )−θ1(e2− sin t )+θ2u + cos t ].</p><p>The matrix K =</p><p>�</p><p>2 3</p><p>�</p><p>assigns the eigenvalues of Am = A−BK at −1,−2. The solu-</p><p>tion of the Lyapunov equation PAm +AT</p><p>m P =−I is</p><p>P = 1</p><p>4</p><p>�</p><p>5 1</p><p>1 1</p><p>�</p><p>and PB = 1</p><p>4</p><p>�</p><p>1</p><p>1</p><p>�</p><p>.</p><p>From (4.9) and (4.10), ψ and ϕ are given by</p><p>ψ=</p><p>−2e1− 3e2+ sin(e1+ cos t )− cos t + θ̂1(e2− sin t )</p><p>θ̂2</p><p>,</p><p>ϕ = 1</p><p>2 (e1+ e2)</p><p>�</p><p>−e2+ sin t</p><p>ψ</p><p>�</p><p>.</p><p>The sets Ω and Ωδ , with δ = 0.05, are given by</p><p>Ω= {θ | θ1 ∈ [0,0.1], θ2 ∈ [0.5,2]},</p><p>Ωδ = {θ | θ1 ∈ [−0.05,0.15], θ2 ∈ [0.45,2.05]}.</p><p>The adaptive law is</p><p>˙̂</p><p>θi =πiγiϕi for i = 1,2, where</p><p>π1(θ̂1,ϕ1) =</p><p></p><p></p><p></p><p>1+(0.1− θ̂1)/0.05 if θ̂1 > 0.1 and ϕ1 > 0,</p><p>1+ θ̂1/0.05 if θ̂1 2 and ϕ2 > 0,</p><p>1+(θ̂2− 0.5)/0.05 if θ̂2</p><p>Feedback Linearization</p><p>0 1 2 3 4 5</p><p>0</p><p>0.2</p><p>0.4</p><p>0.6</p><p>0.8</p><p>1</p><p>Time</p><p>e</p><p>2</p><p>(d)</p><p>Figure 4.5. Simulation of Example 4.3. (a) and (b) show y and u under the adaptive</p><p>controller. (c) and (d) compare the tracking errors under the adaptive controller (solid) with the</p><p>errors under the feedback linearization of Example 3.5 (dashed).</p><p>0 2 4 6 8 10</p><p>−0.05</p><p>0</p><p>0.05</p><p>Time</p><p>θ̂</p><p>1</p><p>0 2 4 6 8 10</p><p>0.3</p><p>0.4</p><p>0.5</p><p>0.6</p><p>0.7</p><p>0.8</p><p>0.9</p><p>1</p><p>Time</p><p>θ̂</p><p>2</p><p>Figure 4.6. Simulation of Example 4.3. Convergence of the parameter estimates.</p><p>Extend the dynamics of the system by adding an integrator at the input and let z = u,</p><p>x1 = y, x2 = ẏ, and v = u̇ to arrive at the extended state model</p><p>ẋ1 = x2,</p><p>ẋ2 = x2+θ[z −σ(x1)−σ</p><p>′(x1)x2]+θv,</p><p>ż = v,</p><p>y = x1.</p><p>Assumption 4.2 is satisfied with η= z − x2/θ, which results in the η̇-equation</p><p>η̇=−η− (2/θ)x2+σ(x1)+σ</p><p>′(x1)x2.</p><p>Let e1 = x1− r and e2 = x2− ṙ . Then, e = col(e1, e2) satisfies the equation</p><p>ė =Ae +B[e2+ ṙ − r̈ +θ(z −σ(e1+ r )−σ ′(e1+ r )(e2+ ṙ )+ v)],</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>4.4. EXAMPLES 95</p><p>- j - -</p><p>�</p><p>6</p><p>u y</p><p>G(s)</p><p>σ(y)</p><p>−</p><p>+</p><p>Figure 4.7. Example 4.4.</p><p>where</p><p>A=</p><p>�</p><p>0 1</p><p>0 0</p><p>�</p><p>and B =</p><p>�</p><p>0</p><p>1</p><p>�</p><p>.</p><p>The system ˙̄η= p(η̄,R ,θ) of Assumption 4.3 is given by</p><p>˙̄η=−η̄− (2/θ) ṙ +σ(r )+σ ′(r ) ṙ .</p><p>Since this is a linear exponentially stable system driven by a periodic input, it has a</p><p>periodic steady-state solution ηs s , which satisfies the same equation, that is,</p><p>η̇s s =−ηs s − (2/θ) ṙ +σ(r )+σ</p><p>′(r ) ṙ .</p><p>Then η̃= η−ηs s satisfies the equation</p><p>˙̃η=−η̃− (2/θ)e2+σ(e1+ r )−σ(r )+ [σ ′(e1+ r )−σ ′(r )] ṙ +σ ′(e1+ r )e2.</p><p>Since σ(y) = tanh(y), σ ′(y) = 1/(cosh(y))2, and σ ′′(y) =−2tanh(y)/(cosh(y))2, it can</p><p>be seen that |σ | ≤ 1, |σ ′| ≤ 1, |σ ′′| ≤ 0.8, |σ(e1 + r )− σ(r )| ≤ |e1|, and |σ ′(e1 + r )−</p><p>σ ′(r )| ≤ 0.8|e1|. For the reference signals r = cos t and r = 1, we have |r | ≤ 1, | ṙ | ≤ 1,</p><p>and | r̈ | ≤ 1. Writing ˙̃η = −η̃+ δη, it can be verified that |δη| ≤ 3.4986‖e‖. With</p><p>V1 =</p><p>1</p><p>2 η̃</p><p>2, we have</p><p>V̇1 ≤−|η̃|</p><p>2+ 3.4986‖e‖ |η̃|.</p><p>Thus, Assumption 4.3 is satisfied with V1 =</p><p>1</p><p>2 η̃</p><p>2, ρ1 = ρ2 =</p><p>1</p><p>2 , ρ3 = 1, andρ4 = 3.4986.</p><p>We take K =</p><p>�</p><p>2 3</p><p>�</p><p>to assign the eigenvalues of Am = A− BK at −1 and −2. The</p><p>solution of the Lyapunov equation PAm +AT</p><p>m P =−I is</p><p>P =</p><p>1</p><p>4</p><p>�</p><p>5 1</p><p>1 1</p><p>�</p><p>and 2eT PB = 1</p><p>2 (e1+ e2).</p><p>From (4.9) and (4.10), ψ and ϕ are given by</p><p>ψ=</p><p>−Ke − e2− ṙ + r̈ − θ̂[z −σ(e1+ r )−σ ′(e1+ r )(e2+ ṙ )]</p><p>θ̂</p><p>=</p><p>1</p><p>θ̂</p><p>(−2e1− 4e2− ṙ + r̈ )− [z −σ(e1+ r )−σ ′(e1+ r )(e2+ ṙ )],</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>96 CHAPTER 4. ADAPTIVE CONTROL</p><p>ϕ = 2eT PB[z −σ(e1+ r )−σ ′(e1+ r )(e2+ ṙ )+</p><p>1</p><p>θ̂</p><p>(−2e1− 4e2− ṙ + r̈ )</p><p>− z +σ(e1+ r )+σ ′(e1+ r )(e2+ ṙ )]</p><p>=</p><p>1</p><p>2θ̂</p><p>(e1+ e2)(−2e1− 4e2− ṙ + r̈ ).</p><p>To determine the saturation level, S, of ψ, suppose the initial error e(0) belongs to</p><p>E0 = {|e1| ≤ 1, |e2| ≤ 1}. Set c1 =maxe∈E0</p><p>eT Pe = 2. Taking γ = 10 and δ = 0.1, the</p><p>sets Ω and Ωδ are defined by Ω= {θ | 1≤ θ≤ 3} and Ωδ = {θ | 0.9≤ θ̂≤ 3.1}, and</p><p>max</p><p>θ∈Ω,θ̂∈Ωδ</p><p>1</p><p>2γ</p><p>(θ− θ̂)2 = 0.2205.</p><p>Take c2 = 0.221 and c3 = 2.25> c1+ c2. Now, e(t ) ∈ E = {eT Pe ≤ c3} for all t ≥ 0 and</p><p>max</p><p>e∈E</p><p>‖e‖ ≤</p><p>√</p><p>√</p><p>√</p><p>c3</p><p>λmin(P )</p><p>=</p><p>s</p><p>2.25</p><p>0.191</p><p>= 3.4322,</p><p>max</p><p>e∈E</p><p>|e2|=</p><p>p</p><p>c3</p><p>�</p><p>0 1</p><p>�</p><p>P−1/2</p><p>= 3.3541,</p><p>max</p><p>e∈E</p><p>�</p><p>�</p><p>�</p><p>2 4</p><p>�</p><p>e</p><p>�</p><p>�=</p><p>p</p><p>c3</p><p>�</p><p>2 4</p><p>�</p><p>P−1/2</p><p>= 13.0977.</p><p>It follows that V̇1 ≤ −η̃2 + 12.1|η̃|, which shows that the set Ξ = {|η̃| ≤ 12.1} is posi-</p><p>tively invariant. Since θ ∈ [1,3] and σ ′(r ) ∈ [0,1], we have −(2/θ)+σ ′(r ) ∈ [−2, 1</p><p>3 ].</p><p>Therefore, | − (2/θ) ṙ + σ(r ) + σ ′(r ) ṙ | ≤ 3. It follows from the η̇s s -equation that</p><p>|ηs s (t )| ≤ 3. Since η̃ = η− ηs s = z − ηs s − (e2 + ṙ )/θ, restricting z(0) to |z(0)| ≤ 7</p><p>ensures that η̃(0) ∈ Ξ. Therefore, |η̃(t )| ≤ 12.1 for all t ≥ 0. Hence, |z | ≤ |η̃|+ |ηs s |+</p><p>(|e2|+ | ṙ |)/θ≤ 19.4541. We can now calculate an upper bound on |ψ| as</p><p>|ψ| ≤</p><p>�</p><p>�</p><p>�</p><p>2 4</p><p>�</p><p>e</p><p>�</p><p>�+ | ṙ |+ | r̈ |+ |z |+ |σ |+ |σ ′|(|e2|+ | ṙ |)≤ 39.5495.</p><p>We take S = 41. To examine the persistence of excitation condition, consider</p><p>wr = zs s −σ(r )−σ</p><p>′(r ) ṙ +</p><p>1</p><p>θ</p><p>(− ṙ + r̈ )− zs s +σ(r )+σ</p><p>′(r ) ṙ</p><p>=</p><p>1</p><p>θ</p><p>(− ṙ + r̈ ),</p><p>where wr is persistently exciting when r = cos t since</p><p>∫ t+π</p><p>t (sinτ − cosτ)2 dτ = π,</p><p>while wr = 0 when r = 1. So, the parameter error θ̂−θ is guaranteed to converge to</p><p>zero only when r = cos t . The output feedback controller is given by</p><p>u =</p><p>∫</p><p>v,</p><p>v = 41 sat</p><p>(−2ê1− 4ê2− ṙ + r̈ )/θ̂− u +σ(ê1+ r )+σ ′(ê1+ r )(ê2+ ṙ )</p><p>41</p><p>!</p><p>,</p><p>˙̂e1 = ê2+(2/ε)(e1− ê1),</p><p>˙̂e2 = (1/ε</p><p>2)(e1− ê1),</p><p>˙̂</p><p>θ= 10 π(θ̂,ϕ)ϕ, ϕ = 1</p><p>2 (ê1+ ê2)(−2ê1− 4ê2− ṙ + r̈ )/θ̂,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>4.5. ROBUSTNESS 97</p><p>0 5 10 15 20</p><p>−1</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>0.2</p><p>Time</p><p>y</p><p>−</p><p>r</p><p>0 5 10 15 20</p><p>1</p><p>1.5</p><p>2</p><p>2.5</p><p>Time</p><p>θ̂</p><p>0 5 10 15 20</p><p>−0.5</p><p>0</p><p>0.5</p><p>1</p><p>1.5</p><p>Time</p><p>u</p><p>0 0.02 0.04 0.06 0.08 0.1</p><p>−20</p><p>−10</p><p>0</p><p>10</p><p>20</p><p>30</p><p>40</p><p>50</p><p>Time</p><p>v</p><p>Figure 4.8. Simulation of Example 4.4 when r = cos t .</p><p>where the observer eigenvalues are assigned at −1/ε and</p><p>π(θ̂,ϕ) =</p><p></p><p></p><p></p><p>1+(3− θ̂)/0.1 if θ̂ > 3 and ϕ > 0,</p><p>1+(θ̂− 1)/0.1 if θ̂</p><p>also explored the use of time-varying high-</p><p>gain observers. Khalil and his coworkers continued to investigate high-gain observers</p><p>in nonlinear feedback control for about twenty-five years converging a wide range of</p><p>1</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>2 CHAPTER 1. INTRODUCTION</p><p>problems; cf. [1, 2, 4, 32, 46, 73, 74, 76, 83, 104, 113, 134, 135, 136, 140, 153]. Atassi</p><p>and Khalil [18] proved a separation principle that adds a new dimension to the result</p><p>of Teel and Praly [149]; namely, the combination of fast observer with control satu-</p><p>ration enables the output feedback controller to recover the trajectories of the state</p><p>feedback controller as the observer gain is made sufficiently high.</p><p>1.2 Motivating Examples</p><p>We use a few examples to motivate the design of high-gain observers and discuss their</p><p>main features. Example 1.1 illustrates the robustness of the observer and its peaking</p><p>phenomenon. Example 1.2 shows that the interaction of peaking and nonlinearities</p><p>can lead to a finite escape time. It shows also how saturating the controller overcomes</p><p>the destabilizing effect of peaking. An important feature of high-gain observers in feed-</p><p>back control is illustrated in this example, where it is shown that the output feedback</p><p>controller recovers the performance of the state feedback controller as the observer’s</p><p>dynamics become sufficiently fast. This performance recovery property is due to the</p><p>combined effect of a fast observer and a saturated control. Example 1.3 shows that</p><p>this property does not hold by designing a fast observer without saturated control.</p><p>Example 1.4 introduces the reduced-order high-gain observer and compares it with</p><p>the full-order one.</p><p>Example 1.1. Consider the two-dimensional system</p><p>ẋ1 = x2, ẋ2 =φ(x, u), y = x1,</p><p>where x = col(x1, x2), φ is locally Lipschitz, and x(t ) and u(t ) are bounded for all</p><p>t ≥ 0. To estimate x, we use the observer</p><p>˙̂x1 = x̂2+ h1(y − x̂1),</p><p>˙̂x2 =φ0(x̂, u)+ h2(y − x̂1),</p><p>where φ0(x, u) is a nominal model φ(x, u). We can take φ0 = 0, which simplifies the</p><p>observer to a linear one. Whatever the choice of φ0 is, we assume that</p><p>|φ0(z, u)−φ(x, u)| ≤ L‖x − z‖+M</p><p>for some nonnegative constants L and M , for all (x, z, u) in the domain of interest.1</p><p>In the special case when φ0 =φ and φ is Lipschitz in x uniformly in u, the foregoing</p><p>inequality holds with M = 0. The estimation error x̃ = x − x̂ satisfies the equation</p><p>˙̃x =Ao x̃ +Bδ(x, x̃, u), where Ao =</p><p>�</p><p>−h1 1</p><p>−h2 0</p><p>�</p><p>, B =</p><p>�</p><p>0</p><p>1</p><p>�</p><p>,</p><p>and δ(x, x̃, u) = φ(x, u)−φ0(x̂, u). We view this equation as a perturbation of the</p><p>linear system ˙̃x =Ao x̃. In the absence of δ, asymptotic error convergence is achieved</p><p>by designing H = col(h1, h2) such that Ao is Hurwitz. In the presence of δ, we need</p><p>to design H with the additional goal of rejecting the effect of δ on x̃. This is ideally</p><p>achieved, for any δ, if the transfer function from δ to x̃,</p><p>Go(s) =</p><p>1</p><p>s2+ h1 s + h2</p><p>�</p><p>1</p><p>s + h1</p><p>�</p><p>,</p><p>1Throughout the book, ‖x‖=</p><p>p</p><p>xT x.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>1.2. MOTIVATING EXAMPLES 3</p><p>is identically zero. While this is not possible, we can make supω∈R ‖Go( jω)‖ arbitrarily</p><p>small by choosing h2� h1� 1. In particular, taking</p><p>h1 =</p><p>α1</p><p>ε</p><p>, h2 =</p><p>α2</p><p>ε2</p><p>for some positive constants α1, α2, and ε, with ε� 1, it can be shown that</p><p>Go(s) =</p><p>ε</p><p>(εs)2+α1εs +α2</p><p>�</p><p>ε</p><p>εs +α1</p><p>�</p><p>.</p><p>Hence, limε→0 Go(s) = 0. This disturbance rejection property of the high-gain ob-</p><p>server can be seen in the time domain by scaling the estimation error. Let</p><p>η1 =</p><p>x̃1</p><p>ε</p><p>, η2 = x̃2. (1.1)</p><p>Then</p><p>εη̇= F η+ εBδ, where F =</p><p>�</p><p>−α1 1</p><p>−α2 0</p><p>�</p><p>. (1.2)</p><p>The matrix F is Hurwitz because α1 and α2 are positive. The matrices Ao and F /ε</p><p>are related by the similarity transformation (1.1). Therefore, the eigenvalues of Ao are</p><p>1/ε times the eigenvalues of F . From (1.2) and the change of variables (1.1), we can</p><p>make some observations about the behavior of the estimation error. Using the bound</p><p>|δ| ≤ L‖x̃‖+M ≤ L‖η‖+M and the Lyapunov function V = ηT Pη, where P is the</p><p>solution of P F + F T P =−I , we obtain</p><p>εV̇ =−ηT η+ 2εηT PBδ ≤−‖η‖2+ 2εL‖PB‖ ‖η‖2+ 2εM‖PB‖ ‖η‖.</p><p>For εL‖PB‖ ≤ 1</p><p>4 ,</p><p>εV̇ ≤− 1</p><p>2‖η‖</p><p>2+ 2εM‖PB‖ ‖η‖.</p><p>Therefore (see [80, Theorem 4.5]), ‖η‖, and consequently ‖x̃‖, is ultimately bounded</p><p>by εcM for some c > 0, and</p><p>‖η(t )‖ ≤max</p><p>¦</p><p>b e−at/ε‖η(0)‖, εcM</p><p>©</p><p>∀ t ≥ 0</p><p>for some positive constants a and b . This inequality and the scaling (1.1) show that</p><p>|x̃1| ≤max</p><p>¦</p><p>b e−at/ε‖x(0)‖, ε2cM</p><p>©</p><p>, |x̃2| ≤max</p><p>§ b</p><p>ε</p><p>e−at/ε‖x(0)‖, εcM</p><p>ª</p><p>.</p><p>Hence, ‖x̃(t )‖ approaches the ultimate bound exponentially fast, and the smaller ε,</p><p>the faster the rate of decay, which shows that for sufficiently small ε the estimation</p><p>error x̃ will be much faster than x. The ultimate bound can be made arbitrarily small</p><p>by choosing ε small enough. If M = 0, which is the case when φ0 = φ, then x̃(t )</p><p>converges to zero as t tends to infinity. Notice, however, that whenever x1(0) 6=</p><p>x̂1(0), η1(0) = O(1/ε).2 Consequently, the solution of (1.2) will contain a term of</p><p>the form (1/ε)e−at/ε for some a > 0. While this exponential mode decays rapidly for</p><p>small ε, it exhibits an impulsive-like behavior where the transient response peaks to</p><p>O(1/ε) values before it decays rapidly toward zero. In fact, the function (a/ε)e−at/ε</p><p>2A function f (ε) is O(εn) if | f (ε)| ≤ kεn for some k > 0, where n can be positive or negative.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>4 CHAPTER 1. INTRODUCTION</p><p>approaches an impulse function as ε tends to zero. This behavior is known as the</p><p>peaking phenomenon. It has a serious impact when the observer is used in feedback</p><p>control, as we shall see in the next example. We use numerical simulation to illustrate</p><p>the foregoing observations. Consider the system</p><p>ẋ1 = x2, ẋ2 =−x1− 2x2+ ax2</p><p>1 x2+ b sin2t , y = x1,</p><p>with a = 0.25 and b = 0.2. It can be shown that for all x(0) ∈ Ω = {1.5x2</p><p>1 + x1x2 +</p><p>0.5x2</p><p>2 ≤</p><p>p</p><p>2}, x(t ) is bounded [80, Example 11.1]. We use the high-gain observer</p><p>˙̂x1 = x̂2+</p><p>2</p><p>ε</p><p>(y − x̂1),</p><p>˙̂x2 =−x̂1− 2x̂2+ â x̂2</p><p>1 x̂2+ b̂ sin2t +</p><p>1</p><p>ε2</p><p>(y − x̂1),</p><p>with two different choices of the pair (â, b̂ ). When a = 0.25 and b = 0.2 are known,</p><p>we take â = 0.25 and b̂ = 0.2. This is a case with no model uncertainty and φ0 = φ.</p><p>The other case is when the coefficients a and b are unknown. In this case we take</p><p>â = b̂ = 0. Figure 1.1 shows simulation results for both cases. Figures 1.1(a) and (b)</p><p>show the estimation errors x̃1 and x̃2 in the no-uncertainty case for different values</p><p>of ε. The estimation error x̃2 illustrates the peaking phenomenon. We note that the</p><p>peaking phenomenon is not present in x̃1. While peaking is induced by x̃1(0), it does</p><p>not appear in x̃1 because x̃1 = εη1. Figures 1.1(c) and (d) show x̃2 for the uncertain</p><p>model case when â = b̂ = 0. Comparison of Figures 1.1(b) and (c) shows that the</p><p>presence of uncertainty has very little effect on the performance of the observer when</p><p>ε is sufficiently small. Figure 1.1(d) demonstrates the fact that the ultimate bound on</p><p>x̃2 is O(ε). 4</p><p>0 0.1 0.2 0.3 0.4 0.5</p><p>−0.2</p><p>0</p><p>0.2</p><p>0.4</p><p>0.6</p><p>0.8</p><p>1</p><p>Time</p><p>x̃</p><p>1</p><p>(a)</p><p>0 0.1 0.2 0.3 0.4 0.5</p><p>−40</p><p>−30</p><p>−20</p><p>−10</p><p>0</p><p>(b)</p><p>Time</p><p>x̃</p><p>2</p><p>0 0.1 0.2 0.3 0.4 0.5</p><p>−40</p><p>−30</p><p>−20</p><p>−10</p><p>0</p><p>(c)</p><p>Time</p><p>x̃</p><p>2</p><p>5 6 7 8 9 10</p><p>−0.04</p><p>−0.02</p><p>0</p><p>0.02</p><p>0.04</p><p>(d)</p><p>Time</p><p>x̃</p><p>2</p><p>ε = 0.1</p><p>ε = 0.01</p><p>Figure 1.1. Simulation of Example 1.1. Figures (a) and (b) show the estimation errors x̃1</p><p>and x̃2 in the case â = a and b̂ = b . Figures (c) and (d) show the transient and steady-state behavior</p><p>of x̃2 in the case</p><p>convergence of e to zero; the best we can hope for is to show that, after</p><p>a transient period, e will be small in some sense. Therefore, we replace Assumption 4.3</p><p>with the less restrictive assumption, given next.29</p><p>Assumption 4.5. There is a continuously differentiable function V0(η), possibly depen-</p><p>dent on θ, classK∞ functions ϑ1 and ϑ2, and classK functions ϑ3 and ϑ4, all indepen-</p><p>dent of θ, such that</p><p>ϑ1(‖η‖)≤V0(η)≤ ϑ2(‖η‖), (4.37)</p><p>∂ V0</p><p>∂ η</p><p>p(η, x,θ)≤−ϑ3(‖η‖) ∀ ‖η‖ ≥ ϑ4(‖x‖) (4.38)</p><p>for all η ∈ Rm , x ∈ Rn , and t ≥ 0.</p><p>29Assumption 4.5 is equivalent to input-to-state stability of η̇= p(η, x,θ) with input x [145].</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>4.5. ROBUSTNESS 99</p><p>We start by determining the compact set of operation under state feedback when</p><p>∆= 0. As in Section 4.2, suppose θ̂(0) ∈Ω, e(0) ∈ E0, and z(0) ∈ Z0 and define c1, c2,</p><p>and c3 > c1+ c2 as before. Let E = {eT Pe ≤ c3} and cx =maxe∈E ,R∈Yr</p><p>‖e+R‖. Then,</p><p>‖x‖ ≤ cx . For any c4 ≥ ϑ2(ϑ4(cx )), {V0(η)≤ c4} is positively invariant because</p><p>V0 = c4⇒ ϑ2(‖η‖)≥ c4⇒ ϑ2(‖η‖)≥ ϑ2(ϑ4(cx ))⇒‖η‖ ≥ ϑ4(cx )⇒ V̇0 0 such that for all d ≤ d ∗ the sets {V ≤ c3} and {V0 ≤ c4}</p><p>remain positively invariant under the perturbation∆. It follows that the compact set</p><p>of operation under state feedback remains as e ∈ E , z ∈ Z , R ∈ Yr , |r (n)| ≤ Mr , and</p><p>θ̂ ∈Ωδ .</p><p>The analysis of the output feedback controller proceeds as in the proof of</p><p>Theorem 4.1 to show that the set</p><p>Σ= {V ≤ c3} ∩ {θ̂ ∈Ωδ}× {V0 ≤ c4}× {V2 ≤ c5ε</p><p>2}</p><p>is positively invariant and there is time T (ε), with limε→0 T (ε) = 0, such that the</p><p>trajectories of the closed-loop system are confined to Σ for all t ≥ T (ε). The main</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>100 CHAPTER 4. ADAPTIVE CONTROL</p><p>difference arises in showing that V̇ ≤ 0 on the boundary V = c3. In the presence of</p><p>the disturbance∆, V̇ satisfies the inequality</p><p>V̇ ≤−eT Qe + ka d + kcε</p><p>≤−c0V + 1</p><p>2 c0θ̃</p><p>T Γ−1θ̃+ ka d + kcε</p><p>≤−c0V + c0c2+ ka d + kcε</p><p>for some positive constant ka and kc . When d ≤ 1</p><p>2 c0(c3−c2)/ka and ε</p><p>ph</p><p>p</p><p>102 CHAPTER 4. ADAPTIVE CONTROL</p><p>5 10 15</p><p>−0.02</p><p>−0.01</p><p>0</p><p>0.01</p><p>0.02</p><p>0.03</p><p>0.04</p><p>0.05</p><p>Time</p><p>e 1</p><p>(a)</p><p>5 10 15</p><p>−0.02</p><p>−0.01</p><p>0</p><p>0.01</p><p>0.02</p><p>0.03</p><p>0.04</p><p>0.05</p><p>Time</p><p>e 1</p><p>(b)</p><p>Figure 4.10. Simulation of Example 4.5. (a) shows the tracking error e1 at steady state</p><p>for d = 0.1 (dashed) and d = 0.01 (solid) when no robustifying control is used. (b) shows e1 for</p><p>d = 0.1 in three cases: no robustifying control (dashed), robustifying control with µ = 0.02 (solid),</p><p>and robustifying control with µ= 0.002 (dotted).</p><p>whereφ1 toφN are known basis functions while θ1 to θN are adjustable weights to be</p><p>determined by an adaptive law.32 Different approximation structures are available in</p><p>the literature, including polynomials, splines, neural networks with radial basis func-</p><p>tions, fuzzy approximation, and wavelets. All these methods share the universal ap-</p><p>proximation property; namely, in a given compact set of x, the approximation error</p><p>∆ f (x) = f (x)−</p><p>N</p><p>∑</p><p>i</p><p>θiφi (x)</p><p>can be made arbitrarily small by choosing N sufficiently large.</p><p>In this section, we extend the adaptive controller of Sections 4.2 and 4.3 to</p><p>approximation-based control of a single-input–single-output nonlinear system repre-</p><p>sented globally by the input-output model</p><p>y (n) = F (·)+G(·)u (m), (4.43)</p><p>where u is the control input, y is the measured output, m 0 ∀ x ∈ Rn , z ∈ Rm .</p><p>Assumption 4.7. There exists a global diffeomorphism</p><p>�</p><p>η</p><p>x</p><p>�</p><p>=</p><p>�</p><p>T1(x, z)</p><p>x</p><p>�</p><p>def= T (x, z) (4.45)</p><p>with T1(0,0) = 0, which transforms (4.44) into the global normal form</p><p>η̇= p(η, x),</p><p>ẋi = xi+1, 1≤ i ≤ n− 1,</p><p>ẋn = F (x, z)+G(x, z)v,</p><p>y = x1.</p><p></p><p></p><p></p><p></p><p></p><p>(4.46)</p><p>Assumption 4.8. There is a continuously differentiable function V0(η), classK∞ func-</p><p>tions ϑ1 and ϑ2, and classK functions ϑ3 and ϑ4 such that</p><p>ϑ1(‖η‖)≤V0(η)≤ ϑ2(‖η‖), (4.47)</p><p>∂ V0</p><p>∂ η</p><p>p(η, x)≤−ϑ3(‖η‖) ∀ ‖η‖ ≥ ϑ4(‖x‖) (4.48)</p><p>for all η ∈ Rm and x ∈ Rn .</p><p>The functions F (x, z) and G(x, z) will be approximated for x ∈ X and z ∈ Z , for</p><p>some compact sets X ⊂ Rn and Z ⊂ Rm . The construction of X and Z builds on</p><p>our earlier discussion and makes use of Assumption 4.8. With e ,R , A, B , K , and P as</p><p>defined earlier, suppose e(0) and z(0) belong to the compact sets E0 ⊂ Rn and Z0 ⊂ Rm ,</p><p>respectively. Take c1 ≥ maxe∈E0</p><p>eT Pe and c3 > c1. Define E = {eT Pe ≤ c3} and</p><p>X = {e +R | e ∈ E , R ∈ Yr }. Let cx =maxx∈X ‖x‖ and choose c4 > 0 large enough</p><p>such that c4 ≥ ϑ2(ϑ4(cx )) and for all e(0) ∈ E0 and z(0) ∈ Z0, η(0) ∈ {V0(η) ≤ c4}.</p><p>Define a compact set Z such that for all x ∈ X and η ∈ {V0(η) ≤ c4}, z ∈ Z . The</p><p>definition of X and Z is done in such a way that if we can ensure that e(t ) ∈ E for all</p><p>t ≥ 0, then x(t ) ∈X and z(t ) ∈ Z for all t ≥ 0.</p><p>Using a universal approximator, we represent the functions F (x, z) and G(x, z)</p><p>over X ×Z as</p><p>F (x, z) = θ∗Tf f (x, z)+∆ f (x, z),</p><p>G(x, z) = θ∗Tg g (x, z)+∆g (x, z),</p><p>where the weight vectors θ∗f ∈ R`1 and θ∗g ∈ R`2 are the optimal weights, defined by</p><p>θ∗f = arg min</p><p>θ f ∈R`1</p><p>�</p><p>sup</p><p>x∈X ,z∈Z</p><p>|F (x, z)−θT</p><p>f f (x, z)|</p><p>�</p><p>,</p><p>θ∗g = arg min</p><p>θg∈R`2</p><p>�</p><p>sup</p><p>x∈X ,z∈Z</p><p>|G(x, z)−θT</p><p>g g (x, z)|</p><p>�</p><p>,</p><p>and∆ f and∆g are the approximation errors.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>104 CHAPTER 4. ADAPTIVE CONTROL</p><p>Assumption 4.9. The vectors θ∗f and θ∗g belong to known compact convex sets Ω f ⊂ R`1</p><p>and Ωg ⊂ R`2 , respectively.</p><p>Assumption 4.10. |θT</p><p>g g (x, z)| ≥ ḡ > 0 for all x ∈X , z ∈ Z, and θg ∈ Ω̂g , where Ω̂g is</p><p>a compact convex set that contains Ωg in its interior.</p><p>In view of the robustness results of the previous section, the adaptive controller</p><p>can be designed as in Section 4.2 using the Lyapunov function</p><p>V = eT Pe + 1</p><p>2 θ̃</p><p>T Γ−1θ̃,</p><p>where θ̃= col(θ̂ f −θ∗f , θ̂g −θ∗g ), the feedback control</p><p>v =</p><p>−Ke + r (n)− θ̂T</p><p>f f (x, z)</p><p>θ̂T</p><p>g g (x, z)</p><p>def= ψ(e , z,R , r (n), θ̂),</p><p>and the adaptive law</p><p>˙̂</p><p>θ=Π Γ ϕ,</p><p>where ϕ = col( f , gψ), θ̂ = col(θ̂ f , θ̂g ), and the parameter projection Π is defined as</p><p>in Section 4.2 to ensure that θ̂ ∈ Ωδ . The only missing piece from the analysis of the</p><p>previous sections is the requirement c3 > c1+ c2, where</p><p>c2 ≥ max</p><p>θ∗∈Ω,θ̂∈Ωδ</p><p>1</p><p>2 θ̃</p><p>T Γ−1θ̃.</p><p>In Section 4.2, c1 and c2 were determined first; then c3 was chosen to satisfy c3 > c1+c2.</p><p>In the current case c1 and c3 are chosen first because the choice of c3 determines the sets</p><p>X and Z , which are needed to define the function approximation. The condition c3 ></p><p>c1+ c2 is imposed by choosing the adaptation gain large enough to make c2 k2/(c3− c1), we can ensure that c2 0</p><p>(dependent on γ ) such that for all d ≤ d ∗, the sets {V ≤ c3} and {V0 ≤ c4} are positively</p><p>invariant under the perturbation∆ f +∆gψ.</p><p>The output feedback control is given by</p><p>v =ψs (e , z,R , r (n), θ̂),</p><p>where ê is provided by the high-gain observer (4.18), ψs = S sat(ψ/S), and S ≥</p><p>max |ψ(e , z,R , r (n), θ̂)| with the maximization taken over all e ∈ E , z ∈ Z , R ∈ Yr ,</p><p>|r (n)| ≤Mr and θ̂ ∈Ωδ . The analysis of the output feedback controller proceeds as in</p><p>the proof of Theorem 4.1 to show that the set</p><p>Σ= {V ≤ c3} ∩ {θ̂ ∈Ωδ}× {V0 ≤ c4}× {V2 ≤ c5ε</p><p>2}</p><p>is positively invariant and there is time T (ε), with limε→0 T (ε) = 0, such that the</p><p>trajectories of the closed-loop system are confined to Σ for all t ≥ T (ε). The main</p><p>difference arises in showing that V̇ ≤ 0 on the boundary V = c3. In the presence of</p><p>the disturbance∆ f +∆gψ, V̇ satisfies the inequality</p><p>V̇ ≤−eT Qe + ka d + kcε</p><p>≤−c0V + 1</p><p>2 c0θ̃</p><p>T Γ−1θ̃+ ka d + kcε</p><p>≤−c0V +</p><p>c0k2</p><p>γ</p><p>+ ka d + kcε</p><p>for some positive constant ka and kc . When d ≤ 1</p><p>2 c0(c3− k2/γ )/ka and ε</p><p>d + kcε,</p><p>which shows that ‖e‖ is ultimately bounded with an ultimate bound of the order</p><p>O(</p><p>p</p><p>d + ε+ 1/γ ). For sufficiently large γ and sufficiently small ε, the ultimate bound</p><p>on ‖e‖will be of the order of O(</p><p>p</p><p>d ). As a point of comparison with Section 4.5, note</p><p>that by choosing γ large, we establish an ultimate bound on the error rather than a</p><p>bound on the mean-square error.</p><p>When d is not small enough, we can use Lyapunov redesign to reduce the tracking</p><p>error. The state feedback control is modified to</p><p>ψ(e , z,R , r (n), θ̂) =</p><p>−Ke + r (n)− θ̂T</p><p>f f (x, z)+ψr (e , z,R)</p><p>θ̂T</p><p>g g (x, z)</p><p>, (4.49)</p><p>where ψr is the robustifying control component. Because ∆ f +∆g v depends on v,</p><p>we assume that it satisfies the inequality</p><p>|∆ f (x, z)+∆g (x, z)ψ(e , z,R , r (n), θ̂)| ≤β(e , z,R , r (n), θ̂)+ κ |ψr |</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>106 CHAPTER 4. ADAPTIVE CONTROL</p><p>for a known locally Lipschitz function β and a known positive constant κ</p><p>explains the abundance of integral</p><p>control in automatic control systems.</p><p>The internal model principle has been extended to classes of nonlinear systems.35</p><p>The results of this chapter present one particular extension that is built around the</p><p>use of high-gain observers. In this approach the exosystem is a linear model that has</p><p>distinct eigenvalues on the imaginary axis; thus, it generates constant or sinusoidal</p><p>signals. There are three basic ingredients of the approach. First, by studying the dy-</p><p>namics of the system on an invariant manifold at which the regulation error is zero,</p><p>called the zero-error manifold, a linear internal model is derived. The internal model</p><p>generates not only the modes of the exosystem but also a number of higher-order har-</p><p>monics induced by nonlinearities. Second, a separation approach is used to design a</p><p>robust output feedback controller where a state feedback controller is designed first,</p><p>1</p><p>- l - - - -</p><p>6</p><p>66</p><p>r u y</p><p>−</p><p>+</p><p>Stabilizing</p><p>Controller</p><p>Servo-</p><p>Compensator</p><p>Plant</p><p>Measured</p><p>Signals</p><p>?</p><p>d</p><p>Figure 5.1. Schematic diagram of feedback control with servocompensator.</p><p>34See, for example, [33], [34], and [44].</p><p>35See [28], [30], [60], [69], [70], and [115], and the references therein.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.2. INTEGRAL CONTROL 109</p><p>and then a high-gain observer that estimates the derivatives of the regulation error is</p><p>used to recover the performance achieved under state feedback. As with earlier chap-</p><p>ters, the control is saturated outside a compact set of interest to overcome the effect</p><p>of peaking. Third, to achieve regional or semiglobal stabilization of the augmented</p><p>system (formed of the plant and the servocompensator), the state feedback design uses</p><p>a strategy whereby a robust controller is designed as if the goal were to stabilize the</p><p>origin. This controller brings the trajectories of the system to a small neighborhood</p><p>of the origin in finite time. Near the origin, the robust controller, acting as a high-</p><p>gain feedback controller, and the servocompensator stabilize the zero-error manifold.</p><p>Three different robust control techniques have been used in the literature, namely,</p><p>high-gain feedback, Lyapunov redesign, and sliding mode control.36 The results of</p><p>this chapter use a continuous implementation of sliding mode control.</p><p>5.2 Integral Control</p><p>Consider a single-input–single-output nonlinear system modeled by</p><p>ẋ = f (x, w)+ g (x, w)u, e = h(x, w), (5.1)</p><p>where x ∈ Rn is the state, u is the control input, e is the output, and w is a constant</p><p>vector of reference, disturbance, and system parameters that belongs to a compact set</p><p>W ⊂ Rl . The functions f , g , and h are sufficiently smooth in a domain Dx ⊂ Rn . The</p><p>goal is to design an output feedback controller such that all state variables are bounded</p><p>and the output e is asymptotically regulated to zero.</p><p>Assumption 5.1. For each w ∈ W , the system (5.1) has relative degree ρ ≤ n, for all</p><p>x ∈Dx , that is,</p><p>Lg h = Lg L f h = · · ·= Lg Lρ−2</p><p>f h = 0, Lg Lρ−1</p><p>f h 6= 0,</p><p>and there is a diffeomorphism</p><p>�</p><p>η</p><p>ξ</p><p>�</p><p>= T (x)</p><p>in Dx , possibly dependent on w, that transforms (5.1) into the normal form37</p><p>η̇= f0(η,ξ , w),</p><p>ξ̇i = ξi+1, 1≤ i ≤ ρ− 1,</p><p>ξ̇ρ = a(η,ξ , w)+ b (η,ξ , w)u,</p><p>e = ξ1.</p><p>Moreover, b (η,ξ , w)≥ b0 > 0 for all (η,ξ ) ∈ T (Dx ) and w ∈W .</p><p>The relative degree assumption guarantees the existence of the change of variables</p><p>(5.2) locally for each w ∈W [64]. Assumption 5.1 goes beyond that by requiring the</p><p>change of variables to hold in a given region.</p><p>36See [80] for an introduction to these three techniques.</p><p>37For ρ= n, η and the η̇-equation are dropped.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>110 CHAPTER 5. REGULATION</p><p>For the system to maintain the steady-state condition e = 0, it must have an equi-</p><p>librium point (η,ξ ) = (η̄, 0) and a steady-state control ū that satisfy the equations</p><p>0= f0(η̄, 0, w),</p><p>0= a(η̄, 0, w)+ b (η̄, 0, w)ū.</p><p>Assumption 5.2. For each w ∈W , the equation 0= f0(η̄, 0, w) has a unique solution η̄</p><p>such that (η̄, 0) ∈ T (Dx ).</p><p>Because b 6= 0, the steady-state control that maintains equilibrium is</p><p>ū =−</p><p>a(η̄, 0, w)</p><p>b (η̄, 0, w)</p><p>def= φ(w).</p><p>The change of variables z = η− η̄ transforms the system into the form</p><p>ż = f0(z + η̄,ξ , w) def= f̃0(z,ξ , w), (5.2)</p><p>ξ̇i = ξi+1 for 1≤ i ≤ ρ− 1, (5.3)</p><p>ξ̇ρ = a0(z,ξ , w)+ b (η,ξ , w)[u −φ(w)], (5.4)</p><p>where f̃0(0,0, w) = 0 and a0(0,0, w) = 0. Let Γ ⊂ Rn be a compact set, which contains</p><p>the origin in its interior, such that (z,ξ ) ∈ Γ implies that x ∈Dx for each w ∈W . The</p><p>size of W may have to be restricted in order for Γ to exist.</p><p>The controller is designed using a separation approach. First, a partial state feed-</p><p>back controller is designed in terms of ξ to regulate the output. Then, a high-gain</p><p>observer is used to recover the performance of the state feedback controller. Due to</p><p>the uncertainty of the system, the state feedback controller is designed using sliding</p><p>mode control.38 The restriction to partial state feedback is possible when the system</p><p>is minimum phase. The next two assumptions imply the minimum phase property.</p><p>Assumption 5.3. For each w ∈W , there is a Lyapunov function V1(z), possibly depen-</p><p>dent on w, and classK functions γ1 to γ4, independent of w, such that</p><p>γ1(‖z‖)≤V1(z)≤ γ2(‖z‖),</p><p>∂ V1</p><p>∂ z</p><p>f̃0(z,ξ , w)≤−γ3(‖z‖) ∀ ‖z‖ ≥ γ4(‖ξ ‖)</p><p>for all (z,ξ ) ∈ Γ .</p><p>Assumption 5.4. z = 0 is an exponentially stable equilibrium point of ż = f̃0(z, 0, w),</p><p>uniformly in w.</p><p>Assumption 5.3 implies that, with ξ as the driving input, the system ż = f̃0(z,ξ , w)</p><p>is regionally input-to-state stable, uniformly in w.39</p><p>Augmenting the integrator</p><p>σ̇ = e (5.5)</p><p>38Other robust control techniques such as Lyapunov redesign and high-gain feedback can be used. See</p><p>[80, Chapter 10].</p><p>39See [144] or [80] for the definition of input-to-state stability.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.2. INTEGRAL CONTROL 111</p><p>with the system (5.2)–(5.4) yields</p><p>ż = f̃0(z,ξ , w),</p><p>σ̇ = ξ1,</p><p>ξ̇i = ξi+1 for 1≤ i ≤ ρ− 1,</p><p>ξ̇ρ = a0(z,ξ , w)+ b (η,ξ , w)[u −φ(w)],</p><p>which preserves the normal-form structure with a chain of ρ+ 1 integrators. Let</p><p>s = k0σ + k1ξ1+ · · ·+ kρ−1ξρ−1+ ξρ,</p><p>where k0 to kρ−1 are chosen such that the polynomial</p><p>λρ+ kρ−1λ</p><p>ρ−1+ · · ·+ k1λ+ k0</p><p>is Hurwitz. Then</p><p>ṡ =</p><p>ρ−1</p><p>∑</p><p>i=0</p><p>kiξi+1+ a0(z,ξ , w)+ b (η,ξ , w)[u −φ(w)] def= ∆(z,ξ , w)+ b (η,ξ , w)u.</p><p>Let %(ξ ) be a known locally Lipschitz function such that</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>∆(z,ξ , w)</p><p>b (η,ξ , w)</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤ %(ξ )</p><p>for all (z,ξ , w) ∈ Γ ×W . The state feedback controller is taken as</p><p>u =−β(ξ ) sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>, (5.6)</p><p>whereβ(ξ ) is a locally Lipschitz function that satisfiesβ(ξ )≥ %(ξ )+β0 withβ0 > 0.</p><p>The functions% andβ are allowed to depend only on ξ rather than the full state (z,ξ ).</p><p>This is possible because the inequality |∆/b | ≤ % is required to hold over the compact</p><p>set Γ where the z-dependent part of ∆/b can be bounded by a constant. Under the</p><p>controller (5.6),</p><p>s ṡ = s∆− bβs sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>≤ b%|s | − bβs sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>.</p><p>For |s | ≥µ,</p><p>s ṡ ≤ b (%|s | −β|s |)≤−b0β0|s |.</p><p>The closed-loop system is given by</p><p>ż = f̃0(z,ξ , w),</p><p>ζ̇ =A1ζ +B s ,</p><p>ṡ =−b (η,ξ , w)β(ξ ) sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>+∆(z,ξ , w),</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>112 CHAPTER 5. REGULATION</p><p>where ζ = col(σ ,ξ1, . . . ,ξρ−1), ξ =A1ζ +B</p><p>s ,</p><p>A1 =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 · · · · · · 0</p><p>0 0 1 · · · 0</p><p>...</p><p>...</p><p>0 · · · · · · 0 1</p><p>−k0 · · · · · · · · · −kρ−1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, and B =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>...</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>.</p><p>The matrix A1 is Hurwitz by design. The inequality s ṡ ≤−b0β0|s | shows that the set</p><p>{|s | ≤ c}with c >µ is positively invariant because s ṡ</p><p>4||P1B‖2λmax(P1) because V̇2 0 exists, and for each µ ∈ (0,µ∗], ε∗ = ε∗(µ) exists</p><p>such that for each µ ∈ (0,µ∗] and ε ∈ (0,ε∗(µ)], all state variables are bounded and</p><p>limt→∞ ξ (t ) = 0. 3</p><p>Proof: The closed-loop system under output feedback is given by</p><p>ż = f̃0(z,ξ , w), (5.11)</p><p>ζ̇ =A1ζ +B s , (5.12)</p><p>ṡ = b (η,ξ , w)ψ(σ , ξ̂ ,µ)+∆(z,ξ , w), (5.13)</p><p>εϕ̇ =A0ϕ+ εB[∆(z,ξ , w)−</p><p>ρ−1</p><p>∑</p><p>i=0</p><p>kiξi+1+ b (η,ξ , w)ψ(σ , ξ̂ ,µ)], (5.14)</p><p>where</p><p>ψ(σ ,ξ ,µ) =−βs (ξ ) sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>, ϕi =</p><p>ξi − ξ̂i</p><p>ερ−i</p><p>for 1≤ i ≤ ρ,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>114 CHAPTER 5. REGULATION</p><p>and</p><p>A0 =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>−α1 1 0 · · · 0</p><p>−α2 0 1 · · · 0</p><p>...</p><p>. . .</p><p>...</p><p>−αρ−1 0 1</p><p>−αρ 0 · · · · · · 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>.</p><p>The matrix A0 is Hurwitz by design. Equations (5.11)–(5.13) with ψ(σ , ξ̂ ,µ) replaced</p><p>byψ(σ ,ξ ,µ) are the closed-loop system under state feedback. Let P0 be the solution of</p><p>the Lyapunov equation P0A0+AT</p><p>0 P0 =−I , V3(ϕ) = ϕ</p><p>T P0ϕ, andΣε = {V3(ϕ)≤ ρ3ε</p><p>2},</p><p>where the positive constant ρ3 is to be determined. The proof proceeds in four steps:</p><p>Step 1: Show that there exist ρ3 > 0, µ∗1 > 0, and ε∗1 = ε</p><p>∗</p><p>1(µ) > 0 such that for each</p><p>µ ∈ (0,µ∗1] and ε ∈ (0,ε∗1(µ)] the set Ω×Σε is positively invariant.</p><p>Step 2: Show that for any bounded ξ̂ (0) and any (z(0),ζ (0), s(0)) ∈ Ψ, there exists</p><p>ε∗2 > 0 such that for each ε ∈ (0,ε∗2] the trajectory enters the set Ω×Σε in finite</p><p>time T1(ε), where limε→0 T1(ε) = 0.</p><p>Step 3: Show that there exists ε∗3 = ε</p><p>∗</p><p>3(µ)> 0 such that for each ε ∈ (0,ε∗3(µ)] every</p><p>trajectory in Ω×Σε enters Ωµ×Σε in finite time and stays therein for all future</p><p>time.</p><p>Step 4: Show that there existsµ∗2 > 0 and ε∗4 = ε</p><p>∗</p><p>4(µ)> 0 such that for eachµ ∈ (0,µ∗2]</p><p>and ε ∈ (0,ε∗4(µ)] every trajectory inΩµ×Σε converges to the equilibrium point</p><p>(z = 0,ζ = ζ̄ , s = s̄ ,ϕ = 0) at which ξ = 0.</p><p>For the first step, calculate the derivative of V3 on the boundary V3 = ρ3ε</p><p>2:</p><p>εV̇3 =−ϕ</p><p>Tϕ+ 2εϕT P0B</p><p>�</p><p>∆(z,ξ , w)−</p><p>ρ−1</p><p>∑</p><p>i=0</p><p>kiξi+1+ b (η,ξ , w)ψ(σ , ξ̂ ,µ)</p><p>�</p><p>.</p><p>Since ψ(σ , ξ̂ ,µ) is globally bounded in ξ̂ , for all (z,ζ , s) ∈Ω there is `1 > 0 such that</p><p>|∆(z,ξ , w)−</p><p>∑ρ−1</p><p>i=0 kiξi+1+ b (η,ξ , w)ψ(σ , ξ̂ ,µ)| ≤ `1. Therefore</p><p>εV̇3 ≤−‖ϕ‖</p><p>2+ 2ε`1‖PoB‖ ‖ϕ‖ ≤− 1</p><p>2‖ϕ‖</p><p>2 ∀ ‖ϕ‖ ≥ 4ε`1‖P0B‖. (5.15)</p><p>Taking ρ3 = λmax(P0)(4‖P0B‖`1)</p><p>2 ensures that V̇3 ≤ −</p><p>1</p><p>2‖ϕ‖</p><p>2 for all V3 ≥ ρ3ε</p><p>2. Con-</p><p>sequently, V̇3 0, independent of ε, such that (z(t ),ζ (t ), s(t )) ∈ Ω for t ∈ [0,T0]. During this</p><p>time, inequality (5.15) shows that</p><p>V̇3 ≤−</p><p>1</p><p>2ελmax(P0)</p><p>V3.</p><p>Therefore, V3 reduces toρ3ε</p><p>2 within a time interval [0,T1(ε)] in which limε→0 T1(ε) =</p><p>0. For sufficiently small ε, T1(ε)</p><p>, s = s̄ ,ϕ = 0),</p><p>where ζ̄ = col(σ̄ , 0, . . . , 0), and</p><p>s̄ = k0σ̄ =</p><p>−µφ(w)</p><p>β(0)</p><p>.</p><p>Shifting the equilibrium point to the origin by the change of variables ν = ζ − ζ̄ and</p><p>p = s − s̄ , the system takes the singularly perturbed form</p><p>ż = f̃0(z,A1ν +B p, w),</p><p>ν̇ =A1ν +B p,</p><p>µ ṗ =−b (η,ξ , w)β(ξ )p +µ∆a(·)+µ∆b (·)+µb (η,ξ , w)[ψ(σ , ξ̂ ,µ)−ψ(σ ,ξ ,µ)],</p><p>εϕ̇ =A0ϕ+(ε/µ)B{−b (η,ξ , w)β(ξ )p +µ∆a(·)</p><p>+µb (η,ξ , w)[ψ(σ , ξ̂ ,µ)−ψ(σ ,ξ ,µ)]},</p><p>where</p><p>∆a = a0(z,ξ , w)+ b (η,ξ , w)φ(w)</p><p>�</p><p>β(ξ )−β(0)</p><p>β(0)</p><p>�</p><p>and∆b =</p><p>∑ρ−1</p><p>i=0 kiξi+1. There are positive constants `5 to `9 such that</p><p>|∆a | ≤ `5‖z‖+ `6‖ν‖+ `7|p|, |∆b | ≤ `8‖ν‖+ `9|p|.</p><p>Moreover, µ|ψ(σ , ξ̂ ,µ)−ψ(σ ,ξ ,µ)| ≤ (µ`2+ `3)‖ϕ‖.</p><p>By Assumption 5.4, z = 0 is an exponentially stable equilibrium point of ż =</p><p>f̃0(z, 0, w) uniformly in w. By the converse Lyapunov theorem [78, Lemma 9.8], there</p><p>is a Lyapunov function V0(z), possibly dependent on w, and positive constant c̄1 to</p><p>c̄4, independent of w, such that</p><p>c̄1‖z‖</p><p>2 ≤V0(z, w)≤ c̄2‖z‖</p><p>2,</p><p>∂ V0</p><p>∂ z</p><p>f̃0(z, 0, w)≤−c̄3‖z‖</p><p>2,</p><p>∂ V0</p><p>∂ z</p><p>≤ c̄4‖z‖</p><p>in some neighborhood of z = 0. Consider the composite Lyapunov function</p><p>V = αV0+ ν</p><p>T P1ν +</p><p>1</p><p>2</p><p>p2+ϕT P0ϕ</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>116 CHAPTER 5. REGULATION</p><p>with α > 0. It can be shown that V̇ ≤−Y T QY , where</p><p>Y =</p><p></p><p></p><p></p><p></p><p>‖z‖</p><p>‖ν‖</p><p>|p|</p><p>‖ϕ‖</p><p></p><p></p><p></p><p></p><p>, Q =</p><p></p><p></p><p></p><p></p><p>αc1 −αc2 −(αc3+ c4) −c5</p><p>−αc2 1 −c6 −c7</p><p>−(αc3+ c4) −c6 (c8/µ− c9) −(c10+ c11/µ)</p><p>−c5 −c7 −(c10+ c11/µ) (1/ε− c12− c13/µ)</p><p></p><p></p><p></p><p></p><p>,</p><p>and c1 to c13 are positive constants. Choose α 0 such that</p><p>the maximum of 2|ξ1|+4|ξ2|+3 over Ω is less than k. The output feedback controller</p><p>is given by</p><p>u =−k sat</p><p>σ + 2ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>,</p><p>where ξ̂1 and ξ̂2 are provided by the high-gain observer</p><p>˙̂</p><p>ξ1 = ξ̂2+</p><p>2</p><p>ε</p><p>(ξ1− ξ̂1),</p><p>˙̂</p><p>ξ2 =</p><p>1</p><p>ε2</p><p>(ξ1− ξ̂1).</p><p>Simulation results with zero initial conditions, r =π, µ= 0.1, k = 5, a = 0.03, b = 1,</p><p>and d = 0.3 are shown in Figure 5.2. Figures 5.2(a) and (b) show the output x1 when</p><p>ε= 0.01. The convergence of x1 to π is shown in Figure 5.2(b). Figures 5.2(c) and (d)</p><p>illustrate the property that the trajectories under output feedback approach the ones</p><p>under state feedback as ε decreases. They show the differences</p><p>∆xi = xi (under output feedback)− xi (under state feedback)</p><p>for i = 1,2 when ε= 0.01, 0.005, and 0.001.</p><p>For comparison, consider a sliding mode controller without integral action. The</p><p>design proceeds as in Section 3.2.2 with s1 = ξ1+ ξ2,</p><p>ṡ1 = (1− a)ξ2− sin x1+ b u + d cos x1,</p><p>0 2 4 6 8 10</p><p>0</p><p>1</p><p>2</p><p>3</p><p>4</p><p>(a)</p><p>Time</p><p>x</p><p>1</p><p>29 29.2 29.4 29.6 29.8 30</p><p>3.141</p><p>3.1412</p><p>3.1414</p><p>3.1416</p><p>3.1418</p><p>3.142</p><p>(b)</p><p>Time</p><p>x</p><p>1</p><p>0 2 4 6 8 10</p><p>−0.015</p><p>−0.01</p><p>−0.005</p><p>0</p><p>0.005</p><p>0.01</p><p>Time</p><p>∆</p><p>x</p><p>1</p><p>(c)</p><p>ε=0.01</p><p>ε=0.005</p><p>ε=0.001</p><p>0 2 4 6</p><p>−0.1</p><p>−0.08</p><p>−0.06</p><p>−0.04</p><p>−0.02</p><p>0</p><p>0.02</p><p>0.04</p><p>(d)</p><p>Time</p><p>∆</p><p>x</p><p>2</p><p>Figure 5.2. Simulation of Example 5.1. (a) and (b) show the transient and steady-state</p><p>responses of the output. (c) and (d) show the deviation of the state trajectories under output feedback</p><p>from the ones under state feedback as ε decreases.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>118 CHAPTER 5. REGULATION</p><p>0 2 4 6 8 10</p><p>0</p><p>1</p><p>2</p><p>3</p><p>4</p><p>(a)</p><p>Time</p><p>x</p><p>1</p><p>29 29.2 29.4 29.6 29.8 30</p><p>3.13</p><p>3.135</p><p>3.14</p><p>3.145</p><p>(b)</p><p>Time</p><p>x</p><p>1</p><p>Figure 5.3. Simulation of the output feedback controllers of Example 5.1 with (solid) and</p><p>without (dashed) integral action.</p><p>and</p><p>�</p><p>�</p><p>�</p><p>�</p><p>(1− a)ξ2− sin x1+ d cos x1</p><p>b</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤</p><p>|ξ2|+ 1+ 0.5</p><p>0.5</p><p>= 2|ξ2|+ 3.</p><p>Considering a compact set of operation where 2|ξ2|+ 3</p><p>to zero. Similar to Section 5.2,</p><p>the controller is designed using a separation approach. For the state feedback con-</p><p>troller, start with (5.2) to (5.4) and let</p><p>s1 = k1ξ1+ · · ·+ kρ−1ξρ−1+ ξρ,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.3. CONDITIONAL INTEGRATOR 119</p><p>where k1 to kρ−1 are chosen such that the polynomial</p><p>λρ−1+ kρ−1λ</p><p>ρ−2+ · · ·+ k2λ+ k1</p><p>is Hurwitz. Then</p><p>ṡ1 =</p><p>ρ−1</p><p>∑</p><p>i=1</p><p>kiξi+1+ a0(z,ξ , w)+ b (η,ξ , w)[u −φ(w)] def= ∆1(z,ξ , w)+ b (η,ξ , w)u.</p><p>Let %1(ξ ) be a known locally Lipschitz function such that</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>∆1(z,ξ , w)</p><p>b (η,ξ , w)</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤ %1(ξ )</p><p>for all (z,ξ , w) ∈ Γ ×W . A state feedback sliding mode controller can be taken as</p><p>u =−β(ξ ) sgn(s1), (5.16)</p><p>whereβ(ξ ) is a locally Lipschitz function that satisfiesβ(ξ )≥ %1(ξ )+β0 withβ0 > 0</p><p>and the signum function sgn(·) is defined by</p><p>sgn(s1) =</p><p>§</p><p>1 if s1 > 0,</p><p>−1 if s1 µ and s reaches the</p><p>positively invariant set {|s | ≤ µ} in finite time. Let V2(q) = qT P2q , where P2 is the</p><p>solution of the Lyapunov equation P2A2 +AT</p><p>2 P2 = −I . For |s | ≤ c , the derivative of</p><p>V2 satisfies the inequalities</p><p>V̇2 ≤−qT q + 2‖q‖ ‖P2B2‖ (|s |+ |σ |)≤−‖q‖</p><p>2+ 4‖P2B2‖ ‖q‖c ,</p><p>which shows that the set {V2 ≤ ρ̄1c2} × {|s | ≤ c} is positively invariant for ρ̄1 ></p><p>16||P2B2‖2λmax(P2) because V̇2 1 as a PIDρ−1 controller</p><p>with antiwindup.</p><p>integrator when the control saturates. The antiwindup scheme of Figure 5.4 has two special</p><p>features. First, the PI controller has high-gain k/µ. Second, the gain in the antiwindup</p><p>loop µ/k is the reciprocal of the controller’s high-gain. Figure 5.5 shows the block diagram</p><p>when the relative degree is higher than one. The transfer function H represents the high-</p><p>gain observer. When ρ= 2,</p><p>H = k1+</p><p>s</p><p>(εs)2/α2+(εs)α1/α2+ 1</p><p>,</p><p>and the controller takes the form of a PID controller with antiwindup. When ρ= 3,</p><p>H = k1+</p><p>k2 s +(1+ εk2α2/α3)s</p><p>2</p><p>(εs)3/α3+(εs)2α2/α3+(εs)α1/α3+ 1</p><p>,</p><p>and the controller is PID2 with antiwindup. For ρ > 3, the controller is PIDρ−1 with</p><p>antiwindup. 3</p><p>Theorem 5.2. Suppose Assumptions 5.1 to 5.4 are satisfied and consider the closed-loop</p><p>system formed of the system (5.1), the conditional integrator (5.19), the controller (5.20),</p><p>and the observer (5.21)–(5.22). Let Ψ be a compact set in the interior of Ω and suppose</p><p>|σ(0)| ≤ µ, (z(0), q(0), s(0)) ∈ Ψ, and ξ̂ (0) is bounded. Then µ∗ > 0 exists, and for each</p><p>µ ∈ (0,µ∗], ε∗ = ε∗(µ) exists such that for each µ ∈ (0,µ∗] and ε ∈ (0,ε∗(µ)], all state</p><p>variables are bounded and limt→∞ e(t ) = 0. Furthermore, let χ</p><p>= (z,ξ ) be part of the</p><p>state of the closed-loop system under the output feedback controller, and let χ ∗ = (z∗,ξ ∗)</p><p>be the state of the closed-loop system under the state feedback sliding mode controller (5.16),</p><p>with χ (0) = χ ∗(0). Then for every δ0 > 0 there is µ∗1 > 0, and for each µ ∈ (0,µ∗1] there</p><p>is ε∗1 = ε</p><p>∗</p><p>1(µ)> 0, such that for µ ∈ (0,µ∗1] and ε ∈ (0,ε∗1(µ)],</p><p>‖χ (t )−χ ∗(t )‖ ≤ δ0 ∀ t ≥ 0. (5.23)</p><p>3</p><p>Proof: The closed-loop system under output feedback is given by</p><p>σ̇ = γ [−σ +µ sat( ŝ/µ)] , (5.24)</p><p>ż = f̃0(z,ξ , w), (5.25)</p><p>q̇ =A2q +B2(s −σ), (5.26)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.3. CONDITIONAL INTEGRATOR 123</p><p>ṡ = b (η,ξ , w)ψ(σ , ξ̂ ,µ)+∆1(z,ξ , w)+ γ [−σ +µ sat( ŝ/µ)] , (5.27)</p><p>εϕ̇ =A0ϕ+ εB[a0(z,ξ , w)− b (η,ξ , w)φ(w)+ b (η,ξ , w)ψ(σ , ξ̂ ,µ)], (5.28)</p><p>where</p><p>ŝ = σ + k1ξ̂1+ · · ·+ kρ−1ξ̂ρ−1+ ξ̂ρ, ϕi =</p><p>ξi − ξ̂i</p><p>ερ−i</p><p>for 1≤ i ≤ ρ,</p><p>ψ(σ ,ξ ,µ) =−βs (ξ ) sat(s/µ), and</p><p>A0 =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>−α1 1 0 · · · 0</p><p>−α2 0 1 · · · 0</p><p>...</p><p>. . .</p><p>...</p><p>−αρ−1 0 1</p><p>−αρ 0 · · · · · · 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>.</p><p>The matrix A0 is Hurwitz by design. The proof proceeds in four steps similar to the</p><p>proof of Theorem 5.1. The first three steps are almost the same and will not be repeated</p><p>here. For the fourth step, consider the system inside the set {|σ | ≤µ}×Ωµ×Σε, where</p><p>Σε = {ϕT P0ϕ ≤ ρ3ε</p><p>2} and P0 is the solution of the Lyapunov equation P0A0 + AT</p><p>0</p><p>P0 =−I . There is an equilibrium point at (σ = σ̄ , z = 0, q = 0, s = s̄ ,ϕ = 0), where</p><p>s̄ = σ̄ =</p><p>−µφ(w)</p><p>β(0)</p><p>.</p><p>Shifting the equilibrium point to the origin by the change of variables ϑ = σ − σ̄ and</p><p>p = s − s̄ , the system takes the singularly perturbed form</p><p>ϑ̇ = γ {−ϑ+ p +µ[sat( ŝ/µ)− sat(s/µ)]} ,</p><p>ż = f̃0(z, col(q , p −ϑ− Lq), w),</p><p>q̇ =A2q +B2(p −ϑ),</p><p>µ ṗ =−b (η,ξ , w)β(ξ )p +µ∆a(·)+µ∆c (·)+µb (η,ξ , w)[ψ(σ , ξ̂ ,µ)−ψ(σ ,ξ ,µ)]</p><p>+ γµ2[sat( ŝ/µ)− sat(s/µ)]+ γµ(p −ϑ),</p><p>εϕ̇ =A0ϕ+(ε/µ)B{−b (η,ξ , w)β(ξ )p +µ∆a(·)</p><p>+µb (η,ξ , w)[ψ(σ , ξ̂ ,µ)−ψ(σ ,ξ ,µ)]},</p><p>where</p><p>∆a = a0(z,ξ , w)+ b (η,ξ , w)φ(w)</p><p>�</p><p>β(ξ )−β(0)</p><p>β(0)</p><p>�</p><p>and ∆c =</p><p>ρ−1</p><p>∑</p><p>i=1</p><p>kiξi+1.</p><p>There are positive constants `1 to `10 such that</p><p>µ|sat( ŝ/µ)− sat(s/µ)| ≤ `1‖ϕ‖,</p><p>|ψ(σ , ξ̂ ,µ)−ψ(σ ,ξ ,µ)| ≤ (`2+ `3/µ)‖ϕ‖,</p><p>|∆a | ≤ `4|ϑ|+ `5‖z‖+ `6‖q‖+ `7|p|,</p><p>|∆c | ≤ `8|ϑ|+ `9‖q‖+ `10|p|.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>124 CHAPTER 5. REGULATION</p><p>By Assumption 5.4, z = 0 is an exponentially stable equilibrium point of ż = f̃0(z, 0, w).</p><p>By the converse Lyapunov theorem [78, Lemma 9.8], there is a Lyapunov function</p><p>V0(z), possibly dependent on w, that satisfies the inequalities</p><p>c̄1‖z‖</p><p>2 ≤V0(z)≤ c̄2‖z‖</p><p>2,</p><p>∂ V0</p><p>∂ z</p><p>f̃0(z, 0, w)≤−c̄3‖z‖</p><p>2,</p><p>∂ V0</p><p>∂ z</p><p>≤ c̄4‖z‖,</p><p>in some neighborhood of z = 0, where c̄1 to c̄4 are positive constants independent</p><p>of w. Consider the composite Lyapunov function</p><p>V =V0+ κ1 qT P2q +</p><p>κ2</p><p>2γ</p><p>ϑ2+ 1</p><p>2 p2+ϕT P0ϕ,</p><p>with positive constants κ1 and κ2. It can be shown that V̇ ≤ −Y T QY , where Y =</p><p>col(‖z‖, ‖q‖, |ϑ|, |p|, ‖ϕ‖),</p><p>Q =</p><p></p><p></p><p></p><p></p><p></p><p></p><p>c1 −c2 −c3 −c4 −c5</p><p>−c2 κ1 − κ1 c6 −(κ1 c7+ c8) −c9</p><p>−c3 − κ1 c6 κ2 −(κ2 c10+ c11) −(κ2 c12+ c13)</p><p>−c4 −(κ1 c7+ c8) −(κ2 c10+ c11) (c14/µ− c15) −(c16+ c17/µ)</p><p>−c5 −c9 −(κ2 c12+ c13) −(c16+ c17/µ) (1/ε− c18− c19/µ)</p><p></p><p></p><p></p><p></p><p></p><p></p><p>,</p><p>and c1 to c19 are positive constants independent of κ1, κ2, µ, and ε. Choose κ1 large</p><p>enough to make the 2× 2 principal minor of Q positive; then choose κ2 large enough</p><p>to make the 3× 3 principal minor positive; then choose µ small enough to make the</p><p>4× 4 principal minor positive; then choose ε small enough to make the determinant</p><p>of Q positive. Hence, the origin (ϑ = 0, z = 0, q = 0, p = 0,ϕ = 0) is exponen-</p><p>tially stable, and there is a neighborhood N of the origin, independent of µ and ε,</p><p>such that all trajectories in N converge to the origin as t →∞. By choosing µ and</p><p>ε small enough, it can be ensured that for all (σ , z, q , s ,ϕ) ∈ {|σ | ≤ µ} × Ωµ × Σε,</p><p>(ϑ, z, q , p,ϕ) ∈N . Thus, all trajectories in {|σ | ≤µ}×Ωµ×Σε converge to the equi-</p><p>librium point (σ = σ̄ , z = 0, q = 0, s = s̄ ,ϕ = 0). Consequently, all trajectories with</p><p>(z(0),ζ (0), s(0)) ∈ Ψ, |σ(0)| ≤µ, and bounded ξ̂ (0) converge to this equilibrium point</p><p>because such trajectories enter {|σ | ≤µ}×Ωµ×Σε in finite time. This completes the</p><p>proof that all the state variables are bounded and limt→∞ e(t ) = 0. The proof of (5.23)</p><p>is done in two steps. In the first step, it is shown that the trajectories under the continu-</p><p>ously implemented state feedback controller with conditional integrator (5.17)–(5.18)</p><p>are O(µ) close to the trajectories under the state feedback sliding mode controller</p><p>(5.16). In the second step, with fixed µ, it is shown that the trajectories under the</p><p>output feedback controller (5.19)–(5.22) can be made arbitrarily close to the trajecto-</p><p>ries under the state feedback controller (5.17)–(5.18) by choosing ε small enough. The</p><p>argument for the second step is similar to the argument used in the performance recov-</p><p>ery part of the proof of Theorem 3.1 and will not be repeated here. For the first step, let</p><p>χ † = (z†,ξ †) be part of the state of the closed-loop system under the controller (5.17)–</p><p>(5.18), with χ †(0) = χ ∗(0). For the controller (5.16), |s1| is monotonically decreasing</p><p>and reaches zero in finite time t1. For the controller (5.17)–(5.18), |s |= |σ+s1| is mono-</p><p>tonically decreasing and reaches the set {|s | ≤µ} in finite time t2. Let t3 =min{t1, t2}.</p><p>If t3 > 0, using sat(s†(t )/µ) = sgn(s†(t )) = sgn(s∗1 (t )) for t ∈ [0, t3], it can be shown</p><p>that χ †(t ) = χ ∗(t ) for t ∈ [0, t3]. Next, consider χ †(t ) and χ ∗(t ) for t ≥ t3. Since</p><p>χ †(t3) = χ</p><p>∗(t3), it must be true that t3 = t2 ≤ t1 because if t1</p><p>one. 3</p><p>Remark 5.9. If Assumptions 5.1 to 5.3 hold globally, that is, Dx = T (Dx ) = Rn and</p><p>γ1 is class K∞, then the sets Γ and Ω can be chosen arbitrarily large. For any bounded</p><p>(z(0), q(0), s(0)), the conclusion of Theorem 5.2 will hold by choosingβ large enough. 3</p><p>Example 5.2. Reconsider the pendulum regulation problem from Example 5.1. A</p><p>state feedback sliding mode controller is taken as</p><p>u =−k sgn(ξ1+ ξ2).</p><p>A continuously implemented state feedback sliding mode controller with conditional</p><p>integrator is taken as</p><p>σ̇ =−σ +µ sat</p><p>�</p><p>σ + ξ1+ ξ2</p><p>µ</p><p>�</p><p>, u =−k sat</p><p>�</p><p>σ + ξ1+ ξ2</p><p>µ</p><p>�</p><p>,</p><p>and its output feedback version is</p><p>σ̇ =−σ +µ sat</p><p>σ + ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>, u =−k sat</p><p>σ + ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>126 CHAPTER 5. REGULATION</p><p>0 1 2 3 4 5</p><p>0</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>2.5</p><p>3</p><p>3.5</p><p>(a)</p><p>Time</p><p>x</p><p>1</p><p>0 1 2 3 4 5</p><p>0</p><p>0.05</p><p>0.1</p><p>0.15</p><p>0.2</p><p>(b)</p><p>Time</p><p>∆</p><p>x</p><p>1</p><p>µ=1</p><p>µ=0.1</p><p>0 1 2 3 4 5</p><p>−0.06</p><p>−0.04</p><p>−0.02</p><p>0</p><p>0.02</p><p>(c)</p><p>Time</p><p>∆̃</p><p>x</p><p>1</p><p>ε=0.03</p><p>ε=0.01</p><p>0 1 2 3 4 5</p><p>0</p><p>1</p><p>2</p><p>3</p><p>4</p><p>(d)</p><p>Time</p><p>x</p><p>1</p><p>Integrator</p><p>Conditional Integrator</p><p>Figure 5.6. Simulation of Example 5.2. (a) is the response of the sliding mode controller.</p><p>(b) shows the difference between the responses of the sliding mode controller and the state feedback</p><p>controller with conditional integrator. (c) shows the difference between the state and output feedback</p><p>responses. (d) compares the responses of the traditional and conditional integrators.</p><p>˙̂</p><p>ξ1 = ξ̂2+</p><p>2</p><p>ε</p><p>(ξ1− ξ̂1),</p><p>˙̂</p><p>ξ2 =</p><p>1</p><p>ε2</p><p>(ξ1− ξ̂1).</p><p>Simulation results are shown in Figure 5.4 using zero initial conditions and the same</p><p>parameters as in Example 5.1, namely, r = π, k = 5, a = 0.03, b = 1, and d = 0.3.</p><p>Figure 5.6(a) shows the output response of the state feedback sliding mode controller.</p><p>Figure 5.6(b) demonstrates how the response of the continuously implemented slid-</p><p>ing mode controller with conditional integrator approaches the response of the slid-</p><p>ing mode controller as µ decreases; ∆x1 is the difference between the two responses.</p><p>Figure 5.6(c) demonstrates how the response of the output feedback controller with</p><p>conditional integrator approaches its state feedback counterpart as ε decreases with</p><p>µ= 0.1; ∆̃x1 is the difference between the two responses. Finally, Figure 5.6(d) com-</p><p>pares the response of the output feedback controller with conditional controller with</p><p>the one designed in Example 5.1 using the traditional integrator; in both cases µ= 0.1</p><p>and ε = 0.01. The advantage of the conditional integrator over the traditional one is</p><p>clear. It avoids the degradation of the transient response associated with the traditional</p><p>integrator because it recovers the response of the sliding mode controller. 4</p><p>5.4 Conditional Servocompensator</p><p>Consider a single-input–single-output nonlinear system modeled by</p><p>ẋ = f (x,θ)+ g (x,θ)u +δ(x,θ, d ), (5.29)</p><p>y = h(x,θ)+ γ (θ, d ), (5.30)</p><p>where x ∈ Rn is the state, u ∈ R is the control input, y ∈ R is the measured output,</p><p>θ ∈ Θ is a vector of (possibly uncertain) system parameters, and d ∈ Rl is a bounded</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.4. CONDITIONAL SERVOCOMPENSATOR 127</p><p>time-varying disturbance. The functions f , g , h, and δ are sufficiently smooth in x</p><p>in a domain Dx ⊂ Rn . The functions δ and γ vanish at d = 0, i.e., δ(x,θ, 0) = 0 and</p><p>γ (θ, 0) = 0 for all x ∈ Dx and θ ∈ Θ. The output y is to be regulated to a bounded</p><p>time-varying reference signal r :</p><p>lim</p><p>t→∞</p><p>|y(t )− r (t )|= 0.</p><p>Assumption 5.5. For each θ ∈ Θ, the disturbance-free system, (5.29)–(5.30) with d = 0,</p><p>has relative degree ρ≤ n in Dx , and there is a diffeomorphism</p><p>�</p><p>η</p><p>ζ</p><p>�</p><p>= T (x) (5.31)</p><p>in Dx , possibly dependent on θ, that transforms (5.29)–(5.30), with d = 0, into the normal</p><p>form42</p><p>η̇= f0(η,ζ ,θ),</p><p>ζ̇i = ζi+1, 1≤ i ≤ ρ− 1,</p><p>ζ̇ρ = a(η,ζ ,θ)+ b (η,ζ ,θ)u,</p><p>y = ζ1.</p><p>Moreover, b (η,ζ ,θ)≥ b0 > 0 for all (η,ζ ) ∈ T (Dx ) and θ ∈Θ.</p><p>Assumption 5.6. The change of variables (5.31) transforms the disturbance-driven system</p><p>(5.29)–(5.30) into the form</p><p>η̇= fa(η,ζ1, . . . ,ζm ,θ, d ),</p><p>ζ̇i = ζi+1+ψi (ζ1, . . . ,ζi ,θ, d ), 1≤ i ≤ m− 1,</p><p>ζ̇i = ζi+1+ψi (η,ζ1, . . . ,ζi ,θ, d ), m ≤ i ≤ ρ− 1,</p><p>ζ̇ρ = a(η,ζ ,θ)+ b (η,ζ ,θ)u +ψρ(η,ζ ,θ, d ),</p><p>y = ζ1+ γ (θ, d ),</p><p>where 1≤ m ≤ ρ− 1. The functions ψi vanish at d = 0.</p><p>For m = 1, the ζ̇i -equations for 1 ≤ i ≤ m − 1 are dropped. In the absence of η,</p><p>Assumption 5.6 is satisfied locally if the system (5.29)–(5.30) is observable uniformly</p><p>in θ and d [49].</p><p>Assumption 5.7. The disturbance and reference signals d (t ) and r (t ) are generated by</p><p>the exosystem</p><p>ẇ = S0w,</p><p>�</p><p>d</p><p>r</p><p>�</p><p>=H0w, (5.32)</p><p>where S0 has distinct eigenvalues on the imaginary axis and w(t ) belongs to a compact</p><p>set W .</p><p>This assumption says that d (t ) and r (t ) are linear combinations of constant and</p><p>sinusoidal signals.</p><p>42For ρ= n, η and the η̇-equation are dropped.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>128 CHAPTER 5. REGULATION</p><p>Define τ1(θ, w) to τm(θ, w) by</p><p>τ1 = r − γ (θ, d ),</p><p>τi+1 =</p><p>∂ τi</p><p>∂ w</p><p>S0w −ψi (τ1, . . . ,τi ,θ, d ), 1≤ i ≤ m− 1.</p><p>Assumption 5.8. There exists a unique mapping τ0(θ, w) that solves the partial differen-</p><p>tial equation</p><p>∂ τ0</p><p>∂ w</p><p>S0w = fa (τ0,τ1, . . . ,τm ,θ, d ) (5.33)</p><p>for all θ ∈Θ and w ∈W .</p><p>Remark 5.10. In the special case when the η̇-equation takes the form</p><p>η̇=Aη+ fb (ζ1, . . . ,ζm ,θ, d )</p><p>with a Hurwitz matrix A, (5.33) is a linear partial differential of the form</p><p>∂ τ0</p><p>∂ w</p><p>S0w =Aτ0+ fc (θ, w),</p><p>and its unique solution is given by</p><p>τ0(θ, w) =</p><p>∫ 0</p><p>−∞</p><p>e−At fc (θ, e S0 t w) d t . 3</p><p>Using τ0(θ, w), define τm+1(θ, w) to τρ(θ, w) by</p><p>τi+1 =</p><p>∂ τi</p><p>∂ w</p><p>S0w −ψi (τ0,τ1, . . . ,τi ,θ, d ), m ≤ i ≤ ρ− 1.</p><p>The steady-state zero-error manifold is given by {η= τ0(θ, w), ζ = τ(θ, w)} because</p><p>η and ζ satisfy the equations of Assumption 5.6 and ζ1 = τ1 = r −γ (θ, d ) implies that</p><p>y = r . The steady-state value of the control input u on this manifold is given by</p><p>φ(θ, w) =</p><p>1</p><p>b (τ0,τ,θ)</p><p>�</p><p>(∂ τρ/∂ w)S0w − a(τ0,τ,θ)−ψρ(τ0,τ,θ, d )</p><p>�</p><p>. (5.34)</p><p>Assumption 5.9. There are known real numbers c0, . . . , cq−1 such that the polynomial</p><p>pq + cq−1 pq−1+ · · ·+ c1 p + c0</p><p>has distinct roots on the imaginary axis and φ(θ, w) satisfies the differential equation</p><p>φ(q)+ cq−1φ</p><p>(q−1)+ · · ·+ c1φ</p><p>(1)+ c0φ= 0. (5.35)</p><p>Therefore, φ(θ, w) is generated by the linear internal model</p><p>Φ̇= SΦ, φ=HΦ, (5.36)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.4. CONDITIONAL SERVOCOMPENSATOR 129</p><p>where</p><p>S =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 · · · · · · 0</p><p>0 0 1 · · · 0</p><p>...</p><p>...</p><p>0 · · · · · · 0 1</p><p>−c0 · · · · · · · · · −cq−1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, Φ=</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>φ</p><p>φ(1)</p><p>...</p><p>φ(q−2)</p><p>φ(q−1)</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, and H T =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>1</p><p>0</p><p>...</p><p>...</p><p>0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>.</p><p>This assumption reflects the fact that for nonlinear systems, the internal model</p><p>must reproduce not only the sinusoidal signals generated by the exosystem but also</p><p>higher-order harmonics induced by the nonlinearities. Because the model has finite</p><p>dimension, there can be only a finite number of harmonics. The assumption is satisfied</p><p>when the system has polynomial nonlinearities.</p><p>Example 5.3. Consider the system</p><p>ẋ1 = x2, ẋ2 = θ1(x1− x3</p><p>1 )+θ2u, y = x1,</p><p>where θ1 and θ2 are unknown. It is required to regulate y to r (t ) = α sin(ω0 t + θ0),</p><p>where α and θ0 are unknown but ω0 is known. The signal</p><p>r is generated by the</p><p>exosystem</p><p>ẇ =</p><p>�</p><p>0 ω0</p><p>−ω0 0</p><p>�</p><p>w, w(0) =</p><p>�</p><p>α sinθ0</p><p>α cosθ0</p><p>�</p><p>, r = w1.</p><p>The steady-state controlφ(θ, w) = [−(θ1+ω</p><p>2</p><p>0)w1+θ1w3</p><p>1 ]/θ2 satisfies the differential</p><p>equation</p><p>φ(4)+ 10ω2</p><p>0φ</p><p>(2)+ 9ω4</p><p>0φ= 0.</p><p>The eigenvalues of</p><p>S =</p><p></p><p></p><p></p><p></p><p>0 1 0 0</p><p>0 0 1 0</p><p>0 0 0 1</p><p>−9ω4</p><p>0 0 −10ω2</p><p>0 0</p><p></p><p></p><p></p><p></p><p>are ± jω0, ±3 jω0. The internal model generates the first and third harmonics. 4</p><p>The change of variables</p><p>z = η−τ0(θ, w), ξi = y (i−1)− r (i−1) for 1≤ i ≤ ρ, (5.37)</p><p>transforms the system (5.29)–(5.30) into the form</p><p>ż = f̃0(z,ξ ,θ, w), (5.38)</p><p>ξ̇i = ξi+1, 1≤ i ≤ ρ− 1, (5.39)</p><p>ξ̇ρ = a0(z,ξ ,θ, w)+ b (η,ζ ,θ)[u −φ(θ, w)], (5.40)</p><p>e = ξ1, (5.41)</p><p>where e = y− r is the measured regulation error, f̃0(0,0,θ, w) = 0, and a0(0,0,θ, w) =</p><p>0. In these new variables the zero-error manifold is {z = 0, ξ = 0}. Let Γ ⊂ Rn be a</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>130 CHAPTER 5. REGULATION</p><p>compact set, which contains the origin in its interior, such that (z,ξ ) ∈ Γ implies that</p><p>(η,ζ ) ∈ T (Dx ) for θ ∈ Θ and w ∈W . The existence of Γ may require restricting the</p><p>sizes of Θ and W .</p><p>Assumption 5.10. There is a Lyapunov function V1(z,θ, w) for the system ż = f̃0(z,ξ ,</p><p>θ, w) that satisfies the inequalities</p><p>γ1(‖z‖)≤V1(z,θ, w)≤ γ2(‖z‖),</p><p>∂ V1</p><p>∂ z</p><p>f̃0(z,ξ ,θ, w)+</p><p>∂ V1</p><p>∂ w</p><p>S0w ≤−γ3(‖z‖) ∀ ‖z‖ ≥ γ4(‖ξ ‖)</p><p>for all (z,ξ ,θ, w) ∈ Γ ×Θ×W , where γ1 to γ4 are classK functions independent of θ</p><p>and w.</p><p>Assumption 5.11. In some neighborhood of z = 0, there is a Lyapunov function</p><p>V0(z,θ, w) that satisfies the inequalities</p><p>c̄1‖z‖</p><p>2 ≤V0(z,θ, w)≤ c̄2‖z‖</p><p>2,</p><p>∂ V0</p><p>∂ z</p><p>f̃0(z, 0,θ, w)+</p><p>∂ V0</p><p>∂ w</p><p>S0w ≤−c̄3‖z‖</p><p>2,</p><p>∂ V0</p><p>∂ z</p><p>≤ c̄4‖z‖</p><p>for some positive constants c̄1 to c̄4, independent of θ and w.</p><p>As in the conditional integrator of the previous section, the state feedback control</p><p>design starts by considering the sliding mode controller</p><p>u =−β(ξ ) sgn(s1), (5.42)</p><p>where</p><p>s1 = k1ξ1+ · · ·+ kρ−1ξρ−1+ ξρ</p><p>and k1 to kρ−1 are chosen such that the polynomial</p><p>λρ−1+ kρ−1λ</p><p>ρ−2+ · · ·+ k2λ+ k1</p><p>is Hurwitz. From the equation</p><p>ṡ1 =</p><p>ρ−1</p><p>∑</p><p>i=1</p><p>kiξi+1+ a0(z,ξ ,θ, w)+ b (η,ζ ,θ)[u −φ(θ, w)]</p><p>def= ∆1(z,ξ ,θ, w)+ b (η,ζ ,θ)u</p><p>it can be seen that the condition s ṡ 0, where %1(ξ ) is an upper</p><p>bound on |∆1(z,ξ ,θ, w)/b (η,ζ ,θ)| for all (z,ξ ,θ, w) ∈ Γ ×Θ×W . A continuously</p><p>implemented sliding mode controller with a conditional servocompensator is taken as</p><p>u =−β(ξ ) sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>, (5.43)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.4. CONDITIONAL SERVOCOMPENSATOR 131</p><p>σ̇ = Fσ +µG sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>, (5.44)</p><p>where</p><p>s =ΛT σ + s1 =Λ</p><p>T σ + k1ξ1+ · · ·+ kρ−1ξρ−1+ ξρ, (5.45)</p><p>F is Hurwitz, the pair (F ,G) is controllable, andΛ is the unique vector that assigns the</p><p>eigenvalues of (F +GΛT ) at the eigenvalues of S.43 The closed-loop system is given by</p><p>ẇ = S0w,</p><p>σ̇ = Fσ +µG sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>,</p><p>ż = f̃0(z,ξ ,θ, w),</p><p>q̇ =A2q +B2(s −Λ</p><p>T σ),</p><p>ṡ =−b (η,ζ ,θ)β(ξ ) sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>+∆1(·)+Λ</p><p>T</p><p>�</p><p>Fσ +µG sat</p><p>�</p><p>s</p><p>µ</p><p>��</p><p>,</p><p>where q = col(ξ1, . . . ,ξρ−1), ξ = col(q , s −ΛT σ − Lq), L=</p><p>�</p><p>k1 . . . kρ−1</p><p>�</p><p>,</p><p>A2 =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 · · · · · · 0</p><p>0 0 1 · · · 0</p><p>...</p><p>...</p><p>0 · · · · · · 0 1</p><p>−k1 · · · · · · · · · −kρ−1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, and B2 =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>...</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>.</p><p>The matrices F and A2 are Hurwitz by design. Let Vσ = σ</p><p>T Pσ , where P is the solu-</p><p>tion of the Lyapunov equation P F + F T P =−I . From the inequality</p><p>V̇σ =−σ</p><p>T σ + 2µσT PG sat</p><p>�</p><p>s</p><p>µ</p><p>�</p><p>≤−‖σ‖2+ 2µ‖PG‖ ‖σ‖,</p><p>it follows that the set Ξ= {Vσ ≤ ρ0µ</p><p>2}with ρ0 = 4‖PG‖2λmax(P ) is positively invari-</p><p>ant because V̇σ ≤ 0 on the boundary Vσ = ρ0µ</p><p>2. Therefore, σ(0) ∈ Ξ implies that</p><p>σ(t ) =O(µ) for all t ≥ 0. For |s | ≥µ,</p><p>s ṡ ≤−bβ|s |+ |∆1||s |+ |s | ‖Λ‖(µ‖F ‖</p><p>Æ</p><p>ρ0/λmin(P )+µ‖G‖)</p><p>= b</p><p>�</p><p>−β+</p><p>|∆1|</p><p>b</p><p>+</p><p>kµ</p><p>b0</p><p>�</p><p>|s | ≤ b</p><p>�</p><p>−β0+</p><p>kµ</p><p>b</p><p>�</p><p>|s |,</p><p>where k = ‖Λ‖(‖F ‖</p><p>p</p><p>ρ0/λmin(P )+ ‖G‖). For µ≤ 1</p><p>2 b0β0/k,</p><p>s ṡ ≤− 1</p><p>2 bβ0|s | ≤ −</p><p>1</p><p>2 b0β0|s |,</p><p>which shows that the set {|s | ≤ c} is positively invariant for c > µ, and s reaches the</p><p>positively invariant set {|s | ≤ µ} in finite time. Let V2(q) = qT P2q , where P2 is the</p><p>solution of the Lyapunov equation P2A2+AT</p><p>2 P2 =−I . For |s | ≤ c ,</p><p>V̇2 ≤−qT q + 2‖q‖ ‖P2B2‖ (|s |+ |Λ</p><p>T σ |)≤−‖q‖2+ 2‖P2B2‖ ‖q‖(c + `µ),</p><p>43Λ is unique because G has one column. See [9, Lemma 9.10].</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>132 CHAPTER 5. REGULATION</p><p>where `µ=maxσ∈Ξ ‖ΛT σ‖. For µ≤ c/`,</p><p>V̇2 ≤−‖q‖</p><p>2+ 4‖P2B2‖ ‖q‖c ,</p><p>which shows that the set {V2 ≤ ρ̄1c2} × {|s | ≤ c} is positively invariant for ρ̄1 ></p><p>16||P2B2‖2λmax(P2) because V̇2</p><p>â = b̂ = 0.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>1.2. MOTIVATING EXAMPLES 5</p><p>Example 1.2. Consider the two-dimensional system</p><p>ẋ1 = x2, ẋ2 =φ(x, u), y = x1,</p><p>where x = col(x1, x2). Suppose u = γ (x) is a locally Lipschitz state feedback controller</p><p>that stabilizes the origin of the closed-loop system</p><p>ẋ1 = x2, ẋ2 =φ(x,γ (x)).</p><p>To implement this control with output feedback, we use the high-gain observer</p><p>˙̂x1 = x̂2+(α1/ε)(y − x̂1),</p><p>˙̂x2 =φ0(x̂, u)+ (α2/ε</p><p>2)(y − x̂1),</p><p>whereφ0 is a nominal model ofφ, and α1, α2, and ε are positive constants with ε� 1.</p><p>We saw in the previous example that if</p><p>|φ0(z, u)−φ(x, u)| ≤ L ‖x − z‖+M</p><p>over the domain of interest, then for sufficiently small ε, the estimation errors x̃1 =</p><p>x1− x̂1 and x̃2 = x2− x̂2 satisfy the inequalities</p><p>|x̃1| ≤max</p><p>¦</p><p>b e−at/ε‖x̃(0)‖, ε2cM</p><p>©</p><p>, |x̃2| ≤</p><p>§ b</p><p>ε</p><p>e−at/ε‖x̃(0)‖, εcM</p><p>ª</p><p>for some positive constants a, b , c . These inequalities show that reducing ε dimin-</p><p>ishes the effect of model uncertainty and makes x̃ much faster than x. The bound on</p><p>x̃2 demonstrates the peaking phenomenon; namely, x̃2 might peak to O(1/ε) values</p><p>before it decays rapidly toward zero. The peaking phenomenon might destabilize the</p><p>closed-loop system. This fact is illustrated by simulating the system</p><p>ẋ1 = x2, ẋ2 = x3</p><p>2 + u, y = x1,</p><p>which can be globally stabilized by the state feedback controller</p><p>u =−x3</p><p>2 − x1− x2.</p><p>The output feedback controller is taken as</p><p>u =−x̂3</p><p>2 − x̂1− x̂2, ˙̂x1 = x̂2+(2/ε)(y − x̂1),</p><p>˙̂x2 = (1/ε</p><p>2)(y − x̂1),</p><p>where we take φ0 = 0. Figure 1.2 shows the performance of the closed-loop system</p><p>under state and output feedback. Output feedback is simulated for three different val-</p><p>ues of ε. The initial conditions are x1(0) = 0.1, x2(0) = x̂1(0) = x̂2(0) = 0. Peaking is</p><p>induced by [x1(0)− x̂1(0)]/ε = 0.1/ε when ε is sufficiently small. Figure 1.2 shows a</p><p>counterintuitive behavior as ε decreases. Since decreasing ε causes the estimation error</p><p>to decay faster toward zero, one would expect the response under output feedback to</p><p>approach the response under state feedback as ε decreases. Figure 1.2 shows the oppo-</p><p>site behavior, where the response under output feedback deviates from the response</p><p>under state feedback as ε decreases. This is the impact of the peaking phenomenon.</p><p>The same figure shows the control u on a much shorter time interval to exhibit peak-</p><p>ing. Figure 1.3 shows that as we decrease ε to 0.004, the system has a finite escape time</p><p>shortly after t = 0.07.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>6 CHAPTER 1. INTRODUCTION</p><p>0 1 2 3 4 5 6 7 8 9 10</p><p>−2</p><p>−1.5</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>x 1</p><p>SFB</p><p>OFB ε = 0.1</p><p>OFB ε = 0.01</p><p>OFB ε = 0.005</p><p>0 1 2 3 4 5 6 7 8 9 10</p><p>−3</p><p>−2</p><p>−1</p><p>0</p><p>1</p><p>x 2</p><p>0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1</p><p>−400</p><p>−300</p><p>−200</p><p>−100</p><p>0</p><p>u</p><p>t</p><p>Figure 1.2. Simulation of Example 1.2. Performance under state (SFB) and output (OFB)</p><p>feedback. Reprinted with permission of Pearson Education, Inc., New York, New York [80].</p><p>0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>0.2</p><p>x 1</p><p>0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08</p><p>−600</p><p>−400</p><p>−200</p><p>0</p><p>x 2</p><p>0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08</p><p>−1000</p><p>0</p><p>1000</p><p>2000</p><p>u</p><p>t</p><p>Figure 1.3. Simulation of Example 1.2. Instability induced by peaking at ε = 0.004.</p><p>Reprinted with permission of Pearson Education, Inc., New York, New York [80].</p><p>Fortunately, we can isolate the observer peaking from the plant by saturating the</p><p>control outside a compact set of interest. Writing the closed-loop system under state</p><p>feedback as ẋ =Ax and solving the Lyapunov equation PA+AT P =−I , we have</p><p>A=</p><p>�</p><p>0 1</p><p>−1 −1</p><p>�</p><p>and P =</p><p>�</p><p>1.5 0.5</p><p>0.5 1</p><p>�</p><p>.</p><p>Then V (x) = xT P x is a Lyapunov function for ẋ = Ax and V̇ (x) = −xT x. Suppose</p><p>all initial conditions of interest belong to the set Ω = {V (x) ≤ 0.3}. Because Ω is</p><p>positively invariant, x(t ) ∈ Ω for all t ≥ 0. By maximizing |x1 + x2| and |x2| over</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>1.2. MOTIVATING EXAMPLES 7</p><p>Ω, it can be shown that for all x ∈ Ω, |x1 + x2| ≤ 0.6 and |x2| ≤ 0.6. Hence, |u| ≤</p><p>|x2|3 + |x1 + x2| ≤ 0.816. Saturating u at ±1 results in the globally bounded state</p><p>feedback control</p><p>u = sat(−x3</p><p>2 − x1− x2),</p><p>where sat(y) =min{|y|, 1} sign(y). For all x(0) ∈Ω, the saturated control produces the</p><p>same trajectories as the unsaturated one because for x ∈Ω, |u|</p><p>(5.48), the</p><p>controller (5.49), and the observer (5.50)–(5.51). Let Ψ be a compact set in the interior of</p><p>Ω and suppose σ(0) ∈ Ξ, (z(0), q(0), s(0)) ∈ Ψ, and ξ̂ (0) is bounded. Then µ∗ > 0 exists,</p><p>and for eachµ ∈ (0,µ∗], ε∗ = ε∗(µ) exists such that for eachµ ∈ (0,µ∗] and ε ∈ (0,ε∗(µ)]</p><p>all state variables are bounded and limt→∞ e(t ) = 0. Furthermore, let χ = (z,ξ ) be</p><p>part of the state of the closed-loop system under the output feedback controller, and let</p><p>χ ∗ = (z∗,ξ ∗) be the state of the closed-loop system under the state feedback sliding mode</p><p>controller (5.42), with χ (0) = χ ∗(0). Then, for every δ0 > 0, there isµ∗1 > 0, and for each</p><p>µ ∈ (0,µ∗1], there is ε∗1 = ε</p><p>∗</p><p>1(µ)> 0 such that for µ ∈ (0,µ∗1] and ε ∈ (0,ε∗1(µ)],</p><p>‖χ (t )−χ ∗(t )‖ ≤ δ0 ∀ t ≥ 0. (5.52)</p><p>3</p><p>Proof: The closed-loop system under output feedback is given by</p><p>ẇ = S0w, (5.53)</p><p>σ̇ = Fσ +µG sat</p><p>�</p><p>ŝ</p><p>µ</p><p>�</p><p>, (5.54)</p><p>ż = f̃0(z,ξ ,θ, w), (5.55)</p><p>q̇ =A2q +B2(s −Λ</p><p>T σ), (5.56)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>134 CHAPTER 5. REGULATION</p><p>ṡ = b (η,ζ ,θ)ψ(σ , ξ̂ ,µ)+∆1(z,ξ ,θ, w)+ΛT</p><p>�</p><p>Fσ +µG sat</p><p>�</p><p>ŝ</p><p>µ</p><p>��</p><p>, (5.57)</p><p>εϕ̇ =A0ϕ+ εB[a0(z,ξ ,θ, w)− b (η,ζ ,θ)φ(θ, w)</p><p>+ b (η,ζ ,θ)ψ(σ , ξ̂ ,µ)], (5.58)</p><p>where</p><p>ŝ =ΛT σ + k1ξ̂1+ · · ·+ kρ−1ξ̂ρ−1+ ξ̂ρ, ϕi =</p><p>ξi − ξ̂i</p><p>ερ−i</p><p>for 1≤ i ≤ ρ,</p><p>ψ(σ ,ξ ,µ) =−βs (ξ ) sat(s/µ), and</p><p>A0 =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>−α1 1 0 · · · 0</p><p>−α2 0 1 · · · 0</p><p>...</p><p>. . .</p><p>...</p><p>−αρ−1 0 1</p><p>−αρ 0 · · · · · · 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>.</p><p>The matrix A0 is Hurwitz by design. Equations (5.53) to (5.57) with ŝ and ψ(σ , ξ̂ ,µ)</p><p>replaced by s and ψ(σ ,ξ ,µ), respectively, are the closed-loop system under state feed-</p><p>back. Let P0 be the solution of the Lyapunov equation P0A0 +AT</p><p>0 P0 = −I , V3(ϕ) =</p><p>ϕT P0ϕ, and Σε = {V3(ϕ)≤ ρ3ε</p><p>2}, where the positive constant ρ3 is to be determined.</p><p>The proof proceeds in four steps:</p><p>Step 1: Show that there exist ρ3 > 0, µ∗1 > 0 and ε∗1 = ε</p><p>∗</p><p>1(µ) > 0 such that for each</p><p>µ ∈ (0,µ∗1] and ε ∈ (0,ε∗1(µ)] the set Ξ×Ω×Σε is positively invariant.</p><p>Step 2: Show that for σ(0) ∈ Ξ, (z(0), q(0), s(0)) ∈ Ψ, and any bounded ξ̂ (0), there</p><p>exists ε∗2 > 0 such that for each ε ∈ (0,ε∗2] the trajectory enters the set Ξ×Ω×Σε</p><p>in finite time T1(ε), where limε→0 T1(ε) = 0.</p><p>Step 3: Show that there exists ε∗3 = ε</p><p>∗</p><p>3(µ)> 0 such that for each ε ∈ (0,ε∗3(µ)] every</p><p>trajectory in Ξ×Ω×Σε enters Ξ×Ωµ×Σε in finite time and stays therein for</p><p>all future time.</p><p>Step 4: Show that there existsµ∗2 > 0 and ε∗4 = ε</p><p>∗</p><p>4(µ)> 0 such that for eachµ ∈ (0,µ∗2]</p><p>and ε ∈ (0,ε∗4(µ)] every trajectory in Ξ × Ωµ × Σε approaches the invariant</p><p>manifold {σ = σ̄ , z = 0,ξ = 0,ϕ = 0} as t →∞, where e = 0 on the manifold.</p><p>For the first step, calculate the derivative of V3 on the boundary V3 = ρ3ε</p><p>2:</p><p>εV̇3 =−ϕ</p><p>Tϕ+ 2εϕT P0B[a0(z,ξ ,θ, w)− b (η,ζ ,θ)φ(θ, w)+ b (η,ζ ,θ)ψ(σ , ξ̂ ,µ)].</p><p>Since ψ(σ , ξ̂ ,µ) is globally bounded in ξ̂ , for all (σ , z, q , s) ∈ Ξ×Ω there is `1 > 0</p><p>such that</p><p>|a0(z,ξ ,θ, w)− b (η,ζ ,θ)φ(θ, w)+ b (η,ζ ,θ)ψ(σ , ξ̂ ,µ)| ≤ `1.</p><p>Therefore,</p><p>εV̇3 ≤−‖ϕ‖</p><p>2+ 2ε`1‖PoB‖ ‖ϕ‖ ≤− 1</p><p>2‖ϕ‖</p><p>2 ∀ ‖ϕ‖ ≥ 4ε`1‖P0B‖. (5.59)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.4. CONDITIONAL SERVOCOMPENSATOR 135</p><p>Taking ρ3 = λmax(P0)(4‖P0B‖`1)</p><p>2 ensures that V̇3 ≤ −</p><p>1</p><p>2‖ϕ‖</p><p>2 for all V3 ≥ ρ3ε</p><p>2. Con-</p><p>sequently, V̇3 0, independent of ε, such that (z(t ), q(t ), s(t )) ∈ Ω for t ∈ [0,T0].</p><p>During this time, (5.59) shows that</p><p>V̇3 ≤−</p><p>1</p><p>2ελmax(P0)</p><p>V3.</p><p>Therefore, V3 reduces toρ3ε</p><p>2 within a time interval [0,T1(ε)] in which limε→0 T1(ε) =</p><p>0. For sufficiently small ε, T1(ε)</p><p>(z(0), q(0), s(0)), the conclusion of Theorem 5.3 will hold by choosingβ large enough. 3</p><p>Example 5.4. Consider the system</p><p>ẋ1 =−θ1x1+ x2</p><p>2 + d , ẋ2 = x3, ẋ3 =−θ2x1x2+ u, y = x2,</p><p>where θ1 > 0 and θ2 are unknown parameters, d is a constant disturbance, and the ref-</p><p>erence signal is r = α sin(ω0 t+θ0)with known frequencyω0 and unknown amplitude</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.4. CONDITIONAL SERVOCOMPENSATOR 137</p><p>and phase. Start by verifying the assumptions. Assumptions 5.5 and 5.6 are satisfied</p><p>globally with</p><p>η= x1, ζ1 = x2, ζ2 = x3.</p><p>The exosystem of Assumption 5.7 is given by</p><p>ẇ =</p><p></p><p></p><p>0 ω0 0</p><p>−ω0 0 0</p><p>0 0 0</p><p></p><p>w, w(0) =</p><p></p><p></p><p>α sinθ0</p><p>α cosθ0</p><p>d</p><p></p><p> , r = w1, d = w3.</p><p>With τ1 = w1 and τ2 =ω0w2, Assumption 5.8 is satisfied with</p><p>τ0(θ, w) =</p><p>θ2</p><p>1 + 2ω2</p><p>0</p><p>θ1(θ2</p><p>1 + 4ω2</p><p>0)</p><p>w2</p><p>1 −</p><p>2ω0</p><p>θ2</p><p>1 + 4ω2</p><p>0</p><p>w1w2+</p><p>2ω2</p><p>0</p><p>θ1(θ2</p><p>1 + 4ω2</p><p>0)</p><p>w2</p><p>2 +</p><p>1</p><p>θ1</p><p>w3.</p><p>The steady-state control is given by</p><p>φ(θ, w) =−ω2</p><p>0 w1+θ2τ0(θ, w)w1</p><p>=−ω2</p><p>0 w1+</p><p>θ2</p><p>θ1</p><p>w1w3</p><p>+</p><p>θ2</p><p>θ1(θ2</p><p>1 + 4ω2</p><p>0)</p><p>�</p><p>(θ2</p><p>1 + 2ω2</p><p>0)w</p><p>3</p><p>1 − 2θ1ω0w2</p><p>1 w2+ 2ω2</p><p>0 w1w2</p><p>2</p><p>�</p><p>.</p><p>Assumption 5.9 is satisfied, as φ satisfies the differential equation</p><p>φ(4)+ 10ω2</p><p>0φ</p><p>(2)+ 9ω4</p><p>0φ= 0.</p><p>Hence, the internal model (5.36) is given by</p><p>S =</p><p></p><p></p><p></p><p></p><p>0 1 0 0</p><p>0 0 1 0</p><p>0 0 0 1</p><p>−9ω4</p><p>0 0 −10ω2</p><p>0 0</p><p></p><p></p><p></p><p></p><p>, H =</p><p>�</p><p>1 0 0 0</p><p>�</p><p>.</p><p>It is worthwhile to note that while finding the internal model goes through the elabo-</p><p>rate procedure of finding τ0 and φ and verifying the differential equation satisfied by</p><p>φ, the model can be intuitively predicted. If x2 is to be a sinusoidal signal, then it can</p><p>be seen from the ẋ1-equation that the steady state of x1 will have constant and second</p><p>harmonic terms. Then the product x1x2 will have first and third harmonics. Finally,</p><p>the ẋ3-equation shows that the steady-state control will have first and third harmonics,</p><p>which results in the internal model. With the change of variables</p><p>z = η−τ0 = x1−τ0, ξ1 = y − r = x2−w1, ξ2 = ẏ − ṙ = x3−ω0w2,</p><p>the system is represented by</p><p>ż =−θ1z + ξ 2</p><p>1 + 2ξ1w1,</p><p>ξ̇1 = ξ2,</p><p>ξ̇2 =−θ2[(z +τ0)ξ1+ zw1]+ u −φ(θ, w).</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>138 CHAPTER 5. REGULATION</p><p>Assumption 5.10 is satisfied globally with V1 =</p><p>1</p><p>2 z2. Assumption 5.11 is satisfied</p><p>with V0 =</p><p>1</p><p>2 z2. The output feedback controller with conditional servocompensator is</p><p>taken as45</p><p>σ̇ = Fσ +µG sat</p><p>ΛT σ + ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>,</p><p>u =−20 sat</p><p>ΛT σ + ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>,</p><p>˙̂</p><p>ξ1 = ξ̂2+(2/ε)(e − ξ̂1),</p><p>˙̂</p><p>ξ2 = (1/ε</p><p>2)(e − ξ̂1),</p><p>where µ = 0.1, ε = 0.001, and e = y − r . The pair (F ,G) is taken in the controllable</p><p>canonical form as</p><p>F = ς</p><p></p><p></p><p></p><p></p><p>0 1 0 0</p><p>0 0 1 0</p><p>0 0 0 1</p><p>−1.5 −6.25 −8.75 −5</p><p></p><p></p><p></p><p></p><p>, G = ς</p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p>,</p><p>where ς is a positive parameter. The eigenvalues of F /ς are −0.5, −1, −1.5, and −2.</p><p>The vectorΛ that assigns the eigenvalues of F +GΛT at the eigenvalues of S is given by</p><p>ΛT =</p><p>�</p><p>−9(ω0/ς)</p><p>4+ 1.5, 6.25, −10(ω0/ς)</p><p>2+ 8.75, 5</p><p>�</p><p>.</p><p>The scaling parameter ς is chosen such that Λ does not have large values for large</p><p>ω0. Assumingω0 ≤ 3, ς is taken as ς = 3. The simulation results of Figure 5.7 are for</p><p>θ1 = 3, θ2 = 4, ω0 = 2.5 rad/sec, d = 0.1, and ε = 0.001. The initial conditions are</p><p>x1(0) = x2(0) = 1, x3(0) = ξ̂1(0) = ξ̂2(0) = 0. Figure 5.7(a) shows the regulation error e</p><p>under the output feedback controller. It shows also the error under the state feedback</p><p>sliding mode controller</p><p>u =−20 sgn(ξ1+ ξ2).</p><p>0 2 4 6 8 10</p><p>−1</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>Time</p><p>E</p><p>rr</p><p>or</p><p>(a)</p><p>Conditional Servo</p><p>Sliding Mode</p><p>45 46 47 48 49 50</p><p>−0.025</p><p>−0.02</p><p>−0.015</p><p>−0.01</p><p>−0.005</p><p>0</p><p>0.005</p><p>0.01</p><p>Time</p><p>E</p><p>rr</p><p>or</p><p>(b)</p><p>Conditional Servo</p><p>No Servo</p><p>Figure 5.7. Simulation of Example 5.4. (a) compares the regulation error e under the</p><p>output feedback controller with conditional servocompensator to the error under the state feedback</p><p>sliding mode controller. (b) shows the difference between the steady-state errors of the controllers</p><p>with and without servocompensator.</p><p>45The gain 20 is determined by simulation.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.5. INTERNAL MODEL PERTURBATION 139</p><p>As expected, the two responses are very close. Figure 5.7(b) shows the advantage of</p><p>including a servocompensator by comparing with the controller</p><p>u =−20 sat</p><p>ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>,</p><p>which does not include a servocompensator. This controller does not achieve zero</p><p>steady-state error. It can only guarantee O(µ) steady-state error. 4</p><p>5.5 Internal Model Perturbation</p><p>The servocompensator design of the previous section requires precise knowledge of</p><p>the internal model (5.36), which is equivalent to knowing the frequency components</p><p>of the steady-state control. In this section we study the effect of internal model per-</p><p>turbations on the steady-state error. Let φ̄(θ, w) be the steady-state control on the</p><p>zero-error manifold, as defined by (5.34). The function φ̄(θ, w)may not satisfy (5.35),</p><p>but an approximation of it, φ(θ, w), could do so with known coefficients c0 to cq−1,</p><p>which define the internal model that is used in the design. There are two sources for</p><p>such perturbation. First, φ̄(θ, w)may not have a finite number of harmonics. Second,</p><p>it may have a finite number of harmonics, but the frequencies are not precisely known.</p><p>Assumption 5.12.</p><p>b (λ,π,θ)|φ(θ, w)− φ̄(θ, w)| ≤ δ ∀ (θ, w) ∈Θ×W . (5.60)</p><p>Theorem 5.4. Under the assumptions of Theorem 5.3, if Assumption 5.12 is satisfied,</p><p>then there exist positive constants µ∗, δ∗, `, and T , and for each µ ∈ (0,µ∗], there is a</p><p>positive constant ε∗ = ε∗(µ) such that for each µ ∈ (0,µ∗], ε ∈ (0,ε∗], and δ ∈ (0,δ∗],</p><p>|e(t )| ≤ `µδ ∀ t ≥ T . 3</p><p>Proof: The closed-loop system is a perturbation of equations (5.53) to (5.58) in which</p><p>(5.57) and (5.58) are perturbed by b (η,ζ ,θ)φ̃(θ, w) and εB b (η,ζ ,θ)φ̃(θ, w), respec-</p><p>tively, where φ̃= φ− φ̄. Provided δ is sufficiently small, the four steps of the proof</p><p>of Theorem 5.3 can be repeated to show that every trajectory in Ξ×Ω×Σε enters</p><p>Ξ×Ωµ×Σε in finite time and stays therein for all future time. Inside Ξ×Ωµ×Σε the</p><p>system can be represented in the form</p><p>ẇ = S0w,</p><p>Ż = f1(Z , p,θ, w)+G1h1(p,N (ε)ϕ,µ),</p><p>µ ṗ =−b (η,ζ ,θ)β(ξ )p +µ f2(Z , p,θ, w)+ h2(Z , p,N (ε)ϕ,θ, w)</p><p>+µb (λ,π,θ)φ̃(θ, w),</p><p>εϕ̇ =A0ϕ− (ε/µ)B b (η,ζ ,θ)β(ξ )p +(ε/µ)h3(Z , p,N (ε)ϕ,θ, w)</p><p>+ ε f3(Z , p,θ, w)+ εB b (λ,π,θ)φ̃(θ, w),</p><p>whereZ = col(z, q ,ϑ), ϑ = σ− σ̄ , p = s− s̄ , and N (ε) is a polynomial function of ε.</p><p>All functions on the right-hand side are locally Lipschitz in their arguments, and the</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>140 CHAPTER 5. REGULATION</p><p>functions h1 to h3 satisfy the inequalities |hi | ≤ `i‖ϕ‖with some positive constants `1</p><p>to `3, independent of ε and µ. The function f1 is given by</p><p>f1 =</p><p></p><p></p><p>f̃0(z, col(q , p −ΛTϑ− Lq),θ, w)</p><p>A2q ++B2(p −ΛTϑ)</p><p>Fϑ+G p</p><p></p><p> .</p><p>Because A2 and F are Hurwitz and the origin of ż = f̃ (z, 0,θ, w) is exponentially sta-</p><p>ble, the origin of Ż = f1(Z , 0,θ, w) is exponentially stable, and a Lyapunov function</p><p>for it can be constructed in the form</p><p>V4(Z ,θ, w) =κ1 V0(z,θ, w)+ κ2 qT P2q +ϑT Pϑ</p><p>with sufficiently small κ2 and κ1 / κ2 [80, Appendix C]. It can be verified that V4</p><p>satisfies the inequalities</p><p>c̃1‖Z‖</p><p>2 ≤V4(Z ,θ, w)≤ c̃2‖Z‖</p><p>2,</p><p>∂ V4</p><p>∂ Z</p><p>f1(Z , 0,θ, w)+</p><p>∂ V4</p><p>∂ w</p><p>S0w ≤−c̃3‖Z‖</p><p>2,</p><p>∂ V4</p><p>∂ Z</p><p>≤ c̃4‖Z‖</p><p>in a neighborhood of Z = 0 with positive constants c̃1 to c̃4. Let W1 =</p><p>p</p><p>V4, W2 =</p><p>p</p><p>p2, and W3 =</p><p>p</p><p>ϕT P0ϕ. By calculating upper bounds of D+W1, D+W2, and</p><p>D+W3,46 it can be shown that</p><p>D+W1 ≤−b1W1+ k1W2+ k2W3,</p><p>D+W2 ≤−(b2/µ)W2+ k3W1+ k4W2+(k5/µ)W3+δ,</p><p>D+W3 ≤−(b3/ε)W3+ k6W1+ k7W2+(k8/µ)W2+(k9/µ)W3+ k10δ,</p><p>where b1 to b3 and k1 to k10 are positive constants independent of ε and µ. Rewrite</p><p>the foregoing scalar differential inequalities as the vector differential inequality</p><p>D+W ≤AW +Bδ,</p><p>where</p><p>W=</p><p></p><p></p><p>W1</p><p>W2</p><p>W3</p><p></p><p>, A=</p><p></p><p></p><p>−b1 k1 k2</p><p>k3 −(b2/µ− k4) k5/µ</p><p>k6 (k7+ k8/µ) −(b3/ε− k9/µ)</p><p></p><p>, and B=</p><p></p><p></p><p>0</p><p>1</p><p>k10</p><p></p><p>.</p><p>For sufficiently small µ and ε/µ, the matrixA is Hurwitz and quasi monotone. Ap-</p><p>plication of the comparison method shows that47</p><p>W (t )≤U (t ) ∀ t ≥ 0,</p><p>46See [78, Section 3.4] for the definition of D+W and [78, Section 9.3] for an example of calculating an</p><p>upper bound on D+W .</p><p>47See [125, Chapter IX] for the definition quasi-monotone matrices and the comparison method for vec-</p><p>tor differential inequalities.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.5. INTERNAL MODEL PERTURBATION 141</p><p>where U is the solution of the differential equation</p><p>U̇ =AU +Bδ, U (0) =W (0). (5.61)</p><p>BecauseA is Hurwitz, equation (5.61) has an exponentially stable equilibrium point</p><p>at Ū = col(ū1, ū2, ū3) that satisfies the algebraic equation</p><p>0=A Ū +Bδ.</p><p>It can be verified that there is a constant k11 > 0, independent of ε and µ, such that for</p><p>sufficiently small µ and ε/µ, |ū1| ≤ k11δ. Therefore, W1(t ) is ultimately bounded by</p><p>k12δ for k12 > k11. The proof is completed by noting that e is a component of Z and</p><p>‖Z‖ ≤</p><p>p</p><p>V4/c̃1 =W1/</p><p>p</p><p>c̃1 2</p><p>Remark 5.13. The proof uses linear-type Lyapunov functions and the vector Lyapunov</p><p>function method rather than a quadratic-type Lyapunov function, which could have been</p><p>constructed as a linear combination of V4, p2, and ϕT P0ϕ. The quadratic-type function</p><p>would yield a bound on the error of the order O(</p><p>p</p><p>µδ), which is more conservative than</p><p>O(µδ) for small µδ . 3</p><p>The next two examples illustrate the O(µδ) bound on the steady-state error. In the</p><p>first example the internal model perturbation is due to uncertainty in the parameters</p><p>of the model, while in the second example the steady-state control does not satisfy</p><p>(5.35), but an approximation of it does so.</p><p>Example 5.5. Reconsider Example 5.4 where the conditional servocompensator is</p><p>designed assuming the frequency of the reference signal is 2.5 rad/sec when the ac-</p><p>tual frequency ω0 6= 2.5. The feedback controller is the same as in Example 5.4, and</p><p>the simulation is carried out using the same parameters and initial conditions. The</p><p>simulation results are shown in Figure 5.8. Figures 5.8(a) and (b) compare the regu-</p><p>lation error e for ω0 = 3 and ω0 = 2.7 rad/sec when µ = 0.1. It is observed first</p><p>that the frequency error has a little effect on the transient response. This is expected</p><p>because the transient response is basically the response under the sliding mode con-</p><p>troller u = −20 sgn(ξ1 + ξ2). As for the steady-state response, the error decreases as</p><p>the frequency approaches the nominal frequency of 2.5 rad/sec. Figures 5.8(c) and (d)</p><p>compare the regulation error e for µ= 0.1 and µ= 0.01 with fixed frequency ω0 = 3</p><p>rad/sec. Once again, the change in µ has a little effect on the transient response, as</p><p>both values are small enough to bring the transient response close to that of the sliding</p><p>mode control. The steady-state error decreases with decreasing µ. Both cases demon-</p><p>strate the fact that the steady-state error is of the order O(µδ). 4</p><p>Example 5.6. Consider the system</p><p>ẋ1 =−θ1x1+ x2</p><p>2 + d , ẋ2 = x3, ẋ3 =−θ2x1x2+θ3 sin x2+ u, y = x2,</p><p>where θ1 > 0, θ2, and θ3 are unknown parameters, d is a constant disturbance, and the</p><p>reference signal is r = α sin(ω0 t + θ0) with known frequency ω0 and unknown am-</p><p>plitude and phase. This is the same problem considered in Examples 5.4 and 5.5 with</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>142 CHAPTER 5. REGULATION</p><p>0 2 4 6 8 10</p><p>−1</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>Time</p><p>e</p><p>(a)</p><p>90 92 94 96 98 100</p><p>−6</p><p>−4</p><p>−2</p><p>0</p><p>2</p><p>4</p><p>6</p><p>x 10</p><p>−3</p><p>Time</p><p>e</p><p>(b)</p><p>0 2 4 6 8 10</p><p>−1</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>Time</p><p>e</p><p>(c)</p><p>90 92 94 96 98 100</p><p>−6</p><p>−4</p><p>−2</p><p>0</p><p>2</p><p>4</p><p>6</p><p>x 10</p><p>−3</p><p>Time</p><p>e</p><p>(d)</p><p>Figure 5.8. Simulation of Example 5.5. (a) and (b) show the transient and steady-state</p><p>regulation error e for ω0 = 3 rad/sec (dashed) and ω0 = 2.7 rad/sec (solid) when µ = 0.1. (c) and</p><p>(d) show the transient and steady state regulation error e for ω0 = 3 rad/sec when µ= 0.1 (dashed)</p><p>and µ= 0.01 (solid).</p><p>the additional term θ3 sin x2 in the ẋ3-equation. The only change from Example 5.4 is</p><p>in the steady-state control on the zero-error manifold, which is given by</p><p>φ̄(θ, w) =−θ3 sin w1−ω</p><p>2</p><p>0 w1+</p><p>θ2</p><p>θ1</p><p>w1w3</p><p>+</p><p>θ2</p><p>θ1(θ2</p><p>1 + 4ω2</p><p>0)</p><p>�</p><p>(θ2</p><p>1 + 2ω2</p><p>0)w</p><p>3</p><p>1 − 2θ1ω0w2</p><p>1 w2+ 2ω2</p><p>0 w1w2</p><p>2</p><p>�</p><p>.</p><p>The function φ̄ does not satisfy Assumption 5.33 due to transcendental function sin(·),</p><p>which generates an infinite number of harmonics of the sinusoidal reference signal.</p><p>The sinusoidal function can be approximated by its truncated Taylor series</p><p>sin w1 ≈</p><p>n</p><p>∑</p><p>i=1</p><p>(−1)i−1w2i−1</p><p>1</p><p>(2i − 1)!</p><p>,</p><p>and the approximation error decreases as n increases. The approximate function</p><p>φ(θ, w) =−θ3</p><p>n</p><p>∑</p><p>i=1</p><p>(−1)i−1w2i−1</p><p>1</p><p>(2i − 1)!</p><p>−ω2</p><p>0 w1+</p><p>θ2</p><p>θ1</p><p>w1w3</p><p>+</p><p>θ2</p><p>θ1(θ2</p><p>1 + 4ω2</p><p>0)</p><p>�</p><p>(θ2</p><p>1 + 2ω2</p><p>0)w</p><p>3</p><p>1 +−2θ1ω0w2</p><p>1 w2+ 2ω2</p><p>0 w1w2</p><p>2</p><p>�</p><p>satisfies Assumption 5.33 because it is a polynomial function of w. Two approxima-</p><p>tions are considered with n = 3 and n = 5. In the first case, φ satisfies the equation</p><p>φ(4)+ 10ω2</p><p>0φ</p><p>(2)+ 9ω4</p><p>0φ= 0</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.5. INTERNAL MODEL PERTURBATION 143</p><p>as in Example 5.4. In this case, the same controller of Example 5.4 is used, that is,</p><p>σ̇ = Fσ +µG sat</p><p>ΛT σ + ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>,</p><p>u =−20 sat</p><p>ΛT σ + ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>,</p><p>˙̂</p><p>ξ1 = ξ̂2+(2/ε)(e − ξ̂1),</p><p>˙̂</p><p>ξ2 = (1/ε</p><p>2)(e − ξ̂1),</p><p>F = ς</p><p></p><p></p><p></p><p></p><p>0 1 0 0</p><p>0 0 1 0</p><p>0 0 0 1</p><p>−1.5 −6.25 −8.75 −5</p><p></p><p></p><p></p><p></p><p>, G = ς</p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p>,</p><p>ΛT =</p><p>�</p><p>−9(ω0/ς)</p><p>4+ 1.5, 6.25, −10(ω0/ς)</p><p>2+ 8.75, 5</p><p>�</p><p>,</p><p>and ς = 3. In the case n = 5, φ satisfies the equation</p><p>φ(6)+ 35ω2</p><p>0φ</p><p>(4)+ 259ω4</p><p>0φ</p><p>(2)+ 225ω6</p><p>0 = 0,</p><p>and the internal model (5.36) is given by</p><p>S =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 0 0 0 0</p><p>0 0 1 0 0 0</p><p>0 0 0 1 0 0</p><p>0 0 0 0 1 0</p><p>0 0 0 0 0 0</p><p>−225ω6</p><p>0 0 −259ω4</p><p>0 0 −35ω2</p><p>0 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, H =</p><p>�</p><p>1 0 0 0 0 0</p><p>�</p><p>.</p><p>The controller is the same as in the previous case except for F , G, and Λ, which are</p><p>taken as</p><p>F = ς</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 0 0 0 0</p><p>0 0 1 0 0 0</p><p>0 0 0 1 0 0</p><p>0 0 0 0 1 0</p><p>0 0 0 1 0 1</p><p>−11.25 −55.125 −101.5 −91.875 −43.75 −10.5]</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, G = ς</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>0</p><p>0</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>,</p><p>ΛT =</p><p>�</p><p>Λ1, 55.125, Λ3, 91.875, Λ5, 10.5</p><p>�</p><p>,</p><p>Λ1 = 11.25− 225 ∗ (ω0/ς)</p><p>6, Λ3 = 101.5− 259 ∗ (ω0/ς)</p><p>4, Λ5 = 43.75− 35 ∗ (ω0/ς)</p><p>2,</p><p>and ς = 3. The eigenvalues of F /ς are −0.5, −1, −1.5, −2, −2.5, and −3. Simu-</p><p>lation results are shown in Figures 5.9 and 5.10 for the parameters θ1 = 3, θ2 = 4,</p><p>θ3 = 1, ω0 = 2.5, d = 0.1, and ε = 10−4 and the initial conditions x1(0) = x2(0) = 1,</p><p>x3(0) = ξ̂1(0) = ξ̂2(0) = 0. In Figure 5.9, µ is fixed at 0.1, while δ is reduced by go-</p><p>ing from n = 3 to n = 5. In Figure 5.10, n is fixed at 3, while µ is reduced from 0.1</p><p>to 0.01. The first observation is that in all cases the transient response of the regu-</p><p>lation error is almost the same. For the steady-state error, it is seen that</p><p>reducing δ</p><p>reduces the error. It is interesting to note that in the case n = 3, the approximation</p><p>of the sinusoidal function maintains the first and third harmonics and neglects the</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>144 CHAPTER 5. REGULATION</p><p>0 2 4 6 8 10</p><p>−1</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>Time</p><p>e</p><p>(a)</p><p>95 96 97 98 99 100</p><p>−4</p><p>−2</p><p>0</p><p>2</p><p>4</p><p>x 10</p><p>−6</p><p>Time</p><p>e</p><p>(b)</p><p>0 2 4 6 8 10</p><p>−1</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>Time</p><p>e</p><p>(c)</p><p>95 96 97 98 99 100</p><p>−4</p><p>−2</p><p>0</p><p>2</p><p>4</p><p>x 10</p><p>−8</p><p>Time</p><p>e</p><p>(d)</p><p>Figure 5.9. Simulation of Example 5.6. (a) and (b) show the transient and steady-state</p><p>regulation error e for the case n = 3, while (c) and (d) show the error for the case n = 5. In both</p><p>cases, µ= 0.1.</p><p>0 2 4 6 8 10</p><p>−1</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>Time</p><p>e</p><p>(a)</p><p>95 96 97 98 99 100</p><p>−4</p><p>−2</p><p>0</p><p>2</p><p>4</p><p>x 10</p><p>−6</p><p>Time</p><p>e</p><p>(b)</p><p>0 2 4 6 8 10</p><p>−1</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>Time</p><p>e</p><p>(c)</p><p>95 96 97 98 99 100</p><p>−4</p><p>−2</p><p>0</p><p>2</p><p>4</p><p>x 10</p><p>−7</p><p>Time</p><p>e</p><p>(d)</p><p>Figure 5.10. Simulation of Example 5.6. (a) and (b) show the transient and steady-state</p><p>regulation error e for the case n = 3 when µ= 0.1, while (c) and (d) show the error when µ= 0.01.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.6. ADAPTIVE INTERNAL MODEL 145</p><p>higher-order harmonics, with the fifth harmonic being the most significant neglected</p><p>one. Figure 5.9(b) shows that the error oscillates at the fifth harmonic frequency (12.5</p><p>rad/sec.). In the case n = 5, the approximation maintains the first, third, and fifth</p><p>harmonics and neglects the higher-order harmonics, with the seventh harmonic being</p><p>the most significant neglected one. Figure 5.9(d) shows that the error oscillates at the</p><p>seventh harmonic frequency (17.5 rad/sec.). With n fixed, reducing µ decreases the</p><p>steady-state error. 4</p><p>5.6 Adaptive Internal Model</p><p>The controller of Section 5.4 requires precise knowledge of the eigenvalues of S be-</p><p>cause they are used to calculate Λ. When they are not known, Λ is replaced by Λ̂, and</p><p>an adaptive law is used to adjust Λ̂ in real time. Two cases will be considered: full-</p><p>parameter adaptation and partial-parameter adaptation. In the first case, all q elements</p><p>of Λ̂ are adapted. In the second case, the number of adapted parameters is equal to the</p><p>number of complex modes of S, which is q/2 when q is even and (q − 1)/2 when it</p><p>is odd. This case corresponds to a special choice of the pair (F ,G) in the controllable</p><p>canonical form</p><p>Fc =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 · · · · · · 0</p><p>0 0 1 · · · 0</p><p>...</p><p>...</p><p>0 · · · · · · 0 1</p><p>∗ · · · · · · · · · ∗</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, Gc =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>...</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>,</p><p>where Fc is Hurwitz. Because the matrix S is in the companion form and has simple</p><p>eigenvalues on the imaginary axis, the vector Λ that assigns the eigenvalues of (Fc +</p><p>GcΛ</p><p>T ) at those of S will have only q/2 (or (q − 1)/2) elements that depend on the</p><p>eigenvalues of S. For example, if S has eigenvalues at ± jω1 and ± jω2, the last row</p><p>of S is</p><p>�</p><p>−ω2</p><p>1ω</p><p>2</p><p>2 0 −(ω2</p><p>1 +ω</p><p>2</p><p>2) 0</p><p>�</p><p>.</p><p>One concern with this choice of F is that some components of Λ could be very large</p><p>when the eigenvalues of S are large,48 as it can be seen from the foregoing example of</p><p>S when the frequencies ω1 and ω2 are large. To address this concern, F and G are</p><p>chosen as F = ςFc and G = ςGc for some positive constant ς . It can be seen that</p><p>the coefficient of s q−i in the characteristic equation of (F +GΛT ) is −ς i ( fi + Λi ),</p><p>where fi is the i th element in the last row of Fc and Λi the i th element of Λ. If βi</p><p>is the coefficient of s q−i in the characteristic equation of S, then Λi = − fi −βi/ς</p><p>i .</p><p>Knowing the range of values of the eigenvalues of S, the scaling factor ς can be chosen</p><p>to control the range of the elements of Λ. For example, when the eigenvalues of S are</p><p>± jω1 and ± jω2, its characteristic equation is</p><p>s4+(ω2</p><p>1 +ω</p><p>2</p><p>2)s</p><p>2+ω2</p><p>1ω</p><p>2</p><p>2 = 0.</p><p>Choosing ς2 =max(ω2</p><p>1 +ω</p><p>2</p><p>2) ensures that</p><p>ω2</p><p>1ω</p><p>2</p><p>2</p><p>ς4</p><p>0. With the change of variables</p><p>ϑ = (σ − σ̄)/µ+G%, (5.63)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.6. ADAPTIVE INTERNAL MODEL 147</p><p>the closed-loop system takes the form49</p><p>ẇ = S0w,</p><p>ϑ̇ = Fϑ+ f1(z, q , s1,θ, w),</p><p>ż = f̃0(z, 0,θ, w)+ f2(z, q , s1,θ, w),</p><p>q̇ =A2q +B2 s1,</p><p>ṡ1 =−b (η,ζ ,θ)β(ξ )s1/µ− b (η,ζ ,θ)β(ξ )λ̃T ν/µ+ f3(ϑ, z, q , s1,θ, w),</p><p>in which f1 to f3 are locally Lipschitz functions that satisfy f1(0,0,0,θ, w) = 0,</p><p>f2(z, 0, 0,θ, w) = 0, and f3(0,0,0,0,θ, w) = 0. Consider the Lyapunov function candi-</p><p>date</p><p>Va = ϑ</p><p>T Pϑ+ κ1 V0+ κ2 qT P2q +V5+</p><p>µ</p><p>2γ</p><p>λ̃T λ̃, (5.64)</p><p>where V0, P , and P2 are defined in Section 5.4; γ , κ1, and κ2 are positive constants;</p><p>and V5 is defined by</p><p>V5 =</p><p>∫ s1</p><p>0</p><p>y</p><p>β(ξ1, . . . ,ξρ−1, y −</p><p>∑ρ−1</p><p>i=1 kiξi )b̃ (z,ξ1, . . . ,ξρ−1, y −</p><p>∑ρ−1</p><p>i=1 kiξi ,θ, w)</p><p>d y.</p><p>It can be seen that</p><p>s2</p><p>1</p><p>2bmβm</p><p>≤V5 ≤</p><p>s2</p><p>1</p><p>2b0β0</p><p>,</p><p>where bm and βm are upper bounds on b and β in the set Ωµ. The derivative of Va</p><p>satisfies the inequality</p><p>V̇a ≤−Y</p><p>T</p><p>a QaYa −</p><p>1</p><p>µ</p><p>s1λ̃</p><p>T ν +</p><p>µ</p><p>γ</p><p>λ̃T λ̇,</p><p>where</p><p>Ya = col(‖ϑ‖,‖z‖,‖q‖, |s1|), (5.65)</p><p>Qa =</p><p></p><p></p><p></p><p></p><p>1 −c2 −c3 −c4</p><p>−c2 κ1 c5 − κ1 c6 −(κ1 c7+ c8)</p><p>−c3 − κ1 c6 κ2 −(κ2 c9+ c10)</p><p>−c4 −(κ1 c7+ c8) −(κ2 c9+ c10) 1/µ− c11</p><p></p><p></p><p></p><p></p><p>, (5.66)</p><p>and c1 to c11 are positive constants independent of κ1, κ2, and µ. Choose κ1 large</p><p>enough to make the 2×2 principal minor of Qa positive; then choose κ2 large enough</p><p>to make the 3×3 principal minor positive; then chooseµ small enough to make the de-</p><p>terminant of Qa positive. The adaptive law λ̇= (γ/µ2)s1ν results in V̇a ≤−Y T</p><p>a QaYa ,</p><p>which shows that limt→∞Ya(t ) = 0 [78, Theorem 8.4]; hence, limt→∞ e(t ) = 0 be-</p><p>cause e = ξ1 = q1. Under output feedback,</p><p>s1 is replaced by its estimate ŝ1 provided by</p><p>a high-gain observer; therefore, parameter projection is used, as in Chapter 4, to ensure</p><p>that λ̂ remains bounded. Knowing upper bounds on the unknown frequencies of the</p><p>internal model, it is possible to determine constants ai and bi such that λ belongs to</p><p>the set</p><p>Υ = {λ | ai ≤ λi ≤ bi , 1≤ i ≤ m}.</p><p>49Equations (5.46) and (5.47) are used in arriving at these equations.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>148 CHAPTER 5. REGULATION</p><p>Let</p><p>Υδ = {λ | ai −δ ≤ λi ≤ bi +δ, 1≤ i ≤ m},</p><p>where δ > 0. The adaptive law is taken as</p><p>˙̂</p><p>λi = (γ/µ</p><p>2)Pi (λ̂i , s1νi )s1νi , where</p><p>Pi (λ̂i , s1νi ) =</p><p></p><p></p><p></p><p>1+(bi − λ̂i )/δ if λ̂i > bi and s1νi > 0,</p><p>1+(λ̂i − ai )/δ if λ̂i 0 exists, and for each µ ∈ (0,µ∗], ε∗ = ε∗(µ) exists</p><p>such that for each µ ∈ (0,µ∗] and ε ∈ (0,ε∗(µ)], all state variables are bounded and</p><p>limt→∞ e(t ) = 0. Furthermore, let χ = (z,ξ ) be part of the state of the closed-loop system</p><p>under the output feedback controller, and let χ ∗ = (z∗,ξ ∗) be the state of the closed-loop</p><p>system under the state feedback sliding mode controller (5.42), with χ (0) = χ ∗(0). Then,</p><p>for every δ0 > 0, there is µ∗1 > 0, and for each µ ∈ (0,µ∗1], there is ε∗1 = ε</p><p>∗</p><p>1(µ) > 0 such</p><p>that for µ ∈ (0,µ∗1] and ε ∈ (0,ε∗1(µ)],</p><p>‖χ (t )−χ ∗(t )‖ ≤ δ0 ∀ t ≥ 0. (5.74)</p><p>Finally, if ν̄ is persistently exciting, then limt→∞</p><p>λ̂(t ) = λ. 3</p><p>Proof: Similar to the case of no adaption (Theorem 5.3), it can be shown that, for</p><p>sufficiently small µ and ε, the set Ξ×Ω×Σε is positively invariant and the trajectory</p><p>(ϑ(t ), z(t ), q(t ), s(t ),ϕ(t )) enters Ξ×Ωµ×Σε in finite time. On the other hand, due</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.6. ADAPTIVE INTERNAL MODEL 151</p><p>to parameter projection λ̂(t ) ∈ Υδ for all t ≥ 0. Inside Ξ×Ωµ ×Σε the closed-loop</p><p>system is represented by51</p><p>ẇ = S0w,</p><p>σ̇ = Fσ +G(Λ̂T σ + s1)+µG[sat( ŝ/µ)− sat(s/µ)],</p><p>ż = f̃0(z,ξ ,θ, w),</p><p>q̇ =A2q +B2 s1,</p><p>ṡ1 =−b (η,ζ ,θ)β(ξ )(Λ̂T σ + s1)/µ+∆1(·)</p><p>+ b (η,ζ ,θ)[−βs (ξ̂ ) sat( ŝ/µ)+βs (ξ ) sat(s/µ)],</p><p>εϕ̇ =A0ϕ+ εB[a0(z,ξ ,θ, w)− b (η,ζ ,θ)φ(θ, w)− b (η,ζ ,θ)βs (ξ̂ ) sat( ŝ/µ)],</p><p>˙̂</p><p>λ= (γ/µ2)Π( ŝ ,µ)P (λ̂, ŝ1ν) ŝ1ν,</p><p>whereP is a diagonal matrix whose diagonal elements areP1 toPm . Inside Ξ×Ωµ×</p><p>Σε, |s | ≤µ and |s − ŝ |=O(ε). For sufficiently small ε, | ŝ | ≤ 1.5µ; hence, Π( ŝ ,µ) = 1.</p><p>Applying the change of variables (5.63), the system can be represented in the form</p><p>ẇ = S0w,</p><p>ϑ̇ = Fϑ+ f1(z, q , s1,θ, w)+ h1(z, q , s1,ϕ,θ, w)/µ,</p><p>ż = f̃0(z, 0,θ, w)+ f2(z, q , s1,θ, w),</p><p>q̇ =A2q +B2 s1,</p><p>ṡ1 =−b (η,ζ ,θ)β(ξ )s1/µ− b (η,ζ ,θ)β(ξ )λ̃T ν/µ+ f3(ϑ, z, q , s1,θ, w)</p><p>+ h2(z, q , s1,ϕ, λ̃,θ, w)/µ,</p><p>εϕ̇ =A0ϕ+(ε/µ)B[−b (η,ζ ,θ)β(ξ )s1/µ− b (η,ζ ,θ)β(ξ )λ̃T ν/µ]</p><p>+ f4(ϑ, z, q , s1,θ, w)+ h3(ϑ, z, q , s1,ϕ, λ̃,θ, w)/µ,</p><p>˙̃</p><p>λ= (γ/µ2)P (λ̂, ŝ1ν) ŝ1ν = (γ/µ</p><p>2)P (λ̂, s1ν)s1ν + h4(ϑ, q , s1,ϕ, λ̃,θ, w)/µ,</p><p>where h1 to h4 are locally Lipschitz functions that vanish at ϕ = 0, and f1 to f4 are lo-</p><p>cally Lipschitz functions that satisfy f1(0,0,0,θ, w) = 0, f2(z, 0, 0,θ, w) = 0,</p><p>f3(0,0,0,0,θ, w) = 0, and f4(0,0,0,0,θ, w) = 0. By repeating the state feedback analy-</p><p>sis, it can be seen that the derivative of Va of (5.64) satisfies</p><p>V̇a ≤−Y</p><p>T</p><p>a QaYa +(`1+µ`2)‖Ya‖ ‖ϕ‖/µ+ `3‖λ̃</p><p>T ν‖ ‖ϕ‖/µ, (5.75)</p><p>whereYa and Qa are defined by (5.65) and (5.66), respectively, and `1 to `3 are positive</p><p>constants. In arriving at (5.75) the term −s1λ̃</p><p>T ν ++λ̃TP ŝ1ν is written as</p><p>−s1λ̃</p><p>T ν + λ̃TP (λ̂, ŝ1ν) ŝ1ν =− ŝ1λ̃</p><p>T ν ++λ̃TP (λ̂, ŝ1ν) ŝ1ν +( ŝ1− s1)λ̃</p><p>T ν</p><p>≤ ( ŝ1− s1)λ̃</p><p>T ν ,</p><p>where − ŝ1λ̃</p><p>T ν + λ̃TP (λ̂, ŝ1ν) ŝ1ν ≤ 0 according to the adaptive law. As it was shown</p><p>earlier, the matrix Qa is positive definite for sufficiently large κ1 and κ2 and sufficiently</p><p>small µ.</p><p>51See the proof of Theorem 5.3 for the definition of variables that are not defined here.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>152 CHAPTER 5. REGULATION</p><p>By Lemma 5.1 and (5.63),</p><p>λ̃T ν = λ̃T L−1LEσ = λ̃T L−1LE(µϑ+ σ̄ −µG%]</p><p>=</p><p>�</p><p>λ̃T</p><p>a λ̃T</p><p>b</p><p>�</p><p>�</p><p>µϑa − ν̄a −µGa%</p><p>µϑb −µGb%</p><p>�</p><p>, (5.76)</p><p>where</p><p>�</p><p>ϑa</p><p>ϑb</p><p>�</p><p>= LEϑ,</p><p>�</p><p>Ga</p><p>Gb</p><p>�</p><p>= LEG, and</p><p>�</p><p>λ̃a</p><p>λ̃b</p><p>�</p><p>= L−T λ̃.</p><p>The ṡ1- and</p><p>˙̃</p><p>λ-equations can be written as</p><p>�</p><p>ṡ1</p><p>˙̃</p><p>λa</p><p>�</p><p>= (bβ/µ)</p><p>�</p><p>−1 −ν̄T</p><p>a</p><p>γ ν̄a 0</p><p>�</p><p>�</p><p>s1</p><p>λ̃a</p><p>�</p><p>+</p><p>�</p><p>−bβλ̃T</p><p>a (ϑa −Ga%)− bβλ̃T</p><p>b (ϑb −Gb%)+ f3+ h2/µ</p><p>−γ bβν̄a s1/µ+ γNaP s1ν/µ</p><p>2+Na h4/µ</p><p>�</p><p>, (5.77)</p><p>where b = b (η,ζ ,θ), β = β(ξ ), and L−T = col(Na ,Nb ). The right-hand side of</p><p>(5.77) vanishes at (ϑ = 0, z = 0, q = 0, s1 = 0,ϕ = 0, λ̃a = 0) regardless of λ̃b , which</p><p>is bounded due to parameter projection. Therefore, λ̃b is treated as bounded time-</p><p>varying disturbance. Because ν̄a is persistently exciting, the origin of the systems</p><p>�</p><p>ṡ1</p><p>˙̃</p><p>λa</p><p>�</p><p>= (bβ/µ)</p><p>�</p><p>−1 −ν̄T</p><p>a</p><p>γ ν̄a 0</p><p>�</p><p>�</p><p>s1</p><p>λ̃a</p><p>�</p><p>is exponentially stable [75, Section 13.4]. By the converse Lyapunov theorem [78,</p><p>Theorem 4.14], there is a Lyapunov function V6 whose derivative along the system</p><p>(5.77) satisfies the inequality</p><p>V̇6 ≤−k1|s1|</p><p>2− k2‖λ̃a‖</p><p>2+ k3‖Ya‖</p><p>2+ k4‖Ya‖ ‖λ̃a‖+ k5‖Ya‖ ‖ϕ‖+ k6‖λ̃a‖ ‖ϕ‖,</p><p>where, from now on, the positive constants ki could depend onµ but are independent</p><p>of ε. Using (5.76) in (5.75) shows that</p><p>V̇a ≤−k7‖Ya‖</p><p>2+ k8‖Ya‖ ‖ϕ‖+ k9‖λ̃a‖ ‖ϕ‖.</p><p>For the ϕ̇-equation, it can be shown that the derivative of V3 = ϕ</p><p>T P0ϕ satisfies the</p><p>inequality</p><p>V̇3 ≤−‖ϕ‖</p><p>2/ε+ k10‖ϕ‖</p><p>2+ k11‖λ̃a‖ ‖ϕ‖+ k12‖Ya‖ ‖ϕ‖.</p><p>Using W = αVa+V3+V6 with α > 0 as a Lyapunov function candidate for the closed-</p><p>loop system, it can be shown that</p><p>Ẇ ≤−Y T QY ,</p><p>where</p><p>Y =</p><p></p><p></p><p>‖Ya‖</p><p>‖λ̃a‖</p><p>‖ϕ‖</p><p></p><p> and Q = 1</p><p>2</p><p></p><p></p><p>2(αk7− k3) −k4 −(k13+αk8)</p><p>−k4 2k2 −(k14+αk9)</p><p>−(k13+αk8) −(k14+αk9) 2(1/ε− k10)</p><p></p><p> ,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.6. ADAPTIVE INTERNAL MODEL 153</p><p>where k13 = k5 + k12 and k14 = k6 + k11. Choose α large enough to make the 2× 2</p><p>principal minor of Q positive; then choose ε small enough to make Q positive definite.</p><p>Thus, limt→∞ ‖Y (t )‖= 0 [78, Theorem 8.4], which implies that limt→∞ e(t ) = 0 and</p><p>limt→∞ λ̃a(t ) = 0. If ν̄ is persistently exciting, it follows from Lemma 5.1 that ν̄a = ν̄.</p><p>Hence, λ̃ = λ̃a . Therefore, limt→∞ λ̂(t ) = λ. The proof of (5.74) is done as in the</p><p>proof of Theorem 5.2, whereµ is reduced first to bring the trajectories under the state</p><p>feedback controller with conditional servocompensator close to the trajectories under</p><p>the sliding mode controller, then ε is reduced to bring the trajectories under output</p><p>feedback close to the ones under state feedback. �</p><p>Remark 5.15. The proof shows that when ν̄ is not persistently exciting, partial parameter</p><p>convergence is achieved as limt→∞ λ̃a(t ) = 0. 3</p><p>The performance of the conditional servocompensator with adaptive internal</p><p>model is illustrated by two examples. Example 5.7 reconsiders Example 5.4 and com-</p><p>pares the adaptive internal model with the known one. Example 5.8 illustrates the</p><p>effect of the persistence of excitation condition. Both examples use partial parameter</p><p>adaptation.</p><p>Example 5.7. Consider the system</p><p>ẋ1 =−θ1x1+ x2</p><p>2 + d , ẋ2 = x3, ẋ3 =−θ2x1x2+ u, y = x2</p><p>from Example 5.4, where θ1 > 0 and θ2 are unknown parameters, d is a constant</p><p>disturbance, and the reference signal r = α sin(ω0 t +θ0) has unknown frequencyω0,</p><p>in addition to unknown amplitude and phase. Building on Example 5.4, the output</p><p>feedback controller is taken as</p><p>σ̇ = Fσ +µG sat</p><p>Λ̂T σ + ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>,</p><p>u =−20 sat</p><p>Λ̂T σ + ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>,</p><p>˙̂</p><p>ξ1 = ξ̂2+(2/ε)(e − ξ̂1),</p><p>˙̂</p><p>ξ2 = (1/ε</p><p>2)(e − ξ̂1),</p><p>where the only change is that ΛT is replaced by</p><p>Λ̂T =</p><p>�</p><p>Λ̂1, 6.25, Λ̂3, 5</p><p>�</p><p>.</p><p>Assuming that ω0 ∈ [1.2,3], the scaling factor ς = 3 remains the same as in Exam-</p><p>ple 5.4. Using the expressions Λ1 =−9(ω0/ς)</p><p>4+ 1.5 and Λ3 =−10(ω0/ς)</p><p>2+ 8.75, it</p><p>can be verified that</p><p>−7.5≤Λ1 ≤ 1.27 and − 1.25≤Λ2 ≤ 7.15.</p><p>The sets Υ and δ are taken as</p><p>Υ = {(Λ1,Λ3) | − 7.5≤Λ1 ≤ 1.3, −1.25≤Λ3 ≤ 7.7}, δ = 0.1.</p><p>With λ̂1 = Λ̂1 and λ̂2 = Λ̂3, the matrix E of (5.62) is given by</p><p>E =</p><p>�</p><p>1 0 0 0</p><p>0 0 1 0</p><p>�</p><p>.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>154 CHAPTER 5. REGULATION</p><p>Therefore ν1 = σ1 and ν2 = σ3. The adaptive law is given by</p><p>˙̂</p><p>λi = (γ/µ</p><p>2)ΠPi ŝ1νi for i = 1,2,</p><p>where ŝ1 = ξ̂1+ ξ̂2,</p><p>P1 =</p><p></p><p></p><p></p><p>1+(1.3− λ̂1)/0.1 if λ̂1 > 1.3 and ŝ1ν1 > 0,</p><p>1+(λ̂1+ 7.5)/0.1 if λ̂1 7.7 and ŝ1ν2 > 0,</p><p>1+(λ̂2+ 1.25)/0.1 if λ̂2</p><p>model (solid)</p><p>and the known internal model (dashed). It can be seen that the error trajectories are</p><p>very close and that the control trajectories are almost indistinguishable. Recall from</p><p>Example 5.4 that the trajectories under the known internal model are very close to the</p><p>ones under sliding mode control, so the same observation applies to the trajectories</p><p>under the adaptive internal model. Figures 5.11(c) and (d) show the convergence of</p><p>the parameter errors to zero, which is valid because all modes of the internal model</p><p>are excited; hence the signal ν̄ is persistently exciting. 4</p><p>0 2 4 6 8 10</p><p>−1</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>Time</p><p>(a)</p><p>e</p><p>Adaptive</p><p>known</p><p>0 2 4 6 8 10</p><p>−10</p><p>−5</p><p>0</p><p>5</p><p>10</p><p>Time</p><p>(b)</p><p>u</p><p>0 10 20 30</p><p>0</p><p>1</p><p>2</p><p>3</p><p>4</p><p>Time</p><p>(c)</p><p>Λ̂</p><p>1</p><p>−</p><p>Λ</p><p>1</p><p>0 10 20 30</p><p>−4</p><p>−2</p><p>0</p><p>2</p><p>4</p><p>6</p><p>Λ̂</p><p>3</p><p>−</p><p>Λ</p><p>3</p><p>Time</p><p>(d)</p><p>Figure 5.11. Simulation of Example 5.7. (a) and (b) show the regulation error e and the</p><p>control signal u for the adaptive internal model (solid) and the known internal model (dashed). (c)</p><p>and (d) show the parameter errors for the adaptive internal model.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>5.6. ADAPTIVE INTERNAL MODEL 155</p><p>Example 5.8. Consider the system</p><p>ẋ1 =−θ1x1+ x2</p><p>2 + d , ẋ2 = x3, ẋ3 =−θ2x1+ u, y = x2,</p><p>where θ1 > 0 and θ2 are unknown parameters, and d = α1 sin(ω1 t+φ1)+α2 sin(ω2 t+</p><p>φ2) is a disturbance input. The output y is to be regulated to a constant reference r .</p><p>Assumptions 5.5 and 5.6 are satisfied globally with</p><p>η= x1, ζ1 = x2, ζ2 = x3.</p><p>The exosystem of Assumption 5.7 is given by</p><p>ẇ =</p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 0 0 0 0</p><p>0 0 ω1 0 0</p><p>0 −ω1 0 0 0</p><p>0 0 0 0 ω2</p><p>0 0 0 −ω2 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p>w, r = w1, d = w2+w4.</p><p>It can be verified that τ1 = w1, τ2 = 0, the solution of (5.33) is</p><p>τ0 =</p><p>1</p><p>θ1</p><p>w2</p><p>1 +</p><p>1</p><p>θ2</p><p>1 +ω</p><p>2</p><p>1</p><p>(θ1w2−ω1w3)+</p><p>1</p><p>θ2</p><p>1 +ω</p><p>2</p><p>2</p><p>(θ1w4−ω2w5),</p><p>and φ= θ2τ0 satisfies the differential equation</p><p>φ(5)+(ω2</p><p>1 +ω</p><p>2</p><p>2)φ</p><p>(3)+ω2</p><p>1ω</p><p>2</p><p>2φ= 0.</p><p>Thus, the internal model is given by</p><p>S =</p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 0 0 0</p><p>0 0 1 0 0</p><p>0 0 0 1 0</p><p>0 0 0 0 1</p><p>0 −ω2</p><p>1ω</p><p>2</p><p>2 0 −(ω2</p><p>1 +ω</p><p>2</p><p>2) 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p>, H =</p><p>�</p><p>1 0 0 0 0</p><p>�</p><p>.</p><p>With the change of variables</p><p>z = x1−τ0, ξ1 = x2−w1, ξ2 = x3,</p><p>the system is represented by</p><p>ż =−θ1z + ξ 2</p><p>1 + 2ξ1w1,</p><p>ξ̇1 = ξ2,</p><p>ξ̇2 =−θ2z + u −φ(θ, w).</p><p>With known frequenciesω1 andω2, the output feedback controller is taken as52</p><p>σ̇ = Fσ +µG sat</p><p>ΛT σ + ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>,</p><p>u =−20 sat</p><p>ΛT σ + ξ̂1+ ξ̂2</p><p>µ</p><p>!</p><p>,</p><p>˙̂</p><p>ξ1 = ξ̂2+(2/ε)(e − ξ̂1),</p><p>˙̂</p><p>ξ2 = (1/ε</p><p>2)(e − ξ̂1),</p><p>52The gain 20 is determined by simulation.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>156 CHAPTER 5. REGULATION</p><p>where e = y − r ,</p><p>F = ς</p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 0 0 0</p><p>0 0 1 0 0</p><p>0 0 0 1 0</p><p>0 0 0 0 1</p><p>−3.75 −17.125 −28.125 −21.25 −7.5</p><p></p><p></p><p></p><p></p><p></p><p></p><p>and G = ς</p><p></p><p></p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>0</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p></p><p></p><p>.</p><p>The eigenvalues of F /ς are −0.5, −1, −1.5, −2, and −2.5. The vector Λ that assigns</p><p>the eigenvalues of F +GΛT at the eigenvalues of S is</p><p>ΛT =</p><p>�</p><p>3.75 (17.125−ω2</p><p>1ω</p><p>2</p><p>2/ς</p><p>4) 28.125 (21.25− (ω2</p><p>1 +ω</p><p>2</p><p>2)/ς</p><p>2) 7.5</p><p>�</p><p>.</p><p>Assuming that ω1 and ω2 belong to the interval [1,3], ς is taken as 3. When ω1 and</p><p>ω2 are unknown, ΛT is replaced by</p><p>Λ̂T =</p><p>�</p><p>3.75 Λ̂2 28.125 Λ̂4 7.5</p><p>�</p><p>.</p><p>It can be verified that</p><p>16.125≤Λ2 ≤ 17.1 and 19.25≤Λ4 ≤ 21.</p><p>The sets Υ and δ are taken as</p><p>Υ = {(Λ2,Λ4) | 16.1≤Λ2 ≤ 17.2, 19.1≤Λ4 ≤ 21}, δ = 0.1.</p><p>With λ̂1 = Λ̂2 and λ̂2 = Λ̂4, the matrix E of (5.62) is given by</p><p>E =</p><p>�</p><p>0 1 0 0 0</p><p>0 0 0 1 0</p><p>�</p><p>.</p><p>Therefore, ν1 = σ2 and ν2 = σ4. The adaptive law is given by</p><p>˙̂</p><p>λi = (γ/µ</p><p>2)ΠPi ŝ1νi for i = 1,2,</p><p>where ŝ1 = ξ̂1+ ξ̂2,</p><p>P1 =</p><p></p><p></p><p></p><p>1+(17.2− λ̂1)/0.1 if λ̂1 > 17.2 and ŝ1ν1 > 0,</p><p>1+(λ̂1− 16.1)/0.1 if λ̂1 21 and ŝ1ν2 > 0,</p><p>1+(λ̂2− 19.1)/0.1 if λ̂2</p><p>on Seshagiri and Khalil [135], while</p><p>the conditional servocompensator is based on Seshagiri and Khalil [136]. A key ele-</p><p>ment in the servocompensator design is the observation that the internal model should</p><p>generate not only the modes of the exosystem but also higher-order harmonics in-</p><p>duced by nonlinearities. This finding was reported, independently, by Khalil [73],</p><p>Huang and Lin [61], and Priscoli [122]. The internal model perturbation result of</p><p>Section 5.5 is based on Li and Khalil [96], which improves over an earlier result by</p><p>Khalil [76]. The adaptive internal model of Section 5.6 is based on Li and Khalil [95],</p><p>which builds on earlier work by Serrani, Isidori, and Marconi [133]. Another ap-</p><p>proach to deal with internal model uncertainty without adaptation is given in Isidori,</p><p>Marconi, and Praly [69].</p><p>The results of this chapter assume that the exogenous signals are generated by the</p><p>exosystem. It is intuitively clear that it is enough to assume that the exogenous signals</p><p>asymptotically approach signals that are generated by the exosystem since the error</p><p>is shown to converge to zero asymptotically. This relaxed assumption is used in [76,</p><p>135, 136]. The robust control technique used in the chapter is sliding mode control.</p><p>Results that use Lyapunov redesign are given in [104, 105, 108, 140].</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>Chapter 6</p><p>Extended Observer</p><p>For the normal form</p><p>η̇= f0(η,ξ ),</p><p>ξ̇i = ξi+1 for 1≤ i ≤ ρ− 1,</p><p>ξ̇ρ = a(η,ξ )+ u,</p><p>y = ξ1,</p><p>the ρ-dimensional high-gain observer estimates x1 and its derivatives x (1)1 to x (ρ−1)</p><p>1 . If</p><p>an estimate of the ρth derivative x (ρ)1 is available and u is known, it will be possible</p><p>to estimate the function a, which could play an important role in control design. An</p><p>observer that estimates a is called extended high-gain observer. If a is a disturbance</p><p>term, the observer acts as a disturbance estimator.53 Alternatively, an estimate of a</p><p>could be used to build an observer that estimates the state η of the internal dynamics</p><p>η̇ = f0(η,ξ ). In this case, the extended high-gain observer acts as a soft sensor of the</p><p>internal dynamics.</p><p>The chapter starts in Section 6.1 with two motivating examples that illustrate the</p><p>use of the extended observer as a disturbance estimator and a soft sensor of the in-</p><p>ternal dynamics, respectively. Section 6.2 shows how the disturbance estimator can</p><p>be used to achieve feedback linearization in the presence of uncertainty. The section</p><p>starts with single-input–single-output systems, followed by multi-input–multi-output</p><p>systems. It ends with a dynamic inversion algorithm that can handle systems with un-</p><p>certain control coefficient or systems that depend nonlinearly on the control input.</p><p>Section 6.3 shows how the use of the extended observer as a soft sensor of the internal</p><p>dynamics allows for relaxation of the minimum-phase assumption, which has been a</p><p>common factor of the output feedback control designs in the earlier chapters. Two</p><p>ideas are presented. The first one uses the soft sensor to implement a control designed</p><p>for an auxiliary system that includes the internal dynamics. The second idea uses the</p><p>soft sensor to build an observer for the internal dynamics to estimate η, which together</p><p>with the estimate of ξ from the extended high-gain observer provide a full-order ob-</p><p>server for the system. Notes and references are given in Section 6.4.</p><p>53Disturbance estimators are widely used in feedback control; see [97] for a comprehensive coverage of the</p><p>topic; see also [48] and [58] for the use of disturbance estimators in Active Disturbance Rejection Control</p><p>(ADRC).</p><p>159</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>160 CHAPTER 6. EXTENDED OBSERVER</p><p>6.1 Motivating Examples</p><p>Example 6.1. Consider the system</p><p>ẋ1 = x2, ẋ2 =φ(t , x)+ u, y = x1,</p><p>where φ(t , x) is unknown. Suppose φ(t , x) is continuously differentiable and φ and</p><p>its partial derivatives are bounded on compact sets of x for all t ≥ 0. The goal is to</p><p>design an output feedback controller to stabilize the origin using the measured output</p><p>y. Set σ =φ(t , x) and rewrite the system as</p><p>ẋ1 = x2, ẋ2 = σ + u, y = x1.</p><p>Had x and σ been measured, the control could have been taken as</p><p>u =−σ − x1− x2,</p><p>which would result in the linear system</p><p>ẋ =</p><p>�</p><p>0 1</p><p>−1 −1</p><p>�</p><p>x def= Ax.</p><p>We refer to this system as the target system and to its solution as the target response.</p><p>But x and σ are not measured as the only measured signal is y. To design an observer</p><p>that estimates x and σ , we extend the dynamics of the system by treating σ as an</p><p>additional state. The extended system is given by</p><p>ẋ1 = x2, ẋ2 = σ + u, σ̇ =ψ(t , x, u), y = x1,</p><p>where</p><p>ψ(t , x, u) =</p><p>∂ φ</p><p>∂ t</p><p>+</p><p>∂ φ</p><p>∂ x1</p><p>x2+</p><p>∂ φ</p><p>∂ x2</p><p>[φ(t , x)+ u].</p><p>The functionψ is unknown, but, due to the robustness of high-gain observers, we can</p><p>build a third-order observer to estimate x and σ . The observer is taken as</p><p>˙̂x1 = x̂2+(α1/ε)(y − x̂1),</p><p>˙̂x2 = σ̂ + u +(α2/ε</p><p>2)(y − x̂1),</p><p>˙̂σ = (α3/ε</p><p>3)(y − x̂1),</p><p>where α1 to α3 are chosen such that the polynomial</p><p>s3+α1 s2+α2 s +α3</p><p>is Hurwitz. We take α1 = α2 = 3 and α3 = 1 to assign the three roots of the polynomial</p><p>at −1. With the estimates x̂ and σ̂ at hand, the output feedback controller is taken as</p><p>u =M sat</p><p>�</p><p>−σ̂ − x̂1− x̂2</p><p>M</p><p>�</p><p>.</p><p>The control is saturated to deal with the peaking phenomenon. The saturation level M</p><p>satisfies M >maxx∈Ω,t≥0 |−φ(t , x)−x1−x2|, whereΩ is a compact positively invariant</p><p>set of the target system. To analyze the closed-loop system, let</p><p>η1 =</p><p>x1− x̂1</p><p>ε2</p><p>, η2 =</p><p>x2− x̂2</p><p>ε</p><p>, η3 = σ − σ̂ ,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>6.1. MOTIVATING EXAMPLES 161</p><p>to obtain</p><p>εη̇1 =−α1η1+η2,</p><p>εη̇2 =−α2η1+η3,</p><p>εη̇3 =−α3η1+ εψ(t , x, u),</p><p>where ψ(t , x, u) is bounded on compact sets of x. Let x(0) ∈Ω0, a compact set in the</p><p>interior of Ω; then there exists T (ε)> 0, with limε→0 T (ε) = 0, such that x(t ) ∈Ω for</p><p>all t ≥ 0 and η(t ) =O(ε) for all t ≥ T (ε). Therefore, for t ≥ T (ε),</p><p>ẋ1 = x2, ẋ2 =−x1− x2+O(ε),</p><p>which shows that the closed-loop system under output feedback recovers the perfor-</p><p>mance of the target system for sufficiently small ε. To illustrate the performance re-</p><p>covery, reconsider the system</p><p>ẋ1 = x2, ẋ2 = x3</p><p>2 + u, y = x1,</p><p>from Example 1.2 but suppose that the term x3</p><p>2 is unknown to the designer. The target</p><p>system ẋ =Ax has a Lyapunov function V (x) = xT P x, where P is the solution of the</p><p>Laypunov equation PA+AT P =−I . Recall from Example 1.2 that Ω= {V (x)≤ 0.3}</p><p>is a positively invariant set of the target system and maxx∈Ω | − x3</p><p>2 − x1− x2| ≤ 0.816.</p><p>Taking M = 1, the output feedback controller is given by</p><p>u = sat (−σ̂ − x̂1− x̂2).</p><p>Simulation results of the target system and the system under output feedback for three</p><p>values of ε are shown in Figure 6.1 for the initial conditions x1(0) = 0.1 and x2(0) =</p><p>x̂1(0) = x̂2(0) = σ̂(0) = 0. The figure shows that the response of the closed-loop system</p><p>under output feedback approaches the target response as ε decreases. As a variation</p><p>on the theme of this simulation, let the system be</p><p>ẋ1 = x2, ẋ2 = x3</p><p>2 + sin t + u, y = x1.</p><p>The observer and output feedback controller remain the same, except that the satura-</p><p>tion level M is chosen to satisfy M >maxx∈Ω |− x3</p><p>2 − x1− x2|+ 1= 1.816. The choice</p><p>M = 2 yields the controller</p><p>u = 2 sat</p><p>�</p><p>−σ̂ − x̂1− x̂2</p><p>2</p><p>�</p><p>.</p><p>0 2 4 6 8 10</p><p>−0.02</p><p>0</p><p>0.02</p><p>0.04</p><p>0.06</p><p>0.08</p><p>0.1</p><p>0.12</p><p>Time</p><p>x</p><p>1</p><p>Target</p><p>ε=0.1</p><p>ε=0.01</p><p>ε=0.001</p><p>0 2 4 6 8</p><p>−0.2</p><p>−0.1</p><p>0</p><p>0.1</p><p>Time</p><p>x</p><p>2</p><p>Figure 6.1. Simulation of</p><p>the output feedback controller of Example 6.1 when φ(t , x) = x3</p><p>2 .</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>162 CHAPTER 6. EXTENDED OBSERVER</p><p>0 2 4 6 8 10</p><p>−0.4</p><p>−0.2</p><p>0</p><p>0.2</p><p>0.4</p><p>Time</p><p>x</p><p>1</p><p>0 2 4 6 8 10</p><p>−0.4</p><p>−0.2</p><p>0</p><p>0.2</p><p>0.4</p><p>Time</p><p>x</p><p>2</p><p>Target</p><p>ε=0.1</p><p>ε=0.01</p><p>ε=0.001</p><p>Figure 6.2. Simulation of the output feedback controller of Example 6.1 when φ(t , x) =</p><p>x3</p><p>2 + sin t .</p><p>0 2 4 6 8 10</p><p>−0.02</p><p>0</p><p>0.02</p><p>0.04</p><p>0.06</p><p>0.08</p><p>0.1</p><p>Time</p><p>x</p><p>1</p><p>0 2 4 6 8 10</p><p>−0.06</p><p>−0.04</p><p>−0.02</p><p>0</p><p>0.02</p><p>Time</p><p>x</p><p>2</p><p>Figure 6.3. Simulation of the output feedback controller of Example 6.1 when φ(t , x) =</p><p>x3</p><p>2 (dashed) and φ(t , x) = x3</p><p>2 + sin t (dotted), with ε= 0.001. The solid line is the target response.</p><p>The simulation results of Figure 6.2 are carried out under the same initial conditions.</p><p>They confirm the same trend in Figure 6.1. Figure 6.3 shows that when ε= 0.001 the</p><p>response is almost the same whether the unknown function is x3</p><p>2 or x3</p><p>2 + sin t . 4</p><p>Example 6.2. Consider the system</p><p>η̇= θ1η</p><p>�</p><p>1+</p><p>η2</p><p>1+η2</p><p>�</p><p>+ ξ , ξ̇ = θ2η+ u, y = ξ ,</p><p>where θ1 ∈ [0.1,0.2] and θ2 ∈ [1,2] are unknown parameters. The goal is to design an</p><p>output feedback controller to stabilize the origin. This is a relative-degree-one system</p><p>in the normal form, and the zero dynamics η̇ = θ1η[1+ η</p><p>2/(1+ η2)] have unstable</p><p>origin; hence the system is nonminimum phase. A design that uses the extended high-</p><p>gain observer is based on a procedure introduced by Isidori that aims at stabilizing</p><p>the zero dynamics.54 Had η been measured, in addition to ξ , the system could have</p><p>been stabilized by a backstepping procedure that treats ξ as a virtual input to the η̇-</p><p>equation. After designing ξ as a feedback function of η to stabilize the zero dynamics,</p><p>it would be backstepped to the ξ̇ -equation to find the actual control u that stabilizes</p><p>the overall system.55 While η is not measured, the term θ2η on the right-hand side</p><p>of the ξ̇ -equation is proportional to η. This term can be estimated by an extended</p><p>high-gain observer. Treat σ = θ2η as an additional state variable, to obtain</p><p>ξ̇ = σ + u, σ̇ =ψ(η,ξ ), y = ξ ,</p><p>54See [67] or [66, Section 12.6].</p><p>55See [90] or [80] for an introduction to backstepping.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>6.1. MOTIVATING EXAMPLES 163</p><p>where ψ(η,ξ ) = θ2θ1η[1+η</p><p>2/(1+η2)]+θ2ξ . The extended high-gain observer</p><p>˙̂</p><p>ξ = σ̂ + u +(α1/ε)(y − ξ̂ ), ˙̂σ = (α2/ε</p><p>2)(y − ξ̂ )</p><p>with positive α1, α2, and ε will robustly estimate σ for sufficiently small ε. Based</p><p>on the properties of high-gain observers, it is expected that a feedback controller that</p><p>uses σ̂ can recover the performance of a controller that uses σ . Therefore, we proceed</p><p>to design a state feedback controller that assumes that both ξ and σ are available for</p><p>feedback. The design starts by considering the auxiliary problem</p><p>η̇= θ1η</p><p>�</p><p>1+</p><p>η2</p><p>1+η2</p><p>�</p><p>+ ua , ya = θ2η,</p><p>where ua = ξ is a virtual control and ya = σ is a virtual output. The feedback control</p><p>ua =−kya results in</p><p>η̇= θ1η</p><p>�</p><p>1+</p><p>η2</p><p>1+η2</p><p>�</p><p>− kθ2η.</p><p>The derivative of V0 =</p><p>1</p><p>2η</p><p>2 is</p><p>V̇0 = θ1η</p><p>2</p><p>�</p><p>1+</p><p>η2</p><p>1+η2</p><p>�</p><p>− kθ2η</p><p>2 ≤ 0.4η2− kη2.</p><p>Hence, the origin η = 0 is stabilized by taking k > 0.4. Let k = 0.5. Next, we use</p><p>backstepping to compute u. Let</p><p>s = ua + kya = ξ + kσ = ξ + kθ2η</p><p>and V = 1</p><p>2η</p><p>2+ 1</p><p>2 s2. The derivative of V is</p><p>V̇ = θ1η</p><p>2</p><p>�</p><p>1+</p><p>η2</p><p>1+η2</p><p>�</p><p>− kθ2η</p><p>2+ηs</p><p>+ s</p><p>�</p><p>θ2[1+ k(θ1− kθ2)]η+ kθ1θ2</p><p>η3</p><p>1+η2</p><p>�</p><p>+ kθ2 s2+ s u</p><p>≤−0.1η2+ 2.9|η| |s |+ s2+ s u.</p><p>Take u =−β sat(s/µ), whereβ and µ are positive constants, and let Ω= {V ≤ 1</p><p>2 c2}.</p><p>For (η,ξ ) ∈Ω×{|s | ≥µ}</p><p>V̇ ≤−0.1η2+ 2.9c |s |+ c |s | −β|s |.</p><p>Taking β≥ 0.1+ 3.9c yields V̇ ≤−0.1(η2+ |s |). On the other hand,</p><p>s ṡ = s</p><p>�</p><p>θ2[1+ k(θ1− kθ2)]η+ kθ1θ2</p><p>η3</p><p>1+η2</p><p>�</p><p>+ kθ2 s2+ s u</p><p>≤ 1.9|η| |s |+ s2−β|s | ≤ 2.9c |s | −β|s |=−(β− 2.9c)|s |.</p><p>The choice β ≥ 0.1+ 3.9c ensures that s ṡ ≤ −(c + 0.1)|s |, which shows that |s | de-</p><p>creases monotonically reaching the positively invariant set {|s | ≤ µ} in finite time.</p><p>For (η,ξ ) ∈Ω×{|s | ≤µ},</p><p>V̇ ≤−0.1η2+ 2.9|η| |s |+ s2− (β/µ)s2</p><p>=−</p><p>�</p><p>|η|</p><p>|ξ |</p><p>�T � 0.1 −1.45</p><p>−1.45 (β/µ− 1)</p><p>��</p><p>|η|</p><p>|ξ |</p><p>�</p><p>0 for all (η,ξ , w) ∈</p><p>Dη×Dξ ×W .</p><p>The goal is to design an output feedback controller to asymptotically regulate the</p><p>output e(t ) to zero while meeting certain requirements on the transient response. The</p><p>design is pursued for systems where η is bounded for bounded ξ and w, as will be</p><p>stated precisely in Assumption 6.3. Had (η,ξ , w) been available for feedback and the</p><p>functions a and b been known, we could have used the feedback linearization control</p><p>u =</p><p>−a(η,ξ , w)+ v</p><p>b (η,ξ , w)</p><p>to reduce the input-output model to</p><p>ξ̇ =Aξ +Bv, e =Cξ , (6.5)</p><p>where the triple (A,B ,C ) represents a chain of ρ integrators, that is,</p><p>A=</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 0 · · · 0</p><p>0 0 1 · · · 0</p><p>...</p><p>...</p><p>. . . . . .</p><p>...</p><p>0 0 · · · 0 1</p><p>0 0 · · · · · · 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, B =</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0</p><p>0</p><p>...</p><p>0</p><p>1</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, C =</p><p>�</p><p>1 0 · · · · · · 0</p><p>�</p><p>.</p><p>The auxiliary control v would then be designed as a feedback function of ξ to stabilize</p><p>the origin ξ = 0 and meet the transient response specifications. Let v =−Kξ , where</p><p>K is chosen using a linear control design method, such as pole placement or LQR, to</p><p>make (A−BK)Hurwitz and shape the transient response. Hence, the origin of</p><p>ξ̇ = (A−BK)ξ (6.6)</p><p>is exponentially stable,</p><p>and Vs (ξ ) = ξ</p><p>T Psξ is a Lyapunov function, where Ps = P T</p><p>s ></p><p>0 is the solution of the Lyapunov equation Ps (A−BK)+(A−BK)T Ps =−Qs for some</p><p>Qs =QT</p><p>s > 0. Let c be a positive constant such that {Vs (ξ )≤ c} ⊂Dξ .</p><p>While such design is not realizable, it will be shown that its performance can be re-</p><p>covered by using an extended high-gain observer. Toward that end, we call the system</p><p>(6.6) the target system and its response the target response.</p><p>Assumption 6.3. There is a continuously differentiable function V0(η), classK functions</p><p>%1 and %2, and a nonnegative continuous nondecreasing function χ such that</p><p>%1(‖η‖)≤V0(η)≤ %2(‖η‖),</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>166 CHAPTER 6. EXTENDED OBSERVER</p><p>∂ V0</p><p>∂ η</p><p>f0(η,ξ , w)≤ 0 ∀ ‖η‖ ≥ χ (‖ξ ‖+ ‖w‖)</p><p>for all (η,ξ , w) ∈Dη×Dξ ×W . Moreover, there is c0 ≥ %2(χ (κ1 (c)+ κ2)) such that the</p><p>set {V0(η)≤ c0} is compact and contained in Dη, where κ1 (c) =maxξ∈{Vs (ξ )≤c} ‖ξ ‖ and</p><p>κ2=maxw∈W ‖w‖.</p><p>Assumption 6.3 ensures that the compact set Ω = {V0(η) ≤ c0} × {Vs (ξ ) ≤ c} is</p><p>positively invariant with respect to the system</p><p>η̇= f0(η,ξ , w), ξ̇ = (A−BK)ξ ,</p><p>because on the boundary {V0 = c0}</p><p>%2(‖η‖)≥ c0 ≥ %2(χ (κ1 (c)+ κ2))⇒‖η‖ ≥ χ (κ1 (c)+ κ2)</p><p>⇒‖η‖ ≥ χ (‖ξ ‖+ ‖w‖) ⇒ V̇0 ≤ 0</p><p>and on the boundary {Vs (ξ ) = c}, V̇s 0 be twice continuously differentiable, globally bounded</p><p>functions that model a(η,ξ , w) and b (η,ξ , w), respectively. It is allowed to take â = 0</p><p>and b̂ > 0 as a constant. The ξ̇ρ-equation can be written as</p><p>ξ̇ρ = σ + â(ξ )+ b̂ (ξ )u,</p><p>where</p><p>σ = a(η,ξ , w)− â(ξ )+ [b (η,ξ , w)− b̂ (ξ )]u.</p><p>Augmenting σ as an additional state to the chain of integrators (6.2)–(6.4), a high-gain</p><p>observer for the extended system is taken as</p><p>˙̂</p><p>ξi = ξ̂i+1+(αi/ε</p><p>i )(e − ξ̂1) for 1≤ i ≤ ρ− 1, (6.7)</p><p>˙̂</p><p>ξρ = σ̂ + â(ξ̂ )+ b̂ (ξ̂ )u +(αρ/ε</p><p>ρ)(e − ξ̂1), (6.8)</p><p>˙̂σ = (αρ+1/ε</p><p>ρ+1)(e − ξ̂1), (6.9)</p><p>where α1 to αρ+1 are chosen such that the polynomial</p><p>sρ+1+α1 sρ+ · · ·+αρ+1</p><p>is Hurwitz and ε > 0 is a small parameter. Global boundedness of â and b̂ is required</p><p>to overcome the peaking phenomenon of high-gain observers. It does not exclude</p><p>linear or unbounded functions because global boundedness can be always achieved by</p><p>saturation outside a compact set.</p><p>From the results of Chapter 2, it is anticipated that the terms (αi/ε</p><p>i )(e − ξ̂1) for</p><p>1 ≤ i ≤ ρ will be O(ε) after a short transient period. Then equations (6.7)–(6.8) can</p><p>be viewed as a perturbation of the system</p><p>˙̂</p><p>ξ =Aξ̂ +B</p><p>h</p><p>σ̂ + â(ξ̂ )+ b̂ (ξ̂ )u</p><p>i</p><p>, (6.10)</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>6.2. FEEDBACK CONTROL VIA DISTURBANCE COMPENSATION 167</p><p>which can be made to coincide with the target system (6.6) by taking</p><p>u =</p><p>−σ̂ − â(ξ̂ )−K ξ̂</p><p>b̂ (ξ̂ )</p><p>def= ψ(ξ̂ , σ̂). (6.11)</p><p>To protect the system from peaking in the observer’s transient response, the control</p><p>is saturated outside the compact set Ω. Let</p><p>M > max</p><p>(η,ξ )∈Ω, w∈W</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>−a(η,ξ , w)−Kξ</p><p>b (η,ξ , w)</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>. (6.12)</p><p>Saturating the foregoing expression of u at ±M using the saturation function sat(·)</p><p>yields the output feedback controller</p><p>u =M sat</p><p>ψ(ξ̂ , σ̂)</p><p>M</p><p>!</p><p>, (6.13)</p><p>where ψ is defined by (6.11). Saturating the control at ±M does not restrict the func-</p><p>tions a and b because we can find M for any a, b , Ω, andW . However, the calculation</p><p>of M might not be straightforward, and one might end up with a conservative bound.</p><p>Let</p><p>ku = max</p><p>(η,ξ )∈Ω,w∈W</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>b (η,ξ , w)− b̂ (ξ )</p><p>b̂ (ξ )</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>,</p><p>Γ (s) =</p><p>αρ+1</p><p>sρ+1+α1 sρ+ · · ·+αρ+1</p><p>,</p><p>and ‖Γ‖∞ = maxω |Γ ( jω)|. Because Γ (0) = 1, ‖Γ‖∞ ≥ 1. It can be verified that</p><p>‖Γ‖∞ = 1 if all the poles of Γ (s) are real.</p><p>Theorem 6.1. Consider the closed-loop system formed of the plant (6.1)–(6.4), the observer</p><p>(6.7)–(6.9), and the controller (6.13). Suppose Assumptions 6.1 to 6.3 are satisfied,</p><p>ku 0 such that for every 0 0, there exist ε∗2 > 0, dependent on µ, such that for every 0 0, there exist ε∗3 > 0 and T1 > 0, both dependent on µ, such that for</p><p>every 0 bm ensures that ku</p><p>asymptotically approach its trajecto-</p><p>ries under state feedback as ε tends to zero. This leads to recovery of the performance</p><p>achieved under state feedback. The global boundedness of γ (x) can be always achieved</p><p>by saturating the state feedback control, or the state estimates, outside a compact set</p><p>of interest. 4</p><p>The foregoing example shows that the design of the output feedback controller is</p><p>based on a separation procedure, whereby the state feedback controller is designed as if</p><p>the whole state was available for feedback, followed by an observer design that is inde-</p><p>pendent of the state feedback design. By choosing ε small enough, the output feedback</p><p>controller recovers the stability and performance properties of the state feedback con-</p><p>troller. This is the essence of the separation principle that is presented in Section 3.1.</p><p>The separation principle is known in the context of linear systems where the closed-</p><p>loop eigenvalues under an observer-based controller are the union of the eigenvalues</p><p>under state feedback and the observer eigenvalues; hence stabilization under output</p><p>feedback can be achieved by solving separate eigenvalue placement problems for the</p><p>state feedback and the observer. Over the last two decades there have been several re-</p><p>sults that present forms of the separation principle for classes of nonlinear systems.</p><p>It is important to emphasize that the separation principle in the case of high-gain ob-</p><p>servers has a unique feature that does not exist in other separation-principle results,</p><p>linear systems included, and that is the recovery of state trajectories by making the</p><p>observer sufficiently fast. This feature has significant practical implications because it</p><p>allows the designer to design the state feedback controller to meet transient response</p><p>specification and/or constraints on the state or control variables. Then, by saturating</p><p>the state estimate x̂ and/or the control u outside compact sets of interest to make γ (x̂)</p><p>and φ0(x̂, u) globally bounded in x̂, the designer can proceed to tune the parameter ε</p><p>by decreasing it monotonically to bring the trajectories under output feedback close</p><p>enough to the ones under state feedback. This feature is achieved not only by making</p><p>the observer fast but also by combining this property with the global boundedness of</p><p>γ and φ0. We illustrate this point in the example to follow.</p><p>Example 1.3. Consider the linear system</p><p>ẋ1 = x2, ẋ2 = u, y = x1,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>1.2. MOTIVATING EXAMPLES 9</p><p>0 1 2 3 4 5 6 7 8 9 10</p><p>−0.4</p><p>−0.2</p><p>0</p><p>0.2</p><p>0.4</p><p>0.6</p><p>0.8</p><p>1</p><p>Time</p><p>x 1</p><p>State FB</p><p>Output FB ε = 0.1</p><p>Output FB ε = 0.01</p><p>0 1 2 3 4 5 6 7 8 9 10</p><p>−2</p><p>−1.5</p><p>−1</p><p>−0.5</p><p>0</p><p>0.5</p><p>x 2</p><p>Time</p><p>Figure 1.7. Simulation of Example 1.3. The state trajectories under state and output</p><p>feedback for linear example with unsaturated control. Reprinted with permission from John Wiley</p><p>and Sons, Ltd [84].</p><p>which is a special case of the system of Example 1.1 withφ= u. A linear state feedback</p><p>controller that assigns the eigenvalues at−1± j is given by u =−2x1−2x2. A high-gain</p><p>observer is taken as</p><p>˙̂x1 = x̂2+(3/ε)(y − x̂1),</p><p>˙̂x2 = u +(2/ε2)(y − x̂1).</p><p>It assigns the observer eigenvalues at −1/ε and −2/ε. The observer-based controller</p><p>assigns the closed-loop eigenvalues at−1± j ,−1/ε, and−2/ε. The closed-loop system</p><p>under output feedback is asymptotically stable for all ε > 0. As we decrease ε, we make</p><p>the observer dynamics faster than the state dynamics. Will the trajectories of the sys-</p><p>tem under output feedback approach those under state feedback as ε approaches zero?</p><p>The answer is shown in Figure 1.7, where the state x is shown under state feedback and</p><p>under output feedback for ε = 0.1 and 0.01. The initial conditions of the simulation</p><p>are x1(0) = 1 and x2(0) = x̂1(0) = x̂2(0) = 0. Contrary to what intuition may suggest,</p><p>we see that the trajectories under output feedback do not approach the ones under state</p><p>feedback as ε decreases. In Figure 1.8 we repeat the same simulation when the control</p><p>is saturated at ±4, that is, u = 4 sat((−2x̂1− 2x̂2)/4). The saturation level 4 is chosen</p><p>such that 4 > maxΩ | − 2x1 − 2x2|, where Ω = {1.25x2</p><p>1 + 0.5x1x2 + 0.375x2</p><p>2 ≤ 1.4} is</p><p>an estimate of the region of attraction under state feedback control that includes the</p><p>initial state (1,0) in its interior. This choice of the saturation level saturates the con-</p><p>trol outside Ω. Figure 1.8 shows a reversal of the trend we saw in Figure 1.7. Now the</p><p>trajectories under output feedback approach those under state feedback as ε decreases.</p><p>This is a manifestation of the performance recovery property of high-gain observers</p><p>when equipped with a globally bounded control. Figure 1.9 shows the control signal</p><p>u with and without saturation during the peaking period for ε= 0.01. It demonstrates</p><p>the role of saturation in suppressing the peaking phenomenon. 4</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>10 CHAPTER 1. INTRODUCTION</p><p>0 1 2 3 4 5 6 7 8 9 10</p><p>−0.2</p><p>0</p><p>0.2</p><p>0.4</p><p>0.6</p><p>0.8</p><p>1</p><p>1.2</p><p>Time</p><p>x 1</p><p>State FB</p><p>Output FB ε = 0.1</p><p>Output FB ε = 0.01</p><p>0 1 2 3 4 5 6 7 8 9 10</p><p>−1.2</p><p>−1</p><p>−0.8</p><p>−0.6</p><p>−0.4</p><p>−0.2</p><p>0</p><p>0.2</p><p>x 2</p><p>Time</p><p>Figure 1.8. Simulation of Example 1.3. The state trajectories under state and output</p><p>feedback for linear example with saturated control. Reprinted with permission from John Wiley and</p><p>Sons, Ltd [84].</p><p>0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2</p><p>−100</p><p>−80</p><p>−60</p><p>−40</p><p>−20</p><p>0</p><p>u</p><p>Time</p><p>Without saturation</p><p>0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2</p><p>−5</p><p>−4</p><p>−3</p><p>−2</p><p>−1</p><p>0</p><p>1</p><p>u</p><p>Time</p><p>With saturation</p><p>Figure 1.9. Simulation of Example 1.3. The control signal for the linear example with</p><p>and without control saturation when ε = 0.01. Reprinted with permission from John Wiley and</p><p>Sons, Ltd [84].</p><p>The full-order observer of Example 1.1 estimates both states x1 and x2. Since y = x1</p><p>is measured, we can design a reduced-order observer of the form</p><p>ż =− h (z + hy)+φo(x̂, u), x̂2 = z + hy,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>1.2. MOTIVATING EXAMPLES 11</p><p>to estimate x̂2 only. When the observer gain h is chosen as h = α/ε for some positive</p><p>constants α and εwith ε� 1, it becomes a high-gain observer that has features similar</p><p>to the full-order high-gain observer. In particular, the estimation error decays rapidly</p><p>while exhibiting the peaking phenomenon. The example to follow compares the full-</p><p>and reduced-order observers.</p><p>Example 1.4. Consider the van der Pol oscillator</p><p>ẋ1 = x2, ẋ2 =−x1+(1− x2</p><p>1 )x2,</p><p>and suppose we want to estimate x2 from a measurement of x1. We design a full-</p><p>order high-gain observer of dimension two and a reduced-order high-gain observer of</p><p>dimension one. In both cases we do not include a model of the function x1+(1−x1)</p><p>2x2</p><p>so that the observers would be linear. The full-order observer is taken as</p><p>˙̂x1 = x̂2+(2/ε)(y − x̂1),</p><p>˙̂x2 = (1/ε</p><p>2)(y − x̂1),</p><p>and its transfer function from y to x̂2 is</p><p>G2(s) =</p><p>s</p><p>(εs + 1)2</p><p>.</p><p>The reduced-order observer is taken as</p><p>ż1 =−(1/ε)[z1+(1/ε)y], x̂2 = z1+(1/ε)y,</p><p>and its transfer function from y to x̂2 is</p><p>G1(s) =</p><p>s</p><p>εs + 1</p><p>.</p><p>Comparison of the two transfer functions shows that the magnitude of the frequency</p><p>response of G2(s) rolls off beyond the cutoff frequency of 1/ε, while that of G1(s)</p><p>remains constant beyond the cutoff frequency. If the measurement of x1 is corrupted</p><p>by noise, that is, y = x1 + v, the full-order observer will have better attenuation of</p><p>high-frequency noise. For example, if v(t ) = N sinωt and ω = k/ε with k � 1,</p><p>ψ(ξ , σ̂)</p><p>M</p><p>�</p><p>+∆b g ′ε</p><p>�</p><p>ψ(ξ , σ̂)</p><p>M</p><p>�</p><p>∂ ψ(ξ , σ̂)</p><p>∂ ξ</p><p>ξ̇</p><p>�</p><p>,</p><p>where</p><p>∆0(η,ξ , ξ̂ , σ̂ , w,ε) = â(ξ )− â(ξ̂ )</p><p>+ b (η,ξ , w)M</p><p></p><p>gε</p><p>ψ(ξ̂ , σ̂)</p><p>M</p><p>!</p><p>− gε</p><p>�</p><p>ψ(ξ , σ̂)</p><p>M</p><p>�</p><p></p><p></p><p>+ b̂ (ξ )M gε</p><p>�</p><p>ψ(ξ , σ̂)</p><p>M</p><p>�</p><p>− b̂ (ξ̂ )M gε</p><p>ψ(ξ̂ , σ̂)</p><p>M</p><p>!</p><p>+</p><p>h</p><p>b (η,ξ , w)− b̂ (ξ̂ )</p><p>i</p><p>M</p><p></p><p>sat</p><p>ψ(ξ̂ , σ̂)</p><p>M</p><p>!</p><p>− gε</p><p>ψ(ξ̂ , σ̂)</p><p>M</p><p>!</p><p></p><p> .</p><p>Using (6.17), it is not hard to see that the bracketed term on the right-hand side of</p><p>εϕ̇ρ+1 is a continuous function of (η,ξ ,ϕ, w, ẇ,ε). Because â, b̂ , and gε are contin-</p><p>uously differentiable with locally Lipschitz derivatives and globally bounded, (6.17)</p><p>and the definition of gε can be used to show that∆0/ε is a locally Lipschitz function.</p><p>For example,</p><p>1</p><p>ε</p><p>h</p><p>â(ξ )− â(ξ̂ )</p><p>i</p><p>=</p><p>1</p><p>ε</p><p>∫ 1</p><p>0</p><p>∂ â</p><p>∂ ξ</p><p>(ξ̂ +λ(ξ − ξ̂ )) dλ (ξ − ξ̂ )</p><p>=</p><p>∫ 1</p><p>0</p><p>∂ â</p><p>∂ ξ</p><p>(ξ̂ +λ(ξ − ξ̂ )) dλ</p><p>�</p><p>ερ−1ϕ1, . . . , εϕρ−1, ϕρ</p><p>�T</p><p>.</p><p>Using the foregoing expressions it can be shown that, as long as (η,ξ ) ∈ Ω, the fast</p><p>subsystem is given by</p><p>εϕ̇ =Λϕ− B̄1∆1(·)αρ+1ϕ1+ ε[B̄1 ∆2(·)+ B̄2∆3(·)], (6.19)</p><p>where</p><p>Λ=</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>−α1 1 0 · · · 0</p><p>−α2 0 1</p><p>... 0</p><p>...</p><p>...</p><p>. . . . . .</p><p>...</p><p>−αρ 0 · · · 0 1</p><p>−αρ+1 0 · · · · · · 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>, B̄1 =</p><p>�</p><p>0</p><p>B</p><p>�</p><p>, B̄2 =</p><p>�</p><p>B</p><p>0</p><p>�</p><p>,</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>170 CHAPTER 6. EXTENDED OBSERVER</p><p>∆1(η,ξ ,ϕ, w,ε) =</p><p>∆b (η,ξ , w)</p><p>b̂ (ξ )</p><p>g ′ε</p><p>�</p><p>ψ(ξ , σ̂)</p><p>M</p><p>�</p><p>,</p><p>∆2(η,ξ ,ϕ, w, ẇ,ε) = ∆̇a + ∆̇b M gε</p><p>�</p><p>ψ(ξ , σ̂)</p><p>M</p><p>�</p><p>+∆b g ′ε</p><p>�</p><p>ψ(ξ , σ̂)</p><p>M</p><p>�</p><p>∂ ψ(ξ , σ̂)</p><p>∂ ξ</p><p>ξ̇ ,</p><p>∆3(η,ξ ,ϕ, w,ε) =</p><p>∆0(η,ξ , ξ̂ , σ̂ , w,ε)</p><p>ε</p><p>.</p><p>The functions ∆1, ∆2, and ∆3 are locally Lipschitz in their arguments and bounded</p><p>from above by ka + kb‖ϕ‖, where ka and kb are positive constants independent of ε.</p><p>The matrix Λ is Hurwitz by design. If it were not for the term B̄1∆1(·)αρ+1ϕ1 on the</p><p>right-hand side of (6.19), it would have been the usual fast dynamics equation as in</p><p>the previous chapters, and it could have been easily concluded that ϕ would be O(ε)</p><p>after a short transient period. The new term is dealt with by application of the circle</p><p>criterion [80, Section 7.3]. Consider (6.19) without the O(ε) terms on the right-hand</p><p>side, that is,</p><p>εϕ̇ =Λϕ− B̄1∆1(·)αρ+1ϕ1. (6.20)</p><p>Equation (6.20) can be represented as a negative feedback connection of the transfer</p><p>function</p><p>Γ (εs) =</p><p>αρ+1</p><p>(εs)ρ+1+α1(εs)ρ+ · · ·+αρ+1</p><p>and the time-varying gain ∆1(·), where |∆1(·)| ≤ ku . The transfer function Γ (εs) has</p><p>maxω |Γ ( jεω)|=maxω |Γ ( jω)|= ‖Γ‖∞. The circle criterion shows that the origin of</p><p>(6.20) will be globally exponentially stable if ku‖Γ‖∞ 0 can be chosen such that the set</p><p>Ω×{W (ϕ)≤ ε2c1} is positively invariant, then it is shown thatϕ enters {W (ϕ)≤ ε2c1}</p><p>in a finite time T (ε), where limε→0 T (ε) = 0.</p><p>While (η,ξ ,ϕ) ∈ Ω×{W (ϕ) ≤ ε2c1}, we can use ϕρ+1 =O(ε) and ξ − ξ̂ =O(ε)</p><p>to show that</p><p>ψ(ξ̂ , σ̂) =ψ(ξ , σ̂)+O(ε),</p><p>σ̂ =∆a(η,ξ , w)+∆b (η,ξ , w)M gε</p><p>�</p><p>ψ(ξ , σ̂)</p><p>M</p><p>�</p><p>+O(ε).</p><p>Thus, up to an O(ε) error, ψ(ξ , σ̂) satisfies the equation</p><p>ψ+</p><p>∆b (η,ξ , w)</p><p>b̂ (ξ )</p><p>M gε</p><p>�</p><p>ψ</p><p>M</p><p>�</p><p>=</p><p>−a(η,ξ , w)−Kξ</p><p>b̂ (ξ )</p><p>.</p><p>This equation has a unique solution because |∆b/b̂ | ≤ ku 0 there is a finite time T1 such that</p><p>‖ξr (t )‖ ≤ µ/2 for all t ≥ T1. On the other hand, (6.26) shows that for sufficiently</p><p>small ε, ‖ξ (t )− ξr (t )‖ ≤µ/2 for all t ≥ T1. Hence,</p><p>‖ξ (t )‖= ‖ξ (t )− ξr (t )+ ξr (t )‖ ≤ ‖ξr (t )‖+ ‖ξ (t )− ξr (t )‖ ≤µ ∀ t ≥ T1,</p><p>which proves (6.16). 2</p><p>Remark 6.4. The control is saturated by using the saturation function sat(·), which is</p><p>locally Lipschitz but not continuously differentiable. Because the proof uses a change of</p><p>variables that requires differentiation of the control variable, in (6.18) the function sat(·)</p><p>is replaced by the continuously differentiable function gε(·), which is O(ε) close to sat(·).</p><p>The proof could have been simplified by saturating the control using a smooth saturation</p><p>function. However, the saturation function sat(·) is easier to implement. 3</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>172 CHAPTER 6. EXTENDED OBSERVER</p><p>Theorem 6.1 ensures only practical regulation. The next theorem shows that if</p><p>w is constant, then, under additional conditions on the zero dynamics, the controller</p><p>provides integral action that ensures regulation of e(t ) (in fact the whole vector ξ (t ))</p><p>to zero even when the model uncertainty does not vanish at ξ = 0. The following</p><p>assumption states the additional requirements on η̇= f0(η,ξ , w)when w is constant.</p><p>Assumption 6.4.</p><p>• f0(η, 0, w) is continuously differentiable in η.</p><p>• For each w ∈ W , the system η̇ = f0(η, 0, w) has a unique equilibrium point ηs s =</p><p>ηs s (w) ∈ {V0(η)≤ c0}.</p><p>• With ζ = η− ηs s , there is a continuously differentiable Lyapunov function Va(ζ ),</p><p>possibly dependent on w, and class K functions γ1 to γ4, independent of w, such</p><p>that</p><p>γ1(‖ζ ‖)≤Va(ζ )≤ γ2(‖ζ ‖),</p><p>∂ Va</p><p>∂ ζ</p><p>f0(ζ +ηs s ,ξ , w)≤−γ3(‖ζ ‖) ∀ ‖ζ ‖ ≥ γ4(‖ξ ‖)</p><p>for all (η,ξ , w) ∈Ω×W .</p><p>• The origin of ζ̇ = f0(ζ +ηs s ,ξ , w) is exponentially stable.</p><p>The third bullet requires the system ζ̇ = f0(ζ + ηs s ,ξ , w) to be regionally input-</p><p>to-state stable with respect to the</p><p>the</p><p>magnitude of the steady-state component of x̂2 due to noise will be (kN/ε)/(1+k2)≈</p><p>N/(εk) for the full-order observer and (kN/ε)/</p><p>p</p><p>(1+ k2) ≈ N/ε for the reduced-</p><p>order observer. The high-frequency noise is attenuated much better by the full-order</p><p>observer. The same observation can be seen from simulation. Figure 1.10 shows the es-</p><p>timation error x̃2 = x2− x̂2 for the two observers. The simulation is run with ε= 0.001</p><p>and initial conditions x1(0) = x2(0) = 1, x̂1(0) = x̂2(0) = z1(0) = 0. The measurement</p><p>noise is a uniformly distributed random signal with values between ±0.0001, gener-</p><p>ated by the Simulink icon “Uniform Random Number” with sample time 0.00001.</p><p>The estimation error is plotted over the time period [0,0.01] to show the peaking be-</p><p>havior of x̃2 and over the period [9.9,10] to show the steady-state behavior where the</p><p>effect of noise is significant. In both cases x̃2 peaks to O(1/ε) values, but peaking is</p><p>larger in the reduced-order observer due to the fact that x̂2 has a component propor-</p><p>tional to y/ε. What is significant in this simulation is the effect of measurement noise.</p><p>Comparison of Figures 1.1(b) and (d) shows that the full-order observer attenuates the</p><p>effect of measurement noise better by an order of magnitude. Moreover, it filters out</p><p>the high-frequency content of the noise. 4</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>12 CHAPTER 1. INTRODUCTION</p><p>0 0.002 0.004 0.006 0.008 0.01</p><p>−400</p><p>−300</p><p>−200</p><p>−100</p><p>0</p><p>Time</p><p>x̃</p><p>2</p><p>(a)</p><p>9.9 9.92 9.94 9.96 9.98 10</p><p>−0.01</p><p>−0.005</p><p>0</p><p>0.005</p><p>0.01</p><p>0.015</p><p>0.02</p><p>Time</p><p>x̃</p><p>2</p><p>(b)</p><p>0 0.002 0.004 0.006 0.008 0.01</p><p>−1000</p><p>−800</p><p>−600</p><p>−400</p><p>−200</p><p>0</p><p>Time</p><p>x̃</p><p>2</p><p>(c)</p><p>9.9 9.92 9.94 9.96 9.98 10</p><p>−0.2</p><p>−0.1</p><p>0</p><p>0.1</p><p>0.2</p><p>Time</p><p>x̃</p><p>2</p><p>(d)</p><p>Figure 1.10. Simulation of Example 1.4. Figures (a) and (b) are for the full-order observer.</p><p>Figures (c) and (d) are for the reduced-order observer. All figures show the estimation error x̃2 =</p><p>x2− x̂2.</p><p>1.3 Challenges</p><p>1.3.1 High-Dimensional Observers</p><p>A high-gain observer for the system</p><p>ẋi = xi+1 for 1≤ i ≤ ρ− 1,</p><p>ẋρ =φ(x, u),</p><p>y = x1</p><p>is given by</p><p>˙̂xi = x̂i+1+</p><p>αi</p><p>εi</p><p>(y − x̂1) for 1≤ i ≤ ρ− 1,</p><p>˙̂xρ =φ0(x̂, u)+</p><p>αρ</p><p>ερ</p><p>(y − x̂1),</p><p>where φ0 is a nominal model of φ, ε is a sufficiently small positive constant, and α1</p><p>to αρ are chosen such that the polynomial</p><p>sρ+α1 sρ−1+ · · ·+αρ−1 s +αρ</p><p>is Hurwitz; that is, its roots have negative real parts. This observer faces a numerical</p><p>challenge if its dimension, ρ, is high. The observer gains are proportional to pow-</p><p>ers of 1/ε, with 1/ερ as the highest power. During the transient period, the internal</p><p>states of the observer could peak to large values, which are proportional to powers of</p><p>1/ε, with 1/ερ−1 as the highest power. These features pose a challenge in the numeri-</p><p>cal implementation of the observer when ρ is high because in digital implementation</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>1.3. CHALLENGES 13</p><p>both parameters and signals have to be represented within the finite word length of</p><p>the digital system. It is worthwhile to note that saturating the state estimates or the</p><p>control signal before applying the control to the plant does not prevent peaking in the</p><p>observer’s internal variables. To address this numerical challenge, modified high-gain</p><p>observers are presented in Chapter 7.</p><p>1.3.2 Measurement Noise</p><p>The most serious challenge to the application of high-gain observers is measurement</p><p>noise. This is not surprising because the observer estimates the derivatives of the out-</p><p>put. When the output is corrupted by measurement noise, it is expected that noise will</p><p>have a serious effect on the accuracy of the estimates. The following example explores</p><p>the effect of measurement noise on a second-order observer.</p><p>Example 1.5. Reconsider the system of Example 1.1 and suppose the measurement y</p><p>is corrupted by bounded noise v; that is,</p><p>ẋ1 = x2, ẋ2 =φ(x, u), y = x1+ v,</p><p>where |v(t )| ≤N . Equation (1.2) and the inequality satisfied by V̇ change to</p><p>εη̇= F η+ εBδ − 1</p><p>ε</p><p>Ev, where E =</p><p>�</p><p>α1</p><p>α2</p><p>�</p><p>,</p><p>and</p><p>εV̇ ≤− 1</p><p>2‖η‖</p><p>2+ 2εM‖PB‖ ‖η‖+ 2N</p><p>ε</p><p>‖P E‖ ‖η‖.</p><p>Therefore, the ultimate bound on ‖x̃‖ takes the form</p><p>‖x̃‖ ≤ c1Mε+</p><p>c2N</p><p>ε</p><p>(1.3)</p><p>for some positive constants c1 and c2. This inequality shows a trade-off between model</p><p>uncertainty and measurement noise. An illustration of the ultimate bound in</p><p>Figure 1.11 shows that decreasing ε reduces the ultimate bound until we reach the</p><p>value ε1 =</p><p>p</p><p>c2N/(c1M ). Reducing ε beyond this point increases the ultimate bound.</p><p>εε</p><p>1</p><p>ε c</p><p>1</p><p>M + c</p><p>2</p><p>N/ε</p><p>Figure 1.11. Illustration of the ultimate bound (1.3).</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>14 CHAPTER 1. INTRODUCTION</p><p>Thus, the presence of measurement noise puts a lower bound on ε. Another trade-off</p><p>exists between the speed of convergence of the observer and the ultimate bound on</p><p>the estimation error. For the separation properties we saw in Example 1.2, we need to</p><p>choose ε sufficiently small to make the observer dynamics sufficiently faster than the</p><p>dynamics of the closed-loop system under state feedback. A lower bound on ε limits</p><p>the speed of convergence. For the high-gain observer to be effective, the ratio N/M</p><p>should be relatively small so that ε can be chosen to attenuate the effect of uncertainty</p><p>and make the observer sufficiently fast. Even if there was no model uncertainty, that</p><p>is, M = 0, we still need N to be relatively small so that we can design the observer</p><p>to be sufficiently fast without bringing the ultimate bound on the estimation error to</p><p>unacceptable levels. It is worthwhile to note that the bound (1.3) does not take into</p><p>consideration the low-pass filtering characteristics of the observer at frequencies higher</p><p>than 1/ε, as we saw in Example 1.4. 4</p><p>Ideas to reduce the effect of measurement noise are presented in Chapter 8.</p><p>1.4 Overview of the Book</p><p>Chapter 2 starts by describing the class of nonlinear systems for which high-gain ob-</p><p>servers are designed. The observer design is then presented in terms of a small positive</p><p>parameter ε, which can be thought of as the observer’s time constant or the recip-</p><p>rocal of the observer’s bandwidth when the observer is linear. The observer gains</p><p>are proportional to negative powers of ε. Theorem 2.1 gives an upper bound on the</p><p>estimation error, which illustrates three key features of the observer: the peaking phe-</p><p>nomenon, the fast decay of the error, and the small ultimate bound on the error. The</p><p>main step in the observer design is a pole placement problem where the eigenvalues</p><p>of a matrix are assigned in the left-half plane. This step can be carried out by solving</p><p>Lyapunov or Riccati equations, as shown in Section 2.3. An interesting observation</p><p>about the Lyapunov-equation design is that it assigns all the eigenvalues of the observer</p><p>at −1/ε. For a system with a chain of ρ integrators, the dimension of the observer is</p><p>ρ. Since the first state of the chain is the output, it is possible to design an observer of</p><p>dimension ρ−1. This reduced-order observer is described in Section 2.4. The chapter</p><p>concludes by presenting the observer design for a special class of multi-output systems,</p><p>which covers important physical problems where the measured variables are positions</p><p>and the estimated variables are velocities, accelerations, etc. References are given for</p><p>more general classes of multi-output systems.</p><p>Chapter 3 deals with stabilization and tracking problems. Theorem 3.1 is the sep-</p><p>aration principle for the stabilization of time-invariant systems. Its proof is a must-</p><p>read for anyone who wants to understand how high-gain observers work in feedback</p><p>control. Elements of that proof are repeated in several proofs throughout the book.</p><p>While the separation principle is presented only for time-invariant systems, extensions</p><p>to time-varying systems are referenced. For minimum-phase systems, robust stabiliza-</p><p>tion and tracking problems are presented in Sections 3.2 and 3.3, respectively. In both</p><p>cases the design of output feedback control follows a separation approach where state</p><p>feedback control is design first; then its performance is recovered by a high-gain ob-</p><p>server. A key idea of Section 3.2 is the use of a high-gain observer to reduce a relative</p><p>degree-higher-than-one system to a relative-degree-one system, for which the control</p><p>design is straightforward.</p><p>Chapter 4 considers adaptive control of nonlinear systems modeled globally by an</p><p>nth-order differential equation. The dynamics of the system are extended by adding</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>1.4. OVERVIEW OF THE BOOK 15</p><p>integrators at the input side, which results in a state model in which the state variables</p><p>are either states of the added integrators or derivatives of the output. By using a high-</p><p>gain observer to estimate the derivatives of the output, the output feedback controller</p><p>can recover the performance of the state feedback controller. A Lyapunov-based state</p><p>feedback adaptive control is designed in Section 4.2, and its output feedback implemen-</p><p>tation is given in Section 4.3. Convergence of the tracking error to zero is shown with-</p><p>out additional conditions, but convergence of the parameter errors to zero is shown</p><p>under a persistence of excitation condition. After illustrating the performance of the</p><p>adaptive controller and the persistence of excitation condition by examples in Sec-</p><p>tion 4.4, robustness to bounded disturbance is shown in Section 4.5. It is shown that</p><p>the ultimate bound on the tracking error can be made arbitrarily small by adding a ro-</p><p>bust control component. The robustness properties of the adaptive controller allows</p><p>its application to nonlinear systems where a nonlinear function is approximated by a</p><p>function approximator, such as neural network, with a small bounded residual error;</p><p>this is the subject of Section 4.6.</p><p>The nonlinear regulation problem is the subject of Chapter 5. The controller in-</p><p>cludes a servocompensator that implements an internal model of the exogenous (refer-</p><p>ence and disturbance) signals. The exogenous signals are generated by a linear exosys-</p><p>tem that has simple eigenvalues on the imaginary axis. The problem is treated first for</p><p>the special case of constant exogenous signals in Sections 5.1 and 5.2; then the more</p><p>general cases is treated in the rest of the chapter. In addition to the usual tool of using</p><p>a high-gain observer to recover the performance of state feedback control, the chapter</p><p>has a number of results that are unique to the regulation problem. First, it shows that</p><p>for nonlinear systems the internal model is not simply a model of the exosystem; it</p><p>has to include harmonics of the sinusoidal signals, which are induced by nonlineari-</p><p>ties. Second, it deals with impact the servocompensator has on the transient response</p><p>of the system by introducing the conditional servocompensator, which ensures that</p><p>the transient response of the system under output feedback can be made arbitrarily</p><p>close to the transient response of a state feedback sliding mode controller that has no</p><p>servocompensator. Third, the chapter deals with the case when the internal model is</p><p>uncertain or unknown. The effect of uncertainty is studied in Section 5.5, and adap-</p><p>tation is used in Section 5.6 to estimate the unknown model.</p><p>Chapter 6 presents the extended high-gain observer. It shows two uses of the ex-</p><p>tended observer, one as a disturbance estimator and the other as a soft sensor of the</p><p>internal dynamics. As a disturbance estimator, it is shown in Section 6.1 how to im-</p><p>plement feedback linearization by estimating the uncertain terms and compensating</p><p>for them. This is shown for single-input–single-output systems, then for multi-input-</p><p>multi-output systems. It is also shown how to combine the extended high-gain ob-</p><p>server with dynamic inversion to deal with nonaffine control or uncertain control</p><p>coefficient. Most of the high-gain observer results in Chapters 3 to 5 apply only to</p><p>minimum-phase systems. This is mainly because the high-gain observer does not esti-</p><p>mate the states of the internal dynamics. In Section 6.3 it is shown that the extended</p><p>high-gain observer can provide information about the internal dynamics by sensing</p><p>a term in the external dynamics that can be viewed as an output for the internal dy-</p><p>namics. This is used for stabilization of a nonminimum phase system or for designing</p><p>an observer that estimates the full state of the system. In the later case, two observers</p><p>are used: the extended high-gain observer is used for the external dynamics, and an</p><p>extended Kalman filter is used for the internal dynamics.</p><p>Chapters 7 and 8 address the challenges described in Section 1.3. The challenge</p><p>with a ρ-dimensional observer when ρ is high is that the observer gain is proportional</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>16 CHAPTER 1. INTRODUCTION</p><p>to 1/ερ and the peak of its internal states during the transient period is the order of</p><p>1/ερ−1 . Chapter 7 addresses this challenge by introducing observers where the gain</p><p>and the peaking signal are limited to the order of 1/ε. Such observers are constructed</p><p>as cascade connections of first- or second-order observers where the gains are of the or-</p><p>der of 1/ε. Peaking is limited by inserting saturation functions in between the cascaded</p><p>observers. Two such observers are designed. The first observer is a simple cascade con-</p><p>nection of one second-order observer and ρ−2 first-order observers. This observer is</p><p>robust to model uncertainty, and its performance in feedback control is comparable</p><p>to the standard high-gain observer, but the steady-state estimation errors are orders</p><p>of magnitude higher than the standard observer’s errors. Moreover, the estimation</p><p>error does not converge to zero under conditions where the standard observer’s error</p><p>converges to zero. The second observer is a cascade connection of ρ− 1 second-order</p><p>observers with feedback injection. This observer has the same steady-state and con-</p><p>vergence properties of the standard observer.</p><p>Chapter 8 starts by characterizing the effect of measurement noise on the estima-</p><p>tion error. The general result for bounded noise shows that the ultimate bound on</p><p>the estimation error is of the order of N/ερ−1, where N is the bound on the noise and</p><p>ρ is the dimension of the observer. While the rest of the chapter is concerned with</p><p>this case, Section 8.1 shows less conservative bounds when the frequency of the noise</p><p>is much lower or much higher than 1/ε. The effect of noise on feedback control is</p><p>studied in Section 8.2, where it is shown that a result similar to the separation prin-</p><p>ciple of Section 3.1 can be shown if the amplitude of the noise is restricted. Even in</p><p>this case, the presence of noise puts a lower bound on ε. In the tracking problem of</p><p>Section 8.3 it is shown that the component of the tracking error due to measurement</p><p>noise does not depend on a negative power of ε. Its ultimate bound is of the order of</p><p>N . In Section 8.4 two techniques are discussed for reducing the effect of measurement</p><p>noise. In the first technique, the high-frequency content of the noise is filtered out</p><p>before feeding the measurement into the observer. The second technique uses a non-</p><p>linear gain to adjust ε so</p><p>that a smaller ε is used during the transient period to achieve</p><p>fast convergence and a larger one is used at steady state to reduce the effect of noise.</p><p>Digital implementation of high-gain observers is discussed in Chapter 9. The ob-</p><p>server is discretized with a sampling period proportional to ε. The nonlinear observer</p><p>is discretized using the Forward Difference method, while other discretization meth-</p><p>ods can be used with linear observers. Digital control with zero-order hold is shown</p><p>to have properties similar to continuous-time controllers when ε and the sampling</p><p>period are sufficiently small. Finally, a multirate digital control scheme is presented.</p><p>This scheme is useful for computationally demanding controllers where the control</p><p>sampling period cannot be reduced beyond a certain value, which might not be small</p><p>enough to implement the fast high-gain observer. The multirate scheme allows the</p><p>observer to run with a sampling period shorter than the controller’s sampling period.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>Chapter 2</p><p>High-Gain Observers</p><p>2.1 Class of Nonlinear Systems</p><p>Consider a class of single-output nonlinear systems represented by</p><p>ẇ = f0(w, x, u), (2.1)</p><p>ẋi = xi+1+ψi (x1, . . . , xi , u) for 1≤ i ≤ ρ− 1, (2.2)</p><p>ẋρ =φ(w, x, u), (2.3)</p><p>y = x1, (2.4)</p><p>where w ∈ R` and x = col(x1, x2, . . . , xρ) ∈ Rρ form the state vector, u ∈ Rm is the</p><p>input, and y ∈ R is the measured output. Equations (2.1)–(2.4) cover two important</p><p>classes of nonlinear systems. The normal form of a single-input–single-output non-</p><p>linear system having relative degree ρ is a special case of (2.1)–(2.4) where ψi = 0 for</p><p>1 ≤ i ≤ ρ− 1, f0 is independent of u and φ(w, x, u) = φ1(w, x) + φ2(w, x)u.3 A</p><p>nonlinear single-input–single-output system of the form</p><p>χ̇ = f (χ )+ g (χ )u, y = h(χ ), (2.5)</p><p>has relative degree ρ in an open setR if, for all χ ∈R ,</p><p>Lg Li−1</p><p>f h(χ ) = 0 for i = 1,2, . . . ,ρ− 1; Lg Lρ−1</p><p>f h(χ ) 6= 0,</p><p>where L f h(χ ) = ∂ h</p><p>∂ χ f is the Lie derivative of h with respect to the vector field f .</p><p>Under appropriate smoothness conditions on f , g , and h, there is a change of vari-</p><p>ables that transforms a relative degree ρ system into the normal formal, at least lo-</p><p>cally. Under some stronger conditions, the change of variables will hold globally [64,</p><p>Chapter 9].</p><p>When w is absent, equations (2.2)–(2.4) reduce to</p><p>ẋi = xi+1+ψi (x1, . . . , xi , u) for 1≤ i ≤ ρ− 1, (2.6)</p><p>ẋρ =φ(x, u), (2.7)</p><p>y = x1. (2.8)</p><p>3See [64, Chapter 4] or [80, Chapter 8].</p><p>17</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>18 CHAPTER 2. HIGH-GAIN OBSERVERS</p><p>This system is observable, uniformly in u, in the sense of differential observability [49,</p><p>Chapter 2]. It is not hard to verify that, if the functions ψi are sufficiently smooth,</p><p>the state x can be uniquely expressed as a function y, ẏ, . . . , y (ρ−1), u, u̇, . . . , u (ρ−2).</p><p>A system whose state can be expressed as a function of its output, input, and their</p><p>derivatives up to order ρ−1 is said to be differentially observable of order ρ. As a spe-</p><p>cial case of (2.6)–(2.8), consider a ρ-dimensional, nonlinear, single-input–single-output</p><p>system of the form (2.5) with sufficiently smooth functions f , g , and h. Suppose the</p><p>system is observable uniformly in u in the sense that on any infinitesimally small time</p><p>interval [0,T ] and for any measurable bounded input u(t ) defined on [0,T ], the ini-</p><p>tial state x(0) is uniquely determined from the trajectories of the output y(t ) and the</p><p>input u(t ) over [0,T ]. Then the map</p><p>Φ(χ ) = col</p><p>�</p><p>h(χ ), L f h(χ ), . . . , Lρ−1</p><p>f h(χ )</p><p>�</p><p>is a local diffeomorphism, and the change of variables x = Φ(χ ) transforms the system</p><p>(2.5), at least locally, into the form4</p><p>ẋi = xi+1+ gi (x1, . . . , xi )u for 1≤ i ≤ ρ− 1, (2.9)</p><p>ẋρ =φ1(x)+φ2(x)u, (2.10)</p><p>y = x1. (2.11)</p><p>2.2 Observer Design</p><p>In the motivating examples of Section 1.2 we saw that high-gain observers have two</p><p>features. First, the estimation error decays rapidly toward small (order O(ε)) values</p><p>within a time interval that decreases toward zero as ε decreases. Second, the observer</p><p>is robust with respect to uncertain nonlinear functions. The task of this section is to</p><p>design a high-gain observer for a nonlinear system of the form (2.1)–(2.4), that is,</p><p>ẇ = f0(w, x, u), (2.12)</p><p>ẋi = xi+1+ψi (x1, . . . , xi , u) for 1≤ i ≤ ρ− 1, (2.13)</p><p>ẋρ =φ(w, x, u), (2.14)</p><p>y = x1, (2.15)</p><p>where f0, ψ1,. . . , ψρ−1, and φ are locally Lipschitz in their arguments, and w(t ), x(t ),</p><p>and u(t ) are bounded for all t ≥ 0.5 In particular, let w(t ) ∈W ⊂ R`, x(t ) ∈X ⊂ Rρ,</p><p>and u(t ) ∈ U ⊂ Rm for all t ≥ 0, for some compact sets W , X , and U . The fast</p><p>decay of the estimation error can be achieved only when we estimate x. Therefore,</p><p>the high-gain observer is a partial-state observer that estimates only x, not the full</p><p>state (w, x). The robustness feature of the observer can be achieved only with respect</p><p>to the nonlinearityφ; therefore, we assume thatψ1 toψρ−1 are known. Furthermore,</p><p>we require that for any compact set S ⊂ Rρ, there are positive constants L1 to Lρ−1</p><p>such that the functions ψ1 to ψρ−1 satisfy the Lipschitz conditions</p><p>|ψi (x1, . . . , xi , u)−ψi (z1, . . . , zi , u)| ≤ Li</p><p>i</p><p>∑</p><p>k=1</p><p>|xk − zk | (2.16)</p><p>4See [49, Theorem 4.1 of Chapter 3] or [50, Theorem 2].</p><p>5This assumption is needed because the system is operated in open loop. In later chapters when the</p><p>observer is studied within a feedback loop, the boundedness of the state under feedback will be established</p><p>in the analysis.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>2.2. OBSERVER DESIGN 19</p><p>for all x, z ∈ S, and u ∈U . The observer is taken as</p><p>˙̂xi = x̂i+1+ψ</p><p>s</p><p>i (x̂1, . . . , x̂i , u)+</p><p>αi</p><p>εi</p><p>(y − x̂1) for 1≤ i ≤ ρ− 1, (2.17)</p><p>˙̂xρ =φ0(x̂, u)+</p><p>αρ</p><p>ερ</p><p>(y − x̂1), (2.18)</p><p>where φ0 is a nominal model of φ, ε is a sufficiently small positive constant, and α1</p><p>to αρ are chosen such that the polynomial</p><p>sρ+α1 sρ−1+ · · ·+αρ−1 s +αρ (2.19)</p><p>is Hurwitz.6 The functionsψs</p><p>1 toψs</p><p>ρ−1 are locally Lipschitz and satisfy the conditions</p><p>ψs</p><p>i (x1, . . . , xi , u) =ψi (x1, . . . , xi , u) ∀ x ∈X (2.20)</p><p>and</p><p>|ψs</p><p>i (x1, . . . , xi , u)−ψs</p><p>i (z1, . . . , zi , u)| ≤ Li</p><p>i</p><p>∑</p><p>k=1</p><p>|xk − zk | (2.21)</p><p>for all x, z ∈ Rρ and all u ∈ U . The difference between (2.16) and (2.21) is that (2.16)</p><p>holds for x and z in a compact set, while (2.21) holds for all x and z. If the Lipschitz</p><p>condition (2.16) holds globally, we can take ψs</p><p>i = ψi . Otherwise, we define ψs</p><p>i by</p><p>saturating the x1 to xi arguments of ψi outside the compact set X . Choose Mi ></p><p>maxx∈X |xi | and define ψs</p><p>i by</p><p>ψs</p><p>i (x1, . . . , xi , u) =ψi</p><p>�</p><p>M1 sat</p><p>�</p><p>x1</p><p>M1</p><p>�</p><p>, . . . , Mi sat</p><p>�</p><p>xi</p><p>Mi</p><p>�</p><p>, u</p><p>�</p><p>, (2.22)</p><p>where sat(·) is the standard saturation function defined by</p><p>sat(y) =min{|y|, 1} sign(y) =</p><p></p><p></p><p></p><p>y if |y| ≤ 1,</p><p>−1 if y ≤−1,</p><p>1 if y ≥ 1.</p><p>The function ψs</p><p>i satisfies (2.21) because</p><p>|ψs</p><p>i (x1, . . . , xi , u)−ψs</p><p>i (z1, . . . , zi , u)| ≤ Li</p><p>i</p><p>∑</p><p>k=1</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>Mk sat</p><p>�</p><p>xx</p><p>Mk</p><p>�</p><p>−Mk sat</p><p>�</p><p>zk</p><p>Mk</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>�</p><p>≤ Li</p><p>i</p><p>∑</p><p>k=1</p><p>|xk − zk |,</p><p>where the first inequality follows from (2.16) and the second inequality holds because</p><p>the saturation function is globally Lipschitz with Lipschitz constant equal to one.</p><p>We assume that φ0 is locally Lipschitz in its arguments and</p><p>|φ(w, x, u)−φ0(z, u)| ≤ L ‖x − z‖+M (2.23)</p><p>for all w ∈W , x ∈X , z ∈ Rρ, and u ∈U . Because</p><p>φ(w, x, u)−φ0(z, u) =φ(w, x, u)−φ0(x, u)+φ0(x, u)−φ0(z, u)</p><p>6A polynomial is Hurwitz when all its roots have negative real parts.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>20 CHAPTER 2. HIGH-GAIN OBSERVERS</p><p>and φ0(x, u) can be chosen to be globally Lipschitz in x by saturating its x-argument</p><p>outside the compact set X , (2.23) requires the modeling error φ(w, x, u)−φ0(x, u) to</p><p>be bounded. We can chooseφ0 = 0, which would be a natural choice if no information</p><p>were available on φ. In this case, (2.23) holds with L = 0. On the other hand, if φ is</p><p>known and either it is not function of w or w is measured, we can take φ0 = φ with</p><p>the x-argument of φ0 saturated outside X . In this case, (2.23) holds with M = 0.</p><p>Theorem 2.1. Under the stated assumptions, there is ε∗ ∈ (0,1] such that for 0 0 is the</p><p>solution of the Lyapunov equation P F + F T P =−I . Then</p><p>εV̇ =−ηT η+ 2εηT Pδ.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>2.3. LYAPUNOV AND RICCATI EQUATION DESIGNS 21</p><p>Using (2.28) we obtain</p><p>εV̇ ≤−‖η‖2+ 2ε‖P‖Lδ‖η‖</p><p>2+ 2ε‖P‖M‖η‖.</p><p>For ε‖P‖Lδ ≤</p><p>1</p><p>4 ,</p><p>εV̇ ≤− 1</p><p>2‖η‖</p><p>2+ 2ε‖P‖M‖η‖.</p><p>Hence,</p><p>εV̇ ≤−1</p><p>4</p><p>‖η‖2 ∀ ‖η‖ ≥ 8εM‖P‖.</p><p>We conclude that [80, Theorem 4.5]</p><p>‖η(t )‖ ≤max</p><p>¦</p><p>b e−at/ε‖η(0)‖,εcM</p><p>©</p><p>for some positive constants a, b , c . From (2.25) we see that ‖η(0)‖ ≤ ‖x̃(0)‖/ερ−1 and</p><p>|x̃i |= ερ−i |ηi |, which yield (2.24). 2</p><p>The two terms on the right-hand side of (2.24) show bounds on the estimation er-</p><p>ror due to two sources. The term (b/εi−1) e−at/ε is due to the initial estimation error</p><p>x̃(0) = x(0)− x̂(0). It exhibits the peaking phenomenon of the high-gain observer. It</p><p>can be seen from (2.25) that peaking will not happen if the initial estimation error sat-</p><p>isfies xi (0)− x̂i (0) =O(ερ−i ) for i = 1, . . . ,ρ. This term shows also that the estimation</p><p>error decays rapidly to O(ε) values. In particular, given any positive constant K , it can</p><p>be seen that</p><p>b</p><p>ερ−1</p><p>e−at/ε ≤Kε ∀ t ≥ T (ε) def=</p><p>ε</p><p>a</p><p>ln</p><p>� b</p><p>Kερ</p><p>�</p><p>. (2.29)</p><p>Using l’Hôpital’s rule, it can be seen that limε→0 T (ε) = 0. The second term ερ+1−i cM</p><p>is due to the error in modeling the functionφ. This error will not exist if the observer</p><p>is implemented with φ=φ0. Thus, limt→∞ x̃(t ) = 0 when φ=φ0.</p><p>2.3 Lyapunov and Riccati Equation Designs</p><p>In the observer (2.17)–(2.18), the observer gain can be written as</p><p>H =</p><p>1</p><p>ερ</p><p>D(ε)Ho , (2.30)</p><p>where D(ε) = diag[ερ−1, . . . , 1] and Ho = col(α1, α2, . . . ,αρ) assigns the eigenvalues of</p><p>the matrix F of (2.27) at desired locations in the open left-half plane. The matrix F</p><p>can be written as F =A−HoC , where</p><p>A=</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>0 1 0 · · · 0</p><p>0 0 1 · · · 0</p><p>...</p><p>. . .</p><p>...</p><p>0 0 1</p><p>0 0 · · · · · · 0</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>ρ×ρ</p><p>and C =</p><p>�</p><p>1 0 · · · · · · 0</p><p>�</p><p>1×ρ .</p><p>In this section we present two special designs of Ho , which are obtained by solving</p><p>Lyapunov and Riccati equations. The Lyapunov equation–based design comes from</p><p>early work on high-gain observers [50], while the Riccati equation–based design has</p><p>its roots in loop transfer recovery techniques of linear systems.7</p><p>7See [127] and the reference therein.</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>22 CHAPTER 2. HIGH-GAIN OBSERVERS</p><p>2.3.1 Lyapunov Equation Design</p><p>Let P be the positive definite symmetric solution of the Lyapunov equation</p><p>P (A+ 1</p><p>2 I )+ (A+ 1</p><p>2 I )T P −C T C = 0 (2.31)</p><p>and take</p><p>Ho = P−1C T . (2.32)</p><p>The existence and uniqueness of P is guaranteed by the fact that−(A+ 1</p><p>2 I ) is Hurwitz</p><p>and (A,C ) is observable.8 The fact that A−HoC = A− P−1C T C is Hurwitz can be</p><p>seen from the Lyapunov equation</p><p>P (A− P−1C T C )+ (A− P−1C T C )T P +C T C + P = 0, (2.33)</p><p>which is obtained from (2.31) by adding and subtracting C T C . Equation (2.33) shows</p><p>that A− P−1C T C is Hurwitz because C T C + P is positive definite. What is more</p><p>interesting is the fact that all the eigenvalues of A− P−1C T C are assigned at −1. For</p><p>example, for ρ= 2, 3, and 4, the matrix A− P−1C T C is given by</p><p>�</p><p>−2 1</p><p>−1 0</p><p>�</p><p>,</p><p></p><p></p><p>−3 1 0</p><p>−3 0 1</p><p>−1 0 0</p><p></p><p> , and</p><p></p><p></p><p></p><p></p><p>−4 1 0 0</p><p>−6 0 1 0</p><p>−4 0 0 1</p><p>−1 0 0 0</p><p></p><p></p><p></p><p></p><p>,</p><p>respectively.</p><p>2.3.2 Riccati Equation Design</p><p>Let Q be the positive definite symmetric solution of the Riccati equation</p><p>AQ +QAT −QC T C Q +BBT = 0, (2.34)</p><p>where B = col(0, 0, . . . , 0, 1) is a ρ× 1 matrix, and take</p><p>Ho =QC T . (2.35)</p><p>The existence of Q and the fact that A−HoC = A−QC T C is Hurwitz follow from</p><p>the properties of the solution of Riccati equations because (A,B) is controllable and</p><p>(A,C ) is observable [91]. The eigenvalues of A−QC T C are assigned on the unit circle</p><p>in a Butterworth pattern, that is,</p><p>λk = e jθk , where θk =</p><p>(2k +ρ− 1)π</p><p>2ρ</p><p>for k = 1,2, . . . ,ρ. (2.36)</p><p>Table 2.1 shows the eigenvalues of A−HoC for four different values of ρ.</p><p>2.3.3 Simulation Comparison</p><p>The choice of the observer eigenvalues determines the shape of the transient response</p><p>of the estimation error. In the example to follow we compare a Lyapunov equation</p><p>design, a Riccati equation design, and a pole-placement design.</p><p>8See [78, Theorem 4.6 and Exercise 4.22].</p><p>D</p><p>ow</p><p>nl</p><p>oa</p><p>de</p><p>d</p><p>07</p><p>/0</p><p>9/</p><p>17</p><p>to</p><p>1</p><p>32</p><p>.2</p><p>36</p><p>.2</p><p>7.</p><p>11</p><p>1.</p><p>R</p><p>ed</p><p>is</p><p>tr</p><p>ib</p><p>ut</p><p>io</p><p>n</p><p>su</p><p>bj</p><p>ec</p><p>t t</p><p>o</p><p>SI</p><p>A</p><p>M</p><p>li</p><p>ce</p><p>ns</p><p>e</p><p>or</p><p>c</p><p>op</p><p>yr</p><p>ig</p><p>ht</p><p>; s</p><p>ee</p><p>h</p><p>ttp</p><p>://</p><p>w</p><p>w</p><p>w</p><p>.s</p><p>ia</p><p>m</p><p>.o</p><p>rg</p><p>/jo</p><p>ur</p><p>na</p><p>ls</p><p>/o</p><p>js</p><p>a.</p><p>ph</p><p>p</p><p>2.4. REDUCED-ORDER OBSERVER 23</p><p>Table 2.1. The eigenvalues of A−HoC in the Riccati equation design.</p><p>ρ Eigenvalues θ</p><p>2 −0.7071+ 0.7071 j</p><p>−0.7071− 0.7071 j</p><p>3π/4</p><p>5π/4</p><p>3</p><p>−0.5+ 0.8660 j</p><p>−1</p><p>−0.5− 0.8660 j</p><p>2π/3</p><p>π</p><p>4π/3</p><p>4</p><p>−0.3827+ 0.9239 j</p><p>−0.9239+ 0.3827 j</p><p>−0.9239− 0.3827 j</p><p>−0.3827− 0.9239 j</p><p>5π/8</p><p>7π/8</p><p>9π/8</p><p>11π/8</p><p>5</p><p>−0.3090+ 0.9511 j</p><p>−0.8090+ 0.5878 j</p><p>−1</p><p>−0.8090− 0.5878 j</p><p>−0.3090− 0.9511 j</p><p>3π/5</p><p>4π/5</p><p>π</p><p>6π/5</p><p>7π/5</p><p>Example 2.1. The states of the three-dimensional system</p><p>ẋ1 = x2, ẋ2 = x3, ẋ3 = cos(t ), y = x1,</p><p>are estimated using the high-gain observer</p><p>˙̂x1 = x̂2+(α1/ε)(y − x̂1),</p><p>˙̂x2 = x̂3+(α2/ε</p><p>2)(y − x̂1),</p><p>˙̂x2 = cos(t )+ (α3/ε</p><p>3)(y − x̂1).</p><p>Three different designs of Ho = col(α1, α2, α3) are considered. The first one, Ho =</p><p>col(3, 3, 1), is a Lyapunov equation design that assigns all three eigenvalues at −1.</p><p>The second one, Ho = col(2, 2, 1), is a Riccati equation design that assigns com-</p><p>plex eigenvalues on the unit circle in a Butterworth pattern. The third design, Ho =</p><p>col(3, 2.75, 0.75), assigns the eigenvalues at −0.5, −1,−1.5. Figure 2.1 compares the</p><p>estimation errors obtained by the three designs when ε = 0.01. It can be seen that</p><p>the two designs</p>