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http<span class="_1 blank"></span>s://www.facebook.com/AllAb<span class="_1 blank"></span>outChemc<span class="_1 blank"></span>alEngineering <span class="fs3 ls4"> <span class="ls7"> </span></span></div><div class="t m0 x3 h6 y11 ff2 fs2 fc1 sc0 ls7 ws7">@facebook: http<span class="_1 blank"></span>s://www.facebook.com/g<span class="_1 blank"></span>roups/10436265<span class="_1 blank"></span>147/ <span class="fs3 ls4"> <span class="ls7"> </span></span></div><div class="t m0 x2 h8 y12 ff2 fs3 fc1 sc0 ls7 ws7"> </div><div class="t m0 x5 h9 y13 ff4 fs5 fc3 sc0 ls7 ws6">ADMIN:<span class="ff2 fs3 fc1 ws7"> </span></div><div class="t m0 x6 h9 y14 ff4 fs5 fc3 sc0 ls7 ws6">I.W<span class="ff5 ls6 ws7"> <span class="ff2 fs3 fc1 ls7"> </span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf3" class="pf w4 ha" data-page-no="3"><div class="pc pc3 w4 ha"><div class="t m0 x7 hb y15 ff6 fs6 fc1 sc0 ls7 ws8">Control Theory for Engineers</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf4" class="pf w4 ha" data-page-no="4"><div class="pc pc4 w4 ha"><div class="t m0 x7 hb y15 ff6 fs6 fc1 sc0 lsb ws9">Brigitte d\u2019Andréa-Novel</div><div class="t m0 x7 hb y16 ff6 fs6 fc1 sc0 ls7 wsa">Michel De Lara</div><div class="t m0 x7 hc y17 ff6 fs7 fc1 sc0 lsa wsb">Control Theory for Engineers</div><div class="t m0 x7 hd y18 ff6 fs8 fc1 sc0 ls7 wsc">A Primer</div><div class="t m0 x8 he y19 ff7 fs9 fc1 sc0 ls7">123</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf5" class="pf w4 ha" data-page-no="5"><div class="pc pc5 w4 ha"><div class="t m0 x7 hf y1a ff6 fsa fc1 sc0 ls7 wsd">Brigitte d\u2019Andréa-Novel</div><div class="t m0 x7 hf y1b ff6 fsa fc1 sc0 ls7 wse">Centre de Robotique</div><div class="t m0 x7 hf y1c ff6 fsa fc1 sc0 ls7 wsf">MINES ParisTech</div><div class="t m0 x7 hf y1d ff6 fsa fc1 sc0 ls7 ws10">Paris Cedex 06</div><div class="t m0 x7 hf y1e ff6 fsa fc1 sc0 ls7">France</div><div class="t m0 x9 hf y1f ff6 fsa fc1 sc0 ls7 ws11">Michel De Lara</div><div class="t m0 x9 hf y20 ff6 fsa fc1 sc0 ls7">CERMICS</div><div class="t m0 x9 hf y21 ff6 fsa fc1 sc0 ls7 ws12">Ecole des Ponts ParisTech</div><div class="t m0 x9 hf y22 ff6 fsa fc1 sc0 ls7 ws13">Marne la Vallée Cedex 2</div><div class="t m0 x9 hf y23 ff6 fsa fc1 sc0 ls7">France</div><div class="t m0 x7 hf y24 ff6 fsa fc1 sc0 ls7 ws14">ISBN<span class="_6 blank"> </span>978-3-642-34323-0<span class="_7 blank"> </span>ISBN 978-3-642-34324-7<span class="_8 blank"> </span>(eBook)</div><div class="t m0 x7 hf y25 ff6 fsa fc1 sc0 ls7 ws15">DOI 10.1007/978-3-642-34324-7</div><div class="t m0 x7 h10 y26 ff6 fsb fc1 sc0 ls7 ws16">Springer Heidelberg New York Dordrecht London</div><div class="t m0 x7 h10 y27 ff6 fsb fc1 sc0 ls7 ws17">Library of Congress Control Number: 2013931751</div><div class="t m0 x7 h11 y28 ff8 fsb fc1 sc0 lsc">Ó<span class="ff6 ls7 ws18">Springer-Verlag Berlin Heidelberg 2013</span></div><div class="t m0 x7 h10 y29 ff6 fsb fc1 sc0 ls7 ws19">This work is subject to copyright. All rights<span class="_9 blank"> </span>are reserved by the Publisher, whether the whole or<span class="_9 blank"> </span>part of</div><div class="t m0 x7 h10 y2a ff6 fsb fc1 sc0 ls7 ws1a">the material is concerned, speci\ufb01cally the rights of translation, reprinting, reuse of illustrations,</div><div class="t m0 x7 h10 y2b ff6 fsb fc1 sc0 ls7 ws1b">recitation, broadcasting, reproduction on micro\ufb01lms or in any other<span class="_a blank"> </span>physical way, and transmission or</div><div class="t m0 x7 h10 y2c ff6 fsb fc1 sc0 ls7 ws1c">information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar</div><div class="t m0 x7 h10 y2d ff6 fsb fc1 sc0 ls7 ws1d">methodology now known or hereafter developed. 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Violations are liable to prosecution under the respective Copyright Law.</div><div class="t m0 x7 h10 y34 ff6 fsb fc1 sc0 ls7 ws24">The use of general descriptive names, registered names, trademarks, service marks, etc.<span class="_c blank"> </span>in this</div><div class="t m0 x7 h10 y35 ff6 fsb fc1 sc0 ls7 ws25">publication does not imply, even in the absence<span class="_a blank"> </span>of a speci\ufb01c statement, that such names<span class="_a blank"> </span>are exempt</div><div class="t m0 x7 h10 y36 ff6 fsb fc1 sc0 ls7 ws26">from the relevant protective laws and regulations and therefore free for general use.</div><div class="t m0 x7 h10 y37 ff6 fsb fc1 sc0 ls7 ws27">While the advice and information in this book are believed to be true<span class="_d blank"> </span>and accurate at the date of</div><div class="t m0 x7 h10 y38 ff6 fsb fc1 sc0 ls7 ws28">publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for</div><div class="t m0 x7 h10 y39 ff6 fsb fc1 sc0 ls7 ws29">any errors or omissions that may be made. The<span class="_e blank"> </span>publisher makes no warranty, express or implied<span class="_1 blank"></span>, with</div><div class="t m0 x7 h10 y3a ff6 fsb fc1 sc0 ls7 ws2a">respect to the material<span class="_a blank"> </span>contained herein.</div><div class="t m0 x7 h10 y3b ff6 fsb fc1 sc0 ls7 ws2b">Printed on acid-free paper</div><div class="t m0 x7 h10 y3c ff6 fsb fc1 sc0 ls7 ws2c">Springer is part of Springer Science+Business Media (www.springer.com)</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf6" class="pf w4 ha" data-page-no="6"><div class="pc pc6 w4 ha"><img fetchpriority="low" loading="lazy" class="bi x7 y3d w5 h5" alt="" src="https://files.passeidireto.com/7cbc2ebb-d4ff-49e8-ad59-610154e3a5b4/bg6.png"><div class="t m0 x7 h12 y15 ff9 fs6 fc1 sc0 ls7">Foreword</div><div class="t m0 x7 h13 y3e ff6 fsc fc1 sc0 ls7 ws2d">We are surrounded by electronic devi<span class="_5 blank"> </span>ces: mobile phones, compute<span class="_5 blank"> </span>rs, traf<span class="_1 blank"></span>\ufb01c light</div><div class="t m0 x7 h13 y3f ff6 fsc fc1 sc0 ls7 ws2e">regulators, etc. Many of them work automatically with inputs provided by sensors</div><div class="t m0 x7 h13 y40 ff6 fsc fc1 sc0 ls7 ws2f">scrutinizing<span class="_f blank"> </span>the<span class="_f blank"> </span>environment.<span class="_f blank"> </span>Plane<span class="_6 blank"> </span>trips<span class="_f blank"> </span>contain<span class="_f blank"> </span>less<span class="_f blank"> </span>and<span class="_6 blank"> </span>less<span class="_f blank"> </span>manual<span class="_f blank"> </span>drive.</div><div class="t m0 x7 h13 y41 ff6 fsc fc1 sc0 ls7 ws30">Electrical systems are<span class="_10 blank"> </span>managed with automatic<span class="_10 blank"> </span>circuits ensuring stabil<span class="_5 blank"> </span>ity. Exam-</div><div class="t m0 x7 h13 y42 ff6 fsc fc1 sc0 ls7 ws31">ples abound, and our technol<span class="_5 blank"> </span>ogical society will use mor<span class="_5 blank"> </span>e and more such auto<span class="_5 blank"> </span>matic</div><div class="t m0 x7 h13 y43 ff6 fsc fc1 sc0 ls7 ws32">devices<span class="_d blank"> </span>under<span class="_10 blank"> </span>the<span class="_a blank"> </span>pressure<span class="_10 blank"> </span>of<span class="_d blank"> </span>various<span class="_d blank"> </span>factors.<span class="_d blank"> </span>Technol ogical<span class="_10 blank"> </span>innovation<span class="_d blank"> </span>and<span class="_d blank"> </span>costs</div><div class="t m0 x7 h13 y44 ff6 fsc fc1 sc0 ls7 ws33">reduction increase the<span class="_11 blank"> </span>penetration of so-called<span class="_11 blank"> </span>\u2018<span class="_2 blank"></span>\u2018smart\u2019<span class="_2 blank"></span>\u2019 devices. The world</div><div class="t m0 x7 h13 y45 ff6 fsc fc1 sc0 ls7 ws34">increase in demand for<span class="_a blank"> </span>communications tec<span class="_5 blank"> </span>hnologies and for energy require<span class="_5 blank"> </span>s more</div><div class="t m0 x7 h13 y46 ff6 fsc fc1 sc0 ls7 ws35">and more coordination in<span class="_d blank"> </span>a globa<span class="_5 blank"> </span>l economy. Environm<span class="_5 blank"> </span>ental protection fosters<span class="_12 blank"> </span>the</div><div class="t m0 x7 h13 y47 ff6 fsc fc1 sc0 ls7 ws36">need<span class="_10 blank"> </span>for<span class="_b blank"> </span>soberness<span class="_b blank"> </span>in<span class="_b blank"> </span>the<span class="_10 blank"> </span>use<span class="_b blank"> </span>of<span class="_b blank"> </span>resources,<span class="_b blank"> </span>hence<span class="_b blank"> </span>of<span class="_10 blank"> </span>an<span class="_b blank"> </span>optimized<span class="_b blank"> </span>management.</div><div class="t m0 x7 h13 y48 ff6 fsc fc1 sc0 ls7 ws37">Control Theory is at the hear<span class="_5 blank"> </span>t of Information and Comm<span class="_5 blank"> </span>unication Technologies of</div><div class="t m0 x7 h13 y49 ff6 fsc fc1 sc0 ls7 ws38">complex systems; it<span class="_10 blank"> </span>can contribute to answer<span class="_10 blank"> </span>such challenges.</div><div class="t m0 x1 h13 y4a ff6 fsc fc1 sc0 ls7 ws39">We aim at providing students and engineers with basic tools and methods<span class="_b blank"> </span>to</div><div class="t m0 x7 h13 y4b ff6 fsc fc1 sc0 ls7 ws3a">tackle dynamical system<span class="_5 blank"> </span>s control problems. The<span class="_d blank"> </span>book is organ<span class="_5 blank"> </span>ized as an engineer</div><div class="t m0 x7 h13 y4c ff6 fsc fc1 sc0 ls7 ws3b">classically proceeds to solve a cont<span class="_5 blank"> </span>rol problem, that is elabora<span class="_5 blank"> </span>tion of a mathe-</div><div class="t m0 x7 h13 y4d ff6 fsc fc1 sc0 ls7 ws3c">matical model capturing the process behavi<span class="_5 blank"> </span>or, analysis of this model, and design of</div><div class="t m0 x7 h13 y4e ff6 fsc fc1 sc0 ls7 ws3d">a<span class="_12 blank"> </span>controller<span class="_12 blank"> </span>to<span class="_12 blank"> </span>accompl ish<span class="_10 blank"> </span>desired<span class="_12 blank"> </span>objectives.</div><div class="t m0 x1 h13 y4f ff6 fsc fc1 sc0 ls7 ws3e">Central<span class="_a blank"> </span>to<span class="_a blank"> </span>Control<span class="_a blank"> </span>Theory<span class="_a blank"> </span>are<span class="_a blank"> </span>the<span class="_e blank"> </span>notions<span class="_a blank"> </span>of <span class="ffa ws3f">fee<span class="_5 blank"> </span>dback </span><span class="ws37">and of <span class="ffa ws40">closed-loop</span><span class="ws41">. In<span class="_a blank"> </span>his</span></span></div><div class="t m0 x7 h13 y50 ff6 fsc fc1 sc0 ls7 ws41">autobiography, <span class="ffa ws42">Eye of the Hurricane </span><span class="ws43">(see<span class="_a blank"> </span>[29]),<span class="_a blank"> </span>the<span class="_a blank"> </span>famous<span class="_a blank"> </span>applied<span class="_a blank"> </span>mathematician</span></div><div class="t m0 x7 h13 y51 ff6 fsc fc1 sc0 ls7 ws44">Richard Bellman recalls how he was<span class="_6 blank"> </span>led from <span class="ffa ws45">open-loop </span><span class="ws46">solutions<span class="_6 blank"> </span>of control</span></div><div class="t m0 x7 h13 y52 ff6 fsc fc1 sc0 ls7 ws47">problems (\u2018<span class="_2 blank"></span>\u2018functions of time\u2019<span class="_2 blank"></span>\u2019) to closed-<span class="_5 blank"> </span>loop solutions (\u2018<span class="_2 blank"></span>\u2018policies<span class="_5 blank"> </span>\u2019<span class="_2 blank"></span>\u2019): <span class="ffa ws48">Again the</span></div><div class="t m0 x7 h13 y53 ffa fsc fc1 sc0 ls7 ws49">intriguing thought<span class="ff6 lsd">:</span><span class="ws4a">A<span class="_10 blank"> </span>solution<span class="_12 blank"> </span>is<span class="_12 blank"> </span>not<span class="_12 blank"> </span>merely<span class="_12 blank"> </span>a<span class="_10 blank"> </span>set<span class="_d blank"> </span>of functio<span class="_5 blank"> </span>ns<span class="_12 blank"> </span>of<span class="_12 blank"> </span>time<span class="ff6 lse">,</span><span class="ws38">or<span class="_10 blank"> </span>a set of</span></span></div><div class="t m0 x7 h13 y54 ffa fsc fc1 sc0 ls7 ws40">numbers<span class="ff6 lsf">,</span><span class="ws4b">but a rule tell<span class="_5 blank"> </span>ing the decision mak<span class="_5 blank"> </span>er what to do<span class="ff6 ls10">;</span><span class="ws4c">a policy.</span></span></div><div class="t m1 xa h14 y55 ff6 fsd fc4 sc0 ls7">1</div><div class="t m0 xb h13 y56 ff6 fsc fc1 sc0 ls7 ws4d">The notion of</div><div class="t m0 x7 h13 y57 ffa fsc fc1 sc0 ls7 ws40">feedback<span class="ff6 ws4e">, as a rule mapping the state space (and the time) onto the control space, is</span></div><div class="t m0 x7 h13 y58 ff6 fsc fc1 sc0 ls7 ws4f">developed throughout this text.</div><div class="t m0 x1 h13 y59 ff6 fsc fc1 sc0 ls7 ws50">As with heat regulation by a thermo<span class="_5 blank"> </span>stat, many systems are regulat<span class="_5 blank"> </span>ed by feed-</div><div class="t m0 x7 h13 y5a ff6 fsc fc1 sc0 ls7 ws51">back and the way to elabora<span class="_5 blank"> </span>te such feedback<span class="_10 blank"> </span>controllers needs<span class="_10 blank"> </span>some approp<span class="_5 blank"> </span>riate</div><div class="t m0 x7 h13 y5b ff6 fsc fc1 sc0 ls7 ws52">mathematical tools which<span class="_c blank"> </span>are highlighted here. This is why we are particula<span class="_5 blank"> </span>rly</div><div class="t m2 x7 h15 y5c ff6 fse fc1 sc0 ls7">1</div><div class="t m0 xc h16 y5d ff6 fsf fc1 sc0 ls7 ws53">To make these notions<span class="_a blank"> </span>more concrete,<span class="_a blank"> </span>consider<span class="_d blank"> </span>traf<span class="_3 blank"></span>\ufb01c regula<span class="_5 blank"> </span>tion. Fixed cycle gears<span class="_a blank"> </span>for<span class="_a blank"> </span>traf<span class="_1 blank"></span>\ufb01c</div><div class="t m0 x7 h16 y5e ff6 fsf fc1 sc0 ls7 ws54">lights are examples of open-loop controls, whereas closed-loop traf<span class="_1 blank"></span>\ufb01c lights controls adapt their</div><div class="t m0 x7 h16 y5f ff6 fsf fc1 sc0 ls7 ws55">timing and phasing according to current traf<span class="_1 blank"></span>\ufb01c conditions identi\ufb01ed by means of sensors.</div><div class="t m0 xd h16 y60 ff6 fsf fc1 sc0 ls7">v</div><a class="l" href="#pf6" data-dest-detail='[6,"FitR",0,385,440,0]'><div class="d m3" style="border-style:none;position:absolute;left:324.397000px;bottom:195.364000px;width:3.741000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf7" class="pf w4 ha" data-page-no="7"><div class="pc pc7 w4 ha"><img fetchpriority="low" loading="lazy" class="bi x7 y61 w5 h5" alt="" src="https://files.passeidireto.com/7cbc2ebb-d4ff-49e8-ad59-610154e3a5b4/bg7.png"><div class="t m0 x7 h13 y62 ff6 fsc fc1 sc0 ls7 ws56">interested in syst<span class="_5 blank"> </span>ems which can b<span class="_5 blank"> </span>e represented by<span class="_b blank"> </span>a mathematical<span class="_b blank"> </span>model which</div><div class="t m0 x7 h13 y63 ff6 fsc fc1 sc0 ls7 ws57">embodies the reactions<span class="_10 blank"> </span>to external inputs.</div><div class="t m0 x1 h13 y64 ff6 fsc fc1 sc0 ls7 ws58">The book is divided into three<span class="_10 blank"> </span>Parts.</div><div class="t m0 x1 h13 y65 ff6 fsc fc1 sc0 ls7 ws59">In Part I, we start by expl<span class="_5 blank"> </span>oring a graded approach<span class="_d blank"> </span>of mode<span class="_5 blank"> </span>ling in <span class="fc4 ws5a">Chap. 1</span><span class="ls11 ws5b">.W<span class="_13 blank"></span>e</span></div><div class="t m0 x7 h13 y66 ff6 fsc fc1 sc0 ls7 ws5c">shine the spotlight on<span class="_d blank"> </span>som<span class="_5 blank"> </span>e general principles<span class="_d blank"> </span>such<span class="_d blank"> </span>as<span class="_d blank"> </span>mass<span class="_d blank"> </span>and<span class="_d blank"> </span>energy<span class="_d blank"> </span>conse<span class="_5 blank"> </span>rva-</div><div class="t m0 x7 h13 y67 ff6 fsc fc1 sc0 ls7 ws5d">tion<span class="_c blank"> </span>laws,<span class="_c blank"> </span>illustrated<span class="_c blank"> </span>by<span class="_c blank"> </span>man y<span class="_c blank"> </span>examples<span class="_c blank"> </span>com ing<span class="_c blank"> </span>from<span class="_14 blank"> </span>various<span class="_c blank"> </span>\ufb01elds<span class="_c blank"> </span>(electrical</div><div class="t m0 x7 h13 y68 ff6 fsc fc1 sc0 ls7 ws5e">engineering, mechanics,<span class="_c blank"> </span>chemistry, biological processes, etc.). In<span class="_c blank"> </span>the process of</div><div class="t m0 x7 h13 y69 ff6 fsc fc1 sc0 ls7 ws5f">elaborating<span class="_c blank"> </span>a<span class="_c blank"> </span>mathematical<span class="_c blank"> </span>model<span class="_c blank"> </span>for<span class="_b blank"> </span>control<span class="_c blank"> </span>purposes,<span class="_c blank"> </span>we<span class="_c blank"> </span>highlight<span class="_c blank"> </span>particular</div><div class="t m0 x7 h13 y6a ff6 fsc fc1 sc0 ls7 ws60">dynamical structures and<span class="_a blank"> </span>related classes<span class="_a blank"> </span>of variables, like<span class="_a blank"> </span>state, control, and<span class="_a blank"> </span>output</div><div class="t m0 x7 h13 y6b ff6 fsc fc1 sc0 ls7 ws61">variables. The notion of <span class="ffa ws62">state </span><span class="ws63">of a system<span class="_a blank"> </span>is emphasized, namely a \ufb01nite numb<span class="_5 blank"> </span>er of</span></div><div class="t m0 x7 h13 y6c ff6 fsc fc1 sc0 ls7 ws64">quantities which, bein<span class="_5 blank"> </span>g known at a given<span class="_10 blank"> </span>time instant<span class="_5 blank"> </span>, allow us to determ<span class="_5 blank"> </span>ine the</div><div class="t m0 x7 h13 y6d ff6 fsc fc1 sc0 ls7 ws65">future evolution of the system. A state is a summary of past histor<span class="_5 blank"> </span>y, suf<span class="_3 blank"></span>\ufb01c<span class="_5 blank"> </span>ient for</div><div class="t m0 x7 h13 y6e ff6 fsc fc1 sc0 ls7 ws43">prediction.<span class="_11 blank"> </span>From<span class="_15 blank"> </span>a<span class="_15 blank"> </span>mathem atical<span class="_11 blank"> </span>point<span class="_15 blank"> </span>of<span class="_11 blank"> </span>view,<span class="_15 blank"> </span>we<span class="_11 blank"> </span>focus<span class="_15 blank"> </span>our<span class="_11 blank"> </span>attention<span class="_15 blank"> </span>on</div><div class="t m0 x7 h13 y6f ff6 fsc fc1 sc0 ls7 ws66">dynamical systems whose main<span class="_b blank"> </span>properties are<span class="_b blank"> </span>highlighte<span class="_5 blank"> </span>d in <span class="fc4 ws67">Chap. 2</span><span class="ws68">. Then, we</span></div><div class="t m0 x7 h13 y70 ff6 fsc fc1 sc0 ls7 ws69">shift our gaze from intern<span class="_5 blank"> </span>al to external mathemati<span class="_5 blank"> </span>cal descriptions. <span class="fc4 ws6a">Chapter<span class="_c blank"> </span>3 </span>is</div><div class="t m0 x7 h13 y71 ff6 fsc fc1 sc0 ls7 ws6b">devoted to the <span class="ffa ws6c">frequency-domai<span class="_5 blank"> </span>n approach </span>and the associated <span class="ffa ws6d">input\u2013out<span class="_5 blank"> </span>put rep-</span></div><div class="t m0 x7 h13 y72 ffa fsc fc1 sc0 ls7 ws6e">resentation<span class="ff6 ws6f">, in which the system is embrac<span class="_5 blank"> </span>ed as a sort-of \u2018<span class="_2 blank"></span>\u2018black box\u2019<span class="_2 blank"></span>\u2019 reacting to a</span></div><div class="t m0 x7 h13 y73 ff6 fsc fc1 sc0 ls7 ws70">set of input signals. Graphical representations<span class="_e blank"> </span>are<span class="_e blank"> </span>used in the case<span class="_e blank"> </span>of a scalar syst<span class="_5 blank"> </span>em</div><div class="t m0 x7 h13 y74 ff6 fsc fc1 sc0 ls7 ws71">(one input, one output), highlighting natural de\ufb01nitions of <span class="ffa ws72">robustnes<span class="_5 blank"> </span>s </span><span class="ws73">degrees such</span></div><div class="t m0 x7 h13 y75 ff6 fsc fc1 sc0 ls7 ws74">as gain or phase margi<span class="_5 blank"> </span>ns. Let us emphasize that<span class="_d blank"> </span>the input\u2013output approach<span class="_d blank"> </span>is well</div><div class="t m0 x7 h13 y76 ff6 fsc fc1 sc0 ls7 ws75">adapted to the case of systems for which it is dif<span class="_1 blank"></span>\ufb01cult, or sometimes impossible, to</div><div class="t m0 x7 h13 y77 ff6 fsc fc1 sc0 ls7 ws76">obtain mathematical dynam<span class="_5 blank"> </span>ical models from physical laws.</div><div class="t m0 x1 h13 y78 ff6 fsc fc1 sc0 ls7 ws42">Part II is devoted to system analysi<span class="_5 blank"> </span>s to ensure stability in the neighborh<span class="_5 blank"> </span>ood of a</div><div class="t m0 x7 h13 y79 ff6 fsc fc1 sc0 ls7 ws77">set point, a classical<span class="_c blank"> </span>problem in control<span class="_c blank"> </span>science (take of<span class="_1 blank"></span>f of a rocket, unst<span class="_5 blank"> </span>able</div><div class="t m0 x7 h13 y7a ff6 fsc fc1 sc0 ls7 ws78">chemical reactor. . .). An equi<span class="_5 blank"> </span>librium point of a dynamical system<span class="_c blank"> </span>represents a</div><div class="t m0 x7 h13 y7b ff6 fsc fc1 sc0 ls7 ws79">steady state of the system<span class="_5 blank"> </span>\u2019s model, and <span class="fc4 ws7a">Chap. 4 </span><span class="ws7b">is dedi<span class="_5 blank"> </span>cated to their stability</span></div><div class="t m0 x7 h13 y7c ff6 fsc fc1 sc0 ls7 ws7c">properties. From a practical<span class="_d blank"> </span>point of view, the local stability<span class="_d blank"> </span>property is expresse<span class="_5 blank"> </span>d</div><div class="t m0 x7 h13 y7d ff6 fsc fc1 sc0 ls7 ws7d">from a simpler model than the original one, namely the linearized or tangent model</div><div class="t m0 x7 h13 y7e ff6 fsc fc1 sc0 ls7 ws76">around the equilibrium<span class="_12 blank"> </span>point, which const<span class="_5 blank"> </span>itutes the \ufb01rst-order<span class="_12 blank"> </span>approximation. We</div><div class="t m0 x7 h13 y7f ff6 fsc fc1 sc0 ls7 ws7e">will show how control laws<span class="_10 blank"> </span>can be elaborated from this linear<span class="_10 blank"> </span>model.</div><div class="t m0 x1 h13 y80 ff6 fsc fc1 sc0 ls7 ws7f">The dif<span class="_1 blank"></span>ference between a <span class="ffa ws80">knowledge model<span class="_5 blank"> </span></span><span class="ws81">\u2014devoted to better understanding,</span></div><div class="t m0 x7 h13 y81 ff6 fsc fc1 sc0 ls7 ws82">at the price of a detailed descr<span class="_5 blank"> </span>iption of the \u2018<span class="_2 blank"></span>\u2018piece of reality\u2019<span class="_2 blank"></span>\u2019<span class="_12 blank"> </span>under scrutiny,<span class="_12 blank"> </span>and</div><div class="t m0 x7 h13 y82 ff6 fsc fc1 sc0 ls7 ws83">generally nonlinear\u2014and a control model\u2014simpler, often linear and used to</div><div class="t m0 x7 h13 y83 ff6 fsc fc1 sc0 ls7 ws84">elaborate a control<span class="_10 blank"> </span>law\u2014is emphasized.</div><div class="t m1 xe h14 y84 ff6 fsd fc4 sc0 ls7">2</div><div class="t m2 x7 h15 y85 ff6 fse fc1 sc0 ls7">2</div><div class="t m0 xc h16 y86 ff6 fsf fc1 sc0 ls7 ws85">As an illustration of the dif<span class="_1 blank"></span>ferences between knowledge and<span class="_a blank"> </span>control models, let us consider the</div><div class="t m0 x7 h16 y87 ff6 fsf fc1 sc0 ls7 ws86">climate change issue. Climatologists develop large<span class="_10 blank"> </span>models to improve their knowledge of the</div><div class="t m0 x7 h16 y88 ff6 fsf fc1 sc0 ls7 ws87">climate mechanisms under<span class="_10 blank"> </span>increases in green<span class="_5 blank"> </span>house gas emissions. Such<span class="_10 blank"> </span>models, embodied in</div><div class="t m0 x7 h16 y89 ff6 fsf fc1 sc0 ls7 ws88">computer codes, integrate a huge number<span class="_b blank"> </span>of physical and chemical relations. They are<span class="_b blank"> </span>often</div><div class="t m0 x7 h16 y8a ff6 fsf fc1 sc0 ls7 ws89">obtained from discretizing partial dif<span class="_1 blank"></span>ferential equations on a planetary grid: the \ufb01ner the grid, the</div><div class="t m0 x7 h16 y8b ff6 fsf fc1 sc0 ls7 ws8a">sharper the model. The mythical 1/1 map may be seen as the ultimate form of such knowledge</div><div class="t m0 x7 h16 y8c ff6 fsf fc1 sc0 ls7 ws89">models. Now, climate change economists also develop models. However, the most basic of these</div><div class="t m0 x7 h16 y8d ff6 fsf fc1 sc0 ls7 ws8b">models have a few lines and are written on a spreadsheet.<span class="_a blank"> </span>Such a dif<span class="_1 blank"></span>ference stems from dif<span class="_1 blank"></span>ferent</div><div class="t m0 x7 h16 y8e ff6 fsf fc1 sc0 ls7 ws8c">perspectives. Economists do not care to understand and embrace<span class="_d blank"> </span>the physics of climate change,</div><div class="t m0 x7 h16 y8f ff6 fsf fc1 sc0 ls7 ws8d">but they do care in how to decide about the issue: should we start reducing now (at a sure cost</div><div class="t m0 x7 h16 y90 ff6 fsf fc1 sc0 ls7 ws8e">now, for possible, but uncertain, bene\ufb01ts<span class="_a blank"> </span>tomorrow), or should we wait for more information<span class="_a blank"> </span>and</div><div class="t m0 x7 h16 y91 ff6 fsf fc1 sc0 ls7 ws8f">vi Foreword</div><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_1"><div class="d m3" style="border-style:none;position:absolute;left:334.545000px;bottom:561.260000px;width:31.578000px;height:9.921000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_2"><div class="d m3" style="border-style:none;position:absolute;left:308.466000px;bottom:441.694000px;width:32.315000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_3"><div class="d m3" style="border-style:none;position:absolute;left:334.261000px;bottom:429.732000px;width:40.649000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_4"><div class="d m3" style="border-style:none;position:absolute;left:226.545000px;bottom:298.205000px;width:33.052000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="#pf7" data-dest-detail='[7,"FitR",0,465,440,0]'><div class="d m3" style="border-style:none;position:absolute;left:217.701000px;bottom:206.986000px;width:3.741000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf8" class="pf w4 ha" data-page-no="8"><div class="pc pc8 w4 ha"><img fetchpriority="low" loading="lazy" class="bi x7 y92 w5 h5" alt="" src="https://files.passeidireto.com/7cbc2ebb-d4ff-49e8-ad59-610154e3a5b4/bg8.png"><div class="t m0 x1 h13 y62 ff6 fsc fc1 sc0 ls7 ws90">Both <span class="ffa ws91">knowledge models </span><span class="ws92">and <span class="ffa ws91">control<span class="_12 blank"> </span>models </span><span class="ws93">have their<span class="_12 blank"> </span>part<span class="_12 blank"> </span>in Control Theory<span class="_5 blank"> </span>.</span></span></div><div class="t m0 x7 h13 y63 ff6 fsc fc1 sc0 ls7 ws94">Whereas control laws<span class="_a blank"> </span>or policies are designed<span class="_a blank"> </span>on the basis of a cont<span class="_5 blank"> </span>rol model, they</div><div class="t m0 x7 h13 y64 ff6 fsc fc1 sc0 ls7 ws7e">are tried and simulated on<span class="_10 blank"> </span>the more precise knowled<span class="_5 blank"> </span>ge model.</div><div class="t m0 x1 h13 y65 ff6 fsc fc1 sc0 ls7 ws95">To solve a local stabilization probl<span class="_5 blank"> </span>em of a system described by a model h<span class="_5 blank"> </span>aving</div><div class="t m0 x7 h13 y66 ff6 fsc fc1 sc0 ls7 ws96">input variables (controls or<span class="_10 blank"> </span>disturbances) and<span class="_10 blank"> </span>output variables (m<span class="_5 blank"> </span>easurements), a</div><div class="t m0 x7 h13 y67 ff6 fsc fc1 sc0 ls7 ws97">useful approach consists in computing \ufb01rst its tangent linearization at the desired</div><div class="t m0 x7 h13 y68 ff6 fsc fc1 sc0 ls7 ws50">equilibrium set point. Doing so, we reveal<span class="_10 blank"> </span>the model skeleton tailored<span class="_10 blank"> </span>for stabil-</div><div class="t m0 x7 h13 y69 ff6 fsc fc1 sc0 ls7 ws98">ization purpose. In <span class="fc4 ws99">Chap. 5</span><span class="ws9a">, the<span class="_10 blank"> </span>structural<span class="_10 blank"> </span><span class="ffa ws9b">controllability<span class="_10 blank"> </span>and<span class="_12 blank"> </span>obser vability<span class="_10 blank"> </span></span>prop-</span></div><div class="t m0 x7 h13 y6a ff6 fsc fc1 sc0 ls7 ws9c">erties of this tangent linear system are highlighted. The control<span class="_5 blank"> </span>lability property</div><div class="t m0 x7 h13 y6b ff6 fsc fc1 sc0 ls7 ws9d">means that the system can<span class="_c blank"> </span>be driven from an arbitrary state to anot<span class="_5 blank"> </span>her one by</div><div class="t m0 x7 h13 y6c ff6 fsc fc1 sc0 ls7 ws9e">means of the control input<span class="_5 blank"> </span>s. The observability<span class="_a blank"> </span>property expresse<span class="_5 blank"> </span>s that it is possible</div><div class="t m0 x7 h13 y6d ff6 fsc fc1 sc0 ls7 ws9f">to reconstruct the entire state of the system only from the past history of partial</div><div class="t m0 x7 h13 y6e ff6 fsc fc1 sc0 ls7 wsa0">knowledge of the state vector, that is, the output variables or mea<span class="_5 blank"> </span>surements. We</div><div class="t m0 x7 h13 y6f ff6 fsc fc1 sc0 ls7 wsa1">prove that a controllable and observabl<span class="_5 blank"> </span>e system can be locally asym<span class="_5 blank"> </span>ptotically</div><div class="t m0 x7 h13 y70 ff6 fsc fc1 sc0 ls7 wsa2">stabilized by means of a <span class="ffa wsa3">linear feedback control law<span class="_5 blank"> </span></span><span class="ws60">. This study is done in <span class="fc4 wsa4">Chap. 5</span></span></div><div class="t m0 x7 h13 y71 ff6 fsc fc1 sc0 ls7 wsa5">for continuous-time dynamical systems and<span class="_10 blank"> </span>in <span class="fc4 wsa6">Chap. 6 </span><span class="ws32">for<span class="_12 blank"> </span>discrete-time<span class="_12 blank"> </span>systems.</span></div><div class="t m0 x7 h13 y72 ff6 fsc fc1 sc0 ls7 wsa7">The latter, represe<span class="_5 blank"> </span>nted by dif<span class="_1 blank"></span>ference equations, are of grea<span class="_5 blank"> </span>t importance from<span class="_c blank"> </span>a</div><div class="t m0 x7 h13 y73 ff6 fsc fc1 sc0 ls7 wsa8">practical point of view, since<span class="_b blank"> </span>the digital character<span class="_b blank"> </span>of computers implies<span class="_b blank"> </span>that the</div><div class="t m0 x7 h13 y74 ff6 fsc fc1 sc0 ls7 wsa9">control is fed into the system only<span class="_d blank"> </span>at discrete instants.<span class="_d blank"> </span>We show how<span class="_d blank"> </span>to discretize</div><div class="t m0 x7 h13 y75 ff6 fsc fc1 sc0 ls7 wsaa">continuous-time linear<span class="_d blank"> </span>syst<span class="_5 blank"> </span>ems at a<span class="_d blank"> </span>given<span class="_d blank"> </span>sampling<span class="_d blank"> </span>period<span class="_5 blank"> </span>. In <span class="fc4 ws76">Chaps. 5 </span><span class="wsab">and <span class="fc4 ls12">6</span><span class="ls13 wsac">,w<span class="_13 blank"></span>e</span></span></div><div class="t m0 x7 h13 y76 ff6 fsc fc1 sc0 ls7 wsad">illuminate the links between<span class="_16 blank"> </span>state space and input\u2013output represe<span class="_5 blank"> </span>ntations,</div><div class="t m0 x7 h13 y77 ff6 fsc fc1 sc0 ls7 wsae">respectively, in continuou<span class="_5 blank"> </span>s and discrete time<span class="_5 blank"> </span>.</div><div class="t m0 x1 h13 y78 ff6 fsc fc1 sc0 ls7 wsaf">The<span class="_a blank"> </span>closed-loop<span class="_a blank"> </span>system<span class="_a blank"> </span>is<span class="_a blank"> </span>characterized<span class="_a blank"> </span>by<span class="_a blank"> </span>its<span class="_a blank"> </span><span class="ffa wsb0">stability </span><span class="wsb1">and <span class="ffa wsb2">precision </span><span class="ws40">properties</span></span></div><div class="t m0 x7 h13 y79 ff6 fsc fc1 sc0 ls7 wsb3">and some sort of compromise must be achieved. The quadratic synthesis casted in</div><div class="t m0 x7 h13 y7a ff6 fsc fc4 sc0 ls7 ws64">Chap. 7 <span class="fc1 ws2d">proposes a benchm<span class="_5 blank"> </span>ark to display the tradeof<span class="_1 blank"></span>fs, under the form of a qua-</span></div><div class="t m0 x7 h13 y7b ff6 fsc fc1 sc0 ls7 ws36">dratic<span class="_10 blank"> </span>intertemporal<span class="_10 blank"> </span>criterion.<span class="_10 blank"> </span>Moreover,<span class="_10 blank"> </span>we<span class="_10 blank"> </span>introdu ce<span class="_10 blank"> </span>a<span class="_10 blank"> </span>probabilistic<span class="_10 blank"> </span>frame work</div><div class="t m0 x7 h13 y7c ff6 fsc fc1 sc0 ls7 wsb4">to take into accoun<span class="_5 blank"> </span>t perturbations and measurem<span class="_5 blank"> </span>ent errors as<span class="_d blank"> </span>random variables, so</div><div class="t m0 x7 h13 y7d ff6 fsc fc1 sc0 ls7 wsb5">that the estimation problem can<span class="_14 blank"> </span>be solved as a linear \ufb01lt<span class="_5 blank"> </span>ering problem whose</div><div class="t m0 x7 h13 y7e ff6 fsc fc1 sc0 ls7 ws38">solution is the well<span class="_5 blank"> </span>-known <span class="ffa wsb6">Kalman-Bucy </span><span class="ws40">\ufb01lt<span class="_5 blank"> </span>er.</span></div><div class="t m0 x1 h13 y7f ff6 fsc fc1 sc0 ls7 wsb7">At the end of Parts I and II, the local stabilizat<span class="_5 blank"> </span>ion problem is solved, at least</div><div class="t m0 x7 h13 y80 ff6 fsc fc1 sc0 ls7 wsb8">theoretically.<span class="_d blank"> </span>However,<span class="_a blank"> </span>some<span class="_d blank"> </span>degrees<span class="_d blank"> </span>of freedom<span class="_a blank"> </span>are<span class="_d blank"> </span>left<span class="_a blank"> </span>in<span class="_d blank"> </span>the<span class="_a blank"> </span>cont<span class="_5 blank"> </span>rol<span class="_a blank"> </span>law<span class="_d blank"> </span>which</div><div class="t m0 x7 h13 y81 ff6 fsc fc1 sc0 ls7 ws4e">could be used to deal with transient behaviors. Indeed, besides stability, the control</div><div class="t m0 x7 h13 y82 ff6 fsc fc1 sc0 ls7 wsb9">should also guarantee some robustness. In other words, the cont<span class="_5 blank"> </span>rol should preserve</div><div class="t m0 x7 h13 y83 ff6 fsc fc1 sc0 ls7 wsba">the stability property in spite of modeling errors or unknown disturbances, such as</div><div class="t m0 x7 h13 y93 ff6 fsc fc1 sc0 ls7 wsbb">bias on sensors or actuat<span class="_5 blank"> </span>ors, oldness of components,<span class="_d blank"> </span>etc. Some answers have<span class="_d blank"> </span>been</div><div class="t m0 x7 h13 y94 ff6 fsc fc1 sc0 ls7 wsbc">given in <span class="fc4 wsbd">Chap. 3 </span><span class="wsbe">in the scalar<span class="_10 blank"> </span>case and will be developed<span class="_10 blank"> </span>in Part III.</span></div><div class="t m0 x1 h13 y95 ff6 fsc fc1 sc0 ls7 wsbe">More precisely, to take into accou<span class="_5 blank"> </span>nt the ef<span class="_3 blank"></span>fect<span class="_10 blank"> </span>of disturbances, the state space</div><div class="t m0 x7 h13 y96 ff6 fsc fc1 sc0 ls7 wsbf">approach consists of<span class="_c blank"> </span>adding some<span class="_c blank"> </span>state variabl<span class="_5 blank"> </span>es to estimate the<span class="_c blank"> </span>perturbations;</div><div class="t m0 x7 h16 y97 ff6 fsf fc1 sc0 ls7 wsc0">(Footnote 2 continued)</div><div class="t m0 x7 h16 y98 ff6 fsf fc1 sc0 ls7 wsc1">postpone costly decisions? Such<span class="_b blank"> </span>a<span class="_b blank"> </span>formulation<span class="_b blank"> </span>is<span class="_b blank"> </span>a caricature<span class="_c blank"> </span>of the climate change debate,</div><div class="t m0 x7 h16 y99 ff6 fsf fc1 sc0 ls7 wsc2">discarding details, and emphasizing other traits: economists say their models are fables. Many</div><div class="t m0 x7 h16 y9a ff6 fsf fc1 sc0 ls7 wsc3">\u2018<span class="_2 blank"></span>\u2018fables models\u2019<span class="_3 blank"></span>\u2019 have been developed to try and grasp the key economic features of the climate</div><div class="t m0 x7 h16 y9b ff6 fsf fc1 sc0 ls7 wsc4">change issue, thus shedding light onto<span class="_b blank"> </span>decision making: time preferences, discount<span class="_b blank"> </span>rate, risk</div><div class="t m0 x7 h16 y9c ff6 fsf fc1 sc0 ls7 ws8a">aversion, uncertainty, learning, etc.</div><div class="t m0 x7 h16 y9d ff6 fsf fc1 sc0 ls7 wsc5">Foreword vii</div><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_5"><div class="d m3" style="border-style:none;position:absolute;left:133.965000px;bottom:513.411000px;width:32.032000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_5"><div class="d m3" style="border-style:none;position:absolute;left:354.444000px;bottom:429.732000px;width:31.068000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_6"><div class="d m3" style="border-style:none;position:absolute;left:245.764000px;bottom:417.770000px;width:26.872000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_6"><div class="d m3" style="border-style:none;position:absolute;left:245.764000px;bottom:417.770000px;width:31.861000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_5"><div class="d m3" style="border-style:none;position:absolute;left:307.502000px;bottom:369.978000px;width:35.433000px;height:9.921000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_6"><div class="d m3" style="border-style:none;position:absolute;left:363.401000px;bottom:369.978000px;width:4.989000px;height:9.921000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_7"><div class="d m3" style="border-style:none;position:absolute;left:53.858000px;bottom:310.167000px;width:31.975000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_3"><div class="d m3" style="border-style:none;position:absolute;left:90.539000px;bottom:178.696000px;width:31.918000px;height:9.921000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf9" class="pf w4 ha" data-page-no="9"><div class="pc pc9 w4 ha"><div class="t m0 x7 h13 y62 ff6 fsc fc1 sc0 ls7 ws4e">however, this increases the complexity of the control law.<span class="_a blank"> </span>In <span class="fc4 wsc6">Chap. 8</span><span class="wsc7">, we introduce</span></div><div class="t m0 x7 h13 y63 ff6 fsc fc1 sc0 ls7 wsc8">the <span class="ffa wsc9">polynomial approach </span><span class="wsca">enabl<span class="_5 blank"> </span>ing us to obtain suitable polynomial<span class="_a blank"> </span>representations</span></div><div class="t m0 x7 h13 y64 ff6 fsc fc1 sc0 ls7 wscb">which are well adapted to express the links between<span class="_b blank"> </span>the outpu<span class="_5 blank"> </span>ts and the distur-</div><div class="t m0 x7 h13 y65 ff6 fsc fc1 sc0 ls7 wscc">bances.<span class="_a blank"> </span>In<span class="_a blank"> </span>the<span class="_d blank"> </span>multivariable<span class="_a blank"> </span>case,<span class="_d blank"> </span>after<span class="_a blank"> </span>stabilization,<span class="_d blank"> </span>degrees<span class="_a blank"> </span>of freedom<span class="_a blank"> </span>are<span class="_a blank"> </span>lef<span class="_5 blank"> </span>t<span class="_a blank"> </span>in</div><div class="t m0 x7 h13 y66 ff6 fsc fc1 sc0 ls7 wscd">the control law, wherea<span class="_5 blank"> </span>s they could<span class="_14 blank"> </span>be used for<span class="_14 blank"> </span>disturbance<span class="_14 blank"> </span>rejecti<span class="_5 blank"> </span>on without</div><div class="t m0 x7 h13 y67 ff6 fsc fc1 sc0 ls7 wsce">increasing the complexity of the feedback control<span class="_10 blank"> </span>law.</div><div class="t m0 x1 h13 y68 ff6 fsc fc1 sc0 ls7 wscf">Throughout the text, dif<span class="_1 blank"></span>ferent examples are developed, both<span class="_d blank"> </span>in the chapters and</div><div class="t m0 x7 h13 y69 ff6 fsc fc1 sc0 ls7 ws97">in the exercises. The<span class="_12 blank"> </span>inverted<span class="_12 blank"> </span>pendul<span class="_5 blank"> </span>um on a cart deser<span class="_5 blank"> </span>ves special mention. This</div><div class="t m0 x7 h13 y6a ff6 fsc fc1 sc0 ls7 wsd0">toy problem runs throughout the book<span class="_c blank"> </span>becau<span class="_5 blank"> </span>se it is a prototype of a naturally</div><div class="t m0 x7 h13 y6b ff6 fsc fc1 sc0 ls7 ws95">unstable system that can be stabilized with the dif<span class="_1 blank"></span>ferent techniques that we expose</div><div class="t m0 x7 h13 y6c ff6 fsc fc1 sc0 ls7 wsbe">in the book.</div><div class="t m0 x1 h13 y6d ff6 fsc fc1 sc0 ls7 ws4a">We would like to acknowledge our institutions, Mines ParisTech and École des</div><div class="t m0 x7 h13 y6e ff6 fsc fc1 sc0 ls7 wsd1">Ponts ParisTech, respective<span class="_5 blank"> </span>ly, for the con\ufb01dence and the liberty that they grant us.</div><div class="t m0 x7 h13 y6f ff6 fsc fc1 sc0 ls7 wsd2">This book has bene\ufb01ted from valuable comments from<span class="_b blank"> </span>close<span class="_17 blank"> </span>colleagues, with a</div><div class="t m0 x7 h13 y70 ff6 fsc fc1 sc0 ls7 wsd3">particular mention to Jean-Philippe Chanceli<span class="_5 blank"> </span>er. Thanks to him, we have been able</div><div class="t m0 x7 h13 y71 ff6 fsc fc1 sc0 ls7 wsa8">to elaborate a series<span class="_b blank"> </span>of computer practical<span class="_b blank"> </span>works <span class="fc4 ws5d">http://cermics.e npc.fr/ scicoslab</span></div><div class="t m0 x7 h13 y72 ff6 fsc fc1 sc0 ls7 wsd4">with the free<span class="_f blank"> </span>scienti\ufb01c<span class="_f blank"> </span>software<span class="_f blank"> </span>Scicoslab that<span class="_f blank"> </span>he<span class="_f blank"> </span>has largel<span class="_5 blank"> </span>y contributed to</div><div class="t m0 x7 h13 y73 ff6 fsc fc1 sc0 ls7 wsd5">develop [16]. We feel honore<span class="_5 blank"> </span>d that Professor Jean-Michel Coron<span class="_10 blank"> </span>has accepted to</div><div class="t m0 x7 h13 y74 ff6 fsc fc1 sc0 ls7 wsd6">introduce our book, and we<span class="_10 blank"> </span>warmly thank him.</div><div class="t m0 x7 h17 y9e ff9 fs10 fc1 sc0 ls7">References</div><div class="t m0 x7 h18 y9f ff6 fsf fc1 sc0 ls7 wsd7">1. S. Dreyfus, Richard Bellman on the birth of dynamic programming. Oper. Res. <span class="ff9 wsd8">50</span><span class="wsd9">(1), 48\u201351</span></div><div class="t m0 x1 h16 ya0 ff6 fsf fc1 sc0 ls7">2002</div><div class="t m0 x7 h16 ya1 ff6 fsf fc1 sc0 ls7 wsda">2. S. Campbell, J.-P. Chancelier, R.<span class="_d blank"> </span>Nikoukhah, <span class="ffa wsdb">Modeling and Simulation in Scilab/Scicos<span class="_d blank"> </span>with</span></div><div class="t m0 x1 h16 ya2 ffa fsf fc1 sc0 ls7 wsdc">ScicosLab 4.4<span class="ff6 wsdd">, 2 edn. (Springer-Verlag, New York, 2010)</span></div><div class="t m0 x7 h16 ya3 ff6 fsf fc1 sc0 ls7 wsde">viii Foreword</div><a class="l" href="http://dx.doi.org/10.1007/978-3-642-34324-7_8"><div class="d m3" style="border-style:none;position:absolute;left:297.694000px;bottom:597.090000px;width:31.011000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" href="http://cermics.enpc.fr/scicoslab"><div class="d m3" style="border-style:none;position:absolute;left:260.220000px;bottom:417.770000px;width:125.292000px;height:9.978000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pfa" class="pf w4 ha" data-page-no="a"><div class="pc pca w4 ha"><div class="t m0 x7 h12 y15 ff9 fs6 fc1 sc0 ls7">Preface</div><div class="t m0 x7 h13 y3e ff6 fsc fc1 sc0 ls7 ws70">Control Theory is one of the most import<span class="_5 blank"> </span>ant branches of engineering science<span class="_5 blank"> </span>. There</div><div class="t m0 x7 h13 y3f ff6 fsc fc1 sc0 ls7 wsdf">are already good books on this theory. But this book is an outstanding one. The</div><div class="t m0 x7 h13 y40 ff6 fsc fc1 sc0 ls7 wse0">authors, within less than 280 pages, successfully cover the key concepts and results</div><div class="t m0 x7 h13 y41 ff6 fsc fc1 sc0 ls7 wscf">of the theory. This includes, in particular, the input\u2013output representation, Rou<span class="_5 blank"> </span>th\u2019s</div><div class="t m0 x7 h13 y42 ff6 fsc fc1 sc0 ls7 wse1">criteria, the PID compensa<span class="_5 blank"> </span>tor, gain and phase margi<span class="_5 blank"> </span>ns, stability and asymp<span class="_5 blank"> </span>totic</div><div class="t m0 x7 h13 y43 ff6 fsc fc1 sc0 ls7 wse2">stability, Lyapunov funct<span class="_5 blank"> </span>ions, controllability, linear<span class="_12 blank"> </span>state feedback, obser<span class="_5 blank"> </span>ver, out-</div><div class="t m0 x7 h13 y44 ff6 fsc fc1 sc0 ls7 ws32">put<span class="_d blank"> </span>regulat or,<span class="_12 blank"> </span>quadratic<span class="_12 blank"> </span>optimizati on,<span class="_12 blank"> </span>Kalman\u2014Bucy<span class="_12 blank"> </span>\ufb01lter,<span class="_12 blank"> </span>polynomial<span class="_12 blank"> </span>repr esen-</div><div class="t m0 x7 h13 y45 ff6 fsc fc1 sc0 ls7 wse3">tation, and disturbance rejection. The book deals with linear and nonlinea<span class="_5 blank"> </span>r control</div><div class="t m0 x7 h13 y46 ff6 fsc fc1 sc0 ls7 wse4">systems and the cont<span class="_5 blank"> </span>rol systems may<span class="_d blank"> </span>be<span class="_d blank"> </span>continuo<span class="_5 blank"> </span>us or discrete in<span class="_d blank"> </span>time.</div><div class="t m0 x1 h13 y47 ff6 fsc fc1 sc0 ls7 ws9f">I appreciate very much the excellent balance between the state space and the</div><div class="t m0 x7 h13 y48 ff6 fsc fc1 sc0 ls7 wse5">frequency-domain appro<span class="_5 blank"> </span>aches. These two appro<span class="_5 blank"> </span>aches are very important<span class="_5 blank"> </span>, but, in</div><div class="t m0 x7 h13 y49 ff6 fsc fc1 sc0 ls7 wse6">all the books I know<span class="_c blank"> </span>in Control<span class="_c blank"> </span>Theory,<span class="_c blank"> </span>one of the<span class="_c blank"> </span>two<span class="_c blank"> </span>approache<span class="_5 blank"> </span>s is always</div><div class="t m0 x7 h13 y4a ff6 fsc fc1 sc0 ls7 wse7">essentially ignored. Here,<span class="_17 blank"> </span>one clearly<span class="_17 blank"> </span>sees the usef<span class="_5 blank"> </span>ulness of each approach<span class="_17 blank"> </span>and</div><div class="t m0 x7 h13 y4b ff6 fsc fc1 sc0 ls7 ws50">how they can be used togethe<span class="_5 blank"> </span>r. A special emphasis is<span class="_10 blank"> </span>put on modeling, which<span class="_10 blank"> </span>is</div><div class="t m0 x7 h13 y4c ff6 fsc fc1 sc0 ls7 wse8">indeed a crucial step<span class="_d blank"> </span>for the appl<span class="_5 blank"> </span>ication of Control<span class="_d blank"> </span>Theory to<span class="_d blank"> </span>real life.<span class="_d blank"> </span>The proofs</div><div class="t m0 x7 h13 y4d ff6 fsc fc1 sc0 ls7 wse3">of the theorems and propositions are very clearly detailed. The notions and results</div><div class="t m0 x7 h13 y4e ff6 fsc fc1 sc0 ls7 wse9">are driven from applications. Their importanc<span class="_5 blank"> </span>e is illustrated with numerous and</div><div class="t m0 x7 h13 y4f ff6 fsc fc1 sc0 ls7 wsea">tutorial examples borr<span class="_5 blank"> </span>owed from various doma<span class="_5 blank"> </span>ins of engineering<span class="_12 blank"> </span>and science, as,</div><div class="t m0 x7 h13 y50 ff6 fsc fc1 sc0 ls7 wseb">for<span class="_17 blank"> </span>example,<span class="_c blank"> </span>\ufb02uid<span class="_17 blank"> </span>mechanics,<span class="_c blank"> </span>thermodynamics,<span class="_17 blank"> </span>cla ssical<span class="_c blank"> </span>mechanics,<span class="_17 blank"> </span>cont inuum</div><div class="t m0 x7 h13 y51 ff6 fsc fc1 sc0 ls7 ws3d">mechanics,<span class="_10 blank"> </span>chemistry,<span class="_12 blank"> </span>electricity,<span class="_10 blank"> </span>and<span class="_10 blank"> </span>biology.<span class="_12 blank"> </span>The<span class="_10 blank"> </span>inverted<span class="_12 blank"> </span>pendulum<span class="_10 blank"> </span>on<span class="_12 blank"> </span>a<span class="_10 blank"> </span>cart,</div><div class="t m0 x7 h13 y52 ff6 fsc fc1 sc0 ls7 wsec">which is present throughout the whole book, allows the reader to grasp quickly the</div><div class="t m0 x7 h13 y53 ff6 fsc fc1 sc0 ls7 wsed">big picture on man<span class="_5 blank"> </span>y control issues. At<span class="_10 blank"> </span>the end<span class="_10 blank"> </span>of each<span class="_10 blank"> </span>chapt<span class="_5 blank"> </span>er, there are illumi-</div><div class="t m0 x7 h13 y54 ff6 fsc fc1 sc0 ls7 wsa8">nating exercises.<span class="_b blank"> </span>They are also<span class="_b blank"> </span>issued from vario<span class="_5 blank"> </span>us domains of engineering<span class="_b blank"> </span>and</div><div class="t m0 x7 h13 ya4 ff6 fsc fc1 sc0 ls7 wsee">science. They show the power of the conce<span class="_5 blank"> </span>pts, results, and tools described in this</div><div class="t m0 x7 h13 ya5 ff6 fsc fc1 sc0 ls7">book.</div><div class="t m0 x1 h13 ya6 ff6 fsc fc1 sc0 ls7 wsef">This book is primarily intende<span class="_5 blank"> </span>d for engineers. Howeve<span class="_5 blank"> </span>r, it is also very inter-</div><div class="t m0 x7 h13 ya7 ff6 fsc fc1 sc0 ls7 ws43">esting<span class="_10 blank"> </span>for<span class="_10 blank"> </span>many<span class="_10 blank"> </span>other<span class="_10 blank"> </span>scientists,<span class="_b blank"> </span>including,<span class="_10 blank"> </span>mathematicians,<span class="_b blank"> </span>physicists,<span class="_10 blank"> </span>chemists,</div><div class="t m0 x7 h13 ya8 ff6 fsc fc1 sc0 ls7 wsf0">and biologists.</div><div class="t m0 x1 h13 ya9 ff6 fsc fc1 sc0 ls7 wsf1">In conclusion, I cannot prai<span class="_5 blank"> </span>se this book too much. Th<span class="_5 blank"> </span>is is a must have text.</div><div class="t m0 x7 h13 yaa ff6 fsc fc1 sc0 ls7 ws84">It will become soon a classi<span class="_5 blank"> </span>cal textbook in Con<span class="_5 blank"> </span>trol Theory.</div><div class="t m0 xf h13 yab ff6 fsc fc1 sc0 ls7 wsf2">Professor Jean-Michel Coron</div><div class="t m0 x10 h13 yac ff6 fsc fc1 sc0 ls7 ws84">Senior member of the Institut<span class="_10 blank"> </span>universitaire de Franc<span class="_5 blank"> </span>e</div><div class="t m0 x11 h16 yad ff6 fsf fc1 sc0 ls7">ix</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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