Motor

Prévia do material em texto

Extended State Observer Based Output Feedback Asymptotic 
Tracking Control of DC Motors 
DENG WenxiangˈLUO ChengyangˈYAO Jianyong*ˈMA DaweiˈLE Guigao 
School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China 
 
Abstract: Modeling uncertainties and measurement noise in physical servo systems severely deteriorate their tracking 
performance. In this paper, an output feedback control strategy which integrates a discontinuous robust control with extended 
state observer via feed-forward cancellation technique is proposed for high precision motion control of a dc motor. The designed 
extended state observer can estimate the unmeasured system state and the modeling uncertainties simultaneously. Since the vast 
majority of the modeling uncertainties can be estimated by the extended state observer and then compensated in the controller 
design, the task of robust control can be greatly alleviated which prevents the discontinuous output feedback controller from 
chattering. In addition, the proposed controller theoretically achieves an asymptotic output tracking performance even in the 
presence of modeling uncertainties, which is rather significant for high precision control of motion systems. Extensive 
comparative simulation results are provided to verify the high performance nature of the proposed control strategy. 
Key Words: uncertainties, robust control, output feedback, extended state observer, dc motor, motion control 
 
 
� 
1 Introduction 
Motion systems driven by electrical motors have been 
widely employed in industries, such as machine tools [1], 
robot manipulators [2], gantry systems [3], micro actuators 
[4-5], beam systems [6] and so on. With the increasing 
precision requirements, there is an urgent demand on 
developing high performance controllers for motor motion 
systems. However, various modeling uncertainties, 
including structured uncertainties (e.g., uncertain payloads) 
and unstructured uncertainties (e.g., unmodeled nonlinear 
friction, unmodeled dynamics, external disturbance, and so 
on ) always exist in physical systems. These modeling 
uncertainties often bring adverse effects to the achievable 
control performance and system stability [7], which make 
the high performance controllers design for motion systems 
be a challenging problem. In order to tackle modeling 
uncertainties in motion systems driven by electrical motors 
and improve the tracking performance, various advanced 
nonlinear controllers have been proposed. To mention a few, 
adaptive control can effectively handle structured 
uncertainties and guarantee the output tracking error 
asymptotically converge to zero. However, adaptive 
controller is rather sensitive to unstructured uncertainties. 
Once the system is subject to large unstructured 
uncertainties, parameter drifting may happen and then 
deteriorate the achievable performance. In order to handle 
both structured and unstructured uncertainties, adaptive 
robust control (ARC) was proposed by Yao and Tomizuka in 
[8] and widely employed in motion control[9-10]. The 
adaptive robust controller can obtain uniformly bounded 
output tracking error in general and asymptotic tracking 
performance in the absence of unstructured uncertainties. 
 
* Corresponding Author: Yao Jianyong, jerryyao.buaa@gmail.com 
This work was supported in part by the National Natural Science 
Foundation of China (Grant no. 51305203), in part by the Specialized 
Research Fund for the Doctoral Program of Higher Education (Grant no. 
20133219120026), China, and in part by Jiangsu Innovational Planned 
Projects for Postgraduate Research Funds (Grant no. KYLX_0334). 
However, all aforementioned control strategies are 
full-state feedback, which indicates that the velocity signal 
is also needed in the controller design for motor motion 
systems. But many mechanical systems are not equipped 
with velocity sensors, so it is impossible to get full access to 
system states. Even if the velocity sensors are installed, 
heavy measurement noise will accompany, which may also 
deteriorate the achievable performance of full-state 
feedback controllers[11]. Hence, output feedback controller 
should be designed to further improve the tracking 
performance. In [12], an output feedback adaptive robust 
controller was proposed by L. Xu et al by integrating 
adaptive robust control with a Kreisselmeier observer. But 
the output feedback controller can only guarantee uniformly 
ultimate boundedness of tracking errors in the presence of 
unstructured uncertainties. A discontinuous output feedback 
controller proposed in [13] achieves semi-global asymptotic 
tracking even in the presence of various bounded modeling 
uncertainties. However, the discontinuous function 
contained in the controller may cause severely chattering, 
which limits the employment of the controller in the 
engineering practice. In addition, the discontinuous output 
feedback controller in [13] needs the knowledge of the upper 
bound of the modeling uncertainties which is quite difficult 
to acquire. 
In this paper, an output feedback control strategy which 
integrates a discontinuous robust control with extended 
state observer via feed-forward cancellation technique is 
proposed for high precision motion control of a dc motor. 
The proposed controller theoretically achieves global 
asymptotic output tracking performance only with position 
measurement even in the presence of various bounded 
modeling uncertainties. Furthermore, with the self-tuning 
control gain, the knowledge of the bound of modeling 
uncertainties is not required in the controller design. To 
verify the high performance of the proposed controller, 
extensive comparative simulation results have been 
obtained for the motion control of a dc motor. 
Proceedings of the 34th Chinese Control Conference
July 28-30, 2015, Hangzhou, China
4262
 
2 Problem formulation and dynamic models 
The motor considered here is a current-controlled 
permanent magnet dc motor with a commercial servo 
electrical driver directly driving an inertial load. The goal is 
to have the inertial load to track any smooth motion 
trajectory as closely as possible only with position 
measurement. To satisfy the high performance 
requirements, the model is obtained to include most 
nonlinear effects such as nonlinear friction. Since the 
electric response is much faster than the mechanical 
dynamics, the current dynamic is neglected. The 
mathematical model of the system can be described as [14]: 
 ( , , )fmy k u By F t y y � ��� � � (1) 
1 2 3 4 5
( , , ) ( ) ( , , )
( ) tanh( ) [tanh( ) tanh( )]
f
f
F t y y F y f t y y
F y l l y l l y l y
 �
 � �
� � �
� � � � (2) 
where y, y� , y�� represent the position, velocity, and 
acceleration of the inertial load, respectively; m represents 
the inertial of the load; kf is the torque constant; u is the 
control input; B represents the viscous friction coefficient; 
( , , )F t y y� represents other nonlinear effects like static 
friction ( )fF y� , and the lumped disturbance ( , , )f t y y� 
which cannot be modeled precisely; the term 
1 2tanh( )l l y� approximates the Coulomb friction and the term 
3 4 5[tanh( ) tanh( )]l l y l y�� � captures the Stribeck effect; l1 and 
l3 represent different friction levels; and l2, l4, l5 are various 
shape coefficients to approximate various friction effects. 
Define the state variables as 1 2[ , ] [ , ]
T Tx x x y y � , so the 
system model (1) and (2) can be expressed in the state space 
form as 
1 2
2 2( ) ( , )
f
x x
k
x u x d x t
m
M
 
 � �
�
� (3) 
where31
2 2 2 2
2 2 2 2 4 2 5 2
( ) ( ) ( )
( , ) ( , ) /
( ) tanh( ), ( ) tanh( ) tanh( )
f f
f f
ll Bx S x P x x
m m m
d x t f x t m
S x l x P x l x l x
M � � �
 �
 �
 (4) 
Assumption 1: The desired position trajectory x1d(t) 
belongs to 3C and bounded. 
The objective is to design a bounded control input u such 
that the output y=x1 tracks x1d(t) as closely as possible only 
with position measurement in the presence of modeling 
uncertainties. 
3 Output feedback tracking controller design 
3.1 Extended state observer design 
In order to estimate the unmeasured state and handle the 
modeling uncertainty, an extended state observer (ESO)[15] 
is utilized. Before the observer and controller design, it is 
necessary to illustrate that the nominal values of physical 
parameters are used in the design, and the parameter 
deviations are lumped to the modeling uncertainty, i.e., d(t). 
According to the procedure of ESO design, the modeling 
uncertainty d(t) is extended as an additional state variable, 
i.e., define x3=d(t).Therefore, the system state x is extended 
to x=[x1, x2, x3]T, assume d(t) is differentiable and let h d � , 
(3) can be rewritten in an augmented state space form as 
follows 
1 2
2 2 3
3
( )
( )
f
x x
k
x u x x
m
x h t
M
 
 � �
 
�
�
�
 (5) 
Assumption 2: The function h(t) is bounded but its upper 
bound does not need to be known. 
Let xˆ denote the estimate of state x and define x� =x- xˆ 
as the estimation error, then a linear extended state observer 
can be designed as [16] 
1 2 0 1 1
2
2 2 0 1 1
3
3 0 1 1
ˆ ˆ ˆ3 ( )
ˆ ˆ ˆ( ) 3 ( )
ˆ ˆ( )
f
x x x x
k
x u x x x
m
x x x
Z
M Z
Z
 � �
 � � �
 �
�
�
�
 (6) 
where Ȧ0 is the bandwidth of the extended state observer 
which can be tuned by designers. 
The dynamics of the state estimation error can be 
obtained by (5) and (6) as 
1 2 0 1
2
2 3 0 1
3
3 0 1
3
3
( )
x x x
x x x
x h t x
Z
M Z
Z
 �
 � �
 �
�� � �
� �� � �
�� �
 (7) 
where 2 2ˆ( ) ( )x xM M M �� . Since the function ij(x2) is 
continuously differentiable, it must be Lipschitz with 
respect to x2. Thus we have 
2c xM d� � (8) 
 where c is a known constant. 
Define 10/ , 1, 2,3
i
i ix iH Z � � as the scaled estimation 
error, then (7) can be expressed as 
0 1 2 2
0 0
( )h tA B BMH Z H Z Z � �
�� (9) 
where 1 2 3[ , , ]
TH H H H and 
1 2
3 1 0 0 0
3 0 1 , 1 , 0
1 0 0 0 1
A B B
�ª º ª º ª º« » « » « » � « » « » « »« » « » « »�¬ ¼ ¬ ¼ ¬ ¼
 (10) 
and from its definition, we know that A is Hurwitz. 
Lemma 1 [16]: With the bounded function h(t) and (8), 
the estimated states are always bounded and there exist a 
constant 0iV ! and a finite time T1>0 such that i ix Vd� , 
i=1,2,3, 1 0t T� t ! , in addition, 0(1/ )i NV R Z , for some 
positive integer ɤ. 
Remark 1: According to the results in Lemma 1, the 
extended state observer can guarantee bounded state 
estimation error for any time t>0. Furthermore, more 
excellent results are obtained after a finite time T1, in which 
the state estimation error can be made arbitrarily small by 
increasing the bandwidth Ȧ0. That is to say, the estimated 
state 2xˆ and 3xˆ can be used in the controller design to 
approximate the unmeasured velocity signal x2 and 
compensate the modeling uncertainty x3 respectively. 
4263
 
3.2 Controller design 
Define a set of filter like quantities as[13] 
, (0) 0f f fz z g z � � � (11) 
1( 1)g p zD � � (12) 
1 1( 1)( ) fp g z g z zD � � � � � �� (13) 
where z1=x1d(t)-x1 is the output tracking error; zf and g are 
filtered signals; p is an auxiliary variable used in the filter 
implementation and Į is a positive constant. 
Define the auxiliary variable Ș(t) as 
1 1z z gK � �� (14) 
It is worth to note that the auxiliary variable Ș(t) is not 
measureable since it contains the acceleration signal. And it 
is just used to facilitate the controller design. Then the 
dynamics of the output tracking error can be described as 
1 1z z g K � � �� (15) 
Taking derivative of (12) and utilizing (13)and (14), the 
following equation can be obtained 
1( 1) fg g z zD K � � � � �� (16) 
Differentiating (14) and utilizing (5), (15) and (16), we 
have 
31
1 2 2 2
3
( ) ( )
2
f
d f f
f
k ll Bx u S x P x x
m m m m
x g z
K
DK
 � � � �
� � � �
� ��
 (17) 
Based on the design of linear extended state observer, the 
discontinuous robust controller can be given as 
3 1 1
31
2 2 2
ˆˆ[ 2 sgn( ) ( 1) ]
ˆ ˆ ˆ( ) ( )
f f
f
f f
f f f
mu x g z z z g z
k
ll BS x P x x
k k k
E D � � � � � � � �
� � �
(18) 
Here we use Eˆ to denote the estimate of the controller 
gain ȕ which will be utilized later, and define ˆE E E �� as 
the estimation error. Then design the adaptation law as 
1
ˆ sgn( )fz zE JK �� (19) 
™Š‡”‡�Ȗ is a positive constant.�
	”‘�–Š‡�ƒ†ƒ’–ƒ–‹‘�Žƒ™�‹�ሺͳͻሻǡ�•‹…‡�–Š‡�•‹‰ƒŽ�Șሺ–ሻ�
‹•�—‘™ǡ� ‹–� …‘—Ž†� ‘–� „‡� ‹’Ž‡‡–‡†�†‹”‡…–Ž›Ǥ� �—–��
ˆ”‘�ሺͳͳሻǡ�ሺͳͶሻ�ƒ†�ሺͳͷሻǡ�™‡�Šƒ˜‡�
K P P � � ����������������������������������� (20)�
™Š‡”‡�ȝ=z1+zf .�
�‹‹Žƒ”�–‘�ሾͳ͹ሿǡ�–Š‡�ƒ†ƒ’–ƒ–‹‘�Žƒ™�…ƒ�„‡�‹–‡‰”ƒ–‡†�
„›�’ƒ”–•�ƒ•�ˆ‘ŽŽ‘™•�
0
ˆ ˆ( ) (0) sgn( ) sgn( )
t
t dE E JP P J P P W � � ³ ������(21)�
so that the parameter estimate of the control gain 
implemented in (18) does not depend on the unmeasurable 
signal Ș(t) . 
Substituting (18) into (17), we can obtain 
1 2 3
1 2
ˆ sgn( ) ( 1) Bg z x x
m
N N
K DK E P D � � � � � � �
� �
� � �
 (22) 
where 
31
1 2 2 2 2 2ˆ ˆ[ ( ) ( )], [ ( ) ( )]f f f f
llN S x S x N P x P x
m m
 � � (23) 
Assumption 3: Define N= 2 /Bx m� - 3x� +N1+N2, it is 
smooth enough such that 
1 2,N NG Gd d� (24) 
where į1, į2 are unknown constant. 
3.3 Main results 
Before presenting the main results, we state the following 
lemma to be invoked in the stability analysis. 
Lemma 2 [19]: Let the auxiliary function L(t) be defined 
as follows 
( ) ( sgn( ))L t NK E P � (25) 
If the control gain is selected to satisfy the following 
condition 
1 2E G Gt � (26) 
then the following defined function P(t) is always positive 
( ) (0) (0) (0) ( )
t
o
P t N L dE P P W W � � ³ (27) 
Proof: See Appendix A. 
Theorem 1:With the adaptation law (21), the proposed 
controller (18) guarantees that all the system signals are 
bounded and asymptotic output tracking performance is 
obtained only with position measurement even in the 
presence of various modeling uncertainty, i.e., z1ĺ0 as 
tĺ’. 
Proof: See Appendix B. 
Remark 2: Results of theorem 1 indicates that the output 
feedback controller proposed in this paper can achieve 
asymptotic output tracking performance only with the 
system output signal, which can apparently eliminate the 
adverse effect caused by measurement noise. Furthermore, 
the proposed controller does not need the knowledge of the 
bound of the modeling uncertainty. Instead, the control gain 
corresponding to the bound of the modeling uncertainty is 
self-tuning with the adaptation law (21). 
4 Simulation results 
To verify the high performance of the proposed controller, 
simulation results are obtained for a dc motor system. First, 
the dc motor system parameters are listed in Table 1. 
Table 1: System Parameters 
Parameter Value 
m 0.02 kg·m2 
kf 5 N·m·V-1 
B 10 N·m·rad-1·s-1l1 0.1 N·m 
l2 700 s·rad-1 
l3 0.06 N·m 
l4 15 s·rad-1 
l5 1.5 s·rad-1 
Sampling Time 0.2ms 
 
The following three controllers are compared to verify the 
effectiveness of the proposed control strategy in this paper. 
4264
 
1) PID: This is the proportional-integral-derivative 
controller which is widely used. The controller needs 
position measurement only and makes the tracking error 
small by tuning three parameters, i.e., P-gain kp, I-gain ki, 
D-gain kd. The control gains tuned via trail- and-error 
approach are kp=100, ki=500, kd=0.1. 
2) DOFC: This is a discontinuous output feedback 
controller which integrates the controller in [13] with 
desired compensation control via feed-forward cancellation 
technique. Since the model compensation term 
corresponding to the desired trajectory is injected to the 
controller to compensate the friction effects, the DOFC 
controller designed here should alleviate the chattering 
phenomenon and improve the tracking performance. 
However, explicit knowledge of the upper bound of the 
modeling uncertainties is still required similar to [13] in this 
design. The controller is designed as follows 
1 1
31
1 1 1
[ 2 sgn( ) ( 1) ]
( ) ( )
f f
f
f d f d d
f f f
mu g z z z g z
k
ll BS x P x x
k k k
E D � � � � � � �
� � �� � �
 (28) 
where the control gain ȕ must be selected to satisfy 
condition (26), į1, į2 should be known constants. The 
control gains are presented later. 
3) OFCESO: The ESO based output feedback controller 
proposed in this paper. The control gains are given as 
follows: Į=100, Ȗ=200. The observer bandwidth parameter 
Ȧ0=1000. The initial estimate of ȕ is chosen as ˆ(0) 0E . 
The desired trajectory is a sinusoidal-like curve given by 
x1d(t)=arctan(sint)·[1-exp(-0.01t3)] rad. The desired motion 
trajectory is shown in Fig. 1. Two cases are given to show 
the performance of three controllers. 
Case 1: A constant lumped disturbance f(x,t)=1N·m is 
simulated for the dc motor system. In this case, the control 
gains of the DOFC are given as follows: Į=100, ȕ=55. The 
corresponding tracking errors under three controllers are 
shown in Fig. 2 respectively. As seen, the proposed 
OFCESO has better tracking performance than the other 
two controllers. Although the transient tracking error of 
OFCESO is a bit larger than that of DOFC, the tracking 
performance of DOFC is obtained at the sacrifice of smooth 
control input. As shown in Fig. 3, the control input of DOFC 
is severely chattering. However, with the small control gain 
ȕ adapted by (21), the control input of OFCESO is rather 
smooth, which is easier to be implemented in practice. The 
estimation of control gain ȕ is shown in Fig. 4 and the 
control input of OFCESO is shown in Fig. 5. 
0 20 40 60 80 100
-1
-0.5
0
0.5
1
times(s)
D
es
ir
ed
 T
ra
je
ct
or
y(
ra
d)
 
Fig. 1: The desired trajectory 
0 10 20 30 40 50
-5
0
5
x 10
-3
times(s)
PI
D
(r
ad
)
0 10 20 30 40 50
0
0.5
1
x 10
-3
times(s)
D
O
FC
(r
ad
)
0 10 20 30 40 50
-1
0
1
2
x 10
-3
times(s)O
FC
ES
O
(r
ad
)
 
Fig. 2: Tracking errors of three controllers for Case 1 
0 10 20 30 40 50
-3
-2
-1
0
1
2
3
times(s)
D
O
FC
 C
on
tr
ol
 In
pu
t(
V
)
 
Fig. 3: Control input of DOFC for Case 1 
0 10 20 30 40 50
0
0.05
0.1
0.15
times(s)
E E
st
im
at
io
n
 
Fig. 4: Estimation of control gain ȕ of OFCESO for Case 1 
0 10 20 30 40 50
-2
-1
0
1
2
3
times(s)
O
FC
ES
O
 C
on
tr
ol
 In
pu
t(
V
) 
 
 Fig. 5: Control input of OFCESO for Case 1 
4265
 
Case 2: A time-varying lumped disturbance f(x,t)=sint 
(N·m) is injected to the dc motor system. In this case, the 
control gain of DOFC are given as follows: Į=100, ȕ=105. 
The comparative tracking errors of three controllers are 
shown in Fig. 6. As shown, the proposed OFCESO also 
achieves perfect asymptotic tracking performance even in 
the presence of time-varying modeling uncertainties. The 
PID controller has the largest tracking error, and smaller 
tracking error is obtained of DOFC via high gain feedback, 
which leads to the chattering of the control input. The 
control input of DOFC is similar to Case 1, hence is omitted 
here. The estimation of control gain ȕ of OFCESO is shown 
in Fig. 7. Control input of OFCESO is shown in Fig. 8. 
0 20 40 60 80 100
-5
0
5
x 10
-3
times(s)
PI
D
(r
ad
)
0 20 40 60 80 100
-1
0
1
x 10-3
times(s)
D
O
FC
(r
ad
)
0 20 40 60 80 100
-1
0
1
x 10-3
times(s)
O
FC
ES
O
(r
ad
)
 
 Fig. 6: Tracking errors of three controllers for Case 2 
0 20 40 60 80 100
0
0.05
0.1
0.15
0.2
times(s)
E 
Es
tim
at
io
n
 
Fig. 7: Estimation of control gain ȕ of OFCESO for Case 2 
0 20 40 60 80 100
-3
-2
-1
0
1
2
3
times(s)
O
FC
ES
O
 C
on
tr
ol
 In
pu
t(
V
) 
 
Fig. 8: Control input of OFCESO for Case 2 
5 Conclusion 
In the paper, an output feedback controller is proposed for 
high precision motion control of a dc motor. The proposed 
output feedback controller is based on an extended state 
observer. Since the vast majority of the modeling 
uncertainties can be estimated by the observer and then 
compensated in the controller design, the task of robust 
control can be greatly alleviated which prevents the 
discontinuous output feedback controller from chattering. 
The proposed controller achieves an output tracking 
performance only with position measurement even in the 
presence of various bounded modeling uncertainties. In 
addition, the knowledge of the upper bound of the modeling 
uncertainties is not required in the controller design. 
Appendix A 
Proof of Lemma 2. Integrating (25) in time and applying 
(20), we have 
0 0
0 0
( ) ( )[ ( ) sgn( ( ))]
( ) ( )( ) sgn( ( ))
t t
t t
L d N d
d dN d d
d d
W W P W W E P W W
P W P WW W E P W WW W
 �
� �
³ ³
³ ³
 (29) 
After integrating the second and third integrals on the 
right hand of (28), we can obtain 
0 0
0 00
0
( ) ( )[ ( ) sgn( ( ))]
( )( ) ( ) ( ) ( )
( )( )[ ( ) sgn( ( ))]
( ) ( ) (0) (0) ( ) (0)
| |
t t
tt t
t
L d N d
dNN d
d
dNN d
d
t N t N t
W W P W W E P W W
WP W W P W W E P WW
WP W W E P W WW
P P E P E P
 �
� � �
 � � �
� � �
³ ³
³
³
 (30) 
Then get the upper bound of the right side of (29) as 
follows 
0 0
( ) ( ) [ ( ) ( ) ]
( ) ( ( ) ) (0) (0) (0)
t t
L d N N d
t N t N
W W P W W W E W
P E E P P
d � �
� � � �
³ ³ � (31) 
Apparently, With (24) of the assumption 3, if the control 
gain ȕ satisfies the condition (26), we can conclude from (30) 
that the function P(t) defined in (27) is positive, which 
completes the proof. 
Appendix B 
Proof of Theorem 1: Define a Lyapunov function as 
follows 
2 2 2 2 2
1
1 1 1 1 1
2 2 2 2 2f
V z z g PK EJ � � � � �
� (32) 
After taking derivative of (31) and substituting (11), (15), 
(16), (22) and (25), we have 
1 1
1 1
( ) ( ) [ ( 1)
ˆ] [ sgn( ) ( 1) ]
1 ˆ[ sgn( )]
f f
f
V z z g z z g g g
z z N g z
N
K D K
K DK E P D
K E P EEJ
 � � � � � � � � � �
� � � � � � � � � �
� �
�
��
(33) 
With the adaptation law (19), we have 
2 2 2 2
1 0fV z z g WDK � � � � � d� (34) 
Therefore, 0t� ! , V(t)”V(0), which leads to z1, zf, g, 
Ș, E�ęL’. Furthermore, integrating (33) in time, 
0 0
( ) ( ) (0) ( ) (0)
t t
W d V d V V t VW W W W � � d³ ³ � (35) 
which implies that WęL2. From (11), (15), (16) and (22), 
the derivativeof W is bounded thus W is uniformly 
continuous. By Barbalat’s Lemma [18],Wĺ0 as tĺ’ and 
then z1ĺ0 as tĺ’, i.e., asymptotic output tracking is 
achieved. 
4266
 
References 
[1] W.-S. Huang, C.-W. Liu, P.-L. Hsu, and S.-S. Yeh, Precision 
control and compensation of servomotors and machine tools 
via the disturbance observer, IEEE Trans. on Industrial 
Electronics, 57(1): 420-429, 2010. 
[2] A. Z. Shukor, and Y. Fujimoto, Direct drive position control 
of a spiral motor as a monoarticular actuator, IEEE Trans. on 
Industrial Electronics, 61(2): 1063-1071, 2014. 
[3] C. Hu, B. Yao, and Q. Wang, Coordinated adaptive robust 
contouring controller design for an industrial biaxial 
precision gantry, IEEE/ASME Trans. on Mechatronics, 
15(5): 728-735, 2010. 
[4] H.-W. Chow, and N. C. Cheung, Disturbance and response 
time improvement of submicrometer precision linear motion 
system by using modified disturbance compensator and 
internal model reference control, IEEE Trans. on Industrial 
Electronics, 60(1): 139-150, 2013. 
[5] P. A. Sente, F. M. Labrique, and P. J. Alexandre, Efficient 
control of a piezoelectric linear actuator embedded into a 
servo-valve for aeronautic applications, IEEE Trans. on 
Industrial Electronics, 59(4): 1971-1979, 2012. 
[6] L. Lu, Z. Chen, B. Yao, and Q. Wang, A two-loop 
performance oriented tip-tracking control of a 
linear-motor-driven flexible beam system with experiments, 
IEEE Trans. on Industrial Electronics, 60(3): 1011-1022, 
2013. 
[7] J. Yang, A. Zolotas, W-H. Chen, K. Michail, S. Li, Robust 
control of nonlinear MAGLEV suspension system with 
mismatched uncertainties via DOBC approach, ISA Trans., 
50(3): 389-396, 2011. 
[8] B. Yao, and M. Tomizuka, Adaptive robust control of SISO 
nonlinear systems in a semi-strict feedback form, Automatica, 
33(5): 893–900, 1997. 
[9] J. Yao, Z. Jiao, and D. Ma, Adaptive robust control of DC 
motors with extended state observer, IEEE Trans. on 
Industrial Electronics, 61(7): 3630–3637, 2014. 
[10] Z. Chen, B. Yao, and Q. Wang, Adaptive robust precision 
control of linear motors with integrated compensation of 
nonlinearities and bearing flexible modes, IEEE Trans. on 
Industrial Informatics, 9(2): 965-973, 2013. 
[11] J. Yao, Z. Jiao, and D. Ma, Extended state observer based 
output feedback nonlinear robust control of hydraulic 
systems with backstepping, IEEE Trans. on Industrial 
Electronics, 61(11): 6285-6293, 2014. 
[12] L. Xu, and B. Yao, Output feedback adaptive robust 
precision motion control of linear motors, Automatica, 37(7): 
1029–1039, 2001. 
[13] B. Xian, M. S. de Queiroz, D. M. Dawson, and M. L. 
McIntyre, A discontinuous output feedback controller and 
velocity observer for nonlinear mechanical systems, 
Automatica, 40(4): 695-700, 2004. 
[14] J. Yao, Z. Jiao, and D. Ma, RISE-based precision motion 
control of dc motors with continuous friction compensation, 
IEEE Trans. on Industrial Electronics, 61(12): 7067-7075, 
2014. 
[15] J. Han, From PID to active disturbance rejection control, 
IEEE Trans. on Industrial Electronics, 56(3): 900-906, 
2009. 
[16] Q. Zheng, L. Gao, and Z. Gao, On stability analysis of active 
disturbance rejection control for nonlinear time-varying 
plants with unknown dynamics, in Proceedings of IEEE 
Conference on Decision and Control, 2007: 3501-3506. 
[17] P. M. Patre, W. MacKunis, C. Makkar, and W. E. Dixon, 
asymptotic tracking for systems with structured and 
unstructured uncertainties, IEEE Trans. on Control System 
Technology, 16(2): 373-379, 2008. 
[18] M. Kristic, I. Kanellakopoulos, and P. V. Kokotovic, 
Nonlinear and adaptive control design, New York: Wiley, 
1995. 
[19] B. Xian, D. M. Dawson, M. S. de Queiroz, and J. Chen, A 
continuous asymptotic tracking control strategy for uncertain 
nonlinear systems, IEEE Trans. on Automatic Control, 49(7): 
1206-1211, 2004. 
 
4267