Scientific Journal of Control Engineering June 2013, Volume 3, Issue 3, PP.83-93
Reanalysis of Linear Systems with Time-delay and Actuator Saturation Xinghua Liu#, Yu Kang, Hongsheng Xi Department of Automation, School of Information Science and Technology, University of Science and Technology of China, Anhui 230027, China #
Email: salxh@mail.ustc.edu.cn
Abstract Linear systems with constant time-delay and actuator saturation are investigated in this paper. Auxiliary functions are presented based on additive decomposition approach and the relationship among them is discussed. The sufficient conditions are obtained for asymptotic stability of these systems. Furthermore, the paper gives optimal auxiliary function to make the domain of attraction larger and puts forward the algorithm to solve the problem. Finally, two numerical examples are implemented to show the effectiveness of the results. Keywords: Additive Decomposition Approach; Time-delay and Actuator Saturation; Domain of Attraction; Asymptotic Stability
1 INTRODUCTION It is well-known that time-delay occurs in many real-world control systems owing to measurement, transmission and computational delays and it is often a source of poor performance. The problem of stability and stabilization of timedelay systems has been received considerable attention over the decades. These results can be found in [2], [4], [7], [8], [10], [13]. In addition, actuator saturation is another source of system instability or performance degradation in many physical and industrial systems. To estimate the domain of attraction for control systems with actuator saturation, the methods are mainly based on Lyapunov stability theory. There has been a subject of extensive research in [6], [17], [18], [19]. In the matter of systems containing both delay and input saturation, J.M.G da Silva dealt with the saturated item through anti-windup method and obtained good conservative result in [5]. The stabilization of linear systems with time-delay and actuator saturation has been investigated by many researchers in [1], [9], [12], [14], [16]. The problem of estimating asymptotic stability regions for linear systems subjected to timedelay and actuator saturation has been studied in [11]. In this paper, we reconsider linear systems containing both delay and input saturation. We do not need the exact value of time-delay and will show that the proposed method can reduce the conservatism of result. We present auxiliary functions W2 ( ) , W3 ( ) , W4 ( ) , Wn ( ) based on additive decomposition approach mentioned in [6]. According to Lyapunov stability theory, the paper gives the algorithm to obtain the optimal auxiliary function Wi ( ) in order to optimize the domain of attraction. The remainder of this paper is organized as follows. Problem statement and the preliminaries are given in Section II; the main results and the algorithm to obtain the optimal auxiliary function are presented in Section III; two numerical examples will be given in Section IV to illustrate the effectiveness of the proposed method; the paper will be concluded in Section V.
1.1 Notations m n
n
In this paper, denotes the n dimensional Euclidean space and is for the set of all m n matrices. The notation X < Y (X > Y ), where X and Y are both symmetric matrices, means that X−Y is negative(positive) definite. I denotes the identity matrix with proper dimensions. stands for the magnitude of the domain of attraction, in the paper v means the Euclidean norm of vector v, max (min) (A) means the maximum (minimum) eigenvalue of the matrix - 83 http://www.sj-ce.org/
A , while ( )= max ( ) means the condition number. [1, n] where n
+
min ( )
for the set of all the integers from 1 to n.
Let us define B( )= y R n : y . Finally, diag{ } denotes a block-diagonal matrix determined by the corresponding elements in the brace.
2 PROBLEM STATEMENT AND PRELIMINARIES In this paper, the investigated linear systems with time-delay and actuator saturation are described by the following:
x(t )=Ax(t )+Bx(t -h)+CSat (u(t ))
(1)
where h is a constant time-delay and we do not need to know its value. Linear state feedback is described by the following: (2) u(t ) Kx(t ) n where x(t ) is the system state, u(t) is the control input, A, B and C are real constant matrices of definitively proper dimension, the linear state feedback gain K is denoted as T T T T K [ K1 : K 2 : : K m ]
K i ( i 1, 2, m ) is an n-dimension vector. The notation Sat ( ) is to denote the standard saturation function defined as u =[u1 , u2 , ,um ] : Sat (u(t )) sat (u1 ), sat (u2 ),
ui sat (ui )= ui u i
sat (um )
T
if ui ui (3)
otherwise if ui ui
We have also assumed a unity saturation level for the saturation function without loss of generality. Let us decompose the saturation nonlinear function as the sum of a linear part and another nonlinear part. Definition 2.1: Nonlinear part Dz(u(t)) = u(t) − Sat(u(t)), where Dz (u ) [dz (u1 ), dz (u2 ),
, dz (um )]T
ui ui if ui ui dz (ui )= 0 if -ui <ui ui u u if ui ui i i Making use of nonlinear part Dz(u(t)), we rewrite the system in the following form: x(t )=Ac x(t)+Bx(t -h)+CDz (Kx(t )) where Ac =A CK . Lemma 2.1: Let U
nn
and V
nn
, and let x
n
(4)
(5)
. Then we have
x UVx UV x x U V xT x T
T T 1 T T and 2 x UVx x UU + V V x, Lemma 2.2: Let W
nn
T
>0 .
be a symmetric matrix, and let x n . Then the following inequality holds: min (W )xT x xTWx max (W )xT x
The above lemmas can be seen in [3]. We need them to prove the theorems about asymptotic stability for the system. Definition 2.2: (Kim [6]) Auxiliary functions W1 ( ) , W2 ( ) are described in the following:
m ui 2 T (1 ) Ki Ki if K i ui W1 ( ) 1 Ki 0 otherwise m 2 T if K i ui ( ) K i K i W2 ( ) 1 4ui
0
otherwise - 84 http://www.sj-ce.org/
(6)
(7)
This paper creates the auxiliary functions W3 ( ) , W4 ( ) , Wi ( ) W2 ( ) to reduce the conservatism of result.
Wn ( ) , and proves that there exists in
i such that
3 MAIN RESULTS In this section, we firstly construct the auxiliary functions W2 ( ) , W3 ( ) , W4 ( ) , Wn ( ) through geometric method and obtain the sufficient condition of the asymptotic stability for the systems with time-delay and actuator saturation by those auxiliary functions. In order to reduce the conservatism of result, the paper gives the algorithm to solve the optimal Wi ( ) .
A. The Creation of Auxiliary Functions and the Law of Asymptotical Stability
We consider the polynomial function y ax which is tangent with saturated function y x ui in positive axis. Firstly, we can determine parameter a by this geometric relation. Then we create auxiliary functions and prove the law of asymptotic stability of the systems based on the polynomial function. n
y ax n Let y x ui
Then the slope of polynomial function y ax equals to 1 at the tangent point, so we have n
y nax n 1 =1 It can be determined by solving the resulting equations simultaneously and we obtain
n 1 n1 1 , n 1 1 ) y ( )n1 n uin ui nn ui n Noting in the polynomial function, we replace x with ui here. For simplicity, we consider the cases n = 2 and n = 3. We show their graphs in the following figure. The polynomial function has similar characteristic. We can obtain the following inequality in the Fig.1 obviously. a=(
dz (ui ) (
n 1 n 1 1 n ) ui ui nn
(8)
where n=2,3,4,…
FIG. 1 TWO POLYNOMIAL FUNCTIONS ARE
2
y
1 2 AND 4 1 3 WHEN ui y ui 4ui 27 ui
ui =2 AND THE PIECEWISE LINEAR FUNCTION IS dz (ui ) .
Theorem 3.1: Let us consider Dz(K(x)) and Wn ( ) which have been defined in this paper, they are satisfied Dz ( Kx) Dz （Kx) x Wn ( ) x T
where x B( ) = y R : y .
T
n
Proof: From (8) we obtain that - 85 http://www.sj-ce.org/
(9)
m
Dz T ( Kx ) Dz （Kx ) dz 2 (ui ) 1 m
( 1 m
( 1 m
( 1 m
( 1
n 1 u
i
n 1 u
i
n 1 u
i
n 1 u
i
)2n2 )2n2 )2n2 )2n2
1 n2n 1 n2n 1 n
2n
1 n
2n
ui2 n xT K iT K i x n 1
xxT
xT K iT K i x
xT K iT K i
K iT K i x
2 n 2 xT K iT K i x n
xT Wn ( ) x
This completes the proof. We define
( n 1) Wn ( ) n 1 n 1 n ui
2n2
K
m
T i
Ki
n
where n = 2, 3, 4,… , specially when n equals to 2, W2 ( ) is exactly the same with Kim in [6]. As a matter of fact, Wn ( ) is a piecewise function and it can be described by 2 n2 m n (n 1) KiT Ki n Wn ( ) 1 n 1 n ui 0
if Ki ui otherwise
According to the Lyapunov stability theorem in references [15], we prove the two following theorems which guarantee the asymptotic stability of the system through Wi ( ) . Theorem 3.2: Given the linear systems with time-delay and actuator saturation which can be described by (1) and (2), the system (1) with the control (2) is asymptotic stability for the initial state 1
x(0) B( / 2 ( P)) if there exists a positive definite matrix P with real scalars 0 and 0 such that 1
PCC T P Wi ( )+PBQ 1BT P I n 0 where Q is a given positive definite matrix and Wi ( ) , i=1,2,… ,n are defined as before. AcT P PAc +Q
Proof: The positive definite matrix Q is given, then let
V ( x ) xT Px + x T ( )Qx ( )d t
t -h
We find the time derivative of V (x) along the trajectories of the loop system (5) and deduce by lemma 2.1 and lemma 2.2.
V ( x) xT Px xPxT +xT Qx xT (t h)Qx(t h) = xT [AcT P +PAc +Q]x +xT (t h)BT Px(t )+xT (t )PBx(t h) 1 xT AcT P +PAc +Q + PCC T P x +xT (t h)BT Px +xT PBx (t h )+ Dz T (K (x))Dz (Kx ) x T (t h )Qx (t h )
1 xT AcT P +PAc +Q + PCC T P + Wi ( ) x +xT (t h)BT Px +xT PBx(t h) xT (t h)Qx(t h) - 86 http://www.sj-ce.org/
1 AcT P +PAc +Q+ PCC T P + Wi ( ) PB x x(t -h) BT P Q
= xT xT (t h)
If we want V ( x) <0, then 1 T T Ac P +PAc +Q + PCC P + Wi ( ) PB <0 BT P Q By the Schur complement lemma and choose scalar >0, i.e. 1 AcT P PAc +Q PCC T P Wi ( )+PBQ 1BT P I n 0 This completes the proof.
(10)
Furthermore, let us consider the following uncertain linear system with time-delay and actuator saturation x(t )= A+A(t ) x(t )+Bx(t -h)+ C +C (t ) Sat (u (t ))
(11)
Where A(t ) and C (t ) stand for the uncertainties. For simplicity, the uncertainties are not known but the norm bound is known and can be described as A(t ) and C (t )
t 0
Using the defined nonlinear function Dz(· ), we can also rewrite the uncertain saturated system in the following form x (t )= A+A(t ) C (t )K x (t )+Bx (t -h )
+ C +C (t ) Dz (Kx(t ))
(12)
where Ac is defined as the same with certain saturated system. We present the asymptotic stability condition of the uncertain system though this paper mainly concentrates on the certain system which can be described by (1) and (2). It is easy to note that, when A(t ) = 0 and C (t ) = 0 the uncertain system becomes the certain system. With regard to the uncertain system, we can also prove the following theorem. Theorem 3.3: Given the linear system with time-varying actuator saturation which can be described by uncertain case, the system is asymptotic stability for the initial state 1
x(0) B( / 2 ( P)) if there exists a positive definite matrix P with real scalars 0 and 0 such that 1 1 AcT P PAc +Q (1+ 2 +2 C )P 2 + PCC T P ( 2 +2 K )I + 2 K T K + Wi ( )+PBQ 1BT P I n 0
where Q is a given positive definite matrix and , , Proof: In the same way, we choose
and Wi ( ) , i=1,2,3,…,n are defined as before. t
V ( x) xT Px+ xT ( )Qx( )d t -h
Then we have V ( x)= xT [AcT P +PAc +2P(A CK )+Q]x +xT (t h)BT Px +xT PBx(t h)+2xT P (C +C )Dz (Kx) xT (t h)Qx(t h) 1 1 xT AcT P+PAc +Q+ PP+ (AT A-2AT CK )+ PCC T P
1 + K T CT CK + P(C C T +2C C T )P +xT PBx(t h)+xT (t h)BT Px+ DzT (K (x))Dz (Kx) xT (t h)Qx(t h)
1 xT AcT P+PAc +Q+ Wi ( )+ P(1+ 2 +2 C )P
1 + ( 2 +2 K )I + 2 K T K + PCC T P x+xT (t h)BT Px+xT PBx(t h) xT (t h)Qx(t h)
x B P Q x(t -h)
= xT xT (t h)
PB
T
- 87 http://www.sj-ce.org/
where stands for AcT P PAc +Q
1
(1+ 2 +2 C )P 2 + Wi ( )+
At the same with proof of (11), we obtain
1
PCC T P ( 2 +2 K )I + 2 K T K
PB <0 BT P Q By the Schur complement lemma and choose scalar >0, i.e.
1 1 AcT P PAc +Q (1+ 2 +2 C )P 2 + PCC T P ( 2 +2 K )I + 2 K T K + Wi ( )+PBQ 1BT P I n 0 (13)
This completes the proof.
B. The Size of Relation Between the Auxiliary Functions Wi ( ) In this subsection, we replace W2 ( ) with Wi ( ) and can realize the objective to reduce the conservatism of the result. By theorem 3.1 we know that the domain of attraction can be larger if Wn ( ) W2 ( ) . So we analyze the size of relation between W2 ( ) , W3 ( ) , W4 ( ) , Wn ( ) and deduce the conditions which guarantee i, s.t. Wi ( ) W2 ( ) . Wn ( ) as before, then i, s.t. Wi ( ) W2 ( ) if the following inequality is
Theorem 3.4: Given W2 ( ) , W3 ( ) , W4 ( ) , satisfied
Ki j
1
f j 1 (n)
ui
(14)
where j=2,3,… ,n and n=3,4,…, j stands for the domain attraction which is computed by W j ( ) . f j 1 (n) is described in the following
f j 1 (n)=
( j 1) j
j 1 n j
j n j
n
n n j
(n 1)
n 1 n j
Proof: Firstly, we consider Wi ( ) and W2 ( ) (n 1) Wn ( ) n 1 n 1 n ui
2 n2
K
m
T i
Ki
n
2
2 n2 16(n 1)2 n 2 = KiT Ki KiT Ki ( )2 n 4 ui n2n 1 4ui 2 n2 2 n 4 2 n 4 16( n 1) Ki ( ) W2 ( ) 2n ui n m
Suppose Wn ( ) W2 ( ) , then we can obtain n
K 1 n n2 1 i f1 (n)= 1 n 1 ui 4 n 2 (n 1) n 2 where n=3,4,…. Then we compute Wi ( ) and W3 ( ) (n 1) Wn ( ) n 1 n 1 n ui m
2 = 1 3 3ui m
Ki
2 n 6
(
2n2
K
T i
Ki
n
4
2 2n2 3 n 3 2 n 6 27 ( n 1) T T Ki Ki Ki Ki ( ) 2n ui 16n
2 n6 272 (n 1)2 n2 ) W3 ( ) ui 16n2 n
- 88 http://www.sj-ce.org/
(15)
Similarly, suppose 1
n
K 4 n 3 n n 3 1 i 3 f 2 (n)= 1 n 1 ui 27 n 3 (n 1) n 3 Then the inequality Wn ( ) W3 ( ) is true. By the recursive method, we easily obtain that 1
Ki j ui
j 1
f j 1 (n)=
( j 1) n j j
j n j
n
n n j
(n 1)
W j ( ) W2 ( )
n 1 n j
Thus we know i, s.t. Wi ( ) W2 ( ) if the condition is satisfied. Next we seek to find the optimal Wi ( ) , s.t. Wi ( ) W2 ( ) .
C. The Algorithm to Solve the Optimal Wi ( ) In the subsection B, we have deduced the condition which can reduce the conservatism of the system. And this paper will show how to find the optimal Wi ( ) in this part. We have m 2 4 3 T Ki Ki W3 ( ) 1 3 3ui 0 m 3 4 W4 ( ) 1 3 4 ui 0
Remark: The subscript of
if Ki ui otherwise
6
KT K 4 i i
if Ki ui otherwise
has omitted here for simplicity. We compute f1 (3)= 27 , f (4)= 8 and we have 2 16
1< 1<
Ki i
u
Ki i
u
f1 (3)= f 2 (4)=
3 3
27 W3 ( ) W2 ( ) 16 8
3 3
W4 ( ) W2 ( )
In fact, f j 1 (n) is a monotone decreasing function and lim f j 1 (n)=1 . That is to say, the optimal Wi ( ) must be found n in these N terms. Now we present the main idea of algorithm: Given the time-delay saturating actuator system, we can obtain W2 ( ) through stability analysis algorithm which has mentioned in [6]. Let us compute K i 2 (it is determined when ui
2 has obtained).
If the following inequality is satisfied 1<
Ki 2 27 f1 (3)= ui 16
then we can enlarge the domain of attraction of the system by taking advantage of Wi ( ) which is among WN ( ) . Otherwise the optimal auxiliary function is W2 ( ) and 2 is largest. We summarize the W3 ( ) , W4 ( ) , algorithm as follows: step1) Solve 2 based on the stability analysis algorithm mentioned in [6]. step2) Compute K i 2 , if 1< Ki 2 f (3)= 27 , then go to step3); otherwise 2 is the solution and stop. 1 ui
ui
16
step3) Set 2 , repeat the stability analysis algorithm with W3 ( ) , then obtain 3 iteratively. step4) Compute Ki 3 , if 1< Ki 3 f (4)=1.4047 , then go to step5); otherwise 3 is the solution and stop. 2 ui
ui
- 89 http://www.sj-ce.org/
step5) Set 3 , repeat the stability analysis algorithm with W4 ( ) , then obtain 4 iteratively. step6) Compute K i 4 , if 1< Ki 4 f 2 (4)=1.2875 , then go on the next step; otherwise 4 is the solution and stop. ui
ui
…… We have mentioned that the algorithm will be end in N times.
4 NUMERICAL EXAMPLES In this section, we will implement two numerical examples to show the validness of our results. Example 4.1: We consider the linear system with time-delay and actuator saturation which can be described as follows: 0 0.1 0.1 0.001 5 0 x(t )= x(t )+ x(t h)+ Sat (u (t )) 0.001 0.1 3.0 0 0 1
(16)
Then we can judge that the system (16) is unstable without control since the eigenvalues of the system matrix are 0.0968, 2.9968 . So we consider the linear state feedback control which can be described as
0.7283 0.0338 (17) x 0.0135 1.3583 0.2 0.1 . We define We determine the positive definite matrix Q = u1 =10 , u2 =30 , =1 and =0.25 . According 0.1 0.2 to the algorithm, we obtain 2 = 16.60 at first step. Next we compute u = Kx =
K1 2 1
u
=1.2103 ,
K 2 2 u2
=0.7516
That is to say 1<
Ki 2 i
u
f1 (3)=
27 1.6875 is satisfied. (i=1, 2) 16
So the algorithm will go on, set =2 , repeat the stability analysis algorithm with W3 ( ) , then obtain 3 =19.61 iteratively. Similarly, we compute K1 3 1
u
We can see
K1 3 u1
=1.4297 ,
K 2 3 u2
=0.8879
>f 2 (4)=1.4047 , so the algorithm will be stopped. But if we set
= 3 repeat the stability analysis
algorithm with W4 ( ) , then obtain 4 =19.49 iteratively. Similarly, we compute K1 4 u1
We judge that
K1 4 u1
=1.4210 ,
K 2 4 u2
=0.8825
>f3 (5)=1.2875 . In order to verify the conclusion, we still obtain
5 = 19.01 through W5 ( ) in
the stability analysis algorithm. Obviously we have 5 <4 <3 . The function W3 ( ) is the optimal auxiliary function among W2 ( ) , W3 ( ) , W4 ( ) ,
Wn ( ) .
3 is the largest one among 2 , 3 … n .
Finally, we have found the following initial state bound guaranteeing the asymptotic stability of the system (16) with the control (17). x(0) B(2.8078) with W2 (16.60) x(0) B(3.1163) with W3 (19.61) x(0) B(3.1629) with W4 (19.49) - 90 http://www.sj-ce.org/
x(0) B(3.0160) with W5 (19.01)
We show the result in the following figure:
FIG. 2 THE DOMAIN OF ATTRACTION WITH W2 ( )
IS IN RED, AND OTHER ONE WITH W ( ) IS IN BLUE. 3
So we can enlarge the domain of attraction by replacing W2 ( ) with Wi ( ) . The optimal auxiliary function is W3 ( ) . By theorem 3.4, we must verify W3 ( ) W2 ( ) . In fact, 0.0486 0.0025 and 0.0486 0.0029 W3 ( )= W2 ( )= 0.0025 0.0253 0.0029 0.0652
Obviously, it is correct. Example 4.2: Let us consider the following system 1 1 0.01 0 5 0 x(t )= x(t )+ x(t h)+ Sat (u (t )) 1 3 0 0.01 0 1
(18)
where the saturation limits are u1 =5 and u2 =2 . Then we can easily know that the above system is unstable without control since the eigenvalues of the system matrix are 1+ 3, 1 3 . The linear state feedback control can be described as
0.7283 0.0338 x 0.0135 1.3583
u = Kx =
0.2 0.1 . Besides we choose scalars We determine the positive definite matrix Q = 0.1 0.2 Firstly, we get 2 = 2.80 by the stability analysis algorithm in [6]. Then we compute K1 2 K 2 2 =0.4083 , =1.9017 u1 u2
(19)
=1 and =0.05 .
We know 1<
K1 2 u1
f1 (3)=
27 1.6875 16
f1 (3)=
27 1.6875 16
can be satisfied; but 1<
K 2 2 u2
can’t be satisfied. According to the algorithm, we know that W2 ( ) is the optimal auxiliary function among W2 ( ) , W3 ( ) ,
Wn ( ) and 2 is the largest one among
2 , 3 … n . Now we verify the conclusion, so we compute
3 =2.77 through W3 ( ) in the stability analysis algorithm. Based on theorem 3.2, we find the following initial state bound guaranteeing the asymptotic stability of the system (18) with the control (19). x(0) B(0.6154) with W2 (2.80) - 91 http://www.sj-ce.org/
x(0) B(0.6092) with W2 (2.77)
We show the result in the following figure:
FIG. 3 THE DOMAIN OF ATTRACTION WITH W2 ( )
IS IN RED, AND OTHER ONE WITH W ( ) IS IN BLUE. 3
In this case, we can’t enlarge the domain of attraction by replacing W2 ( ) with Wi ( ) (where i 2 ), because W2 ( ) has already been the optimal auxiliary function among those.
5 CONCLUSIONS In this paper, we mainly consider asymptotic stability of the linear systems with constant time-delay and actuator saturation. At the same time we estimate and enlarge the domain of attraction for the system. We construct the similar auxiliary functions W3 ( ) , W4 ( ) .... Wn ( ) . Applying additive decomposition approach, we present two theorems which guarantee the asymptotic stability of the system. In order to reduce the conservatism and obtain the larger domain of attraction for the system with time-delay and actuator saturation, we deduce the condition as a theorem which can guarantee the effectiveness for the method. Based on this, we present the algorithm to obtain the optimal Wi ( ) and i . In the end, we carry on two numerical examples to show the validness of the results. So it is a perfect result that we can enlarge the domain of attraction for the system in that condition.
ACKNOWLEDGMENT This work was supported in part by the National Natural Science Foundation of China (60935001, 61174061 and 61074033), the Doctoral Fund of Ministry of Education of China (20093402110019) and Anhui Provincial Natural Science Foundation (11040606M143) and the Fundamental Research Funds for the Central Universities and the Program for New Century Excellent Talents in University. The author would like to thank the associate editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper.
REFERENCES [1]
A.H. Glattfelder, W. Schaufelberger. Stability Analysis of Single Loop Control Systems with Saturation and Antireset-Windup Circuits. IEEE Transactions on Automatic Control, vol.28, no.12, pp.1074-1081, 2003
[2]
D. Yue and S.Won. An Improvement on Delay and Its Time-derivative Dependent Robust Stability of Time-delayed Linear Systems with Uncertainty. IEEE Transactions on Automatic Control, vol. 47, pp. 407-408, Feb. 2002
[3] H.B. Ji. Algebra Foundation of Control Theory. University of Science and Technology of China Press, Hefei, 2008 [4]
H. Zhang, D. Zhang and L. Xie. An Innovation Approach to H prediction with application to systems with delayed measurements.Automatica,Vol.40, No.7, pp.1253-1261, 2004
[5]
J.M. G da Silva, S. Tarbouriech. Antiwindup Design with Guaranteed Regions of Stability: An LMI-based Approach. IEEE Transactions on Automatic Control, vol.50, no.1, pp.106-111, 2005
[6]
J.H. Kim, Z. Bien. Robust Stability of Uncertain Linear Systems with Saturating Actuators. IEEE Transactions on Automatic - 92 http://www.sj-ce.org/
Control, vol.49, no.1, pp.202-207, 1994 [7]
J. Zhang, P. Tsiotras and C. Knopse. Stability Analysis of LPV Time-delayed Systems. International Journal of Control, vol. 75, pp.538-558, 2002
[8] K. Gu, V. Kharitonv and J. Chen. Stability of Time-Delay Systems. Boston: Brikhauser, 2003 [9]
K. Yakoubi and Y. Chitour. Linear Systems Subject to Input Saturation and Time Delay: Stabilization and Lp -stability. Proceedings of the 2nd Symposiumon Structure, System and Control, Oaxaca, Mexico, 2004
[10] L. Xie, E. Fridman and U. Shaked. Robust control of
distributed delay systems with application to combustion control. IEEE
Transactions on Automatic Control, Vol. 46, No.12, pp.1930- 1935, 2001 [11] L.X. Zhang, E.K. Boukas and A. Haidar. Delay-range-dependent Control Synthesis for Time-delay Systems with Actuator Saturation. Automatic, vol.44, pp.2691-2695, 2008 [12] M. Klai, S. Tarbouriech and C. Burgat. Some Independent-Time-Delay Stabilization of Linear Systems with Saturating Controls. Proceedings of IEEE Controlâ&#x20AC;&#x2122;94, pp.1358-1363, 1994 [13] M. Wu, Y. He and J.-H. She. New delay-dependent stability criteria and stabilizing method for neutral systems. IEEE Transactions on Automatic Control, Vol.49, pp.2266-2271, 2004 [14] P.L. Liu, H.L. Hung. Stability for Bilinear Time-Delay Systems with Saturating Actuators. IEEE International Symposium on Industrial Electronics, pp.1082-1086, 1999 [15] R.M. Dolphus, W.E. Schmitendorf. Stability Analysis for A Class of Linear Controllers under Control Constraints. Dynamics and control, vol(5), no.2, pp.191-203, 1992 [16] T.J. Su, P.L. Liu and J.T. Tsay. Stabilization of Delay-Dependence for Saturating Actuator Systems. Proceedings of the 30th Conference on Decision and Controlâ&#x20AC;&#x2122;91, pp.2891-2892, 1991 [17] T.S. Hu, Z.L. Lin. Composite Quadratic Lyapunov Functions for Constrained Control Systems. IEEE Transactions on Automatic Control, vol.48, no.3, pp.440-450, 2003 [18] Y.Y. Cao, Z.L. Lin and T.S. Hu. Stability Analysis of Linear Time-delay Systems Subject to Input Saturation. IEEE Transsactions on Circuits and Systems, vol.49, no.2, pp.233-240, 2002 [19] Y.Y. Cao, Z.L. Lin. Stability Analysis of Discrete-time Systems with Saturation by A Saturation-dependent Lyapunov Function. Automatic, vol.39, no.7, pp.1235-1241, 2003
AUTHORS 1
2
Department of Automation, University of
the B.S. and M.S. degrees in applied
Science and Technology of China (USTC).
mathematics from University of Science
He received his bachelor degree from
and Technology of China (USTC), Hefei,
Computational Mathematics of Math
China, in 1980 and 1985, respectively. He
College, Jilin University in 2009. In his
is currently the Dean of the School of
undergraduate
Information Science and Technology,
Xinghua Liu: The Ph.D. candidate at the
study,
he
got
Hongsheng Xi Professor: He received
the
scholarship for four consecutive years and took charge of a
USTC.
College Students' Innovative Plan. In his postgraduate stage, his
Communication System and Control. His research interests
research interests include stochastic estimation and control,
include stochastic control systems, discrete-event dynamic
continuous time markov decision processes, optimization and
systems, network performance analysis and optimization, and
operations research.
wireless communications.
- 93 http://www.sj-ce.org/
He
also
directs
the
Laboratory
of
Network