- Email: [email protected]
Delay-dependent passive control of linear systems with nonlinear perturbation
Journal of Systems Engineering and Electronics Vol. 19, No. 2, 2008, pp.346-350
Delay-dependent passive control of linear systems with nonlinear perturbation∗ Li Caina & Cui Baotong Research Center of Control Science and Engineering, Southern Yangtze Univ., Wuxi 214122, P. R. China (Received October 4, 2006)
Abstract: The problem of delay-dependent passive control of a class of linear systems with nonlinear perturbation and time-varying delay in states is studied. The main idea aims at designing a state-feedback controller such that for a time-varying delay in states, the linear system with nonlinear perturbation remains robustly stable and passive. In the system, the delay is time-varying. And the derivation of delay has the maximum and minimum value. The time-varying nonlinear perturbation is allowed to be norm-bounded. Using the effective linear matrix inequality methodology, the sufficient condition is primarily obtained for the system to have robust stability and passivity. Subsequently the existent condition of a state feedback controller is given, and the explicit expression of the controller is obtained by means of the solution of linear matrix inequalities (LMIs). In the end, a numerical example is given to demonstrate the validity and applicability of the proposed approach.
Keywords: passive control, delay-dependent, time-varying delay.
1. Introduction In recent years, increasing attention has been paid to the passive control of time-delay systems. Using the classical definition of passivity and positive realness, the problem of passive control for a class of linear systems with delay in states is considered in Refs. [2, 3, 8, 10]. Passive control of nonlinear systems is considered in Ref. [4]. The problem of passive control for a class of discrete systems with time-delay is investigated in Refs. [6,11]. Of late, the definition of passivity is also used in singular systems to research on the problem for passive and stable of systems[7] . In Ref. [9] the delay-dependent passive control of linear systems with delay, is discussed. Ref. [12] deals with the problem of observer-based passive control of a class of uncertain linear systems with time-delay in states. However, little attention is paid to the delaydependent passive control of the linear systems with nonlinear perturbation. This motivates the present research on designing a robust, time-dependent state feedback controller for linear systems, with time varying delay in states. In the delay-dependent approach,
the delay is taken into consideration in the process of designing the desired controllers. Hence, the delaydependent approach is generally regarded as being less conservative than the delay-independent, especially in a situation where delays are small. This article mainly deals with the problem of delaydependent passive control of a class of linear systems with nonlinear perturbation and time-varying state delay. The main aim is to design a state-feedback controller such that, for time-varying in states, the linear system remains robustly stable and passive. In this system, the delay is the function of time, and the derivation of time-delay has the maximum and minimum value. The time-varying nonlinear functions are allowed to satisfy norm-bounded. Using the effective matrix inequality methodology, the conditions for the existence of a solution to the earlier mentioned problem are obtained. In the end, a numerical example and the simulation for the trajectory of states are given to demonstrate the validity and applicability of the proposed approach. Notation Throughout the article, AT , A−1 denote the transpose, the inverse; n denotes n-dimens-
* This project was supported by the National Natural Science Foundation of China (60674026; 60574051).
Delay-dependent passive control of linear systems with nonlinear perturbation ional Euclidean space, n×m is the set of all the n× m real matrices. I denotes the identity matrix with appropriate order. The notation X > 0(, <, 0) denotes a symmetric positive definite (positive semidefinite, negative, negative semidefinite) matrix X.
Definition 1 The dynamical systems (Σ 0 ) is called strictly passive if there exists a scalar β 0 such that for all tp 0 and for all x (t, 0)
(2)
x(t) = ϕ(t),
t ∈ [−τ, 0]
(3)
d˙ (t) µ < 1
⎡ φ1
⎢ ⎢ BTP − C ⎢ 1 ⎢ ⎢ P ⎢ ⎢ ⎢ τ BT P ⎢ ⎢ ⎢ τ BT P ⎢ ⎢ ⎢ τ BT P ⎣ τ BT P
0
2AT B ε−1 AT A + εB T B
⎤ P
τP B
τP B
τP B
τP B
τ B1T Q3 B1 − B2T − B2
0
0
0
0
0
0
−ε1 I
0
0
0
0
0
0
−τ Q1
0
0
0
0
0
0
−τ Q2
0
0
0
0
0
0
−τ Q3
0
0
0
0
0
0
−τ δI
φ1 = P (A + B) + (A + B) P + ε1 α2 I+ τ B T Q2 B τ AT Q1 A + τ δα2 I + 1−µ
(8)
To design the expected controller, a lemma is first given to offer the theoretical basis for achieving the desired design goal. Lemma 1 The system (Σ0 ) is said to be strictly passive, if and only if B2 + B2T > 0 and there exist the symmetric positive definite matrixes P, Q1 , Q2 , Q3 , and scalars ε1 > 0, δ > 0, satisfying the linear matrix inequality (LMI)
P B1 − C T
T
(6)
3. Main results
(4)
(5)
where
ω T (s) ω (s)ds
Fact 2 For any matrices with appropriate dimensions A, B and a scalar ε > 0, it follows
τ, µ are the known constants. f (x (t) , t) ∈ n is the unknown nonlinear time-varying function, which satisfies f (0, t) = 0. Assumption 1 The system matrix A is asymptotically stable. Assumption 2 f (x (t) , t) α x (t)
tp
where β is a constant with regard to the initial condition of the system. Fact 1 (Schur complement) Given the constant matrices Ω1 , Ω2 , Ω3 , where Ω1 = Ω1T and 0 < Ω2 = Ω2T , then, Ω1 + Ω3T Ω2−1 Ω3 < 0 holds if and only if ⎤ ⎤ ⎡ ⎡ Ω1 Ω3T −Ω2 Ω3 ⎦ < 0, ⎣ ⎦<0 ⎣ (7) Ω3 −Ω2 Ω3T Ω1
where x(t) ∈ n is the state vector, z(t) ∈ p is the controlled output, ω (t) ∈ l is the disturbing input, ϕ(t) is a continuous vector-valued initial function. A, B, B1 , B2 , C are the known constant matrices with appropriate dimensions. d (t) is the delay function, which satisfies 0 d (t) τ,
∀ω ∈ L2 [0, ∞]
0
z(t) = Cx(t) + B2 ω(t)
ω T (s) z (s) ds −β
o
Consider the following linear system with time-delay states x(t) ˙ = Ax(t) + Bx(t − d(t))+ (1)
tp
2
2. Description of the problem and facts
f (x(t), t) + B1 ω(t)
347
Proof lows
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Define a Lyapunov functional V (x) as folV (x) = V1 (x) + V2 (x)
where
(9)
(10)
348
Li Caina & Cui Baotong V1 (x) = xT (t) P x (t)
0 t V2 (x) = xT (s)P1 x (s) dsdθ+
0
−τ
−τ
0
t
h (x) = −2xT (t) P B
−τ
ω T (s) P3 ω (s) dsdθ+ xT (s)P2 x (s) dsdθ
−d(t)
2 T
δα x (t + s) x (t + s) + xT (t + s − d (t + s))
t−d−θ
For all t d(t), the following is obtained
t x (t − d (t)) = x (t) − x˙ (s)ds =
T
T
B Q2 Bx (t + s − d (t + s)) + ω (t + s) B1T Q3 B1 ω (t
t−d(t)
x (t) −
[Ax(s) + Bx(s − d(s))+
f (x (s) , s) + B1 ω (s)]ds −1 −1 −1 I B T P x (t) + τ xT (t) P B Q−1 1 + Q2 + Q3 + δ
0 [xT (t + s) AT QAx (t + s) +
t+θ
0
t t−d(t)
t+θ
+ s)]ds
·
· (15)
Here it is set that
t
[Ax (s) + Bx (s − d (s)) +
P1 = AT Q1 A + δα2 I, P3 = B1T Q3 B, 1 B T Q2 B P2 = 1−µ
t−d(t)
f (x (s) , s) + B1 ω (s)]ds Thus, it follows that
It follows that x˙ (t) = (A + B) x (t) + f (x (t) , t) +
t B1 ω (t) − B [Ax (s) + f (x (s) , s) +
V˙ (x (t)) − 2z T (t) ω (t) = V˙ 1 (x (t)) + V˙ 2 (x (t)) − 2z T (t) ω (t)
t−d(t)
T
Bx (s − d (s)) + B1 ω (s)]ds
(11)
t ∈ [−2τ, 0]
(12)
x (t) = ϕ (t) , Then
xT (t) P B1 ω (t) + ω T (t) B1T P x (t) +
V˙ 1 (x) = x˙ T (t) P x (t) + xT (t) P x˙ (t) T
T
x (t) [P (A + B)+(A + B)
2 2 P +ε−1 1 P +ε1 α I]x(t)+
T
xT (t)P B1 ω (t) + ω (t) B1T P x (t) + h (x)
(13)
V˙ 2 (x) = τ xT (t) (P1 + P2 ) x (t) + τ ω T (t) P3 ω (t) −
0 xT (t + θ)P1 x (t + θ) dθ−
−τ 0
−τ
ω T (t + θ)P3 ω (t + θ) dθ−
1 − d˙ (t)
0
−τ
·
xT (t − d + θ)
(1 − µ)
−τ
0
−τ 0
−τ
ω T (t + θ) P3 ω (t + θ) dθ−
xT (t − d + θ) P2 x (t − d + θ) dθ
τ ω T (t) B1T Q3 B1 ω (t) − xT (t) C T ω (t) − ω T (t) Cx (t) − ω T (t) B + B T ω (t) = ⎤T ⎡ ⎡ ⎤ x (t) φ2 P B1 − C T ⎦ ⎣ ⎣ ⎦ ω (t) B1T P − C τ B1T Q3 B1 − B2T − B2 ⎤ ⎡ x (t) ⎦ ⎣ ω (t) (16) where
·
T
P2 x (t − d + θ) dθ τ xT (t) (P1 + P2 ) x (t) +
0 T τ ω (t) P3 ω (t) − xT (t + θ) P1 x (t + θ) dθ−
2 2 xT (t) [P (A + B) + (A + B) P + ε−1 1 P + ε1 α I+ −1 −1 −1 τ P B Q−1 I BTP + 1 + Q2 + Q3 + δ τ B T Q2 B]x (t) + τ AT Q1 A + τ δα2 I + 1−µ
(14)
2 2 φ2 = P (A + B) + (A + B) P + ε−1 1 P + ε1 α I+ −1 −1 −1 τ P B Q−1 I BTP + 1 + Q2 + Q3 + δ τ B T Q2 B τ AT Q1 A + τ δα2 I + 1−µ ⎤ ⎡ P B1 − C T φ2 ⎦<0 If ⎣ B1T P − C τ B1T Q3 B1 − B2T − B2 then V˙ (x (t)) − 2z T (t) ω (t) < 0, as V (x) > 0 for
x = 0, it follows that as tp → ∞ the system, (Σ1 ) is
Delay-dependent passive control of linear systems with nonlinear perturbation
349
strictly passive. Using the famous Schur complement, the Lemma can easily be obtained. Remark 1 The main method used in the proof of Lemma 1 is that, suppose the nonlinear function satisfies the norm-bounded, then the nonlinear function can be transformed into the linear function. Next, a state-feedback controller will be designed for the system. Consider the system as follows
(19)
u (t) = Kx (t)
where K is an appropriate dimension matrix to be determined. Substituting (19) into (Σ1 ), the following closed-loop system can be obtained (Σ2 )x˙ (t) = (A + B3 K)x (t) + Bx (t − d (t)) +
(Σ1 )x˙ (t) = Ax (t) + Bx (t − d (t)) + f (x (t) , t) + B1 ω (t) + B3 u(t)
(17)
z (t) = Cx (t) + B2 ω (t)
(18)
where u(t) ∈ m is the controlled input. The statefeedback passive controller to be designed is as follows ⎡
(20)
z (t) = Cx (t) + B2 ω (t)
(21)
Theorem 1 The system (Σ2 ) is said to be strictly passive, if and only if B2 + B2T > 0 and there exist the symmetric positive definite matrices X, Q1 , Q2 , Q3 , scalars ε1 > 0, δ > 0, and matrix W , which satisfy the LMI (22).
B1 − XC T
X
XAT
X
XB T
τ B1T Q3 B1 − B2T − B2
0
0
0
0
0
0
0
0
0
φ3
⎢ ⎢ B T − CX ⎢ 1 ⎢ ⎢ X ⎢ ⎢ ⎢ ⎢ AX ⎢ ⎢ ⎢ X ⎢ ⎣ BX
f (x (t) , t) + B1 ω (t)
0
−
1 −1 ε I α2 1 0
0
0
1 − Q−1 τ 1 0
0
0
0
0
where φ3 = (A + B) X + B3 W + X (A + B) + W T B3T + −1 −1 −1 −1 I)B T ε−1 1 I + τ B(Q1 + Q2 + Q3 + δ
W = KX,
X = P −1
(23)
Proof Observing the form of the system (Σ2 ), it can be seen that it is similar to the form of the system, hence, the proof of Theorem 1 is similar to Lemma 1 too. Substituting the parameters of the system (Σ2 ) into LMI (9), and then through the basic matrix transition and using the Schur complement twice, the LMI can easily be obtained LMI (22). Remark 2 This article only gives the state-feed controller, but a dynamic state-feed controller is desired. Therefore, a lot of work on the researches for the dynamic instance are still to be done.
4. Example In this section, the obtained theoretical result is illustrated through a simulation example. Consider the
−
1 −1 δ I τ α2 0
⎤
0 1 − µ −1 Q2 − τ
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(22)
system (Σ2 ) with the parameters given by A =
1 −1.2 0.1 −0.6 0.7 , , B = , B1 = 1 −0.1 −1.0 −1.0 −0.8
8 , B2 = 10, C = 1 1 , α = 0.1, τ = 1, B3 = 8
1 0 , is chosen, and then using the µ = 0.5.Q3 = 0 1 Matlab LMI Control Toolbox to solve the LMIs (22) and (23), the solution are obtained as follows ⎤ ⎤ ⎡ ⎡ 0.088 4 0.078 7 2.953 5 −1.808 2 ⎦ ⎦ , Q1=⎣ X=⎣ 0.078 7 0.223 9 −1.808 2 5.350 0 W= −1.784 6
−2.229 3
⎡ , Q2=⎣
⎤ 0.051 5 0.056 8 0.056 8 0.290 9
δ = 4.921 5, ε1 = 4.898 0 K = W ∗ X −1 = −1.083 5 −0.782 9 . The desired controller is u (t) = −1.083 5 −0.782 9 x (t).
⎦
350
Li Caina & Cui Baotong
Choose the perturbation f (x(t), t) = e−x(t) . The simulation for the system states are as follows. From Figs. 1 and 2, it can be easily seen that the designed controller is able to guarantee that the closedloop systems are stable and passive.
[4] Feng C B, Zhang Kanjian, Fei Shumin. Passivity-based design of robust control systems. Acta Automatica Sinica, 1999, 25 (5): 577–582. [5]
Shieh C S. H∞ control via output feedback for linear systems with time-varying delays in state and control input. Technical Commmunication, 2002, 28: 649–357.
[6]
Guan X P, Long C N, Duan Guangren. Robust passive control for discrete time-delay systems. Acta Automatica Sinica, 2002, 28 (1): 1–4.
[7] Dong X Z, Zhang Q L, Guo Kai. Passive control for class of nonlinear singular systens. Computing Technology and Automation, 2003, 22(3): 4–6 [8] Magdi S M, Abdulla I. Passivity and passification of timedelay systems. Mathematical Analysis and Applications, 2004, 292: 247–258. Fig. 1
Curve of the system states when ω (t) = 0
[9] Zhang X M, Wu Min. Delay-depentent passive control for linear systems with delay. Control Theory & Applications, 2005, 22 (3): 391–401. [10] Li C N, Cui B T. Passive control of uncertain linear systems with time-varying delay in states. Proc. of the 25th Chinese Control Conference, 2006: 16–20. [11] Cui B T, Hua M G. Robust passive control for uncertain discrete-time systems with time-varying delays.
Chaos,
Solitons and Fractals, 2006, 29: 331–341. [12] Cui B T, Hua M G. Observer-based passive control of linear time-delay systems with parametric uncertainty. Chaos, Fig. 2
Curve of the system states when ω (t) = 0
5. Conclusions The problem of delay-dependent passive control of linear systems with nonlinear perturbation and timevarying state delay is investigated in this article. Using the LMI, the static state-feedback controller is obtained. In the end, a numerical example and the simulation are provided, to show the applicability of the developed result.
References [1] Yu L. Robust control-LMI. Beijing: Tsinghua University Press, 2002; 174–178. [2] Magdi S M, Xie Lihua. Stability and positive realness of time-delay systems. Journal of Mathematical Analysis and Applications, 1999, 239: 7–19. [3] Yu L, Chen G D. Passive control of linear time-delay systems. Control Theory and Applications, 1999, 16 (1): 130– 133 (in Chinese).
Solitons and Fractals, 2007, 32: 160–167.
Li Caina was born in 1981. She is now pursuing the M. S. degree in the Research Center of Control Science and Engineering, Southern Yangtze University, China. Her current research interest includes robust passive control of linear system. E-mail: licaina [email protected] Cui Baotong was born in 1960. He received the Ph. D. degree in control theory and control engineering from the College of Automation Science and Engineering, South China University of Technology, China, in 2003. He was a post-doctoral fellow at Shanghai Jiaotong University, China, from 2003 to 2005. He joined the Southern Yangtze University, China in 2003, where he is a full professor for College of Communication and Control Engineering. His current research interests include systems analysis, stability theory, artificial neural networks, impulsive control, and chaos synchronization. E-mail: [email protected]
Delay-dependent passive control of linear systems with nonlinear perturbation∗ Li Caina & Cui Baotong Research Center of Control Science and Engineering, Southern Yangtze Univ., Wuxi 214122, P. R. China (Received October 4, 2006)
Abstract: The problem of delay-dependent passive control of a class of linear systems with nonlinear perturbation and time-varying delay in states is studied. The main idea aims at designing a state-feedback controller such that for a time-varying delay in states, the linear system with nonlinear perturbation remains robustly stable and passive. In the system, the delay is time-varying. And the derivation of delay has the maximum and minimum value. The time-varying nonlinear perturbation is allowed to be norm-bounded. Using the effective linear matrix inequality methodology, the sufficient condition is primarily obtained for the system to have robust stability and passivity. Subsequently the existent condition of a state feedback controller is given, and the explicit expression of the controller is obtained by means of the solution of linear matrix inequalities (LMIs). In the end, a numerical example is given to demonstrate the validity and applicability of the proposed approach.
Keywords: passive control, delay-dependent, time-varying delay.
1. Introduction In recent years, increasing attention has been paid to the passive control of time-delay systems. Using the classical definition of passivity and positive realness, the problem of passive control for a class of linear systems with delay in states is considered in Refs. [2, 3, 8, 10]. Passive control of nonlinear systems is considered in Ref. [4]. The problem of passive control for a class of discrete systems with time-delay is investigated in Refs. [6,11]. Of late, the definition of passivity is also used in singular systems to research on the problem for passive and stable of systems[7] . In Ref. [9] the delay-dependent passive control of linear systems with delay, is discussed. Ref. [12] deals with the problem of observer-based passive control of a class of uncertain linear systems with time-delay in states. However, little attention is paid to the delaydependent passive control of the linear systems with nonlinear perturbation. This motivates the present research on designing a robust, time-dependent state feedback controller for linear systems, with time varying delay in states. In the delay-dependent approach,
the delay is taken into consideration in the process of designing the desired controllers. Hence, the delaydependent approach is generally regarded as being less conservative than the delay-independent, especially in a situation where delays are small. This article mainly deals with the problem of delaydependent passive control of a class of linear systems with nonlinear perturbation and time-varying state delay. The main aim is to design a state-feedback controller such that, for time-varying in states, the linear system remains robustly stable and passive. In this system, the delay is the function of time, and the derivation of time-delay has the maximum and minimum value. The time-varying nonlinear functions are allowed to satisfy norm-bounded. Using the effective matrix inequality methodology, the conditions for the existence of a solution to the earlier mentioned problem are obtained. In the end, a numerical example and the simulation for the trajectory of states are given to demonstrate the validity and applicability of the proposed approach. Notation Throughout the article, AT , A−1 denote the transpose, the inverse; n denotes n-dimens-
* This project was supported by the National Natural Science Foundation of China (60674026; 60574051).
Delay-dependent passive control of linear systems with nonlinear perturbation ional Euclidean space, n×m is the set of all the n× m real matrices. I denotes the identity matrix with appropriate order. The notation X > 0(, <, 0) denotes a symmetric positive definite (positive semidefinite, negative, negative semidefinite) matrix X.
Definition 1 The dynamical systems (Σ 0 ) is called strictly passive if there exists a scalar β 0 such that for all tp 0 and for all x (t, 0)
(2)
x(t) = ϕ(t),
t ∈ [−τ, 0]
(3)
d˙ (t) µ < 1
⎡ φ1
⎢ ⎢ BTP − C ⎢ 1 ⎢ ⎢ P ⎢ ⎢ ⎢ τ BT P ⎢ ⎢ ⎢ τ BT P ⎢ ⎢ ⎢ τ BT P ⎣ τ BT P
0
2AT B ε−1 AT A + εB T B
⎤ P
τP B
τP B
τP B
τP B
τ B1T Q3 B1 − B2T − B2
0
0
0
0
0
0
−ε1 I
0
0
0
0
0
0
−τ Q1
0
0
0
0
0
0
−τ Q2
0
0
0
0
0
0
−τ Q3
0
0
0
0
0
0
−τ δI
φ1 = P (A + B) + (A + B) P + ε1 α2 I+ τ B T Q2 B τ AT Q1 A + τ δα2 I + 1−µ
(8)
To design the expected controller, a lemma is first given to offer the theoretical basis for achieving the desired design goal. Lemma 1 The system (Σ0 ) is said to be strictly passive, if and only if B2 + B2T > 0 and there exist the symmetric positive definite matrixes P, Q1 , Q2 , Q3 , and scalars ε1 > 0, δ > 0, satisfying the linear matrix inequality (LMI)
P B1 − C T
T
(6)
3. Main results
(4)
(5)
where
ω T (s) ω (s)ds
Fact 2 For any matrices with appropriate dimensions A, B and a scalar ε > 0, it follows
τ, µ are the known constants. f (x (t) , t) ∈ n is the unknown nonlinear time-varying function, which satisfies f (0, t) = 0. Assumption 1 The system matrix A is asymptotically stable. Assumption 2 f (x (t) , t) α x (t)
tp
where β is a constant with regard to the initial condition of the system. Fact 1 (Schur complement) Given the constant matrices Ω1 , Ω2 , Ω3 , where Ω1 = Ω1T and 0 < Ω2 = Ω2T , then, Ω1 + Ω3T Ω2−1 Ω3 < 0 holds if and only if ⎤ ⎤ ⎡ ⎡ Ω1 Ω3T −Ω2 Ω3 ⎦ < 0, ⎣ ⎦<0 ⎣ (7) Ω3 −Ω2 Ω3T Ω1
where x(t) ∈ n is the state vector, z(t) ∈ p is the controlled output, ω (t) ∈ l is the disturbing input, ϕ(t) is a continuous vector-valued initial function. A, B, B1 , B2 , C are the known constant matrices with appropriate dimensions. d (t) is the delay function, which satisfies 0 d (t) τ,
∀ω ∈ L2 [0, ∞]
0
z(t) = Cx(t) + B2 ω(t)
ω T (s) z (s) ds −β
o
Consider the following linear system with time-delay states x(t) ˙ = Ax(t) + Bx(t − d(t))+ (1)
tp
2
2. Description of the problem and facts
f (x(t), t) + B1 ω(t)
347
Proof lows
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Define a Lyapunov functional V (x) as folV (x) = V1 (x) + V2 (x)
where
(9)
(10)
348
Li Caina & Cui Baotong V1 (x) = xT (t) P x (t)
0 t V2 (x) = xT (s)P1 x (s) dsdθ+
0
−τ
−τ
0
t
h (x) = −2xT (t) P B
−τ
ω T (s) P3 ω (s) dsdθ+ xT (s)P2 x (s) dsdθ
−d(t)
2 T
δα x (t + s) x (t + s) + xT (t + s − d (t + s))
t−d−θ
For all t d(t), the following is obtained
t x (t − d (t)) = x (t) − x˙ (s)ds =
T
T
B Q2 Bx (t + s − d (t + s)) + ω (t + s) B1T Q3 B1 ω (t
t−d(t)
x (t) −
[Ax(s) + Bx(s − d(s))+
f (x (s) , s) + B1 ω (s)]ds −1 −1 −1 I B T P x (t) + τ xT (t) P B Q−1 1 + Q2 + Q3 + δ
0 [xT (t + s) AT QAx (t + s) +
t+θ
0
t t−d(t)
t+θ
+ s)]ds
·
· (15)
Here it is set that
t
[Ax (s) + Bx (s − d (s)) +
P1 = AT Q1 A + δα2 I, P3 = B1T Q3 B, 1 B T Q2 B P2 = 1−µ
t−d(t)
f (x (s) , s) + B1 ω (s)]ds Thus, it follows that
It follows that x˙ (t) = (A + B) x (t) + f (x (t) , t) +
t B1 ω (t) − B [Ax (s) + f (x (s) , s) +
V˙ (x (t)) − 2z T (t) ω (t) = V˙ 1 (x (t)) + V˙ 2 (x (t)) − 2z T (t) ω (t)
t−d(t)
T
Bx (s − d (s)) + B1 ω (s)]ds
(11)
t ∈ [−2τ, 0]
(12)
x (t) = ϕ (t) , Then
xT (t) P B1 ω (t) + ω T (t) B1T P x (t) +
V˙ 1 (x) = x˙ T (t) P x (t) + xT (t) P x˙ (t) T
T
x (t) [P (A + B)+(A + B)
2 2 P +ε−1 1 P +ε1 α I]x(t)+
T
xT (t)P B1 ω (t) + ω (t) B1T P x (t) + h (x)
(13)
V˙ 2 (x) = τ xT (t) (P1 + P2 ) x (t) + τ ω T (t) P3 ω (t) −
0 xT (t + θ)P1 x (t + θ) dθ−
−τ 0
−τ
ω T (t + θ)P3 ω (t + θ) dθ−
1 − d˙ (t)
0
−τ
·
xT (t − d + θ)
(1 − µ)
−τ
0
−τ 0
−τ
ω T (t + θ) P3 ω (t + θ) dθ−
xT (t − d + θ) P2 x (t − d + θ) dθ
τ ω T (t) B1T Q3 B1 ω (t) − xT (t) C T ω (t) − ω T (t) Cx (t) − ω T (t) B + B T ω (t) = ⎤T ⎡ ⎡ ⎤ x (t) φ2 P B1 − C T ⎦ ⎣ ⎣ ⎦ ω (t) B1T P − C τ B1T Q3 B1 − B2T − B2 ⎤ ⎡ x (t) ⎦ ⎣ ω (t) (16) where
·
T
P2 x (t − d + θ) dθ τ xT (t) (P1 + P2 ) x (t) +
0 T τ ω (t) P3 ω (t) − xT (t + θ) P1 x (t + θ) dθ−
2 2 xT (t) [P (A + B) + (A + B) P + ε−1 1 P + ε1 α I+ −1 −1 −1 τ P B Q−1 I BTP + 1 + Q2 + Q3 + δ τ B T Q2 B]x (t) + τ AT Q1 A + τ δα2 I + 1−µ
(14)
2 2 φ2 = P (A + B) + (A + B) P + ε−1 1 P + ε1 α I+ −1 −1 −1 τ P B Q−1 I BTP + 1 + Q2 + Q3 + δ τ B T Q2 B τ AT Q1 A + τ δα2 I + 1−µ ⎤ ⎡ P B1 − C T φ2 ⎦<0 If ⎣ B1T P − C τ B1T Q3 B1 − B2T − B2 then V˙ (x (t)) − 2z T (t) ω (t) < 0, as V (x) > 0 for
x = 0, it follows that as tp → ∞ the system, (Σ1 ) is
Delay-dependent passive control of linear systems with nonlinear perturbation
349
strictly passive. Using the famous Schur complement, the Lemma can easily be obtained. Remark 1 The main method used in the proof of Lemma 1 is that, suppose the nonlinear function satisfies the norm-bounded, then the nonlinear function can be transformed into the linear function. Next, a state-feedback controller will be designed for the system. Consider the system as follows
(19)
u (t) = Kx (t)
where K is an appropriate dimension matrix to be determined. Substituting (19) into (Σ1 ), the following closed-loop system can be obtained (Σ2 )x˙ (t) = (A + B3 K)x (t) + Bx (t − d (t)) +
(Σ1 )x˙ (t) = Ax (t) + Bx (t − d (t)) + f (x (t) , t) + B1 ω (t) + B3 u(t)
(17)
z (t) = Cx (t) + B2 ω (t)
(18)
where u(t) ∈ m is the controlled input. The statefeedback passive controller to be designed is as follows ⎡
(20)
z (t) = Cx (t) + B2 ω (t)
(21)
Theorem 1 The system (Σ2 ) is said to be strictly passive, if and only if B2 + B2T > 0 and there exist the symmetric positive definite matrices X, Q1 , Q2 , Q3 , scalars ε1 > 0, δ > 0, and matrix W , which satisfy the LMI (22).
B1 − XC T
X
XAT
X
XB T
τ B1T Q3 B1 − B2T − B2
0
0
0
0
0
0
0
0
0
φ3
⎢ ⎢ B T − CX ⎢ 1 ⎢ ⎢ X ⎢ ⎢ ⎢ ⎢ AX ⎢ ⎢ ⎢ X ⎢ ⎣ BX
f (x (t) , t) + B1 ω (t)
0
−
1 −1 ε I α2 1 0
0
0
1 − Q−1 τ 1 0
0
0
0
0
where φ3 = (A + B) X + B3 W + X (A + B) + W T B3T + −1 −1 −1 −1 I)B T ε−1 1 I + τ B(Q1 + Q2 + Q3 + δ
W = KX,
X = P −1
(23)
Proof Observing the form of the system (Σ2 ), it can be seen that it is similar to the form of the system, hence, the proof of Theorem 1 is similar to Lemma 1 too. Substituting the parameters of the system (Σ2 ) into LMI (9), and then through the basic matrix transition and using the Schur complement twice, the LMI can easily be obtained LMI (22). Remark 2 This article only gives the state-feed controller, but a dynamic state-feed controller is desired. Therefore, a lot of work on the researches for the dynamic instance are still to be done.
4. Example In this section, the obtained theoretical result is illustrated through a simulation example. Consider the
−
1 −1 δ I τ α2 0
⎤
0 1 − µ −1 Q2 − τ
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(22)
system (Σ2 ) with the parameters given by A =
1 −1.2 0.1 −0.6 0.7 , , B = , B1 = 1 −0.1 −1.0 −1.0 −0.8
8 , B2 = 10, C = 1 1 , α = 0.1, τ = 1, B3 = 8
1 0 , is chosen, and then using the µ = 0.5.Q3 = 0 1 Matlab LMI Control Toolbox to solve the LMIs (22) and (23), the solution are obtained as follows ⎤ ⎤ ⎡ ⎡ 0.088 4 0.078 7 2.953 5 −1.808 2 ⎦ ⎦ , Q1=⎣ X=⎣ 0.078 7 0.223 9 −1.808 2 5.350 0 W= −1.784 6
−2.229 3
⎡ , Q2=⎣
⎤ 0.051 5 0.056 8 0.056 8 0.290 9
δ = 4.921 5, ε1 = 4.898 0 K = W ∗ X −1 = −1.083 5 −0.782 9 . The desired controller is u (t) = −1.083 5 −0.782 9 x (t).
⎦
350
Li Caina & Cui Baotong
Choose the perturbation f (x(t), t) = e−x(t) . The simulation for the system states are as follows. From Figs. 1 and 2, it can be easily seen that the designed controller is able to guarantee that the closedloop systems are stable and passive.
[4] Feng C B, Zhang Kanjian, Fei Shumin. Passivity-based design of robust control systems. Acta Automatica Sinica, 1999, 25 (5): 577–582. [5]
Shieh C S. H∞ control via output feedback for linear systems with time-varying delays in state and control input. Technical Commmunication, 2002, 28: 649–357.
[6]
Guan X P, Long C N, Duan Guangren. Robust passive control for discrete time-delay systems. Acta Automatica Sinica, 2002, 28 (1): 1–4.
[7] Dong X Z, Zhang Q L, Guo Kai. Passive control for class of nonlinear singular systens. Computing Technology and Automation, 2003, 22(3): 4–6 [8] Magdi S M, Abdulla I. Passivity and passification of timedelay systems. Mathematical Analysis and Applications, 2004, 292: 247–258. Fig. 1
Curve of the system states when ω (t) = 0
[9] Zhang X M, Wu Min. Delay-depentent passive control for linear systems with delay. Control Theory & Applications, 2005, 22 (3): 391–401. [10] Li C N, Cui B T. Passive control of uncertain linear systems with time-varying delay in states. Proc. of the 25th Chinese Control Conference, 2006: 16–20. [11] Cui B T, Hua M G. Robust passive control for uncertain discrete-time systems with time-varying delays.
Chaos,
Solitons and Fractals, 2006, 29: 331–341. [12] Cui B T, Hua M G. Observer-based passive control of linear time-delay systems with parametric uncertainty. Chaos, Fig. 2
Curve of the system states when ω (t) = 0
5. Conclusions The problem of delay-dependent passive control of linear systems with nonlinear perturbation and timevarying state delay is investigated in this article. Using the LMI, the static state-feedback controller is obtained. In the end, a numerical example and the simulation are provided, to show the applicability of the developed result.
References [1] Yu L. Robust control-LMI. Beijing: Tsinghua University Press, 2002; 174–178. [2] Magdi S M, Xie Lihua. Stability and positive realness of time-delay systems. Journal of Mathematical Analysis and Applications, 1999, 239: 7–19. [3] Yu L, Chen G D. Passive control of linear time-delay systems. Control Theory and Applications, 1999, 16 (1): 130– 133 (in Chinese).
Solitons and Fractals, 2007, 32: 160–167.
Li Caina was born in 1981. She is now pursuing the M. S. degree in the Research Center of Control Science and Engineering, Southern Yangtze University, China. Her current research interest includes robust passive control of linear system. E-mail: licaina [email protected] Cui Baotong was born in 1960. He received the Ph. D. degree in control theory and control engineering from the College of Automation Science and Engineering, South China University of Technology, China, in 2003. He was a post-doctoral fellow at Shanghai Jiaotong University, China, from 2003 to 2005. He joined the Southern Yangtze University, China in 2003, where he is a full professor for College of Communication and Control Engineering. His current research interests include systems analysis, stability theory, artificial neural networks, impulsive control, and chaos synchronization. E-mail: [email protected]