Delay-dependent L2–L∞ control of linear systems with multiple time-varying state and input delays

Nonlinear Analysis: Real World Applications 13 (2012) 486–496

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Delay-dependent L2 –L∞ control of linear systems with multiple time-varying state and input delays Aimin Chen a,b,∗ , Junwei Wang c a

Institute of Applied Mathematics, Henan University, Kaifeng 475004, China

b

School of Mathematics and Information Science, Henan University, Kaifeng 475004, China

c

School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510006, China

article

info

Article history: Received 28 January 2011 Accepted 8 August 2011 Keywords: L2 –L∞ control Lyapunov–Krasovskii functional Jensen integral inequality Linear matrix inequality

abstract This paper is concerned with the L2 –L∞ controller design problem for delayed linear systems. We first propose a novel class of continuous-time multiple delayed linear systems. The time-delays are assumed to be time-varying continuously differentiable functions, which appear in the state and control input simultaneously. By using the Lyapunov–Krasovskii functional approach and Jensen integral inequality technique, a sufficient condition for the existence of an L2 –L∞ controller is formulated in the form of linear matrix inequalities (LMIs). When these LMIs are feasible, the L2 –L∞ controller is explicitly presented. Finally, a numerical example is provided to demonstrate the effectiveness and feasibility of the developed controller design procedures. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction When information or material is transmitted from one place to another, time-delays inevitably exist associated with the transmission process. In fact, time-delays are common in many biological and engineering systems [1]. For example, in the context of modern digital power control systems, time-delays are introduced due to measurement processing and control signal computation. In the elementary processes of genetic circuits (such as mRNA and protein synthesis), time-delays arise from the series of steps required between the expression of individual genes to the production of the corresponding proteins. The existence of time-delays is usually considered as a source of instability, and in some cases, neglecting time-delays may deteriorate the closed-loop system performance [2]. On the other hand, it is well known that time-delay systems are infinite dimensional in nature, which imposes difficulties and restrictions on the design of a stabilizing controller even for a simple linear system with a single constant delay. Therefore, controller design issue of time-delay linear and nonlinear systems, in the past few decades, has grown as one of the most important concerns of researchers in addition to stability analysis; see, e.g., [3–9] and the references therein. Recently, much effort has been made in the analysis and synthesis of dynamical systems with time-delays so as to stabilize systems or reject disturbance. For linear systems with state delays, H∞ control problem of linear systems with time-varying delays is considered, and the delay-dependent sufficient conditions are obtained with the help of Lyapunov–Krasovskii approach and Finsler’s projection lemma [10]. By introducing some relaxation matrices, Jiang and Han [11] investigated the delay-dependent robust stability problem for uncertain linear systems with interval time-varying delay. With the free weighting matrices method, Li and Jia [12] have designed a non-fragile H∞ and L2 –L∞ dynamic output feedback controller for a class of linear systems with time-varying state delay. For linear systems with input delays, many stabilization

∗

Corresponding author at: Institute of Applied Mathematics, Henan University, Kaifeng 475004, China. Tel.: +86 013707617918. E-mail address: [email protected] (A.M. Chen).

1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.08.006

A.M. Chen, J. Wang / Nonlinear Analysis: Real World Applications 13 (2012) 486–496

487

methods including the descriptor system approach [13], the memoryless controller design method [14], the model reduction method [15,16], and the state transformation method [17], has been developed. These previous studies have greatly advanced the analysis and synthesis of time-delay systems. However, most of the above research focused on the stability analysis or the control problem of linear systems with either state delay [10–12] or with input delay [13–17]. It should be noted that in reality, time-delays often appear simultaneously in both state and control input, and the presence of them may cause serious deterioration in the performance of the resulting control systems. Hence it is of great significance to consider the stability and control problem of linear systems with both state and input delays, which also represents one of the open problems of current research of time-delay systems summarized by Richard [18]. Up to now, only a few research works were reported to deal with the delay-dependent stability or/and stabilization for the linear systems with both state and input time-delays [19,20,4,5,21]. Employing the integralinequality method, Zhang et al. [19] investigated the delay-dependent stabilization problem of such kinds of systems under the assumption that the derivative bound of the time-varying delays is required to be less than one. By combining an augmentation approach and the delay partitioning technique, the static and integral output feedback stabilization problem of continuous-time linear systems with an time-invariant state/input delay are addressed in [21]. To the best of our knowledge, the delay-dependent control problem of linear systems with multiple time-varying state and input delays has not been fully investigated, which remains to be challenging. Motivated by the above concerns, we will consider the problem of L2 –L∞ controller design for multiple time-delayed linear systems. The time-delays are assumed to be time-varying continuously differentiable functions and appear in both state and control input simultaneously. The derivative bound of the time-varying delays is not needed to less than 1. By developing an appropriate Lyapunov–Krasovskii functional and employing Jensen integral inequality, sufficient conditions for the existence of L2 –L∞ controllers are formulated in terms of a set of linear matrix inequalities (LMIs), upon which the L2 –L∞ controller can be obtained from the solution of a convex optimization problem. A numerical example is provided to illustrate the effectiveness of the proposed controller design procedures. The rest of this paper is organized as follows. In Section 2, model description and preliminaries are given. Section 3 is devoted to the L2 –L∞ controller design for the proposed model. In Section 4, numerical simulations are given to show the effectiveness of the proposed control method. Finally, conclusions in Section 5 close the paper. Notations. Throughout this paper, R n denotes the n-dimensional Euclidean space, R n×m represents the set of all n × m real matrices, and I is the identity matrix of appropriate dimensions. The superscript ‘‘T ’’ represents the transpose of a matrix. The shorthand diag{· · ·} denotes the block diagonal matrix. For real symmetric matrices X and Y , the notation X > Y (respectively, X ≥ Y ) means that the matrix X − Y is positive definite (respectively, semi-positive). In symmetric block matrices, the notation ‘‘*’’ is used as an ellipsis for terms induced by symmetry. L2 [0, ∞) denotes the space of square integrable vector functions on [0, ∞), and ‖ · ‖2 is the L2 -norm. Matrices, if the dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. 2. Model description and preliminaries 2.1. Model description In the following, we focus on a class of continuous-time systems with multiple time-varying state and input delays: x˙ (t ) = Ax(t ) +

r1 −

Ai x(t − τi (t )) + Bu(t ) +

i=1

r2 −

Bj u(t − σj (t )) + Bω ω(t )

j =1

z (t ) = Cx(t ) + Du(t ) x(t ) = φ(t ),

(1)

∀t ∈ [−d, 0]

where x(t ) ∈ R is the state vector, u(t ) ∈ R m is the control input, ω(t ) ∈ R p is the exogenous disturbance signal belonging to L2 [0, ∞), z (t ) ∈ R q is the controlled output; A, Ai , B, Bj , Bω , C and D are known constant matrices of appropriate dimensions. The time-varying continuous functions τi (t ) and σj (t ) represent state and input delays satisfying n

0 ≤ τi (t ) ≤ τ1i , 0 ≤ σj (t ) ≤ σ1j ,

τ˙i (t ) ≤ τ2i σ˙ j (t ) ≤ σ2j

(2)

where τ1i , τ2i (i = 1, 2, . . . , r1 ), σ1j , σ2j (j = 1, 2, . . . , r2 ) are known constants and d = max{τ1i , σ1j }. For convenience and clarity, here we only consider the model (1) with two state delays and two input delays, i.e., r1 = r2 = 2. However, we point out that the results presented in this paper are still valid for the general system (1) with r1 state delays and r2 input delays. In this paper, we are interested in designing the following state-feedback controller u(t ) = Kx(t ) where K is the fixed controller gain matrix to be determined.

(3)

488

A.M. Chen, J. Wang / Nonlinear Analysis: Real World Applications 13 (2012) 486–496

By substituting the feedback control law (3) to the system (1), we obtain the closed-loop system x˙ (t ) = (A + BK )x(t ) +

2 −

Ai x(t − τi (t )) +

i=1

2 −

Bj Kx(t − σj (t )) + Bω ω(t )

(4)

j =1

z (t ) = (C + DK )x(t ). The purpose of this paper is to design a L2 –L∞ controller (3) for the system (1), such that for a prescribed level of noise attenuation γ > 0, the closed-loop system (4) with ω(t ) ≡ 0 is asymptotically stable and satisfies ‖z (t )‖∞ < γ ‖ω‖2 for any nonzero ω(t ) ∈ L2 [0, ∞) under the zero initial conditions. 2.2. Preliminaries To get the main results of this paper, the following lemmas will be used during the development of our main results. Lemma 1 (Jensen Integral Inequality [1]). For any positive symmetric constant matrix Q ∈ R n×n , scalars h1 , h2 satisfying h1 < h2 , a vector function φ : [h1 , h2 ] → R n such that the integrations concerned are well defined, then h2

∫

φ(s)ds

T

h2

∫ Q

h1

φ(s)ds

T

≤ (h2 − h1 )

h1

∫

h2

T

φ (s)Q φ(s)ds T

(5)

h1

Lemma 2 (Schur Complement [22]). For a given symmetric matrix S =



S11 S21

S12 S22



T T T , where S11 = S11 , S22 = S22 , S12 = S21 , the

linear matrix inequality (LMI):

[ S=

S11 S21

S12 S22

]

<0

is equivalent to one of the following conditions: −1 T • S11 < 0, S22 − S12 S11 S12 < 0. −1 T • S22 < 0, S11 − S12 S22 S12 < 0.

3. Main result 3.1. L2 –L∞ controller design In this section, we will establish a delay-dependent sufficient condition on the existence of L2 –L∞ controller (3) for system (1) in the following theorem. Theorem 1. For some given constants γ > 0, τij > 0, σij > 0 (i, j = 1, 2), if there exist matrices X > 0, Pij > 0, Sij > 0 and Y , such that the following inequalities (6) and (7) hold:

[ [

Π1 ∗

] Π3 <0 Π2

(6)

−X ∗

] (CX + DY )T <0 −I

(7)

where

 Θ11  ∗  Π1 =  ∗  ∗ ∗  Θ66  ∗  Π2 =  ∗  ∗ ∗

Θ12 Θ22 ∗ ∗ ∗

Θ13

Θ14

Θ15



0

0 0

0 0 0

   

0

0 0

Θ77 ∗ ∗ ∗

Θ33 ∗ ∗

Θ88 ∗ ∗

Θ44 ∗ 0 0 0

Θ99 ∗

Θ55 0 0 0 0

−γ 2

    

A.M. Chen, J. Wang / Nonlinear Analysis: Real World Applications 13 (2012) 486–496

Θ17 (A1 X )T (A2 X )T (B1 Y )T (B2 Y )T

 Θ 16 T (A1 X )  Π3 = (A2 X )T (B Y )T 1 (B2 Y )T Θ11 Θ12 Θ16 Θ22 Θ44 Θ66

Θ18 (A1 X )T (A2 X )T (B1 Y )T (B2 Y )T

Θ19 (A1 X )T (A2 X )T (B1 Y )T (B2 Y )T

489

 Bω 0  0  0 0

= (AX + BY ) + (AX + BY ) + P21 + P22 + S21 + S22 , = A1 X , Θ13 = A2 X , Θ14 = B1 Y , Θ15 = B2 Y , T = Θ17 = Θ18 = Θ19 = (AX + BY ) , = (τ21 − 1)P21 , Θ33 = (τ22 − 1)P22 , = (σ21 − 1)S21 , Θ55 = (σ22 − 1)S22 , −2 −2 = τ11 (P11 − 2X ), Θ77 = τ12 (P12 − 2X ), T

−2 Θ88 = σ11 (S11 − 2X ),

−2 Θ99 = σ12 (S12 − 2X ).

then system (4) is asymptotically stable with L2 –L∞ performance γ . Moreover, the control gain matrix K is given by K = YX −1 .

(8)

Proof. First, assuming that ω(t ) = 0, we consider the asymptotic stability of the system (4). Select a Lyapunov–Krasovskii functional candidate as V (t ) = V0 (t ) +

2 − (V1i (t ) + V2i (t ) + V1i (t ) + V2i (t ))

(9)

i =1

with

ˆ V0 (t ) = xT Px 0

∫

V1i (t ) = τ1i

−τ1i

V2i (t ) =

∫

t −τi (t )

xT (s)Pˆ 2i x(s)ds

0

∫

t

∫

−σ1i

∫

x˙ T (v)Pˆ 1i x˙ (v)dv ds t +s

t

V1i (t ) = σ1i V2i (t ) =

t

∫

x˙ T (v)Sˆ1i x˙ (v)dv ds t +s

t t −σi (t )

xT (s)Sˆ2i x(s)ds

ˆ Pˆij , Sˆij (i, j = 1, 2) are positive symmetric matrices to be determined. where matrices P, Then along the trajectory of system (4) and using Lemma 1, the derivative of (9) is given by V˙ (t ) = 2xT (t )Pˆ x˙ (t ) +

2 −

x˙ T (t )[τ1i2 Pˆ 1i + σ1i2 Sˆ1i ]˙x(t ) +

i=1

2 −

xT (t )(Pˆ 2i + Sˆ2i )x(t )

i =1

2 2 − − − (1 − τ˙i (t ))xT (t − τi (t ))Pˆ2i x(t − τi (t )) − (1 − σ˙ i (t ))xT (t − σi (t ))Sˆ2i x(t − σj (t )) i=1

−

2 −

i =1

τ1i

t

∫

x˙ T (s)Pˆ 1i x˙ (s)ds − t −τ1i

i=1

≤ 2xT (t )Pˆ x˙ (t ) +

2 −

σ1i

t

x˙ T (s)Sˆ1i x˙ (s)ds

t −σ1i

i =1

2 −

∫

x˙ T (t )[τ1i2 Pˆ 1i + σ1i2 Sˆ1i ]˙x(t ) +

i=1

2 −

xT (t )(Pˆ 2i + Sˆ2i )x(t )

i =1

2 2 − − + (τ2i − 1)xT (t − τi (t ))Pˆ2i x(t − τi (t )) + (σ2i − 1)xT (t − σi (t ))Sˆ2i x(t − σi (t )) i=1

+

i =1

]T [ x(t ) −Pˆ1i x(t − τ1i ) ∗

2 [ − i=1

˜ ξ (t ) = ξ T (t )Θ

Pˆ 1i −Pˆ1i

][

x( t )

]

x(t − τ1i )

+

2 [ − i=1

x( t )

]T [

x(t − σ1i )

−Sˆ1i ∗

Sˆ1i −Sˆ1i

][

x(t )

]

x(t − σ1i ) (10)

490

A.M. Chen, J. Wang / Nonlinear Analysis: Real World Applications 13 (2012) 486–496

where

ξ (t ) = [xT (t ), xT (t − τ1 (t )), xT (t − τ2 (t )), xT (t − σ1 (t )), xT (t − σ2 (t )), xT (t − τ11 ), xT (t − τ12 ), xT (t − σ11 ), xT (t − σ12 )]T ˜  ˜ 12 Θ ˜ 13 Θ ˜ 14 Θ ˜ 15 Pˆ11 Θ11 Θ Pˆ 12 Sˆ11 Sˆ12 ˜ 22 Θ ˜ 23 Θ ˜ 24 Θ ˜ 25  ∗ 0 0 0 0  Θ   ˜ 33 Θ ˜ 34 Θ ˜ 35  ∗ ∗ Θ 0 0 0 0    ˜ 44 Θ ˜ 45  ∗ ∗ ∗ Θ 0 0 0 0    ˜ 55 ˜ = ∗ ∗ ∗ ∗ Θ 0 0 0 0  Θ    ∗ ∗ ∗ ∗ ∗ −Pˆ11 0 0 0     ∗ ∗ ∗ ∗ ∗ ∗ −Pˆ12 0 0     ∗ ∗ ∗ ∗ ∗ ∗ ∗ −Sˆ11 0  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −Sˆ12 with

˜ 11 = Pˆ (A + BK ) + (A + BK )T Pˆ + Θ

2 2 − − [Pˆ2i + Sˆ2i − Pˆ1i − Sˆ1i ] + (A + BK )T [τ1i2 Pˆ1i + σ1i2 Sˆ1i ](A + BK ), i =1

ˆ 1+ ˜ 12 = PA Θ

2 −

(A + BK )T [τ1i2 Pˆ1i + σ1i2 Sˆ1i ]A1 ,

i=1 2

ˆ 2+ ˜ 13 = PA Θ

−

(A + BK )T [τ1i2 Pˆ1i + σ1i2 Sˆ1i ]A2 ,

i=1 2

ˆ 1K + ˜ 14 = PB Θ

− (A + BK )T [τ1i2 Pˆ1i + σ1i2 Sˆ1i ]B1 K , i=1

˜ 15 Θ

2 − ˆ 2K + = PB (A + BK )T [τ1i2 Pˆ1i + σ1i2 Sˆ1i ]B2 K , i=1

˜ 22 = (τ21 − 1)Pˆ21 + Θ

2 −

AT1 [τ1i2 Pˆ 1i + σ1i2 Sˆ1i ]A1 ,

i=1

˜ 23 = Θ

2 −

AT1 [τ1i2 Pˆ 1i + σ1i2 Sˆ1i ]A2 ,

i =1

˜ 24 = Θ

2 −

AT1 [τ1i2 Pˆ 1i + σ1i2 Sˆ1i ]B1 K ,

i =1 2

˜ 25 = Θ

−

AT1 [τ1i2 Pˆ 1i + σ1i2 Sˆ1i ]B2 K ,

i =1

˜ 33 = (τ22 − 1)Pˆ22 + Θ

2 −

AT2 [τ1i2 Pˆ 1i + σ1i2 Sˆ1i ]A2 ,

i=1

˜ 34 = Θ

2 −

AT2

[τ ˆ + σ1i2 Sˆ1i ]B1 K , 2 1i P1i

i =1

˜ 35 = Θ

2 −

AT2 [τ1i2 Pˆ 1i + σ1i2 Sˆ1i ]B2 K ,

i =1

˜ 44 = (σ21 − 1)Sˆ21 + Θ

2 −

(B1 K )T [τ1i2 Pˆ1i + σ1i2 Sˆ1i ]B1 K ,

i=1

˜ 45 = Θ

2 − (B1 K )T [τ1i2 Pˆ1i + σ1i2 Sˆ1i ]B2 K , i =1

˜ 55 = (σ22 − 1)Sˆ22 + Θ

2 − i=1

(B2 K )T [τ1i2 Pˆ1i + σ1i2 Sˆ1i ]B2 K .

i =1

(11)

A.M. Chen, J. Wang / Nonlinear Analysis: Real World Applications 13 (2012) 486–496

491

By using Lemma 2, it is easy to check that the matrix inequality

˜ <0 Θ

(12)

is equivalent to

ˆ Θ11  ∗   ∗   ∗   ∗   ∗   ∗   ∗ ∗

ˆ 12 Θ ˆ 22 Θ ∗ ∗ ∗ ∗ ∗ ∗ ∗

ˆ 13 Θ

ˆ 14 Θ

ˆ 15 Θ

ˆ 16 Θ

ˆ 17 Θ

ˆ 18 Θ

0

0 0

0 0 0

AT1 AT2 (B1 K )T (B2 K )T

AT1 AT2 (B1 K )T (B2 K )T 0

AT1 AT2 (B1 K )T (B2 K )T 0 0

ˆ 33 Θ ∗ ∗ ∗ ∗ ∗ ∗

ˆ 44 Θ ∗ ∗ ∗ ∗ ∗

ˆ 55 Θ ∗ ∗ ∗ ∗

ˆ 66 Θ ∗ ∗ ∗

ˆ 77 Θ ∗ ∗

ˆ 88 Θ ∗

ˆ 19  Θ AT1   AT2   (B1 K )T   (B2 K )T  <0 0   0   0 

(13)

ˆ 99 Θ

with

ˆ 11 = Pˆ (A + BK ) + (A + BK )T Pˆ + Pˆ21 + Pˆ22 + Sˆ21 + Sˆ22 , Θ ˆ 1, ˆ 2, ˆ 1K , ˆ 2K , ˆ 12 = PA ˆ 13 = PA ˆ 14 = PB ˆ 15 = PB Θ Θ Θ Θ T ˆ ˆ ˆ ˆ Θ16 = Θ17 = Θ18 = Θ19 = (A + BK ) , ˆ 22 = (τ21 − 1)Pˆ21 , ˆ 33 = (τ22 − 1)Pˆ22 , Θ Θ ˆ 44 = (σ21 − 1)Sˆ21 , ˆ 55 = (σ22 − 1)Sˆ22 , Θ Θ − 1 −2 ˆ −1 − 2 ˆ ˆ 77 = −τ12 ˆ 66 = −τ11 P11 , P12 , Θ Θ −2 ˆ −1 ˆ 88 = −σ11 Θ S11 ,

−2 ˆ −1 ˆ 99 = −σ12 Θ S12 .

By pre- and post-multiplying the (13) by diag{I , I , I , I , I , Pˆ , Pˆ , Pˆ , Pˆ } and its transpose respectively, and then using the following inequality [23]: −1 ˆ P ≥ 2Pˆ − Pˆ 1i Pˆ Pˆ 1i

(14)

−1 ˆ P ≥ 2Pˆ − Sˆ1i Pˆ Sˆ1i

it can be verified that (13) holds if the following inequality (15) is satisfied

Θ ˆ 11  ∗   ∗   ∗   ∗   ∗   ∗  ∗ ∗

ˆ 12 Θ ˆ 22 Θ

ˆ 13 Θ

ˆ 14 Θ

ˆ 15 Θ

0

∗ ∗ ∗ ∗ ∗ ∗ ∗

ˆ 33 Θ ∗ ∗ ∗ ∗ ∗ ∗

0 0

0 0 0

ˆ 44 Θ ∗ ∗ ∗ ∗ ∗

ˆ 55 Θ ∗ ∗ ∗ ∗

¯ 16 Θ AT1 Pˆ AT2 Pˆ

(B1 K )T Pˆ (B2 K )T Pˆ ¯ 66 Θ ∗ ∗ ∗

¯ 17 Θ AT1 Pˆ AT2 Pˆ

¯ 18 Θ AT1 Pˆ AT2 Pˆ

(B1 K )T Pˆ (B2 K )T Pˆ

(B1 K )T Pˆ (B2 K )T Pˆ

0

0 0

¯ 77 Θ ∗ ∗

¯ 88 Θ ∗

¯ 19 Θ AT1 Pˆ AT2 Pˆ



    T ˆ (B1 K ) P  <0 (B2 K )T Pˆ    0   0 

(15)

0

¯ 99 Θ

with

¯ 16 = Θ ¯ 17 = Θ ¯ 18 = Θ ¯ 19 = (A + BK )T Pˆ , Θ − 2 −2 ˆ ¯ 66 = τ11 (Pˆ11 − 2Pˆ ), ¯ 77 = τ12 Θ Θ (P12 − 2Pˆ ), −2 ˆ ¯ 88 = σ11 Θ (S11 − 2Pˆ ),

−2 ˆ ¯ 99 = σ12 Θ (S12 − 2Pˆ ).

By defining X = Pˆ −1 , Y = KX , P1i = X T Pˆ 1i X , P2i = X T Pˆ 2i X , S1i = X T Sˆ1i X , S2i = X T Sˆ2i X , then pre- and post-multiplying (15) by diag{X T , X T , X T , X T , X T , X T , X T , X T , X T }

492

A.M. Chen, J. Wang / Nonlinear Analysis: Real World Applications 13 (2012) 486–496

and its transpose, respectively, we can obtain

Θ 11  ∗  ∗   ∗  Θ=  ∗  ∗   ∗  ∗ ∗

Θ12 Θ22 ∗ ∗ ∗ ∗ ∗ ∗ ∗

Θ13

Θ14

Θ15

0

0 0

0 0 0

Θ33 ∗ ∗ ∗ ∗ ∗ ∗

Θ44 ∗ ∗ ∗ ∗ ∗

Θ55 ∗ ∗ ∗ ∗

Θ16 (A1 X )T (A2 X )T (B1 Y )T (B2 Y )T Θ66 ∗ ∗ ∗

Θ17 (A1 X )T (A2 X )T (B1 Y )T (B2 Y )T

Θ18 (A1 X )T (A2 X )T (B1 Y )T (B2 Y )T

0

0 0

Θ77 ∗ ∗

Θ88 ∗

Θ19  (A1 X )T  (A2 X )T   (B1 Y )T   (B2 Y )T  <0 0   0  

(16)

0

Θ99

with

Θ11 Θ12 Θ16 Θ22 Θ44 Θ66

= (AX + BY ) + (AX + BY )T + P21 + P22 + S21 + S22 , = A1 X , Φ13 = A2 X , Φ14 = B1 Y1 , Φ15 = B2 Y2 , = Θ17 = Θ18 = Θ19 = (AX + BY )T , = (τ21 − 1)P21 , Θ33 = (τ22 − 1)P22 , = (σ21 − 1)S21 , Θ55 = (σ22 − 1)S22 , −2 −2 = τ11 (P11 − 2X ), Θ77 = τ12 (P12 − 2X ),

−2 Θ88 = σ11 (S11 − 2X ),

−2 Θ99 = σ12 (S12 − 2X ).

Hence, by applying the Lemma 2, the inequality (6) implies Θ < 0. Therefore, V˙ (t ) < 0 for all ξ (t ) ̸= 0. Thus system (4) with ω(t ) = 0 is asymptotically stable under the condition of theorem. Next, assuming that φ(t ) = 0, t ∈ [−d, 0], we will show that the closed-loop system (4) has L2 –L∞ performance γ . Define a functional Jt as t

∫

[V˙ (t ) − γ 2 ωT (t )ω(t )]ds

Jt =

(17)

0

where t > 0 is a scalar, and V (t ) is defined in (9). By some calculations, it can be verified that

˜ ω ζ (t ) V˙ (t ) − γ 2 ωT (t )ω(t ) ≤ ζ T (t )Θ

(18)

where

ζ (t ) = [xT (t ), xT (t − τ1 (t )), xT (t − τ2 (t )), xT (t − σ1 (t )), xT (t − σ2 (t )), ω(t ), xT (t − τ11 ), xT (t − τ12 ), xT (t − σ11 ), xT (t − σ12 )]T  ˜ ω ˜ 14 Θ ˜ 15 Θ ˜ 16 ˜ 12 Θ ˜ 13 Θ Θ11 Θ Pˆ 11 Pˆ 12 Sˆ11 Sˆ12 ω ˜ 22 Θ ˜ 23 Θ ˜ 24 Θ ˜ 25 Θ ˜ 26  ∗ 0 0 0 0  Θ   ω ˜ 33 Θ ˜ 34 Θ ˜ 35 Θ ˜ 36  ∗ ∗ Θ 0 0 0 0    ω ˜ 44 Θ ˜ 45 Θ ˜ 46  ∗ ∗ ∗ Θ 0 0 0 0    ω  ∗ ˜ 55 Θ ˜ 56 ∗ ∗ ∗ Θ 0 0 0 0  ω   ˜ Θ = ω ˜ 66 ∗ ∗ ∗ ∗ Θ 0 0 0 0    ∗  ∗ ∗ ∗ ∗ ∗ ∗ −Pˆ11 0 0 0     ∗ ∗ ∗ ∗ ∗ ∗ ∗ −Pˆ12 0 0     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −Sˆ11 0  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −Sˆ12

(19)

A.M. Chen, J. Wang / Nonlinear Analysis: Real World Applications 13 (2012) 486–496

ω ˆ ω+ ˜ 16 Θ = PB

493

2 − (A + BK )T [τ1i2 Pˆ1i + σ1i2 Sˆ1i ]Bω , i =1

ω

2 −

ω

2 −

ω

2 −

ω

2 −

ω

2 −

˜ 26 = Θ

AT1

[τ1i2 Pˆ1i + σ1i2 Sˆ1i ]Bω ,

i=1

˜ 36 = Θ

AT2 [τ1i2 Pˆ 1i + σ1i2 Sˆ1i ]Bω ,

i=1

˜ 46 = Θ

(20)

(B1 K )T [τ1i2 Pˆ1i + σ1i2 Sˆ1i ]Bω ,

i=1

˜ 56 = Θ

(B2 K )T [τ1i2 Pˆ1i + σ1i2 Sˆ1i ]Bω ,

i=1

˜ 66 = Θ

BTω [τ1i2 Pˆ 1i + σ1i2 Sˆ1i ]Bω − γ 2 .

i=1

˜ ω < 0 which leads to Similar to the conclusion as in (16), if (6) holds, then Θ ∫ Jt =

t

[V˙ (t ) − γ 2 ωT (t )ω(t )]ds < 0.

(21)

0

Using Lemma 2, the LMI (7) guarantees

(C + DK )T (C + DK ) < Pˆ . Then it can be easily established that for all t > 0 z T (t )z (t ) = xT (t )(C + DK )T (C + DK )x(t )

ˆ (t ) ≤ V (t ) ≤ γ 2 < x (t )Px T

t

∫

ωT (t )ω(t )ds. 0

Taking the supremum over t > 0 yields

‖z (t )‖∞ < γ ‖ω(t )‖2

(22)

for any nonzero ω(t ) ∈ L2 [0, ∞). This completes the proof.



3.2. Two special cases In practical problems, we often encounter the following two types of systems, which are two special cases of the general time-delay system described by system (4). Case I. τi (t ) ≡ 0, (i = 1, 2). In this case, system (4) becomes x˙ (t ) = (A + BK )x(t ) +

2 −

Bj Kx(t − σj (t )) + Bω ω(t )

j =1

(23)

z (t ) = (C + DK )x(t ). Following the proof process of Theorem 1, we have Corollary 1. For some given constants γ > 0, σij > 0 (i, j = 1, 2), if there exist matrices X > 0, Sij > 0 and Y , such that the following inequalities (24) and (25) hold:

Ψ11 Ψ12 Ψ13 Ψ14  ∗ Ψ22 0 Ψ24  ∗ Ψ33 Ψ34  ∗  ∗ ∗ ∗ Ψ44   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ [ ] −X (CX + DY )T <0 ∗ −I



Ψ15 Ψ25 Ψ35 0

Ψ55 ∗

Bω 0 0 0 0

−γ 2

    <0  

(24)

(25)

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A.M. Chen, J. Wang / Nonlinear Analysis: Real World Applications 13 (2012) 486–496

where

Ψ11 Ψ12 Ψ22 Ψ33 Ψ44

= (AX + BY ) + (AX + BY )T + S21 + S22 , = B1 Y , Ψ13 = B2 Y , Ψ14 = Ψ15 = (AX + BY )T , = (σ21 − 1)S21 , Ψ24 = Ψ25 = (B1 Y )T , = (σ22 − 1)S22 , Ψ34 = Ψ35 = (B2 Y )T , −2 −2 = σ11 (S11 − 2X ), Ψ55 = σ12 (S12 − 2X )

then system (23) is asymptotically stable with L2 –L∞ performance γ . Moreover, the control gain matrix K is given by K = YX −1 .

(26)

Case II. σj (t ) ≡ 0, (j = 1, 2). In this case, system (4) becomes x˙ (t ) = (A + BK )x(t ) +

2 −

Ai x(t − τi (t )) + Bω ω(t )

i=1

(27)

z (t ) = (C + DK )x(t ). Following the proof process of Theorem 1, we have. Corollary 2. For some given constants γ > 0, τij > 0 (i, j = 1, 2), if there exist matrices X > 0, Pij > 0 and Y , such that the following inequalities (28) and (29) hold:

 Υ11 Υ12 Υ13 Υ14  ∗ Υ22 0 Υ24  ∗ Υ33 Υ34  ∗  ∗ ∗ ∗ Υ44   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ [ ] −X (CX + DY )T <0 ∗ −I

Υ15 Υ25 Υ35

Bω 0 0 0 0

0

Υ55 ∗

−γ 2

    <0  

(28)

(29)

where

Υ11 Υ12 Υ22 Υ33 Υ44

= (AX + BY ) + (AX + BY )T + P21 + P22 , = A1 X , Υ13 = A2 X , Υ14 = Υ15 = (AX + BY )T , = (τ21 − 1)P21 , Υ24 = Υ25 = (A1 X )T , = (τ22 − 1)P22 , Υ34 = Υ35 = (A2 X )T , −2 −2 = τ11 (P11 − 2X ), Υ55 = τ12 (P12 − 2X )

then system (27) is asymptotically stable with L2 –L∞ performance γ . Moreover, the control gain matrix K is given by K = YX −1 .

(30)

4. Illustrative example To illustrate the applicability of the above controller design method, in this section, we consider the following numerical example which is described by x˙ 1 (t ) = 0.2x1 (t ) − 0.5x2 (t ) + 0.1x1 (t − τ1 (t )) − 0.2x2 (t − τ2 (t )) + 2u1 (t ) + 0.6u2 (t )

− u2 (t − σ1 (t )) + 0.2u1 (t − σ2 (t )) + ω(t ) x˙ 2 (t ) = 1.3x1 (t ) − 0.4x2 (t ) + 0.1x2 (t − τ2 (t )) − 0.5u2 (t ) + u2 (t − σ1 (t )) + 0.3u1 (t − σ2 (t )) z1 (t ) = 0.2x1 (t ) + x2 (t ) − 0.7u1 (t ) + 1.6u2 (t ) z2 (t ) = 0.4x2 (t ) + 2u2 (t ) x(t ) = [3, −4]T ,

t ∈ [−0.8, 0]

(31)

A.M. Chen, J. Wang / Nonlinear Analysis: Real World Applications 13 (2012) 486–496

495

which can be reformulated in the form of (1) with 0.2 1.3

] [ ] [ ] −0.5 0.1 0 0 −0.2 , A1 = , A2 = , −0.4 0 0 0 0.1 [ ] [ ] [ ] 2 0.6 0 −1 0.2 0 B= , B1 = , B2 = , 0 −0.5 0 1 0.3 0 [ ] [ ] [ ] 1 0.2 1 −0.7 1.6 Bω = , C = , D= . 0 0 0.4 0 2 [

A=

(32)

Note that multiple time-varying state delays and the input delays appear in the system dynamics equations. For this example, we let τ1 (t ) = 0.1 + 0.1 sin(t ), τ2 (t ) = 0.2 + 0.2 sin(2t ), σ1 (t ) = 0.3 + 0.3 sin(3t ) and σ2 (t ) = 0.4 + 0.4 sin(4t ). The prescribed L2 –L∞ performance level is chosen γ = 1.6. Then using MATLAB LMI Toolbox to solve the LMIs (6) and (7), we can obtain that 2.8747 −3.0053

[ X =

[ −3.3100 −0.0486 [ 2.8799 P11 = −3.8800 [ 0.7935 P12 = −1.0503 [ 0.5252 P21 = −0.5449 [ 0.5903 P22 = −1.0156 [ 0.3986 S11 = −0.4894 [ 0.2449 S12 = −0.2951 [ 0.2362 S21 = −0.2106 [ 2.4280 S22 = −2.7305 Y =

] −3.0053 6.5067 ] 3.7723 −0.8735 ] −3.8800 9.1976 ] −1.0503 3.5093 ] −0.5449 0.8113 ] −1.0156 2.1811 ] −0.4894 1.8381 ] −0.2951 1.1847 ] −0.2106 1.0362 ] −2.7305 . 3.3323

Therefore, by Theorem 1, the desired L2 –L∞ controller can be constructed as (3) with K =

[ −1.0545 −0.3041

0.0927 . −0.2747

]

(33)

By choosing the exogenous disturbance as ω(t ) = 4e−0.3t cos(5t ), Figs. 1 and 2 show the state responses of the systems (31) without control and with control, respectively. It is seen from Fig. 2 that the closed-loop system is asymptotically stable.

5. Conclusion The problem of the L2 –L∞ controller design for a class of time-varying delayed linear systems has been investigated in this paper. By using the Lyapunov–Krasovskii functional approach and Jensen integral inequality technique, a sufficient condition for the existence of an L2 –L∞ controller is formulated in the form of linear matrix inequalities (LMIs). When these LMIs are feasible, the L2 –L∞ controller is presented. Finally, a numerical example is provided to demonstrate the effectiveness of the developed controller design procedure. Acknowledgments This work is supported by the Natural Science Foundation of the Education Department of Henan, China (Grant No. 2011B110003), the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant No. LYM10074) and Guangdong Natural Science Foundation, China (Grant No. S2011040001704).

496

A.M. Chen, J. Wang / Nonlinear Analysis: Real World Applications 13 (2012) 486–496

Fig. 1. Time response of system (31) without control input.

Fig. 2. Time response of system (31) with control input.

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