Dynamical Output Feedback Control for Distributed-delay Systems
JOURNAL OF COMPUTERS, VOL. 8, NO. 3, MARCH 2013
701
Dynamical Output Feedback Control for Distributed-delay Systems Juan Liu Department of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China Email: [email protected]
Lin Chen Department of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, China Email: [email protected]
Jiaolong Zhang School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo, China Email: [email protected]
Abstract—The dynamical output feedback controller design for systems with distributed delay is concerned in this paper. Firstly, model transformation of the closed-loop system is used and a proper Lyapunov-Krsovskii functional is constructed, by which the stability criterion is obtained in terms of nonlinear matrices inequality, which guarantees the asymptotical stability of closed-loop system. Secondly, the parameterization of controller is used and the delaydependent design condition of the desired controller is established in terms of linear matrices inequality. Finally, a simulation is given to illustrate the effectiveness of the proposed method. Index Terms—distributed delay, dynamical output feedback, Lyapunov-Krsovskii functional, linear matrix inequality (LMI)
I. INTRODUCTION Distributed time-delay is commonly encountered in various physical and engineering systems. It has been shown that the existence of delay is the sources of instability and poor performance of control systems. Therefore, stability analysis and controller design of linear systems with distributed delay have attracted much attention in the last years. To stability analysis of time-delay systems, in order to reduce the conservatism of stability criteria, many researchers developed different approaches, such as constructing discretized Lyapunov functional[1], model transformation[2, 3], integral inequality[4, 5], free matrices[6, 7] and constructing proper LyapunovKrasovskii functional[8, 9]. As we all known, the controller design depend on the stability results, many stability results do well in reducing the conservatism of the stability criteria, but they can not be used to controller design because of their complexity. Therefore, to the control problem of time-delay system, solving the parameters of controller is important as well as reducing the conservatism. In [10], the state feedback control of
© 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.3.701-708
distributed delay system was investigated. In [11], the controller design of linear system with distributed delay is studied via static output feedback. In [12, 13], the dynamical output feedback controllers were designed for distributed delay systems, but the results is delayindependent and more conservative. To the best of our knowledge, few results have been achieved on delaydependent dynamical feedback stabilization for linear systems with distributed delay. Based on above discussion, in this note, we study the stabilization problem for distributed time-delay system via dynamical output feedback. Through neutral model transformation of closed-loop system and constructing a proper Lyapunov-Krasovskii functional, a delaydependent stability criterion is obtained in terms of nonlinear matrix inequality. Then by using parameterization of controller, the design condition of the controller, which guarantees the asymptotical stability of closed-loop system, is established in terms of LMI, which is easy to check. In the end, an example is also given to illustrate the effectiveness of the proposed method. Notations: In this paper, C [− h,0]; ℜ n denotes the
(
)
family of continuous functions from [− h,0] to ℜ n . For symmetric matrices A and B, λmax ( A) and λmin ( A) denote the largest and smallest eigenvalue of A, respectively; the notation A ≥ B (respectively, A > B ) means that the matrix A − B is positive semi-definite (respectively, positive definite). ⋅ denotes the Euclidean
norm for vector or the spectral norm of matrices. In the symmetric block matrices, ∗ denotes a term that is induced by symmetry. Every matrices, if not explicitly mentioned in the paper, are assumed to have compatible dimensions. II. PROBLEM FORMULATION Consider the following system with distributed delay:
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JOURNAL OF COMPUTERS, VOL. 8, NO. 3, MARCH 2013
x& (t ) = Ax(t ) + A1 x(t − h ) + A2
t
∫
Lemma 1 [14]. For given positive scalars α1 , α 2 , α 3 , where α1 + α 2 + α 3 < 1 , if a symmetric positive-definite
x(σ )dσ + Bu (t )
t −d
y (t ) = Cx(t )
(1)
x(t ) = ϕ (t ), t ∈ [−γ ,0]
where x ∈ ℜ n , u ∈ ℜ m , y ∈ ℜ h is state, control input and output respectively; h, d > 0 are constant delays, γ = max{h, d } , A, A1 , A2 , B, C are matrices with appropriate dimension. The aim of the note is to design the following dynamical output feedback controller: x&ˆ (t ) = Ac xˆ (t ) + Bc y (t ) u (t ) = Cc xˆ (t ) + Dc y (t ), t ≥ 0
(2)
which guaranteed the asymptotical stability of the closedloop system: t
ξ&(t ) = A ξ (t ) + A1ξ (t − h ) + A2 ∫ ξ (σ )dσ , t ≥ 0
(3)
matrix M ∈ ℜ 2 n× 2 n exists such that the following LMI holds, then the operator Gξ t is stable: ⎡ ⎢ N ' MN − α1M ⎢ hA ' MN 1 ⎢ ⎢ dA ' MN 2 ⎢⎣
⎡h 2 A1T MA1 − α1M ⎢ ⎢ dhA2T MA1 ⎣
where
⎡ x⎤
⎣ ⎦ ⎡I ⎤ A1 = ⎢ n ⎥[A1 0] ⎣0⎦
dhA2' MA1
⎤ ⎥ dN ' MA2 ' ⎥ 0 , we have:
y (t ) = Cx(t ), t ≥ 0 x(t ) = φ (t ),−h ≤ t ≤ 0
⎡ A + BDC C A=⎢ ⎣ BC C
⎡A ⎤ ∫ ∫ ξ (σ )⎢⎣ 0 ⎥⎦ S [A 0 t
∫
Applying Lemma 2, we have: ⎡ AT ⎤ V&2 (ξ t ) = hξ T (t )⎢ 1 ⎥ S [A1 0]ξ (t ) ⎣0⎦ t ⎡ AT ⎤ − ∫ ξ T (θ )⎢ 1 ⎥ S [A1 0]ξ (θ )dθ ⎣0⎦ t −h ⎡ AT ⎤ ≤ hξ T (t )⎢ 1 ⎥ S [A1 0]ξ (t ) ⎣0⎦ t t ⎡ AT ⎤ 1 − ∫ ξ T (θ )⎢ 1 ⎥ dθ S ∫ [A1 0]ξ (θ )dθ h t −h ⎣0⎦ t −h so, it is easy to get: ⎡ ˆT T ⎡ P1 ⎤ −1 ⎢ A P + PAˆ + hAˆ ⎢ T ⎥ S [P1 ⎣ P3 ⎦ ⎢ V& (ξ t ) ≤ ξ T (t )⎢ T ⎢+ h ⎡⎢ A1 ⎤⎥ S [A 0] ⎢ ⎣0⎦ 1 ⎣
⎤ P3 ]Aˆ ⎥ ⎥ ⎥ξ (t ) ⎥ ⎥ ⎦
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JOURNAL OF COMPUTERS, VOL. 8, NO. 3, MARCH 2013
by Lyapunov stability theorem, we can get the following condition which guarantees the stability of the closedloop system (22) : For given positive scalars h > 0 , 0 < δ < 1 , if there exist symmetric positive-definite matrices S ∈ ℜ n×n , ⎡ P P3 ⎤ 2 n×2 n , M ∈ ℜ 2 n×2 n satisfying (23) and P=⎢ 1 ⎥ ∈ℜ P ∗ 2⎦ ⎣
the following matrix inequality, then the closed-loop system (22) is asymptotically stable. ⎡ T ⎡ A1T ⎤ ⎤ T ⎡ P1 ⎤ ⎢ Aˆ P + PAˆ hAˆ ⎢ T ⎥ h ⎢ ⎥ ⎥ ⎣ P3 ⎦ ⎢ ⎣ 0 ⎦⎥ ⎢ * 0 ⎥ 0, Y > 0 ~ Denoting A := A + A1 and Z := X − Y −1 , the parameterized forms of the compensator are given as: ⎡ Dc ⎢B ⎣ c
⎡ W Cc ⎤ ⎢ = −1 Ac ⎥⎦ ⎢ BW − Y V ⎢ ⎣
(− WCX + U )Z −1
⎤ ⎛ − BWCX + BU + Y VCX ⎞ −1 ⎥ ⎜ ⎟Z ⎥ ~ ⎜ − Y −1 R + A ⎟ ⎥ X ⎝ ⎠ ⎦ −1
(26) ⎡X P −1 = Q = ⎢ ⎣Z
Z⎤ Z ⎥⎦
then we have
⎡Y P=⎢ ⎣− Y
−Y ⎤ Z −1 XY ⎥⎦
Substituting the parameterized compensator to the closed-loop system (22), the parameterized form of closed-loop system is obtained with the coefficient matrices as: ~ ⎤ ⎡ A (− BWCX + BU )Z −1 + BWC ⎥ ⎢ −1 ˆA = ⎢ BWC − Y −1VC ⎛⎜ − BWCX + BU + Y VCX ⎞⎟ Z −1 ⎥ ~ ⎢ ⎜ − Y −1 R + A ⎟ ⎥ X ⎝ ⎠ ⎦ ⎣ Corollary 1. For given positive scalars h > 0 , 0 < δ < 1 , if there exist symmetric positive-definite ⎡ P P3 ⎤ 2 n× 2 n matrices S ∈ ℜ n×n , S ≤ λI n , P = ⎢ 1 ⎥ ∈ℜ ⎣ ∗ P2 ⎦ M ∈ ℜ 2 n×2 n and a parameters set (25) satisfying (23) and the following LMIs:
© 2013 ACADEMY PUBLISHER
⎡Γ1 + Γ1T ⎢ ⎢ * ⎢ * ⎣
hΓ2T − hS * ⎡X ⎢I ⎣ n
where ~ ⎡A X + BU Γ1 = ⎢ R ⎣⎢
[
~ Γ2 = R YA + VC Γ3 = λ [A1 X
hΓ3T ⎤ ⎥ 0 ⎥0 Y ⎥⎦
(27a)
(27b)
~ A + BWC ⎤ ⎥ ~ YA + VC ⎦⎥
]
A1 ] .
then there exists a full order dynamic output feedback controller in form of (2) such that the closed-loop system (22) is asymptotically stable. In this case, the parameters of the desired controller are given in (25). Y ⎤ ⎡I Proof. Denoting T = ⎢ n ⎥ , then it follows: ⎣ 0 −Y⎦ ⎡I T T QT = ⎢ n ⎣Y ⎡I =⎢ n ⎣Y ⎡X So the LMI ⎢ ⎣I n
0 ⎤⎡ X − Y ⎥⎦ ⎢⎣ Z 0 ⎤⎡X − Y ⎥⎦ ⎢⎣ Z
Z ⎤⎡I n Y ⎤ Z ⎥⎦ ⎢⎣ 0 − Y ⎥⎦ In ⎤ ⎡X In ⎤ =⎢ ⎥ 0 ⎥⎦ ⎣ I n Y ⎦
In ⎤ > 0 is equivalent to P > 0 . Y ⎥⎦
Then substituting the parameterized functional matrix and the parameterized closed-loop system matrix into the matrix inequality (24), we can obtain: ⎡ T ⎡ A1T ⎤ ⎤ T ⎡ Y ⎤ ⎢ Aˆ P + PAˆ hAˆ ⎢ ⎥ hλ ⎢ ⎥ ⎥ ⎣− Y ⎦ ⎢ ⎣ 0 ⎦⎥ ⎢ * 0 ⎥
701
Dynamical Output Feedback Control for Distributed-delay Systems Juan Liu Department of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China Email: [email protected]
Lin Chen Department of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, China Email: [email protected]
Jiaolong Zhang School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo, China Email: [email protected]
Abstract—The dynamical output feedback controller design for systems with distributed delay is concerned in this paper. Firstly, model transformation of the closed-loop system is used and a proper Lyapunov-Krsovskii functional is constructed, by which the stability criterion is obtained in terms of nonlinear matrices inequality, which guarantees the asymptotical stability of closed-loop system. Secondly, the parameterization of controller is used and the delaydependent design condition of the desired controller is established in terms of linear matrices inequality. Finally, a simulation is given to illustrate the effectiveness of the proposed method. Index Terms—distributed delay, dynamical output feedback, Lyapunov-Krsovskii functional, linear matrix inequality (LMI)
I. INTRODUCTION Distributed time-delay is commonly encountered in various physical and engineering systems. It has been shown that the existence of delay is the sources of instability and poor performance of control systems. Therefore, stability analysis and controller design of linear systems with distributed delay have attracted much attention in the last years. To stability analysis of time-delay systems, in order to reduce the conservatism of stability criteria, many researchers developed different approaches, such as constructing discretized Lyapunov functional[1], model transformation[2, 3], integral inequality[4, 5], free matrices[6, 7] and constructing proper LyapunovKrasovskii functional[8, 9]. As we all known, the controller design depend on the stability results, many stability results do well in reducing the conservatism of the stability criteria, but they can not be used to controller design because of their complexity. Therefore, to the control problem of time-delay system, solving the parameters of controller is important as well as reducing the conservatism. In [10], the state feedback control of
© 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.3.701-708
distributed delay system was investigated. In [11], the controller design of linear system with distributed delay is studied via static output feedback. In [12, 13], the dynamical output feedback controllers were designed for distributed delay systems, but the results is delayindependent and more conservative. To the best of our knowledge, few results have been achieved on delaydependent dynamical feedback stabilization for linear systems with distributed delay. Based on above discussion, in this note, we study the stabilization problem for distributed time-delay system via dynamical output feedback. Through neutral model transformation of closed-loop system and constructing a proper Lyapunov-Krasovskii functional, a delaydependent stability criterion is obtained in terms of nonlinear matrix inequality. Then by using parameterization of controller, the design condition of the controller, which guarantees the asymptotical stability of closed-loop system, is established in terms of LMI, which is easy to check. In the end, an example is also given to illustrate the effectiveness of the proposed method. Notations: In this paper, C [− h,0]; ℜ n denotes the
(
)
family of continuous functions from [− h,0] to ℜ n . For symmetric matrices A and B, λmax ( A) and λmin ( A) denote the largest and smallest eigenvalue of A, respectively; the notation A ≥ B (respectively, A > B ) means that the matrix A − B is positive semi-definite (respectively, positive definite). ⋅ denotes the Euclidean
norm for vector or the spectral norm of matrices. In the symmetric block matrices, ∗ denotes a term that is induced by symmetry. Every matrices, if not explicitly mentioned in the paper, are assumed to have compatible dimensions. II. PROBLEM FORMULATION Consider the following system with distributed delay:
702
JOURNAL OF COMPUTERS, VOL. 8, NO. 3, MARCH 2013
x& (t ) = Ax(t ) + A1 x(t − h ) + A2
t
∫
Lemma 1 [14]. For given positive scalars α1 , α 2 , α 3 , where α1 + α 2 + α 3 < 1 , if a symmetric positive-definite
x(σ )dσ + Bu (t )
t −d
y (t ) = Cx(t )
(1)
x(t ) = ϕ (t ), t ∈ [−γ ,0]
where x ∈ ℜ n , u ∈ ℜ m , y ∈ ℜ h is state, control input and output respectively; h, d > 0 are constant delays, γ = max{h, d } , A, A1 , A2 , B, C are matrices with appropriate dimension. The aim of the note is to design the following dynamical output feedback controller: x&ˆ (t ) = Ac xˆ (t ) + Bc y (t ) u (t ) = Cc xˆ (t ) + Dc y (t ), t ≥ 0
(2)
which guaranteed the asymptotical stability of the closedloop system: t
ξ&(t ) = A ξ (t ) + A1ξ (t − h ) + A2 ∫ ξ (σ )dσ , t ≥ 0
(3)
matrix M ∈ ℜ 2 n× 2 n exists such that the following LMI holds, then the operator Gξ t is stable: ⎡ ⎢ N ' MN − α1M ⎢ hA ' MN 1 ⎢ ⎢ dA ' MN 2 ⎢⎣
⎡h 2 A1T MA1 − α1M ⎢ ⎢ dhA2T MA1 ⎣
where
⎡ x⎤
⎣ ⎦ ⎡I ⎤ A1 = ⎢ n ⎥[A1 0] ⎣0⎦
dhA2' MA1
⎤ ⎥ dN ' MA2 ' ⎥ 0 , we have:
y (t ) = Cx(t ), t ≥ 0 x(t ) = φ (t ),−h ≤ t ≤ 0
⎡ A + BDC C A=⎢ ⎣ BC C
⎡A ⎤ ∫ ∫ ξ (σ )⎢⎣ 0 ⎥⎦ S [A 0 t
∫
Applying Lemma 2, we have: ⎡ AT ⎤ V&2 (ξ t ) = hξ T (t )⎢ 1 ⎥ S [A1 0]ξ (t ) ⎣0⎦ t ⎡ AT ⎤ − ∫ ξ T (θ )⎢ 1 ⎥ S [A1 0]ξ (θ )dθ ⎣0⎦ t −h ⎡ AT ⎤ ≤ hξ T (t )⎢ 1 ⎥ S [A1 0]ξ (t ) ⎣0⎦ t t ⎡ AT ⎤ 1 − ∫ ξ T (θ )⎢ 1 ⎥ dθ S ∫ [A1 0]ξ (θ )dθ h t −h ⎣0⎦ t −h so, it is easy to get: ⎡ ˆT T ⎡ P1 ⎤ −1 ⎢ A P + PAˆ + hAˆ ⎢ T ⎥ S [P1 ⎣ P3 ⎦ ⎢ V& (ξ t ) ≤ ξ T (t )⎢ T ⎢+ h ⎡⎢ A1 ⎤⎥ S [A 0] ⎢ ⎣0⎦ 1 ⎣
⎤ P3 ]Aˆ ⎥ ⎥ ⎥ξ (t ) ⎥ ⎥ ⎦
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JOURNAL OF COMPUTERS, VOL. 8, NO. 3, MARCH 2013
by Lyapunov stability theorem, we can get the following condition which guarantees the stability of the closedloop system (22) : For given positive scalars h > 0 , 0 < δ < 1 , if there exist symmetric positive-definite matrices S ∈ ℜ n×n , ⎡ P P3 ⎤ 2 n×2 n , M ∈ ℜ 2 n×2 n satisfying (23) and P=⎢ 1 ⎥ ∈ℜ P ∗ 2⎦ ⎣
the following matrix inequality, then the closed-loop system (22) is asymptotically stable. ⎡ T ⎡ A1T ⎤ ⎤ T ⎡ P1 ⎤ ⎢ Aˆ P + PAˆ hAˆ ⎢ T ⎥ h ⎢ ⎥ ⎥ ⎣ P3 ⎦ ⎢ ⎣ 0 ⎦⎥ ⎢ * 0 ⎥ 0, Y > 0 ~ Denoting A := A + A1 and Z := X − Y −1 , the parameterized forms of the compensator are given as: ⎡ Dc ⎢B ⎣ c
⎡ W Cc ⎤ ⎢ = −1 Ac ⎥⎦ ⎢ BW − Y V ⎢ ⎣
(− WCX + U )Z −1
⎤ ⎛ − BWCX + BU + Y VCX ⎞ −1 ⎥ ⎜ ⎟Z ⎥ ~ ⎜ − Y −1 R + A ⎟ ⎥ X ⎝ ⎠ ⎦ −1
(26) ⎡X P −1 = Q = ⎢ ⎣Z
Z⎤ Z ⎥⎦
then we have
⎡Y P=⎢ ⎣− Y
−Y ⎤ Z −1 XY ⎥⎦
Substituting the parameterized compensator to the closed-loop system (22), the parameterized form of closed-loop system is obtained with the coefficient matrices as: ~ ⎤ ⎡ A (− BWCX + BU )Z −1 + BWC ⎥ ⎢ −1 ˆA = ⎢ BWC − Y −1VC ⎛⎜ − BWCX + BU + Y VCX ⎞⎟ Z −1 ⎥ ~ ⎢ ⎜ − Y −1 R + A ⎟ ⎥ X ⎝ ⎠ ⎦ ⎣ Corollary 1. For given positive scalars h > 0 , 0 < δ < 1 , if there exist symmetric positive-definite ⎡ P P3 ⎤ 2 n× 2 n matrices S ∈ ℜ n×n , S ≤ λI n , P = ⎢ 1 ⎥ ∈ℜ ⎣ ∗ P2 ⎦ M ∈ ℜ 2 n×2 n and a parameters set (25) satisfying (23) and the following LMIs:
© 2013 ACADEMY PUBLISHER
⎡Γ1 + Γ1T ⎢ ⎢ * ⎢ * ⎣
hΓ2T − hS * ⎡X ⎢I ⎣ n
where ~ ⎡A X + BU Γ1 = ⎢ R ⎣⎢
[
~ Γ2 = R YA + VC Γ3 = λ [A1 X
hΓ3T ⎤ ⎥ 0 ⎥0 Y ⎥⎦
(27a)
(27b)
~ A + BWC ⎤ ⎥ ~ YA + VC ⎦⎥
]
A1 ] .
then there exists a full order dynamic output feedback controller in form of (2) such that the closed-loop system (22) is asymptotically stable. In this case, the parameters of the desired controller are given in (25). Y ⎤ ⎡I Proof. Denoting T = ⎢ n ⎥ , then it follows: ⎣ 0 −Y⎦ ⎡I T T QT = ⎢ n ⎣Y ⎡I =⎢ n ⎣Y ⎡X So the LMI ⎢ ⎣I n
0 ⎤⎡ X − Y ⎥⎦ ⎢⎣ Z 0 ⎤⎡X − Y ⎥⎦ ⎢⎣ Z
Z ⎤⎡I n Y ⎤ Z ⎥⎦ ⎢⎣ 0 − Y ⎥⎦ In ⎤ ⎡X In ⎤ =⎢ ⎥ 0 ⎥⎦ ⎣ I n Y ⎦
In ⎤ > 0 is equivalent to P > 0 . Y ⎥⎦
Then substituting the parameterized functional matrix and the parameterized closed-loop system matrix into the matrix inequality (24), we can obtain: ⎡ T ⎡ A1T ⎤ ⎤ T ⎡ Y ⎤ ⎢ Aˆ P + PAˆ hAˆ ⎢ ⎥ hλ ⎢ ⎥ ⎥ ⎣− Y ⎦ ⎢ ⎣ 0 ⎦⎥ ⎢ * 0 ⎥
