时滞分布参数控制系统指数镇定的LMI方法
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Abstract A renovating method for distributed parameter control systems with constants, varying-delays, and multi-varying-delays is put forward. By constructing average Lyapunov functions and employing linear matrix inequality (LMI) and other matrix inequality technologies, several sufficient conditions for exponential stabilization are derived. In this method, the conditions are delay-dependent and at the same time, the upper-bound of exponential convergence rate is obtained. In addition, the distinctive advantage of our method is that the criteria mentioned in the paper are easy to check, so it can be applied to practice easily. Finally, a computation example is given to illustrate the proposed method. Key words Distributed parameter, stabilization, delay, linear matrix inequality (LMI)
holds. In this paper, we let W (x , t) u (x , t) = KW (9)
Theorem 1. For arbitrarily given A0 , A, B , and β , if there exist a matrix K and a positive matrix P , such that the following LMI
k=1 n
∂ wi (x , t) + a0 ij wj (x , t)+ ∂x2 k j =1
n
2
n
aij wj (x , t − τ ) +
j =1 j =1
bij uj (x , t) (1)
where Q(x ) = Q (x ), R(x ) = R (x ), and S (x ) is affine on x. Lemma 2[16−17] . Let U1 , U2 , U3 be real matrices, and T U3 = U3 > 0, then for an arbitrary scalar β > 0, the following inequality
The model of distributed parameter control systems is widely applied in heat processing, migration, and other areas, therefore it is significant to research the control of distributed parameter systems. As we know, variable structure control is the main method applied in the parameters control system at present[1−12] . However, it is especially difficult to avoid the wobble phenomenon[6] , and the control that is designed as a tool for operator semigroup theory or a matrix norm theory is based on variable structure control theory[1−12] . Reference [6] has pointed that the controller based on the semigroups operator theory is difficult to use in practice because it is hard to verify compactness, incredulity, and exchangeability of the operator, which are requested. On the other hand, variable structure controller designed on matrix norm theorem is also difficult to apply. Therefore, it is a hot topic to find a practical and effective method for distributed parameter control system. In order to avoid the above-mentioned problems, we recently proposed a way with some useful results[13−14] . However, these results were mainly used by comparative principles. In this method, the stability conditions of the closed-loop system require that all the system parameters should be of absolute value format. This paper is to propose a new method to obtain the stabilization conditions of the distributed parameter control system. By choosing a Lyapunov function, applying distributed control, and using linear matrix inequality (LMI) and the related theory of matrix inequality with the choice of linear state feedback controller, the exponential stabilization of distributed parameter systems with constants, varying delay, and multi-varying-delays is obtained.
>0
(6)
1
Description
Consider the following distributed parameter system with multi-varying-delays ∂wi (x , t) =D ∂t
m
Q(x ) − S (x )R−1 (x )S T (x ) > 0
T
(7)
The matrix form of system (1) is W ∂W W (x , t − τ ) + Bu u(x , t) = D W (x , t) + A0W (x , t) + AW ∂t (2) where (x , t) ∈ Ω × R+ , D > 0, and τ > 0 are constants; A0 = (a0 ij ), A = (aij ), and B = (bij ) are constant matrices with corresponding ranks; Ω = {x , x < l < +∞} ⊂ Rm is the bounded domain with smooth boundary ∂ Ω, and mesΩ > 0 (mes is short for measure). State function W (x, t) = col(w1 (x, t), w2 (x, t), · · · , wn (x, t)) ∈ Rn , ∆ = ∂2 m is the Laplace diffusion operator on Ω. And the k=1 ∂x2 k initial value and boundary value conditions satisfy W (x , t) = 0, (x , t) ∈ ∂ Ω × [−τ, +∞) (3) (4) (5)
W (x, t) ∂W = 0, (x , t) ∈ ∂ Ω × [−τ, +∞) n ∂n W (x , t) = ϕ (x , t), (x , t) ∈ ∂ Ω × [−τ, 0)
where n is the unit outward normal vector of ∂ Ω and ϕ (x , t) is the suitable smooth function.
Vol. 35, No. 3
ACTA AUTOMATICA SINICA
March, 2009
LMI Approach to Exponential Stabilization of Distributed Parameter Control Systems with Delay
LUO Yi-Ping1 XIA Wen-Hua1 LIU Guo-Rong1 DENG Fei-Qi2
T T T −1 T U2 U1 + U1 U2 ≤ β −1 U1 U3 U1 + βU2 U3 U2
i = 1, 2, · · · , n
(8)
Received October 19, 2007; in revised form June 24, 2008 Supported by the Natural Science Foundation of Hunan Province (07JJ6112), the Construct Program of the Key Discipline in Hunan Province (control theory and control engineering), and Scientific Research Fund of Hunan Provincial Education Department (04A012, 07A015) 1. Hunan Institute of Engineering, Xiangtan 411101, P. R. China 2. Institute of System Engineering, South China University of Technology, Guangzhou 510640, P. R. China DOI: 10.3724/SP.J.1004.2009.00299
2
Main results
In order to get our main results, we first give some lemmas. Lemma 1[15] . The inequality Q(x ) S (x ) is equal to R(x ) > 0,
Hale Waihona Puke T TS (x ) R(x )
300
T T A0 + AT 0 + BK + K B + 2kI + βP kτ T e A
ACTA AUTOMATICA SINICA ekτ A −βP <0 (10) According to Lemma 2, it follows that
holds. In this paper, we let W (x , t) u (x , t) = KW (9)
Theorem 1. For arbitrarily given A0 , A, B , and β , if there exist a matrix K and a positive matrix P , such that the following LMI
k=1 n
∂ wi (x , t) + a0 ij wj (x , t)+ ∂x2 k j =1
n
2
n
aij wj (x , t − τ ) +
j =1 j =1
bij uj (x , t) (1)
where Q(x ) = Q (x ), R(x ) = R (x ), and S (x ) is affine on x. Lemma 2[16−17] . Let U1 , U2 , U3 be real matrices, and T U3 = U3 > 0, then for an arbitrary scalar β > 0, the following inequality
The model of distributed parameter control systems is widely applied in heat processing, migration, and other areas, therefore it is significant to research the control of distributed parameter systems. As we know, variable structure control is the main method applied in the parameters control system at present[1−12] . However, it is especially difficult to avoid the wobble phenomenon[6] , and the control that is designed as a tool for operator semigroup theory or a matrix norm theory is based on variable structure control theory[1−12] . Reference [6] has pointed that the controller based on the semigroups operator theory is difficult to use in practice because it is hard to verify compactness, incredulity, and exchangeability of the operator, which are requested. On the other hand, variable structure controller designed on matrix norm theorem is also difficult to apply. Therefore, it is a hot topic to find a practical and effective method for distributed parameter control system. In order to avoid the above-mentioned problems, we recently proposed a way with some useful results[13−14] . However, these results were mainly used by comparative principles. In this method, the stability conditions of the closed-loop system require that all the system parameters should be of absolute value format. This paper is to propose a new method to obtain the stabilization conditions of the distributed parameter control system. By choosing a Lyapunov function, applying distributed control, and using linear matrix inequality (LMI) and the related theory of matrix inequality with the choice of linear state feedback controller, the exponential stabilization of distributed parameter systems with constants, varying delay, and multi-varying-delays is obtained.
>0
(6)
1
Description
Consider the following distributed parameter system with multi-varying-delays ∂wi (x , t) =D ∂t
m
Q(x ) − S (x )R−1 (x )S T (x ) > 0
T
(7)
The matrix form of system (1) is W ∂W W (x , t − τ ) + Bu u(x , t) = D W (x , t) + A0W (x , t) + AW ∂t (2) where (x , t) ∈ Ω × R+ , D > 0, and τ > 0 are constants; A0 = (a0 ij ), A = (aij ), and B = (bij ) are constant matrices with corresponding ranks; Ω = {x , x < l < +∞} ⊂ Rm is the bounded domain with smooth boundary ∂ Ω, and mesΩ > 0 (mes is short for measure). State function W (x, t) = col(w1 (x, t), w2 (x, t), · · · , wn (x, t)) ∈ Rn , ∆ = ∂2 m is the Laplace diffusion operator on Ω. And the k=1 ∂x2 k initial value and boundary value conditions satisfy W (x , t) = 0, (x , t) ∈ ∂ Ω × [−τ, +∞) (3) (4) (5)
W (x, t) ∂W = 0, (x , t) ∈ ∂ Ω × [−τ, +∞) n ∂n W (x , t) = ϕ (x , t), (x , t) ∈ ∂ Ω × [−τ, 0)
where n is the unit outward normal vector of ∂ Ω and ϕ (x , t) is the suitable smooth function.
Vol. 35, No. 3
ACTA AUTOMATICA SINICA
March, 2009
LMI Approach to Exponential Stabilization of Distributed Parameter Control Systems with Delay
LUO Yi-Ping1 XIA Wen-Hua1 LIU Guo-Rong1 DENG Fei-Qi2
T T T −1 T U2 U1 + U1 U2 ≤ β −1 U1 U3 U1 + βU2 U3 U2
i = 1, 2, · · · , n
(8)
Received October 19, 2007; in revised form June 24, 2008 Supported by the Natural Science Foundation of Hunan Province (07JJ6112), the Construct Program of the Key Discipline in Hunan Province (control theory and control engineering), and Scientific Research Fund of Hunan Provincial Education Department (04A012, 07A015) 1. Hunan Institute of Engineering, Xiangtan 411101, P. R. China 2. Institute of System Engineering, South China University of Technology, Guangzhou 510640, P. R. China DOI: 10.3724/SP.J.1004.2009.00299
2
Main results
In order to get our main results, we first give some lemmas. Lemma 1[15] . The inequality Q(x ) S (x ) is equal to R(x ) > 0,
Hale Waihona Puke T TS (x ) R(x )
300
T T A0 + AT 0 + BK + K B + 2kI + βP kτ T e A
ACTA AUTOMATICA SINICA ekτ A −βP <0 (10) According to Lemma 2, it follows that