切换时滞系统稳定性的若干问题研究
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摘要
有着广泛的工程背景和理论研究意义的切换时滞系统是一类重要的混杂系统。切换系统由多个子系统和切换规则组成,每个子系统只在切换规则激活下才成为系统模型,并且系统的状态轨迹在系统模型发生变化的时刻仍然连续。如果子系统或者切换规则包含时滞,就称之为切换时滞系统。本文所关注的内容是有关切换时滞系统Lyapunov意义下的稳定(主要指渐近稳定)和有界输入有界输出稳定(简称BIBO稳定)的判别方法的研究。
首先,针对系数矩阵存在Hurwitz线性凸组合的混合时滞(离散时滞和分布时滞)的切换线性系统,通过线性划分状态空间,构造各子系统渐近稳定的区域,并设计一个依赖于状态的切换信号,利用Lyapunov泛函和不等式分析技巧,讨论了这一类切换混合时滞系统的渐近稳定性。由Lyapunov方程和不等式的形式给出了这一类切换混合时滞系统渐近稳定的与时滞相关的充分条件。
其次,依次讨论了连续时间大系统、连续时间单时滞系统、混合时滞系统、多重混合时滞系统和离散时间单时滞系统的BIBO稳定性质。对于连续时间大系统,本文利用Lyapunov函数和不等式技巧,以线性矩阵不等式(LMI)的形式给出了连续时间多变量反馈控制大系统BIBO稳定的条件。在此基础上通过矩阵分析的技巧给出了状态反馈控制器的设计方法并将其推广到系统结构中存在不确定项的情形。对于连续时间单时滞系统、混合时滞系统、多重混合时滞系统,本文运用Lyapunov泛函理论并结合Riccati方程、线性矩阵不等式,分别给出了系统时滞无关和时滞相关的BIBO稳定性判据,并将其推广到时滞系统结构中存在扰动项和不确定项的情形,合理引入自由矩阵和运用Schur补引理,给出了当不确定项满足范数有界条件时,系统时滞无关和时滞相关的鲁棒BIBO稳定性判据。对于离散时间单时滞系统,采用了将离散时滞系统转化为无时滞的离散系统的方法,设计控制律,运用迭代法和矩阵范数的性质,给出了系统时滞无关的BIBO稳定的条件。
最后,分两种情形讨论了状态反馈控制下切换线性时滞系统BIBO稳定的问题。针对系数矩阵也存在Hurwitz线性凸组合的切换时滞系统,通过对状态空间有效的划分,设计一类状态依赖的切换规则,利用Lyapunov泛函理论和不等式技巧,给出了切换时滞系统的状态反馈控制器的设计方法和在状态反馈控制下BIBO稳定的与时滞相关的充分条件。由线性矩阵不等式的形式给出的结论可由Matlab工具箱快速得到可行解。对于切换域是固定的一类切换时滞系统,通过构造分段连续的二次Lyapunov泛函和Riccati方程,得到了在这类固定切换域的约束下切换时滞系统BIBO稳定的充分条件。系统的切换信号受事件驱动,依赖于系统状态的变化。
Switched systems are important class of hybrid systems,which have been widely applied to the engineering and have great significance of theoretical study. Switched systems consist of more than one subsystem and a switching rule indicating the active subsystem at each instant of time. The state trajectory of system is continuous at the time when the model is changed. If time delays are in each subsystem or in the switching rule, it is called switched systems with delays. Founding the criteria of stability of switched systems with delays in the sense of Lyapunov (focusing on asymptotical stability) and Bounded Input and Bounded Output stability (shorting for BIBO stability) is the main consideration of this paper.
Firstly, assuming that there exists a Hurwitz linear convex combination in the coefficient matrices of switched linear systems with mixed delays (both discrete and distributed time delays), this paper is to propose the uniform asymptotic stability conditions for this class of switched systems by Lyapunov functional and inequality technique. According to the linear partition of state space, we can construct the stable regions and a state dependent switching rule. By a defined Lyapunov equation and certain inequality, sufficient conditions of stability are given.
Secondly, this paper is to discuss the BIBO stability of continuous large scale systems, continuous systems with single delay, mixed delays, multiple mixed delays and discrete systems with single delay. For continuous large scale systems, we use Lyapunov function and inequality to obtain the novel BIBO stabilization criteria expressed in terms of linear matrix inequality (LMI) for continuous multivariable feedback systems. Based on this, the design of state feedback is given by matrix transform. The issue of robust BIBO stabilization for uncertain large scale systems is also addressed. For continuous control systems with single delay, mixed delays, multiple mixed delays, delay-independent and delay-dependent stabilizable criteria for these control systems are presented by theory of Lyapunov functional and Riccati equation, linear matrix inequality technique to guarantee that the bounded input lead to the bounded output. The robust BIBO stability for such systems with perturbation and uncertainty is also discussed. When the norm condition of time-varying uncertain matrices is satisfied, novel robust BIBO stabilization criteria delay-independent and delay-dependent are also established by the introduction of free-weighting matrices and the Schur’s lemma. For the discrete systems with time delay, an equivalent transformation to get systems without delay is presented and a control law is designed. At last, delay-independent BIBO stability criteria are given by iterative relation and theory of matrix norm.
Lastly, this paper is concerned with the problems of BIBO stabilization of the switched linear systems in presence of time delays under feedback control. This part consists of two cases. Under the assumption that there also exists a Hurwitz linear convex combination in the coefficient matrices of delayed switched linear systems, criteria of BIBO stabilization are established by linear partition of state space and a state-dependent switching rule. Stabilization criteria and the design of state feedback controller are derived and proposed by Lyapunov functional and inequality technique. BIBO stabilization criteria are given in terms of LMI and can be easily solved by LMI Toolbox in Matlab. For switched delay systems in fixed switching regions, the main contribution of the paper is the derivation of sufficient conditions of BIBO stabilization in the form of algebraic Riccati matrix equation based on piecewise quadratic Lyapunov functional. The switching rule is driven by event and state-dependent.
首先,针对系数矩阵存在Hurwitz线性凸组合的混合时滞(离散时滞和分布时滞)的切换线性系统,通过线性划分状态空间,构造各子系统渐近稳定的区域,并设计一个依赖于状态的切换信号,利用Lyapunov泛函和不等式分析技巧,讨论了这一类切换混合时滞系统的渐近稳定性。由Lyapunov方程和不等式的形式给出了这一类切换混合时滞系统渐近稳定的与时滞相关的充分条件。
其次,依次讨论了连续时间大系统、连续时间单时滞系统、混合时滞系统、多重混合时滞系统和离散时间单时滞系统的BIBO稳定性质。对于连续时间大系统,本文利用Lyapunov函数和不等式技巧,以线性矩阵不等式(LMI)的形式给出了连续时间多变量反馈控制大系统BIBO稳定的条件。在此基础上通过矩阵分析的技巧给出了状态反馈控制器的设计方法并将其推广到系统结构中存在不确定项的情形。对于连续时间单时滞系统、混合时滞系统、多重混合时滞系统,本文运用Lyapunov泛函理论并结合Riccati方程、线性矩阵不等式,分别给出了系统时滞无关和时滞相关的BIBO稳定性判据,并将其推广到时滞系统结构中存在扰动项和不确定项的情形,合理引入自由矩阵和运用Schur补引理,给出了当不确定项满足范数有界条件时,系统时滞无关和时滞相关的鲁棒BIBO稳定性判据。对于离散时间单时滞系统,采用了将离散时滞系统转化为无时滞的离散系统的方法,设计控制律,运用迭代法和矩阵范数的性质,给出了系统时滞无关的BIBO稳定的条件。
最后,分两种情形讨论了状态反馈控制下切换线性时滞系统BIBO稳定的问题。针对系数矩阵也存在Hurwitz线性凸组合的切换时滞系统,通过对状态空间有效的划分,设计一类状态依赖的切换规则,利用Lyapunov泛函理论和不等式技巧,给出了切换时滞系统的状态反馈控制器的设计方法和在状态反馈控制下BIBO稳定的与时滞相关的充分条件。由线性矩阵不等式的形式给出的结论可由Matlab工具箱快速得到可行解。对于切换域是固定的一类切换时滞系统,通过构造分段连续的二次Lyapunov泛函和Riccati方程,得到了在这类固定切换域的约束下切换时滞系统BIBO稳定的充分条件。系统的切换信号受事件驱动,依赖于系统状态的变化。
Switched systems are important class of hybrid systems,which have been widely applied to the engineering and have great significance of theoretical study. Switched systems consist of more than one subsystem and a switching rule indicating the active subsystem at each instant of time. The state trajectory of system is continuous at the time when the model is changed. If time delays are in each subsystem or in the switching rule, it is called switched systems with delays. Founding the criteria of stability of switched systems with delays in the sense of Lyapunov (focusing on asymptotical stability) and Bounded Input and Bounded Output stability (shorting for BIBO stability) is the main consideration of this paper.
Firstly, assuming that there exists a Hurwitz linear convex combination in the coefficient matrices of switched linear systems with mixed delays (both discrete and distributed time delays), this paper is to propose the uniform asymptotic stability conditions for this class of switched systems by Lyapunov functional and inequality technique. According to the linear partition of state space, we can construct the stable regions and a state dependent switching rule. By a defined Lyapunov equation and certain inequality, sufficient conditions of stability are given.
Secondly, this paper is to discuss the BIBO stability of continuous large scale systems, continuous systems with single delay, mixed delays, multiple mixed delays and discrete systems with single delay. For continuous large scale systems, we use Lyapunov function and inequality to obtain the novel BIBO stabilization criteria expressed in terms of linear matrix inequality (LMI) for continuous multivariable feedback systems. Based on this, the design of state feedback is given by matrix transform. The issue of robust BIBO stabilization for uncertain large scale systems is also addressed. For continuous control systems with single delay, mixed delays, multiple mixed delays, delay-independent and delay-dependent stabilizable criteria for these control systems are presented by theory of Lyapunov functional and Riccati equation, linear matrix inequality technique to guarantee that the bounded input lead to the bounded output. The robust BIBO stability for such systems with perturbation and uncertainty is also discussed. When the norm condition of time-varying uncertain matrices is satisfied, novel robust BIBO stabilization criteria delay-independent and delay-dependent are also established by the introduction of free-weighting matrices and the Schur’s lemma. For the discrete systems with time delay, an equivalent transformation to get systems without delay is presented and a control law is designed. At last, delay-independent BIBO stability criteria are given by iterative relation and theory of matrix norm.
Lastly, this paper is concerned with the problems of BIBO stabilization of the switched linear systems in presence of time delays under feedback control. This part consists of two cases. Under the assumption that there also exists a Hurwitz linear convex combination in the coefficient matrices of delayed switched linear systems, criteria of BIBO stabilization are established by linear partition of state space and a state-dependent switching rule. Stabilization criteria and the design of state feedback controller are derived and proposed by Lyapunov functional and inequality technique. BIBO stabilization criteria are given in terms of LMI and can be easily solved by LMI Toolbox in Matlab. For switched delay systems in fixed switching regions, the main contribution of the paper is the derivation of sufficient conditions of BIBO stabilization in the form of algebraic Riccati matrix equation based on piecewise quadratic Lyapunov functional. The switching rule is driven by event and state-dependent.
引文
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