非受限及Markov随机型线性切换系统的分析与控制
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摘要
切换系统是一类重要的混杂系统,它是由几个连续时间子系统或离散时间子系统及作用在其中的切换规则构成的。许多工程实际问题都可以用这样的系统模型来描述。在过去的近五十年中,切换系统以其广泛的实际应用背景和重大的理论研究意义而受到人们极大的关注,一直是系统科学领域研究的热点之一。
切换系统研究的一个重要分支就是具有Markov跳跃参数的混合系统,也称为Markov切换系统(或Markov跳跃系统),其特征是系统各模态之间的随机切换符合一定的统计特性———Markov过程。正是Markov切换系统所具有的特殊混合信息结构,使得所研究的控制方法并不能用传统的单由时间驱动的动态系统的控制理论或者由单一针对离散事件驱动的动态系统的控制理论来代替,因此Markov切换系统的控制理论研究是一个具有挑战性的课题。
本文以李雅普诺夫稳定性理论为基础,采用线性矩阵不等式方法,研究了几类切换系统分析与控制设计的若干问题,主要内容如下:
(1)介绍了切换系统的基本概念及相关定义,详细综述了切换系统控制理论的研究历史和现状,给出了论文选题的理论意义和实际应用背景,以及本文的主要研究对象与研究内容。
(2)针对一类具有参数不确定性的离散时间切换系统,在任意切换信号情况下,引入鲁棒D-稳定的概念,研究了这类系统的极点配置问题。设计了保证闭环不确定系统同时具有鲁棒D-稳定性和H_∞性能条件的状态反馈控制器。
(3)研究了一类不确定线性离散切换系统带有状态重置的鲁棒H_∞与鲁棒l_2-l_∞动态输出反馈控制问题。通过在切换时刻对控制器的状态重置,得到使闭环系统具有更优H_∞性能与l_2-l_∞性能的低保守性动态输出反馈控制器设计方案。
(4)研究了时变时滞Markov切换系统的随机稳定性问题。基于随机稳定性理论,通过建立不同的Lyapunov-Krasovskii函数,以线性矩阵不等式(LMI)形式,得到具有较小保守性的时滞依赖稳定性条件。同时研究此类系统的H_∞滤波算法。
(5)针对时变时滞模型相关的时滞Markov切换系统,研究了时变时滞Markov切换系统的随机稳定性与镇定问题。同时基于随机有界实引理,研究了此类时滞Markov切换系统的H_∞性能分析问题。依据新的H_∞性能准则,讨论了此类系统的H_∞控制问题。
(6)研究了Ito?型时滞Markov切换系统的随机稳定性问题。通过构造改进的Lyapunov-Krasovskii泛函,并建立其较小的导数上界,得到具有较小保守性的均方指数稳定准则。在此基础上研究了这类系统的H_∞滤波问题。
Switched systems are an important category of hybrid systems, which consist of continuous-time subsystems or discrete-time subsystems and the rule acting among them. Many practical engineering issues can be described by this class of switched systems. Due to the signaficance of both theory and broad practical applications, the study of switched systems has been received much attention, and has become one of the hotspots of research in the field of systems science.
A special class of switched systems is the systems with Markovian jump parameters, i.e. Markovian switching systems (or say Markovian jump systems), in which the transitions among differengt regimes are random and are further supposed as a Markov process. Due to the complicated hybrid information structure, the research of this class of jump systems can not be treated as simple combination of traditional control theories for the continous systems driven by time or discrete events systems. Thus, the invetigation of the control theories for Markovian switching systems is a challenging work.
Based on Lyapunov stability theory together with linear matrix inequility technique, this dissertation investigates the analysis and synthesis problems for the above-mentioned switched systems, and the main contributions are as follows:
(1) The basic conceptions and concerned definitions of switched systems are introduced, then, the internal and overseas research situations for the control theory of switched systems are classified and summarized, where the theoretical significance and the practical application backgrounds are given, and the study objects and contents of this dissertation are presented.
(2) The pole placement problem is studied for a class of linear discrete-time switched systems with parameter uncertainties by introducing the conception of robust D - stability. Then a state-feedback controller is designed to make the closed uncertain system be robust D -stable and have H_∞performance.
(3) The robust H_∞and l_2-l_∞output feedback controllers via state-reset are designed for a class of uncertain linear discrete-time switched systems. We address output feedback stabilization algorithms through resetting the state of the controllers at the switching moments, in which more optimized H_∞and l_2-l_∞performances are required for the closed systems.
(4) The stochastic stability of Markovian switching systems with time-varying delays is investigated. Based on the stochastic stability theory, new stability criteria are obtained in terms of LMIs by constructing different Lyapunov-Krasovskii functional. Then, the H_∞filtering algorithm is proposed.
(5) For a class of Markovian switching systems with mode-dependent time-varying delays, the stochastic stability and stabilization problems are concerned. Meanwhile, based on the stochastic bounded real lemma, the condition is derived from the H_∞performance. According to the new H_∞performance criterion, our attention is focused on the H_∞control problem for this class of systems.
(6) The stability problem for a class of Ito? stochastic Markovian switching systems with time delays is investigated. By constructing an improved Lyapunov-Krasovskii functional, a tighter upper bound of its derivative is established, and the exponential mean-square stability condition with reduced conservativeness is obtained for this class of systems. Then, the H_∞filter is designed for such class of systems.
切换系统研究的一个重要分支就是具有Markov跳跃参数的混合系统,也称为Markov切换系统(或Markov跳跃系统),其特征是系统各模态之间的随机切换符合一定的统计特性———Markov过程。正是Markov切换系统所具有的特殊混合信息结构,使得所研究的控制方法并不能用传统的单由时间驱动的动态系统的控制理论或者由单一针对离散事件驱动的动态系统的控制理论来代替,因此Markov切换系统的控制理论研究是一个具有挑战性的课题。
本文以李雅普诺夫稳定性理论为基础,采用线性矩阵不等式方法,研究了几类切换系统分析与控制设计的若干问题,主要内容如下:
(1)介绍了切换系统的基本概念及相关定义,详细综述了切换系统控制理论的研究历史和现状,给出了论文选题的理论意义和实际应用背景,以及本文的主要研究对象与研究内容。
(2)针对一类具有参数不确定性的离散时间切换系统,在任意切换信号情况下,引入鲁棒D-稳定的概念,研究了这类系统的极点配置问题。设计了保证闭环不确定系统同时具有鲁棒D-稳定性和H_∞性能条件的状态反馈控制器。
(3)研究了一类不确定线性离散切换系统带有状态重置的鲁棒H_∞与鲁棒l_2-l_∞动态输出反馈控制问题。通过在切换时刻对控制器的状态重置,得到使闭环系统具有更优H_∞性能与l_2-l_∞性能的低保守性动态输出反馈控制器设计方案。
(4)研究了时变时滞Markov切换系统的随机稳定性问题。基于随机稳定性理论,通过建立不同的Lyapunov-Krasovskii函数,以线性矩阵不等式(LMI)形式,得到具有较小保守性的时滞依赖稳定性条件。同时研究此类系统的H_∞滤波算法。
(5)针对时变时滞模型相关的时滞Markov切换系统,研究了时变时滞Markov切换系统的随机稳定性与镇定问题。同时基于随机有界实引理,研究了此类时滞Markov切换系统的H_∞性能分析问题。依据新的H_∞性能准则,讨论了此类系统的H_∞控制问题。
(6)研究了Ito?型时滞Markov切换系统的随机稳定性问题。通过构造改进的Lyapunov-Krasovskii泛函,并建立其较小的导数上界,得到具有较小保守性的均方指数稳定准则。在此基础上研究了这类系统的H_∞滤波问题。
Switched systems are an important category of hybrid systems, which consist of continuous-time subsystems or discrete-time subsystems and the rule acting among them. Many practical engineering issues can be described by this class of switched systems. Due to the signaficance of both theory and broad practical applications, the study of switched systems has been received much attention, and has become one of the hotspots of research in the field of systems science.
A special class of switched systems is the systems with Markovian jump parameters, i.e. Markovian switching systems (or say Markovian jump systems), in which the transitions among differengt regimes are random and are further supposed as a Markov process. Due to the complicated hybrid information structure, the research of this class of jump systems can not be treated as simple combination of traditional control theories for the continous systems driven by time or discrete events systems. Thus, the invetigation of the control theories for Markovian switching systems is a challenging work.
Based on Lyapunov stability theory together with linear matrix inequility technique, this dissertation investigates the analysis and synthesis problems for the above-mentioned switched systems, and the main contributions are as follows:
(1) The basic conceptions and concerned definitions of switched systems are introduced, then, the internal and overseas research situations for the control theory of switched systems are classified and summarized, where the theoretical significance and the practical application backgrounds are given, and the study objects and contents of this dissertation are presented.
(2) The pole placement problem is studied for a class of linear discrete-time switched systems with parameter uncertainties by introducing the conception of robust D - stability. Then a state-feedback controller is designed to make the closed uncertain system be robust D -stable and have H_∞performance.
(3) The robust H_∞and l_2-l_∞output feedback controllers via state-reset are designed for a class of uncertain linear discrete-time switched systems. We address output feedback stabilization algorithms through resetting the state of the controllers at the switching moments, in which more optimized H_∞and l_2-l_∞performances are required for the closed systems.
(4) The stochastic stability of Markovian switching systems with time-varying delays is investigated. Based on the stochastic stability theory, new stability criteria are obtained in terms of LMIs by constructing different Lyapunov-Krasovskii functional. Then, the H_∞filtering algorithm is proposed.
(5) For a class of Markovian switching systems with mode-dependent time-varying delays, the stochastic stability and stabilization problems are concerned. Meanwhile, based on the stochastic bounded real lemma, the condition is derived from the H_∞performance. According to the new H_∞performance criterion, our attention is focused on the H_∞control problem for this class of systems.
(6) The stability problem for a class of Ito? stochastic Markovian switching systems with time delays is investigated. By constructing an improved Lyapunov-Krasovskii functional, a tighter upper bound of its derivative is established, and the exponential mean-square stability condition with reduced conservativeness is obtained for this class of systems. Then, the H_∞filter is designed for such class of systems.
引文
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