切换系统的最优控制和稳定性
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摘要
切换系统是一类非常重要的混合系统,它是由若干个连续时间或离散时间或连续和离散时间子系统及一个作用在其上的切换规则组成。在过去的近四十年中,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,一直是控制界研究的热点之一。
本文研究了切换系统的最优控制、稳定性和吸引域的估计等问题。主要研究内容及创新点如下:
(1)在本文中我们提出了一种新的嵌入方法来研究带m个模态的切换系统的最优控制问题。用这种新的嵌入方式我们先把切换系统嵌入到一族新的更大的系统中,对这族系统建立起相应的最优控制问题,接着直接根据李雅普诺夫定理证明如下关键结论:由切换系统轨线组成的集合在由嵌入系统轨线组成的集合中是稠密的,最后基于这个结论通过研究嵌入系统的最优控制问题,我们获得了切换系统最优控制问题的最优解和次优解。
(2)对一类新的切换系统-由连续时间子系统和离散时间子系统构成的切换系统我们研究其稳定性和镇定问题。目前这类由连续子系统和离散子系统构成的切换系统的稳定性结果都是针对线性情形的。于是在本文中对这类切换系统的一个非线性情形我们给出了其状态的估计,获得其全局指数稳定的充分条件。此外,当该切换系统存在不稳定的子系统时我们还设计了使其渐近稳定的切换信号。
(3)我们对一类具有三角结构的连续切换系统的稳定性和吸引域的估计进行了讨论,先给出这类连续切换系统在任意切换下全局渐近稳定的充分条件,然后给出了该切换系统吸引域的估计,并把这些估计应用到几个具体的例子中。
(4)我们研究了一类具有三角结构的离散切换系统的稳定性和吸引域估计,先给出这类离散切换系统在任意切换下全局渐近稳定性的一般结果,接着对该离散切换系统的全局渐近稳定性又给出了更易于验证的充分条件,最后对该离散切换系统的吸引域进行了讨论,给出了其吸引域的估计,并用具体的例子进行了说明。
Switched systems are an important class of hybrid systems, which consist of several subsystems (continuous-time subsystems or discrete-time subsystems or continuous-time and discrete-time subsystems) and a rule that orchestrates the switching among them. In the last four decades a study of switched systems has received much attention due to signaficance in theory and broad practical applications, and it is one of hotspot of research of control science at all times.
In this dissertation we study optimal control, stability and estimates of the region of attraction of switched sytems. The main contributions are as follows.
(1) In this dissertation we propose a new embedding approach to investigate the optimal control problem of the switched system with m modes. By means of the new embedding approach the switched systesm is embedded into a new larger family of systems and the corresponding optimal control problem is formulated for the new embedded system, and we directly use the Lyapunov theorem to prove the following key result: the set of trajectories of the switched system with m modes is dense in the set of trajectories of the embedded system. Based on the above key result we obtain optimal solutions and suboptimal solutions of the optimal control problem of the switched system with m modes by studying the embedded optimal control problem.
(2)We study stability and stabilization of a new type of switched systems (switched systems with continuous-time and discrete-time subsystems) proposed in the literature. Up to now, for stability of such type of nonlinear switched systems there are not any available results. Therefore for a specific class of switched systems composed of continuous-time nonlinear subsystems and discrete-time uncertain subsystems we give an estimation of state of this class of switched systems, and from the estimation we present sufficient conditions for global exponential stability. Moreover, when there are unstable subsystems in this class of switched systems, we obtain some results on stabilization of this class of switched systems.
(3)Stability and estimates of the region of attraction of a class of continuous-time switched systems with triangular forms are addressed. We first give sufficient conditions for global asymptotic stability under arbitrary switching signals of switched systems with triangular forms, and then we estimate the region of attraction of switched systems with triangular forms. Various examples are presented to illustrate these estimates.
(4)Stability and estimates of the region of attraction of a class of discrete-time switched systems with triangular forms are also studied. We first give a result on global asymptotic stability of switched systems under arbitrary switching signals with triangular forms, on the basis of the result we also present suffiecient conditions which are easily checked for its global asymptotic stability under arbitrary switching signals, and then we estimate the region of attraction of this class of switched systems with triangular forms. Some examples are presented to illustrate these estimates.
本文研究了切换系统的最优控制、稳定性和吸引域的估计等问题。主要研究内容及创新点如下:
(1)在本文中我们提出了一种新的嵌入方法来研究带m个模态的切换系统的最优控制问题。用这种新的嵌入方式我们先把切换系统嵌入到一族新的更大的系统中,对这族系统建立起相应的最优控制问题,接着直接根据李雅普诺夫定理证明如下关键结论:由切换系统轨线组成的集合在由嵌入系统轨线组成的集合中是稠密的,最后基于这个结论通过研究嵌入系统的最优控制问题,我们获得了切换系统最优控制问题的最优解和次优解。
(2)对一类新的切换系统-由连续时间子系统和离散时间子系统构成的切换系统我们研究其稳定性和镇定问题。目前这类由连续子系统和离散子系统构成的切换系统的稳定性结果都是针对线性情形的。于是在本文中对这类切换系统的一个非线性情形我们给出了其状态的估计,获得其全局指数稳定的充分条件。此外,当该切换系统存在不稳定的子系统时我们还设计了使其渐近稳定的切换信号。
(3)我们对一类具有三角结构的连续切换系统的稳定性和吸引域的估计进行了讨论,先给出这类连续切换系统在任意切换下全局渐近稳定的充分条件,然后给出了该切换系统吸引域的估计,并把这些估计应用到几个具体的例子中。
(4)我们研究了一类具有三角结构的离散切换系统的稳定性和吸引域估计,先给出这类离散切换系统在任意切换下全局渐近稳定性的一般结果,接着对该离散切换系统的全局渐近稳定性又给出了更易于验证的充分条件,最后对该离散切换系统的吸引域进行了讨论,给出了其吸引域的估计,并用具体的例子进行了说明。
Switched systems are an important class of hybrid systems, which consist of several subsystems (continuous-time subsystems or discrete-time subsystems or continuous-time and discrete-time subsystems) and a rule that orchestrates the switching among them. In the last four decades a study of switched systems has received much attention due to signaficance in theory and broad practical applications, and it is one of hotspot of research of control science at all times.
In this dissertation we study optimal control, stability and estimates of the region of attraction of switched sytems. The main contributions are as follows.
(1) In this dissertation we propose a new embedding approach to investigate the optimal control problem of the switched system with m modes. By means of the new embedding approach the switched systesm is embedded into a new larger family of systems and the corresponding optimal control problem is formulated for the new embedded system, and we directly use the Lyapunov theorem to prove the following key result: the set of trajectories of the switched system with m modes is dense in the set of trajectories of the embedded system. Based on the above key result we obtain optimal solutions and suboptimal solutions of the optimal control problem of the switched system with m modes by studying the embedded optimal control problem.
(2)We study stability and stabilization of a new type of switched systems (switched systems with continuous-time and discrete-time subsystems) proposed in the literature. Up to now, for stability of such type of nonlinear switched systems there are not any available results. Therefore for a specific class of switched systems composed of continuous-time nonlinear subsystems and discrete-time uncertain subsystems we give an estimation of state of this class of switched systems, and from the estimation we present sufficient conditions for global exponential stability. Moreover, when there are unstable subsystems in this class of switched systems, we obtain some results on stabilization of this class of switched systems.
(3)Stability and estimates of the region of attraction of a class of continuous-time switched systems with triangular forms are addressed. We first give sufficient conditions for global asymptotic stability under arbitrary switching signals of switched systems with triangular forms, and then we estimate the region of attraction of switched systems with triangular forms. Various examples are presented to illustrate these estimates.
(4)Stability and estimates of the region of attraction of a class of discrete-time switched systems with triangular forms are also studied. We first give a result on global asymptotic stability of switched systems under arbitrary switching signals with triangular forms, on the basis of the result we also present suffiecient conditions which are easily checked for its global asymptotic stability under arbitrary switching signals, and then we estimate the region of attraction of this class of switched systems with triangular forms. Some examples are presented to illustrate these estimates.
引文
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