几类广义系统的稳定性分析与控制
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摘要
自上个世纪70年代以来,广义系统理论已经逐渐发展成为当今最重要的控制理论分支之一.广义系统以其强大的应用背景,吸引了国内外众多学者的关注.关于广义系统的基本理论、稳定性等方面的研究都有了新的进展,但仍然存在一定的局限性,还有待于进一步发展和完善.
本论文在广义系统以及广义系统控制理论的基础上,对几类广义系统的稳定性和控制问题进行了分析和研究.全文由以下几部分组成：
第一章为绪论,介绍了本文研究工作的背景知识和研究概况.首先对广义系统理论的研究背景、结构特征及研究方法进行了阐述,介绍了广义系统发展状况和实际应用；其次,针对几类带有切换、具有Markov跳变、多自主体、不连续非线性广义系统,介绍其稳定性和控制问题发展的现状、研究的方法以及取得的成果；最后,简要地介绍了本文的工作安排.
第二章研究的是线性切换微分代数方程(DAEs),首先针对系统中既包括稳定的子系统,又包括不稳定的子系统的情形进行讨论,证明了只要平均驻留时间足够大,并且不稳定子系统比稳定子系统的运行时间相对短,就能得到切换DAEs的渐近稳定性.基于合适的Lyapunov函数,给出了切换DAEs稳定的充分条件.主要结果给出了一个额外的协调投影算子满足的条件,继而得到系统在平均驻留时间下的渐近稳定性.其次,讨论在切换信号具有时滞的情况下,借助于协调投影算子、平均驻留时间方法和合并切换信号技巧,给出了闭环切换系统渐近稳定的充分条件.
第三章讨论的是一类具有Markov跳变的广义系统.首先给出的是离散时间马尔科夫跳变广义系统的有界实引理,借助于已有的广义系统随机容许性的定义,得到无激励系统随机容许的一个充分必要条件,进而以严格矩阵不等式形式给出了离散时间马尔科夫跳变广义系统的有界实引理.与已有的结果相比,这里的结果在数值计算中更易处理.
其次,考虑转移概率部分未知的马尔科夫跳变广义系统的稳定和镇定问题,分为连续时间和离散时间情形.充分利用转移概率矩阵的性质,和不确定区域的凸性,得到一个新的严格矩阵不等式形式的充分必要条件,使得系统是正则,无脉冲和随机稳定的.对于连续时间情形,进一步设计了状态反馈控制器,给出了闭环系统是随机容许的充分条件；对于离散时间情形,设计了状态反馈控制器和输出反馈控制器,给出了相应的充分条件,使得闭环系统是随机容许的,最后,针对不同情形的系统,给出算例论证了结果的有效性.
第四章研究了一类具有切换拓扑的广义多自主体系统,将一般的线性系统推广到广义系统,讨论无领导者和有领导者跟随两个一致性问题.讨论的拓扑图是动态图,利用代数拓扑知识和广义系统理论,将广义多自主体系统进行快慢子系统分解,只需通过研究其慢子系统的性质,即设计状态反馈控制协议,得到慢子系统的一致性,就能解决广义多自主体系统相应的两个一致性问题.
第五章针对右端不连续的非线性广义系统,将正常系统的不变原理和稳定性推广到不连续的非线性广义系统.首先,给出了这类系统的稳定性和渐近稳定性,以及不变集的定义,在此基础上,利用E-不变集和广义Lyapunov方法提出了非线性广义系统的不变原理,其次,给出了非线性广义系统稳定和渐近稳定的充分条件.最后给出了算例论证了不变原理的应用.
第六章是全文总结.总结了本文的主要工作和贡献,并展望了进一步的研究.
Since the seventies of the20th century, singular system theory has gradually de-veloped into an most important branches of control theory. Since its strong application background, Singular system has attracted the attention of many researchers at home and abroad. There have been great development on fundamental theory and stability of singular systems, but there are still some limitations, such systems remains to be further developed.
Based on singular systems and the control theory, this dissertation studies the prob-lems of stability and control of several classes of singular systems.
The main contents of this dissertation include:
Chapter1is the preface, the research background and research states of this disserta-tion are introduced. Firstly, the research background of singular systems theory, structure characteristics and research method are summarized, the development status and actual applications are introduced in detail. Then, the research states, methods and obtained results about switched, Markov jump, multi-agent and discontinuous nonlinear singular systems are presented. Finally, the main work of this dissertation is introduced.
Chapter2studies a class of linear switched differential algebraic equations (DAEs). Case1:stable and unstable subsystems coexist. Asymptotic stability of the switched DAEs is obtained if the average dwell time is chosen sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of stable ones. Sufficient conditions for stability of switched DAEs are presented based on the existence of suitable Lyapunov functions. The main result shows that asymptotic stability is preserved under switching with an average dwell time and an additional condition involving consistency projectors holds. Case2:switching signals with time delays. Based on average dwell time method and the merging switching signal technique, the switched DAEs achieve asymptotic stabllity when consistency projectors are involved.
Chapter3discussed a class of singular systems with Makov jump. Firstly, a new nec-essary and sufficient condition is proposed in terms of strict linear matrix inequality, which guarantees the stochastic admissibility of the unforced Markovian jump singular systems. A bounded real lemma for discrete-time Markovian jump singular systems is derived and can be described by a strict matrix inequality. The results are more tractable and reliable in numerical computations than existing conditions.
Then, the problem of the stability and stabilization of Markovian jump singular sys-tems with partly known transition probabilities is concerned, including two cases(continuous time and discrete time). By fully unitized the properties of the the transition rate ma-trix(TRM), and the convexity of the uncertain domains, a new sufficient and necessary condition in terms of strict linear matrix inequalities (LMIs) for the MJSS to be regu-lar, impulsive and stochastically stable is obtained. In continuous time case, based on the proposed stability criterion, a state feedback controller is designed, and a sufficient condition which guarantees the stochastic admissibility of the closed loop systems is given; In the discrete time case, not only a state feedback controller but also an output feedback controller is designed, and a sufficient condition such that the closed loop systems is stochastically admissible is also given.
Finally, according to the different cases of the systems, several examples are given to demonstrate the effectiveness of the obtained results.
Chapter4studies a class of multi-agent singular systems under switching topol-ogy. We extend the general linear systems to the singular systems, and discuss both the leaderless consensus problem and the leader-following consensus problem under switching network topology. Using the algebraic topology and singular systems theory, the multi-agent singular system can be decomposed into fast-slow subsystems, and we just need to consider the property of the slow subsystem, that is to design a state feedback control protocol, and get the consensusability of the slow subsystem. Then, the two consensus problems of the multi-agent singular systems are also solvable.
Chapter5extends the LaSalle invariance principle and stability of conventional systems to discontinuous nonlinear singular systems. Firstly, the definitions of stabil-ity, asymptotical stability, and invariant set for such systems are proposed. Then, the LaSalle invariance principle is presented by citing the notion of E-invariant set and the generalized Lyapunov method, and sufficient conditions about the stability and asymp-totic stability of the discontinuous nonlinear singular systems are given. Furthermore, the application of the invariance principle is demonstrated by the given example.
Chapter6summarizes the main results of this dissertation and point out the further research.
本论文在广义系统以及广义系统控制理论的基础上,对几类广义系统的稳定性和控制问题进行了分析和研究.全文由以下几部分组成：
第一章为绪论,介绍了本文研究工作的背景知识和研究概况.首先对广义系统理论的研究背景、结构特征及研究方法进行了阐述,介绍了广义系统发展状况和实际应用；其次,针对几类带有切换、具有Markov跳变、多自主体、不连续非线性广义系统,介绍其稳定性和控制问题发展的现状、研究的方法以及取得的成果；最后,简要地介绍了本文的工作安排.
第二章研究的是线性切换微分代数方程(DAEs),首先针对系统中既包括稳定的子系统,又包括不稳定的子系统的情形进行讨论,证明了只要平均驻留时间足够大,并且不稳定子系统比稳定子系统的运行时间相对短,就能得到切换DAEs的渐近稳定性.基于合适的Lyapunov函数,给出了切换DAEs稳定的充分条件.主要结果给出了一个额外的协调投影算子满足的条件,继而得到系统在平均驻留时间下的渐近稳定性.其次,讨论在切换信号具有时滞的情况下,借助于协调投影算子、平均驻留时间方法和合并切换信号技巧,给出了闭环切换系统渐近稳定的充分条件.
第三章讨论的是一类具有Markov跳变的广义系统.首先给出的是离散时间马尔科夫跳变广义系统的有界实引理,借助于已有的广义系统随机容许性的定义,得到无激励系统随机容许的一个充分必要条件,进而以严格矩阵不等式形式给出了离散时间马尔科夫跳变广义系统的有界实引理.与已有的结果相比,这里的结果在数值计算中更易处理.
其次,考虑转移概率部分未知的马尔科夫跳变广义系统的稳定和镇定问题,分为连续时间和离散时间情形.充分利用转移概率矩阵的性质,和不确定区域的凸性,得到一个新的严格矩阵不等式形式的充分必要条件,使得系统是正则,无脉冲和随机稳定的.对于连续时间情形,进一步设计了状态反馈控制器,给出了闭环系统是随机容许的充分条件；对于离散时间情形,设计了状态反馈控制器和输出反馈控制器,给出了相应的充分条件,使得闭环系统是随机容许的,最后,针对不同情形的系统,给出算例论证了结果的有效性.
第四章研究了一类具有切换拓扑的广义多自主体系统,将一般的线性系统推广到广义系统,讨论无领导者和有领导者跟随两个一致性问题.讨论的拓扑图是动态图,利用代数拓扑知识和广义系统理论,将广义多自主体系统进行快慢子系统分解,只需通过研究其慢子系统的性质,即设计状态反馈控制协议,得到慢子系统的一致性,就能解决广义多自主体系统相应的两个一致性问题.
第五章针对右端不连续的非线性广义系统,将正常系统的不变原理和稳定性推广到不连续的非线性广义系统.首先,给出了这类系统的稳定性和渐近稳定性,以及不变集的定义,在此基础上,利用E-不变集和广义Lyapunov方法提出了非线性广义系统的不变原理,其次,给出了非线性广义系统稳定和渐近稳定的充分条件.最后给出了算例论证了不变原理的应用.
第六章是全文总结.总结了本文的主要工作和贡献,并展望了进一步的研究.
Since the seventies of the20th century, singular system theory has gradually de-veloped into an most important branches of control theory. Since its strong application background, Singular system has attracted the attention of many researchers at home and abroad. There have been great development on fundamental theory and stability of singular systems, but there are still some limitations, such systems remains to be further developed.
Based on singular systems and the control theory, this dissertation studies the prob-lems of stability and control of several classes of singular systems.
The main contents of this dissertation include:
Chapter1is the preface, the research background and research states of this disserta-tion are introduced. Firstly, the research background of singular systems theory, structure characteristics and research method are summarized, the development status and actual applications are introduced in detail. Then, the research states, methods and obtained results about switched, Markov jump, multi-agent and discontinuous nonlinear singular systems are presented. Finally, the main work of this dissertation is introduced.
Chapter2studies a class of linear switched differential algebraic equations (DAEs). Case1:stable and unstable subsystems coexist. Asymptotic stability of the switched DAEs is obtained if the average dwell time is chosen sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of stable ones. Sufficient conditions for stability of switched DAEs are presented based on the existence of suitable Lyapunov functions. The main result shows that asymptotic stability is preserved under switching with an average dwell time and an additional condition involving consistency projectors holds. Case2:switching signals with time delays. Based on average dwell time method and the merging switching signal technique, the switched DAEs achieve asymptotic stabllity when consistency projectors are involved.
Chapter3discussed a class of singular systems with Makov jump. Firstly, a new nec-essary and sufficient condition is proposed in terms of strict linear matrix inequality, which guarantees the stochastic admissibility of the unforced Markovian jump singular systems. A bounded real lemma for discrete-time Markovian jump singular systems is derived and can be described by a strict matrix inequality. The results are more tractable and reliable in numerical computations than existing conditions.
Then, the problem of the stability and stabilization of Markovian jump singular sys-tems with partly known transition probabilities is concerned, including two cases(continuous time and discrete time). By fully unitized the properties of the the transition rate ma-trix(TRM), and the convexity of the uncertain domains, a new sufficient and necessary condition in terms of strict linear matrix inequalities (LMIs) for the MJSS to be regu-lar, impulsive and stochastically stable is obtained. In continuous time case, based on the proposed stability criterion, a state feedback controller is designed, and a sufficient condition which guarantees the stochastic admissibility of the closed loop systems is given; In the discrete time case, not only a state feedback controller but also an output feedback controller is designed, and a sufficient condition such that the closed loop systems is stochastically admissible is also given.
Finally, according to the different cases of the systems, several examples are given to demonstrate the effectiveness of the obtained results.
Chapter4studies a class of multi-agent singular systems under switching topol-ogy. We extend the general linear systems to the singular systems, and discuss both the leaderless consensus problem and the leader-following consensus problem under switching network topology. Using the algebraic topology and singular systems theory, the multi-agent singular system can be decomposed into fast-slow subsystems, and we just need to consider the property of the slow subsystem, that is to design a state feedback control protocol, and get the consensusability of the slow subsystem. Then, the two consensus problems of the multi-agent singular systems are also solvable.
Chapter5extends the LaSalle invariance principle and stability of conventional systems to discontinuous nonlinear singular systems. Firstly, the definitions of stabil-ity, asymptotical stability, and invariant set for such systems are proposed. Then, the LaSalle invariance principle is presented by citing the notion of E-invariant set and the generalized Lyapunov method, and sufficient conditions about the stability and asymp-totic stability of the discontinuous nonlinear singular systems are given. Furthermore, the application of the invariance principle is demonstrated by the given example.
Chapter6summarizes the main results of this dissertation and point out the further research.
引文
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