离散时间Markov跳变系统的稳定性与鲁棒H_2/H_∞控制
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摘要
由于广泛的实际应用背景和智能控制飞速发展的需要,马尔可夫(Markov)跳变系统的稳定性分析和鲁棒控制问题愈来愈受到学者们的关注。本文采用随机谱方法研究了离散时间带马尔可夫跳参数和乘积噪声随机系统的稳定性,采用纳什(Nash)博弈法解决了离散时间带马尔可夫跳参数和乘积噪声随机时变系统有限时间鲁棒H2/H∞控制问题。本文的主要研究成果如下：
一、给出了离散时间带马尔可夫跳参数和乘积噪声随机系统的谱、不可移动的谱的定义,讨论了它们的性质,并给出了不可移动的谱的判别条件。
二、研究了离散时间带马尔可夫跳参数和乘积噪声随机系统的稳定性。根据系统的谱在复平面上的分布情况区分出三种稳定性：渐近均方稳定、临界稳定和本质不稳定,依次讨论了这三种稳定性的性质,分别给出了它们的谱判据和广义李亚普诺夫(Lyapunov)方程判据。
三、研究了离散时间带马尔可夫跳参数和乘积噪声随机系统的区域能稳性,证明了判定系统的D(0,α)-能稳性(0<α≤1)可以转化为判断一个线性矩阵不等式的可行性。研究了将离散时间随机跳变系统的谱配置在线性矩阵不等式(LMI)区域和广义LMI区域内的条件,揭示了离散时间随机系统的D(0,α;β)-稳定性(0≤α<β≤1)与二阶矩李亚普诺夫指数、系统的收敛速度之间的关系。
四、分别给出了离散时间带马尔可夫跳参数和乘积噪声随机系统的精确能观测、精确能检测的定义和PBH(Popov-Belevitch-Hautus)判据,讨论了它们与系统的稳定性、广义李亚普诺夫方程的解之间的关系。
五、解决了离散时间带马尔可夫跳参数和乘积噪声随机时变系统的混合H2/H∞控制问题,证明了控制器的存在惟一性,并借助四个耦合差分矩阵迭代给出了反馈增益明确的解析式,提供了精确求解耦合差分矩阵迭代的算法。
As one of the most basic dynamics models, Markov jump linear systems can be used to represent random failure processes in manufacturing industry, and some investment portfolio models. Recently, some topics related to Markov jump linear systems have attracted more and more attention. This dissertation devotes to coping with stability and stabilizability of discrete-time Markov jump linear stochastic systems with multiplicative noise by means of the operator spectrum. In addition, using Nash game approach, the finite horizon mixed H2/H∞control problem for discrete-time stochastic linear time-varying systems subject to Markov jump parameters and multiplicative noise is settled thoroughly. The main work and contribution of this dissertation are summarized as follows:
1. The spectra and unremovable spectra of discrete-time Markov jump linear stochastic time-invariant systems with multiplicative noise are well-defined and investigated. Then a criterion is provided for testing unremovable spectra.
2. According to the spectral distribution of an uncontrolled stochastic linear time-invariant system in the complex plane, we distinguish three kinds of stabilities:asymptotical mean square stability-all spectra of the given system belonging to the open left-half complex plane; critical stability-weaker than asymptotical mean square stability and all spectra belonging to the closed left-half complex plane; and essential instability-at least one of the spectra lying in the open right-half complex plane. While dealing with their criteria, two methods are involved: the spectral analysis technique and the generalized Lyapunov equation (GLE) approach, both of which are the most common ways in characterizing system stability.
3. A new concept called "D(0,α)-stabilizability" (0< a≤1) is introduced, for which, a necessary and sufficient condition is also proposed via LMI-based approach. A more general regional stability is discussed extensively with some concrete illustrative examples. As one of the applications, the relationship among D(O,α;β)- stability (0≤α<β≤1) of a discrete-time stochastic system, the decay rate of the system state response and the second-order moment Lyapunov exponent is revealed.
4. The notions of exact observability and exact detectability for discrete-time Markov jump linear stochastic systems with multiplicative noise are put forward. Stochastic PBH criteria for exact observability and exact detectability are obtained respectively. The relationship among stability, exact observability and the solutions of (GLEs) is also considered.
5. The finite horizon mixed H2/H∞control problem for discrete-time stochastic linear time-varying systems subject to Markov jump parameters and multiplicative noise is handled. Based on four coupled difference matrix-valued recursions (CDMRs), we derive a necessary and sufficient condition for the solvability of H2/H∞control problem and the controller can be designed explicitly according to the solutions of CDMRs. Moreover, a recursive procedure is supplied to solve the CDMRs.
一、给出了离散时间带马尔可夫跳参数和乘积噪声随机系统的谱、不可移动的谱的定义,讨论了它们的性质,并给出了不可移动的谱的判别条件。
二、研究了离散时间带马尔可夫跳参数和乘积噪声随机系统的稳定性。根据系统的谱在复平面上的分布情况区分出三种稳定性：渐近均方稳定、临界稳定和本质不稳定,依次讨论了这三种稳定性的性质,分别给出了它们的谱判据和广义李亚普诺夫(Lyapunov)方程判据。
三、研究了离散时间带马尔可夫跳参数和乘积噪声随机系统的区域能稳性,证明了判定系统的D(0,α)-能稳性(0<α≤1)可以转化为判断一个线性矩阵不等式的可行性。研究了将离散时间随机跳变系统的谱配置在线性矩阵不等式(LMI)区域和广义LMI区域内的条件,揭示了离散时间随机系统的D(0,α;β)-稳定性(0≤α<β≤1)与二阶矩李亚普诺夫指数、系统的收敛速度之间的关系。
四、分别给出了离散时间带马尔可夫跳参数和乘积噪声随机系统的精确能观测、精确能检测的定义和PBH(Popov-Belevitch-Hautus)判据,讨论了它们与系统的稳定性、广义李亚普诺夫方程的解之间的关系。
五、解决了离散时间带马尔可夫跳参数和乘积噪声随机时变系统的混合H2/H∞控制问题,证明了控制器的存在惟一性,并借助四个耦合差分矩阵迭代给出了反馈增益明确的解析式,提供了精确求解耦合差分矩阵迭代的算法。
As one of the most basic dynamics models, Markov jump linear systems can be used to represent random failure processes in manufacturing industry, and some investment portfolio models. Recently, some topics related to Markov jump linear systems have attracted more and more attention. This dissertation devotes to coping with stability and stabilizability of discrete-time Markov jump linear stochastic systems with multiplicative noise by means of the operator spectrum. In addition, using Nash game approach, the finite horizon mixed H2/H∞control problem for discrete-time stochastic linear time-varying systems subject to Markov jump parameters and multiplicative noise is settled thoroughly. The main work and contribution of this dissertation are summarized as follows:
1. The spectra and unremovable spectra of discrete-time Markov jump linear stochastic time-invariant systems with multiplicative noise are well-defined and investigated. Then a criterion is provided for testing unremovable spectra.
2. According to the spectral distribution of an uncontrolled stochastic linear time-invariant system in the complex plane, we distinguish three kinds of stabilities:asymptotical mean square stability-all spectra of the given system belonging to the open left-half complex plane; critical stability-weaker than asymptotical mean square stability and all spectra belonging to the closed left-half complex plane; and essential instability-at least one of the spectra lying in the open right-half complex plane. While dealing with their criteria, two methods are involved: the spectral analysis technique and the generalized Lyapunov equation (GLE) approach, both of which are the most common ways in characterizing system stability.
3. A new concept called "D(0,α)-stabilizability" (0< a≤1) is introduced, for which, a necessary and sufficient condition is also proposed via LMI-based approach. A more general regional stability is discussed extensively with some concrete illustrative examples. As one of the applications, the relationship among D(O,α;β)- stability (0≤α<β≤1) of a discrete-time stochastic system, the decay rate of the system state response and the second-order moment Lyapunov exponent is revealed.
4. The notions of exact observability and exact detectability for discrete-time Markov jump linear stochastic systems with multiplicative noise are put forward. Stochastic PBH criteria for exact observability and exact detectability are obtained respectively. The relationship among stability, exact observability and the solutions of (GLEs) is also considered.
5. The finite horizon mixed H2/H∞control problem for discrete-time stochastic linear time-varying systems subject to Markov jump parameters and multiplicative noise is handled. Based on four coupled difference matrix-valued recursions (CDMRs), we derive a necessary and sufficient condition for the solvability of H2/H∞control problem and the controller can be designed explicitly according to the solutions of CDMRs. Moreover, a recursive procedure is supplied to solve the CDMRs.
引文
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