连续随机时滞系统鲁棒控制和滤波
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摘要
随机系统由于其深刻的实际背景近年来已受到广泛关注。本文在已有随机系统理论基础上，在不确定随机时滞系统鲁棒H_∞镇定、滤波、不确定随机时滞马尔可夫切换系统鲁棒控制、不确定奇异随机时滞系统滤波等方面作了研究，得到了一些较为深刻的研究成果。本文主要研究内容概括如下：
一、研究了不确定随机时滞系统鲁棒H_∞镇定问题。不确定参数分别满足凸多面体结构和线性分式结构。对凸多面体随机时滞系统，通过选取依赖于参数的Lyapunov函数和引入自由变量相结合的方法，以线性矩阵不等式形式(LMI)提出不确定随机时滞系统鲁棒镇定控制器存在的判别条件，并设计适当的状态反馈控制律，使闭环系统对于所有容许的不确定参数鲁棒均方随机渐近稳定且满足给定的H_∞性能指标。进而，提出了不确定参数满足线性分式结构情况下，随机时滞系统鲁棒H_∞镇定的充分条件。
二、针对连续随机时滞系统，研究了L_2-L_∞滤波问题。通过引入多个自由变量，利用线性矩阵不等式方法(LMI)，给出了依赖于时滞的充分条件，从而保证滤波误差方程均方随机渐近稳定且满足L_2-L_∞性能指标，在此基础上给出L_2-L_∞滤波器的设计方法。通过仿真比较，本论文工作较已有结果具有更小保守性。
三、在第一部分结论的基础上，讨论了不确定随机时滞马尔可夫切换系统鲁棒指数镇定及鲁棒H_∞控制问题。利用线性矩阵不等式方法(LMI)，给出了不确定随机时滞马尔可夫切换系统鲁棒指数镇定及鲁棒H_∞控制问题可解的充分条件及控制器的设计方法。在此基础上，设计无记忆的状态反馈控制器，使得闭环系统鲁棒均方随机稳定且满足给定的H_∞性能指标。
四、利用拓展的伊藤微分方程，研究了不确定奇异随机时滞系统及不确定奇异随机时滞马尔可夫切换系统的H_∞滤波问题。分别给出了依赖于时滞和不依赖于时滞的可解性充分条件。通过设计适当的H_∞滤波器，使得误差系统鲁棒均方随机稳定且满足给定的H_∞性能指标。并将问题最终转化为线性矩阵不等式可解性条件(LMI)。
Since stochastic systems have come to play an important role in many branches ofscience and engineering applications, stochastic systems have been widely investigated inrecent years. The dissertation provides some results and concepts on robust H_∞controlfor uncertain stochastic time-delay systems, robust control for the uncertain stochastictime-delay markovian jump systems, and filtering for the uncertain singular stochastictime-delay systems. The main results obtained in this dissertation are as follows:
1、The problems of robust stochastic stabilization and robust H_∞control forstochastic systems with parameter uncertainties and state delays are addressed,respectively. The parameter uncertainties are of the polytopic form and linear fractionalform, respectively. Attention is focused on the design of state feedback controllers. Forthe robust stabilization problem, a state feedback controller is designed such that theclosed-loop system is robustly stable in mean square, while for the robust H_∞controlproblem, a state feedback controller is designed such that the closed-loop system is notonly robustly stable in mean square but also with a prescribed H_∞performance level.
2、The problem of the L_2-L_∞filter design for a class of stochastic time-delaysystems is concerned. By the introductions of slack varialbles, delay-dependent sufficientconditions are presented, which guarantee the existence of a linear filter ensuring that thefiltering error system is stochastically stable in mean square and its L_2-L_∞performance satisfies a prescribed level. A desired filter can be constructed by solvingcertain linear matrix inequalities. A simulation example is given to demonstrate the lessconservatism of the proposed method.
3、Based on the results in the first part, we consider the problems of robustexponential stabilization in mean square and robust H_∞control for time-delaystochastic systems with Markov jump parameters and parameter uncertainties. Attentionis focused on the design of state feedback controllers, which ensure the closed-loopsystem is not only robustly exponential stable in mean square but also satisfies aprescribed H_∞performance level.
4、The robust H_∞filtering problem for a class of uncertain singular stochasticsystems with time-delay and uncertain singular stochastic Markovian jump systems are studied based on the extended Ito stochastic differential formula. By means of thesingular stochastic Lyapunov method and linear matrix inequalities(LMIs),delay-dependent sufficient conditions and delay-independent sufficient conditions areobtained, respectively, such that the filtering error system is robustly stable in meansquare with a prescribed disturbance attenuation level.
一、研究了不确定随机时滞系统鲁棒H_∞镇定问题。不确定参数分别满足凸多面体结构和线性分式结构。对凸多面体随机时滞系统，通过选取依赖于参数的Lyapunov函数和引入自由变量相结合的方法，以线性矩阵不等式形式(LMI)提出不确定随机时滞系统鲁棒镇定控制器存在的判别条件，并设计适当的状态反馈控制律，使闭环系统对于所有容许的不确定参数鲁棒均方随机渐近稳定且满足给定的H_∞性能指标。进而，提出了不确定参数满足线性分式结构情况下，随机时滞系统鲁棒H_∞镇定的充分条件。
二、针对连续随机时滞系统，研究了L_2-L_∞滤波问题。通过引入多个自由变量，利用线性矩阵不等式方法(LMI)，给出了依赖于时滞的充分条件，从而保证滤波误差方程均方随机渐近稳定且满足L_2-L_∞性能指标，在此基础上给出L_2-L_∞滤波器的设计方法。通过仿真比较，本论文工作较已有结果具有更小保守性。
三、在第一部分结论的基础上，讨论了不确定随机时滞马尔可夫切换系统鲁棒指数镇定及鲁棒H_∞控制问题。利用线性矩阵不等式方法(LMI)，给出了不确定随机时滞马尔可夫切换系统鲁棒指数镇定及鲁棒H_∞控制问题可解的充分条件及控制器的设计方法。在此基础上，设计无记忆的状态反馈控制器，使得闭环系统鲁棒均方随机稳定且满足给定的H_∞性能指标。
四、利用拓展的伊藤微分方程，研究了不确定奇异随机时滞系统及不确定奇异随机时滞马尔可夫切换系统的H_∞滤波问题。分别给出了依赖于时滞和不依赖于时滞的可解性充分条件。通过设计适当的H_∞滤波器，使得误差系统鲁棒均方随机稳定且满足给定的H_∞性能指标。并将问题最终转化为线性矩阵不等式可解性条件(LMI)。
Since stochastic systems have come to play an important role in many branches ofscience and engineering applications, stochastic systems have been widely investigated inrecent years. The dissertation provides some results and concepts on robust H_∞controlfor uncertain stochastic time-delay systems, robust control for the uncertain stochastictime-delay markovian jump systems, and filtering for the uncertain singular stochastictime-delay systems. The main results obtained in this dissertation are as follows:
1、The problems of robust stochastic stabilization and robust H_∞control forstochastic systems with parameter uncertainties and state delays are addressed,respectively. The parameter uncertainties are of the polytopic form and linear fractionalform, respectively. Attention is focused on the design of state feedback controllers. Forthe robust stabilization problem, a state feedback controller is designed such that theclosed-loop system is robustly stable in mean square, while for the robust H_∞controlproblem, a state feedback controller is designed such that the closed-loop system is notonly robustly stable in mean square but also with a prescribed H_∞performance level.
2、The problem of the L_2-L_∞filter design for a class of stochastic time-delaysystems is concerned. By the introductions of slack varialbles, delay-dependent sufficientconditions are presented, which guarantee the existence of a linear filter ensuring that thefiltering error system is stochastically stable in mean square and its L_2-L_∞performance satisfies a prescribed level. A desired filter can be constructed by solvingcertain linear matrix inequalities. A simulation example is given to demonstrate the lessconservatism of the proposed method.
3、Based on the results in the first part, we consider the problems of robustexponential stabilization in mean square and robust H_∞control for time-delaystochastic systems with Markov jump parameters and parameter uncertainties. Attentionis focused on the design of state feedback controllers, which ensure the closed-loopsystem is not only robustly exponential stable in mean square but also satisfies aprescribed H_∞performance level.
4、The robust H_∞filtering problem for a class of uncertain singular stochasticsystems with time-delay and uncertain singular stochastic Markovian jump systems are studied based on the extended Ito stochastic differential formula. By means of thesingular stochastic Lyapunov method and linear matrix inequalities(LMIs),delay-dependent sufficient conditions and delay-independent sufficient conditions areobtained, respectively, such that the filtering error system is robustly stable in meansquare with a prescribed disturbance attenuation level.
引文
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