几类随机混杂系统的稳定性分析及其控制
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摘要
随机混杂系统是由连续(或离散)时间变量和离散事件(或逻辑)变量以及随机因素相互作用的复杂系统。在描述和研究许多复杂的物理现象和实际应用的过程中,随机混杂系统能有效地提供一个数学框架(模型)。它在经济系统控制、容错系统控制,柔性制造系统控制、多目标跟踪控制、航空系统控制以及网络控制等实际问题中都有着广泛的应用。因此,对于随机混杂系统的研究具有重要的理论意义和实用价值。
本文主要研究了几类随机混杂系统的稳定性以及相关的控制器设计问题。主要工作概括如下:
1.研究一类随机混杂系统的依概率稳定性及其稳定化问题。首先,应用多Lyapunov函数这一工具给出了系统依概率稳定的充分条件。然后,运用线性矩阵不等式(LMI)和状态划分的方法研究系统的依概率稳定化问题,给出系统的状态反馈增益矩阵和脉冲增益矩阵的求解方法。并将前述结果推广到系统为不确定的情形。
2.研究一类随机混杂系统的几乎必然稳定性及其稳定化问题。首先,应用切换的Lyapunov函数法研究系统的稳定性,获得了判别其几乎必然稳定且独立于P阶稳定的充分条件,同时,将结果应用到线性随机混杂系统上,获得其相应的判别条件。所得的定理条件允许系统在切换时刻状态发生重置,从而将Mao[95]的模型进一步推广。进一步,设计出随机子系统的位移部分以及扩散部分两部分的控制律,使得在P阶矩意义下不稳定的系统达到几乎必然稳定,同时,结果也应用到线性随机混杂系统上,获得其相应的控制律。
3.研究一类随机混杂系统的有限时间随机稳定性及其稳定化的问题。首先,利用多Lyapunov函数,给出若干个非线性随机混杂系统的有限时间随机稳定性条件。同时,将结果应用到线性随机混杂系统,通过多Lyapunov函数和线性矩阵不等式(LMI)相结合的方法,给出系统有限时间随机稳定的充分条件。然后,进一步研究系统的有限时间随机稳定化问题,分别给出非线性和线性随机混杂系统的有限时间控制器的设计方法。值得注意的是:所设计的控制器是包括系统位移和扩散两部分的混合控制器。
4.研究一类广义随机混杂系统的随机稳定性及其稳定化的问题。首先,利用多Lyapunov函数,给出非线性广义随机混杂系统的随机稳定性条件。同时,进一步研究线性广义随机混杂系统的随机稳定性问题。通过多Lyapunov函数和随机广义Lyapunov函数相结合的方法,以耦合广义Lyapunov方程的形式给出系统随机稳定的充要条件,该条件可转化为严格的LMI条件,且无需对原系统作等价变换。然后,将前述结果应用到系统的随机稳定化问题上,给出相应的控制器设计方法。最后,将前述结果推广到系统具有不确定性的情况,分别给出系统随机稳定的判别条件和非脆弱控制器的设计方法。
5.研究不确定随机混杂系统的鲁棒H∞滤波与控制问题。首先,针对非线性Markov切换随机不确定系统研究它的无限维鲁棒H∞状态估计问题。假设系统的状态不仅受到白噪声影响,而且还受到外在扰动信号的影响,同时测量输出也受到外在扰动的影响,构造了一个渐近稳定的滤波器,使得扩充之后的系统依概率稳定,得到了非线性H∞滤波器的一个确切的形式。然后,针对在切换时刻具有脉冲行为的线性Markov切换随机系统,讨论了它的鲁棒H∞控制问题,分别从系统的鲁棒稳定性及鲁棒H∞性能两方面进行分析,利用多Lyapunov函数法对系统的稳定性进行分析,给出了系统鲁棒依概率稳定的几个充分条件,进一步运用线性矩阵不等式(LMI)法对系统的鲁棒H∞性能进行分析,得到了一般系统的状态反馈矩阵和脉冲控制矩阵,并在此基础上得出了一个鲁棒H∞控制律,最后提出了一套基于Matlab软件的鲁棒控制器的设计方法。
Stochastic hybrid systems are the complicated systems that consist of continuous (or discrete) time dynamics, discrete event (or logical) dynamics, randomness, and the interaction among them. Stochastic hybrid systems can provide an effective framework for mathematical modeling and analysis of many complex physical phenomena and practical applications. They have a variety of applications such as economic systems, fault tolerant control systems, flexible manufacturing systems, multiple target tracking, aircraft systems and network control systems etc. Thus, the study of stochastic hybrid systems has important significance both in theory and applications.
In this dissertation, stability analysis and control problem of several classes stochastic hybrid systems are studied. The main contributions and original ideas included in the dissertation are summarized as follows.
1.The stability and stabilization in probability of a class of stochastic hybrid systems are studied. Multiple Lyapunov techniques are used to derive sufficient conditions for stability in probability of the overall system. The conditions are in linear matrix inequalities form, and can be used to solve stabilization synthesis problem with the method of state parttition. The results are extended to the design of a robust-stabilized state-feedback controller as well. A numerical example shows the effectiveness of the proposed approach.
2.The almost sure stability and stabilization of a class of stochastic hybrid systems are studied. By using switched Lyapunov techniques, sufficient conditions for almost sure stability are presented and they do not rely on the moment stability of the system. The conditions permit continuous state reset at the switching instant and extend the modeling in [95] of Mao. The conditions are then specialized to case of linear systems, to solve the stabilization synthesis problem. Moreover, the control structure appears not only in the shift part but also in the diffusion part of the underlying stochastic subsystem. The results are easily checkable. A numerical example illustrates the effectiveness of the proposed approach.
3.The finite time stability and stabilization of a class of stochastic hybrid systems are studied. Multiple Lyapunov techniques are used to derive sufficient conditions for finite-time stochastic stability of the overall system. The results are reduced to feasibi- lity problems involving linear matrix inequalities(LMIs). Furthermore, based on the state partition of continuous parts of systems, hybrid state feedback controllers which stabilize the closed loop nonlinear and linear systems in the finite-time sense, are then addressed respectively. Moreover, the controller appears not only in the shift part but also in the diffusion part of the underlying stochastic subsystem. A numerical example is presented to illustrate the proposed methodology.
4.The problems of stochastic stability and stabilization for a class of discrete-time singular hybrid systems with Markov jump parameters are investigated. Based on multiple Lyapunov function and stochastic generalized Lyapunov function techniques, a necessary and sufficient condition is derived without using the restricted equivalent property of singular systems. The condition is given in terms of coupled generalized Lyapunov equations (CGLEs) such that the solution of the discrete-time singular hybrid systems is stochastic stable with time-homogenous finite state Markov chain. The equations can be solved out by changing into strict linear matrix inequalities (SLMIs). The result is extended to solve stabilization problem and the design of robust state-feedback controller and nonfragile controller. A numerical example shows the effectiveness of the proposed approach.
5. The robust H-infinity filtering and control problem for stochastic hybrid systems are discussed. First, the robust H-infinity estimation for nonlinear perturbed stochastic hybrid systems is investigated. We assume that the state and measurement are corrupted by uncertain exogenous disturbances. The H-infinity filter can be abtained by solving second-order nonlinear Hamilton-Jacobi inequalities. Then, multiple Lyapunov techniq- ues are used to derive some important sufficient conditions for the robust stability in probability of the linear stochastic hybrid systems. Furthermore, by analyzing the robust H-infinity performance of the system with the help of the linear matrix inequality(LMI) method, the state feedback matrix and the impulsive control matrix of the system are obtained, and then a robust H-infinity control rule is derived. Finally, a robust H-infinity control design method based on MATLAB software is presented, and a numerical example shows the effectiveness of the proposed approach.
本文主要研究了几类随机混杂系统的稳定性以及相关的控制器设计问题。主要工作概括如下:
1.研究一类随机混杂系统的依概率稳定性及其稳定化问题。首先,应用多Lyapunov函数这一工具给出了系统依概率稳定的充分条件。然后,运用线性矩阵不等式(LMI)和状态划分的方法研究系统的依概率稳定化问题,给出系统的状态反馈增益矩阵和脉冲增益矩阵的求解方法。并将前述结果推广到系统为不确定的情形。
2.研究一类随机混杂系统的几乎必然稳定性及其稳定化问题。首先,应用切换的Lyapunov函数法研究系统的稳定性,获得了判别其几乎必然稳定且独立于P阶稳定的充分条件,同时,将结果应用到线性随机混杂系统上,获得其相应的判别条件。所得的定理条件允许系统在切换时刻状态发生重置,从而将Mao[95]的模型进一步推广。进一步,设计出随机子系统的位移部分以及扩散部分两部分的控制律,使得在P阶矩意义下不稳定的系统达到几乎必然稳定,同时,结果也应用到线性随机混杂系统上,获得其相应的控制律。
3.研究一类随机混杂系统的有限时间随机稳定性及其稳定化的问题。首先,利用多Lyapunov函数,给出若干个非线性随机混杂系统的有限时间随机稳定性条件。同时,将结果应用到线性随机混杂系统,通过多Lyapunov函数和线性矩阵不等式(LMI)相结合的方法,给出系统有限时间随机稳定的充分条件。然后,进一步研究系统的有限时间随机稳定化问题,分别给出非线性和线性随机混杂系统的有限时间控制器的设计方法。值得注意的是:所设计的控制器是包括系统位移和扩散两部分的混合控制器。
4.研究一类广义随机混杂系统的随机稳定性及其稳定化的问题。首先,利用多Lyapunov函数,给出非线性广义随机混杂系统的随机稳定性条件。同时,进一步研究线性广义随机混杂系统的随机稳定性问题。通过多Lyapunov函数和随机广义Lyapunov函数相结合的方法,以耦合广义Lyapunov方程的形式给出系统随机稳定的充要条件,该条件可转化为严格的LMI条件,且无需对原系统作等价变换。然后,将前述结果应用到系统的随机稳定化问题上,给出相应的控制器设计方法。最后,将前述结果推广到系统具有不确定性的情况,分别给出系统随机稳定的判别条件和非脆弱控制器的设计方法。
5.研究不确定随机混杂系统的鲁棒H∞滤波与控制问题。首先,针对非线性Markov切换随机不确定系统研究它的无限维鲁棒H∞状态估计问题。假设系统的状态不仅受到白噪声影响,而且还受到外在扰动信号的影响,同时测量输出也受到外在扰动的影响,构造了一个渐近稳定的滤波器,使得扩充之后的系统依概率稳定,得到了非线性H∞滤波器的一个确切的形式。然后,针对在切换时刻具有脉冲行为的线性Markov切换随机系统,讨论了它的鲁棒H∞控制问题,分别从系统的鲁棒稳定性及鲁棒H∞性能两方面进行分析,利用多Lyapunov函数法对系统的稳定性进行分析,给出了系统鲁棒依概率稳定的几个充分条件,进一步运用线性矩阵不等式(LMI)法对系统的鲁棒H∞性能进行分析,得到了一般系统的状态反馈矩阵和脉冲控制矩阵,并在此基础上得出了一个鲁棒H∞控制律,最后提出了一套基于Matlab软件的鲁棒控制器的设计方法。
Stochastic hybrid systems are the complicated systems that consist of continuous (or discrete) time dynamics, discrete event (or logical) dynamics, randomness, and the interaction among them. Stochastic hybrid systems can provide an effective framework for mathematical modeling and analysis of many complex physical phenomena and practical applications. They have a variety of applications such as economic systems, fault tolerant control systems, flexible manufacturing systems, multiple target tracking, aircraft systems and network control systems etc. Thus, the study of stochastic hybrid systems has important significance both in theory and applications.
In this dissertation, stability analysis and control problem of several classes stochastic hybrid systems are studied. The main contributions and original ideas included in the dissertation are summarized as follows.
1.The stability and stabilization in probability of a class of stochastic hybrid systems are studied. Multiple Lyapunov techniques are used to derive sufficient conditions for stability in probability of the overall system. The conditions are in linear matrix inequalities form, and can be used to solve stabilization synthesis problem with the method of state parttition. The results are extended to the design of a robust-stabilized state-feedback controller as well. A numerical example shows the effectiveness of the proposed approach.
2.The almost sure stability and stabilization of a class of stochastic hybrid systems are studied. By using switched Lyapunov techniques, sufficient conditions for almost sure stability are presented and they do not rely on the moment stability of the system. The conditions permit continuous state reset at the switching instant and extend the modeling in [95] of Mao. The conditions are then specialized to case of linear systems, to solve the stabilization synthesis problem. Moreover, the control structure appears not only in the shift part but also in the diffusion part of the underlying stochastic subsystem. The results are easily checkable. A numerical example illustrates the effectiveness of the proposed approach.
3.The finite time stability and stabilization of a class of stochastic hybrid systems are studied. Multiple Lyapunov techniques are used to derive sufficient conditions for finite-time stochastic stability of the overall system. The results are reduced to feasibi- lity problems involving linear matrix inequalities(LMIs). Furthermore, based on the state partition of continuous parts of systems, hybrid state feedback controllers which stabilize the closed loop nonlinear and linear systems in the finite-time sense, are then addressed respectively. Moreover, the controller appears not only in the shift part but also in the diffusion part of the underlying stochastic subsystem. A numerical example is presented to illustrate the proposed methodology.
4.The problems of stochastic stability and stabilization for a class of discrete-time singular hybrid systems with Markov jump parameters are investigated. Based on multiple Lyapunov function and stochastic generalized Lyapunov function techniques, a necessary and sufficient condition is derived without using the restricted equivalent property of singular systems. The condition is given in terms of coupled generalized Lyapunov equations (CGLEs) such that the solution of the discrete-time singular hybrid systems is stochastic stable with time-homogenous finite state Markov chain. The equations can be solved out by changing into strict linear matrix inequalities (SLMIs). The result is extended to solve stabilization problem and the design of robust state-feedback controller and nonfragile controller. A numerical example shows the effectiveness of the proposed approach.
5. The robust H-infinity filtering and control problem for stochastic hybrid systems are discussed. First, the robust H-infinity estimation for nonlinear perturbed stochastic hybrid systems is investigated. We assume that the state and measurement are corrupted by uncertain exogenous disturbances. The H-infinity filter can be abtained by solving second-order nonlinear Hamilton-Jacobi inequalities. Then, multiple Lyapunov techniq- ues are used to derive some important sufficient conditions for the robust stability in probability of the linear stochastic hybrid systems. Furthermore, by analyzing the robust H-infinity performance of the system with the help of the linear matrix inequality(LMI) method, the state feedback matrix and the impulsive control matrix of the system are obtained, and then a robust H-infinity control rule is derived. Finally, a robust H-infinity control design method based on MATLAB software is presented, and a numerical example shows the effectiveness of the proposed approach.
引文
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