切换广义系统稳定性分析与控制方法研究
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摘要
作为一类重要的特殊混杂系统，切换广义系统由若干连续时间或离散时间广义子系统构成，根据切换策略在子系统间切换达到控制目的，具有较强的工程应用背景。与正常切换系统相比，由于自身结构的复杂性及正则性、脉冲模消除、状态跳变及一致初始状态等问题的存在，同时在实际控制系统中，非线性扰动、时滞和不确定性等因素的影响不可避免，使得对切换广义系统稳定性分析与控制器设计更复杂和更具挑战性，还有很多控制理论研究和实际应用问题有待解决。
本文利用公共Lyapunov函数方法联合凸组合技术，多Lyapunov函数和平均驻留时间方法等工具，分别针对切换广义系统的H∞记忆状态反馈控制，H∞保性能控制，输入-状态稳定，以及一致有限时间有界性和有限时间稳定等控制问题展开讨论。全文主要研究工作共分六章，概括如下：
1)分别讨论了连续时间和离散时间不确定时滞切换广义系统的鲁棒H∞控制和记忆状态反馈镇定问题。首先，针对一类连续时间不确定时变时滞切换广义系统，在假设不确定参数项范数有界的情况下，通过构造适当的Lyapunov泛函，引入记忆状态反馈的控制思想，设计相应的切换规则，给出了基于线性矩阵不等式表示的使得系统正则、无脉冲且渐近稳定的充分条件及鲁棒H∞记忆状态反馈控制器的设计方法，使得系统具有一定的干扰抑制水平同时又是状态反馈可切换镇定的。然后，将研究结果推广并对一类离散时间不确定时滞切换广义系统进行了讨论。
2)首先，给出了鲁棒H∞保性能控制的定义，并分别针对含有常数时滞和时变时滞的两类不确定切换广义系统进行了讨论，得到了基于线性矩阵不等式表示的鲁棒H∞保性能控制器存在的充分条件。然后以此为研究基础，利用公共Lyapunov函数方法联合凸组合技术，设计相应的切换规则，针对同时含有不确定性参数、时变时滞及非线性外部扰动的切换广义系统的鲁棒H∞保性能控制和记忆状态反馈镇定问题进行了研究。
3)把一致有限时间稳定和有限时间有界的概念在切换广义系统上进行了推广。通过构造多Lyapunov函数，在任意给定的切换规则作用下，利用平均驻留时间方法讨论了连续时间时变时滞切换广义系统的一致有限时间有界、有限时间稳定及状态反馈镇定等控制问题。然后，再通过构造切换Lyapunov函数，同样采用平均驻留时间方法推广研究了一类不确定离散时间时滞切换广义系统的一致有限时间稳定和有限时间有界问题，另外分别给出了状态反馈控制器的设计方案。与实际应用较困难的状态转移矩阵方法相比，该方法在一定程度上降低了控制器的设计难度。
4)采用平均驻留时间方法并结合Gronwall-Bellman不等式技术，按照输入-状态稳定的定义，分别考虑子系统全是输入-状态稳定和子系统不全是输入-状态稳定两类情况，讨论了一类非线性时变时滞切换广义系统的输入-状态稳定问题。在保证控制输入有界的前提下，该方法无需构造控制输入的具体结构和Lyapunov函数，只需要设计相应的切换规则，即可保证系统整体是输入-状态稳定的。
As an important class of special hybrid systems, switched descriptor systems, whichcomprise a collection of continuous-time or discrete-time descriptor subsystems together witha switching rule that specifies the switching among the subsystems to achieve the controlobjectives, can be used to describe a wide range of engineering systems. Compared with thenormal switched systems, the structure of switched descriptor systems is very complicatedbecause of the existences of a large number of practical problems, such as regularity,eliminating the pulse mode, state jump and the compatibility of initial state and so on.Moreover, in many actual systems, there exist random perturbation, time-delay, uncertainty,nonlinear disturbance and other factors, so it is inevitably difficult to analyze and design thecontroller, and a lot of control theory research and practical problems in engineeringapplication need to be dealt with for switched descriptor systems.
Based on the common Lyapunov function method combined with the convex combinationtechnique, multiple Lyapunov function method and the average dwell time method, thecontrol issues of robust H∞memory state feedback control, robust H∞guaranteed costcontrol，input-to-state stability，uniform finite time stability and finite time boundness arediscussed respectively. This paper is divided into six chapters, and the main study contents areas follow:
1) The problems of robust H∞control and memory state feedback stabilization forcontinuous-time and discrete-time uncertain switched descriptor systems with delays arediscussed respectively. First, suppose that the uncertain parameters are norm bounded, and theidea of memory state feedback control is introduced for a class of uncertain time-varyingdelayed switched descriptor systems with nonlinear disturbance. Sufficient conditions for theexistences of robust H∞memory state feedback controller in terms of linear matrixinequalities (LMIS) and switching rules are presented based on the common Lyapunovfunction and linear matrix inequality approach, and the designed controllers ensure that theclosed-loop systems satisfy the index with a presented H∞disturbance attenuation level γ.Then, the results are generalized to study a class of discrete-time uncertain switcheddescriptor systems with delays.
2) First, the concepts of robust H∞guaranteed cost control is proposed. Then, theproblems for uncertain switched descriptor systems with constant time delays andtime-varying delays are addressed respectively. The delay-dependent sufficient conditions androbust H∞guaranteed cost control controller in terms of linear matrix inequalities arepresented. Based on the above research foundation, and the common Lyapunov functionmethod combined with the convex combination technique, designing the correspondingswitching rules, the problems of robust H∞guaranteed cost control and memory statefeedback stabilization for uncertain switched descriptor systems with time-varying delay andnonlinear perturbation are discussed.
3) The problems of uniform finite-time stability, finite-time boundedness and statefeedback stabilization for continuous-time and discrete-time uncertain switched descriptorsystems are addressed based on the multiple Lyapunov funcntions，switched Lyapunovfuncntions approaches and average dwell time technique. First, the concepts of uniformfinite-time stability and finite-time boundedness are generalized to switched descriptorsystems. In addition, sufficient conditions for the existences of state feedback controllers interms of linear matrix inequalities are obtained with arbitrary switching rules, whichguarantee that the continuous-time and discrete-time uncertain switched descriptor systemsare uniform finite-time stable and finite-time bounded respectively. Compared with statetransition matrix method, to some extent, the proposed results reduce the difficulty ofcontroller designing in actual application.
4) Based on the average dwell time approach and Gronwall-Bellman inequality approach,the control issue of input-to-state stability for a class of nonlinear time-delayed switcheddescriptor systems is divided into two cases to discuss. When each subsystem is input-to-statestable, or some subsystems are not input-to-state stable, according to the definition ofinput-to-state stable, suppose that the control inputs are bounded and design thecorresponding switching rules, the sufficient conditions are given dispense with constructingthe Lyapunov function and specific form of control input, which guarantee that the nonlineartime-delayed switched descriptor systems are input-to-state stable.
本文利用公共Lyapunov函数方法联合凸组合技术，多Lyapunov函数和平均驻留时间方法等工具，分别针对切换广义系统的H∞记忆状态反馈控制，H∞保性能控制，输入-状态稳定，以及一致有限时间有界性和有限时间稳定等控制问题展开讨论。全文主要研究工作共分六章，概括如下：
1)分别讨论了连续时间和离散时间不确定时滞切换广义系统的鲁棒H∞控制和记忆状态反馈镇定问题。首先，针对一类连续时间不确定时变时滞切换广义系统，在假设不确定参数项范数有界的情况下，通过构造适当的Lyapunov泛函，引入记忆状态反馈的控制思想，设计相应的切换规则，给出了基于线性矩阵不等式表示的使得系统正则、无脉冲且渐近稳定的充分条件及鲁棒H∞记忆状态反馈控制器的设计方法，使得系统具有一定的干扰抑制水平同时又是状态反馈可切换镇定的。然后，将研究结果推广并对一类离散时间不确定时滞切换广义系统进行了讨论。
2)首先，给出了鲁棒H∞保性能控制的定义，并分别针对含有常数时滞和时变时滞的两类不确定切换广义系统进行了讨论，得到了基于线性矩阵不等式表示的鲁棒H∞保性能控制器存在的充分条件。然后以此为研究基础，利用公共Lyapunov函数方法联合凸组合技术，设计相应的切换规则，针对同时含有不确定性参数、时变时滞及非线性外部扰动的切换广义系统的鲁棒H∞保性能控制和记忆状态反馈镇定问题进行了研究。
3)把一致有限时间稳定和有限时间有界的概念在切换广义系统上进行了推广。通过构造多Lyapunov函数，在任意给定的切换规则作用下，利用平均驻留时间方法讨论了连续时间时变时滞切换广义系统的一致有限时间有界、有限时间稳定及状态反馈镇定等控制问题。然后，再通过构造切换Lyapunov函数，同样采用平均驻留时间方法推广研究了一类不确定离散时间时滞切换广义系统的一致有限时间稳定和有限时间有界问题，另外分别给出了状态反馈控制器的设计方案。与实际应用较困难的状态转移矩阵方法相比，该方法在一定程度上降低了控制器的设计难度。
4)采用平均驻留时间方法并结合Gronwall-Bellman不等式技术，按照输入-状态稳定的定义，分别考虑子系统全是输入-状态稳定和子系统不全是输入-状态稳定两类情况，讨论了一类非线性时变时滞切换广义系统的输入-状态稳定问题。在保证控制输入有界的前提下，该方法无需构造控制输入的具体结构和Lyapunov函数，只需要设计相应的切换规则，即可保证系统整体是输入-状态稳定的。
As an important class of special hybrid systems, switched descriptor systems, whichcomprise a collection of continuous-time or discrete-time descriptor subsystems together witha switching rule that specifies the switching among the subsystems to achieve the controlobjectives, can be used to describe a wide range of engineering systems. Compared with thenormal switched systems, the structure of switched descriptor systems is very complicatedbecause of the existences of a large number of practical problems, such as regularity,eliminating the pulse mode, state jump and the compatibility of initial state and so on.Moreover, in many actual systems, there exist random perturbation, time-delay, uncertainty,nonlinear disturbance and other factors, so it is inevitably difficult to analyze and design thecontroller, and a lot of control theory research and practical problems in engineeringapplication need to be dealt with for switched descriptor systems.
Based on the common Lyapunov function method combined with the convex combinationtechnique, multiple Lyapunov function method and the average dwell time method, thecontrol issues of robust H∞memory state feedback control, robust H∞guaranteed costcontrol，input-to-state stability，uniform finite time stability and finite time boundness arediscussed respectively. This paper is divided into six chapters, and the main study contents areas follow:
1) The problems of robust H∞control and memory state feedback stabilization forcontinuous-time and discrete-time uncertain switched descriptor systems with delays arediscussed respectively. First, suppose that the uncertain parameters are norm bounded, and theidea of memory state feedback control is introduced for a class of uncertain time-varyingdelayed switched descriptor systems with nonlinear disturbance. Sufficient conditions for theexistences of robust H∞memory state feedback controller in terms of linear matrixinequalities (LMIS) and switching rules are presented based on the common Lyapunovfunction and linear matrix inequality approach, and the designed controllers ensure that theclosed-loop systems satisfy the index with a presented H∞disturbance attenuation level γ.Then, the results are generalized to study a class of discrete-time uncertain switcheddescriptor systems with delays.
2) First, the concepts of robust H∞guaranteed cost control is proposed. Then, theproblems for uncertain switched descriptor systems with constant time delays andtime-varying delays are addressed respectively. The delay-dependent sufficient conditions androbust H∞guaranteed cost control controller in terms of linear matrix inequalities arepresented. Based on the above research foundation, and the common Lyapunov functionmethod combined with the convex combination technique, designing the correspondingswitching rules, the problems of robust H∞guaranteed cost control and memory statefeedback stabilization for uncertain switched descriptor systems with time-varying delay andnonlinear perturbation are discussed.
3) The problems of uniform finite-time stability, finite-time boundedness and statefeedback stabilization for continuous-time and discrete-time uncertain switched descriptorsystems are addressed based on the multiple Lyapunov funcntions，switched Lyapunovfuncntions approaches and average dwell time technique. First, the concepts of uniformfinite-time stability and finite-time boundedness are generalized to switched descriptorsystems. In addition, sufficient conditions for the existences of state feedback controllers interms of linear matrix inequalities are obtained with arbitrary switching rules, whichguarantee that the continuous-time and discrete-time uncertain switched descriptor systemsare uniform finite-time stable and finite-time bounded respectively. Compared with statetransition matrix method, to some extent, the proposed results reduce the difficulty ofcontroller designing in actual application.
4) Based on the average dwell time approach and Gronwall-Bellman inequality approach,the control issue of input-to-state stability for a class of nonlinear time-delayed switcheddescriptor systems is divided into two cases to discuss. When each subsystem is input-to-statestable, or some subsystems are not input-to-state stable, according to the definition ofinput-to-state stable, suppose that the control inputs are bounded and design thecorresponding switching rules, the sufficient conditions are given dispense with constructingthe Lyapunov function and specific form of control input, which guarantee that the nonlineartime-delayed switched descriptor systems are input-to-state stable.
引文
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