几类不确定系统的稳定性与镇定研究
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摘要
几乎所有的实际系统都存在着不确定性.这种不确定性可能来源于系统建模误差、时变参数、量测噪声和外部扰动等.通常设计是基于一个确定的系统模型,而实际系统与这个模型有偏差.因此,使针对确定性系统设计的控制律满足不确定或者时变的实际系统的需要,是一个值得研究的课题.此类研究通常包含两个方面：一、估计一个控制律的鲁棒性,二、寻找鲁棒性较强的控制律的特征.本论文将围绕这两方面的问题展开研究.
Lyapunov稳定性理论是本学位论文用到的主要工具.应用Lyapunov理论主要难点在于一、如何构造系统的Lyapunov函数或泛函；二、如何估计所构造的Lyapunov函数或泛函的时间导数.本论文从这两方面着手,研究了几类不确定系统的鲁棒控制问题.主要内容可以分为两个部分：第二章为第一部分,第三、四、五章构成了第二部分.在第一部分,利用控制Lyapunov函数(CLF)研究了一组不确定非线性系统的同时镇定问题.在第二部分,基于Lyapunov-Krasovskii泛函和参数依赖Lyapunov函数,分别研究了不确定区间时滞系统和凸多胞体不确定离散系统的鲁棒控制问题.现谨将各章的主要内容及研究结果概述如下：
第一章为绪论部分.首先综述了不确定系统鲁棒控制研究的一些进展,简要介绍了同时镇定,参数系统和不确定时滞系统的研究中常用的一些方法及其局限性.接着介绍了本论文讨论的几类不确定系统模型以及用到的数学工具.最后概述了本论文的主要工作
第二章研究一组不确定非线性系统的同时镇定问题.首先研究了单输入仿射非线性系统,利用CLF给出了其同时镇定反馈律存在的一个充分条件.随后将所得的结果推广到带有不确定参数和有Brunovsky标准型结构的多输入非线性系统.最后作为应用,研究了一组混沌控制系统的同时镇定问题,以此来说明设计方案的有效性.
第三章研究不确定离散区间时滞系统的时滞相关稳定性判据及其镇定设计问题.主要讨论了3种不确定性：凸多胞体不确定性,线性分式范数有界不确定性,以及非线性扰动引起的不确定性.通过构造新的Lyapunov-Krasovskii泛函,并利用建立的有限和不等式来估计所构造泛函导数的上界,有效地降低了现有文献中相关结论的保守性.此外,基于稳定性判据还给出了这类系统的状态反馈和时滞状态反馈两种控制设计,并利用数值例子进行了仿真.
第四章分析了一类带有非线性扰动的连续区间时滞系统的时滞相关稳定性.引入了一种新的估计所构造Lyapunov-Krasovskii泛函的时间导数上界的方法.在该方法中,保留了一些有用的时间导数项,可以有效地降低已有文献中时滞相关稳定性判据的一些保守性.数值例子说明了本章中所用方法的优越性.
第五章研究了一类凸多胞体不确定性线性离散系统的鲁棒稳定性及其镇定.基于描述系统变换,对标称的离散线性系统,给出了其渐近稳定性的一个充要条件.在该条件中,系统矩阵和Lyapunov矩阵的乘积项分离,因而它可以很方便的应用于凸多胞体不确定离散系统的稳定性分析及其镇定设计.
第六章总结了本文论研究的主要内容和结论,并给出若干值得进一步研究的问题.
本文主要创新点概括如下：
①对一组具有Brunovsky标准型结构的不确定非线性系统,给出了其共同CLF的构造算法.基于该CLF,分别建立了单输入和多输入情形下的系统的同时镇定控制策略.所得结果简化和推广了Wu关于同时镇定的工作.
②为获得保守性更小的区间时滞系统的稳定性判据,构造了一个Lyapunov-Krasovskii泛函,该泛函充分考虑了区间时滞的信息.给出了新的估计所构造泛函时间导数上界的方法,并建立了新的区间时滞相关稳定性判据.在标称离散系统情形下,从理论上证明了所得判据较已有相关结论的计算复杂度和保守性低.
③利用描述系统方法对凸多胞体不确定离散系统进行了分析和设计.所得结果结合Cao和Lin (2004)的关于连续系统部分的结论,建立了一个求解凸多胞体不确定线性系统分析与综合问题的框架.
Uncertainty exists in almost all real systems. It may arise from the modeling error, the measure noise, the varying parameters and environmental disturbance. However, the design always depends on a certain system. A gap then appears. It is necessary for us to study how to make a control law which is designed based on a certain system meets the requirement of an uncertain or time-varying real system. The investigation consists of two aspects. The first one is to estimate the roust margin of a control law. The second is to find the characters of the better robust control law. This thesis will deal with the two problems.
The main tool used in this thesis is the Lyapunov stability theory. Generally, there are two critical problems in the applications of the Lyapunov theory:one is how to construct a Lyapunov function or functional for the system under consideration, another is how to estimate the time derivative of the constructed Lyapunov function or functional along with the system solution. This thesis studies the robust control of kinds of uncertain systems by using the Lyapunov theory. It consists of two parts. Chapter 2 is the first part and the second part contains Chapters 3,4 and 5. The first part considers the simultaneous stabilization problem of uncertain nonlinear systems by using the control Lyapunov function (CLF). The second part considers the robust control problem of kinds of uncertain time-delay systems and polytopic-type linear discrete-time systems based on the Lyapunov-Krasovskii functional and the parameter dependent Lyapunov function, respectively. The contents and results of the thesis are as follows.
Chapter I is an introduction. It firstly sums up the progress of robust control of kinds of uncertain systems. Some methods employed in simultaneous stabilization, robust control of parameter systems and uncertain time-delay systems, and their limitation are briefly introduced. Consequently, we present several system models discussed in this thesis and the used mathematical lemmas. At last, we briefly sum up the main work of this thesis.
Chapter 2 studies the simultaneous stabilization problem of a collection of uncertain nonlinear systems. Firstly, we consider the single-input affine nonlinear systems. A suffi- cient condition for the simultaneous stabilization of these systems is proposed by using the CLF. The obtained results are then extended to the single-input and multi-input nonlinear systems with uncertain parameters, respectively. At the end of this chapter, the simultane-ous stabilization of unified chaotic systems is considered. Numerical examples are provided to illustrate the effectiveness of the proposed scheme.
Chapter 3 studies uncertain discrete-time systems with interval time-varying delay. Un-certainties considered are polytopic-type uncertainty, linear fractal norm-bounded uncer-tainty, and quadratic nonlinear perturbations. An appreciate Lyapunov-Krasovskii func-tional is constructed, and a sum of finite inequalities is applied to estimate the time deriva-tive of the functional. Delay-range-dependent stability criteria are developed in terms of LMIs. It is shown by simulation that the proposed criteria can provide less conservatism than some existing ones. Moreover, based on the criteria, we also design the state feedback and time-delayed feedback to stabilize the system, respectively.
Chapter 4 analyzes the stability of uncertain continuous-time systems which have inter-val time-varying delay and nonlinear perturbations. In the estimation of the time derivative of the constructed Lyapunov-Krassivskii functional, some useful terms are reserved such that the estimation holds less conservatism. The effectiveness of the proposed approach is demonstrated by numerical examples.
Chapter 5 considers the robust stability and stabilization for a class of discrete-time polytypic linear systems. A sufficient and necessary condition for the stability of nominal system is presented by using the descriptor system transformation. This condition can be easily adapted in controller synthesis since it separates the design of Lyapunov function and the control law.
In Chapter 6 the topics of this thesis are summarized and the problems for further study are presented.
The main contributions of this thesis are as follows:
①For a collection of nonlinear uncertain systems which have the Brunovsky canonical form, a systematic algorithm is proposed to construct the common CLF. Based on the CLF, simultaneous stabilization feedback is presented for the cases of single-input and multi-input, respectively. The results simplify and generalize the corresponding works of Wu (2005,2009).
②To reduce the conservatism of the reported delay-range-dependent stability criteria for uncertain interval time-delay systems, a Lyapunov-Krasovskii functional which includes the information of the range of time delay is presented, and the upper bound of the time derivative of the constructed functional is estimated by a new approach. New delay-range-dependent stability criteria are proposed. In the case of nominal discrete time-delay systems, we prove that the obtained result is less conservative than some existing criteria.
③Descriptor model transformation is employed in the robust stability analysis and control design of polytopic-type discrete-time linear systems. The developed result can be viewed as a discrete-time counterpart of the continuous-time results proposed by Cao and Lin in 2004.
Lyapunov稳定性理论是本学位论文用到的主要工具.应用Lyapunov理论主要难点在于一、如何构造系统的Lyapunov函数或泛函；二、如何估计所构造的Lyapunov函数或泛函的时间导数.本论文从这两方面着手,研究了几类不确定系统的鲁棒控制问题.主要内容可以分为两个部分：第二章为第一部分,第三、四、五章构成了第二部分.在第一部分,利用控制Lyapunov函数(CLF)研究了一组不确定非线性系统的同时镇定问题.在第二部分,基于Lyapunov-Krasovskii泛函和参数依赖Lyapunov函数,分别研究了不确定区间时滞系统和凸多胞体不确定离散系统的鲁棒控制问题.现谨将各章的主要内容及研究结果概述如下：
第一章为绪论部分.首先综述了不确定系统鲁棒控制研究的一些进展,简要介绍了同时镇定,参数系统和不确定时滞系统的研究中常用的一些方法及其局限性.接着介绍了本论文讨论的几类不确定系统模型以及用到的数学工具.最后概述了本论文的主要工作
第二章研究一组不确定非线性系统的同时镇定问题.首先研究了单输入仿射非线性系统,利用CLF给出了其同时镇定反馈律存在的一个充分条件.随后将所得的结果推广到带有不确定参数和有Brunovsky标准型结构的多输入非线性系统.最后作为应用,研究了一组混沌控制系统的同时镇定问题,以此来说明设计方案的有效性.
第三章研究不确定离散区间时滞系统的时滞相关稳定性判据及其镇定设计问题.主要讨论了3种不确定性：凸多胞体不确定性,线性分式范数有界不确定性,以及非线性扰动引起的不确定性.通过构造新的Lyapunov-Krasovskii泛函,并利用建立的有限和不等式来估计所构造泛函导数的上界,有效地降低了现有文献中相关结论的保守性.此外,基于稳定性判据还给出了这类系统的状态反馈和时滞状态反馈两种控制设计,并利用数值例子进行了仿真.
第四章分析了一类带有非线性扰动的连续区间时滞系统的时滞相关稳定性.引入了一种新的估计所构造Lyapunov-Krasovskii泛函的时间导数上界的方法.在该方法中,保留了一些有用的时间导数项,可以有效地降低已有文献中时滞相关稳定性判据的一些保守性.数值例子说明了本章中所用方法的优越性.
第五章研究了一类凸多胞体不确定性线性离散系统的鲁棒稳定性及其镇定.基于描述系统变换,对标称的离散线性系统,给出了其渐近稳定性的一个充要条件.在该条件中,系统矩阵和Lyapunov矩阵的乘积项分离,因而它可以很方便的应用于凸多胞体不确定离散系统的稳定性分析及其镇定设计.
第六章总结了本文论研究的主要内容和结论,并给出若干值得进一步研究的问题.
本文主要创新点概括如下：
①对一组具有Brunovsky标准型结构的不确定非线性系统,给出了其共同CLF的构造算法.基于该CLF,分别建立了单输入和多输入情形下的系统的同时镇定控制策略.所得结果简化和推广了Wu关于同时镇定的工作.
②为获得保守性更小的区间时滞系统的稳定性判据,构造了一个Lyapunov-Krasovskii泛函,该泛函充分考虑了区间时滞的信息.给出了新的估计所构造泛函时间导数上界的方法,并建立了新的区间时滞相关稳定性判据.在标称离散系统情形下,从理论上证明了所得判据较已有相关结论的计算复杂度和保守性低.
③利用描述系统方法对凸多胞体不确定离散系统进行了分析和设计.所得结果结合Cao和Lin (2004)的关于连续系统部分的结论,建立了一个求解凸多胞体不确定线性系统分析与综合问题的框架.
Uncertainty exists in almost all real systems. It may arise from the modeling error, the measure noise, the varying parameters and environmental disturbance. However, the design always depends on a certain system. A gap then appears. It is necessary for us to study how to make a control law which is designed based on a certain system meets the requirement of an uncertain or time-varying real system. The investigation consists of two aspects. The first one is to estimate the roust margin of a control law. The second is to find the characters of the better robust control law. This thesis will deal with the two problems.
The main tool used in this thesis is the Lyapunov stability theory. Generally, there are two critical problems in the applications of the Lyapunov theory:one is how to construct a Lyapunov function or functional for the system under consideration, another is how to estimate the time derivative of the constructed Lyapunov function or functional along with the system solution. This thesis studies the robust control of kinds of uncertain systems by using the Lyapunov theory. It consists of two parts. Chapter 2 is the first part and the second part contains Chapters 3,4 and 5. The first part considers the simultaneous stabilization problem of uncertain nonlinear systems by using the control Lyapunov function (CLF). The second part considers the robust control problem of kinds of uncertain time-delay systems and polytopic-type linear discrete-time systems based on the Lyapunov-Krasovskii functional and the parameter dependent Lyapunov function, respectively. The contents and results of the thesis are as follows.
Chapter I is an introduction. It firstly sums up the progress of robust control of kinds of uncertain systems. Some methods employed in simultaneous stabilization, robust control of parameter systems and uncertain time-delay systems, and their limitation are briefly introduced. Consequently, we present several system models discussed in this thesis and the used mathematical lemmas. At last, we briefly sum up the main work of this thesis.
Chapter 2 studies the simultaneous stabilization problem of a collection of uncertain nonlinear systems. Firstly, we consider the single-input affine nonlinear systems. A suffi- cient condition for the simultaneous stabilization of these systems is proposed by using the CLF. The obtained results are then extended to the single-input and multi-input nonlinear systems with uncertain parameters, respectively. At the end of this chapter, the simultane-ous stabilization of unified chaotic systems is considered. Numerical examples are provided to illustrate the effectiveness of the proposed scheme.
Chapter 3 studies uncertain discrete-time systems with interval time-varying delay. Un-certainties considered are polytopic-type uncertainty, linear fractal norm-bounded uncer-tainty, and quadratic nonlinear perturbations. An appreciate Lyapunov-Krasovskii func-tional is constructed, and a sum of finite inequalities is applied to estimate the time deriva-tive of the functional. Delay-range-dependent stability criteria are developed in terms of LMIs. It is shown by simulation that the proposed criteria can provide less conservatism than some existing ones. Moreover, based on the criteria, we also design the state feedback and time-delayed feedback to stabilize the system, respectively.
Chapter 4 analyzes the stability of uncertain continuous-time systems which have inter-val time-varying delay and nonlinear perturbations. In the estimation of the time derivative of the constructed Lyapunov-Krassivskii functional, some useful terms are reserved such that the estimation holds less conservatism. The effectiveness of the proposed approach is demonstrated by numerical examples.
Chapter 5 considers the robust stability and stabilization for a class of discrete-time polytypic linear systems. A sufficient and necessary condition for the stability of nominal system is presented by using the descriptor system transformation. This condition can be easily adapted in controller synthesis since it separates the design of Lyapunov function and the control law.
In Chapter 6 the topics of this thesis are summarized and the problems for further study are presented.
The main contributions of this thesis are as follows:
①For a collection of nonlinear uncertain systems which have the Brunovsky canonical form, a systematic algorithm is proposed to construct the common CLF. Based on the CLF, simultaneous stabilization feedback is presented for the cases of single-input and multi-input, respectively. The results simplify and generalize the corresponding works of Wu (2005,2009).
②To reduce the conservatism of the reported delay-range-dependent stability criteria for uncertain interval time-delay systems, a Lyapunov-Krasovskii functional which includes the information of the range of time delay is presented, and the upper bound of the time derivative of the constructed functional is estimated by a new approach. New delay-range-dependent stability criteria are proposed. In the case of nominal discrete time-delay systems, we prove that the obtained result is less conservative than some existing criteria.
③Descriptor model transformation is employed in the robust stability analysis and control design of polytopic-type discrete-time linear systems. The developed result can be viewed as a discrete-time counterpart of the continuous-time results proposed by Cao and Lin in 2004.
引文
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