摘要
不确定性是自然界普遍存在的现象,用微分包含来描述系统的不确定性是常用的方法之一。微分包含系统比一般微分方程描述的系统更具有广泛性,例如线性时不变系统、区间系统、多胞体系统等都可以看作是微分包含系统的一种特殊形式。针对微分包含系统,如何建立该系统的分析和控制器的设计方法,是现代控制理论研究中的一个重要课题。本文主要研究了微分包含系统的几类控制器的设计问题。
全文分为五章,各章的研究内容叙述如下:
第一章是绪论部分。首先综述了微分包含系统的性质及其研究的进展情况,简要介绍了微分包含系统的几种具有特殊结构的模型以及研究方法,接着介绍了本论文所用到的数学基础理论,最后概述了本论文的主要工作。
第二章利用滑模变结构控制理论研究了微分包含系统的镇定性问题。首先研究了单个系统的滑模控制。涉及的模型有一类非线性系统和不确定时滞系统。针对一类非线性系统,提出了一个新的趋近律,该趋近律可以有效地抑制抖振。对于具有非线性输入的不确定时滞系统,给出了基于观测器的无源滑模控制器的设计,同时得到了使闭环系统渐近稳定且无源的充分条件。然后重点研究了多胞体微分包含系统以及多胞体随机微分包含系统的滑模控制,分别给出了滑模控制器的设计,并且对多胞体随机微分包含系统还做了无源性分析,最后通过例子验证了所提方法的有效性。
第三章研究了多胞体线性时滞微分包含系统和多胞体随机时滞微分包含系统的非线性状态反馈控制问题。利用二次凸包函数方法,首先给出了多胞体线性时滞微分包含系统的非线性状态反馈控制器的设计,然后推广到多胞体随机时滞微分包含系统中,得到了使闭环系统在均方意义下是指数稳定的充分条件。第四章首先采用描述系统方法研究了多胞体线性微分包含系统和多胞体随机微分包含系统的有限时间控制问题。针对多胞体线性微分包含系统,不仅考虑了有限时间控制问题,也考虑了广义H_2性能。对于多胞体随机微分包含系统,给出了有限时间控制器的设计方法。最后考虑了鲁里叶微分包含系统的非脆弱有限时间镇定问题。
第五章总结了本文研究的主要内容并给出若干值得进一步研究的问题。
本文的主要创新点概括如下:
1)针对一类非线性系统,提出了一个新的趋近律,有效地抑制了抖振。针对多胞体微分包含系统和多胞体随机微分包含系统,首次给出了变结构控制器的设计方案。所得结果建立了用变结构控制研究多胞体微分包含系统的理论框架。
2)将二次凸包函数方法运用到多胞体线性时滞微分包含系统的镇定性分析中,并推广到多胞体随机微分包含系统,从而扩大了非线性控制器的设计范围。
3)将描述系统方法和有限时间控制相结合,为多胞体线性微分包含系统和多胞体随机微分包含系统的有限时间控制器设计提供了一种可行的新途径。针对鲁里叶微分包含系统,提出基于观测器的非脆弱有限时间控制器的设计方案,有效地解决了该系统的有限时间控制问题。
Uncertainty usually exists in the nature. Differential inclusions usually can be used todescribe the system with uncertainty. This kinds of systems are more general than systemsdescribed by ordinary differential equations. For example, linear time-invariant systems,interval systems, polytopic systems, etc, can be taken as a special form of the differentialinclusion systems. For the differential inclusion system, how to construct the analysis andthe method of controller design has been an important topic in modern control theory.This paper mainly investigates several controller design methods of differential inclusionsystems.
This paper consists of five chapters. The main contents of every chapter are as fol-lows:
Chapter one is an introduction. It introduces the properties and the recent develop-ment of differential inclusions, gives a brief introduction about the models of differentialinclusion systems with special structures and some research approaches, then presentsbasic mathematical theories used in this paper. Finally, the major work of the paper issummarized.
Chaper two considers the stabilization problem of differential inclusion systems viasliding mode control. First, the common systems are investigated, the models includea class of nonlinear systems and uncertain time-delay systems. For a class of nonlinearsystems, a new reaching law is proposed to reduce the chattering phenomenon. For theuncertain time-delay system with nonlinear input, an observer-based passive sliding modecontroller is proposed and sufficient conditions are obtained such that the looped system isasymptotically stable and passive. For polytopic differential inclusion systems and poly-topic stochastic differential inclusion systems, the sliding mode control approach is usedto design their controllers, respectively. Moreover, passivity is considered for the poly-topic stochastic differential inclusion systems. Finally, examples are given to illustratethe effectiveness of the proposed methods.
Chapter three investigates the nonlinear state feedback controller design methods ofpolytopic linear differential inclusion systems with time delay and polytopic stochasticdifferential inclusion systems with time delay, respectively. Using the convex hull func-tion method, a nonlinear state feedback law is designed to stabilize the polytopic lineardifferential inclusion system with time delay. Then this technology is successfully ex-tended to the polytopic stochastic differential inclusion system with time delay, sufficientconditions are derived to guarantee the closed-loop system is exponentially stable in themean square.
Chapter four proposes the finite time control problems for both polytopic linear dif-ferential inclusion systems and polytopic stochastic differential inclusion systems. Forthe polytopic linear differential differential inclusion systems, a generalized H2 controlleris designed not only to guarantee finite time boundedness of the closed-loop system, butalso to restrict the effect of disturbance on a prescribed level. For the polytopic stochas-tic differential inclusion systems, a state feedback law is designed to guarantee finite-timestochastically boundedness of the closed-loop system. Finally, observer-based non-fragilefinite time stabilization of Lur’e differential inclusion systems is investigated.
Chapter five summarizes the main contents in this thesis and presents some valuableproblems.
The main innovations of this dissertation can be summarized as follows:
1) For a class of nonlinear systems, a new reaching law is proposed to reduce thechattering phenomenon. For the polytopic differential inclusion systems and the polytopicstochastic differential inclusion systems, sliding mode control approach is firstly used todesign the controllers, respectively. These results construct a theoretical framework of thepolytopic differential inclusion systems via sliding mode control.
2) The method of convex hull of quadratics is successfully extended to stabilize thepolytopic linear differential inclusion system with time delay and the polytopic stochasticdifferential inclusion system with time delay. The results greatly enlarge the design scopeof nonlinear controller.
3) A new feasible approach is proposed to design finite time controllers of both the polytopic linear differential inclusion systems and the polytopic stochastic differentialinclusion systems by the way of combining the descriptor system method and finite timecontrol. For the Lur’e differential inclusion systems, the design of observer-based non-fragile finite time controller is considered. By the design, the problem of finite timecontrol of this type of systems can be effectively resolved.
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