基于LMI方法的广义系统的稳定性与保性能控制研究
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摘要
广义系统和时滞广义系统的稳定性与镇定问题是广义系统理论的基本问题之一，正受到越来越多学者的关注，其理论一直随着控制理论的发展而不断完善。一方面因为对广义系统稳定性的研究，不仅需要考虑其稳定性，而且还要考虑正则性和因果性(离散广义系统)及脉冲自由(连续广义系统)，所以对广义系统稳定性与镇定的研究要比正常系统要复杂得多。另一方面，线性矩阵不等式方法在控制领域中几年来应用越来越广泛，但在广义系统尤其是时滞广义系统中的应用还不多。
本文基于广义系统和时滞广义系统的模型，特别是广义系统和时滞广义系统的研究现状，应用线性矩阵不等式(LMI)方法，较系统地分别研究了广义系统和时滞广义系统、参数不确定广义系统和不确定时滞广义系统的稳定性和镇定控制以及保性能控制问题。
本文的主要内容包括以下几个方面：
一、应用线性矩阵不等式(LMI)方法，研究了广义系统镇定问题，给出了广义系统镇定控制器的设计；研究了参数不确定广义系统的鲁棒稳定性和鲁棒镇定问题，给出了参数不确定广义系统的鲁棒稳定性判定定理与鲁棒镇定控制器的设计。
二、应用线性矩阵不等式(LMI)方法，研究了时滞广义系统和关联时滞广义大系统的稳定性和镇定问题，给出了时滞广义系统和关联时滞广义大系统的稳定性判定定理与镇定控制器的设计。
三、研究了参数不确定时滞广义系统和参数不确定关联时滞广义大系统的鲁棒稳定性和鲁棒镇定问题，给出了参数不确定时滞广义系统和参数不确定关联时滞广义大系统的鲁棒稳定性判定定理与鲁棒镇定控制器的设计。
四、对一类不确定项具有数值界的参数不确定广义系统和一个二次型性能指标，研究了其最优保性能状态反馈控制律的设计问题。基于不确定项的表达形式，应用线性矩阵不等式(LMI)方法和Lyapunov函数法，给出了参数不确定广义系统和参数不确定广义大系统存在最优保性能控制器的LMI条件和最优保性能控制器的设计。此外，基于不确定项的表达形式，应用线性矩阵不等式(LMI)方法和Lyapunov函数法，导出了存在保性能控制器的LMI条件和保性能控制器的设计。
The stability and stabilization problem of singular systems or singular systems withtime-delay is one of the fundamental problems in the theory of singular system. The studyof them is much more complicated than that of state-space systems because the singularsystem requires considering not only stability, but also regularity and causality (for discretesingular systems) or impulse immunity (for continuous singular systems).
Recent years, the linear matrix inequality (LMI) approach has been attracting moreand more attention due to its extensive applications in solving various control problems.However, the problem of stability of large-scale singular systems with time-delay via LMIapproach is scarcely reported in the literatures.
On the basis of the recent progress on singular systems models, this dissertationimplements a systematic study on the stability, stabilization and guaranteed cost control ofsingular systems with time-delay models and parameter uncertainty. By means of LMItechniques, the main results obtained in this dissertation are as follows:
ⅰ) The problem of stability and stabilization of singular systems is investigated.Some stabilization controllers are proposed for singular systems. The problems of robuststability and robust stabilization of singular systems with parameter uncertainties areinvestigated via linear matrix inequality (LMI) method. A theorem of robust stability ofsingular systems with parameter uncertainties is obtained and the robust stabilizationcontrollers are proposed.
ⅱ) The problems of stability and stabilization of the continuous singular systemswith delay-time and the singular linear large-scale system with delay-time are studied. Thecorresponding theorems of stability are presented, based on which the stabilizationcontrollers are designed.
ⅲ) The problems of robust stability and robust stabilization of continuous singularsystems with delay-time and parameter uncertainties & singular linear large-scale systemwith delay-time and parameter uncertainties are discussed. The corresponding theoremsand controllers of robust stability are presented.
ⅳ) For a class of singular large scale systems with value bounded parameteruncertainty and a quadratic cost index, the problem of designing an optimal guaranteed cost state feedback controller is considered. A sufficient condition for the existence of theoptimal guaranteed cost controller and the design of optimal guaranteed cost controllers arederived. Moreover, the problem of guaranteed cost state feedback controller design issurveyed. A sufficient condition for the existence of the guaranteed cost controller and thedesign of guaranteed cost controllers are derived.
本文基于广义系统和时滞广义系统的模型，特别是广义系统和时滞广义系统的研究现状，应用线性矩阵不等式(LMI)方法，较系统地分别研究了广义系统和时滞广义系统、参数不确定广义系统和不确定时滞广义系统的稳定性和镇定控制以及保性能控制问题。
本文的主要内容包括以下几个方面：
一、应用线性矩阵不等式(LMI)方法，研究了广义系统镇定问题，给出了广义系统镇定控制器的设计；研究了参数不确定广义系统的鲁棒稳定性和鲁棒镇定问题，给出了参数不确定广义系统的鲁棒稳定性判定定理与鲁棒镇定控制器的设计。
二、应用线性矩阵不等式(LMI)方法，研究了时滞广义系统和关联时滞广义大系统的稳定性和镇定问题，给出了时滞广义系统和关联时滞广义大系统的稳定性判定定理与镇定控制器的设计。
三、研究了参数不确定时滞广义系统和参数不确定关联时滞广义大系统的鲁棒稳定性和鲁棒镇定问题，给出了参数不确定时滞广义系统和参数不确定关联时滞广义大系统的鲁棒稳定性判定定理与鲁棒镇定控制器的设计。
四、对一类不确定项具有数值界的参数不确定广义系统和一个二次型性能指标，研究了其最优保性能状态反馈控制律的设计问题。基于不确定项的表达形式，应用线性矩阵不等式(LMI)方法和Lyapunov函数法，给出了参数不确定广义系统和参数不确定广义大系统存在最优保性能控制器的LMI条件和最优保性能控制器的设计。此外，基于不确定项的表达形式，应用线性矩阵不等式(LMI)方法和Lyapunov函数法，导出了存在保性能控制器的LMI条件和保性能控制器的设计。
The stability and stabilization problem of singular systems or singular systems withtime-delay is one of the fundamental problems in the theory of singular system. The studyof them is much more complicated than that of state-space systems because the singularsystem requires considering not only stability, but also regularity and causality (for discretesingular systems) or impulse immunity (for continuous singular systems).
Recent years, the linear matrix inequality (LMI) approach has been attracting moreand more attention due to its extensive applications in solving various control problems.However, the problem of stability of large-scale singular systems with time-delay via LMIapproach is scarcely reported in the literatures.
On the basis of the recent progress on singular systems models, this dissertationimplements a systematic study on the stability, stabilization and guaranteed cost control ofsingular systems with time-delay models and parameter uncertainty. By means of LMItechniques, the main results obtained in this dissertation are as follows:
ⅰ) The problem of stability and stabilization of singular systems is investigated.Some stabilization controllers are proposed for singular systems. The problems of robuststability and robust stabilization of singular systems with parameter uncertainties areinvestigated via linear matrix inequality (LMI) method. A theorem of robust stability ofsingular systems with parameter uncertainties is obtained and the robust stabilizationcontrollers are proposed.
ⅱ) The problems of stability and stabilization of the continuous singular systemswith delay-time and the singular linear large-scale system with delay-time are studied. Thecorresponding theorems of stability are presented, based on which the stabilizationcontrollers are designed.
ⅲ) The problems of robust stability and robust stabilization of continuous singularsystems with delay-time and parameter uncertainties & singular linear large-scale systemwith delay-time and parameter uncertainties are discussed. The corresponding theoremsand controllers of robust stability are presented.
ⅳ) For a class of singular large scale systems with value bounded parameteruncertainty and a quadratic cost index, the problem of designing an optimal guaranteed cost state feedback controller is considered. A sufficient condition for the existence of theoptimal guaranteed cost controller and the design of optimal guaranteed cost controllers arederived. Moreover, the problem of guaranteed cost state feedback controller design issurveyed. A sufficient condition for the existence of the guaranteed cost controller and thedesign of guaranteed cost controllers are derived.
引文
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