一类T-S模糊广义系统的无源控制
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摘要
耗散性理论在系统分析与控制研究中起着重要的作用.其本质含义是存在一个非负的能量函数(即存储函数),使得系统的能量损耗总小于能量的供给率.而无源性是耗散性的一个重要方面,它将输入输出的乘积作为能量的供给率,体现了系统在有界输入条件下能量的衰减特性.事实上,基于李亚普诺夫函数的稳定性理论,也可从无源性的角度加以解释,可以说,无源性是稳定性的一种更高层次的抽象.在对系统进行镇定时,人们常常需要构造一个李亚普诺夫函数,现有文献表明,这一过程可转化为构造一个使系统无源的存储函数.
T-S模糊广义系统的无源控制研究尚在起步阶段,还有许多亟待解决的问题.本文借鉴线性广义系统理论和现代鲁棒控制理论思想,利用先进的矩阵分析理论和线性矩阵不等式(LMI)技术,在Lyapunov稳定性理论框架下,考虑了一类T-S模糊广义系统无源控制问题与非脆弱H∞控制问题.主要工作概括如下：
(一)研究了T-S模糊系统与T-S模糊广义的无源控制问题.首先对确定性T-S模糊系统与T-S模糊广义系统分别设计了状态反馈控制器、基于观测器的反馈控制器与动态输出反馈控制器,使得闭环系统容许且具有耗散率为η的无源性；然后对于范数有界不确定性T-S模糊系统与T-S模糊广义系统,给出了基于观测器的无源控制器存在条件和设计方法,保证闭环系统是广义二次稳定且具有耗散率为η的无源性.
(二)分别研究了时滞T-S模糊系统与时滞T-S模糊广义系统的无源控制问题.针对时滞模糊系统,利用Lyapunov技术,引入Lyapunov-Krasovskii泛函,以线性矩阵不等式的形式分别给出状态反馈控制器、基于观测器的反馈控制器和动态输出反馈控制器的存在条件与设计方法,使得闭环系统是容许且无源的.本文结论可推广到系统具有范数有界不确定性的情况.
(三)研究了T-S模糊系统与T-S模糊广义系统的非脆弱无源控制问题.主要思想是设计非脆弱控制器,在鲁棒镇定系统的同时,能够保证闭环系统具有耗散率为η的无源性.分别对T-S模糊系统与T-S模糊广义系统设计非脆弱状态反馈控制器,并在状态反馈的基础上,设计基于观测器的非脆弱控制器使得系统满足相应性能.其中基于观测器的控制器控制增益和观测增益均具有两种形式的扰动.所得到的控制器增益可以通过求解线性矩阵不等式来表示.对于时滞不确定T-S模糊系统设计非脆弱状态反馈控制器,使得系统满足无源性.
(四)研究T-S模糊系统与T-S模糊广义系统的非脆弱H∞控制问题.对于H∞控制问题,利用隶属函数的性质,设计非脆弱的状态反馈控制器,得到了使得系统满足H∞性能指标保守性更小的条件.
对文中所得结论,都给出了仿真算例,以说明设计方法的有效性.
Dissipative theory plays an important role in the analysis and control research of systems. The implication is that there exists a nonnegative energy function (namely store function) such that the energy consumption of a control system is always less than the supply rate of the energy. The passivity is an important part of the dissipativeness. It takes the product of the input and the output as the supply rate of the energy, which embodies the attenuation property of a system under bounded exogenous input. In fact, the stabilization theory based on Lyapunov function can also be explained in view of the passivity. Therefore, the passivity is a further abstraction of the stability. When investigating the stabilization of a system, a Lyapunov function is alwagys needed. According to the present literatures, this process can be converted into constructing a store function that makes the system passive.
The research on passive control for T-S fuzzy descriptor system is still in preliminary stage. Many unsolved problems need to do. In this paper, by using the advanced matrix theory and the linear matrix inequality (LMI) technique, the problem of passive control and non-fragile H∞control for a kind of T-S fuzzy descriptor system is studied under the framework of Lyapunov stability theory, which is motivated by the linear descriptor system theory and advanced robust control theory. The main contributions are as follows:
(1) The passive control problems are considered for T-S fuzzy systems and T-S fuzzy descriptor systems. Passive control is firstly discussed for T-S fuzzy systems and T-S fuzzy descriptor systems without uncertainty, and the conditions for the closed-loop systems to be admissible and passive with dissipationηare proposed in the cases of state feedback controller, observer-based feedback controller and dynamic output feedback controller. Then robust passive analysis and control are discussed for T-S fuzzy systems and T-S fuzzy descriptor systems with norm-bounded uncertainties. The design of observe-based controllers is presented for the closed-loop systems to be generalized quadratically stable and passive with dissipation 77.
(2) The passive control problems are discussed for T-S fuzzy systems and T-S fuzzy descriptor systems with time-delay. As for the time-delay fuzzy systems, by using of the Lyapunov technique and introducing the Lyapuniov-Krasovskii functional, the existence conditions of state feedback, observer-based feedback and dynamic output feedback passive controllers with their design methods are derived in terms of linear matrix inequalities, such that the closed-loop systems are admissible and passive. The results gained in this paper can be generalized into the case of systems with norm-bounded uncertainties.
(3) The non-fragile passive control problems are considered for T-S fuzzy systems and T-S fuzzy descriptor systems. The main idea is to design non-fragile controller such that the closed-loop system is robustly stable and passive with dissipationη. The non-fragile state feedback controller is designed firstly to guarantee the relative performance of the systems, and then non-fragile observer-based controller with the perturbations in both the control gain and observer gain is discussed for T-S fuzzy descriptor systems. The observer gain perturbations are in both the additive form and multiplicative form. Moreover, the existence conditions and design methods are presented in terms of LMI. The passive control problem of uncertain T-S fuzzy systems with time-delay is investigated. The non-fragile controller is designed to guarantee the passivity of the systems.
(4) The non-fragile H∞control problems are considered for T-S fuzzy systems and T-S fuzzy descriptor systems. By virtue of the property of membership function, the H∞control conditions are generalized with non-fragile state feedback controller.
As to the conclusions obtained in the paper, simulation examples are presented to show the effectiveness of the proposed methods.
T-S模糊广义系统的无源控制研究尚在起步阶段,还有许多亟待解决的问题.本文借鉴线性广义系统理论和现代鲁棒控制理论思想,利用先进的矩阵分析理论和线性矩阵不等式(LMI)技术,在Lyapunov稳定性理论框架下,考虑了一类T-S模糊广义系统无源控制问题与非脆弱H∞控制问题.主要工作概括如下：
(一)研究了T-S模糊系统与T-S模糊广义的无源控制问题.首先对确定性T-S模糊系统与T-S模糊广义系统分别设计了状态反馈控制器、基于观测器的反馈控制器与动态输出反馈控制器,使得闭环系统容许且具有耗散率为η的无源性；然后对于范数有界不确定性T-S模糊系统与T-S模糊广义系统,给出了基于观测器的无源控制器存在条件和设计方法,保证闭环系统是广义二次稳定且具有耗散率为η的无源性.
(二)分别研究了时滞T-S模糊系统与时滞T-S模糊广义系统的无源控制问题.针对时滞模糊系统,利用Lyapunov技术,引入Lyapunov-Krasovskii泛函,以线性矩阵不等式的形式分别给出状态反馈控制器、基于观测器的反馈控制器和动态输出反馈控制器的存在条件与设计方法,使得闭环系统是容许且无源的.本文结论可推广到系统具有范数有界不确定性的情况.
(三)研究了T-S模糊系统与T-S模糊广义系统的非脆弱无源控制问题.主要思想是设计非脆弱控制器,在鲁棒镇定系统的同时,能够保证闭环系统具有耗散率为η的无源性.分别对T-S模糊系统与T-S模糊广义系统设计非脆弱状态反馈控制器,并在状态反馈的基础上,设计基于观测器的非脆弱控制器使得系统满足相应性能.其中基于观测器的控制器控制增益和观测增益均具有两种形式的扰动.所得到的控制器增益可以通过求解线性矩阵不等式来表示.对于时滞不确定T-S模糊系统设计非脆弱状态反馈控制器,使得系统满足无源性.
(四)研究T-S模糊系统与T-S模糊广义系统的非脆弱H∞控制问题.对于H∞控制问题,利用隶属函数的性质,设计非脆弱的状态反馈控制器,得到了使得系统满足H∞性能指标保守性更小的条件.
对文中所得结论,都给出了仿真算例,以说明设计方法的有效性.
Dissipative theory plays an important role in the analysis and control research of systems. The implication is that there exists a nonnegative energy function (namely store function) such that the energy consumption of a control system is always less than the supply rate of the energy. The passivity is an important part of the dissipativeness. It takes the product of the input and the output as the supply rate of the energy, which embodies the attenuation property of a system under bounded exogenous input. In fact, the stabilization theory based on Lyapunov function can also be explained in view of the passivity. Therefore, the passivity is a further abstraction of the stability. When investigating the stabilization of a system, a Lyapunov function is alwagys needed. According to the present literatures, this process can be converted into constructing a store function that makes the system passive.
The research on passive control for T-S fuzzy descriptor system is still in preliminary stage. Many unsolved problems need to do. In this paper, by using the advanced matrix theory and the linear matrix inequality (LMI) technique, the problem of passive control and non-fragile H∞control for a kind of T-S fuzzy descriptor system is studied under the framework of Lyapunov stability theory, which is motivated by the linear descriptor system theory and advanced robust control theory. The main contributions are as follows:
(1) The passive control problems are considered for T-S fuzzy systems and T-S fuzzy descriptor systems. Passive control is firstly discussed for T-S fuzzy systems and T-S fuzzy descriptor systems without uncertainty, and the conditions for the closed-loop systems to be admissible and passive with dissipationηare proposed in the cases of state feedback controller, observer-based feedback controller and dynamic output feedback controller. Then robust passive analysis and control are discussed for T-S fuzzy systems and T-S fuzzy descriptor systems with norm-bounded uncertainties. The design of observe-based controllers is presented for the closed-loop systems to be generalized quadratically stable and passive with dissipation 77.
(2) The passive control problems are discussed for T-S fuzzy systems and T-S fuzzy descriptor systems with time-delay. As for the time-delay fuzzy systems, by using of the Lyapunov technique and introducing the Lyapuniov-Krasovskii functional, the existence conditions of state feedback, observer-based feedback and dynamic output feedback passive controllers with their design methods are derived in terms of linear matrix inequalities, such that the closed-loop systems are admissible and passive. The results gained in this paper can be generalized into the case of systems with norm-bounded uncertainties.
(3) The non-fragile passive control problems are considered for T-S fuzzy systems and T-S fuzzy descriptor systems. The main idea is to design non-fragile controller such that the closed-loop system is robustly stable and passive with dissipationη. The non-fragile state feedback controller is designed firstly to guarantee the relative performance of the systems, and then non-fragile observer-based controller with the perturbations in both the control gain and observer gain is discussed for T-S fuzzy descriptor systems. The observer gain perturbations are in both the additive form and multiplicative form. Moreover, the existence conditions and design methods are presented in terms of LMI. The passive control problem of uncertain T-S fuzzy systems with time-delay is investigated. The non-fragile controller is designed to guarantee the passivity of the systems.
(4) The non-fragile H∞control problems are considered for T-S fuzzy systems and T-S fuzzy descriptor systems. By virtue of the property of membership function, the H∞control conditions are generalized with non-fragile state feedback controller.
As to the conclusions obtained in the paper, simulation examples are presented to show the effectiveness of the proposed methods.
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