摘要
T-S模糊广义系统模型能够很好地逼近非线性广义系统,是解决非线性广义系统控制问题的一个有效途径。目前,虽然对T-S模糊广义系统稳定性分析与控制的研究已经取得了一些成果,但是与经典控制理论相比还相差甚远。T-S模糊广义系统的研究尚在初级阶段,还有许多有待解决的问题。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(LMIs)技术,在Lyapunov稳定性理论框架下,研究了一类T-S模糊广义系统的控制问题。主要工作有以下几个方面：
1.研究了一类由T-S模糊广义系统模型描述的连续非线性时滞广义系统的保成本控制问题。首先,以LMIs的形式给出了状态反馈保成本控制器存在的充分条件,并在此基础上利用模糊规则权以及引入新的变量获得一个改进的模糊保成本控制器设计方法,在一定程度上降低了设计的保守性。其次,将具有最小J。值的保成本控制器设计问题转化为一个LMIs约束的凸优化问题来求解。数值算例说明所设计的模糊控制器不仅能保证闭环系统渐近稳定,而且满足最小化的性能指标。最后,研究了基于广义观测器的输出反馈保成本控制问题,以LMIs的形式给出了基于广义观测器的输出反馈保成本控制器存在的充分条件。
2.针对一类由T-S模糊广义系统模型描述的参数不确定的连续非线性时滞广义系统,首先利用Lyapunov稳定性理论和线性矩阵不等式(LMIs)技术研究了系统的二次稳定性,并在此基础上提出鲁棒H∞控制设计方法,以LMIs的形式给出系统二次稳定条件及鲁棒H∞控制器存在的充分条件。其次研究了不确定时滞T-S模糊广义系统的鲁棒保成本控制问题,将具有最小J0值的保成本控制器设计问题转化为一个LMIs约束的凸优化问题来求解。数值算例说明所提出的模糊鲁棒H∞控制器及模糊鲁棒保成本控制器的设计方法的有效性。
3.基于时滞相关准则研究了一类由T-S模糊广义系统模型描述的不确定连续非线性时滞广义系统的鲁棒保成本控制问题,讨论了时滞的大小对稳定性的影响,对于使系统保持渐近稳定所允许的时滞上界max的求取,归结为一个LMI约束的凸优化问题来求解。而相应于允许的时滞上界的保成本性能指标的最小上界则转化为一个寻找次优值的优化问题。仿真结果表明了所提出方法的有效性。
4.基于时滞相关的方法研究了一类由T-S模糊广义系统模型描述的不确定连续非线性时滞广义系统的非脆弱H∞控制问题。首先,针对参数确定的标称系统,提出了一个新的时滞相关H。稳定性判据。其次,考虑系统存在范数约束的参数不确定性时,分别提出了基于两种增益变化形式的非脆弱控制器设计方法。以LMIs的形式给出了满足H∞性能指标的非脆弱控制器存在的充分条件,仿真示例表明所提出方法的有效性。
5.提出了一个多时滞的T-S模糊广义系统模型,并且基于Lyapunov稳定性理论和时滞无关的方法研究了一类由多时滞连续T-S模糊广义系统模型描述的多时滞非线性广义系统的保成本控制问题以及H∞控制问题。以LMIs的形式给出了系统的稳定性条件,以及保成本控制器和H∞控制器存在的充分条件。多时滞T-S模糊广义系统,是对单时滞T-S模糊广义系统的推广,具有更广泛的实际意义。目前在其它文献中还没有研究过。
6.研究了一类由离散T-S模糊时滞广义系统模型描述的离散非线性时滞广义系统的控制问题,首先给出了该类系统的一致正则、因果以及稳定性和容许性的定义,并基于Lyapunov函数方法,对系统的稳定性进行分析,以线性矩阵不等式(LMIs)的形式给出系统一致正则、因果和稳定的充分条件。提出了模糊状态反馈控制器和输出反馈控制器的设计方法。
T-S fuzzy descriptor system model which can approximate a non-linear descriptor system perfectly, is thought to be an effective approach to solve the control problem of non-linear descriptor systems. Currently, although much effort has been made in the exploration of stability analysis and control for the T-S fuzzy descriptor system, the results on this topic are far fewer than those on normal systems. The research on T-S fuzzy descriptor system is still in primary stage. Lots of problems need to be solved. In this paper, by using the advanced matrix theory and the linear matrix inequality (LMIs) technique, we study the problems of stability analysis and control design for a class of T-S fuzzy descriptor systems 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 problems of guaranteed cost control of a class of continous time nonlinear descriptor delay systems described by T-S fuzzy descriptor model are studied. First, sufficient conditions of the existence of state feedback guaranteed cost controller are given in terms of LMIs. Then fuzzy rule weights and some new variables are used to obtain an improved guaranteed cost controller design method which is less conservative. And design of guaranteed cost controller with minimized J0 is reduced to a convex optimization problem constrained by LMIs. A numeric example is provided to illustrate the effectiveness of the proposed method. At last, the problem of generalized observer based output feedback guaranteed cost control is discussed. The sufficient conditions of the existence of observer based output feedback guaranteed cost controller are given in terms of LMIs.
2. Aiming at a class of uncertain continous time nonlinear descriptor delay systems described by T-S fuzzy descriptor model, the quadratic stability is studied by using Lyapunov stability theory and LMIs technology. Based on the quadratic stability, the robust H∞control method is proposed. The quadratic stability conditions and the sufficient conditions of the existence of robust H∞controller are given in terms of LMIs. Next, Robust guaranteed cost control for uncertain T-S fuzzy descriptor delay system is discussed. And design of guaranteed cost controller with minimized J0 is reduced to a convex optimization problem constrained by LMIs. numeric examples are provided to illustrate the effectiveness of the proposed methods of robust H∞controller and robust guaranteed cost controller.
3. Based on delay-dependent method, the robust guaranteed cost control of a class of uncertain continous time nonlinear descriptor delay systems described by T-S fuzzy descriptor model is investigated. The relationship between the size of delay and stability is discussed. And the estimation of the upper bound of delayτmax is formulated by a convex optimization problem. The minimized guaranteed cost upper bound corresponding toτmax is reduced to an optimization problem of searching for the suboptimal value. The simulation results indicate the effectiveness of the proposed method.
4. Based on delay-dependent method, the non-fragile H∞control problem of a class of uncertain continous time nonlinear descriptor delay systems described by T-S fuzzy descriptor model is considered. First, a new delay-dependent H∞stability criterion is proposed aiming at the normal system. Then, based on two kinds of gain variations, the design methods of non-fragile controllers satisfying H∞norm bound are provided. The sufficient conditions of the existence of non-fragile H∞controller are given in terms of LMIs. The simulation example indicates the effectiveness of the proposed method.
5. A multiple delay T-S fuzzy descriptor model is proposed. Based on Lyapunov stability theory and delay-independent method, guaranteed cost control and H∞control problems of a class of multiple delay nonlinear descriptor systems described by multiple delay T-S fuzzy descriptor model are studied. The stability conditions and the sufficient conditions of existence of guaranteed cost controller and H∞controller are given in terms of LMIs. I is worth noting that multiple delay T-S fuzzy descriptor system is extension of single delay T-S fuzzy descriptor system. It is of more extensive practical significance, and has not been studied in other literature yet.
6. The control problems of a class of discrete time nonlinear descriptor delay systems described by the discrete time T-S fuzzy descriptor model are considered. First, the definitions of consistent regularity, cause and effect, stability and admissibility are given. Then, The stability of the system is analyzed based on Lyapunov method. The sufficient conditions of the system to be consistent regular, cause and effect and stable are given in terms of LMIs. At last, design methods of state feedback and output feedback controllers are proposed.
引文
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