模糊系统的稳定性分析与控制器设计
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摘要
模糊控制是基于美国学者Zadeh L A教授于1965年提出的“模糊集”概念发展起来的一种智能控制方法。模糊控制方法及其理论研究主要是针对不同类型的模糊系统提出相应的控制方法并对其进行稳定性和鲁棒性分析,从而保证控制系统的性能。日本学者Takagi T和Sugeno M在1985年提出的Takagi-Sugeno (T-S)模糊模型,给模糊控制理论研究及应用带来了深远的影响,使模糊系统稳定性分析上升到新的理论高度。其优点在于它充分运用了Lyapunov稳定性理论来进行系统分析和控制器设计,通过对非线性系统进行T-S模糊建模,然后提供一套系统化的方法来研究非线性系统的稳定性以及控制器设计问题。
本文分别基于T-S线性模型和T-S双线性模型,根据Lyapunov稳定性理论、鲁棒控制理论和H∞控制理论,结合线性矩阵不等式(LMI)技术,深入研究了模糊系统稳定性和稳定化控制问题。
主要工作有以下几个方面:
1.针对现有方法判断不确定离散模糊系统鲁棒H∞稳定性的保守性,利用模糊Lyapunov函数,给出了这一系统鲁棒渐近稳定的充分条件。应用并行分布补偿算法(PDC),设计出使全局渐近稳定的鲁棒控制器。多个附加矩阵变量的引入,使控制器可以通过求解一系列LMI获得。
2.提出了一种新方法,对一类状态和输入矩阵都带有不确定的时滞模糊系统进行了鲁棒稳定性分析和设计,这种方法较现有的方法有着较小的保守性。定义了一种新型的模糊Lyapunov-Krasovskii函数(LKF),得到了系统时滞相关的稳定性条件。在推导过程中,引入多个自由权值矩阵变量来表示系统方程中各项及Leibuiz-Newton公式各项之间的关系,考虑了在以前文献中常被忽略的项,避免使用边界不等式和模型转换所带来的保守性。
3.针对一类带有时变时滞的不确定模糊系统,研究了时滞相关鲁棒非脆弱H∞反馈控制问题。基于模糊Lyapunov-Krasovkii泛函和PDC设计了模糊控制器,使得在控制器存在可加性摄动的情况下,其闭环系统鲁棒渐近稳定。利用线性矩阵不等式,导出了非脆弱鲁棒控制律的存在条件,控制器的设计可由一组LMI的解得到。
4.对一类由T-S双线性模型描述的带有时变时滞的非线性关联大系统,研究了其分散状态反馈控制问题。根据Lyapunov稳定性分析理论和并行分布补偿算法,得到了闭环关联大系统时滞相关渐近稳定的充分条件。相应的分散模糊控制器的设计可转化成一个受LMI约束的凸优化问题。
5.研究了一类基于T-S双线性模型的非线性关联大系统的分散静态输出控制反馈问题。应用Lyapunov稳定性分析理论,得到了闭环关联大系统渐近稳定的充分条件,并把这些条件转换成LMI的形式,相应的分散模糊控制器可由线性矩阵不等式的解得到。
最后,对全文进行了概括性总结,并指出了有待进一步研究和完善的问题。
The strategy of fuzzy control is one of the intelligent control scheme, which is based on the concept of "fuzzy sets" proposed in 1965 by Prof. Zadeh L A in American. The research of fuzzy control theory includes a series of main problems, such as the stability and robustness analysis, the system design approach and the improvement of the system performance etc. In 1985, Takagi T and Sugeno M proposed the Takagi-Sugeno (T-S) fuzzy model, which brings far-researching impact on fuzzy control theory and its application, and makes the stability analysis of fuzzy systems to a new theoretical height. An advantage of T-S fuzzy model is that it can fully use Lyapunov stability theory to analyze stability of systems and design controller,and a systematic method is provided to study the problem of stability of nonlinear systems and controller design by T-S fuzzy modelling for nonlinear systems.
Combining with the Lyapunov stability theory, robust control theory and H∞control theory, using the Linear Matrix Inequality (LMIs), this thesis discussed the stability and stabilization problems of fuzzy systems based on T-S linear model and T-S bilinear model in detail, respectively.
The main research works in this thesis can be described as follows:
1. For the conservation of checking the robust stability of uncertain discerte fuzzy system with the approaches those have been proposed,sufficient conditions for globally asymptotical H-infinity stability of uncertain discrete T-S fuzzy system are presented by using fuzzy Lyapunov function.State-feedback controller is designed by the method of parallel distributed compensation (PDC). The controller design involves solving a set of LMIs by introducing multiply additional matrix variables.
2. A new method for the delay-dependent stability analysis for continuous-time Takagi and Sugeno(T-S) fuzzy systems with a time-varying delay is suggested, which is less conservative than other existing ones.First, based on a fuzzy Lyapunov-Krasovskii functional (LKF), a delay-dependent stability criterion is derived for the fuzzy systems.In the derivation process, some free-weighting matrices are introduced to expressed the relationships among the terms of the systems equation, and among the terms in the Leibuiz-Newton formula, which may avoid the conservation producing by some bounding inequalities for cross products between two vectors and model transformation. At the same time, the subtle difference is payed careful attention to, which is largely ignored in the existing literature.
3. A delay-dependent robust H∞non-fragile control problem is presented for a class of uncertain T-S fuzzy systems with time-varying delay. A delay-dependent robust non-fragile controller is designed via a fuzzy Lyapunov-Krasovkii functional and the PDC approach, such that the closed-loop systems are robust asymptotically stable in the presence of the additive controller gain perturbations. A sufficient condition for the existence of such robust non-fragile controller is drived via the LMI. The feedback controller design involves solving a set of linear matrix inequalities (LMIs).
4. The problem of decentralized state feedback control is presented for a nonlinear interconnected system with time-varying delay in both states and inputs which is composed by a number of T-S fuzzy bilinear subsystems with interconnections. Based on the Lyapunov criterion and the parallel distribute compensation scheme, the delay-dependent stabilization sufficient conditions are derived for the whole close-loop fuzzy interconnected systems. The corresponding decentralized fuzzy controller design is converted into a convex optimization problem with LMI constraints.
5. The problem of decentralized static output feedback control is presented for a nonlinear interconnected system which is composed by a number of T-S fuzzy bilinear subsystems with interconnections. Based on the Lyapunov criterion, some sufficient stabilization conditions are derived for the whole close-loop fuzzy interconnected systems. The stabilization conditions are further formulated into LMIs so that the corresponding decentralized controllers can be easily obtained by using the Matlab LMI toolbox.
Finally, some concluding remarks are given and the future research works are pointed out.
本文分别基于T-S线性模型和T-S双线性模型,根据Lyapunov稳定性理论、鲁棒控制理论和H∞控制理论,结合线性矩阵不等式(LMI)技术,深入研究了模糊系统稳定性和稳定化控制问题。
主要工作有以下几个方面:
1.针对现有方法判断不确定离散模糊系统鲁棒H∞稳定性的保守性,利用模糊Lyapunov函数,给出了这一系统鲁棒渐近稳定的充分条件。应用并行分布补偿算法(PDC),设计出使全局渐近稳定的鲁棒控制器。多个附加矩阵变量的引入,使控制器可以通过求解一系列LMI获得。
2.提出了一种新方法,对一类状态和输入矩阵都带有不确定的时滞模糊系统进行了鲁棒稳定性分析和设计,这种方法较现有的方法有着较小的保守性。定义了一种新型的模糊Lyapunov-Krasovskii函数(LKF),得到了系统时滞相关的稳定性条件。在推导过程中,引入多个自由权值矩阵变量来表示系统方程中各项及Leibuiz-Newton公式各项之间的关系,考虑了在以前文献中常被忽略的项,避免使用边界不等式和模型转换所带来的保守性。
3.针对一类带有时变时滞的不确定模糊系统,研究了时滞相关鲁棒非脆弱H∞反馈控制问题。基于模糊Lyapunov-Krasovkii泛函和PDC设计了模糊控制器,使得在控制器存在可加性摄动的情况下,其闭环系统鲁棒渐近稳定。利用线性矩阵不等式,导出了非脆弱鲁棒控制律的存在条件,控制器的设计可由一组LMI的解得到。
4.对一类由T-S双线性模型描述的带有时变时滞的非线性关联大系统,研究了其分散状态反馈控制问题。根据Lyapunov稳定性分析理论和并行分布补偿算法,得到了闭环关联大系统时滞相关渐近稳定的充分条件。相应的分散模糊控制器的设计可转化成一个受LMI约束的凸优化问题。
5.研究了一类基于T-S双线性模型的非线性关联大系统的分散静态输出控制反馈问题。应用Lyapunov稳定性分析理论,得到了闭环关联大系统渐近稳定的充分条件,并把这些条件转换成LMI的形式,相应的分散模糊控制器可由线性矩阵不等式的解得到。
最后,对全文进行了概括性总结,并指出了有待进一步研究和完善的问题。
The strategy of fuzzy control is one of the intelligent control scheme, which is based on the concept of "fuzzy sets" proposed in 1965 by Prof. Zadeh L A in American. The research of fuzzy control theory includes a series of main problems, such as the stability and robustness analysis, the system design approach and the improvement of the system performance etc. In 1985, Takagi T and Sugeno M proposed the Takagi-Sugeno (T-S) fuzzy model, which brings far-researching impact on fuzzy control theory and its application, and makes the stability analysis of fuzzy systems to a new theoretical height. An advantage of T-S fuzzy model is that it can fully use Lyapunov stability theory to analyze stability of systems and design controller,and a systematic method is provided to study the problem of stability of nonlinear systems and controller design by T-S fuzzy modelling for nonlinear systems.
Combining with the Lyapunov stability theory, robust control theory and H∞control theory, using the Linear Matrix Inequality (LMIs), this thesis discussed the stability and stabilization problems of fuzzy systems based on T-S linear model and T-S bilinear model in detail, respectively.
The main research works in this thesis can be described as follows:
1. For the conservation of checking the robust stability of uncertain discerte fuzzy system with the approaches those have been proposed,sufficient conditions for globally asymptotical H-infinity stability of uncertain discrete T-S fuzzy system are presented by using fuzzy Lyapunov function.State-feedback controller is designed by the method of parallel distributed compensation (PDC). The controller design involves solving a set of LMIs by introducing multiply additional matrix variables.
2. A new method for the delay-dependent stability analysis for continuous-time Takagi and Sugeno(T-S) fuzzy systems with a time-varying delay is suggested, which is less conservative than other existing ones.First, based on a fuzzy Lyapunov-Krasovskii functional (LKF), a delay-dependent stability criterion is derived for the fuzzy systems.In the derivation process, some free-weighting matrices are introduced to expressed the relationships among the terms of the systems equation, and among the terms in the Leibuiz-Newton formula, which may avoid the conservation producing by some bounding inequalities for cross products between two vectors and model transformation. At the same time, the subtle difference is payed careful attention to, which is largely ignored in the existing literature.
3. A delay-dependent robust H∞non-fragile control problem is presented for a class of uncertain T-S fuzzy systems with time-varying delay. A delay-dependent robust non-fragile controller is designed via a fuzzy Lyapunov-Krasovkii functional and the PDC approach, such that the closed-loop systems are robust asymptotically stable in the presence of the additive controller gain perturbations. A sufficient condition for the existence of such robust non-fragile controller is drived via the LMI. The feedback controller design involves solving a set of linear matrix inequalities (LMIs).
4. The problem of decentralized state feedback control is presented for a nonlinear interconnected system with time-varying delay in both states and inputs which is composed by a number of T-S fuzzy bilinear subsystems with interconnections. Based on the Lyapunov criterion and the parallel distribute compensation scheme, the delay-dependent stabilization sufficient conditions are derived for the whole close-loop fuzzy interconnected systems. The corresponding decentralized fuzzy controller design is converted into a convex optimization problem with LMI constraints.
5. The problem of decentralized static output feedback control is presented for a nonlinear interconnected system which is composed by a number of T-S fuzzy bilinear subsystems with interconnections. Based on the Lyapunov criterion, some sufficient stabilization conditions are derived for the whole close-loop fuzzy interconnected systems. The stabilization conditions are further formulated into LMIs so that the corresponding decentralized controllers can be easily obtained by using the Matlab LMI toolbox.
Finally, some concluding remarks are given and the future research works are pointed out.
引文
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