Stability analysis and H/sub /spl infin// controller design of fuzzy large-scale systems based on piecewise Lyapunov functions

This paper presents a novel approach to stability analysis of a fuzzy large-scale system in which the system is composed of a number of Takagi-Sugeno (T-S) fuzzy subsystems with interconnections. The stability analysis is based on Lyapunov functions that are continuous and piecewise quadratic. It is shown that the stability of the fuzzy large-scale systems can be established if a piecewise Lyapunov function can be constructed, and, moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. It is also demonstrated via a numerical example that the stability result based on the piecewise quadratic Lyapunov functions is less conservative than that based on the common quadratic Lyapunov functions. The H/sub /spl infin// controllers can also be designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions.     

Hinfinity controller design of fuzzy dynamic systems based on piecewise Lyapunov functions.

This paper presents a controller design method for fuzzy dynamic systems based on a piecewise smooth Lyapunov function. The basic idea of the proposed approach is to construct controllers for the fuzzy dynamic systems in such a way that a piecewise continuous Lyapunov function can be used to establish the global stability with Hinfinity performance of the resulting closed loop fuzzy control systems. It is shown that the control law can be obtained by solving a set of Linear Matrix Inequalities (LMI) that is numerically feasible with commercially available software. An example is given to illustrate the application of the proposed method.     

Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions

This paper presents a stability analysis method for discrete-time Takagi-Sugeno fuzzy dynamic systems based on a piecewise smooth Lyapunov function. It is shown that the stability of the fuzzy dynamic system can be established if a piecewise Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities that is numerically feasible with commercially available software. It is also demonstrated via numerical examples that the stability result based on the piecewise quadratic Lyapunov functions is less conservative than that based on the common quadratic Lyapunov functions.     

Controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions

This paper presents a kind of controller synthesis method for fuzzy dynamic systems based on a piecewise smooth Lyapunov function. The basic idea of the proposed approach is to construct controllers for the fuzzy dynamic systems in such a way that a piecewise continuous Lyapunov function can be used to establish the global stability with H/sub /spl infin// performance of the resulting closed loop fuzzy control systems. It is shown that the control laws can be obtained by solving a set of linear matrix inequalities that is numerically feasible with commercially available software. An example is given to illustrate the application of the proposed methods.     

基于分段多项式李雅普诺夫函数的模糊系统的稳定性分析

本文采用最大型分段多项式李雅普诺夫函数研究了多项式模糊系统的闭环稳定性问题.首先,本文设计了与分段李雅普诺夫函数对应的切换模糊控制器,提出了多项式模糊模型稳定的平方和条件,同时证明了最大型分段多项式李雅普诺夫函数在函数切换点的稳定性.然后,设计了相应的路径跟踪优化算法,对本文非凸的稳定条件进行迭代求解.最后,通过两个算例进行仿真与比较,说明并验证了本文所提出结论的可行有效性.     

Piecewise H infinity: controller design of discrete time fuzzy systems.

This paper presents a new H infinity controller design method for the discrete time fuzzy systems based on the piecewise Lyapunov functions. The basic idea of the proposed approach is to construct the controller for the fuzzy systems in such a way that a discrete time piecewise Lyapunov function can be used to establish the global stability with H infinity-disturbance attenuation performance of the resulting close loop fuzzy control systems. It is shown that the control laws can be obtained by solving a set of linear matrix inequalities (LMIs) that is numerically tractable with commercially available software. Numerical example is given to demonstrate the advantage of the proposed method.     

H/sub /spl infin// controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions and bilinear matrix inequalities

This work presents an H/sub /spl infin// controller design method for fuzzy dynamic systems based on techniques of piecewise smooth Lyapunov functions and bilinear matrix inequalities. It is shown that a piecewise continuous Lyapunov function can be used to establish the global stability with H/sub /spl infin// performance of the resulting closed-loop fuzzy control systems and the control laws can be obtained by solving a set of bilinear matrix inequalities (BMIs). Two examples are given to illustrate the application of the proposed methods.     

H/sub /spl infin// state-feedback controller design for discrete-time fuzzy systems using fuzzy weighting-dependent Lyapunov functions

For discrete-time Takagi-Sugeno (TS) fuzzy systems, we propose an H/sub /spl infin// state-feedback fuzzy controller associated with a fuzzy weighting-dependent Lyapunov function. The controller, which is designed via parameterized linear matrix inequalities (PLMIs), employs not only the current-time but also the one-step-past information on the time-varying fuzzy weighting functions. Appropriately selecting the structures of variables in the PLMIs allows us to find an LMI formulation as a special case.     

Relaxed delay-dependent stabilization criteria for discrete-time fuzzy systems based on a switching fuzzy model and piecewise Lyapunov functions

This paper presents delay-dependent stability analysis and controller synthesis methods for discrete-time Takagi-Sugeno （T-S） fuzzy systems with time delays. The T-S fuzzy system is transformed to an equivalent switching fuzzy system. Consequently, the delay-dependent stabilization criteria are derived for the switching fuzzy system based on the piecewise Lyapunov function. The proposed conditions are given in terms of linear matrix inequalities （LMIs）. The interactions among the fuzzy subsystems are considered in each subregion, and accordingly the proposed conditions are less conservative than the previous results. Since only a set of LMIs is involved, the controller design is quite simple and numerically tractable. Finally, a design example is given to show the validity of the proposed method.     

Non-fragile control of fuzzy affine dynamic systems via piecewise Lyapunov functions

This paper addresses the robust H_∞ static output feedback (SOF) controller design problem for a class of uncertain fuzzy affine systems that are robust against both the plant parameter perturbations and controller gain variations. More specifically, the purpose is to synthesize a non-fragile piecewise affine SOF controller guaranteeing the stability of the resulting closed-loop fuzzy affine dynamic system with certainH_∞ performance index. Based on piecewise quadratic Lyapunov functions and applying some convexification procedures, two different approaches are proposed to solve the robust and non-fragile piecewise affine SOF controller synthesis problem. It is shown that the piecewise affine controller gains can be obtained by solving a set of linear matrix inequalities (LMIs). Finally, simulation examples are given to illustrate the effectiveness of the proposed methods.     

基于分段模糊Lyapunov方法的T-S模糊系统H∞控制

针对具有外部扰动的T-S模糊系统，利用分段模糊Lyapunov函数和线性矩阵不等式方法研究了模糊系统的镇定问题．给出了闭环系统渐近稳定的充分条件，提出了闭环系统γ一次优H∞控制器的参数化设计方法．仿真实例验证了所提出方法的有效性．     

Stability analysis of fuzzy large-scale systems

This paper is concerned with the stability problem of fuzzy large-scale systems. Each of them consists of J interconnected subsystems which are represented by Takagi-Sugeno fuzzy models. A stability criterion in terms of Lyapunov's direct method is proposed to guarantee the asymptotic stability of fuzzy large-scale systems. Finally, an example is given to demonstrate the results.     

Switching fuzzy controller design based on switching Lyapunov function for a class of nonlinear systems

This paper presents a switching fuzzy controller design for a class of nonlinear systems. A switching fuzzy model is employed to represent the dynamics of a nonlinear system. In our previous papers, we proposed the switching fuzzy model and a switching Lyapunov function and derived stability conditions for open-loop systems. In this paper, we design a switching fuzzy controller. We firstly show that switching fuzzy controller design conditions based on the switching Lyapunov function are given in terms of bilinear matrix inequalities, which is difficult to design the controller numerically. Then, we propose a new controller design approach utilizing an augmented system. By introducing the augmented system which consists of the switching fuzzy model and a stable linear system, the controller design conditions based on the switching Lyapunov function are given in terms of linear matrix inequalities (LMIs). Therefore, we can effectively design the switching fuzzy controller via LMI-based approach. A design example illustrates the utility of this approach. Moreover, we show that the approach proposed in this paper is available in the research area of piecewise linear control.     

基于非脆弱控制器设计的不确定模糊系统稳定性研究

研究不确定动态模糊系统的稳定性问题.提出一类不确定T-S动态模糊系统的非脆弱控制问题,并进行了控制器设计.首先给出不确定T-S动态模糊系统的模型;然后利用Lyapunov函数方法,研究连续不确定动态模糊系统的非脆弱控制器设计,得到基于LMI的不确定动态模糊系统的全局渐近稳定性条件.通过对一级倒立摆的不确定模糊非脆弱控制器设计的实例,表明了设计方法的可行性和有效性.     

Nonlinear state feedback controller for nonlinear systems:Stability analysis and design based on fuzzy plant model

This paper presents the stability analysis of a fuzzy-model-based control system consisting of a nonlinear plant and a nonlinear state feedback controller and the design of the nonlinear gains of the controller. The nonlinear plant is represented by a fuzzy model having p rules. A nonlinear state feedback controller is designed to close the feedback loop. Under this design, the stability condition is reduced to p linear matrix inequalities. An application example on stabilizing a mass-spring-damper system will be given     

Fuzzy pole placement based on piecewise Lyapunov functions

This paper presents a controller design method for fuzzy dynamic systems based on piecewise Lyapunov functions with constraints on the closed‐loop pole location. The main idea is to use switched controllers to locate the poles of the system to obtain a satisfactory transient response. It is shown that the global fuzzy system satisfies the requirements for the design and that the control law can be obtained by solving a set of linear matrix inequalities, which can be efficiently solved with commercially available softwares. An example is given to illustrate the application of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

基于分段连续Lyapunov函数的采样系统鲁棒保性能控制

针对不确定采样控制系统的鲁棒保性能控制问题,首先将采样系统描述为跳变线性系统,基于矩阵凸组合思想构造了分段连续Lyapunov函数,进而在线性矩阵不等式框架内给出了不确定采样系统鲁棒稳定的条件.针对范数有界参数不确定采样系统,提出了鲁棒保性能控制器设计的在线算法,在每个采样周期内通过求解一组线性矩阵不等式的可行解来构造出状态反馈增益矩阵.最后的仿真算例验证了所提设计方法的有效性.     

Relaxed stabilization criteria for discrete-time T-S fuzzy control systems based on a switching fuzzy model and piecewise Lyapunov function.

In this paper, two new relaxed stabilization criteria for discrete-time T-S fuzzy systems are proposed. In the beginning, the operation state space is divided into several subregions, and then, the T-S fuzzy system is transformed to an equivalent switching fuzzy system corresponding to each subregion. Consequently, based on the piecewise Lyapunov function, the stabilization criteria of the switching fuzzy system are derived. The criteria have two features: 1) the behavior of the two successive states of the system is considered in the inequalities and 2) the interactions among the fuzzy subsystems in each subregion Sj are presented by one matrix Xj. Due to the above two features, the feasible solutions of the inequalities in the criteria are much easier to be found. In other words, the criteria are much more relaxed than the existing criteria proposed in other literature. The proposed conditions in the criteria and the fuzzy control design can be solved and achieved by means of linear matrix inequality tools. Two examples are given to present the superiority of the proposed criteria and the effectiveness of the fuzzy controller's design, respectively.     

Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions

This paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems (PLS). This methodology infers global properties of PLS solely by studying the behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface. Such maps are known to be "unfriendly" maps in the sense that they are highly nonlinear, multivalued, and not continuous. We found, however, that an impact map induced by an linear time-invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state. This representation of impact maps allows the search for surface Lyapunov functions (SuLF) to be done by simply solving a semidefinite program, allowing global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS to be efficiently checked. This new analysis methodology has been applied to relay feedback, on/off and saturation systems, where it has shown to be very successful in globally analyzing a large number of examples. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied. This success in globally analyzing certain classes of PLS has shown the power of this new methodology, and suggests its potential toward the analysis of larger and more complex PLS.     

Stability analysis and controller design of interval type-2 fuzzy systems with time delay

The type-2 fuzzy models can handle the system uncertainties directly based on the type-2 fuzzy sets. In this paper, the Takagi–Sugeno fuzzy model approach is extended to the stability analysis and controller design for interval type-2 (IT2) fuzzy systems with time-varying delay. Delay-dependent robust stability criteria are developed in terms of linear matrix inequalities by using the improvement technique of free-weighting matrices. Less conservative results are obtained by considering the information contained in the footprint of uncertainty. Finally, two simulation examples are presented to illustrate the effectiveness of the theoretical results. One is provided to show the merits of the proposed method, the other based on the continuous stirred tank reactor model is given to illustrate the design processes of IT2 fuzzy controller for a nonlinear system with parameter uncertainties.