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.     

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.     

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.     

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.     

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.     

Stability analysis and H infinity 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 infinity controllers can also be designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions.     

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.     

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.     

GH 2 control for uncertain discrete-time-delay fuzzy systems based on a switching fuzzy model and piecewise Lyapunov function

Generalized H2 （GH2） stability analysis and controller design of the uncertain discrete-time Takagi-Sugeno （T-S） fuzzy systems with state delay are studied based on a switching fuzzy model and piecewise Lyapunov function. GH2 stability sufficient conditions are derived in terms of linear matrix inequalities （LMIs）. The interactions among the fuzzy subsystems are considered. Therefore, 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. To illustrate the validity of the proposed method, a design example is provided.     

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.     

Robust control through piecewise Lyapunov functions for discrete time-varying uncertain systems

This paper is concerned with the robust stabilization by state feedback of a linear discrete-time system with time-varying uncertain parameters. An optimization problem involving a set of linear matrix inequalities and scaling parameters provides both the robust feedback gain and the piecewise Lyapunov function used to ensure the closed-loop stability. In the case of linear time-varying systems involving the convex combination of two matrices, only two scaling parameters constrained into the interval [0,?1] are needed, allowing a simple numerical solution as illustrated by means of examples.     

Controller design for nonlinear affine systems by control Lyapunov functions

This paper addresses the stabilization problems for nonlinear affine systems. First of all, the explicit feedback controller is developed for a nonlinear multiple-input affine system by assuming that there exists a control Lyapunov function. Next, based upon the homogeneous property, sufficient conditions for the continuity of the derived controller are developed. And then the developed control design methodology is applied to stabilize a class of nonlinear affine cascaded systems. It is shown that under some homogeneous assumptions on control Lyapunov functions and the interconnection term, the cascaded system can be globally stabilized. Finally, some interesting results of finite-time stabilization for nonlinear affine systems are also obtained.     

Generalized decompositions of dynamic systems and vector Lyapunov functions

The notion of decomposition is generalized to provide more freedom in constructing vector Lyapunov functions for stability analysis of nonlinear dynamic systems. A generalized decomposition is defined as a disjoint decomposition of a system which is obtained by expanding the state-space of a given system. An inclusion principle is formulated for the solutions of the expansion to include the solutions of the original system, so that stabliity of the expansion implies stability of the original system. Stability of the expansion can then be established by standard disjoint decompositions and vector Lyapunov functions. The applicability, of the new approach is demonstrated using the Lotka-Voiterra equations.     

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.     

Computation of piecewise quadratic Lyapunov functions for hybridsystems

This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach     

New Lyapunov functions for power systems based on minimal realizations

In this paper the state models of an n-machine power system for stability studies are obtained in the ‘ minimal state space ’ based on the concept of the degree of a rational function matrix. Lyapunov functions are then constructed for these models in a systematic manner using Anderson's theorem for multi-non-linear systems. These Lyapunov functions are different from those currently obtainable in the literature for power systems.     

A synthesis of Lyapunov functions for non-linear time-varying control systems

The formulation of a class of Lyapunov functions for time-varying nonlinear control systems which occur in the stability analysis of control systems is considered. A new approach is presented, which is an extension of a generation technique for time-invariant non-linear systems. This approach, which uses by analogy the classical theory of Hamilton, permits stability and transient information to be obtained from system equations with time-varying damping as well as time-varying gain. A second and third-order example is used to illustrate the application of this new result.     

Reducing the conservatism of LMI-based stabilisation conditions for TS fuzzy systems using fuzzy Lyapunov functions

In this article, the fuzzy Lyapunov function approach is considered for stabilising continuous-time Takagi-Sugeno fuzzy systems. Previous linear matrix inequality (LMI) stability conditions are relaxed by exploring further the properties of the time derivatives of premise membership functions and by introducing slack LMI variables into the problem formulation. The relaxation conditions given can also be used with a class of fuzzy Lyapunov functions which also depends on the membership function first-order time-derivative. The stability results are thus extended to systems with large number of rules under membership function order relations and used to design parallel-distributed compensation (PDC) fuzzy controllers which are also solved in terms of LMIs. Numerical examples illustrate the efficiency of the new stabilising conditions presented.     

Analysis and synthesis of robust control systems viaparameter-dependent Lyapunov functions

In this paper, the problem of robust stability of systems subject to parametric uncertainties is considered. Sufficient conditions for the existence of parameter-dependent Lyapunov functions are given in terms of a criterion which is reminiscent of, but less conservative than, Popov's stability criterion. An equivalent frequency-domain criterion is demonstrated. The relative sharpness of the proposed test and existing stability criteria is then discussed. The use of parameter-dependent Lyapunov functions for robust controller synthesis is then considered. It is shown that the search for robustly stabilizing controllers may be limited to controllers with the same order as the original plant. A possible synthesis procedure and a numerical example are then discussed