Class Of Discrete-time Fuzzy Systems Research Articles

Convex Stability Analysis of Mamdani-Like Fuzzy Systems With Singleton Consequents

We study the stability of a class of discrete-time fuzzy systems with singleton consequents, called Mamdani-like fuzzy systems. The parametric expressions, specific to this class of fuzzy systems, are leveraged to derive stability analysis conditions via Finsler's lemma and Lyapunov stability tools. This allows avoiding the major challenge in dealing with high-dimensional cases, encountered in the related literature when using the classical state-space representation. Moreover, the information of the piecewise region partition can be fully taken into account in the stability analysis with the well-known <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$S-$</tex-math></inline-formula> procedure to further reduce the stability conservatism. The stability of Mamdani-like fuzzy systems can be checked by solving a set of linear matrix inequalities (LMIs), that is numerically tractable with a suitable semidefinite programming software. Several numerical and physically motivated examples are provided to illustrate the effectiveness of the proposed stability analysis results.

Control Synthesis of Hidden Semi-Markov Uncertain Fuzzy Systems via Observations of Hidden Modes.

This paper investigates the stability analysis and fuzzy control problems for a class of discrete-time fuzzy systems with hidden semi-Markov stochastic uncertainties. The nonlinear plant is described via the Takagi-Sugeno (T-S) fuzzy model, and the parameter uncertainties are represented by a hidden semi-Markov chain. Owing to the semi-Markov kernel (SMK), the probability density functions (PDFs) of sojourn time for different modes in describing the stochastic uncertainties can address different types of distributions according to different target modes. A novel Lyapunov function that depends on the hidden mode and the observed mode with the elapsed time is proposed to analyze the stability and the H∞ performance of the fuzzy system. Then, the sufficient criteria for the elapsed-time-dependent and observed-mode-dependent fuzzy controller are achieved by exploiting the observations of hidden modes, ensuring that the closed-loop system is σ -error mean square stable with guaranteed H∞ performance. A cart-pendulum system is used to demonstrate the effectiveness and applicability of the proposed theoretical results.

Stability and Stabilization of a Class of Discrete-Time Fuzzy Systems With Semi-Markov Stochastic Uncertainties

This paper addresses the problems of stability and stabilization for a class of discrete-time Takagi–Sugeno fuzzy systems with semi-Markov stochastic uncertainties. By means of the semi-Markov kernel, the probability density function of the sojourn time for different modes in describing the underlying semi-Markov stochastic uncertainties can be mode-dependent in contrast with the previous studies. Both the sojourn-time-independent and sojourn-time-dependent Lyapunov functions are proposed, by which the stability criteria are obtained in terms of a new $\boldsymbol {\sigma }$ -error mean square stability concept, and the latter is demonstrated to be less conservative and more practical than some existing results. Then, control synthesis problem is investigated and the sufficient conditions on the existence of admissible mode-dependent state-feedback stabilizing controller are developed. A numerical example and a cart-pendulum system are given to show the effectiveness and potential of the new design techniques.

$H_\infty$ Fuzzy Control With Randomly Occurring Infinite Distributed Delays and Channel Fadings

In this paper, the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> output-feedback control problem is investigated for a class of discrete-time fuzzy systems with randomly occurring infinite distributed delays and channel fadings. A random variable obeying the Bernoulli distribution is introduced to account for the probabilistic infinite distributed delays. The stochastic Rice fading model is employed to simultaneously describe the phenomena of random time delays and channel fadings via setting different values of the channel coefficients. The aim of this paper is to design an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> output-feedback fuzzy controller such that the closed-loop Takagi-Sugeno (T-S) fuzzy control system is exponentially mean-square stable, and the disturbance rejection attenuation is constrained to a given level by means of the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance index. Intensive analysis is carried out to obtain sufficient conditions for the existence of desired output-feedback controllers, ensuring both the exponential mean-square stability and the prescribed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance. The cone-complementarity linearization algorithm is utilized to cast the controller design problem into a sequential minimization: one that is solvable by the semi-definite programming method. A simulation result is exploited to illustrate the usefulness and effectiveness of the proposed design technique.

A new approach to non-fragile H ∞ observer-based control for discrete-time fuzzy systems

In this article, a new approach regarding H ∞ and non-fragile stability analysis and observer-based controller synthesis problem for a class of discrete-time fuzzy systems is investigated. The multiplicative gain variations under consideration are contained in both the controller gain and observer gain. By defining a basis-dependent Lyapunov function and introducing some additional matrix variables, new sufficient conditions are derived to guarantee the stability of the non-fragile H ∞ observer-based control system. Seeking computational convenience, the developed results are cast in the format of linear matrix inequalities and two numerical examples are presented to illustrate the feasibility of theoretical developments.

Homogenous Polynomially Parameter-Dependent $H_{ \infty}$ Filter Designs of Discrete-Time Fuzzy Systems

This paper proposes a novel H(∞) filtering technique for a class of discrete-time fuzzy systems. First, a novel kind of fuzzy H(∞) filter, which is homogenous polynomially parameter dependent on membership functions with an arbitrary degree, is developed to guarantee the asymptotic stability and a prescribed H(∞) performance of the filtering error system. Second, relaxed conditions for H(∞) performance analysis are proposed by using a new fuzzy Lyapunov function and the Finsler lemma with homogenous polynomial matrix Lagrange multipliers. Then, based on a new kind of slack variable technique, relaxed linear matrix inequality-based H(∞) filtering conditions are proposed. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approach.

Robust H2 guaranteed cost fuzzy control for uncertain discrete-time fuzzy systems via poly-quadratic Lyapunov functions

In this paper, new approaches regarding H2 guaranteed cost stability analysis and controller synthesis problems for a class of discrete-time fuzzy systems with uncertainties are investigated. The state-space Takagi-Sugeno fuzzy model with norm-bounded parameter uncertainties is adopted. Based on poly-quadratic Lyapunov functions, sufficient conditions for the existence of the robust H2 fuzzy controller can be obtained in terms of linear matrix inequalities (LMIs). Furthermore, a convex optimization problem with LMI constraints is formulated to design a suboptimal fuzzy controller which minimizes the upper bound on the quadratic cost function. The effectiveness of the proposed design approach is illustrated by two examples. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

New LMI Approach to Fuzzy $H_{\infty}$ Filter Designs

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This brief is concerned with the <formula formulatype="inline"><tex Notation="TeX">$H_{\infty}$</tex></formula> filtering problem for a class of discrete-time fuzzy systems. The objective is to find a new way to design a stable filter guaranteeing the asymptotic stability and a prescribed <formula formulatype="inline"><tex Notation="TeX">$H_{\infty}$</tex> </formula> performance of the filtering error system. By using the fuzzy Lyapunov function approach and adding slack matrix variables, a new condition for <formula formulatype="inline"><tex Notation="TeX">$H_{\infty}$</tex></formula> performance analysis is proposed. Then, a linear matrix inequality (LMI)-based approach is developed for designing the fuzzy <formula formulatype="inline"><tex Notation="TeX">$H_{\infty}$</tex></formula> filter, which can be obtained from the solution of convex optimization problems. When these LMIs are feasible, an explicit expression of a desired <formula formulatype="inline"><tex Notation="TeX">$H_{\infty}$</tex></formula> fuzzy filter is given. Finally, a numerical example is provided to illustrate the effectiveness and less conservatism of the proposed approach. </para>

H ∞ Fuzzy Filtering for Discrete-Time Nonlinear Systems via Fuzzy Lyapunov Function Approach

This paper is concerned with the H ? fuzzy filtering problem for a class of discrete-time fuzzy systems. The objective is to design a stable filter guaranteeing the asymptotic stability and a prescribed H ? performance of the filtering error system. Motivated by the parallel distributed compensation (PDC) technique, a new filter model is proposed in this paper. Both full-order and reduced-order filters are established, and they can be obtained from the solution of convex optimization problems in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired H ? fuzzy filter is given. A numerical example is provided to demonstrate the effectiveness and applicability of the proposed design approach.

Formula omitted] filtering of discrete-time fuzzy systems via basis-dependent Lyapunov function approach

This paper is concerned with the H ∞ filtering problem for a class of discrete-time fuzzy systems. Attention is focused on the design of a stable filter guaranteeing a prescribed noise attenuation level in the H ∞ sense. By using basis-dependent Lyapunov functions, sufficient conditions for the solvability of this problem are obtained. It has been shown that the H ∞ filtering problem can be solved as a linear matrix inequality (LMI) optimization problem. Two examples are provided to demonstrate the applicability of the proposed approach.

Generalised H2 controller synthesis for uncertain discrete-time fuzzy systems via basis-dependent Lyapunov functions

S. Zhou and G. Feng Abstract: The generalised H2 control problem is investigated for a class of discrete-time fuzzy systems with uncertainties. The uncertain fuzzy dynamic model is used to represent a class of uncertain discrete-time complex nonlinear systems which include both linguistic information and system uncertainties. Using basis-dependent Lyapunov functions an H2 control design approach is developed. It is shown that the controller can be obtained by solving a set of linear matrix inequalities. It is also shown that the basis-dependent results are less conservative than the basis-independent ones. Numerical examples including a discrete chaotic Lorenz system are also given to demonstrate the applicability of the proposed approach.

Robust H∞ control for discrete-time fuzzy systems via basis-dependent Lyapunov functions

This paper deals with the robust H ∞ control problem for a class of discrete-time fuzzy systems with uncertainty. The uncertainty is assumed to be of structured linear fractional form. By using basis-dependent Lyapunov function, an H ∞ control design approach is developed. The control design approach is facilitated by introducing some additional instrumental matrix variables. These additional matrix variables decouple the Lyapunov and the system matrices, which makes the control design feasible. The proposed approach leads to some sufficient results in the form of strict linear matrix inequalities (LMIs). It is expected that the basis-dependent results are less conservative than the basis-independent ones due to the introduction of basis-dependent Lyapunov function. Finally, numerical examples including the discrete chaotic Lorenz system are also given to demonstrate the applicability of the proposed approach.