A relaxed stability analysis of T-S fuzzy system

We derive a parameterization of positive definite matrices using the Cholesky decomposition in combination with hyperspherical coordinates. Based on the parameterization we develop a simple and efficient method to randomly generate positive definite matrices with constant or bounded trace according to a uniform distribution. Further, we present an efficient implementation using the inversion method and either rejection sampling or transforming a beta distribution. The matrix parameterization might be of independent interest, whereas the random sampling algorithm finds applications in Monte Carlo simulations, testing of algorithms, and performance studies. With the help of an abstract example we describe how the sampling method can be used to approximate the optimum in a difficult, e.g., nonconvex, optimization problem for which no solution or efficient global optimization algorithm is known. In this paper we consider real as well as complex matrices.

In this paper, we consider generalized delay-dependent stability conditions of Takagi-Sugeno fuzzy systems with discrete and distributed time-delays. For fuzzy systems with discrete time-delays, both delay-independent stability conditions and delay-dependent stability conditions have already been obtained. However, no stability condition for fuzzy systems with distributed time-delays has appeared in the literature. This is also true in case of the robust stability for uncertain fuzzy systems with distributed time-delays. With the aid of the stability theorems of linear systems with distributed time-delays, we employ a similar Lyapunov functional for the stability of fuzzy time-delay systems. Then we introduce many free matrices to such a Lyapunov functional, in order to obtain delay-dependent stability conditions. These techniques lead to generalized and less conservative stability conditions. In fact, delay-dependent stability conditions thus obtained are given in linear matrix inequalities(LMIs) and guarantee a wide stability region. We also consider the robust stability of fuzzy time-delay systems with uncertain parameters. Applying the same techniques made on the stability conditions, we obtain delay-dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed time-delays. Moreover, we consider the state feedback stabilization. Based on stability and robust stability conditions, we obtain conditions for the state feedback controller to stabilize the fuzzy time-delay systems. Finally, we take some examples to illustrate our results.

This paper addresses the problem of sampled-data control in T-S fuzzy systems with time delays. Initially, a negative-definite criterion is developed for the matrix containing membership functions with the help of the idea of a switching mechanism. Building upon this criterion, a membership-function-based looped Lyapunov functional is introduced. Furthermore, a sampling-time-dependent looped Lyapunov functional is constructed, incorporating crucial information about the sampling instance. Notably, this functional matrix is not constrained to satisfy positive-definiteness, and it varies with each sampling point. By utilizing the enhanced looped Lyapunov functional, less conservative stability conditions and a new controller criterion for the T-S fuzzy system with time delays are derived. This demonstrates that our proposed stability conditions allow for a larger sampling interval compared to existing work. Finally, the proposed method’s effectiveness is demonstrated through the validation of two simulation examples.

In this paper, we consider robust stability and stabilization of uncertain Takagi-Sugeno fuzzy time-delay systems where uncertainties come into the state and input matrices. Some stability conditions and robust stability conditions for fuzzy time-delay systems have already been obtained in the literature. However, those conditions are rather conservative and do not guarantee the stability of a wide class of fuzzy systems. This is true in case of designing stabilizing controllers for fuzzy time-delay systems and it thus leads to a conservative fuzzy controller design as well. We first consider rather relaxed robust stability conditions of uncertain fuzzy systems. To this end, we introduce an auxiliary system to the original fuzzy time-delay system to obtain generalized delay-dependent stability conditions. Such an auxiliary system has some arbitrary matrices that generalize not only the system representation but also delay-dependent stability conditions. Conditions we obtain here are delay-dependent conditions that depend on the upper bound of time-delay, and are given in terms of linear matrix inequalities (LMIs). Then, we compare our delay-dependent stability conditions with other conditions in the literature, and show that our conditions guarantee the stability of a wider class of systems than others. Next, we consider the robust stabilization problem with memoryless and delayed state feedback controllers. Based on our generalized robust stability conditions, we obtain delay-dependent sufficient conditions for the closed-loop system to be robustly stable, and give a design method of robustly stabilizing controllers. Finally, we give three examples that illustrate our results.

Optimal, constant I/O similarity scaling for full-information and state-feedback control problems

This paper presents new stability conditions and LMI (linear matrix inequality) based designs for both continuous and discrete fuzzy control systems. First, Takagi and Sugeno's fuzzy models and some stability results are recalled. To design fuzzy control systems, nonlinear systems are represented by Takagi-Sugeno fuzzy models. The concept of parallel distributed compensation is employed to design fuzzy controllers from the Takagi-Sugeno fuzzy models. New stability conditions are obtained by relaxing the stability conditions derived in previous papers. The stability analysis of the feedback system is reduced to a problem of finding a common Lyapunov function for a set of linear matrix inequalities. Convex optimization techniques involving LMIs are utilized to find a common Lyapunov function. A procedure to design the fuzzy controller is constructed using the parallel distributed compensation and the relaxed stability conditions. Some stability issues are remarked. A simple example demonstrates the effects of the derived stability conditions.

New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models

Sparse matrix decompositions and graph characterizations

This article focus on the robustH∞stability analysis and controller design for a class of uncertain and disturbed continuous‐time systems with input time‐varying delays characterized by stochastic Bernoulli distributions. First, robustH∞stability conditions for linear continuous systems with interval input time‐varying delays is investigated. A delay‐distribution‐approach is considered to reduce the conservatism of the stability conditions from the convenient selection of a Lyapunov–Krasovskii functional (LKF), which considers both the lower and upper bounds of the stochastic time‐varying delay to derive new delay‐dependentH∞stability conditions in terms of linear matrix inequalities (LMIs). Then, derived from the proposed stability conditions, LMI‐based conditions for the design of stabilizing robustH∞delayed state feedback controller are proposed. Finally, five numerical examples are considered to show the effectiveness of the proposed stability analysis and controller design conditions, in comparison to previous related results.

A robust controller design method based on the concept of fuzzy region is presented for Takagi-Sugeno (T-S) fuzzy systems with parametric uncertainties. The uncertain T-S fuzzy model is converted into uncertain T-S fuzzy region model. A relaxed sufficient condition is derived for robust stabilization by using Lyapunov function approach in the form of linear matrix inequalities (LMIs). Comparing with other similar conditions, the difficulty of solving LMIs is greatly reduced in this method. The efficiency of the method is illustrated through the simulation of a numerical uncertain fuzzy system.

Robust Stabilizability of Switched Linear Time-Delay Systems with Polytopic Uncertainties

This paper provides a conducive criterion to investigate the stability of unforced continuous-time T-S fuzzy models in case of having second-order state matrices. According to the Tanaka and Sugeno theorem, the stability of a Takagi-Sugeno (T-S) fuzzy model can be proved by finding a common symmetric positive-definite matrix P for all the subsystems satisfying set of Lyapunov inequalities. But in many cases, especially by increasing the number of fuzzy rules, seeking such a common matrix seems to be a daunting task. In this paper, a simple sufficient stability criterion in terms of Lyapunov function candidate is proposed to investigate the global asymptotic stability of continuous-time T-S fuzzy models. In this case, the region of existence for the common positive-definite matrix P is investigated. It will be shown that choosing any arbitrary point from the obtained common region leads to a matrix P which can fulfill the relevant set of Lyapunov inequalities. In other words, finding any common region for such a matrix concludes the asymptotic stability of the system. This approach has the merit of specifying the existence or non-existence of the common matrix P for a fuzzy system. Several examples are given to illustrate the effectiveness of the proposed method.

New upper and lower bounds are given for the arithmetic,geometric and harmonic means of a set of positive definite matrices

The basic motion of the inverted pendulum with a DC motor can be described by a set of nonlinear differential equations. In this paper, on the basis of the method proposed by the Kawamoto et al., first the equations including a control input are rewritten into the form of a fuzzy system without any inference and approximation. Next, by applying Tanaka and Sugeno's (1994) stability theorem to the fuzzy system, the problem of finding a common symmetric positive definite matrix P satisfying two Lyapunov inequalities, that is, the common Lyapunov problem can be derived. Then, the P-region method is introduced for finding the matrix P, and feedback gains are determined under the existence of P. For the inverted pendulum treated here, the stability of both cases of linear and fuzzy control inputs is rigorously analyzed and is guaranteed in the whole region of all the dependent variables. Finally, it is numerically shown that the fuzzy control input gives better control responses than the linear one does.

This paper presents stability analysis of polynomial fuzzy-model-based (FMB) control systems using the sum-of-squares (SOS) approach. Recently, stability analysis of the polynomial fuzzy-control systems, which is a generalized form of the well-known Takagi-Sugeno (T-S) FMB control systems, has been reported in the form of SOS-based stability conditions. Lack of information on the relations between membership functions and premise variables, in the existing stability analysis approaches, causes conservatism of their results. In this paper, to derive relaxed stability conditions for polynomial FMB control systems, membership functions which are approximated with polynomials and carrying relations between membership functions and premise variables are brought into the stability analysis. Considering a polynomial FMB control system and based on the Lyapunov stability theory, stability conditions in the form of fuzzy summations are derived, where each term contains product of membership functions of the polynomial fuzzy model and polynomial fuzzy controller. Each product term is approximated by a polynomial. In order to obtain better approximation, the operating domain of membership functions is partitioned to subregions. Then, SOS-based stability conditions for all subregions are derived. Unlike some published stability-analysis approaches, the proposed one can be employed for stability analysis of polynomial fuzzy-control systems under imperfect premise matching of which the fuzzy model and fuzzy controller do not share the same membership functions. The solution of the SOS-based stability conditions can be found numerically using SOSTOOLS, which is a free third-party MATLAB Toolbox. Numerical examples are given to illustrate the effectiveness of the proposed stability conditions.