INTEGRAL EVALUATIONS FOR A CLASS OF MULTIVARIABLE FUZZY CONTROL SYSTEMS Jacek Kluska 1 and Krzysztof Wiktorowicz 1 Department of Control and Computer Engineering, Rzeszow University of Technology, 35-959 Rzeszow, W. Pola 2, Poland. E-mail: jacklu@prz-rzeszow.pl, kwiktor@prz-rzeszow.pl Abstract: The problem of the integral evaluations of the state and control vectors in continuous multi-input-multi-output fuzzy control systems is investigated. The fuzzy controller is assumed to be nonlinearity in a bounded sector and the controlled plant - a linear system described by the differential or integral equations. The stability problem is investigated and the problem of how to use the integral evaluations for the fuzzy control system design is proposed. Keywords: Fuzzy control, Nonlinear systems; Performance; Absolute stability 1. INTRODUCTION Despite the increasing number of industrial ap- plications of fuzzy control, there are some gaps in the analytical methods of fuzzy control quality as- sessment and stability analysis of the closed-loop systems containing nonlinear fuzzy controllers. Still lagging behind fuzzy controllers applications seem to be the systematic methods of analysis and design of fuzzy controllers, especially in the case of multi-input-multi-output systems, even if the controlled plants are linear. In this work continuous closed-loop systems con- taining multivariable fuzzy controller and the lin- ear or at least linearizable stationary dynamical plant are investigated. The plant can be asymp- totically stable or not, but has to be stabiliz- able by the linear feedback. The fuzzy controller is regarded as a nonlinear function satisfying a sector condition, similarly as in (Ray and Ma- jumder, 1984), (Calcev et al., 1998) and the other works. Although the sector condition is a restric- tion of the class of investigated fuzzy controllers, it applies to a great deal of fuzzy expert systems 1 Partially supported by the grant 7 T11A 018 21 of the Polish State Commitee of Scientific Research as direct control devices in the closed-loop. Most often such systems either exactly are, or resemble the sector bounded nonlinearities in the sense of functions approximation. The main goal of this article is to give an analyti- cal method of the state and control vector norms evaluations. It seems to be a novelty in the fuzzy control literature. The controlled plant can be described by a differential or integral equations. The obtained integral evaluations one can use to quality assessment of the fuzzy control systems. The way of how to use these evaluations for fuzzy controller design is briefly explained. 2. ASSUMPTIONS AND PROBLEM STATEMENT We consider the multi-input-multi-output (MIMO) closed-loop system as in Fig. 1. There are n state variables and m control signals, which constitute the vectors x (t)=[x 1 (t), ... , x n (t)] T ∈ R n and u (t)=[u 1 (t), ... , u m (t)] T ∈ R m in the system. We assume that the controlled plant is linear one or linearizable and described by the state equation ˙ x (t)= Ax (t)+ Bu (t) , x (0) ∈ R n (1) Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain