Fuzzy Control of Multivariable Nonlinear Systems Subject to Parameter Uncertainties: Model Reference - Fuzzy Systems, 2000. FUZZ IEEE 2000. The Ninth IEEE International Conference on

This paper presents the control of nonlinear systems subject to parameter uncertainties under a fuzzy control approach. The nonlinear plant tackled in this paper is an n-order nonlinear systems with n inputs. A design procedure of fuzzy controller will be provided such that the states of the closed-loop system will follow those of a userdefined stable reference model despite the presence of parameter uncertainties. A numerical example will be given to show the design procedures and the merits of the proposed fuzzy controller. I . INTRODUCTION Fuzzy control is a useful control technique for uncertain and ill-defined nonlinear systems. Control actions of the fuzzy controller are described by some linguistic rules. This property makes the control algorithm to be understood easily. The early design of fuzzy controllers is heuristic. It incorporates experiences or knowledge of the designer into the rules of the fuzzy controller, which is fine tuned based on trial and error. A fuzzy controller implemented by neural-fuzzy network was proposed in [3, 41. Through the use of tuning methods, fuzzy rules can be generated automatically. These methodologies make the design simple; however, the design does not guarantee the system stability, robustness and good performance. To facilitate the system analysis of fuzzy control systems and the design of fuzzy controllers, fuzzy plant model was proposed in 11, IO]. The fuzzy plant model represents a nonlinear system as a weighted sum of some linear systems. Based on this structure, fuzzy controllers comprising a number of sub-controllers were proposed. State feedback controllers were proposed as the sub-controllers in 141. The closed-loop system is guaranteed to be asymptotically stable if there exists a common solution for a number of linear matrix inequalities (LMI). Other stability conditions can be found in [9-IO]. For fuzzy plant models subject to parameter uncertainties, robustness analysis was carried in 15, I I ] . In this paper, a design methodology will be proposed to design a fuzzy state feedback controller. An n-order-n-input nonlinear system will be tackled. This system will be represented by a fuzzy plant model. The system states of the closed-loop system will follow those of a stable user-defined reference model. A numerical example will be given to show the design procedures and the merits of the proposed fuzzy controller. 11. REFERENCE MODEL, FUZZY PLANT MODEL AND FUZZY An n-order-n-input nonlinear plant subject to parameter uncertainties will be considered. This plant is represented by a fuzzy plant model. A fuzzy controller will be designed to close the feedback loop such that the closed-loop system states will follow those of a stable reference model. CONTROLLER A. Reference Model A reference model is a stable linear system given by, 2(t) = H n z i ( t ) + B n r r ( t ) (1) where H,, E '3""" is a constant stable system matrix, ' This work was supported by a Research Grant of The Hong Kong Polytechnic University (project number G-SSSS). BnI E '3"""' is a constant input vector, c E '3"" is the system state vector of this reference model and r(t) E '3"' is the bounded reference input. B. F u u y Plant Model with Parameter Uncertainty nonlinear plant. The i-th rule is of the following format, Rule i : IF x l ( t ) is M: and . . . and x,,(t) is M:, Let p be the number of fuzzy rules describing the uncertain THEN X(t) = (A, +AA, )X(t) + B , u ( t ) (2) where Mfi is a fuzzy term of rule i corresponding to the state xk(t), k = 1, 2, ..., n, i = 1, 2, ..., p ; A, E %"'" and B, E '3""'' are the system and input matrix respectively in phase variable canonical form; A A , E 9Inm is the parameter uncertainties of A, ; x( t ) E %""' is the system state vector and u ( t ) E '3''"' is the input vector. The plant dynamics is described by, X(t) = f: w, (x ( t ) ) [ (A , + AA, )x(t) + B F ( t ) l where 5 w, ( x ( t ) ) = 1 , w, ( x ( t ) ) E [O, I] for all i ( 3 )