Proceedings of the American Control Conference Chicago, Ulinois • June 2000 Development and Analysis of a Fuzzy Identifier for Nonlinear Discrete Systems Hassan M. Emara *, A.L Elshafei **, and A. Bahgat* * Faculty of Engineering, Cairo University, Giza, Egypt ** Faculty of Engineering, United Arab Emirates University, E1-Ain, UAE Abstract This paper presents a new algorithm for fuzzy logic identification of nonlinear systems. The proposed algorithm uses a linear in parameters identifier to avoid the problem of local minima. The stabihty of the proposed algorithm is proved and sufficient condition on system excitation to ensure identifiability is presented. The algorithm is validated by comparing its performance with neural networks and back propagation trained fuzzy identifiers. I. Introduction Advanced process control and fault diagnosis schemes often require accurate modeling. Blackbox models are ready-made models that ignores totally any a-priori knowledge about the system. This choice results in an increased number of unknown parameters and a harder learning. Gray box models provide a way of including some a-priori knowledge about the system. This a-priori knowledge is usually the mathematical form of the system equations. Hence the identification problem is reduced to a parameter estimation problem. For many industrial systems, human operator can provide linguistic information about the process rather than mathematical forms. Hence, it is convenient to incorporate such information in the model. Fuzzy systems provide a mean to incorporate such information [1,2]. The use of Sugeno inference based fuzzy models is studied in [3]. Fuzzy identification of continuous systems was studied by Wang [4] using a steepest descent type algorithm. Moreover many other papers [5,6,7] propose backpropagation based learning, in which the fuzzy system is represented as a Neural Network (NN). These algorithms usually have a slow convergence rate. In this paper, an algorithm for the identification of discrete nonlinear systems is developed to overcome the slow convergence rate. Stability of the proposed algorithm is studied. To assist experiment design, sufficient condition on system excitation that ensures identifiability is presented. 2. Problem Formulation Consider the identification of the affine discrete time nonlinear system Yk = f(Xk)+g(Xk)Uk + h(Xk) (1) Where f and g are unknown functions, while h is a known one. As fuzzy identifiers are universal approximators I4], an identification model can be build where f and g are replaced by fuzzy logic systems, such that: • The system has L rules with the jth rule denoted as W • Each rule has n antecedent and one consequent • W has the form: If(xt is Ftj) and (x2 is F2 j) and ...and(Xnm is F i ) nm then y is CCd • Product implication and T norm are used • All membership functions (MFs) are gaussian with J and variance t~ j mean m i Fi In this case the fuzzy system represents a nonlinear mappingfl.) whose output can be expressed as: f(X)= Oj~j(X) / o~j (X) (2) where nm I ctj(X) = i=~lexp - (xi-mi) j2 2CrFi Or in vectorial notation (3) flX)=O P(x) (4) where and 0 =[Or 02... OL] [ ~_l(x) o~2(x) L ...... T c~ L (X) [ (5) P(x)= L ~%(x) J j=l Using two fuzzy systems to estimate f(.) and g(.) in (1), the estimated output ~rk can be given by: Yk =OkP(Xk)+PkPu(Xk)Uk+h(Xk) (6) 0-7803-5519-9•00 $10.00 © 2000 AACC 250