Fuzzy Dynamic Systems: Analysis, Control and Identification

Abstract

This chapter dels with fuzzy dynamic systems paying attention to three important issues, namely analysis, control and identification. Analysis is concernced with the study of input- output descriptions which are based on fuzzy relational equations. One of the models that played a central role in the analysis of classical (crisp) dynamic systems is the state-space model. This model can be extended to cover the case of fuzzy dynamic systems, and is studied in the present chapter. Control is concerned with the problem of determining a control sequence which brings the system state, in one or more time steps, from the present fuzzy value X_k to a desired final fuzzy value X_f. This problem is equivalent ot solving the relevant equation X_f=U_koX_koR with respect to U_k. The fuzzy system identification problem solved here consists in estimating the system fuzzy relational matrix R from a given set of fuzzy input-outpu pairs (X_i, Y_i), i=1,2...,N. The non-noise case is considered, since the case where noise is present is still a challenging open problem. An algorithm is provided for determining good upper and lower bounds that confince R. Also, the state prediction problem of fuzzy state space models is considered. The chapter closes with a review of three practical fuzzy controllers which have been applied with success to several industrial systems. These controllers are: (i) the fuzzy self-organized controller (SOC), the fuzzy PID supervisor, and the fuzzy PID incremental controller.