Design and analysis of fuzzy identifiers of nonlinear dynamic systems

Uses fuzzy systems as identifiers for nonlinear dynamic systems. The author provides a theoretical justification for the fuzzy identifiers by proving that they are capable of following the output of a general nonlinear dynamic system to arbitrary accuracy in any finite time interval. The fuzzy identifiers are constructed from a set of adaptable fuzzy IF-THEN rules and can combine both numerical information (in the form of input-output pairs obtained by exciting the system with an input signal and measuring the corresponding outputs) and linguistic information (in the form of IF-THEN rules about the behavior of the system in terms of vague and fuzzy words) into their designs in a uniform fashion. The author develops two fuzzy identifiers. The first one is designed through the following four steps: 1) define some fuzzy sets in the state space U/spl sub/R/sup n/ of the system; these fuzzy sets do not change; 2) construct fuzzy rule bases of the fuzzy identifier which comprise rules whose IF parts constitute all the possible combinations of the fuzzy sets defined in 1); 3) design the fuzzy systems in the fuzzy identifier based on the fuzzy rule bases of 2); and 4) develop an adaptive law for the free parameters in the fuzzy identifier. The second fuzzy identifier is designed in a similar way as the first one except that: a) the parameters characterizing the fuzzy sets in the state space change during the adaptation procedure; and b) the fuzzy systems and the adaptive law are different. The author proves that: 1) both fuzzy identifiers are globally stable in the sense that all variables in the fuzzy identifiers are uniformly bounded, and 2) under some conditions the identification errors of both fuzzy identifiers converge to zero asymptotically. Finally, the author simulates the fuzzy identifiers for identifying the chaotic glycolytic oscillator, and the results show that: 1) the fuzzy identifiers can approximate the chaotic system at a reasonable speed and accuracy without using any linguistic information, and 2) by incorporating some fuzzy linguistic IF-THEN rules about the behavior of the system into the fuzzy identifiers, the speed and accuracy of the fuzzy identifiers are greatly improved.<
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