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Nash-Game-Oriented Optimal Design in Controlling Fuzzy Dynamical Systems

The robust control design problem for uncertain dynamical systems is considered in this study. The uncertainty is time varying (possibly fast) and bounded, and the bound lies within a prescribed fuzzy set (hence the fuzzy dynamical system). We design the robust control in two steps. First, we propose a class of robust controls based on tunable parameters, which is in deterministic form and not conventionally IF–THEN fuzzy rule based. It is shown that these controls are able to guarantee deterministic system performance, namely uniform boundedness and ultimate uniform boundedness. Second, we seek the optima of tunable parameters in the control by formulating a two-player Nash game, which is based on two performance indexes (i.e., the cost functions). It is shown that the Nash equilibrium (i.e., the optima of tunable parameters) always exists. The procedure of obtaining the Nash equilibrium is provided. Under the proposed control, the system performance is both deterministically guaranteed and fuzzily optimized from the Nash game perspective. The effectiveness of the control design is illustrated by the simulation control of a unicycle robot.