Robust Control Design for Fuzzy Mechanical Systems: A Two-Player Nash Game Approach

Mechanical systems with (possibly fast) time-varying but bounded uncertainty are considered. The exact value of the boundary is unknown. All the designer can get is that the boundary values are in a (known) fuzzy set. On the basis, a robust control method is proposed, which can ensure the uniformly bounded and uniformly ultimately bounded of the controlled mechanical system. The control contains two flexibly selectable design parameters. We seek to choose the optimal parameters. For a superior performance, two fuzzy-set-based performance indices are proposed to reflect the transient performance as well as the steady-state performance. Each performance index can be influenced by two design parameters. However, influences are nonconciliatory. This poses a design dilemma: the increase of a parameter may harness one performance while inflict the other. Therefore, the “optimal” choice of the design parameters is not intuitively clear. To resolve this dilemma, the two-player Nash-based noncooperative game theory is adopted, which is a notable feature of this article. Once the problem is formulated, we show that there is always a Nash-equilibrium solution to the two-player problem. We also show how to find it. The approach is very general, which also can be extended to
$n$
-player Nash game for future research. The control is applied to a compressor powered by permanent magnet synchronous motor (PMSM) as a demonstration. The resulting performance shows that this Nash-based robust control design is both practical and effective.