Optimal Adaptive Robust Control Based on Cooperative Game Theory for a Class of Fuzzy Underactuated Mechanical Systems

While designing control for a class of underactuated mechanical systems (UMSs), the uncertainty and the prescribed nonholonomic tracking trajectories should be taken into consideration. Uncertainty considered in this article is time varying and bounded, and the bound of uncertainty is described using the fuzzy set theory, namely, fuzzy UMSs. An analytical dynamics-based view is taken in which the prescribed tracking trajectories are viewed as servo constraints which can be linear, nonlinear, holonomic, and nonholonomic. Using this view, a novel closed form solution of adaptive robust control is found with leakage type adaptive law to guarantee deterministic system performance, including uniform boundedness and uniform ultimate boundedness. In order to find the optimal dual gain parameters of the designed control, a two-player cooperative game is proposed for which the Pareto optimality can always be guaranteed. The effectiveness of the proposed control is shown through numerical simulation of a two-wheeled inverted pendulum vehicle.