Fuzzy set theory for optimal design in controlling underactuated dynamical systems with uncertainty

The fuzzy set theory for optimal control design of underactuated dynamical systems with uncertainty is proposed. The uncertainty is (possibly fast) time varying whose bound is unknown and lies in a prescribed set expressed by a fuzzy set. Control goals of uncertain dynamical systems are formulated as a series of constraints, which may be holonomic and nonholonomic. From the view of constraint-following control, we design an adaptive robust control based on a creative uncertainty decomposition, which divides the uncertainty into matched and mismatched terms. This decomposition renders the mismatched term to “disappear” in the stability analysis. The proposed control is able to drive the system to approximately follow the constraints and guarantee system performance (uniform boundedness and uniform ultimate boundedness), in the presence of the uncertainty. It is in deterministic form and not “if–then” fuzzy rules based. Because of the expressed fuzzy set of the uncertainty bounds, a performance index, which balances the conflict between system performance and control cost, can be constructed. An optimal design gain can be obtained by minimizing the performance index whose solution is guaranteed to always exist. Numerical results on a planar vertical takeoff and landing aircraft are presented for demonstration.