Game-Theoretic Optimization Toward Diffeomorphism-Based Robust Control of Fuzzy Dynamical Systems With State and Input Constraints

This work investigates a game-theoretic optimization approach toward robust control of uncertain dynamical systems with state and input constraints. The uncertainty involved is possibly rapidly time-varying but bounded within a prescribed fuzzy set. For this, the associated fuzzy dynamical system is appropriately established and constructed based on fuzzy set theory. To cope with the bounded state and input constraints, a novel state-and-input diffeomorphism technique is proposed, where a transformed system is formulated such that the prescribed inequality constraints are innovatively merged into the stabilization and trajectory tracking problems. Furthermore, a diffeomorphism-based robust control strategy is developed to ensure the uniform boundedness and uniform ultimate boundedness of the transformed system. Under this proposed control architecture, the constraint satisfaction of the original system is, thus, always analytically ensured based on the rigorous properties of the diffeomorphism technique. The resulting control parameter optimization problem then has to take into account the multiple considerations (and compromise) amongst the factors of the steady-state performance; the finite convergence time, and the control effort. For this, a two-player Nash game is formulated and solved in an effective manner. The Nash equilibrium is obtained and the existence of the solution is also proved theoretically. With this methodology, and with the resulting attainment of the desired Nash equilibrium, the attendant outcome of superior system performance is achieved. Finally, numerical simulations on a steer-by-wire system demonstrate the effectiveness of the proposed approach.