Optimal Control for Fractional Order Nonlinear Systems Based on Adaptive Dynamic Programming

In this article, an adaptive dynamic programming (ADP)-based optimal control strategy for a series of fractional-order nonlinear systems (FONS) with unknown control directions is investigated. To eliminate the challenges posed by unknown control directions, fractional-order Nussbaum-type functions are introduced for FONS, expanding the range of possible applications. Additionally, since system performance is compromised by disturbances, a fractional-order disturbance observer is designed to counteract the effects of external disturbances and enhance system robustness. Furthermore, differential geometric methods are employed to investigate FONS, constructing appropriate diffeomorphism that provide equivalent systems for decoupled linearization. Then, an optimal control method is studied for a class of strictly feedback FONS, in which Nussbaum-type functions are combined with ADP theory during the backstepping design process. Finally, based on fractional Lyapunov stability theory and backstepping method, it is guaranteed that all signals of the closed-loop FONS are uniformly ultimately bounded (UUB). Numerical simulation and a PMSM model are utilized to verify the effectiveness of the presented method.