Adaptive Fuzzy Inverse Optimal Fixed-Time Control of Uncertain Nonlinear Systems

Most existing methods on optimal finite-time control are restricted to a complex design and learning procedure, and only practical finite-time stable is ensured, which greatly limits the desirable performance of optimal and finite-time control. To solve the problem, an adaptive fuzzy fixed-time inverse approach is first proposed in this article, which achieves the optimized performance without recourse to Hamilton–Jacobi–Bellman equations and improves practical finite/fixed-time stable to fixed-time stable. Technically, to overcome the inverse optimal design difficulty of a nonlinear fixed-time controller, a series of singularity-avoidance functions and a Sontag-type function are incorporated to design a specified form of auxiliary controller, based on which an inverse optimal fixed-time controller is designed. Then, by introducing a two-Lyapunov functions method, it is proved that inverse optimal stabilization is ensured and the tracking error goes to a prescribed interval asymptotically within a fixed-time. Effectiveness of the proposed methods are illustrated by two examples.