Spatiotemporal Adaptive Fuzzy Control for State Profile Tracking of Nonlinear Infinite-Dimensional Systems on a Hypercube

The issue of spatiotemporal adaptive fuzzy state profile tracking (SAFSPT) control is addressed in this study for parabolic partial differential systems (PPDSs) on a hypercube exposed to unknown nonlinear dynamics. Specifically, a spatiotemporal fuzzy set on a hypercube is first constructed from the three-dimensional fuzzy set on a simple 1-D space domain to completely take into account the spatiotemporal coupling property of the PPDS state. Then a spatiotemporal fuzzy system is developed via the spatiotemporal fuzzy set to approximate any real continuous functions on an open subset of a separable Hilbert space. It is shown that any continuous functions can be approximated by the proposed spatiotemporal fuzzy system in arbitrary precision. Both indirect and direct SAFSPT control strategies are put forth with the use of the spatiotemporal fuzzy system. The closed-loop tracking error system's stability is thoroughly examined in terms of Lyapunov stability. Finally, the effectiveness and benefit of theoretical results are illustrated by extensive simulation experiments.