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

In this article, the issue of spatiotemporal adaptive fuzzy state profile tracking control is addressed 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 3-D fuzzy set on a simple 1-D space domain to completely take into account the spatial-temporal property of the PPDS. Then, a spatiotemporal fuzzy system is developed on an open subset to approximate any real continuous functions in a separable Hilbert space based on the spatiotemporal fuzzy set notion. It is shown that any continuous functions can be approximated by the proposed spatiotemporal fuzzy system in arbitrary precision. Both indirect and direct spatiotemporal adaptive fuzzy state profile tracking control strategies are put forth with the use of the spatiotemporal fuzzy system. The resulting 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.