Boundary Output Tracking of Nonlinear Parabolic Differential Systems via Fuzzy PID Control

Abstract

In this article, the problem of output tracking via proportional-integral-derivative (PID) control scheme is discussed for a class of nonlinear infinite-dimensional spatiotemporal dynamic systems modeled by a semilinear parabolic partial differential equation (PDE) with collocated boundary control input and measurement output. To surmount the difficulty caused by the infinite-dimensional spatiotemporal nonlinear dynamics, a Takagi–Sugeno (T–S) fuzzy parabolic PDE model is first constructed to represent the nonlinear spatiotemporal dynamics, and then a fuzzy PID boundary output tracking control (BOTC) scheme is proposed via the obtained T–S fuzzy PDE model and the difference between the boundary measurement output and its desired constant reference signal to achieve the output tracking goal. Utilizing the Lyapunov technique combined with the inequality techniques, a systematic, conceptually simple yet effective parameter tuning method is developed for the fuzzy PID control scheme such that the suggested fuzzy PID-BOTC law drives the measurement output to asymptotically track the desired reference signal and ensures the boundedness of the resulting closed-loop system signals. Such parameter tuning is formulated as a feasibility problem subject to linear matrix inequality constraints. Moreover, two special cases of the proposed PID-BOTC design (i.e., fuzzy PI-BOTC scheme and fuzzy integral BOTC one) are also provided in this article. Extensive simulation results for a numerical example and a chemical axial dispersion tubular reactor are presented to show the effectiveness of the proposed fuzzy PID-BOTC scheme and its merit in the fast response of the preset reference signal and the less overshot is also illustrated by comparing with the fuzzy PI control law and the fuzzy integral one.