Switching Event-Triggered Control of Nonlinear Parabolic PDE Systems via Galerkin/Neural-Network-Based Modeling Approach

TL;DR

This paper develops a switching event-triggered control method for nonlinear parabolic PDE systems, using Galerkin and neural network modeling to ensure stability and performance, demonstrated through simulations.

Contribution

It introduces a novel control scheme combining Galerkin, neural networks, and switching event-triggered control for PDE systems with unknown nonlinearities.

Findings

Effective control of PDE systems demonstrated via simulations.

Conversion of BMIs to LMIs enables practical controller design.

Sub-optimal controllers achieved through iterative LMI optimization.

Abstract

This paper focuses on switching event-triggered output feedback control for a class of parabolic partial differential equation (PDE) systems subject to unknown nonlinearities and external bounded disturbance. Initially, the PDE systems is properly separated into a finite-dimensional ordinary differential equation (ODE) slow system and an infinite-dimensional ODE fast system based on Galerkin technique, especially the slow system can characterize the dominated dynamics. Then, a three-layer neural network is employed to approximate the unknown nonlinearities, and Levenberg-Marquardt algorithm is adopted to get a relative accurate slow system. Subsequentaly, a switching event-triggered control scheme is developed, and a waiting time subject to the triggered condition is implemented to avoid the Zeno behavior and convert the slow system into a switching system. In the following, the stability and $\textbf{\textit{H}}_\infty$ performance issues of the closed-loop system are discussed, and the controllers are displayed in terms of bilinear matrix inequalities (BMIs). Novel algorithms are proposed to convert the BMIs into linear matrix inequalities (LMIs). Additionally, a sub-optimal switching event-triggered controller is obtained using an iterative optimization approach based on LMIs. Finally, simulation on the catalytic rod reaction model and traffic flow model demonstrate the effectiveness of switching event-triggered control strategy.