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Approximate optimal control design for nonlinear one-dimensional parabolic PDE systems using empirical eigenfunctions
1Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University (formerly Beijing University of Aeronautics and Astronautics), Beijing 100191, China. biao.luo@hotmail.com
Summary
This study introduces an approximate optimal control method for parabolic partial differential equation systems using empirical eigenfunctions and neural networks. The approach effectively reduces system complexity and ensures stability for nonlinear systems.
Area of Science:
- Control Theory
- Applied Mathematics
- Computational Science
Background:
- Parabolic partial differential equation (PDE) systems with nonlinear spatial operators present significant challenges for optimal control.
- Existing methods often struggle with high dimensionality and complex dynamics inherent in these systems.
Purpose of the Study:
- To develop an approximate optimal control design method for nonlinear parabolic PDE systems.
- To address the challenges of high dimensionality and ensure closed-loop stability.
Main Methods:
- Karhunen-Loève decomposition to compute empirical eigenfunctions (EEFs) for system representation.
- Singular perturbation (SP) technique to derive a reduced-order model (ROM) from a high-order ordinary differential equation (ODE) system.
- Hamilton-Jacobi-Bellman (HJB) method for optimal controller synthesis, incorporating neural networks (NN) for cost function approximation.
Main Results:
- A reduced-order model accurately captures dominant PDE system dynamics.
- An effective approximate optimal controller is synthesized, guaranteeing closed-loop asymptotic stability.
- A control update strategy based on successive approximation is proposed and its convergence proven.
Conclusions:
- The developed approximate optimal control method, utilizing EEFs and NNs, is effective for nonlinear parabolic PDE systems.
- The method successfully handles complex dynamics and ensures system stability.
- Simulation results on a diffusion-reaction process validate the approach's efficacy.
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