A galerkin/neural-network-based design of guaranteed cost control for nonlinear distributed parameter systems

Area of Science:

• Control Systems Engineering
• Applied Mathematics
• Computational Science

Background:

• Parabolic partial differential equation (PDE) systems often feature complex dynamics with both slow and fast modes.
• Controlling systems with unknown nonlinearities presents significant challenges in achieving stability and performance guarantees.
• Existing control methods may struggle with the infinite-dimensional nature and inherent uncertainties of PDE systems.

Purpose of the Study:

• To develop a robust guaranteed cost control (GCC) strategy for parabolic PDE systems with unknown nonlinearities.
• To effectively approximate the dominant dynamics of the PDE system using a neural network model.
• To design a controller that guarantees closed-loop stability and minimizes a quadratic cost function.

Main Methods:

• Application of the Galerkin method to discretize the parabolic PDE into a finite-dimensional ordinary differential equation (ODE) system.
• Parameterization of the resulting nonlinear ODE system using a multilayer neural network (MNN).
• Design of a linear modal feedback controller based on a neural model and Lyapunov stability theory, formulated as a linear matrix inequality (LMI) problem.

Main Results:

• The proposed method successfully derives an ODE model capturing the slow dynamics of the PDE system.
• A neural network effectively approximates the unknown nonlinearities within the ODE model.
• The resulting controller guarantees stability and provides a minimized cost bound for the closed-loop system, validated on a catalytic rod temperature control problem.

Conclusions:

• The Galerkin/neural-network-based GCC approach offers a viable solution for controlling complex parabolic PDE systems with uncertainties.
• The formulation as an LMI problem allows for efficient computation of suboptimal controllers.
• This methodology demonstrates practical applicability in real-world engineering problems, such as thermal process control.