Backstepping-Based Fuzzy Adaptive Stabilization of Reaction-Diffusion Equation With State Constraints

Area of Science:

• Control Systems Engineering
• Applied Mathematics
• Nonlinear Dynamics

Background:

• Cascaded parabolic partial differential equation (PDE)-ordinary differential equation (ODE) systems present significant control challenges due to nonlinearities and state constraints.
• Existing control methods often struggle with the
• explosion of complexity
• and ensuring state constraints are respected.

Purpose of the Study:

• To propose a novel stabilization scheme for cascaded PDE-ODE systems with state constraints.
• To effectively handle unknown nonlinearities within the ODE subsystem.
• To ensure all closed-loop system signals remain bounded and states converge to zero.

Main Methods:

• Fuzzy-logic system (FLS) technique to eliminate nonlinearities in the ODE subsystem.
• Infinite and finite-dimensional backstepping methods for control design.
• Introduction of a novel first-order filter to mitigate "explosion of complexity".
• Construction of a barrier Lyapunov function (BLF) to enforce state constraints.

Main Results:

• Successful elimination of nonlinear influences using FLS.
• Development of a new PDE subsystem via backstepping transformation for simplified control.
• Controller design that prevents "explosion of complexity" and guarantees constraint satisfaction.
• Demonstration of bounded closed-loop signals and convergence to zero states.

Conclusions:

• The proposed stabilization scheme effectively controls cascaded PDE-ODE systems with state constraints.
• The integration of FLS, backstepping, filtering, and BLF provides a robust and computationally tractable solution.
• Simulation results confirm the satisfactory performance and effectiveness of the developed control strategy.