Backstepping Control of Coupled General Hyperbolic-Parabolic PDE-PDE Systems

TL;DR

This paper develops a backstepping control and observer design methodology for coupled hyperbolic-parabolic PDE systems, simplifying the process by solving standard kernel and decoupling equations, and verifies exponential stability through numerical examples.

Contribution

It introduces a multi-step backstepping approach for coupled hyperbolic-parabolic PDEs, enabling easier stability analysis and numerical implementation.

Findings

Successfully designed controllers and observers for coupled PDE systems.

Proved exponential stability of the closed-loop system.

Demonstrated effectiveness through numerical examples.

Abstract

This paper considers the backstepping state feedback and observer design for hyperbolic and parabolic PDEs, which are bidirectionally interconnected in a general coupling structure. Both PDE subsystems consist of coupled scalar PDEs with the heterodirectional hyperbolic PDE subsystem subject to actuation and sensing. By making use of a multi-step approach to construct the transformation into a stable target system, it is shown that a backstepping state feedback and observer design only requires to solve the well-known kernel equations for the hyperbolic and parabolic subsystems as well as additional decoupling equations. The latter are standard initial boundary value problems for parabolic PDEs. This significantly facilitates the well-posedness analysis and the numerical computation of the backstepping controller. Exponential stability is verified for the state feedback loop, the observer error dynamics, and the closed-loop system using an observer-based compensator. The proposed backstepping design procedures are demonstrated for numerical examples.