Output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients

TL;DR

This paper develops a backstepping-based output feedback control method for complex linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients, including observer design and practical implementation.

Contribution

It introduces a systematic backstepping approach with simple kernel and decoupling equations for controlling coupled PDE-ODE systems with spatially-varying coefficients.

Findings

Successfully designed a state feedback controller for a 4x4 hyperbolic PDE-ODE system.

Developed observer-based compensator with verifiable existence conditions.

Demonstrated the approach on a coupled PDE-ODE example with boundary conditions.

Abstract

This paper presents a backstepping solution for the output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients. Thereby, the coupling in the PDE is in-domain and at the uncontrolled boundary, whereby the ODE is coupled with the latter boundary. For the state feedback design a two-step backstepping approach is developed, that yields the conventional kernel equations and additional decoupling equations of simple form. The latter can be traced back to simple Volterra integral equations of the second kind, which are directly solvable with a successive approximation. In order to implement the state feedback controller, the design of observers for the ODE-PDE systems in question is considered, whereby anticollocated measurements are assumed. Simple conditions for the existence of the resulting observer-based compensator are formulated, that can be evaluated in terms of the plant transfer behaviour. The resulting systematic compensator design is illustrated for a 4x4 heterodirectional hyperbolic system coupled with a third order ODE modelling a dynamic boundary condition.