On the strict-feedback form of hyperbolic distributed-parameter systems

TL;DR

This paper explores the structural properties of hyperbolic distributed-parameter systems, extending backstepping techniques to map controllable hyperbolic PDEs and PDE-ODE systems into strict-feedback form, facilitating control design.

Contribution

It extends existing backstepping results to map controllable hyperbolic PDEs and PDE-ODE systems into strict-feedback form, providing new insights for control design.

Findings

Mapped controllable hyperbolic PDEs into strict-feedback form

Extended backstepping results for PDE-ODE systems

Provided structural insights for control design

Abstract

The paper is concerned with the strict-feedback form of hyperbolic distributed-parameter systems. Such a system structure is well known to be the basis for the recursive backstepping control design for nonlinear ODEs and is also reflected in the Volterra integral transformation used in the backstepping-based stabilization of parabolic PDEs. Although such integral transformations also proved very helpful in deriving state feedback controllers for hyperbolic PDEs, they are not necessarily related to a strict-feedback form. Therefore, the paper looks at structural properties of hyperbolic systems in the context of controllability. By combining and extending existing backstepping results, exactly controllable heterodirectional hyperbolic PDEs as well as PDE-ODE systems are mapped into strict-feedback form. While stabilization is not the objective in this paper, the obtained system structure is the basis for a recursive backstepping design and provides new insights into coupling structures of distributed-parameter systems that allow for a simple control design. In that sense, the paper aims to take backstepping for PDEs back to its ODE origin.