$H^\infty$-control for a class of boundary controlled hyperbolic PDEs

TL;DR

This paper presents a method to solve the suboptimal $H^inity$-control problem for boundary-controlled hyperbolic PDEs by transforming it into a finite-dimensional discrete-time system, simplifying the control design process.

Contribution

It introduces an approach to convert boundary controlled hyperbolic PDEs into a finite-dimensional system for easier $H^inity$-control solution derivation.

Findings

The PDEs can be represented as an infinite-dimensional discrete-time system.

The control problem reduces to a finite-dimensional discrete-time control problem.

The method is demonstrated on a boundary controlled vibrating string.

Abstract

A solution to the suboptimal $H^\infty$-control problem is given for a class of hyperbolic partial differential equations (PDEs). The first result of this manuscript shows that the considered class of PDEs admits an equivalent representation as an infinite-dimensional discrete-time system. Taking advantage of this, this manuscript shows that it is equivalent to solve the suboptimal $H^\infty$-control problem for a finite-dimensional discrete-time system whose matrices are derived from the PDEs. After computing the solution to this much simpler problem, the solution to the original problem can be deduced easily. In particular, the optimal compensator solution to the suboptimal $H^\infty$-control problem is governed by a set of hyperbolic PDEs, actuated and observed at the boundary. We illustrate our results with a boundary controlled and boundary observed vibrating string.