On observer forms for hyperbolic PDEs with boundary dynamics

TL;DR

This paper introduces a new hyperbolic observer canonical form for linear hyperbolic PDEs with boundary dynamics, utilizing observability coordinates derived from a neutral functional differential equation.

Contribution

It presents a systematic transformation to the HOCF based on observability coordinates linked to a neutral FDE, applicable to boundary-controlled hyperbolic PDEs.

Findings

The HOCF is explicitly constructed from the FDE-based observability coordinates.

The transformation simplifies the analysis and design of observers for hyperbolic PDEs.

An illustrative string-mass-spring example demonstrates the approach.

Abstract

A hyperbolic observer canonical form (HOCF) for linear hyperbolic PDEs with boundary dynamics is presented. The transformation to the HOCF is based on a general procedure that uses so-called observability coordinates as an intermediate step. These coordinates are defined from an input-output relation given by a neutral functional differential equation (FDE), which, in the autonomous case, reduces to an autonomous FDE for the output. The HOCF coordinates are directly linked to this FDE, while the state transformation between the original coordinates and the observability coordinates is obtained by restricting the observability map to the interval corresponding to the maximal time shift appearing in the FDE. The proposed approach is illustrated on a string-mass-spring example.