Spectral Boundary Observer for Counter-Flow Heat Exchangers

TL;DR

This paper introduces a spectral boundary observer for counter-flow heat exchangers modeled by coupled hyperbolic PDEs, enabling tunable convergence rates using only boundary temperature measurements.

Contribution

It develops a spectral design approach for boundary observers, proving spectral stability is equivalent to exponential stability in the $L^2$ norm.

Findings

Spectral boundary observer achieves tunable convergence rates.

Spectral stability is equivalent to $L^2$ exponential stability.

Operator satisfies spectral mapping property, ensuring stability.

Abstract

We consider a system of two coupled first-order linear hyperbolic partial differential equations modeling heat transport in a counter-flow heat exchanger: one equation describes the transport of a hot fluid, and the other the transport of a cold fluid in the opposite direction. For this system, we design a boundary observer that uses only the temperature of the cold fluid measured at one boundary. Our approach is spectral: by assigning the spectrum of the operator governing the observation error dynamics to a prescribed region within the open left-half complex plane, we can freely tune the convergence rate of the observation error to zero in the $L^2$ norm. The main technical contribution is the proof that spectral stability, that is, the location of the spectrum in the open left-half plane, is equivalent to $L^2$ exponential stability of the origin for the observation error dynamics. This equivalence is established by showing that the operator governing the observation error dynamics satisfies the so-called spectral mapping property.