Polynomial Convergence of an Observer for an Infinite-Dimensional Oscillating System

TL;DR

This paper analyzes the polynomial convergence rate of an observer for infinite-dimensional linear systems, using spectral theory to explicitly construct the resolvent and characterize the decay of observation error.

Contribution

It introduces a spectral-theoretic approach to explicitly determine the polynomial stability and convergence rate of observers in infinite-dimensional systems.

Findings

Explicit resolvent construction for the observer error system

Polynomial decay rate characterized via spectral analysis

Application to oscillating flexible structures

Abstract

This paper is devoted to analyzing the observer convergence rate for a class of linear control systems in a Hilbert space. To characterize the polynomial stability of the observer error system, we apply the spectral theory of linear operators and explicitly construct the resolvent of the corresponding infinitesimal generator. The asymptotic behavior of the resolvent on the imaginary axis is studied to describe the rate of decay of the observation error. The estimated decay rate is illustrated through an example of an oscillating flexible structure with one-dimensional output.