Extended Kalman filter based observer design for semilinear infinite-dimensional systems

TL;DR

This paper extends the extended Kalman filter to semilinear infinite-dimensional systems, providing theoretical guarantees for well-posedness and local stability, with practical implementation on finite-dimensional approximations.

Contribution

It introduces a novel EKF extension for infinite-dimensional systems and proves its well-posedness and local stability under mild conditions.

Findings

Proved well-posedness of the infinite-dimensional EKF equations

Established local exponential stability of the error dynamics

Demonstrated practical implementation with finite-dimensional approximations

Abstract

In many physical applications, the system's state varies with spatial variables as well as time. The state of such systems is modelled by partial differential equations and evolves on an infinite-dimensional space. Systems modelled by delay-differential equations are also infinite-dimensional systems. The full state of these systems cannot be measured. Observer design is an important tool for estimating the state from available measurements. For linear systems, both finite- and infinite-dimensional, the Kalman filter provides an estimate with minimum-variance on the error, if certain assumptions on the noise are satisfied. The extended Kalman filter (EKF) is one type of extension to nonlinear finite-dimensional systems. In this paper we provide an extension of the EKF to semilinear infinite-dimensional systems. Under mild assumptions we prove the well-posedness of equations defining the EKF. Local exponential stability of the error dynamics is shown. Only detectability is assumed, not observability, so this result is new even for finite-dimensional systems. The results are illustrated with implementation of finite-dimensional approximations of the infinite-dimensional EKF on an example.