Extended Kalman Filter-Based Observer Design for Semilinear Infinite-Dimensional Systems

In many physical applications, the system's state varies with spatial variables as well as time. The state of such systems is modeled by partial differential equations and evolves on an infinite-dimensional space. Systems modeled 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 article 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. Next, 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.