The Polynomial Extended Kalman Filter as an exponential observer for nonlinear discrete-time systems

This paper presents some results on the local exponential convergence of the Polynomial Extended Kalman Filter (PEKF, see [14]) used as a state observer for deterministic nonlinear discrete-time systems (Polynomial Extended Kalman Observer, PEKO). A new compact formalism is introduced for the representation of the so called Carleman linearization of nonlinear discrete time systems, that allows for the derivation of the observation error dynamics in a concise form, similar to the one of the classical Extended Kalman Filter. The stability analysis performed in this paper is also important in the stochastic framework, in that the exponential stability of the error dynamics can be used to prove that the moments of the estimation error, up to a given order, remain bounded over time (stability of the PEKF).