Arbitrarily Fast Robust KKL Observer for Nonlinear Time-Varying Discrete Systems

This work presents the Kazantzis-Kravaris/Luenberger (KKL) observer design for nonlinear time-varying discrete systems. We first give sufficient conditionson the existence of a sequence of functions(Tk)k is an element of Ntrans-forming the given system dynamics into an exponentiallystable filter of the output in some other target coordinates,where an observer is directly designed. Then, we prove thatunder uniform Lipschitz backward distinguishability, themaps(Tk)k is an element of Nbecome uniformly Lipschitz injective after acertain time if the target dynamics are pushed sufficientlyfast. This leads to an arbitrarily fast discrete observer aftera certain time, which exhibits similarities with the famoushigh-gain observer for continuous-time systems. Input-to-state stability of the estimation error with respect to un-certainties, input disturbances, and measurement noise isthen shown. Next, under the milder backward distinguisha-bility, we show the injectivity of the maps(Tk)k is an element of Nafter acertain time for a generic choice of the target filter dynam-ics. Examples including a discretized permanent magnetsynchronous motor illustrate the proposed observer.