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 conditions on the existence of a sequence of functions
$(T_{k})_{k \in \mathbb {N}}$
transforming the given system dynamics into an exponentially stable filter of the output in some other target coordinates, where an observer is directly designed. Then, we prove that under uniform Lipschitz backward distinguishability, the maps
$(T_{k})_{k \in \mathbb {N}}$
become uniformly Lipschitz injective after a certain time if the target dynamics are pushed sufficiently fast. This leads to an arbitrarily fast discrete observer after a certain time, which exhibits similarities with the famous high-gain observer for continuous-time systems. Input-to-state stability of the estimation error with respect to uncertainties, input disturbances, and measurement noise is then shown. Next, under the milder backward distinguishability, we show the injectivity of the maps
$(T_{k})_{k \in \mathbb {N}}$
after a certain time for a generic choice of the target filter dynamics. Examples including a discretized permanent magnet synchronous motor illustrate the proposed observer.