Dealing With Indistinguishability in Gluing KKL Observer Design for Hybrid Systems With Unknown Jump Times

Abstract

In previous work, we presented a novel systematic observer design for hybrid systems with unknown jump times by employing the Kravaris–Kazantzis$\slash$Luenberger (KKL) observer paradigm and the concept of “gluing” the jumps. Specifically, we demonstrated the existence of a transformation that converts a hybrid system with continuous output at jumps into a continuous-time stable filter of this output, for which an observer can always be designed without requiring jump detection. However, the inherent indistinguishability appearing at jumps: First, prevents the convergence of the Euclidean estimation error to zero, thus rendering the definition and analysis of convergence$\slash$stability$\slash$robustness nonstraightforward; Second, prevents the injectivity of the change of coordinates, thus raising issues in the implementation of the observer. In this article, we deal with those two problems by: First, showing asymptotic convergence and robust stability of the gluing KKL observer in the sense of a distance to an appropriate indistinguishability-aware set; Second, proposing a systematic implementation scheme that involves learning numerical models of two inverse maps based on neural networks that are trained on data generated by offline simulations. Our contributions are academically illustrated on a bouncing ball and then applied for state and parameter estimation in a friction oscillator exhibiting stick-slip.