Saltation-Based analysis of estimation error in observers for hybrid systems with unknown jump times

This paper highlights some key conditions for local state estimation of hybrid dynamical systems with unknown jump times, for which the observer jumps cannot be triggered at the same time as the system’s. We provide sufficient conditions – including appropriate transversality conditions of flows with respect to guard maps – to ensure observer solutions with sufficiently small initial error exist at least during one flow-jump cycle of the system, and we study the propagation of infinitesimal estimation errors throughout this cycle. We show that their evolution through non-simultaneous jumps depends on two distinct saltation matrices, depending whether the observer jumps before or after the system: one is similar to the classical saltation matrix derived in the literature, while the other differs due to the mismatch of the derivative of the output around jump times. We illustrate on a linear toy example the importance of ensuring transversality and contraction with both saltation matrices to ensure observer convergence.