Observer Design for Hybrid Dynamical Systems with Approximately Known Jump Times
Résumé
This paper proposes a general framework for the state estimation of plants modeled as hybrid dynamical systems with discrete events (or jumps) occurring at (approximately) known times. A candidate observer consists of a hybrid dynamical system with jumps triggered when the plant jumps. With some information about the time elapsed between successive jumps, a Lyapunov-based analysis allows us to derive sufficient conditions for the design of the observer that renders the zero-estimation set uniformly asymptotically stable. In particular, we show that a high-gain flow-based observer, with innovation during flow only, can be designed when the flow dynamics are strongly differentially observable. On the other hand, a jump-based observer, with innovation at jumps only, should be designed based on an equivalent discrete-time system corresponding to the hybrid system discretized at jump times, and presenting the observability of the combination of both flows and jumps. In the linear context, this reasoning leads us to a constructive hybrid Kalman filter. These designs apply to a large class of hybrid systems, including cases where the time between successive jumps is unbounded or tends to zero -- namely, Zeno behavior--, as well as cases where detectability only holds during flows, at jumps, or neither. Building from these sufficient conditions, we study the robustness of this approach when the jumps of the observer are delayed with respect to those of the plant. Under some regularity and dwell-time conditions, we show that the estimation error remains bounded and satisfies a semi-global practical asymptotic stability-like property. The results are illustrated in several examples and applications, including mechanical systems with impacts, spiking neurons, and switched systems.
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