Observer Design for Hybrid Dynamical Systems with Approximately Known Jump Times
Résumé
This paper proposes a general framework for state estimation of plants modeled as hybrid dynamical systems with jumps occurring at (approximately) known times. A candidate observer is 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 observer design. In particular, a high-gain flow-based observer, with innovation during flow only, can be designed for plants with an average dwell-time when the flow dynamics are strongly differentially observable. On the other hand, when the jumps are persistent, 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. In the linear context, this reasoning leads us to a 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--, and cases where detectability only holds during flows, at jumps, or neither. We also 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 is semiglobally practically asymptotically stable over time intervals after such delays. The results are illustrated in examples and applications, including mechanical systems with impacts, spiking neurons, and switched systems.
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