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Summary: In this paper, we define solutions for hybrid systems with prespecified hybrid inputs. Unlike previous work where solutions and inputs are assumed to be defined on the same domain a priori, we consider the case where intervals of flow and jump times of the input are not necessarily synchronized with those of the state trajectory. This happens in particular when the input is the output of another hybrid system, for instance, in the context of observer design or reference tracking. The proposed approach relies on reparametrizing the jumps of the input in order to write it on a common domain. The solutions then consist of a pair made of the state trajectory and the reparametrized input. Our definition generalizes the notions of solutions of continuous‐time and discrete‐time systems with inputs. We provide an algorithm that automatically performs the construction of solutions for a given hybrid input. In the context of hybrid interconnections, we show how the solutions of the individual systems can be linked to the solutions of a closed‐loop system. Example illustrate the notions and the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper proposes a general framework for state estimation of systems 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 system 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 systems 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 context of linear maps, 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 system. 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. [ABSTRACT FROM AUTHOR]