Semiglobal High-Gain Hybrid Observer for a Class of Hybrid Dynamical Systems with Unknown Jump Times

We study the problem of observer design for hybrid dynamical systems in the challenging setting where the times at which jumps occur are unknown or not detected precisely. We remark that when the solutions of interest are known to remain in a compact set and admit a uniform dwell-time and when the flows are strongly differentially observable, a sufficiently fast high-gain observer can be designed to estimate the state during flow, but using the output of the system near its jump times is counterproductive. We thus propose to “disconnect” the high-gain observer when its estimate gets close to the jump set. More precisely, the proposed observer generates an estimate that, during flow, is obtained via the high-gain observer and, around jump times, is obtained by integrating forward the flow map of the system, until reaching the jump set. Under appropriate assumptions around the jump set of the system, we show that this observer guarantees local uniform asymptotic stability of an appropriately defined zero-error set. Then, we develop a method to turn any such local hybrid observer into a semiglobal hybrid observer. This observer operates sequentially by first employing a continuous-time high-gain observer, and then, after a finite amount of time, solely determined by the current estimate, employing the available local hybrid observer. The capabilities and performance of the proposed hybrid observer are illustrated on a hybrid spiking neuron model.