On Notions of Detectability and Observers for Hybrid Systems

Notions of observer and detectability are well established for continuous-time and discrete-time systems, and are known to be linked, since a system must be detectable to admit an observer. Unfortunately, defining such notions for a hybrid system is not straightforward because solutions do not share the same hybrid time domain. In this paper, we propose to define observers and detectability for hybrid systems, such that detectability is still necessary for the existence of an observer and such that standard definitions are recovered for continuous-time and discrete-time systems, when seen as special cases of hybrid systems. We rely on a recent definition of hybrid systems with hybrid inputs and use jump reparametrizations to define convergence and equality of outputs.