Discrete and continuous structural properties for observability∗

Hybrid systems are powerful abstractions for modeling complex systems so that they have been used in a number of applications to provide models better reflecting the nature of control problems such as the ones related to embedded system design where discrete controls are routinely applied to continuous processes. Their theoretical properties are still the subject of intense research: in particular, observability is one of the fundamental system properties that form the foundations of control. The notion of observability is non trivial and attention must be paid to the implications of the definitions used. In traditional continuous and discrete–time systems, the concept of observability has been studied extensively. Among the seminal papers on the topic, Sontag in [14] introduces a set of observability–related definitions and examines the implications among the various concepts of observability. For hybrid systems, the notion of observability regards both the discrete and the continuous components of the hybrid state. Hence, the verification of the observability property is very involved due to the complex interactions between the discrete and continuous behaviors exhibited by hybrid systems. In particular, in this work we investigate generic final-state asymptotic determinability for hybrid systems (see [14]). Informally, a hybrid system is Generic Final-State Asymptotically Determinable (GFSAD) if any generic input/output experiment permits the identification of the discrete state, after a finite number of transitions, and the asymptotic determination of the continuous state. We consider living hybrid systems with no-multiple transitions, defined as those which admit only executions that are non-Zeno (see [9]) and have an infinite number of transitions interleaved by continuous evolutions. For example, viable hybrid systems with 0-lag nonblocking control strategies and initial set equal to the viability kernel, are living hybrid