Analysis of Zeno behaviors in hybrid systems

Abstract

In this paper, we investigate conditions for existence of Zeno behaviors in hybrid systems. These are behaviors that occur in a hybrid system when the system undergoes an unbounded number of discrete transitions in a finite and bounded length of time. Zeno behavior occurs, for example, when a controller unsuccessfully attempts to satisfy an invariance specification by switching the system among different configurations faster and faster. Two types of Zeno systems will be investigated: (1) strongly Zeno systems where all runs of the system are Zeno; and (2) (weakly) Zeno systems where only some runs of the system are Zeno. We derive necessary and sufficient conditions for both strong Zenoness and Zenoness, under certain assumptions. Our analysis is based on studying the trajectory set of a certain "equivalent" continuous-time system that is associated with the dynamic equations of the hybrid system. We also study the relation between the possibility of existence of Zeno behaviors in a system and the problem of existence of non-Zeno safety controllers that prevent the system from entering a suitably defines illegal region of its operating space. In particular, we show that if the system is Zeno (but not strongly Zeno), then a minimally-interventive safety controller may not exist, even if a safety controller does exist, disproving a conjecture made earlier in the literature. We also argue that any attempt of "regularizing" Zeno systems by forcing delays between successive configuration switches will not be fruitful.

Reasoning About Transfinite Sequences

“…However, Zeno behaviors are usually excluded from the modelling of real-time controllers, which is a reasonable requirement (see e.g. [10]), but also from the modelling of the physical systems, see some exception in [8,30]. This is a quite drastic limitation, since Zeno sequences are often acceptable behaviors for physical systems.…”

Discussion about Sliding Mode Algorithms, Zeno Phenomena and Observability

“…For instance, consider the bouncing ball example (see [2] and [20]). This is a switched system with jumps (actually an impact system [6]) that exhibits a Zeno phenomena.…”

“…It is important to note that S O characterizes the singularity observability considering only y andẏ. However, there exist several different observability definitions for nonlinear systems and hybrid systems [5], [11], [20], [22], [25], involving high order output derivatives with order greater than the dimension of the state space. Moreover, the system satisfies the observability matching condition [31] 8…”

Is there life after Zeno? Taking executions past the breaking (Zeno) point

“…[1]- [6], [10], [11] and [15] for more on Zeno behavior). We then discuss two important classes of Zeno executions: chattering Zeno executions and genuinely Zeno executions.…”