Comparing asynchronous l-complete approximations and quotient based abstractions

Abstract

This paper is concerned with a detailed comparison of two different abstraction techniques for the construction of finite state symbolic models for controller synthesis of hybrid systems. Namely, we compare quotient based abstractions (QBA), e.g., described in [11, Part II] with different realizations of strongest (asynchronous) l-complete approximations (SAlCA) from [4], [8]. Even though the idea behind their construction is very similar, we show that they are generally incomparable both in terms of behavioral inclusion and similarity relations. We therefore derive necessary and sufficient conditions for QBA to coincide with particular realizations of SAlCA. Depending on the original system, either QBA or SAlCA can be a tighter abstraction.

Computing the sampling performance of event-triggered control

“…4 The output map is the next sample alone. The following fact is a direct consequence of Theorems 6 and 7 from [13], and gives the desired simulation refinement properties: Proposition 3. Consider the system S from Eq.…”

“…The present work builds upon the bisimulation-like algorithm of [11], which can be seen as a modified 𝑙-complete abstraction [12,13], to compute the MAIST of PETC for linear time-invariant systems. The main insight is that computing the MAIST once there is a finite-state simulation (abstraction) of the PETC traffic is easy, as it reduces to finding the cycle of minimum average length of the associated weighted graph [14].…”

“…The transition relation is simply what is called in[13] the domino rule: a state associated with a sequence 𝑘 1 𝑘 2 ...𝑘 𝑙 must naturally lead to a state whose next first 𝑙 − 1 samples are 𝑘 2 𝑘 3 ...𝑘 𝑙 , because the system is deterministic, autonomous, and time-invariant. Hence, any state in X 𝑙 that starts with 𝑘 2 𝑘 3 ...𝑘 𝑙 is a possible successor of 𝑘 1 𝑘 2 ...𝑘 𝑙 .…”

Computing the sampling performance of event-triggered control

“…This relation can also be seen as a method to construct the strongest 𝑙-complete approximation [12] of the system (6). As a consequence of [13,Corollary 2], for autonomous deterministic systems, both methods lead to the same abstraction. Hence, here we can use both terms interchangeably, but for purposes of exposition we will use the term 𝑙-complete in what follows.…”

Refinements of behavioural abstractions for the supervisory control of hybrid systems

“…A technical comparison of abstractions by symbolic dynamics and behavioural abstractions is given inSchmuck et al (2015).…”