Towards Traffic Bisimulation of Linear Periodic Event-Triggered Controllers

Abstract

We provide a method to construct finite abstractions exactly bisimilar to linear systems under a modified periodic event-triggered control (PETC), when considering as output the inter-event times they generate. Assuming that the initial state lies on a known compact set, these finite-state models can exactly predict all sequences of sampling times until a specified Lyapunov sublevel set is reached. Based on these results, we provide a way to build tight models simulating the traffic of conventional PETC. These models allow computing tight bounds of the PETC average frequency and global exponential stability (GES) decay rate. Our results are demonstrated through a numerical case study.

Computing the sampling performance of event-triggered control

“…6 provides means to compute a lower bound (or the actual value) for the MAIST. This is promising, because works in [7,11] provides methods to find simulations of a PETC traffic. However, on the one hand, a simulation alone does not provide how conservative the lower bound may be; on the other hand, a finite-state bisimulation of an infinite system is often impossible to be obtained.…”

“…For this we need to be able to (i) build a finite-state simulation of the system; (ii) check if its minimum mean cycle exists in the actual system; if not, (iii) refine the simulation until the cycle breaks; and (iv) repeat the process. This method is essentially the same as the bisimulation algorithm from a quotient model, presented in [10], which was used for PETC in [11], but with a different stopping criterion. Therefore, let us recover the simulation relation in [11], with a simplification that suits our purpose:…”

“…These works are concerned with short-term prediction of inter-sample times in order to develop a scheduler that can, e.g., request sensor data before events are triggered; they do not capture long-term properties of the sampling behavior of ETC like the MAIST, which, we argue, provide a more definitive information about the sampling performance. Still in the same category, [11] has recently given a step towards understanding longer-term traffic patterns of PETC, by proposing the usage of a bisimulation-like algorithm which determines the 𝑚 next inter-sample times from a given state. This allows a very conservative estimate of the MAIST by taking the minimum average of all such 𝑚-length sequences.…”

“…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].…”

Visualisation of Control Software for Cyber-Physical Systems

“…Many papers (e.g., see [10,11,[50][51][52][53][54]) tackling the problem of synthesizing control software (which looks tato quantized states) or control laws (which look at real states) of hybrid systems show pictures of the type we generate in this paper (with r = 1, i.e., only one bit for the actions). However, to the best of our knowledge there are no papers directly focusing on the method to generate such pictures, thus no automatic approach to controllers visualization is described.…”