Formal Analysis of the Sampling Behaviour of Stochastic Event-Triggered Control

TL;DR

This work formally analyze the sampling behavior of stochastic linear periodic ETC (PETC) systems by computing bounds on associated metrics by constructing Interval Markov Chains equipped with suitable reward functions that abstract stochastic PETC's sampling behavior.

Abstract

Analyzing Event-Triggered Control's (ETC) sampling behaviour is of paramount importance, as it enables formal assessment of its sampling performance and prediction of its sampling patterns. In this work, we formally analyze the sampling behaviour of stochastic linear periodic ETC (PETC) systems by computing bounds on associated metrics. Specifically, we consider functions over sequences of state measurements and intersampling times that can be expressed as average, multiplicative or cumulative rewards, and introduce their expectations as metrics on PETC's sampling behaviour. We compute bounds on these expectations, by constructing appropriate Interval Markov Chains equipped with suitable reward structures, that abstract stochastic PETC's sampling behaviour. Our results are illustrated on a numerical example, for which we compute bounds on the expected average intersampling time and on the probability of triggering with the maximum possible intersampling time in a finite horizon.

Figures (6)

• Figure 1: A flowchart showing the steps followed to compute bounds on the expected rewards $\mathrm{E}_{\mathbb{P}^{y_0}_{\mathrm{Y}_N}}[g_{\star,N}(\omega)]$.
• Figure 2: The interplay between sets $\mathcal{R}$, $\mathcal{S}$, $\overline{\Phi}(x)\cap \mathcal{S}$ and $\mathcal{S} \setminus \bigcup_{x\in\mathcal{R}}\Phi(x)$.
• Figure 3: The blue and red lines are the computed lower and upper bounds, respectively, on the expected multiplicative reward from Example 3 in Section \ref{['sec:sampling_behaviour']} starting from any initial condition $x_0\in \mathcal{R}_i$ (initial intersampling time is assumed $s_0=0$), for all regions $\mathcal{R}_i\subset[-1.2,1.2]^2$ in the partition. The yellow (middle) line is the statistical estimate of the expected reward for a random initial condition from each region.
• Figure 4: Surface plot of the obtained lower and upper bounds on the expected multiplicative reward from Example 3 in Section \ref{['sec:sampling_behaviour']} for all regions $\mathcal{R}_i\subset[-2,2]^2$ in the partition (x-axis). The surface on the bottom is the lower bound, the surface at the top is the upper bound, and the one in the middle is the statistical estimate of the expected reward for a random initial condition from each region, as obtained from simulations.
• Figure 5: The blue and red lines are lower and upper bounds, respectively, on the bounded-until probability \ref{['eq:example_until']} starting from any initial condition $x_0\in \mathcal{R}_i$ (initial intersampling time is assumed $s_0=0$), for all regions $\mathcal{R}_i\subset[-1.2,1.2]^2$. The yellow line (the one in the middle) is the statistical estimate of the probability for a random initial condition from each region.

Theorems & Definitions (29)

• Definition 2.1: Interval Markov Chain (IMC)
• Definition 2.2: Adversary
• Remark 1
• Definition 3.1: Sampling Behaviour
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Theorem 4.1
• proof : Proof Sketch