We present ETCetera, a Python library developed for the analysis and synthesis of the sampling behaviour of event triggered control (ETC) systems. In particular, the tool constructs abstractions of the sampling behaviour of given ETC systems, in the form of timed automata (TA) or finite-state transition systems (FSTSs). When the abstraction is an FSTS, ETCetera provides diverse manipulation tools for analysis of ETC's sampling performance, synthesis of communication traffic schedulers (when networks shared by multiple ETC loops are considered), and optimization of sampling strategies. Additionally, the TA models may be exported to UPPAAL for analysis and synthesis of schedulers. Several examples of the tool's application for analysis and synthesis problems with different types of dynamics and event-triggered implementations are provided. @en

A fundamental challenge in networked control systems is reducing the amount of communications of each system in the network, so that bandwidth and energy are used efficiently. To address the challenge, the research community has shifted its focus to Event-Triggered Control (ETC), in which communication between the control system's different components takes place only when a state-dependent condition is satisfied. However, although ETC indeed often reduces communication, its communication times (or sampling times) are unknown beforehand and predictions thereof require intricate mathematical analysis on the system's (perturbed) dynamics. Nonetheless, predicting ETC's sampling is of paramount importance, as it enables: • Self-Triggered Control (STC), which is a more economic implementation of ETC. In STC the controller, at each sampling time, decides the next sampling time, by employing 1-step predictions of ETC's sampling; given a state measurement it predicts ETC's next sampling time. • Traffic scheduling, which is planning bandwidth allocation to each entity using the network and requires multi-step or infinite-step predictions of ETC's communication times. Without scheduling, many systems may access the network at the same time, resulting into network overflow and hindering the systems' stability. • Formal assessment of an ETC-design's performance in terms of sampling and control, e.g. by computing associated long-term metrics such as the expected average intersampling time, which again requires multi-/infinite-step predictions of ETC's sampling. This dissertation studies ETC's sampling behaviour and derives predictions thereof in all three aforementioned contexts. First, we propose a novel STC scheme, termed region-based STC, for nonlinear systems with bounded disturbances and uncertainties. The system's state-space is partitioned into a finite number of regions, and to each region a uniform STC intersampling time is assigned. To decide the next sampling time, at each sampling time the controller simply checks to which region the measured state belongs. To derive the partition and corresponding intersampling times, we use approximations of so-called isochronous manifolds. To derive the approximations, we address theoretical issues of prior works and propose a computational algorithm, and, to account for disturbances/uncertainties, we employ differential inclusions. Regarding traffic scheduling, our work follows the recently proposed abstraction-based approach. The sampling behaviour of a given ETC system is modeled by a finite-state system (the abstraction), offering an infinite-horizon prediction on ETC's sampling. In this work, we construct abstractions of (perturbed) nonlinear ETC systems. The system's state-space is partitioned into finitely many regions, representing the abstraction's states. For each region, a timing interval is determined, containing all intersampling times corresponding to states in the region. These intervals serve as the abstraction's output. Finally, the abstraction's transitions, given a starting region, indicate where the system's trajectories end up after an elapsed intersampling time. To determine the timing intervals and the transitions, we propose algorithms based on reachability analysis. Regarding state-space partitioning, we propose a partition similar to that of region-based STC, aiming at providing control over the timing intervals and improving their tightness. Finally, on the formal-assessment front, 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 (IMCs) equipped with suitable reward structures, that abstract stochastic PETC's sampling behaviour, and employing value iteration over these IMCs.@en

In this work, we derive a region-based self-triggered control (STC) scheme for nonlinear systems with bounded disturbances and model uncertainties. The proposed STC scheme is able to guarantee different performance specifications (e.g. stability, boundedness, etc.), depending on the event-triggered control (ETC) triggering function that is chosen to be emulated. To deal with disturbances and uncertainties, we employ differential inclusions (DIs). By introducing ETC/STC notions in the context of DIs, we extend well-known results on ETC/STC to perturbed uncertain systems. Given these results, and adapting tools from our previous work, we derive inner-approximations of isochronous manifolds of perturbed uncertain ETC systems. These approximations dictate a partition of the state-space into regions, each of which is associated to a uniform inter-sampling time. At each sampling time instant, the controller checks to which region the measured state belongs and correspondingly decides the next sampling instant.@en

Interval Markov Decision Processes (IMDPs) are finite-state uncertain Markov models, where the transition probabilities belong to intervals. Recently, there has been a surge of research on employing IMDPs as abstractions of stochastic systems for control synthesis. However, due to the absence of algorithms for synthesis over IMDPs with continuous action-spaces, the action-space is assumed discrete a-priori, which is a restrictive assumption for many applications. Motivated by this, we introduce continuous-action IMDPs (caIMDPs), where the bounds on transition probabilities are functions of the action variables, and study value iteration for maximizing expected cumulative rewards. Specifically, we decompose the max-min problem associated to value iteration to |Q| max problems, where |Q| is the number of states of the caIMDP. Then, exploiting the simple form of these max problems, we identify cases where value iteration over caIMDPs can be solved efficiently (e.g., with linear or convex programming). We also gain other interesting insights: e.g., in certain cases where the action set A is a polytope, synthesis over a discrete-action IMDP, where the actions are the vertices of A, is sufficient for optimality. We demonstrate our results on a numerical example. Finally, we include a short discussion on employing caIMDPs as abstractions for control synthesis.@en

Scheduling communication traffic in networks of event-triggered control (ETC) systems is challenging, as their sampling times are unknown, hindering application of ETC in networks. In previous work, finite-state abstractions were created, capturing the sampling behavior of linear time-invariant (LTI) ETC systems with quadratic triggering functions. Offering an infinite-horizon look to ETC systems' sampling patterns, such abstractions can be used for scheduling of ETC traffic. Here, we significantly extend this framework, by abstracting perturbed uncertain nonlinear ETC systems with general triggering functions. To construct an ETC system's abstraction: 1) the state space is partitioned into regions; 2) for each region, an interval is determined, containing all intersampling times of points in the region; and 3) the abstraction's transitions are determined through reachability analysis. To determine intervals and transitions, we devise algorithms based on reachability analysis. For partitioning, we propose an approach based on isochronous manifolds, resulting into tighter intervals and providing control over them, thus containing the abstraction's nondeterminism. Simulations showcase our developments.@en

In this article, we propose a region-based self-triggered control (STC) scheme for nonlinear systems. The state space is partitioned into a finite number of regions, each of which is associated to a uniform interevent time. The controller, at each sampling time instant, checks to which region does the current state belong, and correspondingly decides the next sampling time instant. To derive the regions along with their corresponding interevent times, we use approximations of isochronous manifolds, a notion first introduced in Anta and Tabuada (2012). This article addresses some theoretical issues of Anta and Tabuada (2012) and proposes an effective computational approach that generates approximations of isochronous manifolds, thus enabling the region-based STC scheme. The efficiency of both our theoretical results and the proposed algorithm is demonstrated through simulation examples.@en

In this letter, we consider the predecessor-following control problem for a platoon of car-like vehicles moving on a planar surface with cyclic obstacles. Each vehicle is equipped with an on-board camera that detects its preceding vehicle, and a laser scanner that detects the obstacles around it. Within this framework, we design a fully decentralized control scheme, in the sense that each vehicle calculates its own control signal incorporating only local information, acquired by its on-board camera and laser scanner. Collisions with obstacles, collisions between successive vehicles and connectivity breaks owing to the limited field of view of the camera as well as to visual occlusions raised by static obstacles are provably avoided. Moreover, the transient and steady-state response of the closed-loop system is a priori determined by certain designer-specified functions and is fully decoupled by the number of vehicles in the platoon and the control gains selection. Finally, a simulation study is carried out inMATLAB and Coppelia Robotics V-REP to prove the control protocol's efficiency.@en

In this letter, we consider the predecessor-following control problem for a platoon of car-like vehicles moving on a planar surface with cyclic obstacles. Each vehicle is equipped with an on-board camera that detects its preceding vehicle, and a laser scanner that detects the obstacles around it. Within this framework, we design a fully decentralized control scheme, in the sense that each vehicle calculates its own control signal incorporating only local information, acquired by its on-board camera and laser scanner. Collisions with obstacles, collisions between successive vehicles and connectivity breaks owing to the limited field of view of the camera as well as to visual occlusions raised by static obstacles are provably avoided. Moreover, the transient and steady-state response of the closed-loop system is a priori determined by certain designer-specified functions and is fully decoupled by the number of vehicles in the platoon and the control gains selection. Finally, a simulation study is carried out inMATLAB and Coppelia Robotics V-REP to prove the control protocol's efficiency.@en

Recently, there have been efforts towards understanding the sampling behaviour of event-triggered control (ETC), for obtaining metrics on its sampling performance and predicting its sampling patterns. Finite-state abstractions, capturing the sampling behaviour of ETC systems, have proven promising in this respect. So far, such abstractions have been constructed for non-stochastic systems. Here, inspired by this framework, we abstract the sampling behaviour of stochastic narrow-sense linear periodic ETC (PETC) systems via Interval Markov Chains (IMCs). Particularly, we define functions over sequences of state-measurements and interevent times that can be expressed as discounted cumulative sums of rewards, and compute bounds on their expected values by constructing appropriate IMCs and equipping them with suitable rewards. Finally, we argue that our results are extendable to more general forms of functions, thus providing a generic framework to define and study various ETC sampling indicators.@en

In previous work, linear time-invariant eventtriggered control (ETC) systems were abstracted to finite-state systems that capture the original systems’ sampling behaviour. It was shown that these abstractions can be employed for scheduling of communication traffic in networks of ETC loops. In this paper, we extend this framework to the class of nonlinear homogeneous systems, however adopting a different approach in a number of steps. Finally, we discuss how the proposed methodology could be extended to general nonlinear systems@en

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.@en

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.@en