# Guaranteed Self-Triggered Control of Disturbed Systems: A Set Invariance Approach

Florian Pouthier, Sylvain Durand, Nicolas Marchand, Jonathan Dumon, Abdoullah Ndoye, Amaury Nègre, Pierre Susbielle, Jose Castillo-Zamora, J. Fermi Guerrero Castellanos, Franck Ruffier

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## To cite this version:

Florian Pouthier, Sylvain Durand, Nicolas Marchand, Jonathan Dumon, Abdoullah Ndoye, et al. Guaranteed Self-Triggered Control of Disturbed Systems: A Set Invariance Approach. International Journal of Robust and Nonlinear Control, 2025, 35 (15), pp.6429-6443. 10.1002/rnc.8060. hal-05105420

HAL Id: hal-05105420

https://hal.science/hal-05105420v1

Submitted on 10 Jun 2025

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RESEARCH ARTICLE

# Guaranteed Self-Triggered Control of Disturbed Systems: A Set Invariance Approach

Florian Pouthier<fnref n="1" />,<fnref n="2" />

Sylvain Durand<fnref n="1" />

Nicolas Marchand<fnref n="2" />

Jonathan Dumon<fnref n="2" />

Abdoullah Ndoye<fnref n="2" />

Amaury Negre<fnref n="2" />

Pierre Susbielle<fnref n="2" />

Jose J. Castillo-Zamora<fnref n="3" />

Fermi Guerrero-Castellanos<fnref n="4" />

Franck Ruffier<fnref n="3" />

**Correspondence**

Corresponding author Florian Pouthier. This is sample corresponding address.

Email: f.pouthier@unistra.fr

**Present address**

Univ. Strasbourg, CNRS, INSA Strasbourg, ICube, 67000 Strasbourg, France

**Abstract**

This paper introduces a novel self-triggering strategy designed to ensure the control of discrete-time linear systems with guaranteed stability, even in the presence of disturbances and uncertainties. This strategy aims to consistently maintain satisfaction of state constraints while accounting for the uncertainties in the system through a set-membership description. The self-triggering framework primarily relies on reachable and invariant sets. Reachable sets quantify the maximum deviation of the disturbed system from the predicted behavior, while an invariant set establishes triggering bounds for these reachable sets. This control method is intended to minimize the number of measurements required, thereby avoiding network bandwidth saturation. To validate the effectiveness of the proposed strategy, experiments are conducted on an air extractor system, demonstrating a reduction in the number of measurement samples while ensuring stability and satisfying system state constraints.

**KEYWORDS**

Self-triggered control, sparse measurements, robust positively invariance, linear systems.

# 1 | INTRODUCTION

## 1.1 | Fundamental Motivation

Networked Control Systems (NCSs) are systems with numerous constraints connected to a network to execute tasks remotely<fnref n="1" />. In these multi-constraint systems, overuse of sensors can be damaging, mainly for two reasons. The first reason is that sensor data transits through a network that could be shared by numerous agents. Overloading this network can quickly lead to bandwidth saturation, resulting in the loss or drop of data packets. The second reason is that some sensors may be internally energy-consuming (high-power electronics) or brittle (sensitive mechanics). In these cases, limiting the number of measurements becomes crucial, especially for applications where safety is required.

It is then of great interest to make specific measurements on the system in a parsimonious way, by choosing the most relevant measurement instants. These challenges have prompted a departure from the conventional periodic sampling scheme in system actuation and sensing. Specifically, this has given rise to the research field of *event-driven* systems, where sampling occurs not periodically but in response to significant *events*<fnref n="2" />. In this framework, the system is only controlled on the basis of information from the last sampling instant. This requires the design of a triggering criterion to decide whether or not a new system measurement is necessary.

<footnote n="1">¹Univ. Strasbourg, CNRS, INSA Strasbourg, ICube, 67000 Strasbourg, France</footnote>

<footnote n="2">²Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France</footnote>

<footnote n="3">³Univ. Aix Marseille, CNRS, ISM, 13009 Marseille, France</footnote>

<footnote n="4">⁴Facultad de Ciencias de la Electrónica, Benemérita Universidad Autónoma de Puebla, Puebla, 72570, Puebla, Mexico</footnote>
However, one of the challenges of the event-driven paradigm is to account for disturbances that may arise in the controlled system during open-loop phases, i.e., between two successive events. Robustly integrating disturbances during such phases can be accomplished using *set-theoretical* tools³. In contrast to stochastic tools, set-theoretical tools have the advantage of providing guaranteed results, such as constraint satisfaction or safety guarantees.

The issue of controlling disturbed systems while minimizing sensor measurements thus underscores the importance of considering both event-driven and set-theoretical methods. In the present work, these methods are combined with set invariance properties to design a new kind of triggering criterion guaranteeing system safety. This will ensure the control system remains in a safe zone while reducing the frequency of sensor measurements.

## 1.2 | Related Work

Early research has focused on highlighting the importance of event-based sampling techniques. It was initially demonstrated that Event-Triggered Control (ETC) could result in closed-loop performance equal to or better than classical periodic control⁴,⁵. ETC continuously monitors system output measurements and then decides to update the control value (i.e., close the loop) when a *relevant event* occurs. In the literature, the concept of a *relevant event* initially relied on a threshold crossing of the output measurement (referred to as the *Send-on-Delta* technique)⁶. Similarly, it was extended to the threshold crossing of a Lyapunov function instead of the output⁷. To enable formal stability proofs, the next generation of ETC considers an event function typically constructed using the Control Lyapunov Function of the system⁸. This approach has been applied to a wide range of physical linear and nonlinear systems⁹,¹⁰,¹¹,¹². An extension to Partial Differential Equations (PDE) governed systems has also been proposed¹³,¹⁴, as well as application of ETC to other control methods like backstepping approaches¹⁵. Discrete-time versions of ETC (known as Periodic ETC or PETC) have also been extensively investigated for periodically sampled systems (see¹⁶ and references therein).

To minimize unnecessary computations further, the concept of using trajectory prediction was explored through the introduction of Self-Triggered Control (STC)¹⁷, and its counterpart in a periodically sampled context, Periodic STC (PSTC)¹⁸. Instead of continuously monitoring the system for relevant events, STC and PSTC *predict the timing of future events* based on the last measurement and the system's dynamic model knowledge. In essence, *event-triggering* involves measurement triggering at the same instant, while *self-triggering* involves measurement triggering at a future instant¹⁹. STC have been efficiently applied to many control applications, such as control of multiagent systems²⁰,²¹. The challenge in STC and PSTC lies in accurately predicting the future triggering instants, especially when the system is disturbed. Before delving into the works on STC and PSTC for disturbed systems, let's first discuss the contributions in the literature of set-theoretic control for continuous-time and discrete-time disturbed systems.

Without knowledge of the disturbance, the problem cannot be solved. Typically, assumptions are imposed on the disturbances, primarily focusing on their stochastic characteristics or bounds. In cases of deterministic disturbance bounds, many contributions build upon the ideas initiated in the seminal work²² on disturbed linear systems. Such development proves particularly beneficial in providing *guaranteed* results in control strategy²³. Deterministic methods are commonly referred to as *set-membership* methods in the state-of-the-art. Historically, significant efforts have been dedicated to the set-membership estimation of disturbed dynamical systems²⁴. Subsequent advancements in set-membership estimation have predominantly focused on state estimation for nonlinear systems²⁵,²⁶, or more recently, in the context of robotic navigation, as demonstrated in²⁷. All these works are linked to *reachability analysis*, which, in this context, yields *reachable sets*. These sets encompass all reachable states of a disturbed system at a given time, considering all feasible (yet bounded) disturbances occurring in the system. Reachable sets are described using various mathematical objects such as ellipsoids²⁸, zonotopes²⁹, or polytopes³⁰, which are computationally convenient for integration over time.

When analyzing the reachability of disturbed control systems, there is often interest in identifying a limit set wherein, for any bounded disturbance input, the system remains contained. This definition aligns with that of an *invariant set*. Invariant sets possess interesting properties for robust control of systems and have been previously employed for such purposes³¹. The theory of set invariance has been extensively explored in related research, finding applications in robust model predictive control³², fault-tolerant control³³, adaptive fuzzy control³⁴, and various other domains. Additionally, when the system's state is constrained within a defined set, it may be advantageous to consider a maximum invariant set³⁵,³⁶. Numerous studies have concentrated on designing methods to compute invariant sets as efficiently as possible³⁷,³⁸.

Let's revisit the ETC and STC methods, specifically focusing on their robust versions applied to disturbed systems. The emphasis here lies on research proposing PETC and PSTC based on the set-theoretical tools previously discussed. Invariant sets
have already been incorporated into robust PETC, as seen in³⁹. This study introduces a *to-zero controller* triggered within an invariant set combined with a custom event set. Moreover, invariant sets have been leveraged to implement robust self-triggered MPC⁴⁰. Set invariance is typically integrated into the PSTC literature alongside reachability, where reachable sets – such as ellipsoids⁴¹ or polytopes⁴²,⁴³ – are considered. In these works, reachable sets are utilized within the STC framework to trigger system measurements. An algorithm periodically verifies if a point of the reachable set violates a specific triggering condition, typically a set-membership condition ensuring that reachable sets remain within a calculated (invariant) stability set. However, this stability set is recalculated at each periodic instant to define a new triggering limit. Consequently, these latest methods are computationally demanding and necessitate significant storage space.

## 1.3 | Contribution of this Work

The PSTC proposed in this study precisely measures the system state only at predicted event instants, using the boundaries of an invariant set as triggering thresholds. Reachable sets are utilized to represent the worst-case scenarios of the disturbed system, while the invariant set serves as a boundary that these reachable sets must not exceed. All sets considered in this paper are polytopes, allowing for a balance between computational complexity and conservatism. Unlike existing literature⁴¹,⁴²,⁴³, the invariant set employed in this study for triggering remains unchanged between two instants, thereby saving storage space and reducing computational complexity. The triggering criterion therefore boils down to checking the inclusion of one polytope in another (a simple task if the polytopes are in the right representations at all times). The PSTC strategy ensures stability and satisfaction of state constraints despite system disturbances. Additionally, the control signal in the proposed scheme no longer remains constant between triggers (as seen in a sparse controller³⁹), but follows a *model-based* control profile.

Although event-driven control based on set-theoretical methods has already been studied in previous works⁴⁴,⁴⁵,⁴⁶, the proposed strategy does not involve self-triggering of the controller but rather of the sensor measurement. The coupling of these methods with set invariance properties has also been explored in the past⁴⁷,⁴², but this work aims to consider a purely geometric approach (based on set inclusion) for the triggering criterion of a new measurement.

This paper aims to provide the fundamental guidelines for implementing the proposed strategy in real-time control systems. When dealing with a linear control system, the reader can adopt the proposed PSTC strategy through a systematic procedure outlined in the following sections. Additionally, an algorithm is presented as a reference to facilitate the implementation of the strategy on an embedded system with constraints.

This paper ultimately aims to showcase the benefits of the proposed strategy through its application in a real-world scenario: an air extractor system. The air extractor serves as a pertinent benchmark for implementing the PSTC strategy, given its exposure to various disturbances. The achieved results provide assurance of the PSTC strategy's effectiveness. The extractor air system is controlled based on self-triggered measurements, while satisfying rotation speed constraints despite disturbances. The PSTC strategy makes it possible to interrupt position sensor measurement more often, and thus to limit communications on the network to the most relevant information.

## 1.4 | Paper Structure

The remainder of the paper is structured as follows: Section 2 provides some preliminary set-theoretical tools helpful in developing the proposed strategy, while formally stating the control problem to be solved. The proposed self-triggering strategy that solves the problem is revealed and developed in Section 3. This section also synthesizes the strategy in order to reproduce it on any system by providing a procedure, an algorithm, and an academic example. Then, Section 4 validates the proposed strategy by applying it to a real system, namely an air extractor system. Conclusions and future research trajectories are discussed in Section 5.

## 1.5 | Basic Nomenclature and Definitions

In this paper, $\mathbb{N}$ will stand for the set of *natural integers*, with its subsets $\mathbb{N}^* = \mathbb{N} \setminus \{0\}$ and $\mathbb{N}_k = \{0, 1, \dots, k\}$. The set of *real numbers* is denoted $\mathbb{R}$. Given two sets $A \subseteq \mathbb{R}^n$ and $B \subseteq \mathbb{R}^n$, the *Minkowski sum* is defined as $A \oplus B = \{a + b | a \in A, b \in B\}$.
Conversely, the Pontryagin difference of two sets is defined as $A \ominus B = \{a \in A | \{a\} \oplus B \subseteq A\}$. A polyhedron in $\mathbb{R}^n$ is the intersection of a finite number of half-spaces. A polytope in $\mathbb{R}^n$ is a compact polyhedron.

# 2 | PRELIMINARIES & PROBLEM FORMULATION

## 2.1 | System Description

The theory developed hereafter is intended to be applied to any discrete-time linear time-invariant system of the form

$$ \xi_{k+1} = A\xi_k + Bu_k + Ew_k \quad (1) $$

where $\xi_k \in \mathbb{R}^n$ is the system state, $u_k \in \mathbb{R}^m$ is the control input, and $w_k \in \mathbb{R}^d$ is an unknown but bounded disturbance, which all occur at the current time instant $k \in \mathbb{N}$. $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$ and $E \in \mathbb{R}^{n \times d}$ are respectively the state, control and disturbance matrices. System (1) will be referenced as the *disturbed system*. It is assumed that the system variables are subject to given constraints, and therefore

$$ (\xi, u, w) \in (X, U, W) \quad (2) $$

with $X \subset \mathbb{R}^n$, $U \subset \mathbb{R}^m$ and $W \subset \mathbb{R}^d$. The disturbance-free evolution of the system (1), can also be defined as

$$ \tilde{\xi}_{k+1} = A\tilde{\xi}_k + Bu_k \quad (3) $$

where $\tilde{\xi}_k \in \mathbb{R}^n$ is the predicted evolution of the system state without input disturbances. The above system is the so-called *predicted system*, and the evolution of its state is deterministic.

**Assumption 1.** The matrix pair $(A, B)$ is fully known and controllable.

**Assumption 2.** Although the disturbance at each instant is unknown, its bounds, defined set-wise by $W$, are known.

**Assumption 3.** The subsets $X$, $U$ and $W$ are expected to be compact, convex polytopes.

Assumption 1 is necessary to implement the model-based controller proposed in equation (10). As explained above, it is essential to verify Assumption 2 to be able to deploy the self-triggered control strategy. The open-loop integration of bounded admissible disturbances ensures that measurement is triggered at the right instant, thus guaranteeing stability and compliance with system constraints. Given Assumption 3, the state, control, and disturbance are constrained in a particular type of set, namely polytopes. Such a set representation helps synthesize and control constrained systems. In fact, system constraints will define subsets of the state space, which will generally be treated using set-theoretic methods³. The set theory also provides useful tools to deal with disturbances, like invariant sets. The remainder of this section is a theoretical review of set structures and operations that will be used in the sequel.

**Remark 1.** The exposed strategy is aimed at stabilizing the system (1) around an equilibrium point $\{\xi_{eq}, u_{eq}, w_{eq}\}$. However, for the sake of simplicity but without loss of generality, it will be considered in this section and Section 3 that the equilibrium point is the state-space origin, so that $\{\xi_{eq}, u_{eq}, w_{eq}\} = \{0, 0, 0\}$, where 0 is a null vector of appropriate length. All the following results can be derived trivially for a non-zero equilibrium point. Such an adaptation could be done by substituting the open-loop system (1) and the system constraints (2) respectively by the open-loop system of errors

$$ \tilde{\xi}_{k+1} = A\tilde{\xi}_k + Bu_k + E\tilde{w}_k \quad (4) $$

and the system constraints

$$ (\tilde{\xi}, \tilde{u}, \tilde{w}) \in (\tilde{X}, \tilde{U}, \tilde{W}) \quad (5) $$

where $\tilde{\xi} = \xi - \xi_{eq}$, $\tilde{u} = u - u_{eq}$, $\tilde{w} = w - w_{eq}$, $\tilde{X} = X \ominus \{\xi_{eq}\}$, $\tilde{U} = U \ominus \{u_{eq}\}$, and $\tilde{W} = W \ominus \{w_{eq}\}$. All the following results will be valid only if the equilibrium point belongs to the sets $\tilde{X}$, $\tilde{U}$ and $\tilde{W}$. This adaptation will be useful in Section 4 for the stabilization of the air extractor system around a non-zero equilibrium point.

**Assumption 4.** The equilibrium point is included in the system constraints, i.e. $\xi_{eq} \in \tilde{X}$, $u_{eq} \in \tilde{U}$ and $w_{eq} \in W$. In Sections 2 and 3, it is assumed that $0 \in X$, $0 \in U$ and $0 \in W$.
## 2.2 | Reachable Set Definitions

Reachability analysis aims to provide information characterizing the possible future states of a dynamic system, based on information from its past. To perform this task, a *reachable operator* integrates the dynamic equation of the disturbed system (1) over a given discrete horizon. In the case of a disturbed system, the reachable operator integrates the dynamic equation over sets. From an initial set, this operator produces the set of all *reachable* states while considering the set of all *feasible* disturbances.

**Definition 1 (1-step reachable set).** Given a non-empty set $S \subseteq \mathcal{X}$, and a control input $u \in \mathcal{U}$, the one-step reachable set associated to the system (1) and constraints (2) is defined by

$$ 
\begin{aligned}
\mathcal{R}(S, u) &= \{A\xi + Bu + Ew \mid \xi \in S, w \in W\} \\
&= \{A\xi + Bu \mid \xi \in S\} \oplus EW \\
&= AS \oplus Bu \oplus EW
\end{aligned}
$$

Moreover, $\mathcal{R}(\cdot)$ is referred to as the *one-step reachable operator*.

The Minkowski set sum ($\oplus$), introduced in Section 1.5, is the key operator that allows to take into account all the feasible disturbances that may occur on the disturbed system in one discrete-time step. Moreover, Definition 1 can be recursively extended to the notion of *k-step reachable set* defined as follows.

**Definition 2 (k-step reachable set).** Given a non-empty set $S \subseteq \mathcal{X}$, and a control input sequence $\{u_0, \dots, u_k\} \in \mathcal{U}^{k+1}$, the *k-step reachable set* associated to the system (1) and constraints (2) is defined by the recursion

$$ 
\begin{aligned}
\mathcal{R}_0(S, u_0) &= S \\
\mathcal{R}_{i+1}(S, u_{i+1}) &= \mathcal{R}(\mathcal{R}_i(S, u_i), u_{i+1}), \forall i \in \mathbb{N}^{k-1}
\end{aligned}
$$

Moreover, $\mathcal{R}_k(\cdot)$ is referred to as the *k-step reachable operator*.

It is trivially verified that the operator $\mathcal{R}_1(\cdot)$ is equivalent to the one defined by Definition 1. It could also be noted that the reachable operators defined above take an initial set $S$ as an input, as well as a given control input $u$ (or a sequence $\{u_i\}$), which become parameters of the operator. The next step will be to adapt the above definitions of reachable sets to the case of a system controlled with a self-triggered control law. This will be covered in Section 3.2.

## 2.3 | Invariant Set Definitions

The synthesis of robust control laws on dynamical systems can be performed using the set invariance theory$^{31}$. Indeed, the characterization of invariant sets allows the synthesis of controllers guaranteeing stability with respect to given state constraints. More particularly, a region of invariance for a closed-loop system is defined as a *Positively Invariant (PI) set*. That is, since the system state belongs to a PI set at a given time instant, then it is guaranteed to stay in this set for all future *positive instants*. This notion of PI can be further extended to closed-loop systems subject to bounded disturbances. In such a case, the set guaranteeing the invariance property is said to be a *Robust Positively Invariant (RPI) set*.

Considering a linear control law for system (1), on the form $u_k = -K\xi_k$, leads to the closed-loop system

$$ \xi_{k+1} = \underbrace{(A - BK)}_{A_{CL}} \xi_k + Ew_k \quad (6) $$

where the applied control law is *stabilizing* the system, that is $A_{CL}$ is Schur (i.e. all the eigenvalues of $A_{CL}$ are contained within the complex unit circle). This system is then subject to the closed-loop constraints

$$ (\xi, w) \in (\mathcal{X}_K, W) \quad (7) $$

where $\mathcal{X}_K = \{\xi \in \mathcal{X} | -K\xi \in \mathcal{U}\}$. The closed-loop system (6) with its constraints (7) is considered to define the RPI set.

**Definition 3 (Robust Positively Invariant set$^{31}$).** A set $\mathcal{Z} \subseteq \mathcal{X}_K$ is a *RPI set* for the closed-loop system (6) and constraints (7) if for all $\xi \in \mathcal{Z}$, $A_{CL}\xi + Ew \in \mathcal{Z}$, for any disturbance $w \in W$.
**Proposition 1 (Existence of the RPI set³¹).** A RPI set exists for the closed-loop system (6) and constraints (7) if and only if: (i) the $A_{CL}$ matrix is Schur, and (ii) the set $W$ satisfies Assumption 3.

The RPI set thus provides strong properties regarding the stability of a disturbed dynamical system. Whatever the amplitude of the bounded disturbance at each instant, a trajectory initiated inside the RPI set will end inside this set. Therefore the main idea of the authors is to use the boundaries of an RPI set as a triggering threshold for a new state measurement. Such a triggering policy will be addressed in Section 3.3.

## 2.4 | Set-membership Problem for Polytopes

This theoretical review on sets ends by addressing the problem of set-membership with polytopes. To better address this problem, some ways to represent the boundaries of polytopes are first recalled.

**Definition 4 (H-representation⁴⁸).** A polyhedron $\mathcal{P} \subseteq \mathbb{R}^n$, defined as the intersection of $p$ half-spaces, can be defined as $\mathcal{P} = \{\xi \in \mathbb{R}^n | H_p \xi \le b_p\}$, for some particular matrix $H_p \in \mathbb{R}^{p \times n}$ and particular vector $b_p \in \mathbb{R}^p$.

A polyhedron can be represented by an infinite number of H-representations. However, the minimal H-representation, which avoids redundant linear inequalities, is unique and will be denoted $\mathcal{H}(\mathcal{P})$. As particular polyhedra, polytopes necessarily have an H-representation, but as compact polyhedra, polytopes could also be represented by the so-called V-representation.

**Definition 5 (V-representation⁴⁸).** A polytope $\mathcal{P} \subset \mathbb{R}^n$ can be defined as the convex hull of a finite set of points.

Although an infinite number of V-representations of a polytope also exists, the minimal V-representation, by means of extreme points, is unique and is given by the set of the polytope's vertices, denoted $\mathcal{V}(\mathcal{P})$. Finally, the strict membership of one polytope to another could be checked. For instance, let's check whether a polytope $\mathcal{P}_1$ is fully included or not into a polytope $\mathcal{P}_2$. In this case, one may take the V-representation $\mathcal{V}(\mathcal{P}_1)$ of the former, the H-representation $\mathcal{H}(\mathcal{P}_2)$ of the latter, and recursively check for all vertices of $\mathcal{V}(\mathcal{P}_1)$ if they verify all the half-space equations in $\mathcal{H}(\mathcal{P}_2)$. This could be formally written as follows:

$$ 
\begin{aligned}
\forall \mathbf{v} \in \mathcal{V}(\mathcal{P}_1), \mathbf{v} \in \mathcal{P}_2 &\Rightarrow \mathcal{P}_1 \subseteq \mathcal{P}_2 \\
\exists \mathbf{v} \in \mathcal{V}(\mathcal{P}_1), \mathbf{v} \notin \mathcal{P}_2 &\Rightarrow \mathcal{P}_1 \not\subseteq \mathcal{P}_2
\end{aligned}
\qquad (8) $$

Set-membership criterion will be particularly helpful for the inclusion tests that will be performed in Section 3.3.

## 2.5 | Problem Statement

The set-theoretical tools presented above will be useful in addressing the main problem of this paper, which is stated below.

**Problem 1.** Let's consider the disturbed system (1) with a stabilizing periodic control law (6). Then, how to replace this periodic control with a PSTC strategy:

* reducing the amount of times a measurement is performed (i.e., instants when $\xi$ is accurately known);

* ensuring compliance with state constraints (2).

In the sequel, a strategy is designed to solve Problem 1 using the theory of reachable and invariant sets.

## 3 | SELF-TRIGGERED CONTROL STRATEGY DESIGN

The theoretical notions on sets introduced in Section 2 will now be used to build a self-triggered control strategy. As introduced in Section 2.5, our goal is to make the disturbed system (1) satisfy state constraints (2) under a self-triggered control law. In order to do this, the reachable sets will be used to characterize the reachable states of the open-loop disturbed system at a given time. The fundamental idea of this paper is to use the boundaries of the RPI set as a trigger for a new measurement. Some points of a reachable set lying outside the invariant set may result at the next instant in a violation of system constraints, or even lead to system instability. Closing the loop at this moment with a new instantaneous measurement ensures that the system remains stable and still verifies the state constraints. This section first introduces the self-triggered control law as the basis of the proposed
strategy. The control strategy is then formally stated by developing the calculations of the reachable sets in the case of the PSTC, as well as defining the triggering system on the edges of the RPI set. Stability and constraint satisfaction are finally verified through theorems.

## 3.1 Self-Triggered Control Law

The purpose of this paper is to provide a strategy measuring the current state of the system (1) only when needed. More precisely, the system evolves without knowledge of the accurate system state between two update instants. However, it is not desirable to feed the controller with a constant or null state between these instants, as proposed in³⁹. Therefore, the current idea is to feed the linear controller with the state of the predicted system (3) when no measurement is triggered. This remains to say that the periodic control applied to the system (1) is

$$ u_k = -K\bar{\xi}_k \qquad (9) $$

at all time instants. Furthermore, the evolution of the predicted system (3) follows a self-triggered framework which is given by

$$ \left\{ \begin{array}{ll} \bar{\xi}_{k+1} = A_{CL}\bar{\xi}_k & \text{if } \sigma_k = 0 \\ \xi_k = \xi_k^+ & \text{if } \sigma_k = 1 \end{array} \right. \qquad (10) $$

where $\sigma_k \in \{0, 1\}$ is the switching decision trigger which enables ($\sigma_k = 1$) or disables ($\sigma_k = 0$) the measurement of the current real state $\xi_k$. This measurement is denoted $\xi_k^+$ and corresponds to the new initial condition of the predicted system. The self-triggered control law will be referenced in the sequel as the pair (9, 10), where (9) is the linear control law regarding the predicted state and (10) is the evolution of the predicted state following the self-triggered framework. The next step is to characterize the criterion $\sigma$ in a way that state constraints satisfaction (2) is ensured at any time with the applied control (9, 10), but it is first required to compute reachable sets under such a self-triggered control law.

## 3.2 Reachable Sets Computation under Self-Triggered Control

The proposed STC approach relies on the fact that the disturbed system could deviate from the predicted system between two accurate state updates. Using the definitions of the disturbed system (1) and the predicted system (3), one can define the state error, denoted $\delta = \xi - \bar{\xi}$, between these two systems. The evolution of this error is consequently given by

$$ \delta_{k+1} = \xi_{k+1} - \bar{\xi}_{k+1} = A\delta_k + Ew_k, \quad w_k \in W \qquad (11) $$

*Remark 2.* Since the term $Bu_k$ appears both in system (1) and system (3), it cancels out in the state error, making the calculation of reachable sets independent of the applied control.

However, the STC law (9, 10) implies that the disturbed system can evolve freely during multiple time steps. It is therefore needed to compute a sequence of sets that gathers all possible deviations between the disturbed system (1) and the predicted system (3). Such a computation will be based on the definition of the state error and, more precisely, on the expression of its reachable operator. Indeed, applying Definition 1 to the state error dynamics (11) while considering the control (9, 10) leads to

$$ \begin{aligned} \mathcal{R}(S_k) &= \{A(\xi_k - \bar{\xi}_k) + Ew_k | \xi_k \in S_k, w_k \in W\} \\ &= A(S_k \ominus \{\bar{\xi}_k\}) \oplus EW \end{aligned} \qquad (12) $$

Definition 1 also implies that the succeeding state error $\delta_{k+1}$ belongs to the reachable set $\mathcal{R}(S_k)$. If we also assume that the succeeding real state $\xi_{k+1}$ belongs to a succeeding set denoted $S_{k+1}$, the reachable set $\mathcal{R}(S_k)$ could be rewritten in terms of succeeding states and set like

$$ \begin{aligned} \mathcal{R}(S_k) &= \{\delta_{k+1} = \xi_{k+1} - \bar{\xi}_{k+1} | \xi_{k+1} \in S_{k+1}\} \\ &= S_{k+1} \ominus \{\bar{\xi}_{k+1}\} \end{aligned} \qquad (13) $$
Considering both expressions (12) and (13), it is possible to provide a set-based construction of $S_{k+1}$, given by

$$S_{k+1} = A (S_k \ominus \{\bar{\xi}_k\}) \oplus EW \oplus \{\bar{\xi}_{k+1}\} \quad (14)$$

The last remaining construction step is to integrate the switching criterion $\sigma_k$. Indeed, the recurrent construction (14) is only valid while $\sigma_k = 0$. However, when $\sigma_k = 1$, the predicted system state is updated with the disturbed system state ($\bar{\xi}_k^+ = \xi_k$, given (10)) because of the measurement, implying $\delta_k = 0$ (so $S_k = \{\bar{\xi}_k\}$) and consequently, expression (14) becomes in this case

$$S_{k+1} = EW \oplus \{\bar{\xi}_{k+1}\} \quad (15)$$

Finally, Definition 2 makes it possible to define a recurrent calculation of the error reachable sets, assuming that the initial set corresponds to the initial value of the predicted system (i.e. $\sigma_0 = 1$ at the initial instant), and that the succeeding reachable set depends on the measurement triggering policy $\sigma_k$. Hence, for all $k \in \mathbb{N}$,

$$S_0 = \{\bar{\xi}_0\} \\ S_{k+1}(\sigma_k) = \begin{cases} A(S_k \ominus \{\bar{\xi}_k\}) \oplus EW \oplus \{\bar{\xi}_{k+1}\} & \text{if } \sigma_k = 0 \\ EW \oplus \{\bar{\xi}_{k+1}\} & \text{if } \sigma_k = 1 \end{cases} \quad (16)$$

A simple and visual example of the calculation of such sets is given in the following example. Let's consider the system (1) as a discrete-time double integrator with a unitary sampling period, i.e. considering the following disturbed system

$$\xi_{k+1} = \underbrace{\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}}_{A} \xi_k + \underbrace{\begin{bmatrix} 0.5 \\ 1 \end{bmatrix}}_{B} u_k + \underbrace{\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}}_{E} w_k \quad (17)$$

and the disturbance vector $w_k$ belonging to the set $W = [\pm 0.1 \times \pm 0.1]$. The system is linearly controlled, and a pole placement method is chosen for tuning the state-feedback gain $K$. The desired poles ($0.4 \pm 0.49i$) of the closed-loop system are within the unit circle and the proposed feedback then stabilizes the system. By using the Ackermann's formula, a state-feedback gain $K = [0.6 \ 0.9]$ is found. This state-feedback gain implies that the closed-loop dynamics are described by the matrix

$$A_{CL} = A - BK = \begin{bmatrix} 0.70 & 0.55 \\ -0.60 & 0.10 \end{bmatrix} \quad (18)$$

Figure 1 shows an example for the computation steps of $R(S_k)$, where $S_k$ is an arbitrary set centered at $\bar{\xi}_k = [1 \ 1]^T$, and $\sigma_k = 0$ (no measurement triggered).

It is now necessary to demonstrate certain properties of these reachable sets before pursuing the development of the strategy.

**Proposition 2 (Convexity of reachable sets).** The reachable sets $S_k$, given by the recurrence relation (16), are convex, for all $k \in \mathbb{N}$.

*Proof.* By recurrence. The initial reachable set $S_0 = \{\bar{\xi}_0\}$ is a point, which is by convention a convex set. Considering now that $S_k$ is convex, it is necessary to distinguish two cases depending on the value of $\sigma_k$. When $\sigma_k = 1$, $S_{k+1} = EW \oplus \{\bar{\xi}_{k+1}\}$ in (16), where $W$ is a convex (polytopic) set (from Assumption 3), scaled by the matrix $E$, which is then translated around the point $\bar{\xi}_{k+1}$. Such a construction results in a convex set. Finally, when $\sigma_k = 0$, $S_{k+1}$ is given in (16) by an affine transformation of $S_k$ (composed by, in that order, a translation, a linear function, a sum with a convex set, and a translation). Such a transformation has been demonstrated to preserve the convexity of sets (and the reader is referred to 49 §2.3, for deeper explanations). This ends the proof since convexity is verified for all reachable sets $S_k$, $k \in \mathbb{N}$. □

**Proposition 3 (Compactness of reachable sets).** The reachable sets $S_k$, given by the recurrence relation (16), are compact, for all $k \in \mathbb{N}$.

*Proof.* Formally analogous to the proof of Proposition 2, considering that a point is a compact set and the affine transformation between $S_k$ and $S_{k+1}$ preserves compactness. □

Propositions 2 and 3 let us establish an important property of the reachable sets: as convex and compact sets, built from a polytope $W$ (Assumption 3), the reachable sets $S_k$ are *polytopes*, for all $k \in \mathbb{N}$.
<figure><img src="image_3.png" /></figure>

**Step 1:** Let's consider an arbitrary set $S_k$ centered on a predicted state $\{\bar{\xi}_k\}$.

<figure><img src="image_4.png" /></figure>

**Step 2:** The initial reachable set $S_k$ is firstly centered at the origin by subtracting $\{\bar{\xi}_k\}$.

<figure><img src="image_5.png" /></figure>

**Step 3:** This set is afterward transformed at the origin by the state matrix $A$.

<figure><img src="image_6.png" /></figure>

**Step 4a:** The Minkowski sum is computed by virtually placing EW at each vertex...

<figure><img src="image_7.png" /></figure>

**Step 4b:** ... before taking the convex hull of the whole figure construction.

<figure><img src="image_8.png" /></figure>

**Step 5:** Finally, the reachable set $S_{k+1}$ is centered around the next predicted state $\{\bar{\xi}_{k+1}\}$.

<caption>FIGURE 1 Reachable set computation example in the 2-state space.</caption>

## 3.3 Self-Triggering on the RPI Boundaries

Once the reachable sets are computed for the self-triggered control law (9, 10), the adequate switching trigger criterion $\sigma_k \in \{0, 1\}$ at each instant $k$ still remains to be computed. For this purpose, it is necessary to compute a polytopic RPI set for the closed-loop system (6), denoted $Z_\infty$ in the sequel. The criterion $\sigma_k$ will then be based on the boundaries of $Z_\infty$.

The PSTC relies on the characterization of the switching decision trigger $\sigma_k \in \{0, 1\}$. For that, the next reachable set is computed at each instant $k$ as if there was not going to be any measurement at that moment ($S_{k+1}(\sigma_k = 0)$). Then, the method previously discussed in Section 2.4 is used to determine whether $S_{k+1}$ is strictly included inside the RPI set $Z_\infty$. Especially, since $S_{k+1}$ is a polytope from above, it is possible to compute its $V$-representation $V(S_{k+1})$. The $H$-representation of the RPI set $H(Z_\infty)$ can also be computed since it is also a polytope. One can notice that the computation of $H(Z_\infty)$ can be performed only once during the strategy, and can even be computed offline. The only effort at each instant $k$ consists in calculating the set of vertices $V(S_{k+1})$ and checking if all the vertices verify the half-space equations in $H(Z_\infty)$. Then the triggering criterion is obvious: if $S_{k+1}$ is strictly included inside $Z_\infty$, then no measurement is needed and $\sigma_k = 0$. Otherwise, a measurement is needed and the measurement trigger is set to $\sigma_k = 1$. This leads to the following switching trigger criterion:

$$ \sigma_k = \begin{cases} 0 & \text{if } S_{k+1} \subseteq Z_\infty \Leftrightarrow \forall v \in V(S_{k+1}), v \in Z_\infty \\ 1 & \text{if } S_{k+1} \not\subseteq Z_\infty \Leftrightarrow \exists v \in V(S_{k+1}), v \notin Z_\infty \end{cases} \qquad (19) $$

It can now be verified that the controlled system is stable under such a switching trigger criterion for the control law (9, 10).

## 3.4 Stability and Constraints Satisfaction Proofs

The goal of this section is to verify that the control law (9, 10), applied to the system (1) with the switching decision trigger (19), satisfies the constraints (2) at any time instant. It will also be shown that the self-triggering strategy stabilizes the system.
**Assumption 5.** The initial state is assumed to be in the RPI set: $\xi_0 \in \mathbb{Z}_\infty$.

**Theorem 1.** (Asymptotic satisfaction of state constraints). Under Assumption 5, the disturbed system (1), subject to the control (9, 10) with the switching trigger (19), always satisfies state constraints (2), i.e. $\xi_k \in \mathcal{X}, \forall k \in \mathbb{N}$.

*Proof.* By recurrence. The initial state verifies $\xi_0 \in \mathbb{Z}_\infty$, from Assumption 5. Considering now that $\xi_k \in \mathbb{Z}_\infty$, two cases are again to be distinguished depending on the switching criterion $\sigma_k$. If $\sigma_k = 0$, this means from (19) that $S_{k+1} \subseteq \mathbb{Z}_\infty$, and consequently that $\xi_{k+1} \in \mathbb{Z}_\infty$. Otherwise, if $\sigma_k = 1$, then a measurement is captured, i.e. $\bar{\xi}_k = \xi_k^+$, from (10). This implies that $\xi_{k+1} = A_{CL}\xi_k + Ew_k$, $w_k \in W$, which means that $\xi_{k+1} \in \mathbb{Z}_\infty$ by Definition 3. At this point, it has been shown that $\xi_k \in \mathbb{Z}_\infty, \forall k \in \mathbb{N}$. Finally, it is known that $\mathbb{Z}_\infty \subseteq \mathcal{X}_K \subseteq \mathcal{X}$. This ends the proof, since $\xi_k \in \mathcal{X}, \forall k \in \mathbb{N}$. □

**Theorem 2.** (Asymptotic stability of the self-triggered control). Under Assumption 5, the disturbed system (1), subject to the control (9, 10) with the switching trigger (19), is asymptotically stabilizing the system.

*Proof.* The same arguments as in the proof of Theorem 1 hold to show that $\xi_k \in \mathbb{Z}_\infty, \forall k \in \mathbb{N}$. Knowing that $\mathbb{Z}_\infty$, by the RPI properties, is an asymptotically stable set, ends the proof. □

**Remark 3.** In addition to system stabilization, it is important to point out that the PSTC verifies avoidance of the Zeno phenomenon. Since system measurements can only occur on discrete-time instants, the measurement buffer is never saturated, and is at worst filled in the same way as in the classical time-triggered case.

Now that the stability of the proposed self-triggered control law has been proven, a practical procedure to apply the strategy to any dynamical system is hereafter proposed.

## 3.5 | Procedure to Deploy the Self-Triggered Strategy

The strategy developed earlier is now synthesized into a procedure applicable to the stabilization of any system of the form (1). Some steps are required to be performed offline in order to control the system with the self-triggered strategy. From identification to tuning parameters, these preliminary steps are listed as follows.

### 1. Model the targeted system

The system to control with the self-triggered law (9, 10) needs to be modeled on the form (1). This means a linearization of the targeted system is necessary, and the disturbances applied to the system must be combined into a single additive vector $w$.

### 2. Choose a stabilizing control feedback

Given the open-loop system (1), choosing a stabilizing feedback in the form of a linear gain $K$ is necessary. For discrete-time dynamical systems, this means that all the poles of the closed-loop system (6) without disturbance must lie inside the unit circle (or, equivalently for a system with state-space representation, that the state matrix $A_{CL}$ must be Schur). Any classical method for feedback design can be used to select $K$. In the examples presented in this paper, a pole placement method is used, but it is also possible to tune the state-feedback gain with other methods such as LQ tuning method or $H_\infty$ synthesis.

### 3. Identify the state and disturbances bounds

Since the strategy aims to be robust and to guarantee performance, the bounds of the system are required to be identified. The bounds of $\mathcal{X}$ are in general physical constraints such as position or velocity limits, whereas the bounds of $\mathcal{U}$ are mainly actuator limits such as actuator saturations. An accurate identification of the disturbance bounds is particularly crucial for our strategy. The set $\mathcal{W}$ needs to be sufficiently large to take into account all feasible disturbances in the real system, but should not be overestimated at the risk of triggering measurements too frequently.

### 4. Compute an adequate RPI set

The proposed strategy relies on the computation of an RPI set, previously denoted $\mathbb{Z}_\infty$ in Section 3.3. The RPI set $\mathbb{Z}_\infty$ is computed offline using algorithms from the literature$^{37,38}$, and could be computed using the MPT toolbox$^{50}$ available in MATLAB. The calculation of $\mathbb{Z}_\infty$ requires the closed-loop evolution matrix $A_{CL}$ and the previously identified set of disturbances $\mathcal{W}$. Finally, a polytope should result from the algorithm, whose $\mathcal{H}$-representation $\mathcal{H}(\mathbb{Z}_\infty)$ can be determined and used in the evaluation (19) of the switching trigger criterion. Note that the size of the invariant set $\mathbb{Z}_\infty$ can also be reduced by the choice of $K$ (therefore
implicitly modifying $A_{CL}$), allowing to design a PSTC with a greater capacity for disturbance rejection (at the expense of a higher measurement frequency).

### 5. Run the self-triggered strategy

Algorithm 1 synthesizes the self-triggered control strategy previously developed. The strategy is to run during a time T discretized with sampling period $\tau_s$, which corresponds to an execution over $N_s + 1$ discrete samples, where $N_s = T/\tau_s$. The proposed strategy firstly requires the knowledge of the open-loop and closed-loop state matrices, as well as an RPI set computed as above. Next, the predicted state and reachable set sequences take a new initial condition by triggering a measurement of the system state (whose initial condition is subject to Assumption 5). The state measurement is performed with the function `triggerStateMeasurement`. Then, at each discrete-time instant $k$, the predicted system evolves one time step further, which also makes it possible to compute the reachable set $S_{k+1}$ as if no measurement was needed. This set is used to compute $\sigma_k$ with the function `computeSwitchingTrigger`, which implements the inclusion test (19). If a measurement is needed, it is immediately triggered with `triggerStateMeasurement`. As a consequence, $\tilde{\xi}_{k+1}$ and $S_{k+1}$ are updated for this particular case. Finally, the control law (9, 10) is applied to the system input with `sendSystemInput`.

<caption>**Algorithm 1** Self-triggered measurement strategy</caption>

Require: $A, A_{CL}, EW, Z_\infty$
Ensure: $\xi_k \in \mathcal{X}, \forall k \in \mathbb{N}^{N_s+1}$
$k \leftarrow 0$
$\tilde{\xi}_0 \leftarrow \text{triggerStateMeasurement()}$ \quad $\triangleright$ \text{Initial measurement of } $\xi_0$
$S_0 \leftarrow \{\tilde{\xi}_0\}$
for $k \in \mathbb{N}^{N_s}$ do
$\bar{\xi}_{k+1} \leftarrow A_{CL}\tilde{\xi}_k$
$S_{k+1} \leftarrow A(S_k \oplus \{\tilde{\xi}_k\}) \oplus EW \oplus \{\tilde{\xi}_{k+1}\}$ \quad $\triangleright$ \text{Reachable set when } $\sigma_k = 0$
$\sigma_k \leftarrow \text{computeSwitchingTrigger}(S_{k+1}, Z_\infty)$
if $\sigma_k = 1$ then
$\tilde{\xi}_k \leftarrow \text{triggerStateMeasurement()}$ \quad $\triangleright$ \text{Accurate measurement of } $\xi_k$
$\tilde{\xi}_{k+1} \leftarrow A_{CL}\tilde{\xi}_k$
$S_{k+1} \leftarrow EW \oplus \{\tilde{\xi}_{k+1}\}$ \quad $\triangleright$ \text{Reachable set when } $\sigma_k = 1$
end if
$u_k \leftarrow -K\tilde{\xi}_k$
sendSystemInput($u_k$)
end for

Figure 2 shows the block diagram of the overall proposed control strategy. The considered system is depicted in gold while the blocks related to the deployed controller are represented in blue.

## 3.6 | Self-Triggered Control Strategy: Academic Example

This section provides an academic example that shows the behavior of the self-triggered control strategy on a classical second order system. Let's now consider the following discrete-time disturbed system

$$ \xi_{k+1} = \underbrace{\begin{bmatrix} 1 & 0.05 \\ 0 & 1 \end{bmatrix}}_{A} \xi_k + \underbrace{\begin{bmatrix} 0.00625 \\ 0.250 \end{bmatrix}}_{B} u_k + \underbrace{\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}}_{E} w_k \quad (20) $$

which is a second-order system with position and velocity states. This system is sampled with sampling period $\tau_s = 0.05s$ and the disturbance vector $w_k$ belonging to the set $W = [\pm 0.025 \times \pm 0.05]$. To stabilize the system, the state-feedback gain $K$ is selected based on a pole placement method. Specifically, choosing the desired closed-loop poles $0.9450 \pm 0.13i$ leads to $K = [1.6 \ 0.4]$.
<figure><img src="image_9.png" /><figcaption>FIGURE 2 Block diagram of the proposed PSTC.</figcaption></figure>

This state-feedback gain implies the following closed-loop state matrix

$$A_{CL} = A - BK = \begin{bmatrix} 0.990 & 0.0475 \\ -0.400 & 0.900 \end{bmatrix} \quad (21)$$

with $A_{CL}$ Schur. An RPI set $Z_\infty$ is computed as explained above, and results in a polytope made of 88 vertices. Figure 3 graphically shows the results of the self-triggered strategy applied to system (20), especially the evolutions of $\xi, \tilde{\xi}$ and sequence of reachable sets $S$ over 40 discrete time steps.

The initial state of the system is $\xi_0 = [-0.6 \ 1.5]^T$, and is included inside $Z_\infty$. Then, the reachable sets grow (green sequence) given the switching criterion $\sigma = 0$ until the 5th time instant. At this time, the next reachable set $S_6$ (surrounded by red) is computed and appears to come out from $Z_\infty$. Consequently, $\sigma_5 = 1$ and a new measurement is triggered. This measurement is $\tilde{\xi}_5 = \xi_5^+$, which belongs to $S_5 \subset Z_\infty$. The guaranteed next reachable set $S_6$ is computed appropriately given $\sigma_5 = 1$. The strategy continues following the same pattern, with growing (orange sequence) reachable sets until measurement at the 11th time instant, and with growing (blue sequence) reachable sets until measurement at the 23rd time instant. From the 23rd time instant, reachable sets are not represented anymore to keep Figure 3 readable. In the end, there were 5 measurements captured during the whole simulation, including the initial measurement.

## 4 EXPERIMENTAL RESULTS: APPLICATION TO AN AIR EXTRACTOR MOTOR

The self-triggered control strategy developed in the previous sections is applied here to an air extractor system. Air extractors are commonly used in ventilation systems to replace stale air with fresh air. The velocity control of the air extractor's motor is crucial to ensure efficient ventilation, but it can be challenging due to the presence of disturbances and uncertainties in the system. The self-triggered control strategy proposed in this paper offers a promising solution to these challenges by reducing the number of measurements required.

This section first presents the experimental setup used for the air extractor system. It follows a description of the self-triggered control strategy implementation. The results obtained with the self-triggered control strategy are then presented and compared with a traditional time-triggered control strategy.
<figure><img src="image_10.png" /><figcaption>**FIGURE 3** Self-triggered measurement strategy applied to the system (20). The figure shows the evolution of $\xi$ represented in the 2-state (position, velocity) space. Only 5 measurements of the state at time instants $k = 0, 5, 11, 23$ and 33 (represented by the red circles) are needed to stabilize the system within the constraints during the simulation horizon of 40 sampling periods. These measurements are used to generate the growing reachable sets encompassing all possible trajectories of the real system with disturbances (solid lines). The reachable sets are centered on the predicted trajectory of the system without disturbances (dotted lines). A triggering event arises when a reachable set crosses the frontier of the RPI set $Z_{\infty}$ (dashed gray line).</figcaption></figure>

## 4.1 | Experimental Setup

Figure 5a shows the experimental benchmark intended to implement the self-triggered strategy. It can be seen in the foreground the air extraction system, composed of its motor and its blades which are attached to it. It can be noticed in the background that a blower has been installed to produce an air disturbance. The blower is useful to study the reaction of the self-triggered strategy subject to an unknown but bounded air gust. Figure 4 shows the self-triggered control diagram of the air extractor disturbed by the air gusts.

The strategy is developed on a Windows 10 computer with Intel(R) Core(TM) i5 processor and 8Go of installed RAM. The implementation on the air extractor system is done with Simulink Desktop Real-Time. The computer communicates with the electronic hardware interface based on the STM32 microcontroller via UDP communication. The implementation relies on Matlab/Simulink, where sets are computed using the MPT toolbox⁵⁰.
<figure><img src="image_11.png" /><figcaption>FIGURE 4 Self-triggered control diagram of the air extractor disturbed by blower wind gusts. These wind gusts directly affect the rotation speed of the air extractor blades.</figcaption></figure>

## 4.2 Implementation of Self-Triggered Control

The self-triggered control strategy is applied to the air extractor system by taking Section 3.5 as an implementation guideline.

### 1. Model the targeted system

The motor modeling is based on the following differential equation, which corresponds to a mechanical equation accounting for the torques acting on the rotation axis:

$$J\dot{\omega} = \Gamma - \Gamma_r \qquad (22)$$

where $\omega$ is the rotation speed of the rotor, $\Gamma$ is the motor torque, $\Gamma_r$ is the totality of resisting torques applied to the rotation axis (quantity assumed constant at a given rotation speed), and $J = 6 \cdot 10^{-5} \text{ kg} \cdot \text{m}^{-2}$ is the identified rotor moment of inertia. The angular position of the motor is denoted $\theta$, with $\dot{\theta} = \omega$. The benchmark does not provide a direct measurement of $\omega$, but is reconstructed with an encoder observer providing an estimated angular position $\dot{\theta}$ as well as an estimated rotation speed $\dot{\omega}$. The open-loop system can nevertheless be considered as if there were no encoder observer (i.e. without observer dynamics). The motor is controlled by the torque $\Gamma$, and operates around a working point $\{\xi_{eq}, \Gamma_{eq}, w_{eq}\}$. A discretization of the system with period $\tau_s = 0.03\text{s}$ is also considered. By taking into account all the above assumptions, an air extractor open-loop model can be identified as follows

$$\xi_{k+1} = \underbrace{\begin{bmatrix} 1 & 0.03 \\ 0 & 1 \end{bmatrix}}_{A} \xi_k + \underbrace{\begin{bmatrix} 7.5 \\ 500 \end{bmatrix}}_{B} \dot{\Gamma}_k + E\dot{w}_k \qquad (23)$$

where $\xi = [\theta \ \omega]^T$, and $\dot{\xi} = [\dot{\theta} \ \dot{\omega}]^T$ are the position and velocity variations around the equilibrium point.

### 2. Choose a stabilizing control feedback

The stabilizing feedback is applied to the rotation speed $\omega$ since only a speed control of the air extractor is targeted here. That is why the chosen feedback is on the form $\Gamma = -K\omega$, where $K = 1 \cdot 10^{-4}$ is a satisfactory value regarding the closed-loop behavior of the system. However, the assumption neglecting the dynamics of the encoder observer is no longer valid for the closed-loop
system, and the global dynamics are identified on the system as a second order given by

$$ \tilde{\xi}_{k+1} = \underbrace{\begin{bmatrix} 0.9942 & 0.0275 \\ -0.3740 & 0.8387 \end{bmatrix}}_{A_{CL}} \tilde{\xi}_k + E \tilde{w}_k \quad (24) $$

### 3. Identify the disturbances bounds

For the disturbed system (23), the disturbance matrix is defined as $E = [A B I]$. With such a matrix form, the potential disturbances to which the system is subjected can be fully identified as state (A), control (B) and evolution (I) disturbances, where I is the identity matrix of suitable dimensions. For the system (23), this implies that $w \in \mathbb{R}^5$. To identify the bounds of the set W to which the perturbation vector w belongs, an identification phase is performed, during which the errors $\xi - \tilde{\xi}$, $\Gamma - \bar{\Gamma}$, and the recurrent errors $\xi_{k+1} - \xi_k$ are computed with the system running in a classical time-periodic sampling framework. The system is running around a specifically desired working point

$$ \{\omega_{eq}, \Gamma_{eq}\} = \{110 \text{ rad} \cdot \text{s}^{-1}, 1.7 \cdot 10^{-3} \text{ N} \cdot \text{m}\} \quad (25) $$

A bounded air gust is also randomly applied to the extractor during the identification. The disturbance bounds are then taken so that all the state, control, and evolution error values lie inside these bounds. This identification finally leads to the following disturbance set $W \subset \mathbb{R}^5$:

$$ W = \begin{bmatrix} [-0.5998, 0.1763] \\ [-0.0566, 0.0475] \\ [-8.1487, 6.8145] \cdot 10^{-6} \\ [-0.0609, 0.7396] \\ [-0.0890, 0.0799] \end{bmatrix} \quad \Rightarrow \quad EW = \begin{bmatrix} [-0.6625, 0.9175] \\ [-0.1497, 0.1308] \end{bmatrix} \quad (26) $$

### 4. Compute an adequate RPI set

By considering the closed-loop system matrix $A_{CL}$ in (24) and the set $EW$ (26), the RPI set $Z_\infty$ can be computed for the system (24) using algorithms from the literature$^{37,38}$. The resulting set is here made of 320 vertices.

### 5. Run the self-triggered strategy

The self-triggered strategy is then executed on the air extractor system for the desired equilibrium point (25).

## 4.3 | Results and Analysis

Results of the self-triggered control strategy on the air extractor system are given below. For a better understanding and readability, the results of the strategy will be shown on the rotation speed of the system as a function of time. Reachable sets S and RPI set $Z_\infty$ will be projected into the w-state dimension of the state-space to be represented temporally. The self-triggered control strategy is applied while disturbed by variable air gusts. The air extractor is also arbitrarily sent to given velocity setpoints around the equilibrium, i.e. $\omega_{ref} = \omega_{eq} \pm 5 \text{rad} \cdot \text{s}^{-1}$.

Figure 5c shows the results of the strategy for the previously described scenario. Both disturbed ($\omega_{ST}$, orange curve) and predicted ($\bar{\omega}_{ST}$, blue curve) evolutions of the rotation speed are represented, as well as the profile of the reachable sets (S, green zone) and the RPI set ($Z_\infty$, gray zone). The ground truth variable wind disturbance applied to the airflow of the blades is also shown (purple curve). Furthermore, Figure 5c compares the performance of the self-triggered control strategy with a traditional periodic control ($\omega_{TT}$). The proposed strategy applied to the air extractor system allows a 98.75% measurement harvesting compared to a control with periodical measurement sampling. It is also verified experimentally that constraint satisfaction is ensured at any time, which consequently implies stability of the self-triggered control. Zooming in on Figure 5c around a measurement triggering instant then makes it possible to demonstrate that reachable sets provide a good boundary of the disturbed system and its dynamics (see Figure 5b).
<figure><img src="image_12.png" /><figcaption>(a) View of the air extractor's motor with its blades (in the foreground), and the blower system used as a disturbance generator (in the background).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(b) Zoom around a triggered measurement instant to show guaranteed system bounding by reachable sets.</figcaption></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>(c) Self-triggering strategy applied to air extractor (with wind gusts).</figcaption></figure>

**FIGURE 5** Self-triggering strategy applied to the air extractor system shown on subfigure 5a. Measurements are only triggered at self-triggered instants marked on the time axis of the subfigure 5c. Between measurement instants, the motor is controlled solely on the basis of the predicted angular velocity $\bar{\omega}_{ST}$. This prediction also comes as the center of the growing reachable sets $\mathcal{S}$. Reachable sets accurately encompass the disturbed real angular velocity $\omega_{ST}$, as shown on the zoomed-in subfigure 5b. The performance of the self-triggered strategy is furthermore compared to the system controlled under a classical time-triggered framework $\omega_{TT}$.

## 5 | CONCLUSION

This paper proposed a new self-triggering measurement strategy by investigating the combination of reachable and invariant sets. The resulting control strategy leads to significant improvements in measurement harvesting. This makes able to reduce the usage on energy-consuming or sensitive sensors, and also to improve the bandwidth on networked systems. The proposed PSTC has been theoretically developed for any discrete-time linear system. It can therefore be applied to a wide range of control systems,
including networked control systems, or robotic systems, among others. However, a few points of discussion can be put forward concerning the proposed PSTC method.

In fact, as with any self-triggered strategy, there is a trade-off between the number of events triggered (resource savings on the sensor in the present case) and the control performance on the system. It is obvious that during open-loop phases, when the sensor's accurate measurement is not taken into account, the performance of setpoint tracking deteriorates over time. The PSTC technique is still of interest in applications where measurement are costly, and where the slight deviation of the system is not significant to its operation (if tolerated within an interval which will be taken into account in the calculation of the triggering set).

It can also be noted that a good knowledge of the considered control system is necessary for the application of the PSTC strategy. This knowledge can also be the result of a rigorous identification stage aimed at learning the limits of the system, its (linearized) model and the disturbances or uncertainties that can affect it. The generic formulation of the PSTC method is still useful for this purpose ; it is possible to take into account different descriptions of constraints and disturbances, as demonstrated in the concatenation of the *E* matrix in Section 4.

Experimental results showed the efficiency of the PSTC strategy, with a particular emphasis on the reduction of sampled measurements. These early findings strongly encourage further investigations on the coupling of set reachability with invariance methods for self-triggered control. The experiments also demonstrated the accuracy of the application procedure provided in this paper.

Future work will focus on applying the proposed PSTC to other control systems, in particular mobile robots such as UGVs, UAVs or AUVs. Another research direction would be to adapt the proposed PSTC to robotic navigation, i.e. to take into account information from a map to characterize the control limits used to define the triggering set.

## ACKNOWLEDGEMENT

This research is part of the Dark-NAV project funded by the French National Research Agency (ANR-20-CE33-0009). This work is also part of the French-Mexican TOBACCO project funded by the FORDECYT-PRONACES through the joint SEP-CONACYT-ANUIES-ECOS Nord program (MX-296702 & FR-M18M02). The authors would like to thank John J. Martínez for providing access to the GIPSA-lab's *Wind Energy* platform, and in particular for making the practical benchmark of the air extractor available.

## DECLARATION OF INTERESTS

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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