C De Persis
email: c.de.persis@rug.nl.

R Postoyan
email: romain.postoyan@univ-lorraine.fr.

P Tesi

J C Willems

Event-triggered Control From Data

We present a data-based approach to design eventtriggered state-feedback controllers for unknown continuoustime linear systems affected by disturbances. By an event, we mean state measurements transmission from the sensors to the controller over a digital network. By exploiting a sufficiently rich finite set of noisy state measurements and inputs collected off-line, we first design a data-driven state-feedback controller to ensure an input-to-state stability property for the closed-loop system ignoring the network. We then take into account sampling induced by the network and we present robust data-driven triggering strategies to (approximately) preserve this stability property. The approach is general in the sense that it allows deriving data-based versions of various popular triggering rules of the literature. In all cases, the designed transmission policies ensure the existence of a (global) strictly positive minimum inter-event time thereby excluding Zeno phenomenon despite disturbances. These results can be viewed as a step towards plugand-play control for networked control systems, i.e., mechanisms that automatically learn to control and to communicate over a network.

I. INTRODUCTION

E Vent-triggered control is an implementation paradigm, which consists in transmitting data between the plant and its controller whenever a state-or output-dependent criterion is verified. The underlying idea is to generate communications between the plant and its controller only when this is needed to achieve the desired control objectives, as opposed to classical time-triggered (periodic) strategies for which the communication instants depend on the elapsed time and not on the actual system needs. Event-triggered control is motivated by resource aware scenarios where communicating, computing or updating the control input comes with a certain cost, such as networked control systems and embedded systems. Since the results in [START_REF] Årzén | A simple event-based PID controller[END_REF], [START_REF] Astr | Comparison of Riemann and Lebesgue sampling for first order stochastic systems[END_REF] and the pioneering work in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], various event-triggered control strategies have emerged in the literature for a range of set-ups and control problems, see, e.g., [START_REF] Heemels | An introduction to eventtriggered and self-triggered control[END_REF], [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF].

The vast majority of the literature on event-triggered control focuses on model-based approaches, in the sense that both the feedback law and the triggering condition are designed by relying on a model of the plant dynamics. These results are therefore not applicable when first-principle models are not conceivable, or when exact/accurate enough system identification pietro.tesi@unifi.it.

is impossible because of noisy data. In this case, an alternative consists in designing the controller directly based on available input-state/output data, we talk of data-driven control, see e.g., [START_REF] Safonov | The unfalsified control concept and learning[END_REF]- [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF] for earlier contributions and one recent survey on the topic. This paradigm is also appealing as it may ease the controller design step. Few techniques are currently available in the literature to design data-driven event-based controllers see, e.g., [START_REF] Liu | Data-driven selftriggered control via trajectory prediction[END_REF]- [START_REF] Wang | Model-based and data-driven control of event-and self-triggered discrete-time linear systems[END_REF], which consider discrete-time systems, and the recent work in [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF] dedicated to continuous-time systems. In [START_REF] Liu | Data-driven selftriggered control via trajectory prediction[END_REF]- [START_REF] Digge | Data-driven event-triggered control for discrete-time LTI systems[END_REF], no disturbances act on the data collected off-line, which simplifies the learning, in particular, for linear timeinvariant systems, an exact data-based representation can be obtained in this case under mild conditions [START_REF] Willems | A note on persistency excitation[END_REF]. This is not the case of [START_REF] Wang | Model-based and data-driven control of event-and self-triggered discrete-time linear systems[END_REF] where the controller and the triggering policy are designed once and for all with one single batch of noisy data collected from the system. The idea is that, under certain conditions, this data batch returns a (non-parametric) system model. This makes it possible to cast control and triggering rule design as data-dependent problems, in particular via databased linear matrix inequalities (LMI). The authors of [START_REF] Wang | Model-based and data-driven control of event-and self-triggered discrete-time linear systems[END_REF] then derive a dynamic event-triggered control policy inspired by the model-based technique in [START_REF] Girard | Dynamic triggering mechanisms for event-triggered control[END_REF]. However, while off-line data are noisy, disturbances are ignored when implementing the controller. In [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF], on the other hand, disturbances are considered both for learning and during closed-loop operation and a specific triggering strategy is presented to ensure an L 2 -stability property.

There is therefore a strong need for general data-based event-triggered control techniques that are applicable to continuous-time systems, as many real-life processes have a continuous-time nature, and that are robust to disturbances affecting both the data acquisition phase and the closed-loop operations. In this context, we present a data-based approach to design state-feedback event-triggered controllers for unknown stabilizable continuous-time linear systems affected by disturbances. Like in [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF], noisy data is explicitly accounted both off-line, when acquiring data, and on-line, when implementing the designed controller. Noisy data in the acquisition phase prevent exact system identification, and this calls for control design routines that are robust to uncertainty. Noise during the closed-loop operations, on the other hand, is even more problematic because it may lead to Zeno behavior even with exact modelling [START_REF] Borgers | Event-separation properties of event-triggered control systems[END_REF], meaning that an infinite number of transmissions occur in finite time. To address these challenges, we proceed by emulation and, as a first step, we design a statefeedback controller that stabilizes the closed-loop system in absence of sampling, in the sense that an input-to-state stability (ISS) property holds. We then take into account sampling induced by the network and we design data-driven triggering techniques based on the relative threshold technique of [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] that preserves (approximately) the original stability property of the closed-loop system. In particular, we first develop data-driven mixed absolute/relative threshold event-based policies [START_REF] Borgers | Event-separation properties of event-triggered control systems[END_REF]. In this case, the closed-loop system is guaranteed to satisfy a practical ISS property, where the adjustable parameter is the absolute threshold. Further, we prove the existence of a strictly positive uniform minimum inter-event time between any two transmissions and we provide a data-based estimate of it.

To preserve the asymptotic nature of the stability property of the continuous-time closed-loop system in absence of sampling, we present an alternative triggering rule based on timeregularization of the relative threshold, see e.g., [START_REF] De Persis | Input-to-state stabilizing control under denialof-service[END_REF]- [START_REF] Dolk | Output-based and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF]. By doing so, we enforce a given strictly positive minimum inter-event time after which a relative threshold condition is checked thereby excluding Zeno phenomenon by design. The maximum value of this enforced minimum time between any two transmissions is given by an expression, which depends on the data collected off-line only. While the timeregularized triggering condition ensures a stronger stability property compared to the mixed strategy, it may generate more transmissions as suggested by a numerical example we provide. From a technical viewpoint, the proposed design conditions take the form of semi-definite programs (SDP), in particular feasibility of optimization problems with LMI constraints. This is appealing because many results on eventtriggered control developed in a model-based context involve LMI formulations, see [START_REF] Abdelrahim | Robust eventtriggered output feedback controllers for nonlinear systems[END_REF], [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF].

Afterwards, we illustrate the generality of the approach by designing robust data-based versions of several major triggering rules of the literature originally developed in a modelbased setting, namely: (i) quadratic triggering policies, e.g., [START_REF] Heemels | An introduction to eventtriggered and self-triggered control[END_REF]; (ii) dynamic event-triggered control [START_REF] Girard | Dynamic triggering mechanisms for event-triggered control[END_REF], [START_REF] Dolk | Output-based and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF] where we consider a more general form compared to [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF] (see Remark 9 for more details); and (iii) decreasing threshold on a Lyapunov function designed ignoring sampling, like in [START_REF] Wang | On event design in event-triggered feedback systems[END_REF], [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF]. In all cases, ISS properties are established and a global (i.e., independent of the initial conditions of the system) minimum inter-event time is guaranteed to exist despite disturbances acting on the system. In addition, data-based estimates of these minimum inter-event times are provided. These results can be viewed as a step towards plug-and-play control for networked control systems, i.e., mechanisms that automatically learn to control and to communicate over a network.

The remainder of this paper is as follows. We introduce the framework of interest in Section II. We then first concentrate on the noise-free case to put in place the required background and techniques in Section III. The main results dedicated to noisy systems are provided in Section IV, where we establish (practical) ISS properties as well as the existence of a global minimum inter-event time. The extension to other triggering policies is presented in Section V. Section VI ends the paper with concluding remarks, and the appendix contains a couple of additional technical results.

Notation. N 0 :" t0, 1, 2, . . .u denotes the set of non-negative integers and N :" N 0 zt0u. R stands for the set of real numbers, R ě0 :" r0, 8q and R ą0 :" p0, 8q. Given a symmetric matrix M , the notation M ą 0 (M ľ 0) and M ă 0 (M ĺ 0) means that M is positive and, respectively, negative (semi)definite. Given a matrix M , M J denotes the transpose of M .

The notation I and 0 stand for the identity matrix and the zero matrix, respectively, whose dimensions depend on the context. For the sake of convenience, we write r A B ‹ C s for

" A B B J C
‰ where A, B, C are matrices of appropriate dimensions. The induced 2-norm of a matrix is denoted } ¨}. Given x P R n and y P R m with n, m P N, we use the notation px, yq to denote px J , y J q J . Given x, y P R n with n P N, xx, yy " x J y is the dot product. For d : R ě0 Ñ R n with n P N, the supremum of d on r0, ts with t ě 0 is }d} r0,ts :" sup sPr0,ts |dpsq|, and }d} 8 :" sup tě0 |dptq|, where | ¨| denotes the Euclidean norm.

II. FRAMEWORK

Consider the continuous-time linear system

9 xptq " Axptq `Buptq `dptq, (1)
where xptq P R n is the state and uptq P R m is the control input at time t P R ě0 with n, m P N. Input disturbance d P R ě0 Ñ R n is unknown, Lebesgue measurable and bounded in the sense that }d} 8 ă `8. Matrices A and B are real, constant, unknown and assumed to be such that pA, Bq is stabilizable. We will not need to directly check stabilizability of pA, Bq in the sense that this property is necessary for the feasibility of the design data-based programs that we will consider.

We investigate the scenario where plant ( 1) is connected to its controller via a network. In particular, state measurements1 are sent from the sensors to the controller via a digital channel, and the controller is directly connected to the actuators. 2 Our goal is to design a state-feedback event-triggered controller to stabilize, in a sense made precise in the sequel, system [START_REF] Årzén | A simple event-based PID controller[END_REF]. In particular, we aim at designing a linear state-feedback law with gain K P R mˆn and a triggering policy that defines the sequence of transmission (or sampling) instants tt k u kPN with N Ď N 0 for each solution to the system. Without loss of generality, we consider that a transmission occurs at t " 0 so that t 0 " 0. We implement the controller using zero-order-hold devices, which leads to the control input uptq " Kxpt k q, t P rt k , t k`1 q.

(2)

The solutions to (1) in closed-loop with (2) are understood in the Carathéodory sense 3 , i.e., the solution flows on rt k , t k`1 s and experiences a jump at t k`1 for k, k `1 P N . We will establish later in the paper that, for any k P N , t k`1 ´tk is lower bounded by a strictly positive constant independent of k and of the initial condition so that the t k 's do not accumulate (i.e., Zeno phenomenon does not occur). Also, by a solution, we mean a maximal solution, i.e., one that cannot be extended. Notice that the control law (2) can also be computed at the sensor node, in which case the sensor directly transmits upt k q over the network to the actuators. All the results of this paper apply to this scenario without any modifications.

Various solutions to this problem are available in the literature when A and B are known see, e.g., [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]- [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF], [START_REF] Girard | Dynamic triggering mechanisms for event-triggered control[END_REF], [START_REF] De Persis | Input-to-state stabilizing control under denialof-service[END_REF], [START_REF] De Persis | Input-to-state stabilizing control under denialof-service[END_REF]- [START_REF] Dolk | Output-based and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF], [START_REF] Wang | On event design in event-triggered feedback systems[END_REF], [START_REF] Liu | Event-based control of nonlinear systems with partial state and output feedback[END_REF]. Since A and B in (1) are unknown, we cannot apply these results and the challenge is thus to design the event-triggered controller directly based on some available data. In particular, we assume that we are given a set of data collected off-line D :" tuptq, xptq, 9

xptq : t P t0, T s , . . . , pT ´1qT s uu ,

where T s ą 0 is a sampling time used for collecting the data and T ą 0 is the number of samples. Set D consists of input, state and state derivative data collected from the system with an experiment. This means we have access to a set of input-state samples verifying 9 xptq " Axptq `Buptq `dptq for t P t0, T s , . . . , pT ´1qT s u. We consider data collected periodically only to ease the exposition: this is not required for the forthcoming results to hold, in other words aperiodic sampled data can very well be considered.

The problem of interest is to determine, using D, a controller gain K in (2) and a triggering condition policy that defines the transmission instants tt k u kPN (not necessarily periodic) for each solution, such that the induced closed-loop system has suitable stability properties, as formalized in the sequel. We proceed by emulation for this purpose, in the sense that we will first design the stabilizing controller gain K in the absence of sampling. Afterwards, sampled communications due to the network are taken into account and we will derive triggering conditions to (approximately) preserve stability. We first focus on the ideal case of noise-free data to introduce the required background and techniques in Section III. These results are then extended to the case where data are noisy in Section IV. We will explain how the followed approach can be used to develop various other triggering conditions in Section V.

Remark 1 (Data acquisition phase): The question of the selection T s and T is important for the forthcoming results and we will comment on this point later in the paper. Here, we remark that the sampling time T s used for collecting the data does not need to coincide with those generated by the triggering policy to be designed. l Remark 2 (Measurement noise): All the results of this paper can be extended to the case where the off-line collected data are affected by measurement noise, i.e., to the case in which the process output (the measured signal) is z " x`w, with w a noise signal. As shown in [26, Section V-A], this case reduces to input disturbances with suitable manipulations. When the on-line measurements are corrupted by noise, the problem remains challenging even in the model-based setting, see, e.g., [START_REF] Abdelrahim | Robust eventtriggered output feedback controllers for nonlinear systems[END_REF], [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF]- [START_REF]Robustifying event-triggered control to measurement noise[END_REF], and is left for future work. l Remark 3 (State derivative measurements): The off-line computation of 9

x in (3) is error-prone. We can explicitly account for errors in the computation of 9

x by modelling these errors as a measurement noise. Error bounds when 9

x is computed with Euler discretization are provided in [START_REF] Berberich | Datadriven analysis and control of continuous-time systems under aperiodic sampling[END_REF]. Alternatively, we can consider an integral version of the relation 9

xptq " Axptq `Buptq `dptq which permits to construct datasets that do not involve the computation of 9

x, see Appendix A. l

III. LEARNING FROM NOISE-FREE DATA A. Assumption on set D

We consider throughout this section system (1) with d " 0 and a noise-free dataset D in (3). We define U 0 :" " up0q upT s q ¨¨¨uppT ´1qT s q ‰ P R mˆT , (4a) X 0 :" " xp0q xpT s q ¨¨¨xppT ´1qT s q ‰ P R nˆT , (4b)

X 1 :" " 9 xp0q 9 xpT s q ¨¨¨9 xppT ´1qT s q ‰ P R nˆT . (4c)
We assume a condition on the richness of the data. Assumption 1: The matrix " U0 X0 ‰ has full row rank. l Assumption 1 can be easily checked for a given set D. For discrete-time systems, when pu, dq is persistently exciting then Assumption 1 holds [START_REF] Willems | A note on persistency excitation[END_REF] (for an extension of this result to continuous-time systems see [START_REF] Lopez | On a continuous-time version of Willems' lemma[END_REF]). We will elaborate more on Assumption 1 in the discussion following Theorem 1 and at the beginning of Section IV.

B. Learning a feedback controller

To design the feedback law, we follow the approach introduced in [START_REF] De Persis | Formulas for data-driven control: Stabilization, optimality, and robustness[END_REF], which gives simple formulas for controller design and returns a data-based representation of the closedloop dynamics that is useful to determine later on the triggering policy. The next two results are taken from [START_REF] De Persis | Formulas for data-driven control: Stabilization, optimality, and robustness[END_REF]; we recall them for convenience. We start with an auxiliary result.

Lemma 1: Let Assumption 1 hold and consider any matrix K P R mˆn . Then A `BK " X 1 G where G P R T ˆn is any solution to the system of equations

" K I  " " U 0 X 0  G. (5)
l Proof. By Assumption 1, there exists a T ˆn matrix G such that (5) holds. Hence, A `BK " r B A sr K I s " r B A sr U0 X0 sG, where the second identity follows from [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF]. The result follows because the elements of U 0 , X 0 and of X 1 satisfy the relation 9 xptq " Axptq `Buptq, t P t0, T s . . . , pT ´1qT s u, which, in compact form, gives X 1 " AX 0 `BU 0 .

We exploit Lemma 1 to design K such that A `BK is Hurwitz, despite the fact that A and B are unknown. As a result, the origin of system (1) with d " 0 in closed-loop with the continuous-time controller u " Kx will be globally exponentially stable, i.e., there exist c 1 ě 1 and c 2 ą 0 such that any solution x satisfies |xptq| ď c 1 e ´c2t |xp0q| for any t ě 0. We derive for this purpose a convex program, specifically a SDP. As A and B are unknown, we use the fact that for any matrix K P R nˆm , X 1 G " A `BK where X 1 and G are known from D in (3) in view of Lemma 1, to select suitable K based on U 0 , X 0 and X 1 in the next theorem.

Theorem 1: Consider stabilizable system (1) with d " 0 and suppose Assumption 1 holds. Consider the next SDP in the decision variable Y P R T ˆn

X 1 Y `pX 1 Y q J ă 0, (6a) X 0 Y ą 0. (6b)
SDP ( 6) is feasible and any solution Y to ( 6) is such that the matrix K " U 0 Y pX 0 Y q ´1 renders A `BK Hurwitz. As a result, the origin of system 9 xptq " pA `BKqxptq is globally exponentially stable. l Proof. We first show that ( 6) is feasible. Consider any matrix K that makes A `BK Hurwitz, which exists since pA, Bq is stabilizable. By Lyapunov theory, there exists a symmetric matrix S ą 0 such that SpA `BKq `pA `BKq J S ă 0. Furthermore, by Lemma 1 there exists a matrix G that satisfies r K I s " r U0 X0 sG, which implies A `BK " X 1 G. Hence, Y :" GS ´1 satisfies both (6a) and (6b) as X 0 G " I and X 0 Y " S ´1 ą 0. This shows that ( 6) is feasible.

Concerning the second part of the statement, consider any solution Y to (6) and let G :" Y S, where we set S :" pX 0 Y q ´1. Note that X 0 G " I and that K " U 0 Y pX 0 Y q ´1 can also be written as K " U 0 G, as (5) holds. By combining (6a) and (6b), and because G " Y S and A `BK " X 1 G, we conclude that SpA `BKq `pA `BKq J S ă 0. Thus A `BK is Hurwitz as S is symmetric and S ą 0. The global exponential stability of the origin of 9

xptq " pA `BKqxptq then follows, see [START_REF] Hespanha | Linear systems theory[END_REF]Theorem 8.2].

The decision variable in Theorem 1 is Y (and not G) and Y pX 0 Y q ´1 " G. This change of variable is instrumental to arrive at a convex formulation of the design program as shown in the proof of Theorem 1. We also notice from the proof of Theorem 1 that Assumption 1 is not needed for the second part of the result, which implies that there might exist a solution to (6) even though Assumption 1 does not hold. This fact has been pointed out in the discrete-time case in [START_REF] Van Waarde | Data informativity: a new perspective on data-driven analysis and control[END_REF]. Nevertheless, having " U0 X0 ‰ full row rank gives certain advantages as any stabilizing controller can be parametrized through the data. This is useful when we search for a controller that satisfies extra desirable properties.

Theorem 1 provides a way to design the state feedback controller gain K in absence of network (in the noise-free case). We now move to the next step of the emulation approach that is to take sampling into account and to design the triggering condition.

C. Learning a triggering policy

We write the closed-loop system under the control law (2) as 9

xptq " pA `BKqxptq `BKeptq

where eptq :"

xpt k q ´xptq, t P rt k , t k`1 q, (8)
represents the sampling-induced error, that is the mismatch between the last value of the state transmitted to the controller and its current value. As customary in the event-triggered control literature, the idea is to regard e as a disturbance to the nominal dynamics 9 xptq " pA `BKqxptq, which is then controlled via the triggering condition so that stability is preserved despite sampling.

We develop a data-based version of the approach proposed in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] for this purpose; the extension to other triggering conditions is addressed in Section V. We introduce for this purpose the next parameterized matrix and the vector z Ψpσq :"

" ´σ2 I 0 0 I  , z :" " x e  , (9)
with σ ą 0 a design parameter to be determined. The sampling times are defined as follows: t 0 :" 0 and

t k`1 " $ ' & ' % inftt P R : t ą t k and zptq J Ψpσqzptq " 0u if xpt k q ‰ 0, ` 8 otherwise.
(10) This logic ensures by design that z J Ψpσqz ď 0 along the solutions to ( 7), ( 8), [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF] as long as they exist; we shall prove later on that the sequence of sampling instants does not result in an accumulation point, which guarantees that a solution to [START_REF] Hjalmarsson | Iterative feedback tuning: theory and applications[END_REF], ( 8), [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF] exists for all times (and is unique). This logic may provide asynchronous and sporadic control updates. In fact, it owes its popularity thanks to this latter feature besides its conceptual simplicity.

Closed-loop stability depends on σ, which must be chosen sufficiently small to control the norm of error e. How small σ should be is system-dependent in the sense that the matrices A and B determine which values of σ are allowed, see [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]Section IV]. Hereafter, we propose a data-based method to determine σ.

Consider any controller gain K computed via Theorem 1 and let V pxq " x J Sx for any x P R n with S " pX 0 Y q ´1, which is positive definite by Theorem 1. Function V serves as a Lyapunov function for the closed-loop system ignoring sampling. In particular we have, for any z " px, eq P R 2n ,

x∇V pxq, pA `BKqx `BKey " z J " ´Q SX 1 L pSX 1 Lq J 0  loooooooooooomoooooooooooon ": M z, (11) where
Q :" ´pSX 1 Gq J ´SX 1 G, (12)
which is positive definite in view of Theorem 1 and its proof (recall that X 1 G " A `BK by Lemma 1). Matrix L in [START_REF] Liu | Data-driven selftriggered control via trajectory prediction[END_REF] is any solution to the system of equations

" K 0  " " U 0 X 0  L, (13)
which exists under Assumption 1. Note that (13) implies that BK " X 1 L. Recall now that the event-triggering rule [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF] ensures that z J Ψpσqz ď 0 along the solutions to ( 7), ( 8), [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF]. Hence, the global exponential stability of the origin of closed-loop system ( 7), ( 8) is ensured provided we select σ such that the following implication4 is satisfied for some ε ą 0 and any z P R 2n

z J Ψpσqz ď 0 ùñ z J M z ď ´ε|z| 2 . ( 14
)
This implication ensures that V exponentially decreases along any solution to ( 7), ( 8), [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF] in view of [START_REF] Liu | Data-driven selftriggered control via trajectory prediction[END_REF] and the facts that |x| ď |z| for any z " px, eq P R 2n and that V is a positive definite quadratic function. This implies that x " 0 is globally exponentially stable for system [START_REF] Hjalmarsson | Iterative feedback tuning: theory and applications[END_REF], ( 8), (10) (again, provided Zeno phenomenon does not occur). The next theorem provides a data-based condition to select σ in [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF] to ensure the desired stability property. Theorem 2: Suppose Assumption 1 holds and consider system (7), ( 8), [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF] with K " U 0 Y pX 0 Y q ´1, Y being any solution to [START_REF] Safonov | The unfalsified control concept and learning[END_REF]. Let µ, σ ą 0 be such that the next SDP (in the decision variables µ and σ 2 ) is satisfied

µM ´Ψpσq ă 0, (15)
with S " pX 0 Y q ´1 and L as in [START_REF] Digge | Data-driven event-triggered control for discrete-time LTI systems[END_REF]. Then the following holds.

(a) There exists a global minimum inter-event time, in particular, for any solution x to the system, the sequence of transmission times tt k u kPN satisfies t k`1 ´tk ě τ pσq for every k P N " N 0 where τ pσq :"

1 α σ 1`σ with α :" maxt}X 1 G}, }X 1 L}u. (b)
The origin of the system is globally exponentially stable. l Proof. We first show the feasibility of [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF]. Let σ :" a µ{c with c ą 0 any sufficiently large constant such that ´Q `I{c ă 0 where Q comes from [START_REF] Cordovil | Learning event-triggered control based on evolving data-driven fuzzy granular models[END_REF]; clearly, this c exists because Q ą 0. For such a choice of σ, we have ´µQ `σ2 I " µp´Q `I{cq ă 0 for every µ ą 0. Therefore, for this choice of σ, (15) is equivalent by a Schur complement to the two conditions µp´Q `I{cq ă 0 and

´I `µpSX 1 Lq J pQ ´I{cq ´1pSX 1 Lq ă 0 (16)
which are jointly satisfied for µ sufficiently small. We now prove item (a) of Theorem 2. We follow the same steps as [3, Section III] adapted to the data-based context of this work. Let x be a solution to [START_REF] Hjalmarsson | Iterative feedback tuning: theory and applications[END_REF], [START_REF] Campi | Virtual reference feedback tuning: a direct method for the design of feedback controllers[END_REF], [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF]. The claim is trivial when xp0q " 0. When xp0q ‰ 0, xptq ‰ 0 for any t in the domain of the solution, see [START_REF] Postoyan | Explaining the 'mystery' of periodicity in inter-transmission times in two-dimensional eventtriggered controlled system[END_REF]Proposition 4]. Moreover, we also have from [35, Proposition 1] that the number of jumps is infinite, i.e., N " N 0 . Let k P N 0 , we have for any

t P rt k , t k`1 q, d dt |eptq| |xptq| " eptq J 9 eptq |eptq||xptq| ´xptq J 9 xptq|eptq| |xptq| 3 (17a) ď ˆ1 `|eptq| |xptq| ˙| 9 xptq| |xptq| (17b) ď α ˆ1 `|eptq| |xptq| ˙2 (17c)
with α :" maxt}X 1 G}, }X 1 L}u, where the last inequality is obtained by recalling that A `BK " X 1 G and BK " X 1 L, so that 9

x " X 1 Gx `X1 Le which implies | 9 x| ď αp|x| `|e|q. Therefore, the time needed for |e|{|x| to reach σ is lower bounded by τ pσq, the time needed for φ to grow from 0 (the value of |e|{|x| right after each jump) to σ, where φ is the solution to the differential equation 9φ " αp1 `φq 2 . The expression of τ pσq is given by

τ pσq :" 1 α σ 1 `σ . (18)
Thus, t k`1 ´tk ě τ pσq. Since the solution x and the number of jump k have been arbitrarily selected, we have proved that τ pσq is a global minimum inter-event time. This property implies that any solution to ( 7), ( 8), ( 10) is complete, and the linearity of the flow vector field and (10) ensure the uniqueness of the solution for each initial condition, see [START_REF] Cortés | Discontinuous dynamical systems[END_REF]. Finally, as [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF] holds, there exists ε ą 0 such that M ă 1{µ Ψpσq ´εI. Hence, the satisfaction of [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF] implies that (14) holds. Equation ( 10) enforces z J Ψpσqz ď 0 along the solutions to ( 7), ( 8), [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF]. Hence, for any solution x to ( 7), ( 8), [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF], 9

V pxptqq " zptq J M zptq ď ´ε|zptq| 2 for any t ě 0; note that V pxq is not affected by jumps at the triggering instants. Since V is quadratic and positive definite, we conclude that item (b) of Theorem 2 holds by resorting to standard Lyapunov arguments.

The condition in [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF], which is linear in the decision variables µ ą 0 and σ 2 ą 0, provides a data-based condition to design σ so that the induced event-triggered controlled system enjoys a global exponential stability property as well as the existence of global minimum inter-event time τ pσq. We emphasize that the expression of τ pσq in item (a) of Theorem 2 only depends on experimental data. As a result, different experiments and even different solutions to the SDP in [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF] may lead to a different value for τ pσq. This expression of τ pσq can, by the way, be used as a sampling period for periodic implementations.

Remark 4: Condition (15) may be used to maximize σ, which may help enlarging the inter-event times thereby reducing the number of transmissions, see [START_REF] Postoyan | Explaining the 'mystery' of periodicity in inter-transmission times in two-dimensional eventtriggered controlled system[END_REF]Proposition 3] for further insights on the relationship between the inter-event times and σ. In particular, we can implement the next SDP in place of [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF] in this case maximize µ,σ 2 σ 2 (19a) subject to [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF].

In this regard, it is interesting to note that we may achieve parsimonious sampling while controlling the performance. In fact, suppose that we replace [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF] with µM γ ´Ψpσq ă 0, where M γ is equal to M except for Q, which is replaced by p1 ´γqQ where γ P p0, 1q is a design parameter. Theorem 2 remains valid with the difference that instead of x∇V pxq, pA `BKqx `BKey ď ´ε|z| 2 ď ´ε|x| 2 for some ε ą 0, we will now have x∇V pxq, pA `BKqx `BKey ď ´γx J Qx for any z P R 2n . In this way, we can maximize σ while keeping a fixed guaranteed level of performance in terms of Lyapunov decay along the solutions. l Example 1: Consider system (1) with d " 0 and

A " " 0 0 ´1 ´2 , B " " 1 0  ,
which is marginally stable. We collect the data by running an experiment with input uniformly distributed in r´1, 1s and initial state within the same interval. The data are collected by keeping the input constant, in a sample-and-hold fashion, during each interval riT s , pi `1qT s q with i " 0, 1, . . . , T ´1 and T s " 0.1. We collect T " 10 samples which is sufficient for Assumption 1 to hold. Note that theoretically we need T ě n `m, so that even T " 3 samples may be sufficient. We solve SDP (6) and we obtain K " " ´0.9839 1.7489 ‰ . We then solve [START_REF] De Persis | Input-to-state stabilizing control under denialof-service[END_REF], which gives σ " 0.4595. According to (18), the minimum inter-event time τ pσq is equal to 0.1184, which is only an estimate of the actual minimum inter-event times. Figure 1 reports simulations results.

We have not observed critical issues regarding the choice of the sampling time T s which is used for collecting the data, except when it becomes extremely small (T s ă 0.001). This is not surprising: in this case the samples xpiT s q and upiT s q are approximately constant. In particular, (6) becomes infeasible as soon as X 0 (approximately) looses rank. l

IV. LEARNING FROM NOISY DATA

In this section, we consider system (1) with d not identically equal to 0. With noisy data, the analysis and the design of the event-triggered controller become sensibly more involved, and the reason is twofold. First, noisy data prevent us from obtaining an exact data-based representation of the closedloop behavior as in Lemma 1. More generally, noisy data prevent exact identification of the dynamics, whether it is the closed-loop dynamics (Lemma 1) or the open-loop dynamics (matrices A and B). Using the language of set-membership identification, we can say that we now have a set of dynamics consistent with the data and we should find a controller that stabilizes the whole set. In this respect, Assumption 1 makes sure that this uncertainty set is bounded, cf. [37, Lemma 2]. Second, disturbances impact not only the learning phase but, subsequently, also the behavior of the closed-loop system. In particular, Zeno phenomenon may arise in this case with the triggering rule built in Section III-C even for arbitrarily small disturbances, as demonstrated in [START_REF] Borgers | Event-separation properties of event-triggered control systems[END_REF].

Because we do not assume that the disturbance d is vanishing, we can no longer hope to prove that x " 0 is globally exponentially stable for the considered closed-loop system as in Theorems 1 and 2. We will establish instead a (global) practical exponential ISS property, i.e., that there exist c 1 ě 1, c 2 , c 3 , c 4 ą 0 such that any solution x to (1) with input disturbance d in closed-loop with (2) satisfies

|xptq| ď c 1 e ´c2t |xp0q| `c3 }d} r0,ts `c4 ν, (20)
where ν ě 0 is an extra tuning parameter we will introduce to design the triggering rule. When [START_REF] Abdelrahim | Robust eventtriggered output feedback controllers for nonlinear systems[END_REF] holds with ν " 0, we say that the system is exponentially ISS for system (1), [START_REF] Astr | Comparison of Riemann and Lebesgue sampling for first order stochastic systems[END_REF].

As in Section III, we first design the control law ignoring sampling and we then present the design of the triggering policy. Two triggering techniques are presented and discussed in this section, whether these lead to (practical) exponential ISS. Again, the extension of the presented results to other triggering conditions is addressed in Section V.

A. Learning a feedback controller

The first step is to derive an analogue of Lemma 1. Suppose we perform an experiment on the system, and we collect state and input samples5 that satisfy 9

xptq " Axptq `Buptq `dptq, t P t0, T s , . . . , pT ´1qT s u. The samples are then grouped into the data matrices U 0 , X 0 , X 1 as in (4). Let

D 0 :" " dp0q dpT s q ¨¨¨dppT ´1qT s q ‰ (21)
be the unknown data matrix made of the off-line samples of d. We have the next result. Lemma 2: Let Assumption 1 hold and consider any matrix K P R mˆn . Then A `BK " pX 1 ´D0 qG where G P R T ˆn is any solution to the system of equations [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF].

Proof. By Assumption 1, there exists a T ˆn matrix G such that (5) holds. Hence A `BK " r B A sr K I s " r B A sr U0 X0 sG, where the second identity follows from [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF]. The result follows because the elements of U 0 , X 0 , X 1 and D 0 satisfy the relation 9 xptq " Axptq`Buptq`dptq, t P t0, T s , . . . , pT ´1qT s u which, in compact form, gives X 1 " AX 0 `BU 0 `D0 .

By Lemma 2, the closed-loop dynamics can be written as 9 xptq " pA`BKqxptq " pX 1 ´D0 qG 1 xptq, which depends on the unknown matrix D 0 . As a result, [START_REF] Safonov | The unfalsified control concept and learning[END_REF] no longer provides stability guarantees. In fact, the constraint (6a) ensures that X 1 G 1 is Hurwitz. However, the matrix of interest is now Φ :" pX 1 ´D0 qG 1 in view of Lemma 2, and X 1 G 1 Hurwitz does not imply that Φ is Hurwitz. To be able to prove an ISS property for system (1) in closed-loop with u " Kx (ignoring sampling), we need to modify (6a) accounting for the uncertainty induced by D 0 . An effective way to achieve this is to ensure that pX 1 ´DqG 1 satisfies a Lyapunov equation with a common (symmetric) Lyapunov matrix Y ą 0 for all the matrices D in a given set D to which D 0 is deemed to belong, as formalized next. We thus introduce the set

D :" tD P R nˆT : DD J ĺ ∆∆ J u, (22)
where ∆ P R nˆs is a known matrix, and we make the next assumption.

Assumption 2: D 0 P D. l The choice of set D reflects our prior information or guess about disturbance d. For instance, if we know that there exists δ ą 0 such that }d} 8 ď δ like in [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF], then we can take ∆ :" δ ? T I, where we recall that T is the number of samples. Stochastic disturbances can also be accounted for (possibly with other choices of ∆) as advocated in [START_REF] De Persis | Low-complexity learning of linear quadratic regulators from noisy data[END_REF], [START_REF] De Persis | Learning controllers from data via approximate nonlinearity cancellation[END_REF].

We aim at enforcing the next inequality instead of (6a)

pX 1 ´DqY `Y J pX 1 ´Dq J `Ω ă 0 @D P D (23)
where Ω ą 0 is a free design parameter. By ensuring ( 23), we will see that system 9 xptq " pX 1 ´DqGxptq `dptq is exponentially ISS given any D P D, and therefore for D " D 0 in view of Assumption 2. Large sets D make the condition D 0 P D easier to hold but make (23) more difficult to satisfy.

Condition ( 23) cannot be implemented directly as it involves infinitely many constraints. The next result provides a tractable (convex) sufficient condition for [START_REF] Wang | On event design in event-triggered feedback systems[END_REF] to hold, thereby extending Theorem 1 to the case of noisy data.

Theorem 3: Consider stabilizable system (1) and suppose the following holds.

(i) Assumptions 1 and 2 hold with ∆ given. (ii) Let Ω ą 0, there exist ą 0 and Y P R T ˆn such that

« X 1 Y `pX 1 Y q J `Ω ` ∆∆ J Y J Y ´ I ff ă 0, (24a)
X 0 Y ą 0. (24b) Let K " U 0 Y pX 0 Y q ´1, then 9
xptq " pA `BKqxptq `dptq is exponentially ISS. l The proof of Theorem 3 rests on the next result, which we prove in Appendix B to not overload the exposition.

Lemma 3: Suppose there exist a scalar ą 0 and a matrix Y P R T ˆn such that (24a) holds. Then (23) holds. l Proof of Theorem 3. Let G :" Y S where S :" pX 0 Y q ´1. Note that X 0 G " I and that K " U 0 Y pX 0 Y q ´1 can also be written as K " U 0 G. Accordingly, the identity (5) holds and Lemma 2 implies that A`BK " pX 1 ´D0 qG. We now prove that pX 1 ´D0 qG is Hurwitz. By Lemma 3,[START_REF] Wang | On event design in event-triggered feedback systems[END_REF] holds and this implies that SpX 1 ´DqG`G J pX 1 ´Dq J S ă 0 for all D P D. Thus, pX 1 ´DqG is Hurwitz for all D P D. As D 0 P D by Assumption 2, we thus have A `BK Hurwitz, which implies that 9

xptq " pA `BKqxptq `dptq is exponentially ISS, see [START_REF] Isidori | Nonlinear Control Systems[END_REF]Example 10.4.1] Theorem 3 gives a robust approach to controller design, see also [START_REF] Van Waarde | From noisy data to feedback controllers: Non-conservative design via a matrix S-lemma[END_REF], [START_REF] Dean | On the sample complexity of the linear quadratic regulator[END_REF], and [START_REF] Bisoffi | Data-driven control via Petersen's lemma[END_REF]. Approaches that explicitly enforce or prioritize robustness usually outperform certainty-equivalence design when it comes to ensuring closed-loop stability. This observation has been made in various instances, e.g. [START_REF] Dean | On the sample complexity of the linear quadratic regulator[END_REF], [START_REF] Dörfler | On the certainty-equivalence approach to direct data-driven LQR design[END_REF]. We refer the interested reader to these papers for numerical and theoretical comparisons. In the present context, our approach is particularly effective because it gives us information on the decay rate of the closed-loop dynamics even in the presence of uncertainty, see [START_REF] Wang | On event design in event-triggered feedback systems[END_REF]. This allows us to determine triggering policies that ensure a stable and well-defined network implementation, including a guaranteed minimum inter-event time, as detailed in the next section.

B. Learning a triggering policy 1) Mixed triggering condition: We now take into account sampling. The dynamics of the closed-loop system under the control law (2) becomes 9 xptq " pA `BKqxptq `BKeptq `dptq,

where we recall that eptq :" xpt k q ´xptq, t P rt k , t k`1 q.

There are two main obstacles compared to the noise-free case.

First, as we will see shortly, like A`BK depends on D 0 in the data-based representation according to Lemma 2, so do BK and M in [START_REF] Liu | Data-driven selftriggered control via trajectory prediction[END_REF]. This means that we cannot implement the inequality (15) used in Section III-C to design the triggering policy parameter σ. Second, we need to modify [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF] in order to ensure the existence of a global minimum inter-event time.

We start with the last point, and, inspired by the modelbased results in [START_REF] Borgers | Event-separation properties of event-triggered control systems[END_REF], we modify (10) as: t 0 " 0 and t k`1 " inftt P R : t ą t k and |eptq| " σ|xptq| `νu, [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF] where σ ą 0 is a design parameter to be determined, while ν ą 0 is an arbitrary constant. Triggering rule ( 27) is commonly called mixed (relative-absolute) threshold policy in the event-triggered control literature [START_REF] Borgers | Event-separation properties of event-triggered control systems[END_REF].

Next we introduce the following quantities in place of ( 9) and matrix M in [START_REF] Liu | Data-driven selftriggered control via trajectory prediction[END_REF], respectively, Ψpσq :"

" ´2σ 2 I 0 0 I  , (28)
and

M pDq :" » - ´SΩS 2 SpX 1 ´DqL pSpX 1 ´DqLq J 0 fi fl , (29)
where S :" pX 0 Y q ´1 and Y results from Theorem 3 and where L is any solution to [START_REF] Digge | Data-driven event-triggered control for discrete-time LTI systems[END_REF], which exists under Assumption 1.

We are going to show is that, instead of (15), we can consider the condition µM pD 0 q ´Ψpσq ĺ 0, which can be conveniently cast as a SDP. Consider any solution to [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF] and let V pxq :" x J Sx for any x P R n with S " pX 0 Y q ´1. Like in the noise-free case, V is an ISS Lyapunov function for the nominal dynamics, in particular we now have, for any z " px, eq P R 2n and d P R n , x∇V pxq, pA `BKqx `BKe `dy " 2 rpA `BKqx `BKe `ds J Sx (30a)

" 2 rpX 1 ´D0 qGx `pX 1 ´D0 qLe `ds J Sx (30b) ď ´xJ SΩSx `2 rpX 1 ´D0 qLe `ds J Sx (30c) " ´xJ ˆSΩS 2 ˙x `zJ M pD 0 qz `2d J Sx. (30d)
The second equality follows because A `BK " pX 1 ´D0 qG and BK " pX 1 ´D0 qL (see ( 5) and ( 13)), while the inequality is a consequence of [START_REF] Wang | On event design in event-triggered feedback systems[END_REF]. Finally, the last equality follows from the definition of M pDq. Since SΩS " pSΩSq J ą 0, if we ensure that z J M pD 0 qz ď c along the solutions to ( 25)-( 27)

for some positive constant c (and Zeno phenomenon does not occur), then we can conclude that (20) holds for system (25)-( 27). This is formalized in the next theorem. Theorem 4: Suppose the following holds. (i) Assumptions 1 and 2 are verified with ∆ given. (ii) Let Ω ą 0, (24) holds and K " U 0 Y pX 0 Y q ´1 is the resulting controller gain as in Theorem 3. (iii) There exist µ, , σ ą 0 such that

» - - ´µSΩS 2 `2σ 2 I µSX 1 L µS∆ ‹ ´I ` L J L 0 ‹ ‹ ´ I fi ffi fl ĺ 0.
(31) Then, for any ν ą 0, system ( 25)-( 27): (a) admits a global minimum inter-event time, in particular for any solution x with input disturbance d, the sequence of transmission times tt k u kPN satisfies t k`1 tk ě τ pσq for every k P N where τ pσq :" 1 α σ 1`σ , α :" maxt}X 1 G} `}∆}}G}, }X 1 L} `}∆}}L}, σδ{νu and δ :" }d} 8 ; (b) is practically exponentially ISS. l Proof. We first prove item (a) of Theorem 4. We adapt for this purpose the techniques in the proof of [START_REF] Borgers | Event-separation properties of event-triggered control systems[END_REF]Theorem IV.2] carried out in the model-based setting. Let x be a solution to system ( 25)-( 27

ď α ˆ1 `|eptq| |xptq| `ν ˙2 (32c)
with α :" maxt}X 1 G} `}∆}}G}, }X 1 L} `}∆}}L}, δ{νu where δ :" }d} 8 . The last inequality follows from the identities A `BK " pX 1 ´D0 qG and BK " pX 1 ´D0 qL, which imply 9

x " pX 1 ´D0 qGx `pX 1 ´D0 qLe `d. This, in turn, gives | 9 x| ď αp|x| `|e| `νq. Hence, the time needed for |e|{p|x| `νq to reach σ (the event that triggers a new sampling) is not smaller than the time denoted τ pσq needed for φ to reach σ, where φ is the solution to the differential equation 9

φ " αp1 `φq 2 with initial value φp0q " 0. This time is

τ pσq :" 1 α σ 1 `σ . ( 33
)
Hence, t k`1 ´tk ě τ pσq, and item (a) of Theorem 4 holds as x and k have been arbitrarily selected. We then derive that any solution to ( 25)-( 27) is complete. Consider now item (b) of Theorem 4. By a Schur complement, [START_REF] Lopez | On a continuous-time version of Willems' lemma[END_REF]

is equivalent to µ « ´SΩS 2 SX 1 L ‹ 0 ff ´Ψpσq ` " 0 L J  " 0 L ‰ ` ´1 " µS∆ 0  " pµS∆q J 0 ‰ ĺ 0. ( 34
)
By applying Lemma 6 in the Appendix with B " r 0 L J s and C " r ´µS 0 s J we obtain

µ « ´SΩS 2 SX 1 L ‹ 0 ff ´Ψpσq `" 0 L J  D J " ´µS 0  J `"´µS 0  D " 0 L ‰ ĺ 0 @D P D, (35)
which can be compactly written as µM pDq ´Ψpσq ĺ 0 for any D P D. Thus, if [START_REF] Lopez | On a continuous-time version of Willems' lemma[END_REF] feasible then µM pDq´Ψpσq ĺ 0 for any D P D and Assumption 2 implies µM pD 0 q ´Ψpσq ĺ 0.

Let x be a solution to system ( 25)-( 27) with input disturbance d and t ě 0. The triggering rule ( 27) ensures |eptq| 2 ď 2σ 2 |xptq| 2 `2ν 2 . Therefore, zptq J Ψpσqzptq ď 2ν 2 . Since µM pD 0 q ´Ψpσq ĺ 0 then zptq J M pD 0 qzptq ď 2ν 2 {µ. Substituting this inequality into (30d) and resorting to standard Lyapunov arguments lead to the satisfaction of item (b).

In Theorem 4, we have two LMIs: [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF] and [START_REF] Lopez | On a continuous-time version of Willems' lemma[END_REF]. The first one is related to computing the controller, and the second one is related to computing the triggering law. It can be shown that if [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF] is feasible then ( 31) is also feasible. Furthermore, ( 24) is feasible whenever the noise level is sufficiently small, which follows because a solution always exists in the noiseless case (cf. Theorem 1) and the eigenvalues of the matrix in [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF] depend continuously on ∆. These facts are consistent with the intuition: if the noise level is sufficiently small then we can find a stabilizing controller (i.e., we can solve [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF]), in which case we can always find an event-triggered implementation of the control law (i.e., we can solve [START_REF] Lopez | On a continuous-time version of Willems' lemma[END_REF]).

Similarly to σ, the choice of ν involves a trade off between performance and the number of transmissions: by decreasing ν, the neighborhood of the origin to which the solutions to [START_REF] Liu | Event-based control of nonlinear systems with partial state and output feedback[END_REF], [START_REF] De Persis | Formulas for data-driven control: Stabilization, optimality, and robustness[END_REF], [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF] converges to "shrinks" in view of item (ii) of Theorem 4, which typically leads to more transmissions. In particular, it follows from the proof of Theorem 4 that, when d " 0, the state converges to the ball of R n centered at the origin of radius ν b 2κpSq ωµ , where κpSq is the condition number of S and ω is the smallest eigenvalue of SΩS{2. To ensure an exponential ISS property for the event-triggered controlled system, and not a practical exponential ISS property as in Theorem 4, a different triggering rule is needed (see Remark 6 below): this is the purpose of the next section. Before that, we illustrate the results of Theorem 4 for the system considered in Example 1, and we provide two remarks on the triggering rule [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF].

Example 2: Consider the same system as in Example 1 but this time a disturbance d affects the system dynamics, with }d} 8 ď δ. As before, we collect the data by running an experiment with an input uniformly distributed in r´1, 1s, and with initial state within the same interval. We consider two cases, δ " 0.1 and δ " 0.5. For both cases, we first solve the SDP [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF] with Ω :" 10I to find K; all the choices of Ω that we tested in the form Ω " cI, c P r1, 100s, led to τ pσq of similar magnitude. Then we solve the SDP (31) to find σ, in particular, like [START_REF] De Persis | Input-to-state stabilizing control under denialof-service[END_REF], we cast [START_REF] Lopez | On a continuous-time version of Willems' lemma[END_REF] as an optimization problem where we search for a solution maximizing σ. We use ∆ " δ ? T I in both the SDPs. For δ " 0.1 we obtain The minimum inter-sampling time observed in simulation is 0.0184 for δ " 0.1 and 0.0021 for δ " 0.5, which is about one order of magnitude larger than the theoretical lower bound.

K " " ´3.1769 1.8145 ‰ and σ " 0.0624. Using the value ν " 0.01 in the triggering rule [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF] gives the lower bound τ pσq " 0.0135 for the inter-sampling times. For δ " 0.5 we have K " " ´6.8882 1.5924 ‰ and σ " 0.0058. Using ν " 0.01 in the triggering rule [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF] gives the lower bound τ pσq " 3.9618e-04. For both the cases, we report simulation results in Figure 2.

For both values of δ, the sampling is very frequent initially when the state is far from the origin, and becomes sporadic as soon as the state gets smaller. Here, ν " 0.01 gives a good trade-off between performance and number of samplings. We note that ν can also be tuned online since its choice affects neither [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF] nor [START_REF] Lopez | On a continuous-time version of Willems' lemma[END_REF], hence it can be chosen after K and σ are determined. A second remark regards d. The noise level affects the feasibility of both ( 24) and [START_REF] Lopez | On a continuous-time version of Willems' lemma[END_REF], which become infeasible when the noise level becomes too high. In this example, feasibility is preserved as long as the energy of the noise remains about half that of u (δ « 0.5, which corresponds to an input-disturbance signal-to-noise ratio of about 6dB). Once we find a solution, however, practical exponential ISS holds irrespective of the noise level. Obviously, for large noise levels we might need to increase ν in order to reduce the number of samplings. This is evident from [START_REF] Van Waarde | Data informativity: a new perspective on data-driven analysis and control[END_REF] where we see that increasing δ can make α larger, in which case τ pσq decreases.

Remark 5: The proof of Theorem 4 shows that we can consider this triggering rule t k`1 " inftt P R : t ą t k and |eptq| 2 " 2σ 2 |xptq| 2 `2ν 2 u instead of [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF]. This is clear from the stability analysis in the proof of Theorem 4, and we also have that, for any given state, the next transmission time with this rule will occur not before the one generated by [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF] as |e| ď σ|x| `ν implies |e| 2 ď 2σ 2 |x| 2 `2ν 2 for any px, eq P R 2n . As a result, τ pσq is also a global minimum interevent time for the above policy. Note also that, since Theorem 4 applies for any ν ą 0, in view of the definition of Ψpσq in [START_REF] Scheres | Event-triggered control in presence of measurement noise: A space-regularization approach[END_REF], we can equivalently consider the next triggering rule t k`1 " inftt P R : t ą t k and zptq J Ψpσqzptq " νu. [START_REF] Cortés | Discontinuous dynamical systems[END_REF] This formulation will be convenient when addressing the generalized triggering conditions of Section V. l Remark 6: We discuss in this remark two special cases of [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF], which are not covered by Theorem 4. First, when ν " 0 in [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF], it is not possible to guarantee that the intersampling times remain strictly positive in general as shown in [START_REF] Borgers | Event-separation properties of event-triggered control systems[END_REF]. Second, when σ " 0 in [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF], we can also prove that system (25), ( 26), ( 27) is practically exponentially ISS for any ν ą 0, as in item (b) of Theorem 4. However, in this case, the lower-bound on the inter-event times in item (a) of Theorem 4 is no longer global but semiglobal, in the sense that it depends on the ball of initial conditions of the state (and the supremum of the disturbance on r0, 8q) like in [START_REF] Borgers | Event-separation properties of event-triggered control systems[END_REF]Theorem IV.4]. In fact, for any c x ą 0, any solution x to (25)-( 27) with |xp0q| ď c x and input disturbance d, any k P N and almost all t P rt k , t k`1 s, d dt |eptq| ď c Φ |xptq| `ce |eptq| `}d} 8 , where c Φ :" }X 1 G} `}∆}}G} and c e :" }X 1 L} `}∆}}L}. By following similar lines as in the proof of Theorem 4 there exists a constant c x , which depends on c x , ν and }d} 8 , such that d dt |eptq| ď c Φ c x `ce |eptq| `}d} 8 from which we infer that the time needed for |eptq| to grow from 0 to ν is lower bounded by 1 ce ln ´ceν cΦcx`}d}8 `1¯, which indeed depends on the radius of the ball of initial conditions of the state c x and the supremum norm of d. Note that in this case, i.e., σ " 0 and ν ą 0, the design of triggering rule directly follows from Theorem 3 as item (iii) of Theorem 4 always holds when σ " 0. l 2) Time-regularized triggering condition: We propose in this section an alternative triggering condition to ensure an exponential ISS property, as opposed to practical exponential ISS as in Section IV-B1, at the price of potentially more transmissions as suggested by Example 3 provided hereafter.

Our starting point is the characterization of the Lyapunov function given in [START_REF] Berberich | Datadriven analysis and control of continuous-time systems under aperiodic sampling[END_REF], i.e., for any z " px, eq P R 2n and d P R n , x∇V pxq, pA `BKqx `BKe `dy

ď ´xJ SΩSx `2 rpX 1 ´D0 qLe `ds J Sx ď ´ω1 |x| 2 `ω2 |x||e| `ω3 |x||d|, (37)
with ω 1 the smallest eigenvalue of SΩS, ω 2 :" 2}SX 1 L} 2}S}}∆}}L} and ω 3 :" 2}S}. From the above expression it readily follows that a sufficient condition to ensure exponential ISS is to ensure along any solution to [START_REF] Liu | Event-based control of nonlinear systems with partial state and output feedback[END_REF], [START_REF] De Persis | Formulas for data-driven control: Stabilization, optimality, and robustness[END_REF] and any t in the domain of the solution |eptq| ď σp|xptq| `}d} r0,ts q (38)

with σ P p0, ω 1 {ω 2 q , (39)
provided Zeno phenomenon does not occur. When (38) holds, 9

V pxptqq ď ´pω 1 ´σω 2 q|xptq| 2 `pσω 2 `ω3 q|xptq|}d} r0,ts from which exponential ISS follows as ω 1 ´σω 2 ą 0 (again, provided Zeno phenomenon does not occur). We first provide a model-based condition that ensures [START_REF] De Persis | Low-complexity learning of linear quadratic regulators from noisy data[END_REF]. This result is a variant of [19, Lemma 1] in which we consider the logarithmic norm of A instead of the induced 2-norm considered here. We consider the induced 2-norm of A because it is somehow easier to infer from data 6 . Lemma 4 ([19], Lemma 1): Let K be any feedback matrix that makes A `BK Hurwitz and

τ m pσq :" $ ' ' ' ' & ' ' ' ' % 1 }A} log ˆσ 1 `σ }A} maxt}A `BK}, 1u `1i f A ‰ 0, σ 1 `σ 1 maxt}BK}, 1u
otherwise.

(40) Given any triggering policy and any solution x to the corresponding closed-loop system [START_REF] Liu | Event-based control of nonlinear systems with partial state and output feedback[END_REF], [START_REF] De Persis | Formulas for data-driven control: Stabilization, optimality, and robustness[END_REF] with input disturbance d, for any k P N and any t P rt k , t k `τm pσqs X rt k , t k`1 s, |eptq| ď σp|xptq| `}d} r0,ts q. l The expression of τ m pσq in [START_REF] Isidori | Nonlinear Control Systems[END_REF] depends on the model via the terms }A} and }A `BK}. A lower bound of τ m pσq can be derived based on the available data in D. We use for this purpose data-based upper-bounds on }A} and }A BK}, as the expression in ( 40) is monotonically decreasing in }A} and }A `BK}. We notice for this purpose that A`BK " pX 1 ´D0 qG with G satisfying (5), while A satisfies r B A s " pX 1 ´D0 qr U0 X0 s : where M : is the right inverse of the matrix M . Partitioning r U0 X0 s : " r J0 V0 s with V 0 having the same dimension as A, we thus have A " pX 1 ´D0 qV 0 . Then, }A`BK} ď }X 1 G}`}∆}}G} and }A} ď }X 1 V 0 }`}∆}}V 0 }, and the upper bounds are both computable from data alone. We then have the next result.

Lemma 5: Let K be any feedback matrix that makes A BK Hurwitz and

τ d pσq :" 1 c A log ˆσ 1 `σ c A maxtc Φ , 1u `1˙( 41
)
with c A :" }X 1 V 0 } `}∆}}V 0 } and c Φ " }X 1 G} `}∆}}G}.

Given any triggering policy and any solution x to the corresponding closed-loop system [START_REF] Liu | Event-based control of nonlinear systems with partial state and output feedback[END_REF], [START_REF] De Persis | Formulas for data-driven control: Stabilization, optimality, and robustness[END_REF] with input disturbance d, for any k P N and any t P rt k , t k `τd pσqs X rt k , t k`1 s, |eptq| ď σp|xptq| `}d} r0,ts q. l Proof. The result follows from Lemma 4 and the fact that τ d pσq ď τ m pσq. Specifically, for A ‰ 0 the inequality τ d pσq ď τ m pσq follows since τ m pσq decreases monotonically as }A} and/or }A `BK} increase, while for A " 0 the inequality τ d pσq ď τ m pσq follows since logp1 `sq ď s for every s ě 0.

With this result in hands, we derive the next triggering policy: t 0 " 0 and t k`1 " inf t P R : t ě t k `τd pσq and zptq J Ψpσqzptq ě 0 ( (42) with Ψpσq and σ as in ( 9) and [START_REF] De Persis | Learning controllers from data via approximate nonlinearity cancellation[END_REF], respectively. The rule in ( 42) is a time-regularized version of [START_REF] Sznaier | Control oriented learning in the era of big data[END_REF], which prevents arbitrary fast sampling by enforcing global minimum interevent time τ d pσq ą 0 given in [START_REF] Dean | On the sample complexity of the linear quadratic regulator[END_REF]. We next state the main result of this section. Afterwards, we discuss pros and cons of this approach with respect to the one presented in Section IV-B1.

Theorem 5: Suppose the following holds. (i) Assumptions 1 and 2 are verified with ∆ given. (ii) Let Ω ą 0, SDP (24) is feasible and K " U 0 Y pX 0 Y q ´1 is the resulting controller as in Theorem 3. Let σ and τ d pσq as in [START_REF] De Persis | Learning controllers from data via approximate nonlinearity cancellation[END_REF] and [START_REF] Dean | On the sample complexity of the linear quadratic regulator[END_REF], respectively, then system (25), ( 26), ( 42) is exponentially ISS. l Proof. Let x be a solution to [START_REF] Liu | Event-based control of nonlinear systems with partial state and output feedback[END_REF], [START_REF] De Persis | Formulas for data-driven control: Stabilization, optimality, and robustness[END_REF], [START_REF] Dörfler | On the certainty-equivalence approach to direct data-driven LQR design[END_REF] with input disturbance d. We first note that x is complete as it cannot explode in finite time and there exists a strictly positive minimum inter-event time τ d pσq ą 0, which excludes Zeno phenomenon. Let k P N . We know from Lemma 5 that |eptq| ď σp|xptq| `}d} r0,ts q for all t P rt k , t k `τd pσqs X rt k , t k`1 s " rt k , t k `τd pσqs here in view of [START_REF] Dörfler | On the certainty-equivalence approach to direct data-driven LQR design[END_REF]. Moreover, if t k`1 ą t k `τd pσq then |eptq| ă σ|xptq| at t " t k `τd pσq and thus |eptq| ď σp|xptq| `}d} r0,ts q for all t P rt k `τd pσq, t k`1 s. Therefore, (38) holds for t P rt k , t k`1 s. We derive that [START_REF] De Persis | Low-complexity learning of linear quadratic regulators from noisy data[END_REF] holds (with σ in (39)) along x. Since x and d have been chosen arbitrarily, [START_REF] Bisoffi | Data-driven control via Petersen's lemma[END_REF] and [START_REF] De Persis | Low-complexity learning of linear quadratic regulators from noisy data[END_REF] hold along solutions and V pxq is not affected by jumps at the triggering instants, we derive that system (25), ( 26), ( 42) is exponentially ISS.

Compared with the triggering rule ( 27), [START_REF] Dörfler | On the certainty-equivalence approach to direct data-driven LQR design[END_REF] has the merit to guarantee exponential ISS rather than practical exponential ISS under the conditions of Theorem 5. Further, this transmission policy does not require to solve the SDP in [START_REF] Lopez | On a continuous-time version of Willems' lemma[END_REF], thus removing the question of feasibility related to finding a sampling policy. On the other hand, the next example shows that this new triggering rule may have certain disadvantages, as it may result in many more transmissions.

Example 3: We consider again Example 2 under the same setting. As before, we consider two levels for the disturbance: δ " 0.1 and δ " 0.5. The control matrices are the same as before as the design of K relies on (24) also in this case. For δ " 0.1 a feasible value for σ is σ " 0.0933. Using this value and the estimates c A and c Φ , we find τ d pσq " 0.0197. For δ " 0.5 a feasible value for σ is σ " 0.0046. Using this value and the estimates c A and c Φ we find τ d pσq " 3.2511e-04. We report simulation results in Figure 3.

We see that the number of transmissions is much higher that the one obtained with the triggering rule [START_REF] Selivanov | Event-triggered H8 control: A switching approach[END_REF]. We see in particular that after an initial phase the triggering rule [START_REF] Dörfler | On the certainty-equivalence approach to direct data-driven LQR design[END_REF] eventually becomes almost periodic with t k`1 " t k `τd pσq. Note that in Figure 3, we can have |eptq| ą σ|xptq|. This is fully consistent with the triggering rule (42) which only guarantees |eptq| ď σp|xptq| `}d} r0,ts q along solutions as shown in the proof of Theorem 5. l We conclude this section with a remark on the possible combination of ( 27) and ( 42), which will be useful in Section V.

Remark 7: We can combine triggering rules ( 27) and (42) as: t 0 " 0 and

t k`1 " inf ! t P R : t ě t k `τd pσ 1 q and |eptq| ě σ 2 |xptq| `ν) , (43)
or, in view of (36),

t k`1 " inf ! t P R : t ě t k `τd pσ 1 q and zptq J Ψpσ 2 qzptq ě ν ) , (44)
with σ 1 as in (39) and σ 2 given by [START_REF] Lopez | On a continuous-time version of Willems' lemma[END_REF]. It is useful to distinguish parameters σ 1 and σ 2 used in ( 27) and [START_REF] Dörfler | On the certainty-equivalence approach to direct data-driven LQR design[END_REF], respectively, as these may be assigned to different values. The existence of a global minimum inter-event time immediately follows from the fact that τ d pσ 1 q ą 0. Moreover, in this case, the corresponding closed-loop system (25), ( 26), [START_REF] Khalil | Nonlinear control systems[END_REF] or [START_REF] Cortés | Discontinuous dynamical systems[END_REF] is practically exponentially ISS for ν ą 0 under the conditions of Theorem 4; the proof of this property follows similar steps as those in the proofs of Theorems 4 and 5. l

V. EXTENSION TO OTHER TRIGGERING RULES

The approach presented in the previous section opens the door to the design of a range of robust data-based versions of existing (model-based) triggering rules. We illustrate this by presenting data-based versions of quadratic triggering policies [START_REF] Heemels | An introduction to eventtriggered and self-triggered control[END_REF] in Section V-A, of dynamic event-triggered control [START_REF] Girard | Dynamic triggering mechanisms for event-triggered control[END_REF], [START_REF] Dolk | Output-based and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF] in Section V-B, and of the technique, which consists in imposing a desired (typically decreasing) threshold on V , like in [START_REF] Wang | On event design in event-triggered feedback systems[END_REF], [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF], in Section V-C.

A. Quadratic policies

The triggering rule designed for the noise-free case in Section III-C, which has then been "robustified" in Section IV, is a special type of quadratic policies [START_REF] Heemels | An introduction to eventtriggered and self-triggered control[END_REF]. We can extend these results to triggering policies based on more general quadratic forms, which may be advantageous to reduce the number of transmissions. We first explain how this can be done in the noise-free case.

The triggering rule in (10) developed when d " 0 in (1) is a special type of the general quadratic policy: t 0 " 0 and

t k`1 " $ ' & ' % inftt P R : t ą t k and zptq J r Ψzptq " 0u if xpt k q ‰ 0, `8 otherwise, (45)
with r Ψ P R 2nˆ2n symmetric to be designed. In Section III-C, we focused on the special case where r Ψ " Ψpσq and Ψpσq as in [START_REF] Fliess | Model-free control[END_REF] but other choices of r Ψ are possible as suggested by the model-based results in, e.g., [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], [START_REF] Heemels | An introduction to eventtriggered and self-triggered control[END_REF]. To synthesize r Ψ P R 2nˆ2n in (45), we follow a similar approach as in Theorem 2, as formalized next.

Proposition 1: Suppose Assumption 1 holds and consider system ( 7), ( 8), [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF] with K " U 0 Y pX 0 Y q ´1, Y being any solution to [START_REF] Safonov | The unfalsified control concept and learning[END_REF]. Let r Ψ P R 2nˆ2n symmetric, µ, σ ą 0 be such that the next SDP (in the decision variables µ, σ 2 and r

Ψ) is satisfied µM ´r Ψ ă 0 r Ψ ĺ Ψpσq, (46)
with Ψpσq and M as in ( 9) and [START_REF] Liu | Data-driven selftriggered control via trajectory prediction[END_REF], respectively, S " pX 0 Y q ´1, and L as in [START_REF] Digge | Data-driven event-triggered control for discrete-time LTI systems[END_REF]. Then the following holds.

(a) The system admits τ pσq defined in item (a) of Theorem 2 as global minimum inter-event time with σ as in [START_REF] Liberzon | Small-gain theorems of LaSalle type for hybrid systems[END_REF]. (b) The origin of the system is globally exponentially stable. l Proof. We first note that ( 46) is feasible as it is satisfied with r Ψ " Ψpσq for some small enough σ ą 0 according to the proof of Theorem 2.

Because of the second inequality in [START_REF] Liberzon | Small-gain theorems of LaSalle type for hybrid systems[END_REF], given any solution x to system (7), ( 8), [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF], at any transmission time t k with k P N , t k`1 ´tk is greater than or equal the time it takes for z J Ψpσqz to grow from pxpt k q, 0q J Ψpσqpxpt k q, 0q to 0, which corresponds to the inter-transmission time we would obtain with (10) starting at xpt k q. Item (a) of Theorem 2 ensures the latter is lower bounded by τ pσq in [START_REF] Borgers | Event-separation properties of event-triggered control systems[END_REF]. As a consequence, t k`1 ´tk ě τ pσq and since we have taken an arbitrary solution x and an arbitrary k P N , item (a) of Proposition 1 holds. The proof of item (b) of Proposition 1 follows the same steps as the proof of item (b) of Theorem 2 in view of the first inequality in [START_REF] Liberzon | Small-gain theorems of LaSalle type for hybrid systems[END_REF].

The conditions in Proposition 1 provide more flexibility in the design of matrix r Ψ compared to Theorem 2, which becomes a particular case by taking r Ψ " Ψpσq and Ψpσq as in [START_REF] Fliess | Model-free control[END_REF].

For the noisy case, to present the results in a compact way, we take inspiration from Remark 7, in particular [START_REF] Cortés | Discontinuous dynamical systems[END_REF], and consider the next triggering rule, which allows combining the techniques of Sections IV-B1 and IV-B2 in a unified manner: t 0 " 0 and

t k`1 " inf ! t P R : t ě t k `τ d pσ 1 q and zptq J r Ψzptq ě ν ) (47
) where σ 1 is as in [START_REF] De Persis | Learning controllers from data via approximate nonlinearity cancellation[END_REF], τ d pσ 1 q P t0, τ d pσ 1 qu with τ d pσ 1 q in (41), ν ě 0 arbitrary, and r Ψ P R 2nˆ2n symmetric to be designed. We have the next result.

Proposition 2: Suppose the following holds. (i) Assumptions 1 and 2 are verified with ∆ given. (ii) Let Ω ą 0, SDP (24) is feasible and K " U 0 Y pX 0 Y q ´1 is the resulting controller as in Theorem 3. (iii) There exist µ, , σ 2 ą 0 and r

Ψ P R 2nˆ2n symmetric such that » - - ´µSΩS 2 µSX 1 L µS∆ ‹ L J L 0 ‹ ‹ ´ I fi ffi fl ĺ " r Ψ 0 0 0  r Ψ ĺ Ψpσ 2 q, (48)
with L, ∆ and Ψpσ 2 q as in ( 13), ( 22) and ( 28), respectively. If τ d pσ 1 q " τ d pσ 1 q with σ 1 as in [START_REF] De Persis | Learning controllers from data via approximate nonlinearity cancellation[END_REF] or ν ą 0 in (47), system (25), ( 26), ( 47): (a) admits global inter-event time maxtτ d pσ 1 q, r τ pσ 2 qu with r τ pσ 2 q " τ pσ 2 q as in item (a) of Theorem 4 when ν ą 0, and r τ pσ 2 q " 0 when ν " 0;

(b) is practically exponentially ISS when ν ą 0, and is exponentially ISS when ν " 0. l Proof. Item (a) of Proposition 2 is immediate when τ d pσ 1 q " τ d pσ 1 q ą 0 and ν " 0 in view of [START_REF] Teel | On assigning the derivative of a disturbance attenuation control Lyapunov function[END_REF]. When τ d pσ 1 q " 0, then ν ą 0. In this case, we have that for a given state at which a transmission occurs, the next transmission will occur not earlier than the one generated by triggering rule [START_REF] Cortés | Discontinuous dynamical systems[END_REF] in Remark 5 as r Ψ ĺ Ψpσ 2 q, see (48). Since system (25), ( 26), (36) admits global inter-event time τ pσ 2 q according to item (a) of Theorem 4, so does system (25), ( 26), [START_REF] Teel | On assigning the derivative of a disturbance attenuation control Lyapunov function[END_REF] and item (a) of Proposition 2 holds in this case. The last case where τ d pσ 1 q " τ d pσ 1 q and ν ą 0 similarly follows.

To prove item (b) of Proposition 2, we consider a solution x to (25), ( 26), [START_REF] Teel | On assigning the derivative of a disturbance attenuation control Lyapunov function[END_REF] with input disturbance d and k P N . Let t P rt k , t k`1 q. If t ď t k `τ d pσ 1 q, then (38) holds by Lemma 5 and we have from (37)

9

V pxptqq ď ´pω 1 ´σ1 ω 2 q|xptq| 2 `pσ 1 ω 2 `ω3 q|xptq|}d} r0,ts ,

with ω 1 ´σ1 ω 2 ą 0 as σ 1 is such that (39) holds. If t P rt k τ d pσ 1 q, t k`1 s, then zptq J r Ψzptq ď ν according to [START_REF] Teel | On assigning the derivative of a disturbance attenuation control Lyapunov function[END_REF]. We deduce that µM pD 0 q ´r Ψ ĺ 0 from the first inequality in [START_REF] Wang | A unifying event-triggered control framework based on a hybrid small-gain theorem[END_REF] and by following similar developments as in [START_REF] Van Waarde | From noisy data to feedback controllers: Non-conservative design via a matrix S-lemma[END_REF] and [START_REF] Postoyan | Explaining the 'mystery' of periodicity in inter-transmission times in two-dimensional eventtriggered controlled system[END_REF]. Consequently, since zptq J r Ψzptq ď ν, µzptq J M pD 0 qzptq ď ν and we have from (30d) 9 V pxptqq ď ´xptq J ˆSΩS 2 ˙xptq`ν{µ`2dptq J Sxptq, (

where we recall that SΩS ą 0.

Based on (49) and (50), we deduce that system (25), ( 26), ( 47) is practically exponentially ISS; recall that V pxq is not affected by jumps at the triggering instants. Furthermore, when ν " 0, system (25), ( 26), ( 47) is exponentially ISS.

Like before, Proposition 2 extends Theorems 4 and 5 to more general quadratic triggering rules.

Remark 8: In Proposition 2, consistently with Remark 5, there are parameters σ 1 , used to design τ d pσ 1 q, and σ 2 , used to synthesize r Ψ. We can obviously take a single parameter σ by selecting the minimum value between feasible σ 1 , σ 2 . l

B. Dynamic event-triggered control

The triggering rules considered so far are static, in the sense that they rely on algebraic conditions involving x and e (and possibly the elapsed time since the last transmission). We can also add auxiliary variables to define the triggering rules as advocated in e.g., [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF], [START_REF] Girard | Dynamic triggering mechanisms for event-triggered control[END_REF], in order to potentially further reduce the number of transmissions while preserving stability. In this section, we show how the results of Section V-A can be extended to dynamic triggering rules like in [START_REF] Girard | Dynamic triggering mechanisms for event-triggered control[END_REF], [START_REF] Dolk | Output-based and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF] where model-based results are presented.

Consistently with the structure of the paper so far, we first focus on the noise-free case, before taking into account d in (1) again. We introduce for this purpose variable η P R ě0 , whose dynamics is, for any t P R ě0 , 9 ηptq " ´ληptq ´zptq J r Ψzptq, ηp0q ě 0,

where λ ą 0 is arbitrary and r Ψ P R 2nˆ2n is symmetric and has to be designed. The purpose of variable η is to filter the quadratic term considered in [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF] in Section V-A (in the noise-free case). Variable η does not experience jumps at transmission instants. Note that the state vector is now px, ηq and we will establish that the origin is globally asymptotically stable for the augmented system, as formalized in Proposition 3 below and discussed afterwards.

The triggering rule is defined as7 : t 0 " 0 and if pxpt k q, ηpt k qq ‰ 0, `8 otherwise, (52) where θ P R ě0 is an additional arbitrary tuning parameter. We have the next result.

t
Proposition 3: Suppose the conditions of Proposition 1 are satisfied. Then system ( 7), ( 8), (51), ( 52): (a) admits τ pσq in item (a) of Theorem 2 as a global minimum inter-event time with σ as in Proposition 1; (b) is globally asymptotically stable, in particular there exist c 1 ě 1 and c 2 ą 0 such that any solution px, ηq satisfies |pxptq, a ηptqq| ď c 1 e ´c2t |pxp0q, a ηp0qq| for any t ě 0. l Proof. Let px, ηq be a solution to the corresponding system (7), ( 8), (51), (52). We have that ηptq ě 0 for any t in the domain of the solution by following the same reasoning as in the proof of [START_REF] Girard | Dynamic triggering mechanisms for event-triggered control[END_REF]Lemma 2.2]. We also have that, given any k P N , the next transmission will occur not earlier than the one generated by triggering rule (45) by invoking similar arguments as in the proof of [START_REF] Girard | Dynamic triggering mechanisms for event-triggered control[END_REF]Proposition 2.3]. Consequently, item (a) of Proposition 3 holds in view of item (a) of Proposition 1.

Let z " px, eq P R 2n and η P R ě0 . We define U px, ηq " V pxq `η{µ with V pxq " x J Sx for any x P R n in Section III-C and µ as in [START_REF] Liberzon | Small-gain theorems of LaSalle type for hybrid systems[END_REF]. We have from the latter inequality that there exists ε ą 0 independent of z and η such that x∇U px, ηq, p 9

x, 9 ηqy " z J M z ´λη{µ ´1{µ z J r Ψz ď ´ε|z| 2 ´λη{µ,

where we write with some abuse of notation 9

x " pA`BKqxB Ke and 9

η " ´λη ´zJ r Ψz. By integrating (53) along the solutions to ( 7), ( 8), (51), (52), noting that U px, ηq is not affected by jumps at the triggering instants, and exploiting the expression of U , we deduce that item (b) of Proposition 3 holds.

According to Proposition 3, given a matrix r Ψ, which satisfies [START_REF] Liberzon | Small-gain theorems of LaSalle type for hybrid systems[END_REF], we can then design a dynamic version of the triggering condition in Section V-A in the noise-free case. The convergence of the x-component of the solutions is still exponential as the time goes to infinity in view of item (b) of Proposition 3.

In the noisy case, the dynamics of variable η needs to be modified as, similarly to [START_REF] Dolk | Output-based and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF], 9 ηptq P ´ληptq ´ψpt ´tk q ´zptq J r Ψzptq ´ν¯,

Fig. 1 .

 1 Fig. 1. Results for Example 1. Left: State trajectories. Middle: Behavior of |eptq| and σ|xptq|; a new sampling is triggered when |eptq| " σ|xptq|. Right: Behavior of the inter-sampling times t k`1 ´tk . The minimum inter-sampling time observed in simulation is 0.1371, which is not much larger than the theoretical lower bound τ pσq " 0.1184. Interestingly, the inter-sampling times exhibit an oscillatory behavior consistently with [35, Section IV.A].

Fig. 2 .

 2 Fig. 2. Results for Example 2. Top figures report simulation results for δ " 0.1 while bottom figures report the results for δ " 0.5. Left: State trajectories. Middle: Behavior of |eptq| and σ|xptq| `ν; a new sampling is triggered when |eptq| " σ|xptq| `ν. Right: Behavior of the inter-sampling times t k`1 ´tk .The minimum inter-sampling time observed in simulation is 0.0184 for δ " 0.1 and 0.0021 for δ " 0.5, which is about one order of magnitude larger than the theoretical lower bound.

Fig. 3 .

 3 Fig. 3. Results for Example 3. Top figures report simulation results for δ " 0.1 while bottom figures report the results for δ " 0.5. Left: State trajectories. Middle: Behavior of |eptq| and σ|xptq|; a new sampling is triggered when |eptq| " σ|xptq| if the next inter-sampling is not smaller than τ d pσq and after τ d pσq seconds otherwise. Right: Behavior of the inter-sampling times t k`1 ´tk . The minimum inter-sampling time observed in simulation is 0.0197 for δ " 0.1 and 3.2511e-04 for δ " 0.5, which is the same as the theoretical lower bound.

  ) with input disturbance d. Let ν :" ν{σ and k P N . For any t P rt k , t k`1 q

	d dt	|eptq| |xptq| `ν "	eptq J 9 eptq |eptq|p|xptq| `νq	xptq|eptq| ´xptq J 9 |xptq|p|xptq| `νq 2
					(32a)
		ď ˆ1	`|eptq| |xptq| `ν ˙| 9 xptq| |xptq| `ν	(32b)

Typically, we obtain knowledge of the state variables through physical considerations, such as in mechanical or electrical systems. More generally, the results we present could serve as a basis for addressing output-feedback event-triggered control in the future, as mentioned in Section VI.

The results in Section V offer consideration for a scenario where the network is between the controller and the actuators.

At the exception of the second part of Section V-B as specified there.

Relations involving quadratic inequalities are well-known in optimization and control theory, usually with the term "S-procedure". Recently, there has been interest for the S-procedure also in the context of data-driven control, see[START_REF] Van Waarde | From noisy data to feedback controllers: Non-conservative design via a matrix S-lemma[END_REF] for a recent discussion and new results.

See, again, Remarks 2 and 3 for explanations regarding the possible presence of noise affecting these data.

The result below holds for arbitrary σ ą 0. Imposing σ ă ω 1 {ω 2 is needed to make sure that[START_REF] De Persis | Low-complexity learning of linear quadratic regulators from noisy data[END_REF] guarantees ISS.

When θ " 0, triggering instants are only allowed when ηptq ď 0 and 9 ηptq ď 0, which can be computed on-line in the noise-free case. We do not specify this condition in (52) to not overload the corresponding equation. Note that this extra condition, namely 9 ηptq ď 0 when θ " 0, will not be needed when doing time-regularization in the noisy case in (55).

Claudio De Persis is a Professor with the Engineering and Technology Institute, University of Groningen, the Netherlands, since 2011. He received the Laurea and PhD degree in engineering in 1996 and 2000, both from the University of Rome "La Sapienza", Italy. He held postdoc positions at Washington University in St. Louis (2000-2001) and Yale University (2001-2002) and faculty positions at the University of Rome "La Sapienza" (2002-2009) and Twente University, the Netherlands (2009-2011). His main research interest is in automatic control and its applications. Romain Postoyan received the "Ingénieur" degree in Electrical and Control Engineering from EN-SEEIHT (France) in 2005. He obtained the M.Sc. by Research in Control Theory & Application from Coventry University (United Kingdom) in 2006 and the Ph.D. in Control Theory from Université Paris-Sud (France) in 2009. In 2010, he was a research assistant at the University of Melbourne (Australia). Since 2011, he is a CNRS researcher at CRAN (France). He received the "Habilitation à Diriger des Recherches' (HDR)" in 2019 from Université de Lorraine (Nancy, France). Pietro Tesi received the PhD degree in computer and control engineering from University of Florence, Italy, in 2010, where is currently an Associate Professor. Before that, he was an Assistant Professor at the University of Florence, Italy, and the University Groningen, the Netherlands. His research interests include adaptive and learning systems, data-driven control and network systems. Prof. Tesi is a Senior Editor for the IEEE Control Systems Letters and a member in the IFAC Technical Committee on Networked Systems. He is the recipient of the 2021 IEEE Control Systems Letters Outstanding Paper Award.

His work is supported by the project Digital Twin, number P18-03, of the research programme TTW Perspective which is (partly) financed by the Dutch Research Council (NWO). His work is supported by ANR via grant HANDY, number ANR-18-CE40-0010. His work is supported by the European Union under the Italian National Recovery and Resilience Plan (NRRP) of NextGenerationEU, partnership on "Telecommunications of the Future" (PE00000001 -program "RESTART"). E-mail:

with λ ą 0 arbitrary and r Ψ symmetric to be designed. Given τ d pσ 1 q as in Section V-A, namely τ d pσ 1 q P t0, τ d pσ 1 qu, τ d pσ 1 q as in in [START_REF] Dean | On the sample complexity of the linear quadratic regulator[END_REF] and σ 1 as in [START_REF] De Persis | Learning controllers from data via approximate nonlinearity cancellation[END_REF], ψ is defined as follows. When τ d pσ 1 q " τ d pσ 1 q, ψpsq " 0 for s P r0, τ d pσ 1 qq, ψpsq " 1 for s ą τ d pσ 1 q and ψpsq " r0, 1s for s " τ d pσ 1 q. When τ d pσ 1 q " 0, ψpsq " 1 for any s ě 0. In (54), the term zptq J r Ψzptq´ν, which is related to [START_REF] Teel | On assigning the derivative of a disturbance attenuation control Lyapunov function[END_REF], is filtered after τ d pσ 1 q units of time have elapsed. We note that (54) is a differential inclusion when τ d pσq " τ d pσq because ψ is multi-valued at τ d pσq, and solutions to (54) are understood in the Krasovskii sense on rt k , t k`1 s with k P N in this case, see [START_REF] Goebel | Hybrid dynamical systems[END_REF]Chapter 4.5]. The triggering rule becomes: t 0 " 0 and

and ηptq ´θpzptq J r Ψzptq ´νq ď 0

with arbitrary θ ě 0 and, again, σ 1 as in [START_REF] De Persis | Learning controllers from data via approximate nonlinearity cancellation[END_REF]. We have the next result. Proposition 4: Suppose the conditions of Proposition 2 are satisfied. If τ d pσ 1 q " τ d pσ 1 q or ν ą 0, system [START_REF] Liu | Event-based control of nonlinear systems with partial state and output feedback[END_REF], [START_REF] De Persis | Formulas for data-driven control: Stabilization, optimality, and robustness[END_REF], ( 54), (55):

(a) admits global inter-event time maxtτ d pσ 1 q, r τ pσ 2 qu as in item (a) of Proposition 2; (b) is (practically) ISS, in particular there exist c 1 ě 1 and c 2 , c 3 , c 4 ą 0 such that any solution px, ηq with disturbance input d satisfies |pxptq, a ηptqq| ď c 1 e ´c2t |pxp0q, a ηp0qq| `c3 }d} r0,ts `c4 ν. l Proof. Like in the proof of Proposition 3, we have that η ě 0 along the solutions to ( 25), ( 26), (54), (55). Moreover, for any solution px, ηq to (25), ( 26), (54), (55) with input disturbance d, at any k P N , the next inter-transmission time is greater than the corresponding inter-transmission time generated by [START_REF] Teel | On assigning the derivative of a disturbance attenuation control Lyapunov function[END_REF] at the same state. We then invoke item (a) of Proposition 2 to derive that item (a) of Proposition 4 holds.

To prove item (b) of Proposition 4, we consider the same Lyapunov function as in the proof of Proposition 3, namely U px, ηq " V pxq `η{µ with V pxq " x J Sx for any z " px, eq P R 2n and η P R ě0 with µ coming from [START_REF] Wang | A unifying event-triggered control framework based on a hybrid small-gain theorem[END_REF]. Let px, ηq be a solution to [START_REF] Liu | Event-based control of nonlinear systems with partial state and output feedback[END_REF], [START_REF] De Persis | Formulas for data-driven control: Stabilization, optimality, and robustness[END_REF], (54), (55) with input disturbance d, and k P N . We treat the case where τ d pσ 1 q " τ d pσ 1 q; the proof follows similar developments when τ d pσ 1 q " 0.

When t P rt k , t k `τd pσ 1 qq, we have from the definition of τ d pσ 1 q that (38) holds by Lemma 5. Hence, in view of (49) and ( 54), (49) 9 U pxptq, ηptqq ď ´pω 1 ´σ1 ω 2 q|xptq| 2 `pσ 1 ω 2 `ω3 q|xptq|}d} r0,ts ´λ{µ ηptq, (56) as in this case ψpt ´tk q " 0 and ω 1 ´σ1 ω 2 ą 0 for (39) holds.

When t P pt k `τd pσ 1 q, t k`1 q, we have from (37),

9

U pxptq, ηptqq ď ´xptq J `SΩS 2 ˘xptq `zptq J M pD 0 qzptq `2dptq J Sxptq ´λ{µ ηptq ´1{µ zptq J r Ψzptq `ν{µ.

We then follow similar developments as in the proof of Proposition 2 to conclude that

and we proceed as in the case where t P pt k `τd pσ 1 q, t k`1 q to derive (57). Item (b) of Proposition 4 follows by integration from (56), ( 57) and (59), noting that U px, ηq is not affected by jumps at the triggering instants. Remark 9: Compared to [START_REF] Qi | Data-driven L 2 -stability analysis for dynamic event-triggered networked control systems: A hybrid system approach[END_REF], the triggering condition in (55) is more general as: (i) it involves a general quadratic term; (ii) θ can be non-zero: (iii) a fixed threshold-like parameter ν is allowed, which may help reducing the number of transmissions at the price of a practical ISS property. l

C. Decreasing threshold on the Lyapunov function

We present a last triggering rule, which consists in imposing a given (decreasing) threshold on the Lyapunov function V evaluated along the solutions to (1), ( 2), as suggested in e.g., [START_REF] Wang | On event design in event-triggered feedback systems[END_REF], [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF] in model-based settings. Exploiting the fact that the Lyapunov function V can be derived from data as shown in Sections III and IV, we present "robustified" data-based versions of these sampling strategies.

In the noise-free case and in absence of network, we have that V pxq " x J Sx is a Lyapunov function for 9

xptq " pA `BKqxptq where S " pX 0 Y q ´1, Y is a solution to (6) and K " U 0 Y pX 0 Y q ´1, see Section III-C. We then have x∇V pxq, pA `BKqxy ď ´ρ1 V pxq for any x P R n , where ρ 1 ą 0 is any (sufficiently small) constant such that ´ρ1 S ľ pA`BKq J S `SpA`BKq " pX 1 Gq J S `SpX 1 Gq. Note that the condition ´ρ1 S ľ pX 1 Gq J S `SpX 1 Gq is databased and can thus be exploited to find ρ 1 using set D in (3). Based on this observation, we introduce the threshold variable η P R ě0 , whose dynamics is

where ς P p0, 1q is arbitrary. Hence η upper-bounds V pxq along the solutions to the closed-loop system 9 xptq " pA BKqxptq, as ηp0q ě V pxp0qq and 9 ηptq ě 9 V pxptqq in view of the comparison principle [START_REF] Khalil | Nonlinear control systems[END_REF]Chapter 3.4]. The triggering rule is then given by: t 0 " 0 and 8

(62) 8 Like in footnote 7, triggering occurs when V pxptqq " ηptq and 9 V pxptqq ě 9 ηptq, which we do not specify consistently with related modelbased results of the literature and not to overload (62). This extra condition will not be required in the noisy case in (65) thanks to time-regularization.

We have the next result in the noise-free case.

Proposition 5: Suppose Assumption 1 holds. Let Y be any solution to [START_REF] Safonov | The unfalsified control concept and learning[END_REF], S " pX 0 Y q ´1, K " U 0 Y pX 0 Y q ´1 and let ρ 1 ą 0 be such that pX 1 Gq J S `SpX 1 Gq ĺ ´ρ1 S. Furthermore, let L be as in [START_REF] Digge | Data-driven event-triggered control for discrete-time LTI systems[END_REF], ς P p0, 1q arbitrary, and µ, σ ą 0 be such that the following SDP (in the decision variables µ and σ 2 ) is satisfied

where the matrix M ς is the same as matrix M in [START_REF] Liu | Data-driven selftriggered control via trajectory prediction[END_REF] with ´Q replaced by ´Q`ςρ 1 S and Ψpσq is as in [START_REF] Fliess | Model-free control[END_REF]. Let V pxq "

x J Sx, then system ( 7), ( 8), ( 61), ( 62): (a) admits τ pσq in item (a) of Theorem 2 as global minimum inter-event time with σ ensuring (63); (b) is globally asymptotically stable, in particular there exist c 1 ě 1 and c 2 ą 0 such that any solution px, ηq satisfies |pxptq, a ηptqq| ď c 1 e ´c2t |pxp0q, a ηp0qq| for any t ě 0. l Proof. We first show the feasibility of (63). Let σ :" a µ{c with c ą 0 any sufficiently large constant such that ´Q ςρ 1 S `I{c ă 0, where Q is defined in [START_REF] Cordovil | Learning event-triggered control based on evolving data-driven fuzzy granular models[END_REF]; clearly, this c exists because Q ´ςρ 1 S ą 0 (as Q ľ ρ 1 S and ς P p0, 1q). For such a choice of σ, we have µp´Q `ςρ 1 Sq `σ2 I ă 0 for every µ ą 0. Therefore, for this choice of σ, (63) is equivalent by a Schur complement to the two conditions µp´Q `ςρ 1 S Ì{cq ă 0 and ´I `µpSX 1 Lq J pQ ´ςρ 1 S ´I{cq ´1pSX 1 Lq ă 0, which are jointly satisfied for µ sufficiently small.

In view of (63), for any z P R 2n , z J Ψpσqz ă 0 implies x∇V pxq, pA `BKqx `BKey " z J M z ă ´ςρ 1 x J Sx " ´ςρ 1 V pxq. This implies that for any solution x to (7), ( 8), (61), (62), for any t P rt k , t k`1 s, as long as zptq J Ψpσqzptq ă 0 (which is the case right after a transmission whenever xp0q ‰ 0), we have 9

V pxptqq ă ´ςρ 1 V pxptqq and thus V pxptqq ă ηptq in view of (61) and (62). Hence, as the time it takes for z J Ψpσqz to grow from ´σ2 |x| to 0 along (7), ( 8) is lower bounded by τ pσq in view of item (a) of Theorem 2, the time it takes for V pxq to reach η, which is the inter-transmission time generated by (62), is bigger than the former and is thus also lower bounded by τ pσq. Item (a) of Proposition 5 is then satisfied as when xp0q " 0 no transmissions ever occur.

The proof of item (b) follows by considering Lyapunov function U px, ηq " maxtV pxq, ηu for any x P R n and η ě 0, and exploiting the fact that η upper-bounds V along the solutions to ( 7), ( 8), (61), (62) and strictly decreases outside the origin.

We notice that the same type of stability property is stated in items (b) of Propositions 3 and 5.

In the noisy case, the ISS Lyapunov function is given by V pxq " x J Sx for any x P R n with S " pX 0 Y q ´1 and Y a solution to [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF], assuming it exists, see Section IV-B. We modify the dynamics of threshold η in (61) as

where ς d ą 0 and ν ě 0 are both arbitrary. We emphasize that the decay rate of η, namely ς d , can take any value in R ą0 , which was not the case in (61). We then "robustify" the triggering rule (62) as: t 0 " 0 and t k`1 " inf tt P R : t ě t k `τd pσq and V pxptqq ě ηptqu , (65) with σ and τ d pσq as in ( 39) and ( 41), respectively. We consider τ d pσq in (65) and not τ d pσq as in the other techniques covered so far in this section to simplify the exposition; otherwise the selection of ν would need to depend on some known upperbound on }d} 8 . In view of (65), the existence of a global minimum inter-event time for system (25), ( 26), ( 64), ( 65) is immediate and we have the next stability result. Proposition 6: Suppose the following holds. (i) Assumptions 1 and 2 are verified with ∆ given. (ii) Let Ω ą 0, SDP ( 24) is feasible and K " U 0 Y pX 0 Y q ´1 is the resulting controller as in Theorem 3. Let V pxq " x J Sx for any x P R n with S " pX 0 Y q ´1 and Y from (24), then system ( 25), ( 26), ( 64), ( 65) is (practically) exponentially ISS, in particular there exist c 1 ě 1 and c 2 , c 3 , c 4 ą 0 such that any solution px, ηq with disturbance input d, it holds that |pxptq, a ηptqq| ď c 1 e ´c2t |pxp0q, a ηp0qq|c

3 }d} r0,ts `c4 ν. l Proof. We consider the same Lyapunov function as in the proof of Proposition 5, namely U px, ηq " maxtV pxq, ηu for any x P R n and η ě 0. Let px, ηq be a solution to [START_REF] Liu | Event-based control of nonlinear systems with partial state and output feedback[END_REF], ( 26), ( 64), (65) with input disturbance d. The existence of such a solution is ensured as there exists a minimum inter-event time τ d pσq, thereby excluding Zeno phenomenon, and the involved dynamics is linear. Let k P N and t P rt k , t k`1 s. To differentiate U along px, ηq requires care as U is not differentiable everywhere, but almost everywhere. A convenient tool in this context is Clarke's derivative [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]. To avoid introducing too many technicalities, we exploit [46, Lemma 1], which states in our case that the Clarke's derivative of U pxptq, ηptqq is 9 ηptq when ηptq ą V pxptqq, it is 9

V pxptqq when ηptq ă V pxptqq, and it is less than or equal to maxt 9

V pxptqq, 9 ηptqu when V pxptqq " ηptq. By deriving ISS dissipation inequalities for 9 V pxptqq and 9 ηptq, we then obtain the desired result by invoking [47, p.99] and integrating the obtained inequalities as in the other proofs of this work. We now analyse 9

V pxptqq and 9 ηptq. We have that the η-system satisfies a suitable ISS dissipation inequality by design in view of (64). Regarding 9

V pxptqq, if t P rt k , t k `τd pσqs, we proceed as before in the paper and exploit the fact that, by definition of τ d pσq and since σ satisfies (39), (49) holds. If t P rt k `τd pσq, t k`1 q, then V pxptqq ă ηptq, U pxptq, ηptqq " ηptq and the Clarke's derivative is given by 9 ηptq " ´ςd ηptq `ν " ´ςd U pxptq, ηptqq `ν. Based on these properties, and since U is not affected by jumps at triggering instants along the solutions, the result follows.

VI. CONCLUDING REMARKS

We have presented an approach to design robust eventtriggered state-feedback controllers for unknown stabilizable perturbed linear time invariant systems directly based on a collection of noisy off-line data and not a model of the dynamics. In particular, we have derived data-based version of the event-triggered control strategies originally developed in a model-based settings in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], [START_REF] Girard | Dynamic triggering mechanisms for event-triggered control[END_REF], [START_REF] Borgers | Event-separation properties of event-triggered control systems[END_REF], [START_REF] Dolk | Output-based and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF], [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF], [START_REF] Wang | A unifying event-triggered control framework based on a hybrid small-gain theorem[END_REF].

The tools developed in this work may provide foundations to develop data-based event-triggered control designs for other control scenarios including when the plant is nonlinear, when only output measurements (instead of full-state measurements) are available, and when an 'online' learning mechanism is set to handle plants with time-varying dynamics or to recursively improve the inter-event times.

APPENDIX

A. Alternative data acquisition scheme

The results presented in this work rest on X 1 in (4c). The computation of X 1 can be error-prone as it involves computing the derivative of x. We briefly discuss a data collection scheme that can be used when the derivative of x is difficult to compute. The idea is to consider the integral version of the relation 9

x " Ax `Bu `d, for any τ 1 ď τ 2 , ş τ2 τ1 9

xptqdt " ş τ2 τ1 pAxptq `Buptq `dptqqdt, which writes equivalently as

We can therefore choose a sampling time T s ą 0 to obtain xppk `1qT s q ´xpkT s q looooooooooooomooooooooooooon ":ξpkq (66) with k ě 0. Defining X 1 :" " ξp0q . . . ξpT ´1q ‰ , X 0 :" " rp0q . . . rpT ´1q ‰ , U 0 :" " vp0q . . . vpT ´1q ‰ , and D 0 :" " wp0q . . . wpT ´1q ‰ , we get X 1 " AX 0 `BU 0 D0 , which is the integral version of the original relation X 1 " AX 0 `BU 0 `D0 . We can therefore restate all the results with U 0 , X 0 , X 1 , D 0 replaced by U 0 , X 0 , X 1 , D 0 .

B. Proof of Lemma 3

Lemma 3 is a direct consequence of the next result. Lemma 6: Let B P R nˆp and C P R qˆn be given matrices. Then for any ą 0 and any D P D with D in [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF], we have BD J C `CJ DB J ĺ ´1BB J ` C J ∆∆ J C. l Proof. A completion of squares p ? ´1B´? C J Dqp ?

´1B?

C J Dq J ľ 0 gives the result. Proof of Lemma 3. Let (24a) hold. By a Schur complement, this is equivalent to X 1 Y `pX 1 Y q J `Ω` ´1Y J Y ` ∆∆ J ă 0. By applying Lemma 6 with B " ´Y J and C " I, we obtain X 1 Y `pX 1 Y q J `Ω ´DY ´pDY q J ă 0 @D P D, which corresponds to [START_REF] Wang | On event design in event-triggered feedback systems[END_REF].