Aperiodic dynamic event-triggered control for linear systems: A looped-functional approach

Nonlinear Analysis: Hybrid Systems 52 (2024) 101444 1751-570X/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents lists available at ScienceDirect Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs Aperiodic dynamic event-triggered control for linear systems: A looped-functional approach Yihao Xua,b, Alexandre Seuretb,∗, Kun Liua, Senchun Chaia aSchool of Automation, Beijing Institute of Technology, Beijing, 100081, China bDpto de Ingenieria de Sistemas y Automatica, Universidad de Sevilla, Seville, 41092, Spain ARTICLE INFO Keywords: Event-triggered control Looped-functional Sampled-data systems ABSTRACT The recent literature on event-triggered control has demonstrated the potential of dynamic periodic event-triggered control. Compared to continuous-time event-triggering rules, the benefit of considering periodic event-triggered control is to avoid the Zeno phenomenon, which refers to the situation when there are an infinite number of updates in a bounded interval of time. The idea of periodic event-triggered control is to trigger the control law only at known allowable periodic sampling instants. In this paper, our objective is to relax the constraint on the periodicity of the allowable sampling instants and to adapt this framework to the dynamic event-triggered control, which has not been considered in the literature, as far as we are aware of. Following the successful efforts to assess the stability of aperiodic sampled-data control, here we propose a generic framework to emulate aperiodic dynamic event-triggered control law for linear systems, for which the allowable sampling instants are not necessarily equidistant. Such an analysis is made possible thanks to the looped-functionals framework, which gives the flexibility to consider the periodic/aperiodic static/dynamic event-triggered control in a single formulation. Finally, the efficiency of the proposed results is illustrated through the study of two academic examples. 1. Introduction With the rapid development of computer and network technologies, networked control systems (NCSs) have attracted considerable attention. When designing networked control loops, it is essential to consider the limited communication and/or computation capabilities of the system. The event-triggered (ET) strategy offers an effective tool which can balance the available system resources with the closed-loop system performances better than the classical periodic or aperiodic time-triggered control schemes [1,2]. In event-triggered control (ETC) systems, when the control input has to be updated is determined by a triggering rule which is predefined. That means control inputs are updated only when necessary, namely when an event is generated, rather than timetriggered periodically or aperiodically. Thus, as an effective way to balance system performance and limited network resources, ETC systems have attracted increasing interests of researchers in the last decade. Since an ET feedback controller that asymptotically stabilizes the plant is proposed in [3], various analyses and designs for ETC systems have been proposed [3–21]. Most existing ET strategies can be divided into continuous- and discrete-time strategies based on their time sets, and the controlled plant is generally considered to have the same time nature [4]. However, with the increasing popularity of sampled-data systems in NCSs, it is more interesting to find a ‘‘sampled-data’’-like approach to break the parallels between the continuous-time ET approaches ∗Corresponding author. E-mail address: [email protected] (A. Seuret). https://doi.org/10.1016/j.nahs.2023.101444 Received 2 August 2023; Received in revised form 10 November 2023; Accepted 23 November 2023 Nonlinear Analysis: Hybrid Systems 52 (2024) 101444 2 Y. Xu et al. and the discrete-time counterparts. This means that a discrete-time ET scheme based on sampled data operates on a continuoustime plant. To name a few, a periodic event-triggered strategy (PETS) in a quadratic form is proposed in [5], and the approaches of impulsive systems, piece-wise linear systems, perturbed linear systems are presented to analyze the stability and the 2-gain properties of resulting periodic event-triggered control (PETC) systems. More PETC problems for linear and nonlinear systems can be found in [7–11]. Another very popular method for ETC of sampled-data systems is the time-delay system approach [12,13]. By introducing an artificial piece-wise linear delay function, a delay system model can be constructed for the ETC system based on sampled data [12,13]. Nevertheless, almost all the above mentioned results are based on the technique of the Lyapunov–Krasovskii functional and only address related ET problems for periodic samplings without much attention to aperiodic cases. Sampled-data systems have been widely investigated in recent decades [22,23]. The stability of the system under periodic samplings has been extensively discussed and still attracting a lot of attentions [24]. When the difference between two successive sampling instants is time-varying, several methods are used to analyze system dynamics [25–34]. Discrete time models are adopted in [34] for aperiodic samplings and an input-delay approach based on the Lyapunov–Krasovskii theorem is provided in [25,26]. Small gain theorem is adopted to make some improvement [27] and the impulsive systems method is proposed in [28]. Although thanks to the time-dependent Lyapunov–Krasovskii functionals, the input delay approach and the impulsive system approach can deal with time-varying sampling periods as well as with uncertain systems in a simple way, these sufficient stability conditions are still more conservative than the discrete-time methods. However, due to the exponential term of the transition matrix, it is not an easy task to find an efficient extension of the discrete-time approach to include robustness with respect to the uncertainties of sampling periods and parameters. Another popular way to deal with asynchronous samplings of the system is the looped-functional approach [29,30]. By introducing a looping condition, the equivalence between the discrete-time and continuous-time approaches for the stability analysis is disclosed in [29]. The constraint on the positivity of the Lyapunov functional in continuous-time approaches can be relaxed. Based on the discrete-time Lyapunov theory and the proposed looped-functionals method, the asymptotic and exponential stability analyses for the cases of synchronous, asynchronous, and multiple samplings are discussed in a unified framework [29]. It is worth noting that the looped-functional method provides the missing link between discrete-time approaches and continuous counterparts. The main advantages of this framework with respect to the context of this paper is that the inclusion of additional features such as the triggering variable of a dynamical event-triggered control scheme can be performed in a very elegant manner, without invoking an S-procedure. In this paper, a novel framework is proposed to extend static and dynamic periodic ETC to the case of aperiodic sampled-data systems. To do so, we introduce a discrete-time dynamic event-triggering rule, which includes several existing ET strategies as its special cases. This rule is formulated in quadratic form, which only uses the sampled data from the system. The stability analysis of the continuous-time ETC system is performed using the looped-functional framework. Sufficient conditions for asymptotic stability of the closed-loop system are derived in terms of LMIs, which can be applied to the cases of periodic and aperiodic samplings for nominal and polytopic uncertain systems. Notation.Throughout the paper, the sets N,R,R𝑛,R𝑛×𝑛and S𝑛denote the sets of non-negative integers, real numbers, 𝑛-dimensional vectors, 𝑛×𝑛matrices and symmetric matrices of R𝑛×𝑛, respectively. The notation |⋅|and the superscript ⊤stand for the Euclidean norm and for matrix transposition, respectively. The notation 𝑃 ≻ 0for 𝑃∈S𝑛means that 𝑃is positive definite. For any matrix 𝐴∈R𝑛×𝑛, the notation He{𝐴}refers to 𝐴+𝐴𝑇. The symbols 𝐼and 0represent the identity and zero matrices of appropriate dimensions. 2. Problem statement 2.1. System data Consider the following linear sampled-data system 𝑥(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡𝑘),∀𝑡∈ [𝑡𝑘, 𝑡𝑘+1),(1) where 𝑥(𝑡) ∈ R𝑛and 𝑢(𝑡) ∈ R𝑚represent the state and the control input vectors, respectively. Matrices 𝐴and 𝐵, of appropriate dimensions, are assumed to be either constant and known, or possibly uncertain and time-varying. The sampling is assumed to be aperiodic. This means that there exists a minimal and maximal dwell times 1>0and 2≥1such that the distance between two successive sampling instants 𝑡𝑘+1 −𝑡𝑘=𝑇𝑘verifies 1≤𝑇𝑘≤2,∀𝑘∈N.(2) In this paper, the objective is the design of aperiodic event-triggered rules such that a given state feedback control law asymptotically stabilizes the closed-loop system, corresponding to an emulation problem. In order to solve this problem, we will formulate the dynamics of the system using the looped-functional framework, which will be proven to be well suited for this study. To do so, in the following subsections, we define the control law and the event-triggering rule that will be employed hereafter. Nonlinear Analysis: Hybrid Systems 52 (2024) 101444 3 Y. Xu et al. 2.2. Event-triggered state-feedback control law In this paper, we are looking for a linear state feedback control gain 𝐾∈R𝑚×𝑛, such that the control input to be implemented is given by 𝑢(𝑡𝑘) = {𝐾𝑥(𝑡𝑘),if the control law is updated, 𝑢(𝑡𝑘−1),otherwise. (3) One can understand the previous expression as a ‘double’ sampling effect. The continuous-time sampling makes that the control input remains constant during the sampling interval [𝑡𝑘, 𝑡𝑘+1). Then, a discrete-time sampling is performed depending on whether or not the previous control input has to be updated. In order to complete the implementation rules of the control law, one has to provide a specific rule to impose a control update. To do so, let 𝜅be a strictly increasing function from Nto N, with condition 𝜅(0) = 0. This function defines triggering instants, that is, when the control input has to be updated, in a more formal manner. It means that the triggering instants 𝑡𝜅(𝓁)are all instants of control updates such that 𝑢(𝑡𝜅(𝓁)) = 𝐾𝑥(𝑡𝜅(𝓁)). To determine these 𝜅(𝓁), static and dynamic event-triggered control algorithms will be provided in the next subsections. 2.2.1. Static event-triggering rule Let us introduce the static event-triggering rule defined as follows 𝑡𝜅(𝓁+1) = min 𝑚∈N{𝑡𝜅(𝓁)+𝑚≥𝑡𝜅(𝓁)+1 |𝜑(𝑥(𝑡𝜅(𝓁)+𝑚), 𝑥(𝑡𝜅(𝓁)))≥0},(4) where 𝜑∶R𝑛×R𝑛→Ris called the triggering function to be defined. The only requirement so far, is that the function 𝜑verifies the following constraint: 𝜑(𝑥, 𝑥)≤0,∀𝑥∈R𝑛.(5) Remark 1. The previous event-triggering rule excludes the Zeno phenomenon since the minimal distance between two successive sampling instants is at least 1>0. It is also important to mention that the event-triggering rule (4) is only evaluated at the aperiodic sampling instants {𝑡𝑘}𝑘∈N. Compared to the usual periodic event-triggering rules introduced in [5,14,15], the assumption on the periodicity of the sampling instants is not required. 2.2.2. Dynamic event-triggering rule Let us now introduce the following dynamic event-triggering rule defined as an extended version of the previous static rule. This dynamic rule is formulated as follows: 𝑡𝜅(𝓁+1) = min 𝑚∈N{𝑡𝜅(𝓁)+𝑚≥𝑡𝜅(𝓁)+1 |𝜑(𝑥(𝑡𝜅(𝓁)+𝑚), 𝑥(𝑡𝜅(𝓁)))≥𝜌𝜂𝜅(𝓁)+𝑚},(6) where, again, 𝜑∶R𝑛×R𝑛→Ris the triggering function to be defined which must also verify (5). The above rule is called a dynamic event-triggering rule because it relies on the construction of a new dynamic variable 𝜂𝑘, that is driven by the following discrete-time dynamic equation: {𝜂𝑘+1 = (𝜆+𝜌)𝜂𝑘−𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝜅(𝓁))),∀𝑘∈[𝜅(𝓁), 𝜅(𝓁+ 1))N,∀𝓁∈N, 𝜂0≥0,(7) where the parameters 𝜆and 𝜌are such that 𝜌≥0and (𝜆+𝜌) ∈ [0,1). Compared to the classical ‘‘continuous-time’’ dynamic variable provided in [6], the dynamic here is written as a discrete-time equation because we are considering a periodic (or aperiodic) eventtriggered control. Here, Eq. (7) means that the update of 𝜂𝑘is only performed at the sampling instant 𝑡𝑘. Between two successive sampling instants, the value of 𝜂𝑘remains constant, i.e., d d𝑡𝜂𝑘= 0,∀𝑡∈ (𝑡𝑘, 𝑡𝑘+1),∀𝑘∈N.(8) Remark 2. It is worth mentioning that the static event-triggering rule is retrieved by imposing 𝜌= 0 in (6), whatever the value of 𝜆. Following classical methods of the continuous-time dynamic ETC design [6,7], one has to guarantee that the variable 𝜂𝑘is non-negative for all 𝑘in N. The following lemma addresses this issue. Lemma 1. Consider scalar parameters (𝜆, 𝜌)such that 𝜌≥0and (𝜆+𝜌) ∈ [0,1) and assume that 𝜂0>0. Then, the dynamic event-triggered variable is non-negative for any sampling instants 𝑡𝑘, i.e., 𝜂𝑘≥0for all 𝑘∈N. Nonlinear Analysis: Hybrid Systems 52 (2024) 101444 4 Y. Xu et al. Proof. Consider the dynamic of 𝜂𝑘provided in (7) together with the event-triggering rule (6). Assume that 𝜌≥0,(𝜆+𝜌) ∈ [0,1) and 𝜂0>0. The proof is then made by recursion. Initialization (𝑘= 1): Assume that 𝜂0>0. According to the dynamic Eq. (7) with 𝑘= 0, it holds that 𝜂1= (𝜆+𝜌)𝜂0−𝜑(𝑥(𝑡0), 𝑥(𝑡𝜅(0))), where obviously 𝑡𝜅(0) =𝑡0. Recalling condition (5), it yields 𝜑(𝑥(𝑡0), 𝑥(𝑡𝜅(0)))=𝜑(𝑥(𝑡0), 𝑥(𝑡0))≤0. As (𝜆+𝜌) ∈ [0,1),𝜂0>0and 𝜑(𝑥(𝑡0), 𝑥(𝑡𝜅(0)))≤0, it is clear that 𝜂1≥0, which concludes the initialization. Recursion (𝑘∈N): Assume that the variable 𝜂𝑘is positive and the associated value of 𝜅(𝓁), is an integer lower than 𝑘. Then, we may face the following two cases. If the triggering condition is satisfied, that is, 𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝜅(𝓁)))≥𝜌𝜂𝑘, then, the value of the control law as well as 𝜅(𝓁)is updated, i.e., 𝜅(𝓁+ 1) = 𝑘. In this situation, 𝜑(𝑥(𝑡𝜅(𝓁)), 𝑥(𝑡𝜅(𝓁)))≤0still holds since we get the same constraint (5) of the triggering function 𝜑as in the case 𝑘= 0 above. Consequently, 𝜂𝑘+1 ≥0. If the event-triggered condition is violated, that is, 𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝜅(𝓁)))≤𝜌𝜂𝑘, then, according to the discrete-time Eq. (7) it holds 𝜂𝑘+1 = (𝜆+𝜌)𝜂𝑘−𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝜅(𝓁))). Since 𝜆≥0,𝜂𝑘≥0, it holds 𝜂𝑘+1 = (𝜆+𝜌)𝜂𝑘−𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝜅(𝓁))) =𝜆𝜂𝑘 ⏟⏟⏟ ≥0 +(𝜌𝜂𝑘−𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝜅(𝓁)))) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ≥0 . Hence, 𝜂𝑘+1 ≥0holds for all 𝑘∈N, which concludes the recursion. □ Remark 3. In the literature on the dynamic event-triggered control [6,7], an additional dynamic variable 𝜂is usually defined using a differential equation of the form 𝜂 = −  𝜆𝜂 +𝜌𝜑 for some positive scalars  𝜆and 𝜌. Here, we use the particular feature of the periodic (or aperiodic) event-triggered algorithm to notice that 𝜑is constant over the sampling interval. Therefore, it is easy to derive the solution of the differential equation that leads to the discrete-time Eq. (7), with particular values of 𝜆and 𝜌. In this paper, we have adopted a selection of constant parameters 𝜆and 𝜌for simplicity, noting that it is still possible to use the time-varying but known values of 𝜆and 𝜌for the sampling interval 𝑇𝑘. 2.2.3. Definition of the triggering function 𝜑 In this paper, we will consider a triggering function that is quadratic with respect to its arguments and given by 𝜑(𝑥(𝑡𝜅(𝓁)+𝑚), 𝑥(𝑡𝜅(𝓁)))=[𝑥(𝑡𝜅(𝓁)+𝑚) 𝑥(𝑡𝜅(𝓁))]⊤ 𝛷[𝑥(𝑡𝜅(𝓁)+𝑚) 𝑥(𝑡𝜅(𝓁))],∀𝑘∈[𝜅(𝓁), 𝜅(𝓁+ 1))N,∀𝓁∈N,(9) where 𝛷satisfies [𝐼 𝐼]⊤ 𝛷[𝐼 𝐼]⪯ 0.(10) This above inequality is required to ensure that once an update is performed, the event-triggering condition does not hold any longer until the next update, which ensures 𝜑(𝑥, 𝑥)≤0for all 𝑥in R𝑛. The intuition behind this condition is the following. If the triggering condition is violated, this necessarily means that inequality 𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝜅(𝓁)))≥𝜌𝜂𝑘holds. As 𝜂𝑘is positive for all 𝑘thanks to Lemma 1, this means that 𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝜅(𝓁)))is necessarily positive. Imposing condition (10) implies that, once the memory vector 𝑥(𝑡𝜅(𝓁))has been updated, 𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝜅(𝓁+1)))=𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝑘))is negative, which ensures that the triggering condition (before applying the new control input) is not violated anymore. Intuitively, this condition can be interpreted as the avoidance of Zeno and enforce enlarging the interval between two successive control updates. Indeed, by continuity of the systems, the quantity 𝜑(𝑥(𝑡𝜅(𝓁)+𝑚), 𝑥(𝑡𝜅(𝓁))), for small values of 𝑚is intuitively not so far away from 𝜑(𝑥(𝑡𝜅(𝓁)), 𝑥(𝑡𝜅(𝓁))), which is negative. This aims at giving more margin before the next triggering instant. Remark 4. The previous triggering function (9) that was already considered in papers [16–18] has the benefit of encompassing the rules usually considered in the literature as particular cases. For instance, consider the following selections •Case A: Selecting a matrix 𝛷, which is only required to verify 𝛷with [𝐼 𝐼]⊤ 𝛷[𝐼 𝐼]⪯ 0,(11) leads to the generic triggering function (9) in this paper. Nonlinear Analysis: Hybrid Systems 52 (2024) 101444 5 Y. Xu et al. •Case B: Selecting a matrix 𝛷as 𝛷=[𝛷1−𝛷1 −𝛷1(1 − 𝛼)𝛷1]with 𝛷1>0, 𝛼 > 0,(12) leads to the particular triggering function 𝜑(𝑥(𝑡𝜅(𝓁)+𝑚), 𝑥(𝑡𝜅(𝓁)))∶= (𝑥(𝑡𝜅(𝓁)+𝑚) − 𝑥(𝑡𝜅(𝓁)))⊤𝛷1(𝑥(𝑡𝜅(𝓁)+𝑚) − 𝑥(𝑡𝜅(𝓁)))−𝛼𝑥⊤(𝑡𝜅(𝓁))𝛷1𝑥(𝑡𝜅(𝓁)), which was considered in papers [12,13]. •Case C: Selecting a matrix 𝛷as 𝛷=[𝛷2−𝛷2 −𝛷2𝛷2−𝛷1]with 𝛷1>0, 𝛷2>0,(13) leads to the particular triggering function 𝜑(𝑥(𝑡𝜅(𝓁)+𝑚), 𝑥(𝑡𝜅(𝓁)))∶= (𝑥(𝑡𝜅(𝓁)+𝑚) − 𝑥(𝑡𝜅(𝓁)))⊤𝛷2(𝑥(𝑡𝜅(𝓁)+𝑚) − 𝑥(𝑡𝜅(𝓁)))−𝑥⊤(𝑡𝜅(𝓁))𝛷1𝑥(𝑡𝜅(𝓁)), which was considered, for instance, in papers [20,35]. Recalling (10), the constraint of 𝛷is required for all cases of triggering functions mentioned above to guarantee 𝜑(𝑥, 𝑥)nonpositive. The matrix 𝛷of the triggering rule plays a major role both in the solvability of the LMIs and in the event-triggered mechanism. As it thus appears that the proposed event-triggering rule encompasses the above usual rules with parameters (𝛷1, 𝛷2, 𝛼). In the next developments, we will present the conditions for finding a matrix 𝛷as a candidate. 2.2.4. Modeling of the closed-loop system As is discussed in previous sections, the control law (3) can be rewritten as 𝑢(𝑡𝑘) = {𝐾𝑥(𝑡𝑘),if 𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝜅(𝓁)))≥𝜌𝜂𝑘, 𝑘 =𝜅(𝓁+ 1), 𝐾𝑥(𝑡𝜅(𝓁)),if 𝜑(𝑥(𝑡𝑘), 𝑥(𝑡𝜅(𝓁)))≤𝜌𝜂𝑘, 𝑘 ∈[𝜅(𝓁), 𝜅(𝓁+ 1)).(14) The closed-loop system composed of (1),(6),(14), can be reformulated as 𝑥(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝐾𝑥(𝑡𝜅(𝓁)),∀𝑡∈[𝑡𝑘, 𝑡𝑘+1),∀𝑘∈[𝜅(𝓁), 𝜅(𝓁+ 1)),∀𝓁∈N.(15) The plant (15) can be seen as a particular sampled-data system for which several theoretical tools can be applied to study its stability properties, see the survey paper [24,36]. In this paper, we will focus on the looped functional framework introduced in [29,37] to assess the stability of the closed-loop system. 3. Preliminaries and looped-functional 3.1. Preliminary lemmas In the next development of the paper, we will use the following lemmas: Lemma 2 (Wirtinger-based Inequality [38]).For a given matrix 𝑅 ≻ 0, the following inequality holds for all continuously differentiable function 𝜔in [𝑎, 𝑏]→R𝑛: ∫𝑏 𝑎 𝜔𝑇(𝑢)𝑅 𝜔(𝑢)d𝑢≥1 𝑏−𝑎(𝜔(𝑏) − 𝜔(𝑎))𝑇𝑅(𝜔(𝑏) − 𝜔(𝑎)) + 3 (𝑏−𝑎) 𝛺𝑇𝑅 𝛺, (16) where  𝛺=𝜔(𝑏) + 𝜔(𝑎) − 2 𝑏−𝑎∫𝑏 𝑎𝜔(𝑢)d𝑢. Lemma 3. For all matrices 𝑋,𝑌and 𝑅of appropriate dimensions and positive scalar 𝜖 > 0, the following inequality holds: −1 𝜖𝑋𝑇𝑅𝑋 ≤−2𝑋𝑇𝑌+𝜖𝑌 𝑇𝑅−1𝑌 . (17) 3.2. Lifting of sampled-data systems trajectories The looped functional approach [29] is based on the characterization of the system trajectories (1) in a lifted domain [37,39]. Define K𝑛as K𝑛∶= ⋃ ∈[1,2] ([0,]→R𝑛). Then, we therefore view the entire state-trajectory as a sequence of functions {𝑥(𝑡𝑘+𝜏), 𝜏 ∈ (0, 𝑇𝑘]}𝑘∈Nwith elements having a unique continuous extension to [0, 𝑇𝑘]defined as 𝜒𝑘(𝜏) ∶= 𝑥(𝑡𝑘+𝜏)with 𝜒𝑘(0) = lim𝑠↓𝑡𝑘𝑥(𝑠). Such that the system (1) can be rewritten as 𝜒𝑘(𝜏) = 𝐴𝜒𝑘(𝜏) + 𝐵𝐾𝜒𝜅(𝓁)(0),∀(𝜏, 𝑘, 𝓁) ∈ H𝑛 𝜅(𝑇𝑘) ∶= [0, 𝑇𝑘] × [𝜅(𝓁), 𝜅(𝓁+ 1))×N.(18) Nonlinear Analysis: Hybrid Systems 52 (2024) 101444 6 Y. Xu et al. 3.3. Definition and results about looped-functionals Let us first define looped-functionals following the principles of the paper [37]. Definition 1 ([37]).A functional 𝑓∶ [0,] × K𝑛×R𝑛× [1,2]→R, is said to be a looped-functional if 1. the equality 𝑓(0, 𝑧, 𝑧0,) = 𝑓(, 𝑧, 𝑧0,)(19) holds for all functions 𝑧∈K𝑛, for all vectors 𝑧0∈R𝑛and all ∈ [1,2]. 2. it is differentiable with respect to the first variable with the standard definition of the derivative. The set of all these functionals is denoted by 𝑛([1,2]). Note that in the above definition, we have lightly modified the argument of the looped functional to include an additional component 𝑧0, which is a vector in R𝑛. The idea of proving the stability of (18) under the proposed ET scheme (6) is to look now for a Lyapunov function  𝑉such that  𝑉is monotonically decreasing along the sampling instants. The novelty with respect to previous results on looped-functionals and event-triggered control relies on the particular construction of the Lyapunov function  𝑉. In this paper,  𝑉is composed of : (i) the non-negative discrete-time variable 𝜂𝑘, which is the additional state that has been built to include a dynamic event-triggering scheme, (ii) a classical quadratic Lyapunov function 𝑉∶R𝑛→R+. This is formalized through the following functional existence result: Theorem 1. For given 0<1≤2and 𝜌∈ [0, 𝜆)⊂[0,1], consider 𝑉∶R𝑛→R+a quadratic form satisfying 𝜇1|𝑥|2≤𝑉(𝑥)≤𝜇2|𝑥|2,∀𝑥∈R𝑛,(20) for some scalars 0< 𝜇1≤𝜇2and the non-negative discrete-time event-triggering variable 𝜂𝑘defined in (7). Assume further that one of the following equivalent statements hold: 1. The sequence { 𝑉(𝜒𝑘(0), 𝜂𝑘) ∶= 𝑉(𝜒𝑘(0)) + 𝜂𝑘}𝑘∈Nis decreasing along the trajectories of closed-loop system (18) with the dynamic event-triggering rule (6). 2. There exists a looped-functional ∈𝑛()such that the functional 𝑘defined as 𝑘(𝜏, 𝜒𝑘, 𝜒𝜅(𝓁)) ∶= 𝑉(𝜒𝑘(𝜏)) + (𝜏, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘) + 𝜏 𝑇𝑘 [𝜂𝑘+1 −𝜂𝑘],∀(𝜏, 𝑘, 𝓁) ∈ H𝑛 𝜅(𝑇𝑘),(21) is decreasing along the trajectories of the system (18) under the proposed dynamic event-triggering scheme (6), i.e., 𝑑 𝑑𝜏 𝑘(𝜏, 𝜒𝑘, 𝜒𝜅(𝓁)) ∶= 𝑑 𝑑𝜏 𝑉(𝜒𝑘(𝜏)) + 𝑑 𝑑𝜏 (𝜏, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘) + 𝜂𝑘+1 −𝜂𝑘 𝑇𝑘 <0,∀(𝜏, 𝑘, 𝓁) ∈ H𝑛 𝜅(𝑇𝑘).(22) Then, the aperiodic sampled-data system (18) under the proposed dynamic event-triggering scheme (6) is asymptotically stable. Proof. The proof strictly follows the lines of the proof derived in [31]. However a sketch of the proof is provided to better understand that the inclusion of the triggering variable 𝜂𝑘does not imply critical modifications. Assume that the first statement in Theorem 1 holds. Then, functional = −𝑉(𝜒𝑘(𝜏)) + 𝜏 𝑇𝑘𝛥𝑉 (𝑘)verifies the looping condition and the associated functional 𝑘straightforwardly verifies condition (22). Assume now that the second statement in Theorem 1 is satisfied, that is there exists a looped-functional such that verifies condition (22). Then, integrating  over the interval [0, 𝑇𝑘], we get 𝑘(𝑇𝑘, 𝜒𝑘, 𝜒𝜅(𝓁))−𝑘(0, 𝜒𝑘, 𝜒𝜅(𝓁)) = 𝑉(𝜒𝑘(𝑇𝑘))+(𝑇𝑘, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘)+ 𝑇𝑘 𝑇𝑘 [𝜂𝑘+1 −𝜂𝑘]−𝑉(𝜒𝑘(0))−(0, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘) =𝑉(𝜒𝑘(𝑇𝑘)) − 𝑉(𝜒𝑘(0)) + (𝜂𝑘+1 −𝜂𝑘) + (𝑇𝑘, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘) − (0, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘) =𝛥 (𝜒𝑘, 𝜂𝑘) + (𝑇𝑘, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘) − (0, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘). As verifies the looping condition, the previous expression resumes to 𝑘(𝑇𝑘, 𝜒𝑘, 𝜒𝜅(𝓁))−𝑘(0, 𝜒𝑘, 𝜒𝜅(𝓁)) = 𝛥 (𝜒𝑘, 𝜂𝑘)<0, which is exactly the first statement. Hence, both statement are equivalent. The proof is then concluded by noting that the inter-sampling trajectories are uniformly bounded by the 𝜒𝑘(0) and 𝜒𝜅(𝓁)and 2because of the linear differential equation that defines 𝜒𝑘(𝜏). Therefore, the asymptotic convergence of the discrete sequence  𝑉(𝜒𝑘(0), 𝜂𝑘)to the origin as 𝑘tends to infinity ensures the asymptotic convergence of  𝑉(𝜒𝑘(𝜏), 𝜂𝑘)to the origin as 𝑘tends to infinity, for any 𝜏∈ [0,2].□ Nonlinear Analysis: Hybrid Systems 52 (2024) 101444 7 Y. Xu et al. 4. Main results In this section, we will first present the main result of this paper, dealing with the emulation of aperiodic dynamic ETC. Then, several more or less direct corollaries will be derived to address relevant particular cases. 4.1. Aperiodic dynamic event-triggered control Theorem 2. Let 0<1≤2be two positive scalars and consider the scalar parameters (𝜆, 𝜌, 𝜂0)such that 𝜌∈ [0, 𝜆)⊂[0,1] and 𝜂0>0. Assume that there exist matrices 𝑃=𝑃⊤≻0,𝑅=𝑅⊤≻0,𝑆=𝑆⊤,𝛷=𝛷⊤, the controller gain 𝐾, and matrices 𝑄,𝑀of appropriate dimensions such that the following LMIs are satisfied ⎡⎢⎢⎣ 𝛺1(𝑖,1 − 𝜆 𝜌2 , 𝐴, 𝐵, 𝑀)𝑖𝑀⊤ ∗ −𝑖 𝑅⎤⎥⎥⎦ ≺0, 𝛺1(𝑖,1 − 𝜆 𝜌2 , 𝐴, 𝐵, 𝑀)+𝑖𝛺2(𝐴, 𝐵)≺0,[𝐼 𝐼]⊤ 𝛷[𝐼 𝐼]⪯ 0,(23) for 𝑖= 1,2, where 𝛺1(𝑖,1 − 𝜆 𝜌2 , 𝐴, 𝐵, 𝑀)= He (𝐺⊤ 2(𝐴, 𝐵)𝑃 𝐺1−𝐺⊤ 3𝑄𝐻1−𝐺⊤ 𝑤𝑀)−𝑖𝐹⊤ 1𝑆𝐹1−1 − 𝜆 𝜌2 𝐺⊤ 𝜑𝛷𝐺𝜑, 𝛺2(𝐴, 𝐵) = 𝐺⊤ 2(𝐴, 𝐵)𝑅𝐺2(𝐴, 𝐵)+2𝐹⊤ 1𝑆𝐹1+ He (𝐹⊤ 2𝑆𝐹1+𝐺⊤ 2(𝐴, 𝐵)𝑄𝐻1+𝐺⊤ 3𝑄𝐻2(𝐴, 𝐵)), (24) and 𝐺1=[𝐼000], 𝐺2(𝐴, 𝐵) = [𝐴0𝐵𝐾 0], 𝐺3=[𝐼−𝐼0 0], 𝐺𝜑=[0𝐼0 0 0 0 𝐼0], 𝐻1=⎡⎢⎢⎣ 𝐼0 0 0 0𝐼0 0 0 0 𝐼0⎤⎥⎥⎦ , 𝐻2(𝐴, 𝐵) = ⎡⎢⎢⎣ 𝐴0𝐵𝐾 0 0 0 0 0 0 0 0 0⎤⎥⎥⎦ , 𝐹1=⎡⎢⎢⎣ 0𝐼0 0 0 0 𝐼0 0 0 0 𝐼⎤⎥⎥⎦ , 𝐹2=⎡⎢⎢⎣ 0 0 0 0 0 0 0 0 𝐼00−𝐼⎤⎥⎥⎦ , 𝐺𝑤=[𝐼−𝐼0 0 𝐼 𝐼 0 −2𝐼],  𝑅=[𝑅0 0 3𝑅]. (25) Then, the system (18) with the aperiodic dynamic event-triggered control (6) is asymptotically stable. Proof. Consider the Lyapunov function  𝑉(𝑥) = 𝑉(𝑥)+𝜂𝑘, where 𝑉(𝑥) = 𝑥⊤𝑃 𝑥 for ∀𝑥∈R𝑛and 𝜂𝑘is the discrete-time dynamic eventtriggered variable governed by (7). Let us introduce an appropriate looped functional for this problem. The proposed functional finds its origin in paper [38] with a few modifications to account for the memory variable 𝜒𝜅(𝓁). It is defined as follows (𝜏, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘) = 2(𝑇𝑘−𝜏)(𝜒𝑘(𝜏) − 𝜒𝑘(0))⊤𝑄𝜁𝑘(𝜏)+(𝑇𝑘−𝜏)𝜏𝜌⊤ 𝑘(𝜏)𝑆𝜌𝑘(𝜏)+(𝑇𝑘−𝜏)∫𝜏 0 𝜒𝑘⊤(𝑠)𝑅 𝜒𝑘(𝑠)d𝑠, where 𝜁𝑘=⎡⎢⎢⎣ 𝜒𝑘(𝜏) 𝜒𝑘(0) 𝜒𝜅(𝓁)(0)⎤⎥⎥⎦ , 𝜌𝑘=⎡⎢⎢⎣ 𝜒𝑘(0) 𝜒𝜅(𝓁)(0) 1 𝜏∫𝜏 0𝜒𝑘(𝑠)d𝑠⎤⎥⎥⎦ . Note that the argument ‘(𝜏)’ has been deliberately omitted in these notations to ease the reading of the latter developments. It is easy to verify that the functional verifies the conditions of the looped-functional, since (0, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘) = (𝑇𝑘, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘)=0. Therefore, following the statement of Theorem 1, the stability of the closed-loop system can be assessed by studying the constructed functional (𝜏, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘) = 𝑉(𝜒𝑘(𝜏)) + (𝜏, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘) + 𝜏 𝑇𝑘 (𝜂𝑘+1 −𝜂𝑘).(26) Differentiation of along the trajectories of the system leads to  (𝜏, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘) = 2𝜒⊤ 𝑘(𝜏)𝑃 𝜒𝑘(𝜏)+(𝑇𝑘− 2𝜏)𝜌⊤ 𝑘𝑆𝜌𝑘+ 2𝜏(𝑇𝑘−𝜏)𝜌⊤ 𝑘𝑆𝜌𝑘 − 2(𝜒𝑘(𝜏) − 𝜒𝑘(0))⊤𝑄𝜁𝑘+ 2(𝑇𝑘−𝜏)𝜒⊤ 𝑘(𝜏)𝑄𝜁𝑘+ 2(𝑇𝑘−𝜏)(𝜒𝑘(𝜏) − 𝜒𝑘(0))⊤𝑄 𝜁𝑘 + (𝑇𝑘−𝜏)𝜒⊤ 𝑘(𝜏)𝑅 𝜒𝑘(𝜏) − ∫𝜏 0 𝜒𝑘⊤(𝑠)𝑅 𝜒𝑘(𝑠)d𝑠+𝜂𝑘+1 −𝜂𝑘 𝑇𝑘 . Recalling (7), for ∀𝑘∈[𝜅(𝓁), 𝜅(𝓁+ 1))N,∀𝓁∈N, no sampled data is triggered so that 𝜌𝜂𝑘≥𝜑(𝜒𝑘(0), 𝜒𝜅(𝓁)(0)). Further considering 𝜆∈ [0,1), it yields 𝜂𝑘+1 −𝜂𝑘 𝑇𝑘 =𝜆+𝜌− 1 𝑇𝑘 𝜂𝑘−1 𝑇𝑘 𝜑(𝜒𝑘(0), 𝜒𝜅(𝓁)(0)) (27) ≤𝜆+𝜌− 1 𝜌𝑇𝑘 𝜑(𝜒𝑘(0), 𝜒𝜅(𝓁)(0)) − 1 𝑇𝑘 𝜑(𝜒𝑘(0), 𝜒𝜅(𝓁)(0)) (28) Nonlinear Analysis: Hybrid Systems 52 (2024) 101444 8 Y. Xu et al. ≤−(1 − 𝜆 𝜌𝑇𝑘)𝜑(𝜒𝑘(0), 𝜒𝜅(𝓁)(0)) (29) ≤−(1 − 𝜆 𝜌2)𝜑(𝜒𝑘(0), 𝜒𝜅(𝓁)(0)).(30) Furthermore, based on Lemmas 2 and 3, the following inequality holds for any matrix 𝑀of appropriate dimensions −∫𝜏 0 𝜒⊤ 𝑘(𝑠)𝑅 𝜒𝑘(𝑠)d𝑠≤𝜏𝜉⊤ 𝑘(𝜏)𝑀⊤[𝑅0 0 3𝑅]−1 𝑀𝜉𝑘(𝜏)−2[𝜒𝑘(𝜏) − 𝜒𝑘(0) 𝜒𝑘(𝜏) + 𝜒𝑘(0) − 2 𝜏∫𝜏 0𝜒𝑘(𝑠)d𝑠]⊤ 𝑀𝜉𝑘(𝜏),(31) where we define the augmented vector 𝜉𝑘(𝜏)as follows 𝜉𝑘=[𝜒⊤ 𝑘(𝜏)𝜒⊤ 𝑘(0) 𝜒⊤ 𝜅(𝓁)(0) 1 𝜏∫𝜏 0𝜒⊤ 𝑘(𝑠)d𝑠]⊤ ,(32) and by noting that 𝜒𝑘(𝜏) = 𝐺1𝜉𝑘, 𝜒𝑘(𝜏) = 𝐺2(𝐴, 𝐵)𝜉𝑘, 𝜒𝑘(𝜏) − 𝜒𝑘(0) = 𝐺3𝜉𝑘,[𝜒𝑘(0) 𝜒𝜅(𝓁)(0)]=𝐺𝜑𝜉𝑘, 𝜁𝑘=𝐻1𝜉𝑘, 𝜁𝑘=𝐻2(𝐴, 𝐵)𝜉𝑘, 𝜌𝑘=𝐹1𝜉𝑘, 𝜌𝑘=1 𝜏𝐹2𝜉𝑘. (33) In addition, the following identity holds: [𝜒𝑘(𝜏) − 𝜒𝑘(0) 𝜒𝑘(𝜏) + 𝜒𝑘(0) − 2 𝜏∫𝜏 0𝜒𝑘(𝑠)d𝑠]=𝐺𝑤𝜉𝑘. As mentioned above all, an upper bound of the derivative of expression is given by  (𝜏, 𝜒𝑘, 𝜒𝜅(𝓁))≤𝜉⊤ 𝑘(𝜏)(𝛺1(𝑖,1 − 𝜆 𝜌2 , 𝐴, 𝐵, 𝑀)+𝜏𝑀⊤[𝑅0 0 3𝑅]−1 𝑀+ (𝑇𝑘−𝜏)𝛺2(𝐴, 𝐵))𝜉𝑘(𝜏),(34) where matrices 𝛺1and 𝛺2have been defined in (24). Therefore, the derivative of functional is negative definite if the following condition holds: 𝛺1(𝑖,1 − 𝜆 𝜌2 , 𝐴, 𝐵, 𝑀)+𝜏𝑀⊤[𝑅0 0 3𝑅]−1 𝑀+ (𝑇𝑘−𝜏)𝛺2(𝐴, 𝐵)≺0,∀𝜏∈ [0, 𝑇𝑘]. Since the previous condition is convex with respect to 𝜏∈ [0, 𝑇𝑘], an equivalent condition writes ⎡⎢⎢⎣ 𝛺1(𝑇𝑘,1 − 𝜆 𝜌2 , 𝐴, 𝐵, 𝑀)𝑇𝑘𝑀⊤ ∗ −𝑇𝑘 𝑅⎤⎥⎥⎦ ≺0, 𝛺1(𝑇𝑘,1 − 𝜆 𝜌2 , 𝐴, 𝐵, 𝑀)+𝑇𝑘𝛺2(𝐴, 𝐵)≺0.(35) We use a convexity argument with respect to the inter-sampling time 𝑇𝑘that belongs to the interval [1,2]to conclude that Theorem 2 ensures  (𝜏, 𝜒𝑘, 𝜒𝜅(𝓁), 𝑇𝑘)<0, which guarantees the stability of the closed-loop system with the event-triggering rule (6).□ The above theorem presents a stability condition for closed-loop systems under dynamic aperiodic event-triggered control (6). This theorem is based on the use of the Wirtinger-based inequality, which can be further refined by using a less conservative integral inequality such as Bessel’s inequality but this is not the objective of this paper. Similarly, the number of decision variables involved in the previous stability condition is a crucial issue when dealing with LMIs. Here Theorem 2 introduce an LMI condition whose complexity is measured by the total number of decision variables (18.5𝑛2+ 3.5𝑛) and the dimension of the condition is 11𝑛× 11𝑛. Altogether, the complexity of the problem is reasonable for systems of dimension lower than 𝑛= 20. While the parameters of the Lyapunov function and of the looped-functional must remain free to allow flexibility, it has been shown in [40] that particular cases of the slack variable 𝑀, with fewer decision variables, can be considered to reduce the complexity of the LMI. These particular cases refer, for instance, to the (parameter dependent) reciprocally convexity combination lemma but, again, this is not the objective of this paper. If one is dealing with systems of large dimension, it is possible to reduce the complexity by removing the last components of 𝜉𝑘 (e.g 1 𝜏∫𝜏 0𝜒⊤ 𝑘(𝑠)d𝑠), which will reduce the number of decision variables as well as the dimension of the LMI, of course at the price of introducing conservatism. 4.2. Relevant corollaries It is important to stress that particular cases can be stated to demonstrate the generality of our proposed theorem. The first one is for the case of periodic dynamic event-triggered control. The second deals with aperiodic static event-triggered control. The last one presents an extension to uncertain plants. The other possible combinations, for instance the uncertain case with periodic sampling and static rule, can be easily stated and are therefore omitted. Nonlinear Analysis: Hybrid Systems 52 (2024) 101444 9 Y. Xu et al. 4.2.1. Periodic event-triggered control The following corollary addresses the case of periodic event-triggered control, i.e., when 1=2=>0. Corollary 1. Let be a strictly positive scalar and consider scalar parameters (𝜆, 𝜌, 𝜂0)such that 𝜌∈ [0, 𝜆)⊂[0,1] and 𝜂0>0. Assume that there exist matrices 𝑃=𝑃⊤≻0,𝑅=𝑅⊤≻0,𝑆=𝑆⊤,𝛷=𝛷⊤, the controller gain 𝐾and matrices 𝑄,𝑀of appropriate dimensions such that the conditions (23) are satisfied with 1=2=, that is ⎡⎢⎢⎣ 𝛺1(,1 − 𝜆 𝜌, 𝐴, 𝐵, 𝑀)𝑀⊤ ∗ − 𝑅⎤⎥⎥⎦ ≺0, 𝛺1(,1 − 𝜆 𝜌, 𝐴, 𝐵, 𝑀)+𝛺2(𝐴, 𝐵)≺0,[𝐼 𝐼]⊤ 𝛷[𝐼 𝐼]⪯ 0.(36) Then the system (18) with the periodic dynamic event-triggered control (6) is asymptotically stable. Proof. The proof is derived by simply imposing 1=2=, which makes the sampling periodic. □ 4.2.2. Static event-triggered control Following Remark 2, the following corollary is stated to address the case of aperiodic static event-triggered control. Corollary 2. Let 0<1≤ 2be two positive scalars and consider scalar parameters (𝜆, 𝜌) = (1 − 𝜖, 0), with an arbitrarily small 𝜖 > 0 and 𝜂0>0. Assume that there exist matrices 𝑃=𝑃⊤≻0,𝑅=𝑅⊤≻0,𝑆=𝑆⊤,𝛷=𝛷⊤, the controller gain 𝐾, and matrices 𝑄,𝑀of appropriate dimensions such that the following LMIs are satisfied ⎡⎢⎢⎣ 𝛺1(𝑖,1 2 , 𝐴, 𝐵, 𝑀)𝑖𝑀⊤ ∗ −𝑖 𝑅⎤⎥⎥⎦ ≺0, 𝛺1(𝑖,1 2 , 𝐴, 𝐵, 𝑀)+𝑖𝛺2(𝐴, 𝐵)≺0,[𝐼 𝐼]⊤ 𝛷[𝐼 𝐼]⪯ 0,(37) for 𝑖= 1,2, where 𝛺1and 𝛺2are given in (24). Then, system (18) with the aperiodic static event-triggered control (4) is asymptotically stable. Proof. The proof is performed by selecting the particular case of the dynamic event-triggered variable 𝜂𝑘in (7) with 𝜆= 1 − 𝜖 and the event-triggering rule (6) with 𝜌= 0. It is worth noting that selecting 𝜆= 1 − 𝜖is mandatory to guarantee that variable 𝜂𝑘 converges to zero even though the computation of this variable is not required to compute the control law in the static case but is still required for the proof. These selections of (𝜆, 𝜌)lead us to the definition of  𝛺1(𝑖)in (24), by eliminating 𝜂𝑘in (27).□ 4.2.3. Uncertain and time-varying systems Consider now that 𝐴and 𝐵are uncertain and/or time varying. A possible way to represent such uncertain systems is to include 𝐴and 𝐵in a polytope of 𝐽∈Nvertices, which is defined by some positive scalars 𝛼𝑗≥0,𝑗= 1,…, 𝐽 such that ∑𝐽 𝑗=1 𝛼𝑗= 1 and [𝐴 𝐵𝐾] = 𝐽 ∑ 𝑗=1 𝛼𝑗[𝐴𝑗𝐵𝑗𝐾].(38) An extension of Theorem 2 to this class of uncertain systems is proposed below. Corollary 3. Let 0<1≤ 2be two positive scalars and consider scalar parameters (𝜆, 𝜌, 𝜂0)such that 𝜌∈ [0, 𝜆)⊂[0,1] and 𝜂0>0. Assume that there exist matrices 𝑃=𝑃⊤≻0,𝑅=𝑅⊤≻0,𝑆=𝑆⊤,𝛷=𝛷⊤, the controller gain 𝐾, and matrices 𝑄,𝑀𝑗with 𝑗= 1,…, 𝐽 of appropriate dimensions such that the following LMIs are satisfied ⎡⎢⎢⎣ 𝛺1(𝑖,1 − 𝜆 𝜌2 , 𝐴𝑗, 𝐵𝑗, 𝑀𝑗)𝑖𝑀⊤ 𝑗 ∗ −𝑖 𝑅⎤⎥⎥⎦ ≺0, 𝛺1(𝑖,1 − 𝜆 𝜌2 , 𝐴𝑗, 𝐵𝑗, 𝑀𝑗)+𝑖𝛺2(𝐴𝑗, 𝐵𝑗)≺0,[𝐼 𝐼]⊤ 𝛷[𝐼 𝐼]⪯ 0,(39) for 𝑖= 1,2, and 𝑗= 1,…, 𝐽, where 𝛺1and 𝛺2are given in (24). Then the system (18) with uncertain matrices (38) and the aperiodic dynamic event-triggered control (6) is asymptotically stable. Proof. To perform the proof, we first need to introduce the flexibility in the matrix 𝑀to account for the uncertain case. Let us define a time-varying or uncertainty-dependent matrix 𝑀=∑𝐽 𝑗=1 𝛼𝑗𝑀𝑗, where the 𝛼𝑗are the weights of the polytopic representation and where the 𝑀𝑗are 𝐽matrices of the same size as 𝑀. The proof is performed by noting that conditions (23) of Theorem 2 are convex with respect to 𝐴and 𝐵but also with respect to 𝑀. Consequently, it holds that ⎡⎢⎢⎢⎣ 𝛺1(𝑖,1 − 𝜆 𝜌2 , 𝐽 ∑ 𝑗=1 𝛼𝑗𝐴𝑗, 𝐽 ∑ 𝑗=1 𝛼𝑗𝐵𝑗, 𝐽 ∑ 𝑗=1 𝛼𝑗𝑀𝑗)𝑖∑𝐽 𝑗=1 𝛼𝑗𝑀⊤ 𝑗 ∗ −𝑖 𝑅⎤⎥⎥⎥⎦ ≤𝐽 ∑ 𝑗=1 𝛼𝑗⎡⎢⎢⎣ 𝛺1(𝑖,1 − 𝜆 𝜌2 , 𝐴𝑗, 𝐵𝑗, 𝑀𝑗)𝑖𝑀⊤ 𝑗 ∗ −𝑖 𝑅⎤⎥⎥⎦ , and the same calculations hold for the second inequality of (23), which concludes the proof. □