An ℓ 2 -consistent event-triggered control policy for linear systems ✩

In this article, we consider the design of an event-triggered ℓ 2 -control policy, for a setting where a scheduler is arbitrating state transmissions from the sensors to the controller of a discrete-time linear system. We start by introducing a periodic time-triggered ℓ 2 -controller for different transmission time-periods with a given ℓ 2 -gain bound using the minimax game-theoretical approach. After that, we propose an ℓ 2 -consistent event-triggered controller in the sense that it guarantees at least the same ℓ 2 -gain bound as the designed periodic time-triggered ℓ 2 -controller, however with a larger, or at most equal, average inter-transmission time. In practice, for typical disturbances, the proposed event-triggered scheme can lead to significant gains, both in terms of communication savings and disturbance attenuation, compared to periodic time-triggered policies, which is illustrated through a numerical example.


Introduction
The advent of new communication technologies, such as 5G, will further facilitate the rapid expansion of networked control systems (NCS) in many (industrial) branches of our society in the years to come.In NCSs, sensors, controllers and actuators communicate through shared communication networks.Applications include vehicle platooning, cloud-based control, smart grids, and robot swarms.In configurations where communication between agents happens periodically, the well-developed theory of sampled-data control (Chen & Francis, 2012) can be used to guarantee stability and performance of these systems.However, periodic communication for control applications can be rather resource-inefficient.In fact, control applications require large bandwidth for high communication frequencies and, when relying on wireless technologies, can lead to a large power consumption, which can be prohibitive when using battery powered communication devices.Therefore, managing and reducing the communication between sensors, controllers and actuators is crucial in many networked control applications.
Event-triggered controllers (ETCs) have been proposed in the literature as an alternative to periodic time-triggered controllers ✩ This project was funded from the European Union's Horizon 2020 Frame- work Programme for Research and Innovation under grant agreement No. 674875 (oCPS).The material in this paper was not presented at any conference.This paper was recommended for publication in revised form by Associate Editor Dimos V. Dimarogonas under the direction of Editor Christos G. Cassandras. in order to decrease the communication load in NCSs, while at the same time preserving stability and performance requirements see, e.g., Åström and Bernhardsson (2002), Behera et al. (2018), Heemels et al. (2012), Heemels et al. (2008), Lunze and Lehmann (2010), Molin and Hirche (2014), Nowzari et al. (2019) and Tabuada (2007) and the references therein.In a loop with an ETC, data transmissions between agents (sensors, controllers, actuators) are triggered based on well-defined events such as abrupt changes in the value of data or when estimation errors exceed certain thresholds.A large number of studies has been carried out so far in this research area with promising results in reducing the communication burden of the control loops, see, e.g., Antunes and Heemels (2014), Araujo et al. (2014), Mastrangelo et al. (2019), Mazo and Tabuada (2008), Postoyan et al. (2011), Weerakkody et al. (2016) and Wu et al. (2013).In some studies, ETCs are designed in order to guarantee stability of the system (Mamduhi et al., 2017;Mazo & Tabuada, 2008;Postoyan et al., 2011).Others also provide guarantees on an average quadratic cost of the event-triggered control-loops (Antunes & Heemels, 2014;Araujo et al., 2014;Asadi Khashooei et al., 2018;Balaghi I. & Antunes, 2017;Balaghi I. et al., 2018;Brunner et al., 2018;Goldenshluger & Mirkin, 2017).
Another important performance criterion for control-loops is the ℓ 2 -or L 2 -gain, which captures the worst-case disturbance attenuation level from an exogenous input to a performance output of the control loop for discrete-time or continuous-time systems, respectively.In a networked control configuration with communication limitations, the setup as depicted in Fig. 1 is of interest, where a feedback controller K attenuates the effect of the disturbance input w on the performance output z of the https://doi.org/10.1016/j.automatica.2020.1094120005-1098/© 2020 Elsevier Ltd.All rights reserved.plant G. Here, a scheduler S determines the time instances when the measured state should be communicated to the controller through a communication network N. The event-triggered scheduler should be designed together with an appropriate controller to guarantee a certain ℓ 2 -or L 2 -gain bound for the closed-loop system, while the available communication network should be able to handle the required data transmissions.
In recent years, researchers took different approaches in order to design event-triggered ℓ 2 -or L 2 -controllers.In particular, conditions for the L 2 -stability of the proposed event-triggered transmission policies in a sampled-data control system configuration are given in Peng and Han (2013) and Yan et al. (2015), by constructing Lyapunov-Krasovskii functionals.In Kishida et al. (2017), finite-gain L 2 -stability is guaranteed for an uncertain linear system by jointly designing an event-triggered mechanism in updating the control inputs and a self-triggered mechanism in determining the next sampling time of the sensors.The exponential stability and L 2 -gain analysis of a NCS, where the sensor to controller and the controller to actuator communications are both based on event-triggered mechanisms, is studied by using the delay system approach in Hu and Yue (2013).Moreover, there are some other studies establishing the L 2 -stability of the systems with ETCs (Wang & Lemmon, 2009;Yu & Antsaklis, 2013) or providing guaranteed values for the ℓ 2 -gain of discrete-time linear systems with an ETC (Heemels et al., 2013).In addition, an ETC is designed for output-feedback linear systems by considering the L ∞ -gain of the closed-loops in Donkers and Heemels (2010).For nonlinear systems, ETCs are proposed in Abdelrahim et al. (2017) and Dolk et al. (2017) that guarantee a finite L p -gain for closed-loop systems and prevent the Zeno behaviour in data transmissions.
In principle, employing an ETC in NCSs is beneficial only if it results in a better performance in comparison to time-triggered periodic control when both transmit with the same average transmission rate.This concept was first introduced in Antunes and Asadi Khashooei (2016) and referred to as consistency.In recent years, consistent ETCs in the sense of average quadratic cost have been proposed in both centralized and decentralized NCS configurations, see, e.g., Asadi Khashooei et al. (2018), Balaghi I. et al. (2018), Brunner et al. (2018) and Goldenshluger and Mirkin (2017), see also an early result for scalar systems in Åström and Bernhardsson (2002).
We can also extend the notion of consistency to eventtriggered ℓ 2 -or L 2 -control loops.Accordingly, an ETC is called ℓ 2 -or L 2 -consistent if it guarantees the same ℓ 2 -or L 2 -gain bound as any periodic time-triggered ℓ 2 -or L 2 -controller, however, with a smaller or at most the same average transmission rate (Balaghi I. et al., 2019).In spite of all works previously mentioned in the context of event-triggered ℓ 2 -or L 2 -control, the design of an ℓ 2 -or L 2 -consistent ETC has not received much attention so far.In fact, we are only aware of two very recent references related to our work, see Balaghi I. et al. (2019) and Mi and Mirkin (2019).Our previous work (Balaghi I. et al., 2019) differs from the present paper as it focusses on designing a fixed, a priori given, transmission sequence, and not a policy, while Mi and Mirkin (2019) derive an ETC with similar L 2 -consistent properties as the one we present in this paper.However, they are given for continuous-time systems (and thus L 2 -gain), and, most importantly, follow a very different approach based on the Youla parametrization, whereas we consider discrete-time systems and follow a game-theoretical approach.As both results are developed independently and follow different approaches for different settings, they are of independent interest.To be precise, in this work, for a given fixed transmission time period, we design a periodic time-triggered ℓ 2 -controller for any feasible ℓ 2 -gain bound, following a game-theoretical approach.
Then, we design an ETC guaranteeing an equal ℓ 2 -gain bound as that of the designed periodic time-triggered ℓ 2 -controller, however, with a larger (or at least equal) average inter-transmission time.In fact, based on our proposed ETC, when the realization of the disturbance input follows the worst-case scenario, then the proposed ETC triggers data transmissions periodically.However, when the disturbance input deviates from the worst-case scenario, then our proposed ETC is able to skip data transmissions thereby guaranteeing a larger average inter-transmission time than the time period of the periodic controller, while they both guarantee the same ℓ 2 -gain bound for the system.
Implicit in the NCS of interest in the current work it that (possibly large) packets of information are sent to the controller (and there is no error in the transmitted values) and it is the objective of the scheduler to keep the number of transmissions (average communication rate) as small as possible, while guaranteeing certain performance objectives.An alternative perspective, also considered in the literature (see, e.g., Ishii & Francis, 2002), is to keep the communication frequency constant, but reduce the size of the packets to be transmitted (and thus there is a discrepancy between the actual measurements and the transmitted quantized value) and thereby also realize a small bit rate.The problem of interest in this line of research is to determine the accuracy (or the number of bits) of each communicated data packet on the relation of a given control objective or to find the minimal number of bits needed in order to realize a certain objective.
Although not considered in this paper, there are also some recent works, where this idea is jointly employed with event-triggered transmission mechanisms, which can at the same time reduce the communication frequency as is investigated, for instance, in Abdelrahim et al. (2019), Ling (2020) and Tallapragada and Cortés (2016), all also the references therein, to achieve exponential and input-to-state stability for linear systems, respectively.
The remainder of this paper is organized as follows.The problem of interest is introduced in Section 2 and an ℓ 2 -consistent ETC is proposed in Section 3. The effectiveness of the novel ETC in decreasing the communication load is demonstrated through a numerical example in Section 4. Finally, Section 5 presents concluding remarks.The proofs of lemmas and theorems can be found in the Appendix.
Notation.For r, s ∈ N 0 := N ∪ {0}, we define N s r = {t ∈ N 0 |r ⩽ t ⩽ s} and ℓ d 2 as the Hilbert space of square summable sequences w := {w k } k∈N 0 , where w k ∈ R d for all k ∈ N 0 , and Moreover, ⌊x⌋ indicates the floor of an x ∈ R, and for matrices A, and B, we define diag(A, B) for the corresponding block diagonal matrix.

Problem setting
We introduce the NCS with periodic communication in Section 2.1 and the NCS with event-triggered communication in Section 2.2.The problem of interest is stated in Section 2.3.

Networked control system with periodic communication
Consider the system architecture in Fig. 1 in which the plant G is given by a discrete-time linear time-invariant (LTI) system the state, the control input and the disturbance, respectively, at discrete time k ∈ N 0 .Let w ∈ ℓ d 2 and assume that the disturbance generator at every time step has access to all the state vectors from the initial up to the current time-step.Therefore, w k = T k (E k ), where for some mapping otherwise.For the periodic transmission policy with a given time period τ ∈ N, we set δ k = π τ k , where Then any periodic control policy can be formulated as where at every k ∈ N 0 , is the information set available for the controller and R Although we use here this general definition, in practice, the periodic control policies of interest (see Lemmas 1 and 2) will only depend on the last transmitted state.Therefore, the controller does not need to store all the received state vectors in memory (which can possibly require a large memory).The goal of an ℓ 2 -controller is to attenuate the effect of the disturbance input w k on the performance output of the system, where we assume that F has full column rank.
Let E T E = Q and without loss of generality we can now assume We need the following assumptions and the definition of global asymptotic stability in the sequel.
Note that these assumptions are rather standard in the ℓ 2 -control context (see also Point 3. after Theorem 1).
Definition 1 (Global Asymptotic Stability Aliyu, 2017).The system (1) with w = (0, 0, . . . ) and a given control input policy is said to be globally asymptotically stable (at equilibrium point i) the control loop is Lyapunov stable, i.e., for every ζ ≻ 0, there exists a δ ≻ 0 such that for all initial states x 0 ∈ R n with ∥x 0 ∥ ⩽ δ, it holds that ∥x k ∥ ⩽ ζ for every k ∈ N 0 , (ii) the corresponding state trajectory x k converges to x e = 0 as time goes to infinity, i.e., lim k→∞ x k = 0. □ Next, we formally define the concept of τ -periodic ℓ 2 -controller for the system (1).
the system (1) and ( 6), where F πτ k follows (5), such that (i) the closed-loop control system given by ( 1) and ( 4) is globally asymptotically stable when w = (0, 0, . . .), (ii) when holds for some positive ϵ (independent of w), is referred to as a τ -periodic ℓ 2 -controller with ℓ 2 -gain bound γ .Moreover, the infimum value of γ ∈ R ≻0 for which a τ -periodic ℓ 2 -controller exists with ℓ 2 -gain bound γ is called the infimal ℓ 2 -gain of ( 1) and ( 6), and is denoted by γ * τ .□ Let us define for all w ∈ ℓ d 2 .Based on ( 7), when w = (0, 0, . . .), the designed τ -periodic ℓ 2 -controller should result in J to be equal or less than zero for x 0 = 0.However, when w ̸ = (0, 0, . . .), then J should always be strictly less than zero.Before designing a τ -periodic ℓ 2 -controller with ℓ 2 -gain bound γ , one should evaluate the existence of such a controller for the given value of γ ∈ R ≻0 , i.e., decide if γ ≻ γ * τ .However, for a given γ ∈ R ≻0 , a τ -periodic ℓ 2 -controller with ℓ 2 -gain bound γ exists if and only if, for x 0 = 0, the following minimax optimization problem results in a nonpositive value, i.e.J * ⩽ 0, where This can be concluded from the arguments in Başar and Bernhard (2008).Therefore, the infimal ℓ 2 -gain of the closed-loop system with τ -periodic transmission is the infimum value of the set of γ ∈ R ≻0 for which the minimax problem ( 9) has a non-positive value.Moreover, if for a given γ ∈ R ≻0 , J * is nonpositive, then the optimal control policy determined based on ( 9) is a τ -periodic ℓ 2 -controller in the sense of Definition 2. In the following two lemmas, we provide a τ -periodic ℓ 2 -controller with ℓ 2 -gain bound γ by solving the minimax problem (9).Lemma 1 considers the special case τ = 1 and Lemma 2 provides the results for general τ ∈ N.
Lemma 1 (1-Periodic ℓ 2 -Controller).Let Assumption 1 hold.Then (i) there exists a γ ∈ R ≻0 such that for every γ ≻ γ , the Ricatti where H = I + (BB T − γ −2 DD T )M, has a positive definite solution M and γ 2 I − D T MD ≻ 0.Moreover, the infimum value of γ for which the above holds coincides with the infimal ℓ ] , (13) 1 We can easily investigate the condition with an unknown initial condition by adding one extra time to the time-horizon and considering the initial condition as the disturbance of the previous time (Başar & Bernhard, 2008).
Before considering the general condition in Lemma 2, where τ ∈ N, let us introduce another time variable ι ∈ N 0 , where ι = ⌊ k τ ⌋ and define the following augmented control and disturbance inputs at every ι ∈ N 0 , Lemma 2 (τ -Periodic ℓ 2 -controller for τ ∈ N).Let Assumption 1 hold.Then (i) there exists a γ ∈ R ≻0 such that for every γ ≻ γ , (10) has a positive definite solution M and .
(ii) for any γ ≻ γ * τ , the control policy where K follows (12) and where u * k follows (16) for all k ∈ N 0 , and where Φ τ := Y 0 , for all τ ∈ N, and Y 0 is determined based on the following backward iteration and for all h Lemmas 1 and 2 do not only provide a τ -periodic ℓ 2 -controller for (1), ( 6), and a given τ ∈ N but also introduce upper bounds for J in ( 13) and ( 21) that will be useful in the design of a NCS with event-triggered communication in Section 3, as we will see.
Remark 1.It is important to mention that Lemmas 1 and 2 still hold without any change if at every time-step k ∈ N (ι+1)τ −1 ιτ , the disturbance generator has access also to the control inputs from k up to (ι + 1)τ − 1, i.e., the information set available for the disturbance generator follows Moreover, the disturbance input policies w * k given in ( 14) and ( 19) are the worst-case disturbance scenarios, when the disturbance generator has access to the information set ( 24) at all times.

Networked control system with event-triggered communication
The NCS we are interested in has the same plant G as in (1) and the information set of the disturbance generator also follows (2) (or ( 24)).However, data transmission to the controller follows a state-dependent mechanism, which is called an event-triggered transmission policy, and we can formulate it as where is the information set available for the scheduler at k ∈ N 0 .Then, any appropriate control policy is defined as where is the information set available for the controller at k ∈ N 0 based on an event-triggered scheduling policy defined in (25) and R Similarly to the periodic control case, in practice, the event-triggered scheduling and control policies of interest (see, e.g., the proposed one in Section 3) will only depend on a few members H k and F µ k , respectively.Therefore, the controller does not need to store all the received state vectors (and thus does not need a large memory).We call an event-triggered scheduler and its related controller an ETC, which is denoted by η = (µ, R µ ).Furthermore, we introduce the average transmission rate associated with an event-triggered scheduling policy µ and a disturbance sequence w ∈ ℓ d 2 as fη (w) = lim sup T →∞ 1 T ∑ T −1 t=0 µ t (H t ) and the average inter-transmission time as Ωη (w) = 1/ fη (w).Next, we define the concept of an event-triggered ℓ 2 -controller.
Definition 3 (Event-triggered ℓ 2 -Controller Yan et al., 2015).Given γ ∈ R ≻0 , an ETC η = (µ, R µ ) for the system (1) and ( 6) satisfying that (i) the closed-loop control system (1) and ( 27) is globally asymptotically stable when w = (0, 0, . . .), (ii) under the assumption of zero initial condition J ⩽ 0 for all w ∈ ℓ d 2 , where J follows (8), is referred to as an event-triggered ℓ 2 -controller with ℓ 2 -gain bound γ .Moreover, the infimum value of γ ∈ R ≻0 , where (i) and (ii) hold for an ETC η designed for (1) and ( 6) is called the infimal ℓ 2 -gain of the event-triggered control loop and is denoted by γ * η .□ Remark 2. One could define condition (ii) in Definition 2 exactly in the same way as in Definition 3.However, in this case, γ 2 I − DT τ Mτ Dτ ⩾ 0 would be the necessary condition for the existence of a τ -periodic ℓ 2 -controller for a given ℓ 2 -gain bound γ and τ ∈ N, while γ 2 I − DT τ Mτ Dτ ≻ 0 is the sufficient condition for the existence of the proposed τ -periodic ℓ 2 -control policies in (11) and ( 16).The current condition (ii) of Definition 2 is important to determine γ 2 I − DT τ Mτ Dτ ≻ 0 as both the necessary and sufficient conditions for the existence of a τ -periodic ℓ 2 -controller for a given ℓ 2 -gain bound γ and τ ∈ N. It is also important to mention that based on Definitions 2 and 3, event-triggered ℓ 2 -controllers have to satisfy a weaker condition than τ -periodic ℓ 2 -controllers.However, since ϵ in Definition 2 is allowed to be arbitrarily small, this difference in definitions is negligible.

Problem statement
The τ -periodic ℓ 2 -controllers with ℓ 2 -gain bound γ determined in Lemmas 1 and 2 periodically update their state estimates based on the full-state measurements of the sensors.In this way, these controllers can guarantee a desired disturbance attenuation level γ for all disturbance inputs.For every τ -periodic ℓ 2 -controller given in Lemmas 1 and 2 we can propose an eventtriggered ℓ 2 -controller counterpart η, which guarantees the same disturbance attenuation level γ for the system.However, based on the realization of the disturbance inputs, its scheduler can skip some of these periodic data transmissions needed by the τperiodic ℓ 2 -controller, thereby requiring fewer transmissions and thus resulting in larger (or equal) values of Ωη (w) in comparison to τ .This ETC is called ℓ 2 -consistent according to the following definition.

. □
The goal of this work is to propose an ℓ 2 -consistent ETC for the NCS depicted in Fig. 1.

ℓ 2 -consistent event-triggered controller
We propose an ℓ 2 -consistent ETC in this section.For simplicity we start, in Section 3.1, with the case in which τ = 1, since the main ideas can already be conveyed for this case.In Section 3.2, we consider the general case in which τ ∈ N.
We know that the control policy (11) requires the state information at every time-step.However, in our desired ETC setting, the controller does not have the state information at all times and can, therefore, only use an estimation xk|k of the state x k at time k ∈ N 0 .In particular, we select the controller associated with our desired ℓ 2 -consistent ETC policy as where K is given as in ( 12) and xk|k is the state estimate in the controller.We propose three state estimators.Two are described as at all k ∈ N for N ∈ {I, A} and x0|0 = x 0 .The choice N = I boils down to keeping the estimated state constant if data is not transmitted to the controller and N = A boils down to updating the estimated state based on the system dynamics by ignoring the effects of both the control input and the disturbance at all k ∈ N and x0|0 = x 0 , is another (possibly more reasonable for some special disturbance inputs) state estimator in the controller.In the following theorem, we propose an event-triggered scheduling policy, which together with ( 29) and (30) (or (31)), results in an ℓ 2 -consistent ETC in the sense of Definition 4.
1: Based on the event-triggered scheduling policy (32), a deviation of the actual disturbance inputs from the worst-case disturbance scenario given by ( 14) acts as a ''reward'' in order to skip data transmissions and let the control inputs deviate from the one determined for 1-periodic ℓ 2 -controller in (11).This reward can counteract the penalty incurred by skipping data transmissions (as then u k ̸ = u * k ).This is the main intuition behind the proposed ETC in Theorem 1.Moreover, as it can be easily concluded, if the disturbance inputs follow w k = w * k for all k ∈ N 0 , where {w * k |k ∈ N 0 } can be seen as a worst case disturbance input, then the proposed event-triggered scheduling policy (32) always triggers data transmissions, i.e., µ k (H k ) = 1 at all k ∈ N 0 , unless ûk = u * k , which is typically not the case.

2:
We can show that for the 1-periodic ℓ 2 -controller determined in Lemma 1, and for all γ ≻ γ * 1 , J * = x T 0 Mx 0 , where M is given in Lemma 1.As we select a smaller value for γ , M will become larger (in the sense that . Furthermore, based on (10), we can conclude that MH −1 will also become larger.Now, then Φ 1 becomes larger and Ψ 1 becomes smaller.Therefore, the scheduling law ( 32) is expected to trigger more transmissions for smaller values of γ and the same disturbance input sequence w.
3: In order to evaluate the event-triggered condition (32) at every time k ∈ N 0 , the scheduler needs the values of , which can be calculated by using given the condition that the scheduler receives x k at every k ∈ N 0 and knows the control policy, from which the control inputs u k−1 can be replicated.Therefore, (ii) in Assumption 1 helps to calculate the values of the disturbance inputs, needed in the event-triggered scheduling policy, based on the state measurements, and there is no need for measuring them independently.

General case τ ∈ N
For any γ ≻ γ * τ , the τ -periodic ℓ 2 -controller ( 16) requires periodic state transmission after every τ ∈ N time-steps.However, in this section, we propose an event-triggered ℓ 2 -controller with ℓ 2 -gain bound γ ≻ γ * τ , which can skip data transmissions at some of these time-steps.Let us introduce the augmented control policy associated with our desired ETC as where Kτ = B0 in which B0 is determined based on (23) and similar to the previous section, we can either have as the state estimator in the controller for all ι ∈ N and x0|0 = x 0 , depending on the characteristic of the disturbance input.In the following theorem, we propose an event-triggered scheduler which together with ( 33) and (34) (or ( 35)) result in an ℓ 2consistent ETC based on Definition 4.
The ℓ 2 -consistency of the proposed ETC policies in Theorems 1 and 2 indicates that for the same γ ∈ R ≻0 as the disturbance attenuation level where γ ≻ γ * τ , in case the disturbance input does not follow the worst-case scenario given by ( 19), the eventtriggered scheduler can skip data transmissions at some times required by the τ -periodic controller ( 16) and results in a larger average inter-transmission time than τ while guaranteeing the same ℓ 2 -gain bound γ .Moreover, for the proposed ETC, the behaviour of the average inter-transmission time with respect to γ when γ ≻ γ * τ is not necessarily increasing and it highly depends on the actual disturbance input of the system.This can be clearly seen in Fig. 3 corresponding to a numerical example.

Numerical example
Consider a scalar system where A = 1.1, B = 1, D = 1 are the parameters of the linear model (1), and Q = 1.Moreover, we take w k = e − k 200 sin( k 25 ), k ∈ N 0 , as the unknown disturbance input of the system.The infimal ℓ 2 -gain of the system for periodic control with the inter-transmission time-steps τ ∈ {1, 2, 3, 4} are γ * 1 = 1.487, γ * 2 = 2.202, γ * 3 = 2.999 and γ * 4 = 3.871.According to Lemmas 1 and 2, for any τ ∈ N and γ ≻ γ * τ , we can design a τ -periodic ℓ 2 -controller with ℓ 2 -gain bound γ .Then based on Theorems 1 or 2, we can design its ℓ 2 -consistent ETC counterpart for this system.Based on Definition 4, the proposed ℓ 2 -consistent ETC can result in the same attenuation level (ℓ 2 -gain bound) as the PTC (11) or ( 16), however, with a smaller (or at most an equal) average transmission rate.However, we will show that for the given system and the disturbance input, it is even possible to achieve smaller disturbance attenuation levels by following the proposed ETC in comparison to the PTC (11) or (16), while they both have the same average transmission rate.
Firstly, we consider τ = 1 and design an ETC based on Theorem 1.The controller follows (29), where the state estimation in the controller is determined based on (30) for N = A. Fig. 2(a) shows the state trajectory related to the ETC when τ = 1 and γ = 1.630 ≻ γ * 1 is its corresponding ℓ 2 -gain bound.This ETC results in Ωη (w) = 2.033, where Ωη (w) denotes the average inter-transmission time of scheduler.However, if the scheduler triggers transmissions periodically with τ = 2, then we know that the infimal ℓ 2 -gain of the system with periodic control is γ * 2 = 2.202.The state trajectory of the periodic controller ( 16) for γ = γ * 2 + ϵ, where ϵ ≻ 0 is a small real number, and τ = 2 is shown in Fig. 2(a), which indicates the better disturbance attenuation of the ETC while they both have almost the same average transmission rate for the given disturbance input w.where the state estimation in the controller is determined based on (34) for N = A 2 .Again, for this system, the ℓ 2 -gain bound of the system with the ETC is significantly smaller than the minimum value that results from periodic control while they both have almost the same average inter-transmission timesteps to the controller for the given disturbance input w.Fig. 3 is more generic and illustrates the ℓ 2 -consistency of the proposed ETC when τ = 1 and τ = 2.For the given disturbance input w, we find the average inter-transmission time of the system with the ℓ 2 -consistent ETC designed for different values of γ ≻ γ * τ , a time horizon of 250 and zero initial condition.The solid line shows the trade-offs one can achieve by following a periodic time-triggered control strategy.We easily see the better trade-offs for the proposed ℓ 2 -consistent ETC in comparison to PTCs.In principle, based on the theory (Theorems 1 and 2), for every τ ∈ N and γ ∈ [γ * τ , γ * τ +1 ) the trade-offs for the proposed ETC ( 36) and (33) (or ( 32) and ( 29) when τ = 1) are guaranteed to be bellow (or at most on) the stairwise curve of PTCs for any linear system (1) and disturbance input w.

Conclusions
In this work, we investigated the design of event-triggered controllers (ETCs) for discrete-time linear systems by considering the ℓ 2 -gain as a performance criterion of the closed-loop system.Firstly, for every transmission time period, we determined a periodic ℓ 2 -controller for a given ℓ 2 -gain bound, following a game-theoretical approach.Then, we introduced the notion of ℓ 2 -consistency, which refers to any ETC that guarantees the same ℓ 2 -gain bound as that of the designed periodic ℓ 2 -controller, however, with a larger or an equal average inter-transmission time.Next, we proposed the design of an ℓ 2 -consistent ETC with some interesting features.When the disturbance input follows the worst-case scenario at every time, the scheduler triggers transmissions periodically in order to guarantee an ℓ 2 -gain bound for the system.However, when the disturbance input is not equal to the worst-case scenario, the ℓ 2 -gain bound of our designed ETC is still guaranteed and equal to that of the designed periodic ℓ 2 -controller, however, with a (significantly) larger average inter-transmission time.Possible directions for future work include considering linear plants with partial state information, non-linear plants and date bit rate constraints.

A.1. Proof of Lemma 1
Parts i and ii can be proved by the arguments in Theorem 3.8 of Başar and Bernhard (2008), in which the stabilizability of (A, B) and the observability of (Q 1 2 , A) are used to guarantee the existence of γ ∈ R ≻0 , where for all γ ≻ γ the Ricatti equation ( 10) has a positive definite solution M. They are also proved (by making τ = 1) in a more general setting in Lemma 2. The only point that is not proved in Başar and Bernhard (2008) is the Lyapunov stability of the control loop when w = (0, 0, . . .), which we postpone it to the end of the present proof.
).Now by completing the squares for w k , we obtain Now, we complete the squares for u k .

J(x
where and u * k = −B T MH −1 Ax k .Now by using the matrix inversion lemma (Henderson & Searle, 1981, equation ( 18)), we can show that Φ −1 Then by summing all the values of J(x k , x k+1 ) over k ∈ N ν 0 for an arbitrary ν ∈ N and considering From this equation we conclude that Since M is a positive definite matrix and x 0 = 0, then ] , which proves part iii.Now we need to prove the global asymptotic stability of the control loop, when w = (0, 0, . . . ) and the control input follows (11).We take V (x k ) = x T k Mx k as the Lyapunov function candidate, where M is a positive definite solution of (10).Considering ∆V k := V (x k+1 ) − V (x k ), then based on (A.1), for w = (0, 0, . . . ) and Therefore, the control loop is Lyapunov stable.Moreover, based on the observability of (Q 1 2 , A) it can be shown that the state of the control loop converges to zero as time goes to infinity, when w = (0, 0, . . . ) and u k = u * k at every k ∈ N 0 , see Başar and Bernhard (2008, page 62).Thus, the system is globally asymptotically stable.

A.2. Proof of Lemma 2
(i) necessary and sufficient conditions for the existence of a τperiodic ℓ 2 -controller According to Theorem 3.8 of Başar and Bernhard (2008), taking into account the stabilizability of (A, B) and the observability of (Q 1 2 , A), there exists a γ1 ∈ R ≻0 , where for all γ ≻ γ1 the Ricatti equation ( 10) has a positive definite solution M.Moreover, it is clear that there exists γ2 ≻ 0 such that for all γ ≻ γ2 , γ 2 I − DT τ Mτ Dτ ≻ 0 holds.Then we can take γ := max{ γ1 , γ2 }, which establishes the first assertion.Now to prove the second assertion in statement i, we resort to an argument in Başar and Bernhard (2008), which indicates that the conditions needed to find a controller satisfying part ii of Definition 2 are the same as the conditions needed to have J * ⩽ 0, where J * is given in (9).Solving the minimax optimization problem in (9) is equivalent to finding an appropriate value function V(x ℓ ) such that the following Isaacs equation holds for every ℓ = ιτ ∈ N 0 and every x ℓ ∈ R n (Başar & Olsder, 1999, Corollary 6.2), In this minimax game, the information structures of the two players are periodic with given, generally not equal, time periods.As a result of Theorem 6.9 in Başar and Olsder (1999) the value function for this two-player zero-sum minimax game is ℓ Mx ℓ , where M is the positive definite solution of (10).Therefore, we have to find the conditions under which the following equality always holds for every In order to solve the optimization problem in (A.3) for τ ∈ N, first we need to follow τ maximization steps and determine ] .
By substituting (1) into the above equation we get where a bounded w * s exists if γ 2 I − D T MD ≻ 0, and w Therefore, the optimal game value at time s is a function of x s and u s .Now, by an induction argument, let us assume that at an arbitrary optimization step h + 1 ∈ N τ −1 1 the optimal game value is , where Θ h+1 , Y h+1 are known positive definite matrices, r = ℓ + h, and Ûr+1 = [u T r+1 , . . ., u T ℓ+τ −1 ] T is the augmented control input. Then By substituting (1) into the above equation, we get w ] , Y h := ] , (A.7) for E h = γ −2 DD T V −1 h .Moreover, Ĵh takes the same form as the one assumed for the Ĵh+1 , and therefore, the quadratic form assumed for the Ĵh is correct.Finally, consider at the optimization step ℓ, after determining w * ℓ the optimal game value follows Ĵ0 (x ℓ , Ûℓ ) = Based on the event-triggered policy (32), there is a state transmission to the controller at k = 0, therefore u 0 = u * 0 .Moreover, we can change the summation in the above equation into two summations as follows where j ∈ N 0 represents the number of transmissions and s j is the time at which the j-th transmission happens.Then if u k at all k ∈ N 0 follows (29) for the data transmission scheduling policy (32), we have for ûi = K xi|i and ŵ * i = S(Ax i + B ûi ).Based on the scheduling policy (32), and the fact that at data transmission times ûs j = u * at every ν ∈ N 0 .Moreover, based on the event-triggered scheduling law (32), when w = (0, 0, . . .), at every ν ∈ N 0 .Therefore, ∆V (x ν ) ⩽ 0 at every ν ∈ N 0 , which indicates the Lyapunov stability of the control loop for the proposed ETC, when w = (0, 0, . . .).Then similar to the proof of Lemma 1, the observability of (Q 1 2 , A) and the boundedness of the performance index (8), i.e., J ⩽ x T 0 Mx 0 , guarantees the convergence of the state to zero as time goes to infinity.Therefore, we can conclude the global asymptotic stability of the control loop.
Therefore, the proposed ETC is an ℓ 2 -consistent ETC according to Definitions 3 and 4.

A.4. Proof of Theorem 2
The proof procedure is similar to the one presented for Theorem 1.However, here, in order to satisfy J ⩽ 0 for all w ∈ ℓ d 2 , where J is given in (8), we just need to consider (21).We just need to simplify the disturbance input as it is given in (19) when the control policy follows (33).

Fig. 1 .
Fig.1.The state feedback ℓ 2 -controller with a resource-constraint communication network.G, K , S and N refer to the plant, the controller, the scheduler and the network, respectively.
are the actual values of disturbances and control inputs, respectively, where for

Fig. 2 .Fig. 3 .
Fig. 2. Illustration of the improved performance of the proposed ETC in comparison to periodic time-triggered controller (PTC) in disturbance attenuation with the same average transmission rate (for the given disturbance input w) when (a) ETC designed for τ = 1 and PTC designed for τ = 2 (b) ETC designed for τ = 2 and PTC designed for τ = 4.
Fig. 2(b)   compares similar situations when the ETC is designed for τ = 2 and γ = 2.924 based on Theorem 2. The controller follows(33)