An Introduction to Event-triggered and Self-triggered Control

Transcription

1 An Introuction to Event-triggere an Self-triggere Control W.P.M.H. Heemels K.H. Johansson P. Tabuaa Abstract Recent evelopments in computer an communication technologies have le to a new type of large-scale resource-constraine wireless embee control systems. It is esirable in these systems to limit the sensor an control computation an/or communication to instances when the system nees attention. However, classical sample-ata control is base on performing sensing an actuation perioically rather than when the system nees attention. This paper provies an introuction to event- an self-triggere control systems where sensing an actuation is performe when neee. Eventtriggere control is reactive an generates sensor sampling an control actuation when, for instance, the plant state eviates more than a certain threshol from a esire value. Selftriggere control, on the other han, is proactive an computes the next sampling or actuation instance ahea of time. The basics of these control strategies are introuce together with a iscussion on the ifferences between state feeback an output feeback for event-triggere control. It is also shown how event- an self-triggere control can be implemente using existing wireless communication technology. Some applications to wireless control in process inustry are iscusse as well. I. INTRODUCTION In toay s stanar control textbooks, e.g. [1], [2], perioic control is presente as the only choice for implementing feeback control laws on igital platforms. However, questions relate to perioic vs aperioic implementations have gone in an out of fashion since feeback control loops starte being implemente on computers. Some early examples inclue the following references [3], [4], [5], [6], [7], [8]. This paper is concerne with the latest wave of the perioic vs aperioic control ebate or, as we prefer to call it, perioic vs event-base control. There are two funamental reasons for the resurgence of this ebate in the last 5 or 6 years. The first, is the increasing popularity of (share) wire an wireless networke control systems that raise the importance of explicitly aressing energy, computation, an communication constraints when esigning feeback control Maurice Heemels is with the Hybri an Networke Systems group, Department of Mechanical Engineering, Einhoven Univeristy of Technology, the Netherlans; Karl H. Johansson is with ACCESS Linnaeus Center, Royal Institute of Technology, Sween; Paulo Tabuaa is with Department of Electrical Engineering, University of California, Los Angeles, CA, USA. s: m.heemels@tue.nl,kallej@kth.se,tabuaa@ee.ucla.eu The work of the Maurice Heemels was partially supporte by the Dutch Science Founation (STW) an the Dutch Organization for Scientific Research (NWO) uner the VICI grant Wireless controls systems: A new frontier in automation. The work of Karl Johansson was partially supporte by the Knut an Alice Wallenberg Founation an the Sweish Research Council. Maurice Heemels an Karl Johansson were also supporte by the European 7th Framework Programme Network of Excellence uner grant HYCON The work of Paulo Tabuaa was partially supporte by NSF awars an loops. Event-base control offers some clear avantages with respect to perioic control when hanling these constraints but it also introuces some new theoretical an practical problems. The secon reason is the appearance of two papers [9], [1] that highlighte some of the avantages of event-base control an motivate the evelopment of the first systematic esigns of event-base implementations of stabilizing feeback control laws, e.g.,[11], [12], [13], [14]. Since then, several researchers have improve an generalize these results an alternative approaches have appeare. In the meantime, also so-calle self-triggere control [15] emerge. Event-triggere an self-triggere control systems consists of two elements, namely, a feeback controller that computes the control input, an a triggering mechanism that etermines when the control input has to be upate again. The ifference between event-triggere control an selftriggere control is that the former is reactive, while the latter is proactive. Inee, in event-triggere control a triggering conition base on current measurements is continuously monitore an when violate, an event is triggere. In selftriggere control the next upate time is precompute at a control upate time base on preictions using previously receive ata an knowlege on the plant ynamics. Most of the existing event-triggere control approaches employ the assumption that the full state information is available, even though in most practical situations this assumption is violate. As the separation principle oes not hol in general for event-triggere control systems [16], outputbase schemes are inee har to esign an optimize. Recently, some work on output-base event-triggere control emerge an in this paper we will iscuss a few of these solutions. We istinguish these solutions base on their time sets an the aopte control laws. The nature of the time sets will ifferentiate existing event-triggere control strategies base on the use of either continuous-time or iscrete-time controllers/event-triggering mechanisms. This ifferentiation oes not require much explanation, although one comment is relevant. In many works stuying iscrete-time event-triggere control schemes, the plant is also consiere to be of a iscrete-time nature. Clearly, this allows to obtain irect parallels between some of the continuous-time event-triggere control approaches an the iscrete-time counterparts. However, it is more interesting to take a more sample-ata -like approach on iscrete-time event-triggere control schemes in the sense that the behavior is stuie when this controller interacts with a continuoustime plant. Also stability an performance properties have to be consiere then in a continuous-time setting. In this setup the close loop consists of a continuous-time plant

2 an an event-triggere control strategy, which is a iscretetime controller operating in a perioic time-triggere manner. In this context, sometimes the term perioic event-triggere control is use, see [17], [18]. Regaring the ifferentiation on the nature of the outputbase control law, we istinguish the approaches base on whether there is an observer or not. In the former cases we will talk about an observer-base control law, an in the latter about a irect output-base law. Base on an observer, ifferent strategies can be implemente. In most cases the observer reconstructs the plant state using solely event-base information in the sense that it only applies innovation steps exploiting receive measurements at the event times, although some schemes also exploit the information present at synchronous (time-triggere) instants of time at which no events occur. The latter is, for instance, the case in [19], [2], [21]. In fact, [19] states literally absence of an event is however information that can be use by the observer, see also [22]. The observer-base schemes can still be categorize further base on the fact if the corresponing event-triggering conitions use information of the observer such as the estimate state, as, e.g., in [2], [23], [24] or not [21], [25]. The former schemes typically run the observer at the sensor sie using essentially all measurements available, while at the controller sie a preictor-like structure also prouces a state estimate, which is base on only sporaically receive information from the sensor system (incluing the observer) base on the event-triggering mechanisms. Often the event-triggering mechanisms provie new information to the preictor when the ifference between the state estimate of the preictor eviates too much from the state estimate available in the observer. Next to proviing an introuctory overview on some of the works in the area, the main objective of this paper is to emphasize the key ieas in three ifferent aspects of aperioic control: the basics on an ifferences between event-triggere an self-triggere control, the use of outputbase event-triggere control, an event-base control over wireless communication networks. The outline of the paper is as follows. The basic ieas of event-triggere control are introuce in Section II. Self-triggere control is iscusse in Section III. Output-base event-triggere control is surveye in Section IV followe by exemplary approaches for continuous-time irect output-base control (Section V) an iscrete-time observer-base control (Section VI). Eventtriggere transmission in wireless control systems is iscusse in Section VII. Finally, concluing remarks are given in SectionVIII. II. EVENT-TRIGGERED CONTROL In this section we introuce the main ieas of eventtriggere control following [12]. In orer to simplify the presentation we consier the linear case only even though the results in [12] were originally evelope for nonlinear systems. We start with a linear plant t x p = A p x p + B p u, x p R np, u R nu (1) an assume that a linear feeback control law u = Kx p (2) has been esigne renering the ieal close-loop system t x p = A p x p + B p Kx p (3) asymptotically stable, i.e., renering the real part of the eigenvalues of A p + B p K negative. The question that now arises is how to implement the feeback control law (2) on a igital platform. One possibility is to perioically recompute (2) an keep the actuator values constant in between the perioic upates. Rather than using time (the perio) to etermine when (2) shoul be recompute, we are intereste in recomputing (2) only when performance is not satisfactory. One way to efine performance is to use a Lyapunov function for the ieal close-loop system (3). Such a Lyapunov function, that we enote by V (x p ) = x T p P x p for some symmetric an positive-efinite matrix P, satisfies t V (x p(t)) = V (A p + B p K)x p = x T p Qx p, (4) x p where Q is guarantee to be positive-efinite. Since the time erivative of V along the solution of the close-loop system is negative, V ecreases. Moreover, the rate at which V ecreases is specifie by the matrix Q. If we are willing to tolerate a slower rate of ecrease, we woul require the solution of an event-triggere implementation to satisfy the weaker inequality t V (x p(t)) σx T p Qx p (5) for some σ [, 1[. Note that by choosing σ = 1 (5) becomes (4), while for σ < 1 (5) prescribes a slower rate of ecrease for V. The requirement (5) suggests that we only nee to recompute (2) an upate the actuator signals when (5) is about to be violate, i.e., when (5) becomes an equality. In orer to write such an equality in a convenient manner, we assume the inputs to be hel constant in between the successive recomputations of (2). This is often referre to in the literature as sample-an-hol an can be formalize as u(t) = u(t k ) t [t k, t k+1 [, k N, (6) where the sequence {t k } k N represents the instants at which (2) is re-compute an the actuator signals are upate. We refer to these instants as the triggering times or execution times. For simplicity, we assume that the process of collecting sensor measurements, re-computing (2) an upating the actuators can be one in zero 1 time. We now 1 This iealize assumption escribes the fact that in many implementations this time is much smaller than the time elapse between the instants t k an t k+1. This assumption is not essential an the intereste reaer can consult [12] for a specific extension of the results when this assumption oes not hol.

3 introuce the error e efine by e(t) = x p (t k ) x p (t) t [t k, t k+1 [, k N. Using this error we express the evolution of the close-loop system uring the interval [t k, t k+1 [ by t x p(t) = A p x p (t) + B p Kx p (t k ) = A p x p (t) + B p Kx p (t k ) + B p K(x p (t) x p (t)) = A p x p (t) + B p Kx p (t) + B p Ke(t). We can now use this expression to rewrite the time erivative of V (x p (t)) as t V (x p(t)) = V (A p + B p K)x p (t) + V B p Ke(t) x p x p = x T p (t)qx p (t) + 2x T p (t)p B p Ke(t) (7) Substituting (7) in inequality (5) we arrive at [ x T p (t) e T (t) ] [ (σ 1)Q P B p K K T B T p P ] [ xp (t) e(t) ]. (8) The triggering times t k can now be efine as the times at which the following equality hols with Ψ = z T (t k )Ψz(t k ) = (9) [ ] (σ 1)Q P Bp K K T Bp T, z(t P k ) = [ ] xp (t k ). e(t k ) The event-triggere implementation of the feeback control law (2) thus consists in keeping the actuator values constant as long as the triggering conition (9) is not satisfie an recomputing (2) an upating the actuators when the triggering conition (9) is satisfie (assuming z() T Ψz() < ). By changing the matrix Ψ we obtain other quaratic triggering conitions. For instance, in [12] the triggering conition e 2 σ x 2 is use that correspons to the choice [ ] σi Ψ = I where I enotes the ientity matrix. All of these quaratic triggering conitions are esigne so as to guarantee a esire rate of ecay for the Lyapunov function V through an inequality of the form (5). Hence, asymptotical stability an performance, as measure by the rate of ecay of V, are guarantee by the ifferent choices of Ψ in the triggering conition (9). Furthermore, the triggering times implicitly efine by (9) will not be equiistant, in general, an thus event-triggere implementations result in aperioic control. In fact, the set of triggering times {t k } k N can be formally efine by t =, t k+1 = inf{t R t > t k z T (t)ψz(t) = }. Since these instants are only known at execution time, the scheuling of energy, computation, an communication resources for event-triggering control becomes a very challenging problem. Moreover, the implicit efinition of the times raises the question of the existence of a lower boun τ > for t k+1 t k, k N. The largest value τ for which t k+1 t k τ hols for all k N along all trajectories of interest, is calle the minimal inter-event time. If the minimal inter-event time is zero, then an event-triggere implementation will require faster an faster upates an thus cannot be implemente on a igital platform. It was shown in [12] that such minimal inter-event time is guarantee to exist even in the nonlinear case uner suitable assumptions. For linear plants an linear state-feeback controllers, the minimum inter-event time is always guarantee to exist. Theorem 1 ([12]): Consier the linear plant (1) an linear feeback control law (2) renering the close-loop system (3) asymptotically stable. For any triggering conition (9) with σ [, 1[ there exists τ R + such that t k+1 t k τ for every k N. However, in case output-feeback controllers are use in a similar setup, the minimal inter-event time might be zero an accumulations of event-times occur (Zeno behaviour). This was pointe out in [26], see also Section V below. III. SELF-TRIGGERED CONTROL Event-triggere implementations require the constant monitoring of a triggering conition. For some applications this is a reasonable assumption, e.g., when we can use eicate harware for this purpose. Unfortunately, this is not always the case an the relate concept of self-triggere control is an alternative that can be use in such cases. The term self-triggere control was coine by [15] in the context of real-time systems. A self-triggere implementation of the feeback control law (2) has for objective the computation of the actuator values as well as the computation of the next instant of time at which the control law shoul be recompute. When ealing with linear plants an linear controllers we can leverage the close-form expression of the trajectories to evelop self-triggere implementations as we iscuss next. A. ISS self-triggere implementations The results in this section are base on [27], [28], [29]. We start by extening the linear moel (1) with isturbances t x p = A p x p + B p u + B w w, (1) where w R nw is the isturbance. It is well know that if the control law (2) reners the close-loop system (3) asymptotically stable then, in the presence of isturbances, the close-loop system t x p = (A p + B p K)x + B w w (11) is so-calle exponentially input-to-state stable. Definition 1 (EISS an GES): The system (11) is sai to be exponentially input-to-state stable (EISS) if there exist λ R +, κ R +, an γ R + such that for any w L an any x() = x R nx it hols for the corresponing trajectory that x(t) κ x e λt + γ w (12)

4 for all t R +. When this inequality hols for w =, the system (3) is sai to be globally exponentially stable (GES). We now escribe a self-triggere implementation of (2) that results in an EISS close-loop system. A self-triggere implementation of the linear stabilizing controller (2) for the plant (1) is given by a map Γ : R nx R + etermining the triggering time t k+1 as a function of the state x(t k ) at the time t k, i.e., t k+1 = t k + Γ(x(t k )). If we enote by τ k the inter-execution time τ k = t k+1 t k, we have τ k = Γ(x(t k )). Once the map Γ is efine, the expression self-triggere close-loop system refers to the system (1) an control law (2) implemente in a sample-an-hol manner (6) with triggering times t k+1 given by t = an t k+1 = t k + Γ(x(t k )). In Section II we formalize the notion of performance base on the time erivative of a Lyapunov function. In this section we irectly consier the time evolution of a Lyapunov function of the form V (x) = (x T P x) 1 2. If, for the ieal close-loop system (3) we have V (x(t)) V (x )e λot, t R + x R nx, (13) then we woul like to enforce the weaker inequality V (x(t)) V (x(t k ))e λτ, τ [, t k+1 t k [ x R nx (14) for the self-triggere implementation in the absence of isturbances (w = ) where λ [, λ o [. If we enote by h c : R nx R + R the map h c (x(t k ), t) = V (x(t)) V (x(t k ))e λτ, then the inequality in (14) can be expresse as h c (x(t k ), τ). Since no igital implementation can check h c (x(t k ), τ) for all τ [, t k+1 t k [, we consier instea the following iscrete-time version of h c base on a sampling time R + h (x(t k ), n) := h c (x(t k ), n ), [ [ for all n, tk+1 t k an for all k N. This conition results in the following self-triggere implementation where we use N min := τ min /, N max := τ max /, an τ min an τ max are esign parameters. Definition 2: The map Γ : R n R + is efine by Γ (x) := max{τ min, n(x) } with n(x) := max n N {n N max h (x, s), s =,..., n} for x R n. Using this efinition of Γ, a self-triggere implementation of the linear stabilizing controller (2) for plant (1) is prescribe. Note that the role of τ min an τ max is to enforce explicit lower an upper bouns, respectively, for the inter-execution times of the controller. The upper boun enforces robustness of the implementation an limits the computational complexity. Remark 1: Linearity of (1) an (2) enables us to compute h 2 as a quaratic function of x(t k). Moreover, through a Veronese embeing we can implement the self-triggere policy escribe in Definition 2 so that its computation has space complexity q nx(nx+1) 2 an time complexity q + (2q + 1) nx(nx+1) 2 where q := N max N min. For reasons of space we omit these etails. They can be foun in [27]. The following result establishes EISS of the propose selftriggere implementation. Theorem 2: Let τ R + be efine by where τ = inf{τ R + : et M(τ) = } M(τ) := C(e F T τ C T P Ce F τ C T P Ce λτ )C T, [ ] A F := p + B p K B p K, C := [I ]. A p B p K B p K If τ min τ, the self-triggere implementation in Definition 2 reners the self-triggere close-loop system EISS. Remark 2: When implementing self-triggere policies on igital platforms several issues relate to real-time scheuling nee to be aresse. For a iscussion of some of these issues we refer the reaers to [3]. Here, we escribe the minimal computational requirements for the propose selftriggere implementation uner the absence of other tasks. Let us assume that the computation elays ominate the measurement an actuation elays, as is the case sometimes in practice. The computation of Γ is ivie in two steps: a preprocessing step performe once per execution, an a running step performe n times when computing h (x, n). The preprocessing step computes a matrix use to evaluate h an has time complexity (n 2 x + n x )/2. The running step consists of testing the inequality h (x, n) has time complexity n 2 x + n x. If we enote by τ c the time it takes to execute an instruction in a given igital platform, the selftriggere implementation can be execute if: 3 2 (n2 x + n x )τ c τ min, (n 2 x + n x )τ c. The first inequality imposes a minimum processing spee for the igital platform while the secon equality establishes a lower boun for the choice of. Remark 3: Theorem 2 only guarantees EISS of the selftriggere implementation. In [29] the reaers can fin more etaile results explaining how the constants κ an γ appearing in the efinition of EISS epen on the continuous ynamics (1), the control law (2), an the esign parameters τ min an τ max. We refer the intereste reaer to [27] an [28] for numerical examples illustrating the propose technique an the guarantees it provies. An example comparing this implementation with the implementation escribe in the next section appears in Section III-C. B. Minimum attention implementations In Section III-A we starte with a linear controller an constructe a self-triggere implementation. Although the self-triggere implementation was base on the controller

5 an the system ynamics, the controller was esigne in oblivion of the implementation etails. In this section we take a step towars the co-esign of the control laws an its implementations. We consier a ifferent formulation of the minimum attention control problem introuce in [31]: Given the state of the system, compute a set of inputs that guarantee a certain level of performance while maximizing the next time at which the input nees to be upate. In this formulation of the minimum attention control problem we interpret attention as the inverse of the time elapse between consecutive input upates. The approach we will follow is base on the ieas in [32] an consists in computing all the inputs u R nu satisfying inequality (13), which we reprouce here in a version suitable for our nees: V ( e At x + t ) e A(t τ) Bu τ e λt V (x ). (15) We now make the important observation that by using - norm base Lyapunov functions, the computation of all the inputs satisfying (15) reuces to a feasibility problem with linear constraints an thus can be efficiently one online. Specifically, we take V to be a control Lyapunov function of the form V (x) = P x with P R m nx having rank n x an where enotes the infinity norm, i.e., x = max i {1,2,...,nx} x i. Similarly to Section III-A we efine the map h c by h c (x(t k ), u, τ) = P e Aτ x(t k ) tk +τ + P e A(τ s) Bu s t k e λτ P x(t k ). (16) We can now observe that the constraint h c (x(t k ), u, τ), which appears in (16), is equivalent to [ tk +τ ] P e Aτ x(t k ) + P e A(τ s) Bu s t k e λτ P x(t k ), for all i {1,..., m}, which is equivalent to h c (x(t k ), u, τ), where [ P e Aτ x(t k ) + P ] t k +τ e A(τ s) sbu t h c(x(t k ), u, τ) = k P e Aτ x(t k ) P t k +τ e A(τ s) τbu t k 1 e λτ P x(t k ). (17) an the inequality is assume to be taken element-wise, which results in 2n x linear scalar constraints for u. Since the inequality h c (x(t k ), u, τ) cannot be checke for all τ R + we work, similarly as in Section III-A, with its iscrete analogue h (x(t k ), u, n) := h c (x(t k ), u, n ). 1 i We note that while a self-triggere implementation of a linear control law is specifie by the map Γ : R nx R + etermining the next execution time (as the control law is alreay given), a minimum attention implementation aressing the co-esign problem requires the map Γ as well as the map Ω : R nx 2 Rnu specifying any input u Ω(x) that can be use uring the next Γ(x) units of time, i.e., u(t) = u(t k ) Ω(x(t k )), t [t k, t k+1 [ (18a) t k+1 = t k + Γ(x(t k )) (18b) with t :=. In a concrete implementation one uses aitional criteria, e.g., minimum energy, to select a single input among all the possible inputs given by the set Ω(x(t k )). Algorithm 1 computes both Ω an Γ. Input: P R m nx efining an -base control Lyapunov function an x(t k ) Output: Γ(x(t k )) an Ω(x(t k )) n := ; Ω := R nu ; while Ω n an n < N max o n := n + 1; Ω n := Ω n 1 {u R nu h (x(t k ), u, n) }; en if Ω n = then Ω(x(t k )) := Ω n 1 ; Γ(x(t k )) := (n 1) ; else Ω(x(t k )) := Ω n ; Γ(x(t k )) := n ; en Algorithm 1: Algorithm proviing Ω an Γ for a minimum attention implementation. The correctness of Algorithm 1 is guarantee by the following result whose proof can be foun in [32]. Theorem 3 ([32]): The minimum attention implementation efine by Γ an Ω compute by Algorithm 1 reners the minimum attention close-loop system consisting of (1) an (18) GES. Remark 4: Since verifying that Ω n as specifie in Algorithm 1 is a feasibility test for linear constraints, the algorithm can be efficiently implemente online using existing solvers for linear programs. Remark 5: Theorem 3 only states GES of the minimum attention implementation. In [32] the reaers can also fin more etaile results explaining how the constants κ an λ appearing in the efinition of GES (i.e., (12) for w = ) epen on the continuous ynamics (1) an the choice of. Reference [32] also iscusses how -norm base Lyapunov functions can be constructe. A stuy of the robustness properties of this implementation, e.g. EISS, has not yet appeare in the literature.

6 C. Illustrative example In this section, we illustrate the self-triggere an minimum attention implementations using a well-known example from the networke control systems literature, see, e.g., [33], consisting of a linearize moel of a batch reactor. The linearize batch reactor is given by (1) with [ ] A B = We consier the linear control law (2) with [ ] K = , (19) renering the eigenvalues of A + BK real, istinct an smaller than or equal to 2. In orer to compare the selftriggere with the minimum attention aproach we use in both case the -norm base Lyapunov function V = P x with P = Reference [32] offers more etails on how P was compute. To implement Algorithm 1 in MATLAB, we use the routine polytope of the MPT-toolbox [34], to hanle the sets Ω(x(t k )). When we the response of the plant is simulate with the minimum attention implementation for the initial conition x() = [1 1 ], we can observe that the close-loop system is inee GES, see Fig. 1(a) an Fig. 1(c). The selftriggere implementation also reners the close-loop system GES as can be seen from Fig. 1(b) an Fig. 1(c). Note that the ecay rates for both implementations are comparable as expecte. However, when we compare the resulting interexecution times as epicte in Fig. 1(), we observe that the minimum attention implementation yiels much larger inter-execution times than the self-triggere implementation. This can be explaine from the fact that the former solves a co-esign problem thereby optimizing current values of the control inputs with the objective to maximize the next execution time. The consiere self-triggere approach oes not as it has a prescribe (emulation-base) control law. D. Other approaches to self-triggere control Other approaches to self-triggere control have appeare in the literature. In [35], [36] the authors consier linear stabilizing control laws for linear systems enforcing a esire L 2 gain on the close-loop system. In a relate manner to the implementations iscusse in Section III preserving EISS an GES, the authors of [35], [36] propose self-triggere implementations preserving L 2 -gain stability. The interesting trae-off in this case is how much the L 2 -gain egraes as the number of inter-executions is reuce with respect to a perioic implementation. Self-triggere techniques for nonlinear control systems are reporte in [37], [38] base on the notion of homogeneity an isochronous manifols. Although Fig. 2: Configuration with share network only in the sensorto-controller (s-c) channel. the approach is base on homogeneity, it is shown how it is possible to make any smooth control system homogeneous by increasing the imension of the state space by one. A ifferent approach base on polynomial approximations of nonlinear systems is escribe in [39]. All these approaches consier implementations where the input remains constant in between re-computations of the control law. An alternative approach, base on using a moel of the plant at the actuator, is reporte in [4], where it is shown that non-constant inputs further reuce the number of messages that nee to be sent from the controller to the actuator. Reference [41] extens the results in [29] from state feeback to output feeback. Finally [42] applies self-triggere to a coverage control problem for robotic networks thereby reucing the require communication between robots. IV. OUTPUT-BASED EVENT-TRIGGERED CONTROL The approaches on event-triggere an self-triggere control presente previously were all base on full state feeback, although in practice the full state is often not available for feeback. In fact, in the introuction the importance of eveloping output-base event-triggere controllers was alreay inicate. Moreover, a first categorization of the existing output-base event-triggere control schemes was alreay provie base on their time sets (iscrete-time vs continuous-time) an aopte control law (observer-base or not). In this section, we start by iscussing the literature on continuous-an iscrete-time output-base event-triggere control with an without observer in a bit more etail. After that two exemplary approaches will be presente. A. Continuous-time observer-base event-triggere control In [19] one of the first observer-base event-triggere control loops are propose in the context of continuous-time systems, although the analysis an examples in the en focus on the situation where the full state information is available. A formal analysis can be foun in the more recent work [43], which extens the work in [44] that assume availability of the full state. The work in [43] focuses on continuous-time plants perturbe by a boune isturbance an measure outputs affecte by boune measurement noise. A signal generator (containe in the controller system in the setup epicte in Fig. 2) prouces the control input implemente at the actuators using a preictor that runs the unperturbe moel equations in close loop with a state feeback control law, in which the state variable is upate with state estimates receive from the (more accurate) observer situate at the sensor system in Fig. 2. The sensor system has a copy of

7 tempimage temp 211/12/8 15:26 page 1 #1 tempimage temp 211/12/8 15:26 page 1 # x 1 x 2 x 3 x x 1 x 2 x 3 x time t (a) Evolution of the states of the plant using the minimum attention implementation. tempimage temp 211/12/8 15:26 page 1 # time t (b) Evolution of the states of the plant using the self-triggere implementation. tempimage temp 211/12/8 15:26 page 1 #1 3 Minimum Attention Control Self-Triggere Control Desire Decay.6 Minimum Attention Control Self-Triggere Control time t (c) The ecay of the Lyapunov function using the minimum attention an the self-triggere implementations time t () The inter-execution times using the minimum attention an the self-triggere implementations. Fig. 1: Comparison between the minimum attention an the self-triggere implementations. the preictor. Only when the ifference between the state estimate in the preictor an the observer excees in norm an absolute threshol the estimate state in the observer is sent to the controller. The analysis of this scheme shows that a stable behavior of the event-base control loop can still be guarantee in the sense of ultimate bouneness of the plant s state. Moreover, it is shown that the maximum communication frequency within the control loop is boune, i.e., the minimal inter-event time is strictly positive. The size of the absolute threshol can be use to balance the maximum communication frequency an the size of the ultimate boun. Event-base state estimation is consiere in [21]. In that paper a state estimator aopts a hybri upate scheme in the sense that upates take place both when an event occurs that triggers the transmission of new measurements to the estimator (asynchronous times), as well as when a perioic timer expires (synchronous times). In the latter case the principle that absence of an event is however information that can be use by the observer [19] is use. More specifically, events are triggere only when the monitore output variable leaves a boune set (possibly epening on latest transmitte measurement). Hence, receiving no information at a synchronous time instant inicates that the output is still in this boune set, which is information that can be use to guarantee boune estimation error covariances. In fact, in [21] this is formally shown base on a sum-of-gaussians approach that is use to obtain a computationally tractable algorithm. An example of integrating this event-base state estimator with a perioically time-triggere control algorithm is provie in [25]. In [25] the triggering conition oes not use the estimate state as, for instance, in [43]. For time-stampe measurements, one can also aopt a timevarying iscrete-time Kalman filter approach to obtain a goo estimate of the state. However, note that such a scheme oes not exploit potentially valuable information containe in the absence of events. B. Continuous-time irect event-triggere control In contrast to the results iscusse previously, next eventtriggere control is consiere without any intermeiate processing of measurements by an observer or filter. One work belonging to this category is [45], which stuies linear systems without isturbances an measurement noise an with a finite number of control actions. The metho is base on hysteretic quantization. The transmission of output measurements is triggere by reaching the next quantization level. A consequence, in case of single outputs, is that only one bit has to be transmitte in orer to inform the control system about the quantization level reache (assuming that

8 the previous value is store at the controller). The paper proposes two systematic output feeback control esign strategies. The first is an emulation-base strategy starting from an analog controller, an the secon strategy is a irect esign that rives the plant state to the origin in finite time after a total transmission of 2n+2 bits, where n is the orer of the plant. In [26] it is stuie how event-triggere control strategies tailore to static state-feeback control laws along the lines of Section II can be extene to output-base ynamical controllers using both centralize an ecentralize eventtriggering mechanisms. One of the problems ientifie in [26] was that using the output-base extensions of the eventtriggering mechanisms aopte in [12] base on relative thresholing can result in accumulations of event times (Zeno behaviour) an thus a zero minimal inter-event time. Using a mixe event-triggering mechanism, a strictly positive minimum inter-event time coul be guarantee for outputbase event-triggere control, while still guaranteeing ultimate bouneness an L -performance. This work exploite impulsive moels [46], [47] for escribing the closeloop behavior, which resulte in less conservative stability conitions compare to the original work [12]. See Section V for more etails. C. Discrete-time observer-base event-triggere control In [11] a iscrete-time control problem is consiere in which the communication resources are consiere to be scarce. As such, the objective is to reuce the number of communications by using more computations. The paper uses an emulation-base approach in the sense that a wellfunctioning output-base controller is available (assuming a stanar time-triggere perioic implementation). The main iea of the propose event-triggere control strategy is the use of a state estimator framework such that all noes have ientical estimators an thus ientical estimator states. The estimate values of the remote outputs are use in the feeback control. Every sample time, the controller at the ith noe compares the estimate of the ith output to its true values. If the ifference is greater than a preefine threshol, the true value is communicate to the other noes. When there is a communication from ith noe, the estimators in all noes upate their states to reflect the current actual value of the system outputs or states. As a consequence, the error between the estimate ata use in the control algorithm an the actual values is always boune by a threshol, which can be chosen by the control esigner to balance closeness to the original time-triggere close-loop system responses on the one han an the communication usage on the other han. This boun on the error can be use to obtain BIBO stability conitions. A rawback of the scheme is that it uses a global estimator in each noe, which oes not scale to large systems. In [23], [16] the problem of output-base event-triggere control in iscrete-time is consiere from an optimal control perspective in line with the classical Linear Quaratic Gaussian (LQG) setup. It is shown in [16] that such a set-up can lea to a stochastic control problem with a ual effect, so that the optimal event-trigger an controller are har to fin. In [23] an emulation-base approach is consiere in which the observer at the sensor system an the local observer at the controller are fixe by minimizing the error covariance conitione on the receive information. Base on the appene LQG cost the problem aresse in [23] is to synthesize the ETMs in the s-c an c-a channels in a (sub)optimal manner. In [24] an output-base event-triggere control scheme is propose using moel-base triggering schemes in both the sensor-to-controller an the controller-to-actuator communication channels. A preictive control technique is aopte in the controller-to-actuator channel. By sening control packets containing moel-base preictions of future control values an only transmitting new control packets when these preictions eviate from the current control values compute in the control system (accoring to relative bouns), significant savings can be obtaine compare to a basic zeroorer hol strategy. For the sensor-to-controller channel, the triggering mechanism is base on the ifference between the state estimate of a Luenberger observer running in the sensor system with the state estimate of a preictor (calle the local observer in [23]) running in both the sensor an controller systems. If this ifference gets too large, then the estimate of the Luenberger observer is transmitte to the controller system that upates the state estimate of its preictor. LMIbase tools are provie for close-loop stability an l 2 -gain analysis. See Section VI below for more etails on the setup. Recently, in [48] also an output-base scheme exploiting observer-like structures for iscrete-time linear systems was propose for tracking of references signals generate by an exosystem. D. Discrete-time irect event-triggere control In [1], [13] output-base PID controllers are consiere without the consieration of an observer or estimator. Both these approaches use a timer to avoi problems with a zero minimum inter-event time. In [1] the event etector is truly time-triggere, while in [13] a time regularization is aopte by requiring that after an event at least a fixe amount of time no new event is generate. However, [1] oes not provie any analytical results, while [13] only provies them for state-base event-triggere control strategies. Recently, such results were obtaine in [18]. Interestingly, these results apply to both centralise an ecentralise event-triggering mechanisms, an they provie stability an L 2 -gain guarantees of the close-loop system in continuous time, even though the event-triggere control strategy is a iscrete-time controller operating in a perioic time-triggere manner. In fact, as alreay mentione in the introuction, the term perioic event-triggere control is use in this context, cf. [17] [18].

9 V. CONTINUOUS-TIME DIRECT EVENT-TRIGGERED CONTROL In this section, we present an exemplary event-triggere control problem base on continuous-time output-base controllers an moel the event-triggere control system as an impulsive system. This particular setup is base on [26], but connections to relate methos will be mentione using this exposition. A. Problem Formulation Let us consier a linear time-invariant (LTI) plant given by { t x p = A p x p + B p û + B w w, (2) y = C p x p, where x p R np enotes the state of the plant, û R nu the input applie to the plant, w R nw an unknown isturbance an y R ny the output of the plant. The plant is controlle using a continuous-time LTI controller given by { t x c = A c x c + B c ŷ, (21) u = C c x c, where x c R nc enotes the state of the controller, ŷ R ny the input of the controller, an u R nu the output of the controller. We assume that the controller is esigne to rener (2) an (21) with y(t) = ŷ(t) an u(t) = û(t), for all t R +, asymptotically stable. Here, we consier the case where the controller is implemente in a sample-ata fashion, which causes y(t) ŷ(t) an u(t) û(t) for almost all t R +. In particular, we stuy ecentralise event-triggere control which means that the outputs of the plant an controller are groupe into N noes an the outputs of noe i {1,..., N} are only sent at the transmission instants t i k i, k i N. Hence, at transmission instant t i k i, noe i transmits its respective entries in y an u, an the corresponing entries in ŷ an û are upate accoringly, while the other entries in ŷ an û remain the same. Such constraine ata exchange can be expresse as ˆv + (t i k i ) = Γ i v(t i k i ) + (I Γ i )ˆv(t i k i ), (22) in which v = [y u ], ˆv = [ŷ û ], an Γ i = iag(γ 1 i,..., γ ny+nu i ), (23) for all i {1,..., N}. In between transmissions, we use a zero-orer hol, i.e., t ˆv(t) =, for all t R +\ ( N i=1 {ti k i k i N} ). (24) In (23), the elements γ j i, with i {1,..., N} an j {1,..., n y }, are equal to 1 if plant output y j is in noe i an are elsewhere, the elements γ j+ny i, with i {1,..., N} an j {1,..., n u }, are equal to 1 if controller output u j is in noe i an are elsewhere. We assume that for each j {1,..., n y + n u }, it hols that N i=1 γj i >, i.e., we assume that each sensor an actuator is at least in one noe. Furthermore, we assume that at time t =, it hols that Fig. 3: Control system block iagram with inication of the event-triggering mechanism (ETM). ˆv() = v(). This can be accomplishe by transmitting all sensor an actuator ata at the time the system is eploye. In a conventional sample-ata implementation, the transmission times are istribute equiistantly in time an are the same for each noe, meaning that t i k = i+1 ti k i + h, for all k i N an all i {1,..., N}, an for some constant transmission interval h >, an that t i k = tj k, for all k N an all i, j {1,..., N}. In event-triggere control, however, these transmissions are orchestrate by an event-triggering mechanism, as is shown in Fig. 3, which in this case is ecentralise. We consier a ecentralise event-triggering mechanism that invokes transmissions of noe ata when the ifference between the current values of outputs an their previously transmitte values becomes too large in an appropriate sense. In particular, the eventtriggering mechanism consiere in this section results in transmitting the outputs of the plant or the controller in noe i {1,..., N} at times t i k i, satisfying t i k i+1 = inf { t > t i k i e Ji (t) 2 = σ i v Ji (t) 2 + ε i }, (25) an t i =, for some σ i, ε i. In these expressions, e Ji an v Ji enote the subvectors forme by taking the elements of the signals e an v, respectively, that are in the set J i = {j {1,..., n y + n u } γ j i = 1}, an e(t) = ˆv(t) v(t) (26) enotes the error inuce by the event-triggere implementation of the controller at time t R +. Note that J i is the set of inices of sensors/actuators corresponing to noe i. Hence, the event-triggering mechanism (25), which is base on local information available at each noe, is such that when for some i {1,..., N}, it hols that e Ji (t) 2 = σ i v Ji (t) 2 + ε i, i.e., the norm of the error inuce by the event-triggere implementation of the signals in noe i becomes large for the first time, noe i transmits its corresponing signal v Ji (t) in v(t) an, the signal ˆv(t) is upate accoring to (22). This implies that e + (t i k i ) = (I Γ i )e(t i k i ) an thus e + J i (t i k i ) =. Using this upate law, an the aforementione assumption that ˆv() = v(), yieling e() =, we can observe that the error inuce by the event-triggere control scheme satisfies e Ji (t) 2 σ i v Ji (t) 2 + ε i, (27)

10 for all t R + an all i {1,..., N}. The question that arises now is how to etermine σ i an ε i for all i {1,..., N}, such that the close-loop eventtriggere system is stable in an appropriate sense an a certain level of isturbance attenuation is guarantee, while the number of transmissions of the outputs of the plant an the controller is small. Note that for ε i =, i {1,..., N}, the event-triggering conitions in (25) can be seen as an extension of the event-triggering mechanism of [12] for output-base controllers, an for σ i =, i {1,..., N}, it is equivalent to the event-triggering mechanism of [49], [45], [5]. As such, the event-triggering mechanism in (25) unifies two earlier proposals. B. An impulsive system formulation In this section, we reformulate the event-triggere control system as an impulsive system, e.g., [46], [47], of the form t x = Ā x + Bw, when x C (28a) x + = Ḡi x, when x D i, i {1,..., N}, (28b) where x X R nx enotes the state of the system an w R nw an external isturbance. The flow an the jump sets are enote by C R nx an D i R nx, i {1,..., N}, respectively, an X = C ( N i=1 D i). Note that the transmission times t i k i, k i N, as in (25), are now relate to the event times at which the jumps of x, accoring to (28b) for i {1,..., N}, take place. To arrive at a system escription of the event-triggere control system (2), (21), (22), (24), an (25) of the form (28), we combine (2), (21), (22), (24) an (26), an efine x := [x e ] R nx, where x = [x p x c ] an n x := n p + n c + n y + n u, yieling the flow ynamics of the system [ ] [ ] t x = A + BC B E x + w, (29) C(A + BC) CB CE }{{}}{{} =:Ā =: B in which A = [ Ap A c ], B = [ ] [ [ Bp Cp Bw B c, C = C c ], E = ]. (3) The system continuously flows as long as the event-triggering conitions are not met, i.e., as long as (27) hols for all i {1,..., N}, which can be reformulate as x C, with an C = { x R nx x Q i x ε i i {1,..., N}}, (31) [ ] σi C Q i = Γ i C, (32) Γ i because x Q i x ε i is equivalent to Γ i e(t) 2 σ i Γ i v(t) 2 + ε i, as in (27). As mentione before, when noe i transmits its ata, a reset accoring to e + = (I Γ i )e occurs, while x remains the same, i.e., x + = x, see (22). This can be expresse as [ ] x + I = x, (33) I Γ i }{{} =:Ḡi for all x D i, i {1,..., N}, in which D i = { x R nx x Q i x = ε i }, (34) accoring to (25). Combining (29), (31), (33) an (34) yiels an impulsive system of the form (28). C. Analysis methos an iscussion The available analysis techniques given in [26] buil upon the impulsive system framework [47] with a focus on global asymptotic stability of sets A containing the origin in the interior (in absence of isturbances w) an L -performance of the close-loop system. As such, in case of absence of isturbances a form of practical stability, or ultimate bouneness, is obtaine. The conitions guaranteeing global asymptotic stability of sets an upperbouns on the L -gain of the system from isturbance w to performance output z = C x+ Dw are given in terms of LMIs. We refer the intereste reaer to [26] for the etails an the precise statements of the results. To provie some insights in the consequence of the results, we note that the feasibility of the LMIs is relate to the choice of the relative gains σ i, i {1,..., N}, in the event-triggering conitions (25), but is not affecte by the choice of the absolute threshols ε i, i {1,..., N}. Hence, once the LMIs are feasible, practical stability (for w = ) an upper bouns on the L -gain are guarantee. The size of the set A (ultimate boun) (when w = ), is affecte by both σ i an ε i. However, after having a feasible set of LMIs guaranteeing set stability an finite L -gains, the parameters ε i provie full control to ajust the size of the set A. As we can see from (27), this will affect the number of events, enabling the esigner to make trae-offs between the size of the set A (relate to the ultimate boun of x as t for w = ) an the number of transmissions over each communication channel. Inee, larger ε i, i {1,..., N}, result in fewer events, an thus fewer transmissions, but in a larger set A (i.e., a larger ultimate boun), when w =. In fact. if ε i, i {1,..., N}, all approach zero, we have that A {}. Hence, the set A can be mae arbitrary small (at the cost of more transmissions). The naive choice to take ε i =, for all i {1,..., N}, seems appealing as it woul yiel A = {}. However, this might result in zero minimum inter-event times (Zeno behaviour) as Example 2 in [26] illustrates. In some cases, e.g., state-feeback controlle system with centralise event triggering as iscusse in [12], a strictly positive minimum inter-event time can be guarantee even for ε 1 =, an we have that A = {} is globally asymptotically stable, see also Theorem 1. In fact, in this case also finite L p -gains for p < can be given, see Remark III.7 in [26] an Remark IV.3 in [51]. Here, we iscusse an impulsive system formulation (28) with subsequent LMI-base stability an performance analysis. This leas to less conservative values for ε i, σ i, i {1,..., N} guaranteeing stability than the perturbe system approach given in [12], as is formally proven in [26]. The benefit of aopting the impulsive system formulation can be explaine by the fact that the impulsive system truly escribes the behaviour of the event-triggere control

11 Fig. 4: Networke control configuration. system as it inclues the ynamics for the error e inuce by the event-triggere implementation. Besies the exact moelling of the error ynamics as above, also the fact that LMI-base formulations are use is beneficial as this allows to use an optimisation-base proceure to fin better values for σ i an ε i guaranteeing stability an specific levels of L -performance. Larger values of σ i an ε i result in larger minimum inter-event time, see (25). More recently, the impulsive system framework was also use for other eventtriggere an self-triggere controller setups, see, e.g., [52]. VI. DISCRETE-TIME OBSERVER-BASED EVENT-TRIGGERED CONTROL Just as in the previous section, we present an exemplary event-triggere control problem in this section but now for iscrete-time observer-base controllers. This particular setup follows [24]. A. Problem Formulation In [24], the networke control configuration shown in Fig. 4 is stuie, in which the plant is given by a iscretetime linear time-invariant moel of the form { xk+1 = Ax k + Bu k + Ew k P : (35) y k = Cx k, where x k R nx, u k R nu, w k R nw an y k R ny enote the state, control input, isturbance an measure output, respectively, at iscrete time instant k N. The sensors of the plant transmit their measurements to the controller, an the controller transmits the control ata to the actuators over a share, possibly wireless, network, for which communication an energy resources are limite. For this reason, it is esirable to reuce the transmissions over the sensor-to-controller an controller-to-actuator channels as much as possible, while still guaranteeing esirable closeloop behavior. Hence, the problem is now to esign smart sensor, controller an actuator systems for the setup in Fig. 4 such that this objective is realize. B. An observer-base strategy In this section, we present a possible solution as given in [24] for the problem formulate in Section VI-A in the context of Fig. 2 in which the controller-to-actuator channel is remove. The smart sensor system in Fig. 2 consists of a Luenberger observer O, a preictor Pr an an event-triggering mechanism ET M s that etermines when information shoul Fig. 5: Observer-base PETC strategy with only s-c ETM. be transmitte to the controller system, see Fig. 5. The Luenberger observer is given by O : x s k+1 = Ax s k + Bu k + L(y k Cx s k) (36) in which x s k enotes the estimate state at the sensor system at time k N, an the matrix L is a suitable observer gain. The preictor Pr is given by { Ax c Pr : x c k + Bu k, when x s k is not sent k+1 = Ax s k + Bu k, when x s (37) k is sent. Finally, the event-triggering mechanism is given at time k N by the conition ET M s : x s k is sent x s k x c k > σ s x s k, (38) where σ s is a esign parameter. Before explaining the functioning of Pr an ET M s in more etail, it is convenient to introuce also the controller system. The controller system consists of a copy of the preictor Pr, an a controller gain K, see Fig. 5. In fact, the control signal is given by { Kx c k, when x s k is not sent u k = Kx s k, when x s (39) k is sent. As the sensor system also runs a copy of the preictor Pr (both initialize at the same initial estimate), the sensor system is aware of the estimate x c k the controller system has, an, consequently, can etermine u k to compute the next state estimate x s k+1 accoring to (36). Clearly, the estimate x s k of the observer is typically better than the estimate xc k of the preictor, as the observer has access to all measurements, while the preictor only receives sporaic upates. The rationale now is that if the sensors etects at k N that the estimate x s k of the Luenberger observer (36) eviates significantly from the estimate x c k, i.e., xs k xc k > σ s x s k as in (38), the estimate x s k is transmitte to the controller, an corresponing upates of the estimate x c k+1 (cf. the secon case in (37)) an the control signal u k as in (39) are mae. Hence, as long as x s k xc k is sufficiently small, no transmissions between the sensor an controller systems are neee. This observer-base strategy can provie similar stability an l 2 -gain properties, while requiring significantly less transmissions compare to both a stanar perioic timetriggere implementation an a baseline event-triggere implementation as in [53], [12], [26], [17], [18]. See the example presente below.

12 In [24] extensions are provie for the network configuration in Fig. 4 with communication savings both for the sensor-to-controller an the controller-to-actuator communications. In particular, preictive control techniques are aopte computing moel-base preictions of future control values, which are sent in one (or more) control packets to the actuator system. Only when these preicte future control values (known in the controller system) eviate from the current control values compute in the controller system, new control packets with future values are transmitte to the actuator system. In this manner, significant savings can be obtaine compare to a basic zero-orer hol strategy. In [24] also ecentralise observer-base controllers an eventtriggering mechanisms are presente for large-scale weaklycouple plants. Remark 6: Extensions of the observer (36) incluing isturbance estimators (assuming a suitable linear isturbance moel) are possible following the same rationale as in [24]. This extension can enhance further communication savings in the sensor-to-controller channel. C. Analysis methos an iscussion The analysis of the above mentione moel-base strategies are presente in [24] base on perturbe linear an piecewise linear systems. Base on these moeling paraigms, LMI-base conitions for global exponential stability an guarantee l 2 -gains can be erive. The usage of moel-base preictions are quite powerful for the reuction of network resource utilization, as will also be illustrate in the numerical example below. These observations are in line with the results in the networke control literature in which moel-base approaches inee often perform better [54]. It is also worthwhile to mention the connection of the usage of moel-base preictions to the work in [19], where the relevance of generalize hols was mentione, an the work in [44], [43] in which the term signal generator was use base on moel-base preictions (although in absence of a resource-constraine controller-to-actuator channel). D. Illustrative example In this section, the moel-base event-triggere control strategies iscusse previously will be illustrate using a time-iscretization of the batch reactor example iscusse in Section III-C. Proper values for the observer an state feeback gains K an L as in (36) an (39), respectively, are chosen corresponing to the sampling perio h =.15. See [24] for the exact setup. We compare the moel-base event-triggere control scheme with a corresponing perioic time-triggere controller an with the following baseline event-triggere scheme: The baseline implementation uses ieas presente in [53], [12], [26], [17], [18] an leas to a strategy given by the ynamic controller x c k+1 = Ax c k + Bu k + L(ŷ k Cx c k), a certainty-equivalence control law u k = Kx c k, (4a) (4b) an a sensor-to-controller event-triggering mechanism { yk, when ŷ k 1 y k > σ s y k ŷ k = ŷ k 1, when ŷ k 1 y k σ s y k. (41) Hence, in this baseline setup a sensor reaing is transmitte to the controller only when the ifference between the latest transmitte value an the current sensor reaing is large compare to the value of the reaing. In aition, the hol strategy ŷ k = ŷ k 1 is use when no new output measurement is transmitte. To make a fair comparison between the moel-base an the baseline strategies, we select σ s for both cases such that the guarantee upper boun γ on the l 2 -gain of the resulting close-loop system satisfies γ = 1 an use the piecewise linear system approach of [24] to construct the corresponing values for σ s. This results in σ s =.135 for the moel-base strategy. Using similar techniques for the baseline strategy gives σ s =.343. The corresponing perioic time-triggere control strategy results in an (exact) l 2 -gain of γ = The response of the performance output z to the initial conition [ x = an the isturbance satisfying w k = sin 3π k 1 ] 25 1 for k 3 an wk = for k > 3, for the three strategies is shown in Fig. 6a. We can conclue that all three control strategies show almost inistinguishable responses. However, the number of transmissions that are neee is 2 for perioic time-triggere control, 148 for the baseline strategy an only 41 for the moel-base strategy. This emonstrates that the newly propose moel-base event-triggere control strategy nees significantly fewer transmissions than the other two approaches to realize similar responses, at the price of more computations. This is also further illustrate in Fig. 6b showing the inter-transmission times. We will stuy now more closely the influence of the parameter σ s in (38) on the upper boun γ on the l 2 -gain of the moel-base event-triggere control strategy an the number of transmissions that are generate for the aforementione initial conition an isturbance, see Fig. 6c an Fig. 6, respectively. Fig. 6c shows that the upper boun on the l 2 -gain increases as σ s increases, inicating that closeloop performances egraes as σ s increases. This figure also shows that the guarantee upperbouns on the l 2 -gain provie by the piecewise linear (PWL) approach are less conservative than the perturbe linear (PL) approach. From Fig. 6, it can be seen that the increase of the guarantee l 2 - gain, through an increase σ s, leas to fewer transmissions, which emonstrates the traeoff that can be mae between the close-loop performance an the number of transmissions. Note that for σ s approaching zero, the upper boun of the l 2 gain for the moel-base PETC strategy approaches = 12.75, which is the l 2 -gain of the corresponing perioic time-triggere control strategy. This emonstrates, as formally proven in [24], that the l 2 -gain of the moelbase event-triggere control strategy can approach the l 2 - gain of the perioic time-triggere implementation arbitrarily γ

13 tempimage temp 211/11/18 11:12 page 1 #1 tempimage temp 211/11/18 11:12 page 1 #1 tempimage temp 211/11/18 11:12 page 1 #1 tempimage temp 211/11/18 11:12 page 1 #1 outputs z z 1 time trggr z 2 time trggr z 1 baseline z 2 baseline z 1 moel-base z 2 moel-base inter-transmission times baseline moel-base γ PL PWL number of events time k (a) The evolution of the outputs as function of time k for the timetriggere (trggr), the baseline an the observer-base strategies time k (b) The inter-transmission times as function of time k for the baseline an the observer-base strategies..5.1 σ s (c) The upper boun on the l 2 - gain as function of σ s for the moel-base strategy. Fig. 6: Comparison of iscrete-time observer-base event-triggere control strategies σ s () The number of transmissions as function of σ s for the moel-base strategy. close. Interestingly, even for a small σ s, whichonly leas to a minor egraation of the close-loop performance in terms of the l 2 -gain, the amount of ata transmitte over the network, is alreay significantly reuce. For instance, starting from a perioic time-triggere observer-base controller, we can set σ s =.1, which leas to an upper boun on the l 2 -gain of the corresponing moel-base event-triggere control strategy of as guarantee by the PWL approach, see Fig. 6c, inicating an 8% performance egraation, while the number of transmissions reuce from 2 to 8, see Fig. 6, which is a reuction of 6%. Of course, it shoul be note that the actual savings epen heavily on the consiere isturbance (classes). Using isturbance estimators as pointe out in Remark 6 might be beneficial for further reuction of the number of transmissions. VII. WIRELESS EVENT-TRIGGERED CONTROL SYSTEMS In a networke control system, the communication meium is often share between multiple control loops as inicate in Fig. 7. In Fig. 7 a wireless network connects the sensors with the controllers. For such as system, as alreay mentione, it is often esirable to limit the amount of communication, ue to that either the transmission is battery powere or the network might get congeste. An important approach to efficiently utilize the communication network is to let sensors transmit only when their measurements excee a certain value, i.e., to apply an event-triggere sampling rule. Other alternatives inclue having a network manager eciing when each sensor can communicate. That ecision can be base on information available in the scheuler. A challenge in general is to limit not only the communication of sensor ata, but also limit the nee of communication between noes in orer to take communication ecisions. Hence, an architecture in which the ecision-making is istribute can be esirable, but it is then har to provie guarantees of no collision or congestion in the network. In practice, it is often reasonable to have a more hybri Fig. 7: Wireless control systems. approach where some ecisions are mae centrally by the network manager an some by the iniviual sensors an controllers. A. Optimal event-triggere control It is natural to pose the question if it is possible to esign an optimal event-triggering conition for a networke control system as the one epicte in Fig. 7. Unfortunately, the separation principle oes not hol for the optimal controller an the optimal scheuler, as the close-loop system yiels a ual effect in general [16]. However, by a suitable filter on the sensor sie, it is possible to obtain a control architecture for which certainty equivalence hols. Such an architecture suggests an observer-base controller, but with sacrifice optimality.

14 In certain situations, it is possible to fin the optimal eventbase controller. In the work [9] on Lebesgue sampling, a first-orer system with an optimal event-base sampling for an impulse controller was consiere. For zero-orer hol actuation, it was shown in [55] that the optimal threshol in the event-triggering mechanism is time varying. The influence of limite control actions or sensing for optimal event-base control was consiere in [56]. An important issue not touche upon previously, but important in a wireless large-scale control system, is the possible occurrence of ata rops. For event-triggere sensor communication such ata loss might seem to be critical, as fewer transmissions are generate in event-base control systems. In [57], the influence of inepenent an ientically istribute packet rops was consiere. It was shown how the control performance eteriorate as the probability of packet rops tens to one. It was also shown that if the (sensor) transmitter receives a (negative) acknowlegement for each packet the (controller) receiver oes not receive, then the event-triggering conition can be improve. In particular, the threshol shoul be lowere each time a packet has been lost, so that the chance of a new transmission is increase. The influence of such acknowlegements on the close-loop performance can in some cases be explicitly compute. B. Event-triggere control over wireless networks It is important to have accurate an efficient communication moels of the wireless networks for the esign of event-triggere wireless control systems. Here we will briefly iscuss how self- an event-triggere control can be aapte to a common wireless network protocol. There is obviously a vast literature on wireless communication, but fewer stuies have focuse on moels suitable for control purposes. Some exceptions inclue the Markov moel evelope by Bianchi to stuy the performance of the communication protocol IEEE [58]. Similar Markov moels have been evelope also for IEEE , which is one of the ominating protocol stanar for wireless sensor networks, see, e.g., [59]. The superframe time organization of the slotte IEEE is shown in Fig. 8. Each superframe Γ i starts with a beacon. The rest of the superframe is ivie into an active an an inactive perio. During the inactive perio, no evice is suppose to transmit so they can save power by being in a so-calle sleep moe. The active perio is split into a contention access perio (CAP) an a collision free perio (CFP). During the CAP, the meium access control (MAC) scheme is carrier sense multiple access/collision avoiance (CSMA/CA), where the noes in the network sense if the channel is busy before transmitting a message. The CAP is use by noes to sen best effort messages, as packet rops can happen ue to collision or channel congestion. The CFP is intene to provie real-time guarantee service, by allocating guarantee time slots to the noes in a time ivision multiple access (TDMA) scheme. Since uring the CFP there are no packet losses ue to collisions or channel congestion, this mechanism is an attractive perio for control tasks. Fig. 8: Superframe time organization of the slotte IEEE protocol. It was recently shown that event-triggere an selftriggere control can be implemente over IEEE , see [6]. By allocating a guarantee time slot within the CFP of a future superframe, it is possible to approximately sample the system accoring to the time compute by the self-triggere algorithm. In this way, the sensor oes not have to transmit until it is suggeste by the controller. As isturbances might act on the plant, an event-triggere sampler, which reacts if the sensor measurement starts eviating from its preicte value, nees to be ae as well. For large-scale systems with many sensors within the same wireless range, the guarantee time slots of the CFP are not enough, an, as a consequence, also the CAP nees to be use. Even if there is contention, event-triggere an selftriggere control can be utilize. Analyzing these schemes uner a CSMA/CA MAC is however challenging, as the state of the protocol in general will be correlate with plant state. Various ways to tackle this problem have only recently been consiere in the literature, for example, [61]. VIII. CONCLUDING REMARKS In this paper the aim was to provie an introuctory overview of the fiels of event-triggere an self-triggere control. The literature on these classes of aperioic control is rapily expaning. In the paper we i not try to cover all of the most recent results in orer to be comprehensive, but instea focuse on some of the main evelopments in the latest wave of the perioic versus aperioic ebate. Next to introucing the basics on event-triggere an self-triggere control, the emphasis was on the use of output-base control an implementation issues of event-base control over wireless communication networks. This paper can form a goo starting to become acquainte with the research areas of event-triggere an self-triggere control, an in fact several references are provie as suggestions for further reaing. After the enormous growth of the literature on this topic in the past 5 to 6 years, it seems time to take the next steps. Even though many results are currently available, it is fair to say that the system theory for event-triggere an self-triggere control is far from being mature, certainly compare to the vast literature on time-triggere perioic sample-ata control. One possible next step, that is certainly neee, is to evelop the necessary system theoretic results unerlying complete an efficient (co-)esign methoologies for event-triggere an self-triggere control. This shoul enhance the usage of these control strategies in practical applications. In fact, their valiation in practice is an important next step (which will unoubtely raise new theoretical