Output-Based Event-Triggered Control with. Guaranteed L -gain and Improved and Decentralised Event-Triggering

Transcription

1 Output-Based Event-Trggered Control wth 1 Guaranteed L -gan and Improved and Decentralsed Event-Trggerng M.C.F. Donkers and W.P.M.H. Heemels Abstract Most event-trggered controllers avalable nowadays are based on statc state-feedback controllers. As n many control applcatons full state measurements are not avalable for feedback, t s the objectve of ths paper to propose event-trggered dynamcal output-based controllers. The fact that the controller s based on output feedback nstead of state feedback does not allow for straghtforward extensons of exstng event-trggerng mechansms f a mnmum tme between two subsequent events has to be guaranteed. Furthermore, snce sensor and actuator nodes can be physcally dstrbuted, centralsed event-trggerng mechansms are often prohbtve and, therefore, we wll propose a decentralsed event-trggerng mechansm. Ths event-trggerng mechansm nvokes transmsson of the outputs n a node when the dfference between the current values of the outputs n the node and ther prevously transmtted values becomes large compared to the current values and an addtonal threshold. For such event-trggerng mechansms, we wll study closed-loop stablty and L -performance and provde bounds on the mnmum tme between two subsequent events generated by each node, the so-called nter-event tme of a node. Ths enables us to make tradeoffs between closed-loop performance on the one hand and communcaton load on the other hand, or even between the communcaton load of ndvdual nodes. In addton, we wll model the event-trggered control system usng an mpulsve model, whch truly descrbes the behavour of the event-trggered control system. As a result, we wll be able to guarantee stablty and performance for event-trggered controllers wth larger mnmum nter-event tmes than the exstng results n the lterature. We llustrate the developed theory usng three numercal examples. Index Terms Event-Trggered Control Systems, Impulsve Systems, Hybrd Systems, Dynamc Output-Based Control, Decentralsed Control. Ths work s partally supported by the Dutch Scence Foundaton (STW) and the Dutch Organzaton for Scentfc Research (NWO) under the VICI grant Wreless controls systems: A new fronter n automaton, by the European 7th Framework Network of Excellence by the project Hghly-complex and networked control systems (HYCON ), and by the project Decentralsed and Wreless Control of Large- Scale Systems (WIDE ). Tjs Donkers and Maurce Heemels are wth the Hybrd and Networked Systems group of the Department of Mechancal Engneerng, Endhoven Unversty of Technology, PO Box 513, 56 MB Endhoven, the Netherlands, {m.c.f.donkers, m.heemels}@tue.nl. July 18, 211

2 2 I. INTRODUCTION In many control applcatons nowadays, the controller s mplemented on a dgtal platform. In such an mplementaton, the control task conssts of samplng the outputs of the plant and computng and mplementng new actuator sgnals. Typcally, the control task s executed perodcally, snce ths allows the closed-loop system to be analysed and the controller to be desgned usng the well-developed theory on sampled-data systems, see, e.g., 1, 2. Although perodc samplng s preferred from an analyss and desgn pont of vew, t s sometmes less preferable from a resource allocaton pont of vew. Namely, executng the control task at tmes when no dsturbances are actng on the system and the system s operatng desrably s clearly a waste of computaton resources. Moreover, n case the measured outputs and/or the actuator sgnals have to be transmtted over a shared (and possbly wreless) network, unnecessary utlsaton of the network (or power consumpton of the wreless rados) s ntroduced. To mtgate the unnecessary waste of communcaton and computaton resources, an alternatve to perodc control, namely, event-trggered control has been proposed, see 3 6. Event-trggered control s a control strategy n whch the control task s executed after the occurrence of an external event, generated by some well-desgned event-trggerng mechansm, rather than the elapse of a certan perod of tme as n conventonal perodc control. As expermental results show, see, e.g., 3 11, event-trggered control s capable of reducng the number of control task executons, whle retanng a satsfactory closed-loop performance. Although the advantages of ETC are well-motvated and practcal applcatons show ts potental, relatvely few theoretcal results exst that study ETC systems, see, e.g., In these references, several dfferent eventtrggerng mechansms and control strateges are proposed. For nstance, n 12, 13, an mpulsve control acton s appled to the system that resets the state to the orgn every tme the state of the plant exceeds a certan threshold. The analyss s performed for frst-order stochastc systems, as analyss of larger-dmensonal systems s dffcult, and t s shown that the varance of the state s smaller when compared to a sampled-data controller, whle havng approxmately the same number of control updates. Another nterestng approach to event-trggered control s presented n 14 16, n whch the system s controlled n open loop, usng an nput generator that uses a predcton of the states of the plant to produce a control sgnal. These predcted states are only corrected n case the true plant state devates too much from ts predcted value. Such a devaton can be caused by dsturbances, 14, 15, or by the fact that the plant model s ncorrect 16. A more basc emulaton-based approach s taken n By emulaton-based, we mean that the controller s desgned wthout consderng the event-trggered nature of the control system, and, subsequently, an event-trggerng mechansm s desgned to ensure that the event-trggered control system s stable, has some guaranteed lower bound on the performance and some guaranteed upper bound on the number of events wthn a certan tme nterval. The dfferences between the work dscussed n les n the fact that n 17, 18 the nfluence of unknown dsturbances are studed, whereas n only stablsaton s consdered. Another dfference s the condton to generate the events. In 17, events are generated n case the state of the plant s a larger than a certan threshold, n 18, 19 when the relatve dfference between the state of the plant and the prevously sampled state volates a July 18, 211

3 3 certan threshold, and n 2, 21 when the absolute dfference between the state of the plant and the prevously sampled state volates a certan threshold. An mportant observaton to be made about the aforected works s that most of them consder state-feedback controllers, whch assumes that all the plant states can be measured. To the best of the authors knowledge, the only theoretcal result on event-trggered control usng dynamcal output-based controllers s presented n 2. However, an analyss of the mnmum tme between two subsequent events, the so-called nter-event tme, s not avalable for 2 and, thereby, guarantees on the upper bound on the number of events cannot be made. Furthermore, extendng the event-trggerng mechansms n 18, 19 to output-based controllers s not straghtforward, snce for these event-trggerng mechansms, no mnmum nter-event tme can be shown to exst, even though they have a guaranteed mnmum nter-event tme for state-feedback controllers. For any event-trggered control system to be useful, we need such a lower bound on the nter-event tme, as our prmary reason to make control systems event-trggered s to save computaton and communcaton resources. In ths paper, we analyse stablty and L -performance of event-trggered control systems for gven dynamcal output-based controllers. We consder the case where the sensors and actuators, whch can be grouped nto nodes, and controllers can be physcally dstrbuted. Ths causes a centralsed event-trggerng mechansm to be prohbtve, as the condtons that generate events would need access to all the plant and controller outputs at all tmes and wthout any delays, whch would requre transmttng all the node data contnuously. To resolve ths ssue, we wll propose a decentralsed event-trggerng mechansm, n whch events are trggered on the bass of local nformaton only. Inspred by 19, we propose an event-trggerng mechansm that nvokes transmsson of the controller or the plant outputs of a node when the dfference between the current values n the node and ts prevously transmtted values becomes large compared to the current values and an addtonal threshold. Ths addtonal threshold ensures that each node has a nonzero mnmum nter-event tme, whch allows us to guarantee a bound on the total number of transmssons. Interestngly, the event-trggerng mechansm presented n ths paper can be seen as a unfcaton of the event-trggerng mechansms proposed n 18, 19 and As a second contrbuton of ths paper, we propose to model the event-trggered control system as an mpulsve system, see, e.g., 23, 24, whch truly descrbes the behavour of the event-trggered control system. Furthermore, we extend the framework presented n 19 towards output-feedback controllers and L -performance, and we formally show that the mpulsve systems framework provdes stablty guarantees for event-trggerng mechansms that result n larger mnmum nter-event tmes than the extended results of 19. These stablty condtons wll be based on lnear matrx nequaltes (LMIs), so that effcent verfcaton s possble. We wll provde three numercal examples to demonstrate varous aspects of the developed theory. In partcular, we wll llustrate that the guaranteed lower bounds on the mnmum nter-event tmes are ndeed mproved wth respect to exstng results n the lterature and that the ncluson of a nonzero threshold n the event-trggerng mechansm s necessary to guarantee a postve mnmum nter-event tme for each node. The remander of ths paper s organsed as follows. After ntroducng the necessary notatonal conventons, we ntroduce the model of the decentralsed output-based event-trggered control system n Secton II. We analyse ts July 18, 211

4 4 stablty and ts L -gan propertes n Secton III, and n Secton IV we provde a way to compute the lower bound on the mnmum nter-event tme of each node. In Secton V, we extend the work of 19 towards output-based dynamcal controllers and L -performance, and present a theorem that states that the mpulsve system formulaton of the event-trggered control problem allows us to guarantee stablty and performance for event-trggered controllers wth at least the same mnmum nter-event tmes as the results based on the reasonng of 19. Fnally, the presented theory s llustrated by numercal examples n Secton VI and we draw conclusons n Secton VII. The appendx contans the proofs of the more techncal lemmas and theorems. A. Nomenclature For a vector x R n, we denote by x := x x ts 2-norm, and by x J the subvector formed by all components of x n the ndex set J {1,..., n}. For a symmetrc matrx A R n n, λ max (A) and λ mn (A) denote the maxmum and mnmum egenvalue of A, respectvely. For a matrx A R n m, we denote by A R m n the transposed of A, and by A := λ max (A A) ts nduced 2-norm. Furthermore, by A J and A J, we denote the submatrces formed by takng all the rows of A n the ndex set J {1,..., n}, and by takng all the columns of A n the ndex set J {1,..., m}, respectvely. By dag(a 1,..., A N ), we denote a block-dagonal matrx wth the entres A 1,..., A N on the dagonal, and for A B A brevty we wrte symmetrc matrces of the form as. B C For a sgnal w : R + R n, where R + denotes the set of nonnegatve real numbers, we denote by w Lp = ( w(t) p dt) 1/p ts L p -norm for p N, provded that the ntegral s fnte, and by w L = ess sup t R+ w(t) ts L -norm. Furthermore, we defne the set of sgnals wth a fnte L p -norm as L p := {w : R + R n w Lp < } for p N { }. Fnally, for a sgnal w : R + R n we denote the lmt from above at tme t R + by w + (t) = lm s t w(s), provded that t exsts. B C II. EVENT-TRIGGERED CONTROL In ths secton, we present the event-trggered control problem and model the event-trggered control system as an mpulsve system. A. Problem Formulaton Let us consder a lnear tme-nvarant (LTI) plant gven by d dt x p = A p x p + B p û + B w w, y = C p x p, where x p R np denotes the state of the plant, û R nu the nput appled to the plant, w R nw an unknown dsturbance and y R ny gven by the output of the plant. The plant s controlled usng a contnuous-tme LTI controller d dt x c = A c x c + B c ŷ, u = C c x c, (1) (2) July 18, 211

5 5 where x c R nc denotes the state of the controller, ŷ R ny the nput of the controller, and u R nu the output of the controller. We assume that the controller s desgned to render (1) and (2) wth y(t) = ŷ(t) and u(t) = û(t), for all t R +, asymptotcally stable,.e., an emulaton-based approach s taken. In ths paper, however, we consder the case where the controller s mplemented n a sampled-data fashon, whch causes y(t) ŷ(t) and u(t) û(t) for almost all t R +. In partcular, we study decentralsed event-trggered control whch means that the outputs of the plant and controller are grouped nto N nodes and the outputs of node {1,..., N} are only sent at the transmsson nstants t k, k N. Hence, at transmsson nstant t k, node transmts ts respectve entres n y and u, and the correspondng entres n ŷ and û are updated accordngly, whle the other entres n ŷ and û reman the same. Such constraned data exchange can be expressed as n whch v = y u, ˆv = ŷ û, and ˆv + (t k ) = Γ v(t k ) + (I Γ )ˆv(t k ), (3) Γ = dag(γ 1,..., γ ny+nu ), (4) for all {1,..., N}. In between transmssons, we use a zero-order hold,.e., d dt ˆv(t) =, for all t R +\ ( N =1 {t k k N} ). (5) In (4), the elements γ j, wth {1,..., N} and j {1,..., n y}, are equal to 1 f plant output y j s n node and are elsewhere, the elements γ j+ny, wth {1,..., N} and j {1,..., n u }, are equal to 1 f controller output u j s n node and are elsewhere. We assume that for each j {1,..., n y + n u }, t holds that N =1 γj >,.e., we assume that each sensor and actuator s at least n one node 1. Furthermore, we assume that at tme t =, t holds that ˆv() = v(). Ths can be accomplshed by transmttng all sensor and actuator data at the tme the system s deployed 2. In the case that t k = t j k j for some k, k j N and some, j {1,..., N}, we assume that the updates as n (3) take place smultaneously or drectly after one another n a neglgble amount of tme. Obvously, the order of updatng s rrelevant as can be seen from (3). Moreover, note that performng multple successve transmssons at one tme nstant has exactly the same effect as dong these updates smultaneously. In a conventonal sampled-data mplementaton, the transmsson tmes are dstrbuted equdstantly n tme and are the same for each node, meanng that t k +1 = t k + h, for all k N and all {1,..., N}, and for some constant transmsson nterval h >, and that t k = tj k, for all k N and all, j {1,..., N}. In event-trggered control, however, these transmssons are orchestrated by a decentralsed event-trggerng mechansm, as s shown n Fg. 1. We consder a decentralsed event-trggerng mechansm that nvokes transmssons of node data when the dfference between the current values of outputs and ther prevously transmtted values becomes too large n 1 In case a sensor or actuator s not n any node, meanng that ths sensor or actuator s, effectvely, not part of the control loop, we smply remove the correspondng nput or output from the plant and controller model. 2 Ths assumpton could be removed, but t would ntroduce addtonal techncaltes later. For reasons of readablty, we opted to work under ths rather mld and natural assumpton. July 18, 211

6 6 Fg. 1: Event-trggered Control Schematc, ndcatng the event-trggerng mechansms (ETMs). an approprate sense. In partcular, the event-trggerng mechansm proposed n ths paper, results n transmttng the outputs of the plant or the controller n node {1,..., N} at tmes t k, satsfyng t k +1 = nf { t > t k e J (t) 2 = σ v J (t) 2 } + ε, (6) and t =, for some σ, ε. In these expressons, e J and v J denote the subvectors formed by takng the elements of the sgnals e and v, respectvely, that are n the set J = {j {1,..., n y + n u } γ j = 1}, and e(t) = ˆv(t) v(t) (7) denotes the error nduced by the event-trggered mplementaton of the controller at tme t R +. Hence, the eventtrggerng mechansm (6), whch s based on local nformaton avalable at each node, s such that when for some {1,..., N}, t holds that e J (t) 2 = σ v J (t) 2 +ε,.e., the norm of the error nduced by the event-trggered mplementaton of the sgnals n node becomes too large for the frst tme, node transmts ts correspondng sgnal n v(t) and, thus, the sgnal ˆv(t) s updated accordng to (3). Ths mples that e + (t k ) = (I Γ )e(t k ) and thus e + J (t k ) =. Usng ths update law, and the aforementoned assumpton that ˆv() = v(), yeldng e() =, we can observe that the error nduced by the event-trggered control scheme satsfes e J (t) 2 σ v J (t) 2 + ε, (8) for all t R + and all {1,..., N}. The man objectve of ths paper s to determne σ and ε for all {1,..., N}, such that the closed-loop event-trggered system s stable n an approprate sense and a certan level of dsturbance attenuaton s guaranteed, whle the number of transmssons of the outputs of the plant and the controller s mnmsed. Note that for ε =, {1,..., N}, the event-trggerng condtons n (6) can be seen as an extenson of the event-trggerng mechansm of 19 for output-based controllers, and for σ =, {1,..., N}, t s equvalent to the eventtrggerng mechansm of As such, the event-trggerng mechansm n (6) unfes two earler proposals, whle addtonally, output-based controllers and decentralsed event trggerng are consdered. Remark II.1 In ths paper, we assume that the controller s gven n contnuous tme as n (2). To mplement ths controller on a dgtal platform, the followng optons can be consdered: () the controller output s obtaned by July 18, 211

7 7 numercal ntegraton, or () the controller output s obtaned usng a dscrete-tme equvalent of the contnuoustme controller, based on a samplng nterval that s (suffcently) smaller than the smallest nter-event tme (see Theorem IV.1 below). Ths, however, means that the event-trggered control strategy presented n ths paper s partcularly useful when the objectve s to save communcaton resources and/or battery power of wreless rados, whch s mportant for many (wreless) networked control systems, see, e.g., 25 28, and s less useful for savng computaton resources. B. An Impulsve System Formulaton In ths secton, we reformulate the event-trggered control system as an mpulsve system, see, e.g., 23, 24, of the form d dt x = Ā x + Bw, when x C (9a) x + = Ḡ x, when x D, {1,..., N}, (9b) where x X R nx denotes the state of the system and w R nw an external dsturbance. The flow and the jump sets are denoted by C R nx and D R nx, {1,..., N}, respectvely, and X = C ( N =1 D ). Note that the transmsson tmes t k, k N, as n (6), are now related to the event tmes at whch the jumps of x, accordng to (9b) for {1,..., N}, take place. To arrve at a system descrpton of the event-trggered control system (1), (2), (3), (5), and (6) of the form (9), we combne (1), (2), (3), (5) and (7), and defne x := x e R nx, where x = x p x c and n x := n p +n c +n y +n u, yeldng the flow dynamcs of the system d dt x = A + BC B x + E w, (1) C(A + BC) CB CE }{{}}{{} =:Ā =: B n whch A p B p A =, B =, C = A c B c C p C c, E = B w. (11) The system flows as long as the event-trggerng condtons are not met,.e., as long as (8) holds for all {1,..., N}, whch can be reformulated as x C, wth C = { x R nx x Q x ε {1,..., N}}, (12) and Q = σ C Γ C, (13) Γ because x Q x ε s equvalent to Γ e(t) 2 σ Γ v(t) 2 + ε, as n (8). As mentoned before, when node transmts ts data, a reset accordng to e + = (I Γ )e occurs, whle x remans the same,.e., x + = x, see (3). July 18, 211

8 8 Ths can be expressed by for all x D, {1,..., N}, n whch x + = I x, (14) I Γ } {{ } =:Ḡ D = { x R nx x Q x = ε }, (15) accordng to (6). Combnng (1), (12), (14) and (15) yelds an mpulsve system of the form (9). C. Specal Cases: State Feedback and Centralsed Event Trggerng In the exstng lterature, the event-trggered control problem has mostly been consdered for state-feedback controlled systems, see, e.g., 18, 19. In ths case, the controller s gven by u = K ˆx p, (16) where ˆx p R np denotes the most recently sampled state of the plant and s defned n a smlar fashon as ˆv. We can regard ths as a specal case of the setup presented above. Namely, to formulate the event-trggered control system wth controller (16) as an mpulsve system, we combne (1), (3), (5), (6) and (16), where we take C p = I n (1), as all states are measurable, and take v := y = x p, ˆv := ŷ = ˆx p n (3), (5) and (6). In ths case, the resultng mpulsve model s gven by (9), wth (1), (12), (14) and (15), n whch x := x p e and A = A p, B = B p K, C = I, E = B w. (17) To arrve at the event trggerng mechansm that was proposed n 19, we take N = 1 and Γ 1 = I (.e., a centralsed event-trggerng mechansm), and ε 1 =. Remark II.2 Although we study event-trggerng condtons of the form (6), whch s an extenson of the one presented n 19, we can n prncple study every event-trggerng mechansm wth condtons that can be wrtten n the form x Q x = ε, such as the ones presented n 29. III. STABILITY AND L -GAIN In ths secton, we wll study stablty of the event-trggered control system n the sense of Lyapunov and ts L -gan. We wll frst revew some basc stablty and L -gan results for mpulsve systems of the form (9). A. Stablty and L -gan of Impulsve Systems Let us defne the notons of stablty and of Lyapunov functon canddate that can be used to analyse mpulsve systems of the form (9). Defnton III.1 23 Consder the mpulsve system, gven by (9) wth w = and a compact set A X. July 18, 211

9 9 The set A s sad to be stable for the mpulsve system (9) wth w =, f for each ε > there exsts δ >, such that mn x A x() x δ mples mn x A x(t) x ε, for all solutons x to the mpulsve system (9) wth w = and all t for whch the soluton x s defned. The set A s sad to be globally attractve f each soluton x to the mpulsve system (9), wth w =, satsfes mn x A x(t) x as t. The set A s globally asymptotcally stable for (9), wth w =, f t s stable and globally attractve. Defnton III.2 23 Consder the mpulsve system, gven by (9) wth w =, and a compact set A X. The functon W : X R s a Lyapunov functon canddate for the system (9) and the set A f the functon W () s contnuous and nonnegatve on (C N =1 D )\A X, () s locally Lpschtz on an open set O satsfyng C\A O X, () satsfes lm W ( x) =, and x A, x X (v) the sublevel sets of W on X are compact,.e., the sets { x X W ( x) c W } are compact for all c W >. To prove global asymptotc stablty of the set A of the system (9), we wll make use of the followng lemma. Lemma III.3 Consder the mpulsve system (9) wth w = and a compact set A X satsfyng Ḡ x A for all x D A, {1,..., N}. Assume that for w = and for all x X, a mnmum nter-event tme h mn > exsts for each {1,..., N},.e., t k +1 t k h mn for all k N, and assume there exsts a Lyapunov functon canddate W for the mpulsve system (9) wth w = and the set A X, such that dw ( x) d x Ā x <, for almost all x C\A, W (Ḡ x) W ( x), for all x D \A, {1,..., N}. (18a) (18b) Then, A s a globally asymptotcally stable set for the system (9) wth w =. Proof: The proof s gven n the Appendx. Let us now defne the noton of the L -gan of a system, whch was studed for LTI systems n, e.g., 3, for whch we ntroduce a performance varable z R nz gven by z = C x + Dw, (19) for some matrces C and D of approprate dmensons. Defnton III.4 The L -gan from w to z of the system (9), wth (19), s defned as κ = nf{ κ R + δ : X R +, such that z L κ w L + δ( x()), for all x() X, w L }, (2) n whch z s a soluton to (9) and (19) wth ntal condton x() X, and dsturbance w L. July 18, 211

10 1 B. Stablty and L -gan of the Event-Trggered Control System Usng the results presented above for mpulsve systems of the form (9), we now present the man result of ths secton. The man result conssts of condtons for stablty of a set A, and an explct expresson of ths set A, and an upper bound on the L -gan for the event-trggered control system (9), wth (1), (12), (13), (14) and (15), and (19). We wll also present smpler condtons to guarantee that A s a globally asymptotcally stable set for ths event-trggered control system for the case that the dsturbances are absent (.e., for w = ). Theorem III.5 Consder the event-trggered control system (9), wth (1), (12), (13), (14) and (15), and (19). Moreover, assume that for all x() X and all w L, a mnmum nter-event tme h mn > exsts for each {1,..., N},.e., t k +1 t k h mn for all k N. Now suppose there exst a postve defnte matrx P R (np+nc) (np+nc), a postve semdefnte matrx U R nx nx, scalars α, β, κ >, and µ, ν, {1,..., N}, satsfyng N =1 µ Q Ā P P Ā α P, (21a) B P βi α P C C D C (κ 2 β)i D, (21b) D P Ḡ for all {1,..., N}, n whch P := dag(p, ) + U. Then, P Ḡ ν Q, (21c) A = { x C ( N =1 D ) x N P x µ ε =1 α } (22) s a globally asymptotcally stable set for (9) wth w =. Moreover, the L -gan of (9), wth (19), s smaller than or equal to κ and δ( x()) n (2) can be taken as δ( x()) = (α x () P x() + N =1 µ ε ) 1/2 for x() X. Proof: The proof s gven n the Appendx. Let us now comment on the results presented n Theorem III.5. The frst comment s that the assumpton n the hypotheses of Theorem III.5 on the exstence of a strctly postve mnmum nter-event tme for each {1,..., N} s automatcally guaranteed, f ε > for all {1,..., N} and the LMIs n (21) are feasble. Ths wll be shown n Theorem IV.1 below. In case that ε = for some {1,..., N}, the assumpton on the exstence of a strctly postve mnmum nter-event tme for each {1,..., N} can be volated and the nter-event tmes t k +1 t k can converge to zero. In ths case, an nfnte number of events can occur n a fnte-length tme nterval (.e., the mpulsve system (9) exhbts Zeno behavour). Ths can happen at tmes t when v J (t) = and x(t), as we wll show n Example 2 n Secton VI. Therefore, we should generally take ε > for all {1,..., N} to guarantee mnmum nter-event tmes greater than zero. Another comment regardng Theorem III.5 s that feasblty of (21) s only determned by the choce of sutable α, β, κ, and σ, {1,..., N}, as Q depends on σ, and feasblty s not affected by the choce of ε, {1,..., N}. July 18, 211

11 11 Hence, once (21) s feasble, practcal stablty (for w = ) and the upper bound κ on the L -gan are guaranteed. The sze of the set A as n (22) (when w = ), s affected by α, κ and σ, through the resultng P, as well as ε. Hence, after choosng α, κ and σ that render the set A of the event-trggered control system globally asymptotcally stable and that guarantee the desred upper bound κ on the L -gan, the parameters ε can be freely chosen to adjust the sze of the set A. As we can see from (8), ths wll affect the number of events, enablng us to make trade-offs between the sze of the set A (related to the ultmate bound x reaches as t for w = ) and the number of transmssons n each channel. Indeed, larger ε, {1,..., N}, result n fewer events, and thus fewer transmssons, but n a larger set A (.e., a larger ultmate bound), when w =. In fact f ε, {1,..., N}, all approach zero (from above) we have that A {}. Hence, the set A can be made arbtrary small (at the cost of more transmssons). The nave choce to take ε =, for all {1,..., N}, seems appealng as t would yeld A = {}. However, as argued already above, ths mght result n zero mnmum nter-event tmes. In some cases, such as the case of a state-feedback controlled system wth centralsed event trggerng as dscussed n Secton II-C, a strctly postve mnmum nter-event tmes can guaranteed even for ε 1 =, and we have that A = {} s globally asymptotcally stable. We wll further dscuss the mnmum nter-event tmes below. Fnally, note that the functon δ s also affected by ε, also expressng that larger ε wll result n a larger ultmate bound (even for w ). In case dsturbances are absent (w = ), we can arrve at smpler condtons to guarantee that A s a globally asymptotcally stable set for the event-trggered control system (9), wth (1), (12), (13), (14) and (15). Corollary III.6 Consder the event-trggered control system (9), wth (1), (12), (13), (14) and (15), and w =. Moreover, assume that for all x() X, a mnmum nter-event tme h mn > exsts for each {1,..., N},.e., t k +1 t k h mn for all k N. Now suppose there exst a postve defnte matrx P R (np+nc) (np+nc), a postve semdefnte matrx U R nx nx, scalars α >, and µ, ν, {1,..., N}, satsfyng µ Q Ā P P Ā α P, (23a) =1 α P dag(i, ) (23b) and (21c) for all {1,..., N}, where P := dag(p, ) + U and I denotes the dentty matrx of sze (n p + n c ) (n p + n c ). Then, the set A as n (22) s a globally asymptotcally stable set for (9) wth w =. Furthermore, lm sup t x(t) ( N =1 µ ε ) 1/2. Proof: The proof follows the same lnes of reasonng as the proof of Theorem III.5. The fact that x(t) ( N =1 µ ε ) 1/2 as t follows from (52), wth w =, and the fact that (23b), mples that x(t) 2 αv ( x(t)) for all t R +. Remark III.7 In ths paper, we partcularly study the L -gan from w to z of the system (9), wth (19), nstead of the L p -gan from w to z, for some p N, defned as κ = nf{ κ R + δ : X R +, such that z Lp κ w Lp + δ( x()), for all x() X, w L p }, (24) July 18, 211

12 12 n whch z s a soluton to (9) and (19) for ntal condton x() X, and nput w L p. The reason for focussng on L -gans s that we are generally nterested n ε >, {1,..., N}, as ths guarantees nonzero mnmum nter-event tmes (see Theorem IV.1 below). In ths case, A {}, and thus x(t) wll not converge asymptotcally to the orgn for w =, and therefore z(t) wll typcally not converge to zero when t. Hence, z Lp = for all p. Consequently, a fnte L p -gan for p N cannot be guaranteed n case ε >, {1,..., N}. Snce the L -gan does not requre z(t) when t, but merely that z(t) s bounded for all t R +, we can arrve at a fnte L -gan for the event-trggered control system dscussed n ths paper. Note that n case ε =, {1,..., N}, for whch n some crcumstances t s possble to guarantee that h mn > (e.g., the case of a state-feedback controlled system wth centralsed event trggerng as dscussed n Secton II-C), the L p -gan mght be fnte snce n ths case A = {}. In partcular, the L 2 -gan s guaranteed to be smaller than κ for the system (9), wth (19) and ε = for all {1,..., N}, f there exst a postve defnte matrx P R (np+nc) (np+nc), a postve semdefnte matrx U R nx nx, such that P := dag(p, ) + U, scalars α >, and µ, ν, {1,..., N}, satsfyng N =1 µ Q Ā P P Ā α P B P κ 2 I, (25a) C D I P Ḡ P Ḡ ν Q, (25b) for all {1,..., N}. Of course, ths result only holds f all the mnmum nter-event tmes h mn, {1,..., N}, are strctly postve, as was also requred n Theorem III.5. IV. A LOWER BOUND ON THE INTER-EVENT TIMES In ths secton, we wll show that for each node {1,..., N}, the nter-event tmes t k +1 t k, k N, of the event-trggered control system are bounded from below by a strctly postve constant. The exstence of a lower bound on the nter-event tmes for every node means that the total number of transmssons n a fnte tme nterval s bounded from above, whch guarantees a certan maxmum utlsaton of the communcaton resources. We wll show that, although the stablty and L -gan propertes of the system hold globally, the guaranteed lower bound on the nter-event tmes s a local property, n the sense that t depends on the magntude of the ntal condton and the dsturbance. The analyss s based on studyng the solutons of (9), wth (1), (12), (13), (14) and (15), from t k to t k. To +1 do so, we study the solutons of the auxlary system d x I I dt = Ā e J Γ x Γ e J I (Ā + Γ Γ c e J c + Bw ) wth e + J (t k ) =, from t k to t k +1, for each {1,..., N}. In (26), the submatrces Γ := I J and Γ c := I J c, {1,..., N}, are formed by takng the columns of the dentty matrx I that are n the set J = {j {1,..., n y + n u } γ j = 1}, and n the set J c (26) = {1,..., n y + n u }\J, respectvely, and are used to select the July 18, 211

13 13 sgnals n e that correspond to node {1,..., N},.e., e J = Γ e, and that do not correspond to ths node,.e., e J c = ( Γ c ) e, respectvely. The auxlary system (26), {1,..., N}, s obtaned from (1), by consderng e J c as external nputs. Hence, the fact that the dynamcs of e J c depend on x and e J s gnored n (26), yeldng that the solutons to (9), wth (1), (12), (13), (14) and (15), from t k to t k +1 are ncluded n the solutons of (26) and e + J (t k ) = from t k to t k, for each {1,..., N}. Ths fact, and the fact that e +1 J c n (26) satsfes (8), wll be exploted to derve the lower bound on the mnmum nter-event tme. We now present the man result of ths secton. Theorem IV.1 Consder the event-trggered control system gven by (9), wth (1), (12), (13), (14) and (15), wth ε > for all {1,..., N}. For every δ x and every δ w, there exsts a strctly postve lower bound on the mnmum nter-event tmes h mn (δ x, δ w ) for each node {1,..., N},.e., t k +1 t k h mn, for all k N, for every soluton to (9) wth x() δ x, and w L δ w. An explct expresson for a lower bound h mn s gven by { ( I e I Γ mn h > λ max Ā h Q e Ā I h Γ I ) } ζ(h) η, (27) n whch ζ (h) = ε Q ( 2 ηρ (h) Ā e I Γ h I + ρ (h) ), (28) wth ρ (h) = h h ( e ϑs ds B δ w + Ā and η = λmax( P )αδ 2 x +βδ2 w + N =1 µε αλ mn(p ), ϑ = λ max (Ā Proof: The proof s gven n the Appendx. I Γ + Γ c I j I σ j Γ j C 2 η + ε j ) 2, (29) Γ Ā ), and I := {1,..., N}\{}. The mnmum n (27) n Theorem IV.1 can be solved by computng the maxmum egenvalue of the h-dependent matrx n the condton n (27) for ncreasng h > and check when the nequalty s satsfed for the frst tme. Ths determnes for node {1,..., N} the lower bound on the nter-event tmes h mn, as n (27). Even though a mnmum nter-event tmes s guaranteed for each node, no guarantees can be made about the tme between two events n dfferent nodes. Stll, the lower bound on the nter-event tmes for all nodes allows guarantee to be made about the total number of events wthn a certan tme nterval. The obtaned lower bounds decrease as δ x (whch s related to x() ) ncreases or as δ w (whch s related to w L ) ncreases, mplyng that the control task has to be executed more often f the system s ntal state s further away from the orgn and n case the norm of the dsturbance s larger. We wll llustrate ths observaton n Example 3 of Secton VI. In the specal case that C p and C c are nvertble and the event trggerng s centralsed,.e., the number of nodes N = 1 and Γ 1 = I (mplyng that Q 1 has full rank), and no dsturbances are present (mplyng that δ w = ), the mnmum nter-event tme h 1 mn > even for ε 1 =. Furthermore, we have that ρ 1 (h) = (due to δ w = and I 1 = ), and thus that ζ 1 (h) =, for all h R +, meanng that the obtaned bound s ndependent of δ x (and thus ndependent of x() X ). If July 18, 211

14 14 addtonally the controller s gven by a state-feedback controller (16), the resultng condton recovers the one presented n Theorem 5.1 n 31. In ths case, the bounds are tght n the sense that for some k 1 N, we have that t 1 k 1+1 t1 k 1 = h 1 mn. V. IMPROVED EVENT-TRIGGERING CONDITIONS In the prevous sectons, we modelled the event-trggered control system as an mpulsve system and presented condtons to guarantee ts stablty and an upper bound on the L -gan. The reason to take an mpulsve system approach s that t explctly descrbes the behavour of the event-trggered control system. Ths has the favourable consequence that t yelds less conservatve condtons than the (drect extensons of the) exstng results n the lterature, such as 18, 19. To formally demonstrate ths statement, we frst extend the reasonng of 19 towards dynamcal output-based controllers and by ncludng L -performance, and secondly, we show that the obtaned stablty condtons can be seen as a specal case of the condtons n Theorem III.5 (.e., usng the mpulsve system descrpton of the event-trggered control system). To extend the work of 19, let us consder the followng auxlary system: d dt B x = (A + BC)x + E e, w v = C x + e, z C x D w whch s obtaned from (1) and (19) by consderng the error e as an external nput, nstead of as a state varable as n (9), and by assumng that the performance output, as n (19), s gven by z = C x x + Dw, mplyng that C n (19) has the form C = C x (3). As mportant observaton s that n (3), the dynamcs of e, gven by d dt e = d dtv = C(A + BC)x CBe, s gnored n (3), whle t s captured explctly n the mpulsve system (9), wth (1), (12), (13), (14) and (15). System (3), wth e = and w =, s a globally asymptotcally stable LTI system, because of the assumpton made n Secton II that the controller stablses the plant when ˆv(t) = ŷ (t) û (t) = y (t) u (t) = v(t) for all t R +, meanng that A + BC s Hurwtz,.e., t has all the egenvalues n the left-half complex plane. Ths ensures that there exst a postve defnte storage functon of the form V (x) = x P x, see, e.g., 32, a (suffcently small) postve scalar α, (suffcently large) postve scalars β, κ, satsfyng β κ 2, and (suffcently small) postve scalars σ, {1,..., N}, that satsfy the dsspaton nequalty d dt V (x(t)) αv (x(t)) + ( β w(t) σ e J (t) 2 v J (t) 2) (31) and the nequalty =1 z(t) 2 + (β κ 2 ) w(t) 2 αv (x(t)). (32) Now snce (8) holds, for all {1,..., N} and all t R +, we have that ( vj (t) 2 + ε σ 1 σ e J (t) 2), (33) =1 July 18, 211

15 15 and all t R +. Combnng ths expresson wth (31) yelds that d dt V (x(t)) αv (x(t)) + β w(t) 2 + =1 ε σ, (34) whch allows us to show that for w = and for V (x(t)) N ε =1 σ, t holds that d α dtv (x(t)) <, whch means that the state x of (3), wth (8), converges asymptotcally to the set A = {x R np+nc V (x) N =1 ε σ α }. Furthermore, usng (32) and deas from 3, we can show that the system (3), wth (8), has a fnte L -gan from dsturbance w to performance output z. We wll formalse ths dea n the followng theorem. Theorem V.1 Assume that there exst scalars α, β, κ, σ P R (np+nc) (np+nc), satsfyng Z αp N =1 C Γ C B N P =1 1 σ Γ, E P βi αp C x C x D C x (κ 2 β)i D, D wth Z := (A + BC) P + P (A + BC). Then, the set >, {1,..., N}, and a postve defnte matrx (35a) (35b) A = {x R np+nc x P x N ε =1 σ } (36) α s a globally asymptotcally stable set of the system (3) wth (8) and for w =. Furthermore, the L -gan from w to z s smaller than or equal to κ and δ n (2) can be taken as δ(x()) = (αx ()P x() + N ε =1 σ ) 1/2 for all x R np+nc. Proof: The proof follows drectly from the dscusson above and the fact that (35a) and (35b) mply (31) and (32), respectvely. It also follows drectly from Theorem V.2 that we wll present below. In case the system s controlled by a state-feedback controller (as dscussed n Secton II-C), the event trggerng mechansm s centralsed (.e., N = 1 and Γ 1 = I), and when dsturbances are absent (E = and D = ), the condtons presented n Theorem V.1 provde LMI-based stablty condtons that can be used to analyse the stablty of the event-trggered control system studed n 19. Even though the results n 19 are vald for nonlnear systems as well, whle we focus n ths paper on lnear systems, Theorem V.1 provdes a computatonal procedure that allows us to obtan large values for σ and, thus, large values for the nter-event tmes, whereas 19 only presents exstence results and does not provde a constructve (optmsaton-based) way to obtan sutable choces for σ. Note that obtanng a LMI-based stablty analyss for the case studed n 19 s not the man result of ths secton. Namely, the man result of ths secton s presented below. Ths man result states that f Theorem V.1 guarantees global asymptotc stablty of the set A, as n (36), and guarantees an upper bound κ on the L -gan for the system (3) wth (8), for some scalars σ and the scalars ε, {1,..., N}, then, global asymptotc stablty of the same July 18, 211

16 16 set A and the same upper-bound κ on the L -gan can also be guaranteed for the mpulsve system (9) usng Theorem III.5. Theorem V.2 Consder the model of the event-trggered control system (3), wth (8), and the mpulsve system formulaton of the event-trggered control system (9), wth (1), (12), (13), (14) and (15), and (19) wth C = C x. If there exsts a postve defnte matrx P, and scalars α, β, κ, σ >, {1,..., N}, satsfyng the condtons of Theorem V.1, then P := dag(p, ), U =, µ = 1 σ and ν =, for all {1,..., N}, satsfy the condtons of Theorem III.5 for the same α, β, and κ. Proof: The proof s gven n the Appendx. Theorem V.2 formally shows that the condtons based on mpulsve system (9) are never more conservatve than the ones based on system (3), as the matrx P n Theorem III.5 can have a more general form than P := dag(p, ) (whch was used n the hypothess of Theorem V.2). Hence, ths creates the opportunty to guarantee stablty for event-trggerng condtons wth a larger nter-event tme or a smaller upper-bound on the L -gan. Furthermore, Theorem V.2 also guarantees that the condtons n Theorem III.5 can always be satsfed f all σ, {1,..., N}, are chosen suffcently small. Namely, for the auxlary system (3) the exstence of storage functon of the form V (x) = x P x satsfyng (31) for some postve α, β and σ, {1,..., N}, s guaranteed. Hence, the hypothess of Theorem V.1 can always be satsfed for some suffcently small α and σ, {1,..., N}, and some suffcently large β and κ, whch, n turn, mples feasblty of the condtons n Theorem III.5. VI. ILLUSTRATIVE EXAMPLES In ths secton, we llustrate the presented theory usng three numercal examples. The frst example s taken from 19, n whch an unstable plant s stablsed usng an event-trggered mplementaton of a state-feedback controller and a centralsed event-trggerng mechansm. We wll show that by formulatng the event-trggered control system as an mpulsve system and employng the theory as developed n ths paper, we can guarantee stablty for eventtrggered control systems wth larger mnmum nter-event tmes. In the second example, we stablse an unstable plant usng a dynamcal output-based controller and a decentralsed event-trggerng mechansm to llustrate that ndeed output-based controllers and decentralsed event trggerng can be desgned that perform well. In the last example, we consder a stable plant that s subject to dsturbances and show that outputs of the plant and the controller are only transmtted when dsturbances are actng on the system or durng transents, whle no transmssons occur when dsturbances are absent and the system s n steady state. Ths s a favourable property that tradtonal dgtal control systems wth perodc transmssons do not have. Example 1: Let us consder the numercal example taken from 19. The plant (1) s gven by d dt x p = 1 x p + u, (37) July 18, 211

17 the state-feedback controller s gven by (16), wth K = 1 4, and the event trggerng s centralsed,.e., we have that N = 1 and Γ 1 = I. In 19, global asymptotc stablty of the orgn s guaranteed for σ.55 for the event-trggerng condton e = σ x and was obtaned usng an alternatve approach. Ths yelds σ 1 =.55 2 =.3 and ε 1 = f the event-trggerng mechansm s formulated as n (6). For ths event-trggerng mechansm, Theorem IV.1, or ts counterpart Theorem 5.1 n 31, yelds a lower bound on the nter-event tmes 3 of.318. We now compare ths result wth the event-trggerng mechansm obtaned usng the results from Secton V,.e., obtaned by maxmsng σ 1 n the condtons of Theorem V.1, modfed accordng to Remark??. Takng ths approach allows us to guarantee stablty up to σ 1 =.273, resultng, for ε 1 =, n a lower bound on the nter-event tmes of.84. Therefore, we can conclude that takng the approach as n Secton V already ncreases the allowable mnmum nter-event tme wth respect to 19. However, f we analyse stablty usng the result of Theorem III.5, whch s based on the mpulsve system formulaton, we can guarantee stablty of ths event-trggered control system up to σ 1 =.588, whch yelds, for ε 1 =, a lower bound on the nter-event tmes of The ncrease of nter-event tmes s expected due to the formal result establshed n Theorem V.2. We therefore conclude that by modellng the event-trggered control system usng an mpulsve model, whch truly descrbes the behavour of ths event-trggered control system, stablty can be guaranteed for event-trggerng mechansms that yeld larger mnmum nter-event tmes. Example 2: Let us now consder the plant (1) gven by d dt x p = 1 x p + û y = 1 4 x p, and the controller (2) gven by d dt x c = 1 x c + ŷ 5 1 u = 1 4 x c. We assume that no dsturbances act on the plant,.e., B w =, and, therefore, we smply take C = and D =. Furthermore, we assume that the system s equpped wth an event-trggerng mechansms at both the sensorto-controller channel and the controller-to-actuator channel, whch means that we defne Γ 1 17 (38) (39) = dag(1, ) and Γ 2 = dag(, 1). Hence, we have two nodes. Practcal stablty of the event-trggered control system (1), (2), wth event-trggerng mechansm (6), wth σ 1 = σ 2 = 1 3, can be guaranteed usng the mpulsve system formulaton (9) and the results of Corollary III.6. If we take ε 1 = ε 2 =, the nter-event tmes wll become zero when v J (t) =, at some tme t R + and for some {1, 2}, as was dscussed n Secton III and Secton IV. By smulatng the response of the system to the ntal condton x() = 25 2, 25 2, 25 2, 25 2,,, we can observe that ndeed the nter-event tmes converge to 3 Note that for ths example, the obtaned lower-bound on the mnmum nter-event tmes s tght, as also was observed n Secton IV. July 18, 211

18 tempmage temp 211/4/2 tempmage temp 1:21 page 1 #1 211/4/2 1:2 page 1 #1 replacemen 18 outputs y and u y, node = 1 u, node = tme t nter-event tme t k+1 t k node = 1 node = tme t Fg. 2: Evoluton of y (node = 1) and u (node = 2) as a functon of tme (left) and the nter-event tmes for each node as a functon of tme (rght) for Example 2 wth ε 1 = ε 2 =. zero as v J2 (t) = u(t), see Fg. 2. Hence, ths llustrates that ε > s needed for all {1,..., N} to guarantee mnmum nter-event tmes larger than zero. If we now take ε 1 = ε 2 = 1 3, Corollary III.6 yelds that the states x(t) satsfy lm sup t x(t) 6.4. Usng the result of Theorem IV.1, we obtan that f the ntal condtons satsfy, e.g., x() 25, a lower bound on the nter-event tmes h 1 mn = h2 mn = s guaranteed for both nodes. When we compare these results wth a smulaton of the response of the system to the ntal condton x() = 25 2, 25 2, 25 2, 25 2,,, see Fg. 3, we observe that the states of the plant and the controller ndeed converge asymptotcally to a vcnty of the orgn and that the outputs of the plants and controllers have to be transmtted less often when the state approaches the orgn. However, x(t) even satsfes lm sup t x(t).12, whch s sgnfcantly smaller than the predcted upper bound of approxmately 6.4. In addton, the observed mnmum nter-event tme s h 1 mn h2 mn 1 4, whch s larger than the predcted value of Ths seems to hold for many ntal condtons satsfyng x() < 25. Ths shows that, although we can formally prove the exstence of a globally asymptotcally stable compact set and a nonzero lower bound on the mnmal nter-event tmes, the obtaned bounds can stll be mproved. Improvng these bounds s a topc of future research. Example 3: Let us now consder the (stable) plant (1) gven by d dt x p = 1 x p + û + w y = 1 x p, and the controller (2) gven by d dt x c = 2 1 x c + 2 ŷ u = 5 2 x c. Furthermore, we take C = 1 and D = for the performance output z n (19), and assume that the system s equpped wth event-trggerng mechansms at both the sensor-to-controller channel and the controller-to-actuator (4) (41) July 18, 211

19 tempmage temp 211/4/2 1:19 page 1 # tempmage temp 211/4/2 1:2 page 1 #1 states xp and xc tempmage temp 211/4/2 1:18 page 1 # tme t 1 nter-event tme t k+1 t k node = 1 node = Fg. 3: Evoluton of the states of the plant and controller as a functon of tme (top) and the nter-event tmes for each node as a functon of tme (bottom) for Example 2 wth ε 1 = ε 2 = 1 3. Inserted n the top fgure: A close-up tme t of the evoluton of the states of the plant and the controller at tmes t 17.5, 3 channel. Ths means that we, agan, defne Γ 1 = dag(1, ) and Γ 2 = dag(, 1). Asymptotc stablty of the compact set A and an upper bound of the L -gan of the event-trggered control system (1), (2), wth eventtrggerng mechansm (6), wth σ 1 = σ 1 = 1 3, can be guaranteed usng the mpulsve system formulaton (9) and the results of Theorem III.5. Ths leads to the smallest upper bound on the L -gan, gven by κ =.46. When we smulate the response of the system to the ntal condton x() = and a dsturbance satsfyng w(t) 1 for tme t, 1 and w(t) 1 4 for tme t 2, 3, as shown n Fg. 4, we obtan the trajectores of z as also shown n Fg. 4. In ths fgure, we can observe that the performance output z, as n (19), satsfes z(t).3 for tme t, 1 and z(t).9 for tme t 2, 3, whch satsfes z L κ w L + δ() =.46 w L +.17, whch s an upper bound of the L -gan of Defnton III.4. Furthermore, we can also observe that the nter-event tmes are larger than.22 for t, 1 and larger than.44 for t 2, 3. Ths observaton concurs wth the result of Secton IV, whch stated that transmssons occur less often f the magntude of the dsturbance s smaller. Fnally, snce the system (4) s stable, t seems that the outputs of the plant and controller only have to be transmtted when dsturbances are actng on the system or durng transents (.e., approxmately for t < 1 and t > 2), and no transmssons occur when no dsturbances are actng on the system and the systems s close to ts steady state. One could say that event-trggered control only acts when t s July 18, 211

20 tempmage temp 211/4/2 1:2 page 1 #1 2 dsturbance w and output z output z dsturbance w tempmage temp 211/4/2 1:2 page 1 # tme t nter-event tme t k+1 t k node = 1 node = tme t Fg. 4: Evoluton of the dsturbance w and the output z as a functon of tme (top) and the nter-event tmes for each node as a functon of tme (bottom) for Example 3. necessary from a stablty or performance pont of vew, whch s a favourable property that makes event-trggered control of hgh nterest. Tradtonal dgtal control systems wth perodc transmssons do not have ths appealng property. VII. CONCLUSIONS In ths paper, we studed stablty and L -performance of event-trggered control systems for dynamcal outputbased controllers havng decentralsed event-trggerng mechansms. The proposed event-trggerng mechansm unfes earler proposals for event-trggerng mechansms, whch were manly appled to state-feedback controllers. Va an example (Example 2), we showed that drect extensons of exstng event-trggerng mechansms for outputbased controllers and decentralsed event trggerng are not applcable, as they result n nter-event tmes that converge to zero. Such Zeno behavour s obvously undesrable n practcal mplementatons and, therefore, extensons as proposed n ths paper are necessary. To analyse the resultng event-trggered control system, we modelled the event-trggered control system as an mpulsve system that truly descrbes the behavour of the event-trggered control system. The stablty and L - performance are then analysed usng lnear matrx nequaltes. In addton, we provded expressons for lower bounds on the mnmum nter-event tmes and we formally proved that by usng an mpulsve systems approach, July 18, 211