Riccati-Based Design of Event-Triggered Controllers for Linear Systems With Delays

Transcription

1 174 IEEE TRANSACTINS N AUTMATIC CNTRL, VL. 63, N. 1, JANUARY 218 Riccati-Base Design of Event-Triggere Controllers for Linear Systems With Delays Dominicus Paulus Borgers, Stuent Member, IEEE, Victor Sebastiaan Dolk, Stuent Member, IEEE, an W. P. M. H. Heemels, Fellow, IEEE Abstract In event-triggere control ETC) systems, the measure state or output of the plant is sent to the controller at so-calle event times. In many ETC systems, these event times are generate base on a static function of the current state or output measurement of the system an its sample-an-hel version that is available to the controller. Hence, the event-generator oes not inclue any ynamics of its own. In contrast, ynamic event-generators trigger events base on aitional ynamic variables, whose ynamics epen on the state or output of the system. In this paper, we propose new static an ynamic continuous event-generators which require continuous measuring of the plant output) an perioic eventgenerators which only require perioic sampling of the plant output) for linear control systems with communication elays. All event-generators we propose lea to close-loop systems which are globally exponentially stable with a guarantee ecay rate, L 2 -stable with a guarantee L 2 - gain, an have a guarantee positive minimum inter-event time. By using new Riccati-base analysis tools tailore to linear systems, the conservatism in our ecay rate an L 2 -gain estimates is small. The ynamic event-generators even further reuce this conservatism, an as a result typically generate significantly fewer events than their static counterparts, while guaranteeing the same control performance. The benefits of these new event-generators are emonstrate via two numerical examples. Inex Terms Linear matrix inequalities, linear systems, networke control systems, riccati equations. I. INTRDUCTIN IN MST igital control systems, the measure output of the plant is perioically transmitte to the controller, regarless of the state the system is in. This possibly leas to a waste of e.g., computation, communication, an energy) resources, as many of the transmissions are actually not necessary Manuscript receive November 11, 216; revise November 14, 216 an March 16, 217; accepte May 3, 217. Date of publication June 7, 217; ate of current version December 27, 217. This work was supporte by the research programmes Wireless control systems: A new frontier in automation with Project an Integrate esign approach for safety-critical real-time automotive systems with Project 12698, which are partly) finance by the Netherlans rganisation for Scientific Research. Recommene by Associate Eitor P. Bolzern. Corresponing author: Dominicus Paulus Borgers.) The authors are with the Control Systems Technology Group, Department of Mechanical Engineering, Einhoven University of Technology, Einhoven 5612 AZ, The Netherlans .p.borgers@tue.nl; v.s.olk@tue.nl; m.heemels@tue.nl). Color versions of one or more of the figures in this paper are available online at Digital bject Ientifier 1.119/TAC to achieve the esire performance guarantees. To mitigate this potential waste of valuable resources, many event-triggere control ETC) strategies have been propose, which generate the transmission times base on a triggering conition involving the current state or output measurement of the plant an the most recently transmitte measurement ata, see, e.g., 1] 5] an the references therein. This brings a feeback mechanism into the sampling an communication process, such that measurement ata are only transmitte to the controller when neee in orer to guarantee the require stability an performance properties of the system. ETC strategies can be ivie into static an ynamic strategies, or into continuous an perioic strategies. In static ETC strategies, events are base on a static function of the current state or output measurement an its sample-an-hel version that is available to the controller using, e.g., a zero-orer-hol, a first-orer-hol, or even a moel-base-hol function). In contrast, in ynamic ETC strategies, events are base on an aitional ynamic variable with ynamics that epen on the state or output of the system. Continuous ETC CETC) strategies require continuous measuring of the plant output which is sometimes ifficult to implement on igital platforms), while perioic event-triggere control PETC) strategies only require perioic sampling of the plant output. Static CETC strategies have been propose in 1], 2], 4], 6] 9]. However, the event-generators in these works that lea to asymptotic stability of the CETC system can typically also lea to Zeno behavior an infinite number of events in finite time) in the presence an sometimes even in the absence) of isturbances 1], 11], an those that o not exhibit Zeno behavior only lea to practical stability an not to asymptotic stability 1]. To prevent Zeno behavior, static CETC strategies with waiting times also calle time regularization ) have been propose in, e.g., 5], 12] 17]. Dynamic CETC strategies for nonlinear systems have been propose in 18] 21]. In 19], a ynamic strategy is use to exten the inter-event times compare to a time-triggere system, an in 18], 2], an 21], ynamic strategies are use to exten the inter-event times compare to static CETC systems. In these works, the guarantee control performance an minimum inter-event time of the propose ynamic CETC system are ientical to its static or time-triggere) counterpart, while it is emonstrate that the average number of events in the ynamic CETC system is typically much smaller IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See stanars/publications/rights/inex.html for more information.

2 BRGERS et al.: RICCATI-BASED DESIGN F EVENT-TRIGGERED CNTRLLERS FR LINEAR SYSTEMS WITH DELAYS 175 While the results above show that for CETC systems, the inclusion of a ynamic variable in the event-generator can clearly lea to a significant further reuction of the consumption of communication an energy resources, there are currently no ynamic PETC strategies available in literature. Inee, only static PETC strategies have been propose, see, e.g., 3], 5], 22] 27] for linear systems an 28] 3] for nonlinear systems. In this paper, we provie new static an ynamic) CETC an PETC strategies tailore to linear systems with varying) communication elays. We are able to guarantee a positive minimum inter-event time by esign, an we provie tight estimates of the L 2 -gains an exponential ecay rates of the resulting close-loop systems, by making use of analysis tools specific to the omain of linear systems. These analysis tools are base on Lyapunov/storage functions exploiting matrix Riccati ifferential equations an computationally frienly semiefinite programming, using ieas from 22]. In aition, we also provie even less conservative conitions for global exponential stability GES) an L 2 -stability base on piecewise quaratic Lyapunov functionals. Base on these conitions, we are able to provie traeoffs of guarantee control performance versus minimum an average inter-event times. Interestingly, for ientical control performance guarantees, the ynamic ETC strategies prouce significantly larger average inter-event times than their static counterparts, an hence, require much less communication. These results, base on exploiting the linearity of the unerlying plants an controllers, provie significantly better results than the application of the results obtaine for nonlinear systems 2]. In fact, to the best of our knowlege, the propose continuous event-generators are the least conservative in literature, an we are the first to propose ynamic perioic eventgenerators. Preliminary results have appeare in 17] an 31], in which we i not consier communication elays. Moreover, the conitions provie in 17] an 31] are significantly more conservative in guaranteeing GES an L 2 -stability than the novel conitions provie here. This paper is organize as follows. In Section II, we present the control setup that we consier in this paper, for which we present our new static an ynamic CETC strategies in Section III, an our new static an ynamic PETC strategies in Section IV. In both sections, we provie tight bouns on the L 2 -gains an ecay rates of the resulting close-loop systems. In Section V, we provie even tighter bouns base on less conservative conitions using state-space partitioning an piecewise quaratic Lyapunov/storage functionals. Finally, we emonstrate our results in Section VI an provie concluing remarks in Section VII. All proofs are given in the appenix. A. Notation For a vector x R n x, we enote by x : x x its Eucliean norm. For a symmetric matrix A R n n, we enote by λ max A) an λ min A) its maximum an minimum eigenvalue, respectively. For a matrix P R n n, we write P P )if P is symmetric an positive semi-)efinite, an P P ) if P is symmetric an negative semi-)efinite. By I an,we enote the ientity an zero matrix of appropriate imensions, Fig. 1. ETC setup consiere in this paper. respectively. For a measurable signal w : R R n w, we write w L 2 if w L2 <, where w L2 : wt) 2 t ) 1/2 enotes its L 2 -norm, an we write w L if w L <, where w L : ess sup t wt) enotes its L -norm. By N we enote the set of natural numbers incluing zero, i.e., N : {, 1, 2,...}. A function γ : R R is a K-function if it is continuous, strictly increasing an γ), an a K - functionifitisak-function an, in aition, γs) as s. A function β : R R R is a KL-function if for each fixe t R the function β,t) is a K-function an for each fixe s R, βs, t) is ecreasing in t an βs, t) as t. For vectors x i R n i, i {1, 2,...,N}, we enote by x 1,x 2,...,x N ) the vector x 1 x 2 x N ] R n with n N i1 n i. For a vector y R n, we write y if y i for all i {1, 2,...,n}. For brevity, we sometimes write symmetric matrices of the form A B B C ] as A B C ] or A B C ].For a left-continuous signal f : R R n an t R,weuse ft + ) to enote the limit ft + ) lim s t,s>t fs). II. CNTRL SETUP In this paper, we consier the ETC setup of Fig. 1, in which the plant P is given by t x pt) A p x p t)+b p ut)+b pw wt) P : yt) C y x p t)+d y ut) 1) zt) C z x p t)+d z ut)+d zw wt) an the controller C is given by C : t x ct) A c x c t)+b c ŷt) ut) C u x c t)+d u ŷt). Here, x p t) R n x p enotes the state of the plant P, yt) R n y its measure output, zt) R n z the performance output, an wt) R n w the isturbance at time t R. Furthermore, x c t) R n x c enotes the state of the controller C, ut) R n u is the control input, ŷt) R n y enotes the output that is available at the controller, given by 2) ŷt) yt k ),t t k + τ k,t k+1 + τ k+1 ] 3) where the sequence {t k } k N satisfying t,t k+1 t k h 4)

3 176 IEEE TRANSACTINS N AUTMATIC CNTRL, VL. 63, N. 1, JANUARY 218 enotes the event or transmission) times with h R > the minimum inter-event time, an where the sequence {τ k } k N with τ k D: { 1, 2,..., n } for all k N enotes the communication elays. The set D contains the n N possible elays, i R, an satisfies the following small-elay assumption. Assumption II.1: D : { 1, 2,..., n },h]. By making use of Assumption II.1, we ensure that each ata packet has arrive at its estination before a new transmission is triggere. In Sections III V, we will propose a number of methos to generate the transmission times 4), such that the close-loop system is globally exponentially stable with ecay rate ρ an L 2 -stable with L 2 -gain θ. III. MAIN RESULTS FR THE CETC CASE In this section, we will propose continuous event-generators, which require continuous measuring of the plant output y.first, in Section III-A, we will propose a static continuous eventgenerator, an in Section III-B, we will propose a ynamic continuous event-generator, which generates the transmission times base on an aitional ynamic variable η R that is inclue in the event-generator. To escribe the close-loop system, we first have to introuce a number of variables, inspire by, e.g., the works 2], 32]. We introuce the memory variable s R n y, the timer τ R, the integer κ N, an the boolean l {, 1}. The role of these variables will be explaine below. Finally, we efine ζ : y, s) R 2n y an the state ξ : x p,x c, ŷ, s) R n ξ with n ξ n xp + n xc + n y + n y. The ynamic variable η R that will be inclue in the event-generator will evolve accoring to t ηt) Ψot)), t R \{t k } k N 5a) ηt + )η T ot)), t {t k } k N 5b) where the signal o : R R 2n y R N {, 1} R given by ot) :ζt),τt),κt),lt),ηt)) 6) is the information that is available to the continuous eventgenerator, an where the functions Ψ:R 2n y R N {, 1} R R an η T : R 2n y R N {, 1} R R are to be esigne. Now, we can write the close-loop system as the impulsive system 33] t ξt) Aξt)+Bwt) τt) 1 κt), lt) ηt) Ψot)) t R t/ {t k } k N t/ {t k + τ k } k N 7a) where ξt + ) J ξt) τt + ) κt + ) κt), lt + ) 1 ηt + ) η T ot)) ξt + ) J 1 ξt) τt + ) τt) κt + ) κt)+1, lt + ) ηt + ) ηt) zt) Cξt)+Dwt) l t {t k } k N l 1 t {t k + τ k } k N A p B p C u B p D u B pw A c B c A,B C C z D z C u D z D u ],D D zw I I J I C y D y C u D y D u I I J 1 I. I 7b) 7c) 7) In this moel, the memory variable s R n y stores the value of yt k ) that has been transmitte to the controller an which will arrive at the upate time t k + τ k ), the timer τ R keeps track of the time that has elapse since the latest transmission, the integer κ N is use to count the number of receive) transmissions, an the boolean l {, 1} inicates whether the next jump is a transmission when l ) or an upate when l 1). Furthermore, 7b) moels the jumps at transmission times t k, k N, 7c) moels the jumps at upate times t k + τ k, k N when the transmitte ata arrives at the controller), an 7a) moels the flow in between transmissions an upates. The sequence of transmission times {t k } k N will be generate by the ynamic continuous event-generator t,t k+1 inf{t t k + h ηt) ζ t)qζt) } 8) where the timer threshol h R an the matrix Q R 2n y 2n y are esign variables, in aition to the functions Ψ an η T in 5). The time threshol h actsasawaiting time or time regularization, an ensures that 4) hols.

4 BRGERS et al.: RICCATI-BASED DESIGN F EVENT-TRIGGERED CNTRLLERS FR LINEAR SYSTEMS WITH DELAYS 177 With the moel 5), 8) we can also capture static eventgenerators by choosing η) an Ψo) η T o) 9a) 9b) for all o R 2n y R N {, 1} R, as then we have that ηt) for all t R, an the ynamic event-generator 8) reuces to the static event-generator t,t k+1 inf{t t k + h ζ t)qζt) } 1) which only has h an Q as esign parameters. While tuning h is straightforwar as it is a scalar, choosing a suitable Q is more ifficult. However, a possible esign for Q can be erive from 2] an is given by 1 σ 2 )I I Q 11) I I with σ, 1). With this choice of Q, 1) reuces to t,t k+1 inf{t t k + h st) yt) 2 σ 2 yt) 2 } which can be seen as the event-generator propose in 2] with waiting time h. In case h, the setup 7), 1) can exhibit Zeno behavior in the presence of isturbances, as shown in 1]. Therefore, we often take h>, which leas to static event-triggere controllers with time regularization, see, e.g., 5], 13]. ther control setups an other choices of Q are also possible, see e.g., 22]. We will consier the following two notions of stability. Definition III.1: The CETC system 7) 8) is sai to be input-to-state exponentially stable ISES), if there exist functions α K, β KL, an scalars c>, γ>, an ρ> such that for any initial conition ξ) ξ R n ξ, τ), κ), l), η), an any sequence of elays {τ k } k N with τ k Dfor all k N, all corresponing solutions to 7) 8) with w L satisfy ξt) ce ρt ξ + γ w L an ηt) β ξ,t)+α w L ) for all t R.Inthis case, we call ρ a lower boun on the) ecay rate. Note that our efinition of ISES is a variation of the one in 34], which uses the max formulation. Moreover, we only require exponential ecay of the state variable ξ, asweare mainly intereste in the control performance regaring the plant an controller states, which are capture in ξ. In aition, we require that η stays boune by a KL-function for practical implementability. We o not put any constraint on the variables τ, κ, an l, as these are only use for moeling purposes. In case w L,wehave ξt) ce ρt ξ an ηt) β ξ,t), an thus, ISES implies GES in the absence of isturbances. Definition III.2: The CETC system 7) 8) is sai to have an L 2 -gain from w to z smaller than or equal to θ, if there exists a function δ K such that for any initial conition ξ) ξ R n ξ, τ), κ), l), η), an any sequence of elays {τ k } k N with τ k Dfor all k N,all corresponing solutions to 7) 8) with w L 2 satisfy z L2 δ ξ )+θ w L2. Before proceeing, we introuce the matrix Y R 2n y n ξ Cy D y C u D y D u Y : 12) I such that ζ y, s) Yξ, an the transformation matrix T R n ξ n x p +n x c +n y ) such that I I T : I I T x p x c s 13) x p x c s. 14) s A. Static CETC Before esigning the ynamics of 5) an analyzing ISES an L 2 -stability of the system 7) with the ynamic eventgenerator 8), we will first consier static continuous eventgenerators with time regularization of the form 1). To analyze ISES an L 2 -stability of the static CETC system 7) with 9) an 1), we will use the Lyapunov/storage function U given by Uξ,τ,κ,l,η) V ξ,τ,κ,l)+η 15) with V given by ξ P τ κ 1 τ)ξ, τ,τ κ] an l 1 V ξ,τ,κ,l) ξ P τ)ξ, τ τ κ,h] an l ξ P h)ξ, τ h, ) an l. 16) Here, P :,h] R n ξ n ξ is a continuously ifferentiable function with P τ) for τ,h], an for all D, P1 :,] R n ξ n ξ is a continuously ifferentiable function with P1 τ) for τ,]. The functions P,P1, D, will be chosen such that 15) becomes a storage function 35] 37] for the CETC system 7) with 9) an 1), with the supply rate θ 2 z z w w an ecay rate 2ρ. In particular, we will select the functions P,P1, D, to satisfy the Riccati ifferential equations τ P τ) RP τ)) 17) τ P 1 τ) RP1 τ)), D 18) where R enotes the Riccati operator RP ) A P PA Y N F Y 2ρP θ 2 C C PB + θ 2 C D)MB P + θ 2 D C). 19) Here, M : I θ 2 D D) 1 is assume to exist an to be positive efinite, which correspons to θ 2 > λ max D D), an

5 178 IEEE TRANSACTINS N AUTMATIC CNTRL, VL. 63, N. 1, JANUARY 218 N F R 2n y 2n y, N F, is an arbitrary matrix, which we will use as a esign parameter in Section III-B. Note that V given by 16) epens on the elay τ κ of the current transmission, an thus, the Lyapunov/storage function U is parametrize by the elay sequence {τ k } k N. However, as we will see below, base on this parametrize function U we are able to guarantee ISES an L 2 -stability for any sequence of elays {τ k } k N with τ k Dan D satisfying Assumption II.1. For ease of notation, we will not make this epenence on {τ k } k N explicit in 15) an 16). In orer to fin the explicit expressions for the functions P,P1, D, we introuce the Hamiltonian matrix A+ρI+θ 2 BMD C BMB H : C LC Y N F Y A+ρI+θ 2 BMD C ) in which L : θ 2 I DD ) 1, an we efine the matrix exponential F τ) :e Hτ F11 τ) F 12 τ). 2) F 21 τ) F 22 τ) To guarantee that solutions to τ P τ) RP τ)) are well efine on,h] an that solutions to τ P 1 τ) RP1 τ)) are well efine on,], D, we will make use of the following assumption, see also 22]. Assumption III.3: F 11 τ) is invertible for all τ,h]. Assumption III.3 can always be satisfie by choosing h sufficiently small, as F 11 ) I an F 11 is a continuous function. Note that larger h can be allowe by reucing ρ or increasing θ. In the elay-free case with perioic sampling an ρ,the L 2 -gain θ of the system 7) can be etermine exactly so without conservatism) by using a lifting-base approach as in, e.g., 25], 38]. These works also require that Assumption III.3 or an equivalent thereof) hols. Moreover, if an L 2 -gain θ cannot be achieve with perioic sampling with sampling perio h an without communication elays, this will also not be possible with event-triggere sampling with minimum inter-event time h. Hence, Assumption III.3 is not restrictive in that sense. The function P :,h] R n ξ n ξ is now explicitly efine for τ,h] by P τ) F 21 h τ)+f 22 h τ)p h)) F 11 h τ)+f 12 h τ)p h)) 1 21) an the functions P1 :,] R n ξ n ξ, D, are now explicitly efine for τ,] by P1 τ) F 21 τ)+f 22 τ)p1 ) ) F11 τ)+f 12 τ)p1 ) ) 1 22) where P h),p 1 ), D, are to be selecte. See 22], 39] for further etails. Before stating the next theorem, let us introuce the notation P P ), P P ), P h P h), P 1 P 1 ), P 1 P 1 ), the functions G τ) :F 11 τ) P h F 11 τ) 1 + F 21 τ)f 11 τ) 1 23) for τ,h], G 1τ) :F 11 τ) P1F 11 τ) 1 + F 21 τ)f 11 τ) 1 24) for τ,], D, an finally a matrix function S :,h] R n ξ n ξ that satisfies Sτ)Sτ) : F 11 τ) 1 F 12 τ) for τ,h]. A matrix Sτ) exists uner Assumption III.3, because this assumption guarantees that the matrix F 11 τ) 1 F 12 τ) is positive semiefinite 22]. Theorem III.4: If Assumption II.1 hols, an there exist matrices N F,N T,N N R 2n y 2n y with N F,N T,N N, P h,p1 Rn ξ n ξ with P h,p1, an scalars θ, ρ, β, μ R, D, such that Assumption III.3 hols an the matrix inequalities T A P h + P h A + Y N N βq)y )T B P h T T 2ρP h θ 2 C C)T θ 2 T C D 25) I θ 2 D D J G 1)J J F 11 ) P1 S) ] S) P1 S) Ph Y N T + μ Q)Y 26) I an J 1 G h )J 1 J1 F 11 h ) P h Sh ) Sh ) P h Sh ) P 1 27) I hol for all D, then the static CETC system 7) with 9) an 1) is ISES with ecay rate ρ, an has an L 2 -gain from w to z smaller than or equal to θ. Inequalities 25), 26), an 27) epen nonlinearly on the parameters N F, ρ, an θ. However, once these parameters are fixe, also the matrices M, L, F ij τ), an Sτ) become fixe matrices. The matrices G τ) an G 1τ), D, then only epen linearly on the matrices P h an P 1, D. Hence, inequalities 25), 26), an 27) then become linear matrix inequalities LMIs), an the parameters P h, P 1, N N, N T, β, an μ, D, can be synthesize numerically via semiefinite programming e.g., using Yalmip/SeDuMi 4] in MATLAB). The L 2 -gain estimate θ can be optimize via bisection when N F an ρ are fixe. Although the optimization is nonconvex an we shoul expect to fin local optima, goo results can be foun with proper initial estimates. The same hols for the ecay rate ρ when N F an θ are fixe. As the term Y N F Y in 19) leas to an extra ecrease ζ N F ζ in V uring flow 7a) with τ,h], an increase in N F will typically lea to an increase in θ or a ecrease in ρ. Hence, to analyze ISES an L 2 -stability of the static CETC system 7) with 9) an 1), it is often best to choose N F. However, for ynamic CETC it can sometimes be useful to choose N F, as we will see in Section III-B.

6 BRGERS et al.: RICCATI-BASED DESIGN F EVENT-TRIGGERED CNTRLLERS FR LINEAR SYSTEMS WITH DELAYS 179 B. Dynamic CETC In this section, we present our esign for a ynamic continuous event-generator, which follows from the analysis in Section III-A. The iea is as follows. In Section III-A, the function V an, hence, also the Lyapunov/storage function U) has an extra ecrease ζ N F ζ uring flow 7a) with τ,h], an an extra ecrease ζ N N βq)ζ uring flow 7a) with τ h, ) an ζ Qζ. Aitionally, V is often strictly ecreasing along jumps 7b), while we only require that U is nonincreasing along jumps 37]. To get even less conservative results, we can store this unnecessary ecrease of V in the ynamic variable η as much as possible), which acts as a buffer. When a transmission is necessary accoring to the static event-generator i.e., when τ h, ) an ζ Qζ )theterm ζ N N βq)ζ can become positive, in which case the function V will start to increase with the excess amount ζ N N βq)ζ if we o not transmit. However, as long as η>, we can compensate for this excess increase in V by reucing η, an thus, we can postpone the transmission until the buffer η becomes empty η ). As a result, the conservatism in the stability analysis is reuce, an the same L 2 -gain an ecay rate can be guarantee with typically less transmissions, as will also be emonstrate by a numerical example in Section VI-A. In this way, our esign will lea to a ynamic CETC system with the same L 2 -gain θ an ecay rate ρ as the static CETC system 7) with 9) an 1), but with a significant reuction in the number of transmissions. We select the flow ynamics 5a) of η as { 2ρη + ζ N F ζ, τ,h) 28a) Ψo) 2ρη + ζ N N βq)ζ, τ h, ). 28b) For the function η T, we provie the following two ifferent esigns: 1) State-base ynamic CETC: η T o) η + min D ξ P h J P1J ) ξ. 29) 2) utput-base ynamic CETC: η T o) η + ζ N T ζ. 3) Here, the scalars ρ an β, an the matrices N F, N T, N N, P h,p 1 R n ξ n ξ, D, follow from the stability analysis of the static CETC system in Theorem III.4. The first esign requires that the full state ξt k ) is known to the event-generator at transmission time t k. This is the case when Assumption II.1 hols an y x p,x c ) e.g., when C is a static state-feeback controller in which case y x p an n xc ), as then ŷt k )st k ), k N, an ξt k )Tζt k ). Note that when the set of possible elays D contains only a single element, a copy of the controller coul be inclue in the event-generator in orer to track the controller state x c. In case D contains multiple elements, this is not possible, as the exact input ŷ to the controller is then unknown. The secon esign is more conservative, but can also be use in case the event-generator oes not have access to the full state ξ. Hence, this choice can be use for output-base ynamic CETC. Theorem III.5: If the conitions of Theorem III.4 hol, then the ynamic CETC system 7) with 8), 28), an η T given by 29) or 3) is ISES with ecay rate ρ, an has an L 2 -gain from w to z smaller than or equal to θ. While the static continuous event-generator only has esign parameters h an Q, the state-base ynamic event-generator has esign parameters h, Q, ρ, N F, N N, β, P h, an P1, D, an the output-base ynamic event-generator has esign parameters h, Q, ρ, N F, N N, β, an N T. However, these extra esign parameters irectly follow from the ISES an L 2 -gain analysis of the static CETC system in Theorem III.4. Hence, the esign of these extra parameters follows irectly an naturally from the esign an stability analysis of the static event-generator. f course, manual tuning of one or more of these parameters is also possible, but can be ifficult given the large esign space. In contrast to the static CETC case, in the ynamic CETC case it can make sense to choose N F, asη grows with 28a) uring flow with τ,h], an thus, the average inter-event times might become larger when N F is increase. This inicates the presence of a traeoff between control performance an resource utilization, as an increase in N F typically also leas to an increase in θ or a ecrease in ρ. n the other han, as we can give har guarantees on the minimum inter-event time given by h), but not on the average inter-event time, it often makes more sense to choose N F an to make the traeoff between control performance an resource utilization via the parameter h. Remark III.6: Even though we will often choose N F, the parameter N F can be useful in some cases. For example, in our work 17], we chose N F in orer to moel the ynamic continuous event-generator of 2] in our propose new framework. Remark III.7: In 2], we were able to guarantee GAS an L p -stability for all possible elays τ k,τ ma ], where τ ma R is the maximum allowable elay. In contrast, here we guarantee ISES an L 2 -stability for a finite set D of possible elays. As an engineering solution, we can approach the situation of 2] by incluing sufficiently many elays out of the set,τ ma ] in the set D griing). Assuming that the ETC system has a small amount of robustness against eviations of the elays from the set D, this coul also lea to a stability guarantee for all possible elays τ k,τ ma ], k N, cf., 41], which uses a similar approach for the stability analysis of networke control system with varying transmission intervals. IV. MAIN RESULTS FR THE PETC CASE Consier again the control setup of Fig. 1 with plant P given by 1) an controller C given by 2). Instea of continuously monitoring the output y which is sometimes ifficult to realize in igital implementations), we now perioically sample the output y at sample times {s n } n N given by s n nh, where h R > is the sample perio. At each sample time s n, n N,

7 18 IEEE TRANSACTINS N AUTMATIC CNTRL, VL. 63, N. 1, JANUARY 218 the event-generator ecies whether the sample output shoul be transmitte to the controller or not. In the PETC case, the ynamic variable η will evolve accoring to t ηt) Ψôt)), t R \{s n } n N 31a) ηt + )η T ôt)), t {t k } k N 31b) ηt + )η N ôt)), t {s n } n N \{t k } k N 31c) where the functions Ψ, η T, an η N are to be esigne an where the signal ô : R R 2n y,h] N {, 1} R given by ôt) :ζs n ),τt),κt),lt),ηt)), t s n,s n+1 ] 32) is the information that is available to the perioic eventgenerator. We can now escribe the close-loop system as ξt) Aξt)+Bwt) τt) 1 t R t κt), t/ {s n } n N lt) t/ {t k + τ k } k N ηt) Ψôt)) 33a) ξt + ) J ξt) τt + ) l κt + ) κt), lt + t {t k } k N ) 1 33b) ηt + ) η T ôt)) ξt + ) J 1 ξt) τt + ) τt) l 1 κt + ) κt)+1, lt + t {t k + τ k } k N ) 33c) ηt + ) ηt) ξt + ) ξt) τt + ) l κt + ) κt), t {s n } n N lt + ) lt) t/ {t k } k N 33) ηt + ) η N ôt)) zt) Cξt)+Dwt). 33e) In this moel, τ,h] tracks the time that has elapse since the last sample time, in contrast with the CETC case, in which τ tracke the time since the last transmission. All other variables have the same interpretation as in the CETC case. Furthermore, 33b) moels the jumps at transmission times t k, k N, 33c) moels the jumps at upate times t k + τ k, k N when the transmitte ata arrives at the controller), 33) moels the jumps at sample times s n t k, n, k N, at which no transmission occurs, an 7a) moels the flow in between jumps. In the PETC case, the sequence of transmission times {t k } k N will be generate by the ynamic perioic eventgenerator t,t k+1 min{t >t k η N ôt)) ζ t)qζt),t {s n } n N }. 34) As in the CETC case, the moel 31), 34) can also capture static perioic event-generators by choosing η) an Ψô) η T ô) η N ô) 35) for all ô R 2n y,h] N {, 1} R, as then we have that ηt) for all t R, an the ynamic perioic eventgenerator 34) reuces to the static perioic event-generator t,t k+1 min{t >t k ζt) Qζt),t {s n } n N }. 36) Definitions for ISES an L 2 -stability of the PETC system 33), 34) can be given mutatis mutanis, but are omitte for space reasons. A. Static PETC As in the CETC case, we will first consier static perioic event-generators of the form 36). To analyze ISES an L 2 -stability of the static PETC system 33) with 35) an 36), we will again use the Lyapunov/storage function U given by 15), but now with V given by { ξ P τ κ 1 V ξ,τ,κ,l) τ)ξ, τ,τ κ] an l 1 ξ P τ)ξ, τ,h] an l 37) where again P :,h] R n ξ n ξ is a continuously ifferentiable function satisfying 17) an for all D, P1 :,] R n ξ n ξ is a continuously ifferentiable function satisfying 18). Theorem IV.1: If Assumption II.1 hols, an there exist matrices N F,N T,N N R 2n y 2n y with N F,N T,N N, P h,p1 Rn ξ n ξ with P h,p1, an scalars θ, ρ, β, μ R, D, such that Assumption III.3 hols an inequalities 26), 27), an T G h)t T F 11 h) P h Sh) Sh) P h Sh) T P h Y N N βq)y ) ] T 38) I hol for all D, then the static PETC system 33) with 35) an 36) is ISES with ecay rate ρ, an has an L 2 -gain from w to z smaller than or equal to θ. Note that Theorem IV.1 extens 22, Th. III.2] to the case with elays. When particularize to the elay-free case, Theorem IV.1 becomes equivalent to 22, Th. III.2], see also 31].

8 BRGERS et al.: RICCATI-BASED DESIGN F EVENT-TRIGGERED CNTRLLERS FR LINEAR SYSTEMS WITH DELAYS 181 B. Dynamic PETC In this section, we present our esign for a ynamic perioic event-generator, which follows from the analysis in Section IV-A. As in Section III-B, the iea is to store the extra ecrease of V in the ynamic variable η, which is then use to reuce the number of transmissions while maintaining the same L 2 -gain θ an ecay rate ρ as the static PETC system. We select the flow ynamics 31a) of η as Ψô) 2ρη, for τ,h]. 39) Remark IV.2: As η + η at upate times t k + τ k, k N, an Ψ is given by 39), it follows that ηs n+1 )e 2ρh ηs + n ). Thus, since the event-generator only nees to know the value of η at sample times s n, n N, the variable η oes not nee to continuously evolve accoring to 31a) in the event-generator. Instea, we can use the iscrete-time ynamics just escribe. For the functions η T an η N, we will provie the following two ifferent esigns: 1) State-base ynamic PETC: η T ô) η + min D ξ P h J P1J ) ξ η N ô) η + ξ P h P )ξ. 2) utput-base ynamic PETC: η T ô) η + ζ N T ζ η N ô) η + ζ N N βq) ζ. 4a) 4b) 41a) 41b) Here, the scalars ρ an β, an the matrices N T, N N, P,P h,p1 R n ξ n ξ, D, follow from the stability analysis of the static PETC system in Theorem IV.1. The first esign requires that the full state ξs n ) is known to the event-generator at sample times s n, n N. The secon esign is more conservative, but can also be use in case the event-generator oes not have access to the full state ξ. Hence, this choice can be use for output-base ynamic PETC. Theorem IV.3: If the conitions of Theorem IV.1 hol, then the ynamic PETC system 33) with 34), 39), an η T an η N given by 4) or 41) is ISES with ecay rate ρ, an has an L 2 -gain from w to z smaller than or equal to θ. Remark IV.4: Note that in contrast to the CETC case, 39) oes not inclue the term ζ N F ζ, as in the PETC case we o not continuously measure the output y. Hence, our ynamic PETC esigns o not involve the matrix N F, an we can simply let N F in Theorems IV.1 an IV.3. The matrix N N appears linearly in the LMI 38), an thus can be easily synthesize numerically via semiefinite programming. V. REDUCED CNSERVATISM USING STATE-SPACE PARTITINING The ISES an L 2 -gain analysis in Sections III an IV are base on the common timer-epenent) quaratic Lyapunov function V, in the sense that the same matrix functions P an P τ κ 1 are use for all ξ R n ξ. In this section, we present even less conservative conitions to analyze ISES an L 2 -stability of the propose CETC an PETC systems, using a piecewise quaratic Lyapunov/storage functional technique as propose in 23] for static PETC systems without elays. Define the regions X i : {ξ R n ξ X i ξ },i {1, 2,...,N},N N 42) where the matrices X i R n ξ n ξ, i {1, 2,...,N} are such that {X 1, X 2,...,X N } forms a partition of R n ξ, i.e., the sets X i, i {1, 2,...,N} have nonempty interior, N i1 X i R n ξ, an X i X j is of zero measure for all i j, i, j {1, 2,...,N}. A. Continuous Event-Triggere Control Consier the CETC system 7), 8), an efine the functional V ξ,τ,κ,l,w,t) max ξ τ P κ i,1 τ)ξ, τ,τ κ], l1 i {1,2,...,N } s.t. ξt κ +τ κ t,ξ,w t ) X i max ξ P i,τ)ξ, τ τ κ,h], l i {1,2,...,N } s.t. ξt κ +1 t,ξ,w t ) X i max ξ P i,h)ξ, τ h, ), l i {1,2,...,N } s.t. ξt κ +1 t,ξ,w t ) X i 43) where w t : R R n w enotes the time-shifte signal given by w t s) ws + t) for s, an ξt, ξ, w) enotes the solution to ξ t A ξ + Bw at time t with initial conition ξ) ξ an isturbance signal w : R R n w. Furthermore, P i, :,h] R n ξ n ξ, i {1, 2,...,N} are continuously ifferentiable functions satisfying 17) an Pi,1 :,] Rn ξ n ξ, D, i {1, 2,..., N}, are continuously ifferentiable functions satisfying 18). Note that V given by 43) epens on the value of ξt k + τ k ) for t t k,t k + τ k ] when l 1as in this interval the inex i epens on the value ξt k + τ k )), an epens on the value of ξt k+1 ) for t t k,t k+1 ] when l, as in this interval the inex i epens on the value ξt k+1 ). Hence, V epens not only on the elay sequence {τ k } k N, but also on future values of the isturbance w. As such, we have a trajectory/isturbanceepenent Lyapunov/storage functional, which eviates from the common literature on GES an L 2 -gain analysis, as usually the Lyapunov/storage function only epens on the current an sometimes past) values of the state, but typically not on future values. Even though the interpretation of U V + η as a genuine storage function is now less natural, we will see below that we are still able to prove ISES an L 2 -stability for any sequence of elays {τ k } k N with τ k Dan D satisfying Assumption II.1 an any isturbance w : R R n w with w L 2. Before efining the ynamics of η an stating our next theorem, let us introuce the notation P i, P i, ), P i, P i, ), P i,h P i, h), Pi,1 P i,1 ), P i,1 P i,1 ), an the functions G i, τ) :F 11 τ) P i,h F 11 τ) 1 + F 21 τ)f 11 τ) 1 44)

9 182 IEEE TRANSACTINS N AUTMATIC CNTRL, VL. 63, N. 1, JANUARY 218 for τ,h], i {1, 2,...,N}, an G i,1τ) :F 11 τ) Pi,1F 11 τ) 1 + F 21 τ)f 11 τ) 1 45) for τ,], i {1, 2,...,N}, D. For the ynamics of η, we again select Ψ as in 28). For the state-base ynamic continuous event-generator, we now select η T as η T o) η + max ξ P i,h J Pj,1J ) ξ 46) i {1,2,...,N } s.t. ξ X i min D j {1,2,...,N } an for the output-base ynamic continuous event-generator, we again select η T as in 3). To unerstan 46), some comments are in orer. At the transmission time t k, we know which matrix function P i,, i {1, 2,..., N} we shoul have use in the interval t t k 1 + τ k 1,t k ]. However, we o not know yet which matrix function Pj,1, j {1, 2,...,N}, D, we shoul use in the interval t t k,t k + τ k ]. Hence, to make sure that the Lyapunov/storage function U V + η oes not increase along transmissions 7b), we therefore take the minimum over all j {1, 2,...,N} an all Din 46). Theorem V.1: If Assumption II.1 hols, an there exist matrices N F,N T,N N R 2n y 2n y with N F,N T,N N, P i,h,pi,1 Rn ξ n ξ with P i,h,pi,1, U ij,w ij R n ξ n ξ with Uij U ij an W ij Wij, an scalars β an μ ij, D, i, j {1, 2,...,N}, such that Assumption III.3 hols an the matrix inequalities T A P i,h + P i,h A + Y N N βq)y )T B P i,h T T 2ρP i,h θ 2 C C)T θ 2 T C D I θ 2 D D J G i,1 )J ] J F 11 ) P i,1 S) 47) S) Pi,1 S) Pj,h Y N T + μ ij Q)Y X j U ij X ] j 48) I an J 1 G i, h )J 1 J1 F 11 h ) P i,h Sh ) Sh ) P i,h Sh ) P j,1 Xj W ij X ] j 49) I hol for all Dan all i, j {1, 2,...,N}, then the ynamic CETC system 7) with 8), 28), an 46) or 3) is ISES with ecay rate ρ, an has an L 2 -gain from w to z smaller than or equal to θ. Corollary V.2: If the conitions of Theorem V.1 hol, then the static CETC system 7) with 9) an 1) is ISES with ecay rate ρ, an has an L 2 -gain from w to z smaller than or equal to θ. Remark V.3: When increasing the number of regions N,the ISES an L 2 -gain analysis becomes less conservative at the cost of higher computational complexity. Moreover, the upate 46) becomes more computationally intensive, leaing to more complex event-generators. n the other han, 3) is inepenent of N, an thus, o not lea to more complex event-generators when increasing N. B. Perioic Event-Triggere Control Consier the PETC system 33), 34), an efine the functional V ξ,τ,κ,l,w,t) max ξ τ P κ i,1 i {1,2,...,N } τ)ξ, τ,τ κ], l1 s.t. ξτ κ τ,ξ,w t ) X i max i {1,2,...,N } s.t. ξh τ,ξ,w t ) X i ξ P i,τ)ξ, τ,h], l 5) where again P i, :,h] R n ξ n ξ, i {1, 2,...,N} are continuously ifferentiable functions satisfying 17), an Pi,1 :,] R n ξ n ξ, D, i {1, 2,...,N} are continuously ifferentiable functions satisfying 18). For the ynamics of η, we again choose Ψ as in 39). For the state-base ynamic perioic event-generator, we now select η T an η N as η T ô) η + max η N ô) η i {1,2,...,N } s.t. ξ X i + max i {1,2,...,N } s.t. ξ X i min D j {1,2,...,N } ξ P i,h J P j,1j ) ξ 51a) min j {1,2,...,N } ξ P i,h P j, ) ξ 51b) an for the output-base ynamic perioic event-generator, we again select η T an η N as in 41). Theorem V.4: If Assumption II.1 hols, an there exist matrices N F,N T,N N R 2n y 2n y with N F,N T,N N, P i,h,pi,1 Rn ξ n ξ with P i,h,pi,1, U ij,w ij,v ij R n ξ n ξ with Uij U ij, W ij Wij, an Vij V ij, an scalars β an μ ij, D, i, j {1, 2,...,N}, such that Assumption III.3 hols an the matrix inequalities 48), 49), an T P j,h Y N N βq)y Xj V ) ] ijx j T I T G i, h)t T F 11 h) P i,h Sh) 52) Sh) P i,h Sh) hol for all Dan all i, j {1, 2,...,N}, then the ynamic PETC system 33) with 34), 39), an 51) or 41) is ISES with ecay rate ρ, an has an L 2 -gain from w to z smaller than or equal to θ.

10 BRGERS et al.: RICCATI-BASED DESIGN F EVENT-TRIGGERED CNTRLLERS FR LINEAR SYSTEMS WITH DELAYS 183 Corollary V.5: If the conitions of Theorem V.4 hol, then the static PETC system 33) with 35) an 36) is ISES with ecay rate ρ, an has an L 2 -gain from w to z smaller than or equal to θ. VI. NUMERICAL EXAMPLES In 31], we have alreay shown that in the elay-free case our new ynamic PETC esigns provie the same control performance guarantees with less communication than the static PETC esigns of 22], 25]. Here, we will emonstrate our static an ynamic CETC an PETC esigns for the case with elays via two numerical examples. A. Unstable Batch Reactor Consier the unstable batch reactor of 32], 42], 43], with n xp 4, n xc 2, n y n w n u n z 2, an plant an controller ynamics given by 1) an 2) with A p B p ,B 5 pw ] 1 2 ] C y C z 1,C u 8 ] D y D z D zw A c ] 1 ] 2 B c 1, an D u 5 an D {.1,.125,.15,.175,.2}. Note that for this system y x p,x c ), an thus we cannot use 29), 4), 46), or 51), but we have to resort to 3) for the ynamic CETC case an to 41) for the ynamic PETC case. We choose h.1, Q given by 11), N F, an ρ.5. For each choice of σ, we minimize the L 2 -gain θ, using Theorem III.4 for the CETC case an Theorem IV.1 for the PETC case, from which also the matrices N T an N N follow. Fig. 2a) shows the guarantee L 2 -gain θ as a function of σ for both the CETC an PETC approaches. Fig. 2b) shows the average inter-event times τ avg total number of events)/simulation time) for the static an output-base) ynamic event-generators, which have been obtaine by simulating the system 1 times for 4 time units with ξ) an Fig. 2. Guarantee L 2 -gain θ for varying σ a), average inter-event times τ avg for isturbance w given by 53) an ifferent event-generators b), an actual ratio z L2 / w L2 for isturbance w given by 53) c). isturbance w given by wt) e.2t 5 sin3.5t) cos3t) ]. 53) Finally, Fig. 2c) shows the actual ratio z L2 / w L2 for isturbance w given by 53), which has been obtaine from the same simulations. In Fig. 2c) we see that, while the control performance guarantees for the ynamic CETC an PETC systems are ientical to the performance guarantees for their static counterparts, the ynamic event-generators exploit part of) the conservatism in the L 2 -gain analysis of Theorems III.4 an IV.1 to postpone

11 184 IEEE TRANSACTINS N AUTMATIC CNTRL, VL. 63, N. 1, JANUARY 218 the transmissions. This leas to higher ratios z L2 / w L2 but still below the guarantee bouns in Fig. 2b)), but also to consistently larger τ avg, as can be seen in Fig. 2b). To compare these results with 2], note that for given ρ an θ the waiting time h or τ MIET in the terminology of 2]) of the continuous event-generator propose in 2] cannot excee the maximally allowable transmission interval MATI) of 32]. Moreover, for the same example in 32, Sec. IV] we can calculate that when using the sample-ata protocol, no notion of stability can be guarantee for MATI larger than.63, even without elays. In contrast, here we guarantee ISES an L 2 -stability for h.1, in the presence of elays. Hence, our new framework tailore to linear systems is clearly much less conservative than our previous results for nonlinear systems in 2]. See also 17] for a irect comparison between the static an ynamic continuous event-generators in Section III an the event-generators propose in 2], for the case without elays. B. Reuce Conservatism Using Section V In this example, we show how the conservatism of the ISES an L 2 -stability analysis can be further reuce by partitioning the state-space as in Section V. Consier the example from 1, Sec. VI-B], with n xp 2, n xc, n y n w n u n z 1, an matrices 1 A p,b p B pw,d u 3 ] 3 1 C y C z 1 ], an D y D z D zw ]. We will control the system using a perioic event-generator, an choose h 1, D {,.1,.2}, Q as in 11), N F, an ρ.5. We partition the state-space as in 42), with N 1 an matrices sinφ i ) cosφ i ) sinφ X i i+1 ) cosφ i+1 ) 54) with φ i i 1) 2π, for i {1, 2,...,N +1}. 55) N Fig. 3 shows the guarantee L 2 -gain θ as a function of σ for the PETC approach using Theorem IV.1, an using the less conservative conitions of Theorem V.4 with the state-space partition as efine above. The matrices N T an N N an the scalar β were foun numerically base on Theorem IV.1. In Theorem V.4 we then use the same values for N N, N T, an β, such that the resulting esign for the output-base ynamic perioic event-generator is ientical for both theorems, an any ifference in θ can be solely attribute to the partitioning of the state-space an the use of piecewise quaratic Lyapunov functions. Clearly, by using piecewise quaratic Lyapunov/storage functionals even tighter guarantees on the L 2 -gain can be achieve. Fig. 3. Guarantee L 2 -gain θ for varying σ. However, while the performance guarantees become tighter when the number of regions N in the partition is increase, also the computational complexity of the require calculations becomes larger. VII. CNCLUSIN We propose a new metho for the esign of static an ynamic continuous event-generators which require continuous measuring of the plant output) an static an ynamic perioic event-generators which only require perioic sampling of the plant output) for linear control systems with communication elays. All propose event-generators lea to GES an L 2 -stability with guarantee ecay rates an L 2 -gains, an have a guarantee positive minimum inter-event time. ur esigns exploit Riccati-base tools tailore to linear systems, leaing to a significant reuction in conservatism compare to existing results in the literature which focus on more general nonlinear systems). In fact, we showe via a numerical example that the conservatism in the guarantee L 2 -gain for any of the propose event-generators) is small. Moreover, to the best of the authors knowlege, the propose ynamic perioic event-generators are the first in literature that can eal with communication elays. APPENDIX Proof of Theorem III.4: The proof is base on the storage function U given by 15) with V as efine in 16), P satisfying 17), an P1, D, satisfying 18). However, we only nee to consier the function V, as it hols that ηt) for all t R cf., 9)) an thus in this case U V. The proof consists of showing that V is a proper storage function an satisfies for all ξ R n ξ, τ R, κ N, an all l {, 1}, c 1 ξ 2 V ξ,τ,κ,l) c 2 ξ 2 56) with c 2 c 1 >, has a supply rate θ 2 z z w w 35], 36] an ecay rate 2ρ uring flow 7a), an is nonincreasing along jumps 7b) an 7c). The first property follows from Assumption III.3, as this assumption guarantees that P τ) for all τ,h] an P 1 τ) for all τ,], D, see 22], 39]. Hence, 56)

12 BRGERS et al.: RICCATI-BASED DESIGN F EVENT-TRIGGERED CNTRLLERS FR LINEAR SYSTEMS WITH DELAYS 185 hols with c 1 min min λ minp τ)), min τ,h] D τ,] λ min P 1 τ) ) 55a) c 2 max max λ maxp τ)), max λ max P 1 τ) ) τ,h] D τ,] 55b) where c 2 c 1 >. For brevity, we will use the notation V t) V ξt),τt), κt),lt)) in the remainer of the proof. Following the erivations in the proof of 22, Th. III.2], it can be shown that 17), 18), an 19) imply that t V t) 2ρV t) θ 2 zt) zt) + wt) wt) ζt) N F ζt) 58) uring flow 7a) with τ,τ κ ] an l 1)orτ,h] an l ). Aitionally, uring flow 7a) with τ h, ), l, an ζ Qζ, it hols that ŷ s, which implies that x p t) x p t) x c t) ŷt) T x c t) when t t k + h, t k+1 ] st) st) an thus, it follows from 25) that t V t) 2ρV t) θ 2 zt) zt) +wt) wt) ζt) N N βq)ζt). 59) Equations 58) an 59) together with N F,N N, β, an ζt) Qζt) for t t k + h, t k+1 ] show that t V t) 2ρV t) θ 2 zt) zt)+wt) wt) 6) hols uring flow 7a), proving the secon property. Finally, we show that V oes not increase along jumps. In 22], it is shown that 21) evaluate at τ, D, leas to P G h )+F 11 h ) P h Sh ) I Sh ) P h Sh ) ) 1 Sh ) an for D, 22) evaluate at τ, leas to P h ) F11 h ) 1 61) P1 G 1)+F 11 ) P1S) I S) P1S) ) 1 S) P1 ) F11 ) 1. 62) Here, the existence of I Sh ) P h Sh ) ) 1 an I S) P1 S)) 1 is guarantee by Assumption III.3 an P h,p1, D, cf., 22]. By applying a Schur complement it follows from 26), N T, an μ for all D that along transmissions 7b) when τ h, ), l, an ζ Qζ ) wehave V t + )ξt) J P τ κ t ) 1 J ξt) ξt) P h ξt) ζt) N T + μ τ κ t ) Q)ζt) ξt) P h ξt) V t) 63a) 63b) an it follows from 27) that along upates 7c) when τ τ κ an l 1), we have V t + ) ξt) J 1 P τ κ t ) J 1 ξt) ξt) P τ κ t ) 1τ κ t ) ξt) V t). 64) Combining 56), 6), 63b), an 64) establishes the upper boun θ on the L 2 -gain of the ETC system 7), 1) 22]. Furthermore, it follows that t V t) e 2ρt V ) + e 2ρt s) w 2 L s e 2ρt V ) + 1 2ρ 1 e 2ρt ) w 2 L an thus e 2ρt V ) + 1 2ρ w 2 L ξt) ce ρt ξ) +2ρc 1 ) 1/2 w L 65) with c c 1 /c 2, which proves that the system is ISES with ecay rate ρ. Proof of Theorem III.5: Consier again the Lyapunov/ storage function U given by 15) with V as efine in 16). First, we show that U is a proper storage function, by showing that ηt) for all t R, an that U satisfies for all ξ R n ξ, τ R, κ N, l {, 1}, an all η, c 1 ξ 2 + η Uξ,τ,κ,l,η) c 2 ξ 2 + η 66) where c 1 an c 2 are given by 57). As η), it follows from 28a), N F, an the comparison lemma 44, Lemma 3.4] that ηt) for all t,h). Next,η flows accoring to 28b) on h, t 1 ), i.e., as long as η or ζ Qζ see 8)). However, note that η can only become negative when η an ζ Qζ > see 28b) as N N ), in which case a transmission 7b) woul be triggere. Hence, ηt 1 ).Therelation η T ot 1 )) then follows from 63b) when η T is given by 29), or from N T when η T is given by 3). Hence, in both cases it hols that ηt + 1 ). It now follows by inuction that ηt) for all t R. Property 66) then follows by combining 56) an 15). It remains to show that U has a supply rate θ 2 z z w w an ecay rate 2ρ uring flow 7a), an is nonincreasing along jumps 7b) an 7c). For brevity, we will use the notation Ut) Uξt),τt),κt),lt),ηt)) an V t) V ξt),τt),κt),lt)) in the remainer of the proof.

13 186 IEEE TRANSACTINS N AUTMATIC CNTRL, VL. 63, N. 1, JANUARY 218 From 19) an 28a), it follows using 58)) that t Ut) 2ρV t) 2ρηt) θ 2 zt) zt)+wt) wt) 2ρUt) θ 2 zt) zt)+wt) wt) 67) hols uring flow 7a) with τ,h], an from 19) an 28b) it follows using 59)) that 67) hols uring flow 7a) with τ h, ). Finally, we show that Ut + ) Ut) 68) hols along jumps. When using 29), we fin along transmissions 7b) that cf., 63)) Ut + )ξt) J P τ κ t ) 1 J ξt) + ηt) + min D ξt) P h J P1J ) ξt) ξt) P h ξt)+ηt) Ut). Alternatively, when using 3), we fin using 63a) an μ τ κ t ) ζt) Qζt) ) along transmissions 7b) that Ut + )ξt) J P τ κ t ) 1 J ξt)+ηt)+ζt) N T ζt) ξt) P h ξt)+ηt) Ut). Along upates 7c), 68) follows from 64) which follows from 27)) an ηt + )ηt). Equations 66) 68) together prove that the system has an L 2 -gain from w to z smaller than or equal to θ 35], 36]. Furthermore, it follows that Ut) V t)+ ηt) e 2ρt V ) + e 2ρt η) + t e 2ρt V ) + e 2ρt η) + 1 2ρ w 2 L an, since η), it follows that e 2ρt s) w 2 L s ξt) ce ρt ξ) +2ρc 1 ) 1/2 w L, an 69) ηt) c 2 e 2ρt ξ) 2 +2ρ) 1 w 2 L 7) with c c 1 /c 2, which proves that the system is ISES with ecay rate ρ. Proof of Theorem IV.1: Consier the Lyapunov function U given by 15), with V given by 37). As in the proof of Theorem III.4, we only nee to consier the function V,as it hols that ηt) for all t R, an thus, in this case U V. In the proof of Theorem III.4 it is shown that 56) hols with c 1 an c 2 given by 57), that 6) hols uring flow 33a), that 63b) hols along transmissions 33b), an that 64) hols along upates 33c), It remains to show that V is ecreasing along nontransmission jumps 33). From 61), it follows that 21) evaluate at τ leas to P G h)+f 11 h) P h Sh) I Sh) P h Sh) ) 1 Sh) P h ) F11 h) 1.71) By applying a Schur complement it follows from 38) an ŷt + )ŷt) st) st + ) that along jumps 33) when τ h an ζ Qζ ), we have V t + )ξt) P ξt) ξt) P h ξt) ζt) N N βq)ζt) ξt) P h ξt) V t) 72a) 72b) as N N. Similar arguments as in the proof of Theorem III.4 lea to ISES with ecay rate ρ an L 2 -stability with L 2 -gain θ. Proof of Theorem IV.3: Consier again the Lyapunov function U given by 15), with V given by 37). First, we show that U is a proper storage function, by showing that ηt) for all t R, an that U satisfies 66) for all ξ R n ξ, τ,h], κ N, l {, 1}, an all η. As η), it follows from 39) that ηt) for all t,h], an hence, that ηs 1 ). Next, given event-generator 34), a transmission 33b) occurs in case ζs 1 ) Qζs 1 ) an η N ôs 1 )). In this case, η T ôs 1 )) follows from 63b) when η T is given by 4a), or from N T when η T is given by 41a). therwise, if ζs 1 ) Qζs 1 ) < or η N ôs 1 )) >, no transmission occurs, an the state jumps accoring to 33). bserve however that when ζs 1 ) Qζs 1 ) < it hols that η N ôs 1 )), which follows from 72b) when η N is given by 4b), or from N N an β when η N is given by 41b). Hence, in all cases it hols that ηs + 1 ). It now follows by inuction that ηt) for all t R. Property 66) then follows by combining 56) an 15), where c 1 an c 2 are given by 57). From 19) an 39), it follows that 67) hols uring flow 33a), an thus, that U has a supply rate θ 2 z z w w an ecay rate 2ρ uring flow. It remains to show that 68) hols along jumps. For transmissions 33b) an upates 33c), this has alreay been shown in the proof of Theorem III.5, as the functions 4a) an 29), an 41a) an 3) are ientical. Along jumps 33), we fin when using 4b) that Ut + )ξt) P ξt)+ηt)+ξt) P h P ) ξt) ξt) P h ξt)+ηt) Ut). Alternatively, when using 41b), we fin using 72a) which also hols along jumps 33) when τ h an ζ Qζ > ) that Ut + )ξt) P ξt)+ηt)+ζt) N N βq)ζt) ξt) P h ξt)+ηt) Ut). This completes the proof. Proof of Theorem V.1: The proof irectly follows from the proof of Theorem III.5 by using V as efine in 43) an noting that for all D an all i {1, 2,...,N} it hols

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15 188 IEEE TRANSACTINS N AUTMATIC CNTRL, VL. 63, N. 1, JANUARY 218 Niek Borgers S 14) receive the M.Sc. egree cum laue) in mechanical engineering an the Ph.D. egree in control theory from the Einhoven University of Technology TU/e), Einhoven, The Netherlans, in 213 an 217, respectively. His research interests inclue hybri ynamical systems, networke control systems, an event-triggere control. Victor Dolk S 15) receive the M.Sc. egree cum laue) in mechanical engineering from the Einhoven University of Technology TU/e), Einhoven, The Netherlans, in 213, where he is currently working towar the Ph.D. egree. His research interests inclue hybri ynamical systems, networke control systems, intelligent transport systems, an event-triggere control. Maurice Heemels M 7 SM 1 F 16) receive the M.Sc. egree in mathematics an the Ph.D. egree in control theory both cum laue) from the Einhoven University of Technology TU/e), Einhoven, The Netherlans, in 1995 an 1999, respectively. From 2 to 24, he was with the Electrical Engineering Department, TU/e an from 24 to 26 with the Embee Systems Institute. Since 26, he has been with the Department of Mechanical Engineering, TU/e, where he is currently a Full Professor. He hel visiting professor positions at the Swiss Feeral Institute of Technology ETH), Switzerlan in 21 an at the University of California at Santa Barbara in 28. In 24, he worke also at the Company cé, The Netherlans. His current research interests inclue hybri an cyber-physical systems, networke an event-triggere control systems an constraine systems incluing moel preictive control. Dr. Heemels serve/s on the eitorial boars of Automatica, Nonlinear Analysis: Hybri Systems, Annual Reviews in Control, an IEEE TRANSACTINS N AUTMATIC CNTRL. He receive a personal VICI grant aware by STW Dutch Technology Founation).