Sampled-data implementation of derivative-dependent control using artificial delays

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1 1 Sampled-data implementation of derivative-dependent control using artificial delays Anton Selivanov and Emilia Fridman, Senior Member, IEEE arxiv: v1 [mat.oc] 17 Jan 2018 Abstract We study a sampled-data implementation of linear controllers tat depend on te output and its derivatives. First, we consider an LTI system of relative degree r 2 tat can be stabilized using r 1 output derivatives. Ten, we consider PID control of a second order system. In bot cases, te Euler approximation is used for te derivatives giving rise to a delayed sampled-data controller. Given a derivative-dependent controller tat stabilizes te system, we sow ow to coose te parameters of te delayed sampled-data controller tat preserves te stability under fast enoug sampling. Te maximum sampling period is obtained from LMIs tat are derived using te Taylor s expansion of te delayed terms wit te remainders compensated by appropriate Lyapunov-Krasovskii functionals. Finally, we introduce te event-triggering mecanism tat may reduce te amount of sampled control signals used for stabilization. I. INTRODUCTION Control laws tat depend on output derivatives are used to stabilize systems wit relative degrees greater tan one. To estimate te derivatives, wic can ardly be measured directly, one can use te Euler approximation ẏ (y(t) y(t τ))/τ. Tis replaces te derivative-dependent control wit te delaydependent one [2] [5]. It as been sown in [6] tat suc approximation preserves te stability if τ > 0 is small enoug. Similarly, te output derivative in PID controller can be replaced by its Euler approximation. Te resulting controller was studied in [7] and [8] using te frequency domain analysis. In tis paper, we study sampled-data implementation of te delay-dependent controllers. For double-integrators, tis as been done in [9] using complete Lyapunov-Krasovskii functionals wit a Wirtinger-based term and in [10] via impulsive system representation and looped-functionals. Bot metods lead to complicated linear matrix inequalities (LMIs) containing many decision variables. In tis paper, we obtain simpler LMIs for more general systems and prove teir feasibility for small enoug sampling periods. A simple Lyapunov-based metod for delay-induced stabilization was proposed in [11], [12]. Te key idea is to use te Taylor s expansion of te delayed terms wit te remainders in te integral form tat are compensated by appropriate terms in te Lyapunov-Krasovskii functional. Tis leads to simple LMIs feasible for small delays if te derivative-dependent controller stabilizes te system. In tis paper, we study sampled-data implementation of two types of derivative-dependent controllers. In Section II, we consider an LTI system of relative degree r 2 tat can be Te autors are wit te Scool of Electrical Engineering, Tel Aviv University, Israel ( antonselivanov@gmail.com; emilia@eng.tau.ac.il). Supported by Israel Science Foundation (grant No. 1128/1). Preliminary results ave been presented in [1]. stabilized using r 1 output derivatives. In Section III, we consider PID control of a second order system. In bot cases, te Euler approximation is used for te derivatives giving rise to a delayed sampled-data controller. Assuming tat te derivativedependent controller exponentially stabilizes te system wit a decay rate α > 0, we sow ow to coose te parameters of its sampled-data implementation tat exponentially stabilizes te system wit any decay rate α < α if te sampling period is small enoug. Te maximum sampling period is obtained from LMIs tat are derived using te ideas of [11], [12]. Finally, we introduce te event-triggering mecanism tat may reduce te amount of sampled control signals used for stabilization [13] [17]. In te preliminary paper [1], we studied delayed sampled-data control for systems wit relative degree two. Notations: N 0 = N {0}, 1 r = [1, 1,..., 1] T R r, I l R l l is te identity matrix, stands for te Kronecker product, x = max{n N n x} for x R, col{a 1,..., a r } denotes te column vector composed from te vectors a 1,..., a r. For p R, f() = O( p ) if tere exist positive M and 0 suc tat f() M p for (0, 0 ). Auxiliary lemmas Lemma 1 (Exponential Wirtinger inequality [18]): Let f H 1 (a, b) be suc tat f(a) = 0 or f(b) = 0. Ten b a e2αt f T (t)w f(t) dt e 2 α (b a) (b a) 2 π 2 b a e2αt f T (t)w f(t) dt for any α R and 0 W R n n. Lemma 2 (Jensen s inequality [19]): Let ρ: [a, b] [0, ) and f : [a, b] R n be suc tat te integration concerned is well-defined. Ten for any 0 < Q R n n, [ b a ρ(s)f(s) ds ] T Q [ b a ρ(s)f(s) ds ] b a ρ(s) ds b a ρ(s)f T (s)qf(s) ds. II. DERIVATIVE-DEPENDENT CONTROL USING DISCRETE-TIME MEASUREMENTS Consider te LTI system ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t), wit relative degree r 2, i.e., x R n, u R m, y R l (1) CA i B = 0, i = 0, 1,..., r 2, CA r 1 B 0. (2) Relative degree is ow many times te output y(t) needs to be differentiated before te input u(t) appears explicitly. In particular, (2) implies y (i) = CA i x, i = 0, 1,..., r 1. (3)

2 2 To prove (3), note tat it is trivial for i = 0 and, if it as been proved for i < r 1, it olds for i + 1: y (i+1) = ( y (i)) (3) = (CA i x) = CA i [Ax + Bu] (2) = CA i+1 x. For LTI systems wit relative degree r, it is common to look for a stabilizing controller of te form u(t) = K 0 y(t) + K 1 ẏ(t) K r 1 y (r 1) (t) () wit K i R m l for i = 0,..., r 1. Remark 1: Te control law () essentially reduces te system s relative degree from r 2 to r = 1. Indeed, te transfer matrix of (1) as te form W (s) = βrsn r + +β n s n +α 1s n 1 + +α n wit β r = CA r 1 B 0. Taking u(t) = K 0u 0 (t)+k 1 u 0 (t)+ + K r 1u (r 1) 0 (t), one as ỹ(s) = (βrsn r + +β n)(k r 1 sr 1 + +K 0 ) s n +α 1s n 1 + +α n ũ 0 (s), were ỹ and ũ 0 are te Laplace transforms of y and u 0. If β r K r 1 0, te latter system as relative degree one. If it can be stabilized by u 0 = Ky ten (1) can be stabilized by () wit K i = K i K. Te controller () depends on te output derivatives, wic are ard to measure directly. Instead, te derivatives can be approximated by te finite-differences ẏ(t) y(t) y(t τ1) τ 1, ÿ(t) 1 τ 1 ( y(t) y(t τ1) τ 1 Tis leads to te delay-dependent control ) y(t τ1) y(t τ2) (τ 2 τ 1),... u(t) = K 0 y(t) + K 1 y(t τ 1 ) + + K r 1 y(t τ r 1 ), (5) were te gains K 0,..., K r 1 depend on te delays 0 < τ 1 < < τ r 1. If (1) can be stabilized by te derivative-dependent control (), ten it can be stabilized by te delayed control (5) wit small enoug delays [6]. In tis paper, we study te sampled-data implementation of (5): u(t) = K 0 y(t k )+ r 1 i=1 K iy(t k q i ), t [t k, t k+1 ), (6) were > 0 is a sampling period, t k = k, k N 0, are sampling instants, 0 < q 1 < < q r 1, q i N, are discretetime delays, and y(t) = 0 if t < 0. In te next section, we prove tat if (1) can be stabilized by te derivative-dependent controller (), ten it can be stabilized by te delayed sampled-data controller (6) wit a small enoug sampling period. Moreover, we sow ow to coose appropriate sampling period, controller gains K 0,..., K r 1, and discrete-time delays q 1,..., q r 1. A. Stability conditions Introduce te errors due to sampling δ 0 (t) = y(t k ) y(t), δ i (t) = y(t k q i ) y(t q i ), t [t k, t k+1 ), k N 0, were i = 1,..., r 1. Following [12], we employ Taylor s expansion wit te remainder in te integral form: y(t q i ) = r 1 y (j) (t) j=0 j! ( q i ) j + κ i (t), were κ i (t) = ( 1)r t (r 1)! t q (s t + q i i) r 1 y (r) (s) ds. Combining tese representations wit (3), we rewrite (6) as u = [K 0, K]M Cx + K 0 δ 0 + Kδ + Kκ, (7) were δ = col{δ 1,..., δ r 1 }, κ = col{κ 1,..., κ r 1 }, M =. I l ( q I l q 1I 1 ) 2 l 2! I l I l q 2I l ( q 2 ) 2 2! I l. I l q r 1I l ( q r 1 ) 2 2! I l. ( q 1 ) r 1 I (r 1)! l ( q 2 ) r 1 I (r 1)! l.... ( q r 1 ) r 1 I (r 1)! l C CA K = [ ] K 1 K 2 K r 1, C = Te closed-loop system (1), (6) takes te form. CA r 1 ẋ = Dx + BK 0 δ 0 + BKδ + BKκ, D = A + B[K 0, K]M C., Using (3), te closed-loop system (1), () can be written as Coosing. (8) (9) ẋ = Dx, D = A + B[ K0,..., K r 1 ] C. (10) [K 0, K 1,..., K r 1 ] = [ K 0,..., K r 1 ]M 1, (11) we obtain D = D. (Te Vandermonde-type matrix M is invertible, since te delays q i are different.) If (1), () is stable, D must be Hurwitz and (9) will be stable for zero δ0, δ, κ. Te following teorem provides LMIs guaranteeing tat δ 0, δ, and κ do not destroy te stability of (9). Teorem 1: Consider te LTI system (1) subject to (2). (i) For given sampling period > 0, discrete-time delays 0 < q 1 <... < q r 1, controller gains K 0,..., K r 1 R m l, and decay rate α > 0, let tere exist positivedefinite matrices P R n n, W 0, W i, R i R m m (i = 1,..., r 1) suc tat 1 Φ 0, were Φ = {Φ ij } is te symmetric matrix composed from Φ 11 = P D + D T P + 2αP + 2 e 2α r 1 i=0 (K ica) T W i (K i CA), Φ 12 = 1 T r P B, Φ 13 = 1 T r 1 P B, Φ 1 = (CA r 1 D) T H, Φ 2 = 1 r (CA r 1 B) T H, Φ 22 = diag{w { 0, e 2αq1 W 1,..., e 2αqr 1 W r 1 }, Φ 33 = (r!) 2 e diag 2αq 1 (q 1) R r 1,..., e 2αq r 1 (q r 1) R r r 1 }, Φ 3 = 1 r 1 (CA r 1 B) T H, Φ = H wit H = r 1 i=1 (q i) r K T i R ik i and D defined in (9). Ten te delayed sampled-data controller (6) exponentially stabilizes te system (1) wit te decay rate α. (ii) Let tere exist K0,..., K r 1 R m l suc tat te derivative-dependent controller () exponentially stabilizes (1) wit a decay rate α. Ten, te delayed sampleddata controller (6) wit K 0,..., K r 1 given by (11) wit 1 MATLAB codes for solving te LMIs are available at ttps://gitub.com/antonselivanov/tac18

3 3 Fig. 1. Event-triggering wit respect to te control signal M from (8) and q i = i 1 r 1 (i = 1,..., r 1) 2 exponentially stabilizes (1) wit any given decay rate α < α if te sampling period > 0 is small enoug. Proof is given in Appendix A. Remark 2: Teorem 1(ii) explicitly defines te controller parameters K 0, K i, q i (i = 1,..., r 1), wic depend on. To find appropriate sampling period, one sould reduce until te LMIs from (i) start to be feasible. B. Event-triggered control Event-triggered control allows to reduce te number of signals transmitted troug a communication network [13] [17]. Te idea is to transmit te signal only wen it canges a lot. Te event-triggering mecanism for measurements was implemented in [1] for te system (1), (6) wit relative degree r = 2. Here, we consider te system wit r 2 and introduce te event-triggering for control signals, since te output eventtriggering leads to complicated conditions (see Remark 3). Consider te system (Fig. 1) ẋ(t) = Ax(t) + Bû k, t [t k, t k+1 ), k N 0, y(t) = Cx(t), x R n, u R m, y R l, (12) were û k = u(t k ) if u(t k ) from (6) was transmitted and û k = û k 1 oterwise. Te signal u(t k ) is transmitted if its relative cange since te last transmission is large enoug, namely, if (u(t k ) û k 1 ) T Ω(u(t k ) û k 1 ) > σu T (t k )Ωu(t k ), (13) were σ [0, 1) and 0 < Ω R m m are event-triggering parameters. Tus, û 0 = u(t 0 ) and { u(tk ), (13) is true, û k = (1) û k 1, (13) is false. Teorem 2: Consider te system (12) subject to (2). For given sampling period > 0, discrete-time delays 0 < q 1 <... < q r 1, controller gains K 0,..., K r 1 R m l, eventtriggering tresold σ [0, 1), and decay rate α > 0, let tere exist positive-definite matrices P R n n, Ω, W 0, W i, R i R m m (i = 1,..., r 1) suc tat 3 Φ e 0, were Φ e = Φ P B σ([k 0, K]M C) T Ω 0 σ1 r Ω 0 σ1 r 1 Ω HCA r 1 B 0 Ω 0 0 σω 2 Note tat q i = i 1 r 1 N for small > 0 3 MATLAB codes for solving te LMIs are available at ttps://gitub.com/antonselivanov/tac18 wit M, K, C defined in (8) and Φ, H given in Teorem 1. Ten te event-triggered controller (6), (13), (1) exponentially stabilizes te system (12) wit te decay rate α. Proof is given in Appendix B. Remark 3: Te event-triggering mecanism (13), (1) is constructed wit respect to te control signal. Tis allows to reduce te workload of a controller-to-actuator network. To compensate te event-triggering error, we add (29) to V, wic leads to two additional block-columns and block-rows in te LMI (confer Φ of Teorem 1 and Φ e of Teorem 2). One can study te event-triggering mecanism wit respect to te measurements by replacing y(t k ), y(t k q i ) wit ŷ k = y(t k ) + e k, ŷ k qi = y(t k q i ) + e k qi in (6). Tis may reduce te workload of a sensor-to-controller network but would require to add expressions similar to (29) to V for eac error e k, e k q1,..., e k qr 1. Tis would lead to more complicated LMIs wit two additional block-columns and block-rows for eac error. We study te event-triggering mecanism wit respect to te control for simplicity. Remark : Taking Ω = ωi wit large ω > 0, one can sow tat Φ e 0 and Φ 0 are equivalent for σ = 0. Tis appens since te event-triggered control (6), (13), (1) wit σ = 0 degenerates into periodic sampled-data control (6). Terefore, an appropriate σ can be found by increasing its value from zero wile preserving te feasibility of te LMIs from Teorem 2. C. Example Consider te triple integrator... y = u, wic can be presented in te form (1) wit [ ] [ ] 00 A= 0 0 1, B =, C = [ ]. (15) Tese parameters satisfy (2) wit r = 3. Te derivativedependent control () wit K 0 = 2 10, K1 = 0.06, K2 = 0.32 stabilizes te system (1), (15). Te LMIs of Teorem 1 are feasible for = 0.0, q 1 = 30, q 2 = 60, α = 10 3, K , K , K , were K i are calculated using (11). Terefore, te delayed sampled-data controller (6) also stabilizes te system (1), (15). Consider now te system (12), (15). Te LMIs of Teorem 2 are feasible for = 0.02, σ = wit te same control gains K 0, K1, K2, delays q 1, q 2, and decay rate α. Tus, te event-triggered control (6), (13), (1) stabilizes te system (12), (15). Performing numerical simulations for 10 randomly cosen initial conditions x(0) 1, we find tat te eventtriggered control (6), (13), (1) requires to transmit on average 55.6 control signals during 100 seconds. Te amount of transmissions for te sampled-data control (6) is given by + 1 = Tus, te event-triggering mecanism reduces te workload of te controller-to-actuator network by almost 80% preserving te decay rate α. Note tat σ > 0 leads to a smaller sampling period. Terefore, te event-triggering mecanism requires to transmit more measurements troug sensor-to-controller network. However, te total workload of bot networks is reduced by over 37%. 100

4 III. EVENT-TRIGGERED PID CONTROL Consider te scalar system and te PID controller ÿ(t) + a 1 ẏ(t) + a 2 y(t) = bu(t) (16) u(t) = k p y(t) + k i y(s) ds + k d ẏ(t). (17) Here, we study sampled-data implementation of te PID controller (17) tat is obtained using te approximations 0 y(s) ds k 0 y(s) ds k 1 j=0 y(t j), ẏ(t) ẏ(t k ) y(t k) y(t k q ) t [t k, t k+1 ), q, were > 0 is a sampling period, t k = k, k N 0, are sampling instants, q N is a discrete-time delay, and y(t k q ) = 0 for k < q. Substituting tese approximations into (17), we obtain te sampled-data controller u(t) = k p y(t k ) + k i k 1 j=0 y(t j) + k d y(t k q ), t [t k, t k+1 ), k N 0, wit y(t k q ) = 0 for k < q and 0 (18) k p = k p + k d q, k i = k i, k d = k d q. (19) Similarly to Section II-B, we introduce te event-triggering mecanism to reduce te amount of transmitted control signals. Namely, we consider te system ÿ(t)+a 1 ẏ(t)+a 2 y(t) = bû k, t [t k, t k+1 ), k N 0, (20) were û k is te event-triggered control: û 0 = u(t 0 ), { u(tk ), if (22) is true, û k = û k 1, if (22) is false, wit u(t) from (18) and te event-triggering condition (21) (u(t k ) û k 1 ) 2 > σu 2 (t k ). (22) Here, σ [0, 1) is te event-triggering tresold. Remark 5: We consider te event-triggering mecanism wit respect to te control signal, since te event-triggering wit respect to te measurements ŷ k = y(t k ) + e k leads to an accumulating error in te integral term: k 0 y(s) ds k 1 j=0 ŷj = k 1 j=0 y(t j) + k 1 j=0 e j. A. Stability conditions To study te stability of (20) under te event-triggered PID control (18), (21), (22), we rewrite te closed-loop system in te state space. Let x 1 = y, x 2 = ẏ, and x 3 (t) = (t t k )y(t k ) + k 1 j=0 y(t j), t [t k, t k+1 ). Introduce te errors due to sampling v(t) = x(t k ) x(t) δ(t) = y(t k q ) y(t q), t [t k, t k+1 ), k N 0. Using Taylor s expansion for y(t q) wit te remainder in te integral form, we ave y(t k q ) = y(t q) + δ(t) = y(t) ẏ(t)q + κ(t) + δ(t), were κ(t) = (s t + q)ÿ(s) ds. t q Using tese representations in (18), we obtain u(t k ) = k p x 1 (t k ) + k i x 3 (t k ) + k d y(t k q ) = [k p + k d, qk d, k i ]x + [k p, 0, k i ]v + k d (κ + δ). (23) Introduce te event-triggering error e k = û k u(t k ). Ten te system (20) under te event-triggered PID control (21), (22), (23) can be presented as ẋ = Ax + A v v + Bk d (κ + δ) + Be k, y = Cx, for t [t k, t k+1 ), k N 0, were A = a 2 + b(k p + k d ) a 1 qbk d bk i, A v = bk p 0 bk i, B = 0 b, C = [ ] (2) (25) Note tat te integral term in (18) requires to introduce te error due to sampling v tat appears in (2) but was absent in (9). Te analysis of v is te key difference between Teorem 2 and te next result. Teorem 3: Consider te system (20). (i) For given sampling period > 0, discrete-time delay q > 0, controller gains k p, k i, k d, event-triggering tresold σ [0, 1), and decay rate α > 0, let tere exist positivedefinite matrices P, S R 3 3 and nonnegative scalars W, R, ω suc tat Ψ 0, were Ψ = {Ψ ij } is te symmetric matrix composed from Ψ 11 = P A + A T P + 2αP + [ ] W k 2 d 2 e 2α, Ψ 12 = P A v, Ψ13 = Ψ 1 = Ψ 15 = P B, Ψ 16 = k p+k d k p qk d ωσ, Ψ 26 = 0 ωσ, k i Ψ 17 = A T G, Ψ 22 = S, Ψ 27 = A T v G, Ψ 36 = Ψ 6 = ωσ, Ψ 37 = Ψ 7 = Ψ 57 = B T G, Ψ 33 = W π2 e 2αq, Ψ = R (q) e 2αq, 2 Ψ 55 = ω, Ψ 66 = ωσ, ] Ψ 77 = G, G = 2 e 2α S + Rkd 2(q)2 [ wit A, A v, B, C given in (25). Ten, te event-triggered PID controller (18), (21), (22) exponentially stabilizes te system (20) wit te decay rate α. (ii) Let tere exist k p, k i, k d suc tat te PID controller (17) exponentially stabilizes te system (16) wit a decay rate α. Ten, te event-triggered PID controller (18), (21), (22) wit k p, k i, k d given by (11) and q = 1 2 exponentially stabilizes te system (20) wit any given decay rate α < α if te sampling period > 0 and te event-triggering tresold σ [0, 1) are small enoug. Proof is given in Appendix C. MATLAB codes for solving te LMIs are available at ttps://gitub.com/antonselivanov/tac18 k i

5 5 Remark 6: Te event-triggered control (18), (21), (22) wit σ = 0 degenerates into sampled-data control (18). Terefore, Teorem 3 wit σ = 0 gives te stability conditions for te system (16) under te sampled-data PID control (18). Remark 7: Appropriate values of and σ can be found in a manner similar to Remarks 2 and. B. Example Following [8], we consider (16) wit a 1 = 8., a 2 = 0, b = Te system is not asymptotically stable if u = 0. Te PID controller (17) wit k p = 10, k i = 0, k d = 0.65 exponentially stabilizes it wit te decay rate α 10.. Teorem 3 wit σ = 0 (see Remark 6) guarantees tat te sampled-data PID controller (18) can acieve any decay rate α < α if te sampling period > 0 is small enoug. Since α is on te verge of stability, α close to α requires to use small. Tus, for α = 10.3, te LMIs of Teorem 3 are feasible wit 10 7, q = 272, and k p, k i, k d given by (19). To avoid small sampling period, we take α = 5. For σ = 0, α = 5 and eac q = 1, 2, 3,... we find te maximum sampling period > 0 suc tat te LMIs of Teorem 3 are feasible. Te largest corresponds to α = 5, σ = 0, q = 7, = , k p 29.76, k i = 0, k d 19.76, were k p, k i, k d are calculated using (19). Remark 6 implies tat te sampled-data PID controller (18) stabilizes (16). Teorem 3 remains feasible for α = 5, σ = , q = 7, = 10 3, k p 33.21, k i = 0, k d 23.21, were k p, k i, k d are calculated using (19). Tus, te eventtriggered PID control (18), (21), (22) exponentially stabilizes (20). Performing numerical simulations in a manner described in Section II-C, we find tat te event-triggered PID control requires to transmit on average 628. control signals during 10 seconds. Te sampled-data controller (18) requires = 2128 transmissions. Tus, te eventtriggering mecanism reduces te workload of te controllerto-actuator network by more tan 70%. Te total workload of bot networks is reduced by more tan 26%. wit (i) Consider te functional APPENDIX A PROOF OF THEOREM 1 V = V 0 + V δ0 + V δ + V κ (26) V 0 = x T P x, V δ0 = 2 e 2α t t k e 2α(t s) ẏ T (s)k0 T W 0 K 0 ẏ(s) ds t k e 2α(t s) [y(s) y(t k )] T K0 T W 0 K 0 [y(s) y(t k )] ds, V δ = 2 e 2α r 1 r 1 i=1 i=1 t k q e 2α(t s) i ẏ T (s)ki T W ik i ẏ(s) ds qi t k q e 2α(t s) i [y(s) y(t k q i )] T Ki T W ik i [y(s) y(t k q i )] ds, V κ = r 1 i=1 t q e 2α(t s) i (s t + q i ) r (y (r) (s)) T Ki T R ik i y (r) (s) ds. Te term V κ, introduced in [12], compensates Taylor s remainders κ i, wile V δ0 and V δ, introduced in [9], compensate te sampling errors δ 0 and δ. Te Wirtinger inequality (Lemma 1) implies V δ0 0 and V δ 0. Using (9) and (3), we obtain V 0 + 2αV 0 =2x T P [Dx+BK 0 δ 0 +BKδ+BKκ]+2αx T P x, V δ0 + 2αV δ0 = 2 e 2α x T (K 0 CA) T W 0 (K 0 CA)x δt 0 K0 T W 0 K 0 δ 0, V δ + 2αV δ = 2 e 2α r 1 i=1 xt (K i CA) T W i (K i CA)x r 1 i=1 e 2αqi δ T i KT i W ik i δ i. Using y (r) = CA r 1 ẋ (wic follows from (3)) and Jensen s inequality (Lemma 2) wit ρ(s) = (s t + q i ) r 1, we ave V κ + 2αV κ = r 1 i=1 (q i) r (y (r) (t)) T Ki T R ik i y (r) (t) r 1 i=1 r t q e 2α(t s) i (s t + q i ) r 1 (y (r) (s)) T Ki T R ik i y (r) (s) ds r 1 i=1 (q i) r ẋ T (A r 1 ) T C T Ki T R ik i CA r 1 ẋ r 1 (r!) 2 i=1 (q i) e 2αqi κ T r i KT i R ik i κ i. Summing up, we obtain were V + 2αV ϕ T Φϕ + ẋ T (CA r 1 ) T H(CA r 1 )ẋ, (27) ϕ = col{x, K 0 δ 0,..., K r 1 δ r 1, K 1 κ 1,..., K r 1 κ r 1 } (28) and Φ is obtained from Φ by removing te last block-column and block-row. Substituting (9) for ẋ and applying te Scur complement, we find tat Φ 0 guarantees V 2αV. Since V (t k ) V (t k ), te latter implies exponential stability of te system (9) and, terefore, (1), (6). (ii) Since q i = O( 1 r 1 ), [12, Lemma 2.1] guaranties M 1 = O( 1 r 1 ), wic implies K i = O( 1 r 1 ) for i = 0,..., r 1. Since D = D (wit D defined in (10)) and (1), () is exponentially stable wit te decay rate α, tere exists P > 0 suc tat P D + D T P + 2αP < 0 for any α < α. Coose W 0 = O( 1 r ), W i = O( 1 r ), and R i = O( 1 1 r ) for i = 1,..., r 1. Applying te Scur complement to Φ 0, we obtain P D + D T P + 2αP + O( 1 r ) < 0, wic olds for small > 0. Tus, (i) guarantees (ii). APPENDIX B PROOF OF THEOREM 2 Denote e k = û k u(t k ), k N 0. Te event-triggering mecanism (13), (1) guarantees 0 σu T (t k )Ωu(t k ) e T k Ωe k. (29) Substituting û k = u(t k )+e k into (12) and using (7), we obtain (cf. (9)) ẋ = Dx + BK 0 δ 0 + BKδ + BKκ + Be k, t [t k, t k+1 ) (30) wit D given in (9). Consider V from (26). Calculations similar to tose from te proof of Teorem 1 lead to (cf. (27)) V + 2αV (29) V + 2αV + σu T (t k )Ωu(t k ) e T k Ωe k ϕ T e Φ e ϕ e + ẋ T (CA r 1 ) T H(CA r 1 )ẋ + σu T (t k )Ωu(t k ),

6 6 were ϕ e = col{ϕ, e k } (wit ϕ from (28)) and Φ e is obtained from Φ e by removing te blocks Φ ij wit i {, 6} or j {, 6}. Substituting (30) for ẋ and (7) for u(t k ) and applying te Scur complement, we find tat Φ e 0 guarantees V 2αV. Since V (t k ) V (t k ), te latter implies exponential stability of te system (30) and, terefore, (12) under te controller (6), (13), (1). wit (i) Consider te functional V 0 = x T P x, V v = 2 e 2α APPENDIX C PROOF OF THEOREM 3 V = V 0 + V v + V δ + V κ t k e 2α(t s) ẋ T (s)sẋ(s) ds t k e 2α(t s) v T (s)sv(s) ds, V δ = W kd 22 e 2α t t k q e 2α(t s) (ẏ(s)) 2 ds W kd 2 π2 V κ = Rk 2 d q t k q e 2α(t s) [y(s) y(t k q)] 2 ds, t q e 2α(t s) (s t + q) 2 (ÿ(s)) 2 ds. t Te Wirtinger inequality (Lemma 1) implies V v 0 and V δ 0. Using te representation (2), we obtain V 0 + 2αV 0 = 2x T P [Ax(t) + A v v(t) + Bk d (κ + δ) + Be k ] +2αx T P x, V v + 2αV v = 2 e 2α ẋ T (t)sẋ(t) vt (t)sv(t), V δ + 2αV δ = W kd 22 e 2α (x 2 (t)) 2 W kd 2 π2 e 2αq δ 2 (t). Using Jensen s inequality (Lemma 2) wit ρ(s) = (s t+q i ), we obtain V κ + 2αV κ = Rk 2 d (q)2 (ÿ(t)) 2 Rkd 22 t q e 2α(t s) (s t + q)(ÿ(s)) 2 ds Rkd 2(q)2 (ẋ 2 (t)) 2 Rkd 2 (q) e 2αq κ 2 (t). 2 For ω 0, te event-triggering rule (21), (22) guarantees Tus, we ave 0 ωσu 2 (t k ) ωe 2 k. V + 2αV V + 2αV + [ωσu 2 (t k ) ωe 2 k ] ψ T Ψψ + ẋ T (t)gẋ(t) + ωσu 2 (t k ), were ψ = col{x, v/, k d δ, k d κ, e k } and Ψ is obtained from Ψ by removing te last two block-columns and block-rows. Substituting (2) for ẋ and (23) for u(t k ) and applying te Scur complement, we find tat Ψ 0 guarantees V 2αV. Since V (t k ) V (t k ), te latter implies exponential stability of te system (2) and, terefore, (18), (20) (22). (ii) Te closed-loop system (16), (17) is equivalent to ẋ = Āx wit y Ā = a 2 + b k p a 1 + b k d b k i, x = ẏ y(s) ds Since q = O( 1 ), relations (19) imply k p = O( 1 ), k d = O( 1 ). 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