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A Decentralized Event-Triggered Dissipative Control Scheme for Systems With Multiple Sensors to Sample the System Outputs Xian-Ming Zhang and Qing-Long Han, Senior Member, IEEE

Abstract—This paper is concerned with decentralized eventtriggered dissipative control for systems with the entries of the system outputs having different physical properties. Depending on these different physical properties, the entries of the system outputs are grouped into multiple nodes. A number of sensors are used to sample the signals from different nodes. A decentralized event-triggering scheme is introduced to select those necessary sampled-data packets to be transmitted so that communication resources can be saved significantly while preserving the prescribed closed-loop performance. First, in order to organize the decentralized data packets transmitted from the sensor nodes, a data packet processor (DPP) is used to generate a new signal to be held by the zero-order-hold once the signal stored by the DPP is updated at some time instant. Second, under the mechanism of the DPP, the resulting closed-loop system is modeled as a linear system with an interval time-varying delay. A sufficient condition is derived such that the closed-loop system is asymptotically stable and strictly (Q0 , S0 , R0 )-dissipative, where Q0 , S0 , and R0 are real matrices of appropriate dimensions with Q0 and R0 symmetric. Third, suitable output-based controllers can be designed based on solutions to a set of a linear matrix inequality. Finally, two examples are given to demonstrate the effectiveness of the proposed method. Index Terms—Communication networks, decentralized dissipative control, event-triggered control (ETC) schemes.

I. I NTRODUCTION URING the last decade, event-triggered control (ETC) has received increasing attention in real-time control systems. One conspicuous characteristic is that ETC provides a strategy under which the control task is executed only when necessary. Compared with traditional time-triggered control, ETC can efficiently reduce the number of execution of control tasks while preserving the desired closed-loop performance [1]–[4]. The first important issue on ETC is the minimum interevent time (the minimum time interval between two consecutive events). It is necessary to ensure that the minimum interevent

D

Manuscript received July 9, 2015; revised September 15, 2015; accepted September 30, 2015. This work was supported by the Australian Research Council Discovery Project under Grant DP1096780. This paper was recommended by Associate Editor G.-P. Liu. (Corresponding author: Qing-Long Han.) The authors are with the Griffith School of Engineering, Griffith University, Gold Coast, QLD 4222, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2487420

time is larger than a certain positive constant; otherwise, an ETC system exhibits Zeno behavior, which is unexpected in the implementation of the control system. Under an event-triggering scheme, extra hardware is commonly used to monitor the instantaneous system state so that the next event time can be calculated. However, extra hardware incurs extra cost. In order to avoid using dedicated extra hardware to monitor the system state, self-triggered control (STC) is proposed in [5], in which the next event time can be derived by a predefined event-triggering condition related to the last measurement of the system state [6]–[9]. Nevertheless, under an ETC or STC scheme, although several sufficient conditions are obtained to calculate the minimum interevent time, the minimum interevent time may be quite small or may not exist for some real control systems. Recently, a periodic ETC (PETC) has been proposed in [10] and [11]. With the PETC, the system state is sampled with a proper period h > 0. The event-triggering condition in the PETC is verified only at the sampling instants, which means that the PETC directly provides a guaranteed minimum interevent time (at least h > 0) provided that the closed-loop stability can be ensured [11]–[13]. The second important issue on ETC is the feedback controller that computes the control signals. It is natural that the designed controller should reflect the event-triggering nature. However, some existing results follow a so-called emulationbased approach [10], by which the controller is designed using standard period sampled-data controller design tools. This implies that the designed controller just reflects the nature of time-triggering rather than the one of event-triggering. Recently, under the PETC scheme, a co-design approach has been proposed in [11], where both event triggering parameters and state feedback controller can be designed simultaneously provided that a set of linear matrix inequalities (LMIs) are feasible. This idea is then employed to deal with L2 control for sampled-data systems [12], networked Takagi–Sugeno fuzzy systems [14], [15], consensus of multiagents [16], output feedback control for networked control systems [17], and H∞ filtering [13], [18], [19]. Among the results on ETC, STC, and PETC reported in the literature, it is found that most of them are about state feedback control, which requires the system state to be measurable. When the system state is not available, output feedback control is an alternative approach to the implementation of a control system. This important issue has been addressed

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by several researchers. For example, under an STC scheme, Almeida et al. [20] considered the output feedback control using a discrete-time observer. Employing a PETC scheme, Heemels et al. [10] analyzed the dynamical output-based control for a continuous-time linear system. Nevertheless, these results focus only on conditions on the existence of the minimum interevent time, while the design of desired output feedback controllers is not concerned with, which limits the application of the results. The notion of dissipative systems is an important notion in control systems [21], [22]. Dissipativity is a generalization of the notion of passivity in electrical networks and other dynamical systems that dissipate energy in some abstract sense [23]. The theory of dissipative systems includes some basic tools, such as passive theorem, bounded real lemma, and Kalman–Yakubovich lemma, and provides an effective framework for stability analysis and design of control systems. Thus, in the past several decades, dissipativity is applied to a wide range of fields, such as systems, circuits and complex networks [24], and many results are reported in the literature (see [25]–[27] and the references therein). Recently, Yu and Antsaklis [28] have employed the idea of ETC to study output feedback control for networked control systems. Under the assumption that both plant and controller are input feedforward output feedback passive, an event-triggering condition is presented based on the passive theorem so that the minimum interevent time can be ensured. However, the desired controller should be given a priori, and how to design the suitable output feedback controllers is not addressed in [28]. Furthermore, to the best of the authors’ knowledge, few results have been reported on the dissipative control using PETC schemes [29], which is the first motivation of the study. For some practical control systems, the entries of the system outputs are usually of different physical properties. For example, in a satellite control system [30], [31], the system state has four entries, i.e., the yaw angles θ1 (t) and θ2 (t), and the angular velocities θ˙1 (t) and θ˙2 (t) for the main body and the instrumentation module. When information on these four entries is measured as the system outputs for control design, the physical properties of θ1 (t) and θ2 (t) are different from those of θ˙1 (t) and θ˙2 (t). Thus, if an event-triggering scheme is applied to these systems, the entries with different physical properties should be grouped into several nodes so that events can be triggered by different event-triggering conditions related to the physical properties of the signals from the nodes. This leads to a decentralized event-triggering scheme for output-based ETC. However, study on decentralized ETC has not yet received much attention. When the physically distributed sensor nodes have no access to the full state of the plant, a decentralized ETC is studied on wireless sensor/actuator networks [32], [33], networked control systems [34], and distributed networked systems [35], but the event-triggering schemes involved therein are based on system state. When system state is not available, those results are no longer applicable. Although a decentralized PETC strategy is employed in [10] to investigate output-based ETC, entries with different physical properties might be grouped into the same node. Moreover, stability criteria obtained in [10],

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Fig. 1.

Diagram for decentralized ETC.

which are presented in terms of a set of nonlinear matrix inequalities, need to be satisfied for all possible combinations of triggered events. As a result, designing suitable controllers is difficult, as well as, verifying whether or not a chosen controller meets the stability criteria is not an easy task. Therefore, study on decentralized ETC is still challenging, which is the second motivation of the study. This paper focuses on decentralized ETC for systems, where the entries of the system outputs are grouped into m nodes, where m is a positive integer. In each node, a sensor is used to sample the corresponding signals with a period h > 0 and this sampled signal is immediately encapsulated into a data packet with the time stamp. Whether the sampled data packet from the mth sensor should be transmitted or not is determined by the corresponding predefined event-triggering condition. On the side of zero-order-hold (ZOH), a data packet processor (DPP) is introduced to generate a new signal to be held by the ZOH. Under the mechanism of the DPP, the closed-loop system is modeled as a time-delay system, based on which, Lyapunov functional method is employed to derive a sufficient condition such that the closed-loop system is asymptotically stable and strictly (Q0 , S0 , R0 )-dissipative, where Q0 , S0 , and R0 real matrices of appropriate dimensions with Q0 and R0 symmetric. An LMI-based approach is then presented to design desired decentralized event-triggered controllers. Finally, two examples are given to demonstrate the effectiveness of the proposed approach. A. Notation The notations throughout this paper are standard. diag{· · · } and col{· · · } denote a block-diagonal matrix and a blockcolumn vector, respectively. The space of square-integrable vector functions over [0, ∞) is denoted by L2 [0, ∞). We denote W[a, b] with the norm ||ϕ||W = maxs∈[a,b] |ϕ(s)| + b 2 ds)1/2 by the space of functions ϕ : [a, b] → Rn ˙ ( a |ϕ(s)| satisfying: 1) ϕ is absolutely continuous on [a, b); 2) the limitation limθ→b− ϕ(θ ) exists; and 3) ϕ˙ is square integrable on [a, b]. The expression a means the largest integer less than a; and the symbol “” represents the symmetric term in a symmetric matrix. II. P ROBLEM F ORMULATION AND D ECENTRALIZED E VENT-T RIGGERING S CHEME Consider the decentralized ETC problem shown in Fig. 1, where the plant is a continuous-time linear

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ZHANG AND HAN: DECENTRALIZED EVENT-TRIGGERED DISSIPATIVE CONTROL SCHEME FOR SYSTEMS WITH MULTIPLE SENSORS

system described by ⎧ x˙ (t) = Ax(t) + B1 w(t) + B2 u(t) ⎪ ⎪ ⎨ y(t) = Cx(t) + D0 w(t) z(t) = Lx(t) + D1 w(t) + D2 u(t) ⎪ ⎪ ⎩ x(0) = x0

3

(1)

where x(t) ∈ Rn , u(t) ∈ Rnu , y(t) ∈ R , and z(t) ∈ Rp are the system’s state vector, the control input vector, the measurement output vector, and the controlled output vector, respectively; w(t) ∈ Rq is an exogenous disturbance vector belonging to L2 [0, ∞); the system matrices A, B1 , B2 , C, L, D0 , D1 , and D2 are known real matrices with compatible dimensions; and x0 is the initial condition. When the system output y(t) consists of entries with different physical properties, as mentioned in Section I, it is significant to design different event-triggering conditions to trigger the corresponding sampled-data packets. For this reason, we consider a decentralized event-triggering scheme. Assume that the measurement outputs, i.e., the entries of y(t), are grouped into m nodes. The signal from the node i ∈ {1, 2, . . . , m} is denoted by yˆ i (t) ∈ Rli , where m i=1 li = . Note that the entries of col{ˆy1 (t), yˆ 2 (t), . . . , yˆ m (t)} is a permutation of the entries of y(t). Then y(t) = Hcol yˆ 1 (t), yˆ 2 (t), . . . , yˆ m (t) (2) where H is a permutation matrix, and thus H is nonsingular. At the sampling instant kh(k ∈ N), where h > 0 is a sampling period, each node i(i = 1, 2, . . . , m) samples the respective outputs yˆ i (t) by sensor i (see Fig. 1). The sampled signal yˆ i (kh) and its time stamp k are capsulated into a packet (k, yˆ i (kh)). Whether or not the sampled-data packet should be transmitted is determined by the ith event-triggering processor (ETP). The ETP i has the logical capability of choosing those necessary packets to be transmitted. The mechanism of ETP i is precisely described as follows. ETP i(i = 1, 2, . . . , m): Given i > 0 and σi > 0, set t0i = 0, yˆ i (t0i h) = 0, k = 1. 1) At the sampling instant kh, if the event condition

T

yˆ i (kh) − yˆ i t0i h i yˆ i (kh) − yˆ i t0i h

T ≤ σi yˆ i t0i h i yˆ i t0i h (3) is violated, then the ETP i stores the packet (k, yˆ i (kh)) with t0i = k, yˆ i (t0i h) = yˆ i (kh) and releases the packet to the communication channel. 2) Let k = k + 1 and go to step 1). From the above mechanism of ETP i, one can see that ETP i releases the sampled-data packets provided that the event condition (3) is violated by a certain packet, where σi > 0 is a threshold parameter and i > 0 a weighting matrix (i = 1, 2, . . . , m). The data packets released by ETPs are transmitted to the DPP through a communication channel in a singlepacket manner. Due to limited bit rate of a communication channel, time delays are inevitable during the transmission of a sampled-data packet via a communication channel [36], [37]. In this paper, it is assumed that each released packet ( j, yˆ i (jh)) is transmitted to the DPP subject to a time

Fig. 2.

Example to show how the DPP generates the signal y˜ .

delay τji (i = 1, 2, . . . , m, j ∈ N). The transmission delay τji may be constant or time-varying or unknown but upper bounded by h, that is 0 < τji ≤ h, i = 1, 2, . . . , m, j ∈ N.

(4)

Furthermore, during the packet transmission, suppose that no packets are lost and disordered. In Fig. 1, the data packets released by ETP i(i = 1, 2, . . . , m) are transmitted to the DPP. The DPP aims at generating a new signal y˜ to update the store of the ZOH. Once the store of the ZOH is updated, the ZOH immediately actuate the controller. The mechanism of the DPP is given as follows. DPP: Set t0i = 0, yˆ i (t0i h) = 0 (i = 1, 2, . . . , m), flag = 0, k = 1, j = 0, and rj = 0. 1) During the time interval ((k − 1)h, kh], if a packet (sik , yˆ i (sik h)) arrives at the DPP, then flag = 1 and the DPP updates the store of (t0i , yˆ i (t0i h)) with t0i = sik and yˆ i (t0i h) = yˆ i (sik h). 2) If flag = 1, then: a) j = j + 1 and rj = kh; b) the DPP uses the stores to generate a new signal y˜ := Hcol{ˆy1 (t01 h), yˆ 2 (t02 h), . . . , yˆ m (t0m h)} to be sent to the ZOH. 3) Let flag = 0, k = k + 1 and go to step 1). From the description of the DPP, the variables t0i are used to store the time stamps of the latest packets transmitted from ETP i(i = 1, 2, . . . , m). The variables flag indicates that whether or not the DPP generates a new signal y˜ using the current stores of yi (t0i h) (i = 1, 2, . . . , m). Moreover, the time sequence {rj |j = 0, 1, 2, . . .} records the time instants when new signals are generated. Now, we take Fig. 2 as an example to show the mechanism of the DPP. In Fig. 2, during (h, 2h], the DPP receives the released packets (1, yˆ i (h)) (i = 1, 2, 3). At the time instant 2h, the DPP generates a new signal y˜ = Hcol{ˆy1 (h), yˆ 2 (h), yˆ 3 (h)}. During (2h, 3h], no signal is generated because the DPP does not receive any packets during this period. During (3h, 4h], the DPP receives a packet (3, yˆ 1 (3h)). At time instant 4h, the DPP generates a new signal y˜ = Hcol{ˆy1 (3h), yˆ 2 (h), yˆ 3 (h)}; and so on. Moreover, in accordance with the mechanism of the DPP, the time sequence {rj |j = 1, 2, . . .}, which indicates when the DPP generates new signals, can be given by r1 = 2h, r2 = 4h, r3 = 5h, r4 = 8h, r5 = 9h, r6 = 10h, . . . . Remark 1: A significant characteristic is that the DPP generates new signals at the sampling instants, which means that the DPP should be synchronized with the sensors by a clock.



As a result, one does not need to know how the transmission delays τji of released packets ( j, yˆ i (jh)) (i = 1, 2, . . . , m, j ∈ N) vary as long as the assumption in (4) is satisfied. The time sequence {rj |j = 0, 1, 2, . . .} provides an access to the ETC design. Suppose that a new signal y˜ is generated at the time instant rj and the current store of the DPP is (t0i , yˆ i (t0i h)) (i = 1, 2, . . . , m). Thus, the signal y˜ at the time instant rj can be given by y˜ (rj ) := y˜ = Hcol yˆ 1 t01 h , yˆ 2 t02 h , . . . , yˆ m t0m h . (5) Then, this signal is immediately sent to ZOH. Note from the mechanism of DPP that r0 = 0 and yˆ i (0) = 0 (i = 1, 2, . . . , m). Then, without loss of generality we set y˜ (r0 ) = 0. By the property of ZOH, we have

(6) y˜ (t) = y˜ rj , t ∈ rj , rj+1 , j ∈ N ∪ {0}. Denote ρj := (rj+1 − rj )/h ( j = 0, 1, 2, . . .). One can see that the interval [rj , rj+1 ) can be expressed as a union of ρj equidistant subintervals, that is ρj

r r rj , rj+1 = Is j , Is j := rj + (s − 1)h, rj + sh . s=1

Then,

r Is1j

r ∩ Is2j

Let e(t) := Hcol{e1 (t), e2 (t), . . . , em (t)}.

(12)

Thus, one has

y˜ (rj ) = y(t − η(t)) − e(t), t ∈ rj , rj+1 , j ∈ N ∪ {0}.

Therefore, the controller (10) becomes ⎧ x˙ c (t) = AK xc (t) + BK xc (t − η(t)) ⎪ ⎪ ⎨ + CK y(t − η(t))

− C K e(t) u(t) = D x (t), t ∈ rj , rj+1 , j ∈ N ∪ {0} ⎪ K c ⎪ ⎩ xc (θ ) = 0, θ ∈ [ − 2h, 0].

(13)

(14)

Connecting (1) with (14), the closed-loop system is given by ˜ x˙˜ (t) = A0 x˜ (t) + A1 x˜ (t − η(t)) + A2 e(t) + A3 w(t) (15) z(t) = L0 x˜ (t) + L1 w(t), ˜ t ∈ rj , rj+1 , j ∈ N ∪ {0} := col{w(t), where x˜ (t) := col{x(t), xc (t)} and w(t) ˜ w(t − η(t))}; and A B2 DK 0 0 0 , A1 := , A2 := A0 := 0 AK CK C BK −CK A3 := diag{B1 , CK D0 }, L0 := [L D2 DK ], L1 := [D1 0]. The initial condition of the system (15) is supplemented as

= φ(s1 = s2 ) and ρ

j j [0, ∞) = ∪∞ j=0 ∪s=1 Is .

x˜ (θ ) = col{x0 , 0}, θ ∈ [−2h, 0].

In this paper, we are interested in designing a stabilizing controller, whose state-space implementation is give by ⎧ ⎨ x˙ c (t) = AK xc (t) + BK xc rj + (s − 2)h + CK y˜ rj r (7) u(t) = DK xc (t), t ∈ Is j , j ∈ N ∪ {0} ⎩ xc (θ ) = 0, θ ≤ 0

Meanwhile, without loss of generality, it is assumed that w(t) = 0 for t ≤ 0. Remark 3: It is worth pointing out that a delay system model is established in (15) for the closed-loop system of the plant (1) associated with the controller (10) under a decentralized event-triggering scheme. Compared with an impulsive model in [10], on the one hand, transmission delays are taken into account in the delay system model while they are not considered in the impulsive model; on the other hand, the impulsive model depends not only on the system matrices but also explicitly on the matrices indicating which nodes are triggered at a certain time instant while the delay system model only depends on the system matrices. Note that the closed-loop system (15) is related to e(t). By the mechanism of the DPP, it is true that for i = 1, 2, . . . , m T

yˆ i rj + (l − 1)h − yˆ i t0i h i yˆ i rj + (l − 1)h − yˆ i t0i h

T ≤ σi yˆ i t0i h i yˆ i t0i h (16)

r

where AK , BK , CK , and DK are real matrices to be determined. Remark 2: If setting BK = 0, the controller (7) is a dynamic output feedback controller (DOFC). Since rj + (s − 2)h < t r for t ∈ Is j , the controller (7) with BK = 0 is a memory controller. As stated in [38], a memory controller can provide some better system performance than that using a memoryless controller, which is the motivation of the memory controller (7). Define a piecewise affine function η(t) on [rj , rj+1 ) as r η(t) := t − rj + (s − 2)h , t ∈ Is j , s = 1, 2, . . . , ρj . (8) Then, it is clear that

h ≤ η(t) < 2h, t ∈ rj , rj+1 , j ∈ N ∪ {0}.

Moreover, the controller (7) can be rewritten as ⎧ + CK y˜ rj ⎨ x˙ c (t) = AK xc (t) + BK xc (t − η(t)) u(t) = DK xc (t), t ∈ rj , rj+1 , j ∈ N ∪ {0} ⎩ xc (θ ) = 0, θ ∈ [ − 2h, 0].

(9)

(10)

Define for i = 1, 2, . . . , m r ei (t) := yˆ i rj + (s − 2)h − yˆ i t0i h , t ∈ Is j , s = 1, 2, . . . , ρj where yˆ i (rj − h) = 0 for j = 0, i = 1, 2, . . . , m. Then

ei (t) = yˆ i (t − η(t)) − yˆ i t0i h , t ∈ rj , rj+1 , j ∈ N ∪ {0}. (11)

where l = 0, 1, 2, . . . , ρj − 1, which is equivalent to

T [ei (t)]T i [ei (t)] ≤ σi yˆ i (t − η(t)) − ei (t) i

× yˆ i (t − η(t)) − ei (t) , t ∈ rj , rj+1 . (17) The above inequality is also true for j = 0 because yˆ i (t) = 0 for t ∈ [r0 , r1 ) in terms of (6). Thus, e(t) satisfies for t ∈ [rj , rj+1 ), j ∈ N ∪ {0}

T eT (t)H −T 0 H −1 e(t) ≤ y(t − η(t)) − e(t)

× H −T 0 0 H −1 y(t − η(t)) − e(t) (18)


where

0 = diag{1 , 2 , . . . , m } 0 = diag{σ1 I, σ2 I, . . . , σm I}.

(19)

To proceed with, we introduce the following quadratic energy function E associated with the closed-loop system (15): E(w, z, T) = < z, Q0 z >T + 2 < z, S0 w >T + < w, R0 w >T where Q0 , S0 , and R0 are real matrices of appropriate dimensions with Q0 = QT0 and R0 = RT0 ; and for u, v ∈ Rk , T < u, v >T 0 uT vdt. Definition 1 [39]: For given real matrices Q0 , S0 , and R0 with Q0 = QT0 and R0 = RT0 , the closed-loop system (15) subject to (18) is said to be strictly (Q0 , S0 , R0 )-dissipative if, under zero initial conditions, there exists some scalar θ > 0 such that for any T > 0 E(w, z, T) ≥ θ < w, w >T . Remark 4: The notion of strict (Q0 , S0 , R0 )-dissipativity includes some special cases as follows. 1) If Q0 = −γ −1 I, S0 = 0, and R0 = γ I, the strict (Q0 , S0 , R0 )-dissipativity reduces to the H∞ norm constraint [40]. 2) If Q0 = 0, S0 = I, and R0 = γ I, the strict (Q0 , S0 , R0 )dissipativity becomes the passivity [24]. 3) If Q0 = −γ −1 αI, S0 = (1 − α)I, and R0 = γ I, α ∈ (0, 1), the strict (Q0 , S0 , R0 )-dissipativity means mixed H∞ and passive performance [41]. In this case, one can see that the weighting parameter α ∈ (0, 1) defines a tradeoff between H∞ performance and passive performance of the closed-loop system (15) subject to (18). The problem of the decentralized event-triggered dissipative control to be addressed in this paper is described as: for given scalars h > 0, γ > 0, and σi > 0 (i = 1, 2, . . . , m), and real matrices Q0 , S0 , and R0 with Q0 = QT0 and R0 = RT0 , design the controller gain parameters (AK , BK , CK , DK ) and the event-triggered weighting matrices i (i = 1, 2, . . . , m) such that the resultant closed-loop system (15) subject to (18) is asymptotically stable and strictly (Q0 , S0 , R0 )-dissipative, that is: 1) the closed-loop system (15) with w(t) ≡ 0, that is ⎧ ⎨ x˙˜ (t) = A0 x˜ (t) + A 1 x˜ (t − η(t)) + A2 e(t) (20) t ∈ rj , rj+1 , j ∈ N ∪ {0} ⎩ x˜ (θ ) = col{φ0 , 0}, θ ∈ [ − 2h, 0] subject to (18) is asymptotically stable; 2) the closed-loop system (15) subject to (18) is strictly (Q0 , S0 , R0 )-dissipative in the sense of Definition 1. III. D ECENTRALIZED E VENT-T RIGGERED D ISSIPATIVE A NALYSIS In this section, decentralized event-triggered dissipative analysis for the closed-loop system (15) is made. By choosing a discontinuous Lyapunov functional, a sufficient condition is formulated such that the closed-loop system subject to (18) is asymptotically stable and strictly (Q0 , S0 , R0 )-dissipative. We first introduce a useful inequality.

5

Lemma 1 [42]: Let ϕ ∈ W[a, b] and ϕ(a) = 0. Then for a given matrix R > 0 the following inequality holds: b 4(b − a)2 b T ϕ T (s)Rϕ(s)ds ≤ ϕ˙ (s)Rϕ(s)ds. ˙ (21) π2 a a Now we state and establish the following result. Proposition 1: For given scalars h > 0, 0 < β < 1 and σi > 0 and real matrices Q0 , S0 , and R0 with Q0 = QT0 ≤ 0, R0 = RT0 , and i > 0 (i = 1, 2, . . . , m), the closed-loop system (15) subject to (18) is asymptotically stable and strictly (Q0 , S0 , R0 )-dissipative if there exist real matrices P > 0, Q1 > 0, Q2 > 0, Rj > 0 ( j = 1, 2, 3), and S1 of appropriate dimensions such that R2 RS21 ≥ 0 and ⎡

11 ⎢ ⎢ ϒ1 := ⎢ ⎢ ⎣

12 22

h1T hAT3 33

2T H −T ˜ T H −T D 0 0 −( 0 0 )−1

⎤ ˜0 4T Q ˜ 0⎥ L1T Q ⎥ 0 ⎥ ⎥ 0, 0 < β < 1, and σi > 0, and real matrices Q0 , S0 , and R0 with Q0 = QT0 ≤ 0, R0 = RT0 , and i > 0 (i = 1, 2, . . . , m), there exist real matrices P > 0, Q1 > 0, Q2 > 0, Rj > 0( j = 1, 2, 3), and S1 of appropriate dimensions such that R2 RS12 ≥ 0 and (22) are satisfied if and only if there exist real matrices ˜ 2 > 0, R˜ j > 0 ( j = 1, 2, 3), S˜ 1 and ˜ 1 > 0, Q X > 0, Y > 0, Q Ck (k = 1, . . . , 4) of appropriate dimensions such that X I R˜ 2 S˜ 1 := ≥ 0, Z >0 (26) Y R˜ 2 ⎤ ⎡ ˜0 ˜ 11 ˜ 12 h˜ T ˜ 2T H −T ˜ 4T Q 1 ⎢ ˜ 0⎥ ˜ T H −T 22 hT3 L1T Q D 0 ⎥ ⎢ ⎥ < 0 (27) ˜ ϒ2 := ⎢ 0 0 33 ⎥ ⎢ −1 ⎣ − ( 0 0 ) 0 ⎦ −I ˜ 12 := ˜ 3 − ˜ T S˜ 0 and where 22 is defined in Proposition 1, 4 ⎡˜ ⎤ ϑ11 1 0 2 R˜ 1 ⎢ ⎥ ϑ˜ 22 ϑ˜ 23 0 R˜ 2 + S˜ 1T ⎢ ⎥ T ⎢ ⎥ ˜ ˜ ˜ 11 := ⎢ −S1 0 ϑ33 ⎥ ⎣ ⎦ ˜ 2 − R˜ 2 0 −Q −T −1 −H 0 H −1 ˜ 33 := −Z R˜ 1 + R˜ 2 + R˜ 3 Z, ˜ 1 := [0 1 0 0 2 ] ˜ 2 := [0 (CX C) 0 0 − I], ˜ 3 := col{3 , 0, 0, 0, 0}

0 0 0 0 ˜ 4 := LX + D2 C1 L ˜ 1 − R˜ 1 ,ϑ˜ 22 := −S˜ 1 − S˜ T − 2R˜ 2 − with ϑ˜ 11 := 0 + T0 + Q 1 2 ˜ 1 − R˜ 1 − ˜ 2 −Q (π /4)R˜ 3 , ϑ˜ 23 := R˜ 2 + S˜1 + (π 2 /4)R˜ 3 , ϑ˜ 33 := Q R˜ 2 − (π 2 /4)R˜ 3 , and AX + B2 C1 A 0 0 , 1 := 0 := YA C4 C3 C2 C 0 0 B1 , 3 := . 2 := YB1 C2 D0 −C2 Proof: See Appendix B. From Proposition 2, one can see that, if the matrix inequalities (26) and (27) are satisfied, then the controller gain matrices AK , BK , CK , and DK can be given by (64). However, the matrix ˜ 33 . inequality in (27) is nonlinear due to the nonlinear term in The following result presents an LMI-based approach to the design of suitable controllers. Proposition 3: For given scalars h > 0, 0 < β < 1, and σi > 0 (i = 1, 2, . . . , m) and real matrices Q0 , S0 , and R0 with Q0 = QT0 ≤ 0 and R0 = RT0 , the decentralized eventtriggered dissipative control problem is solvable if there exist ˜ 2 > 0, R˜ j > 0 ( j = ˜ 1 > 0, Q real matrices X > 0, Y > 0, Q 1, 2, 3), S˜ 1 , i > 0 (i = 1, 2, . . . , m), and Ck (k = 1, . . . , 4) of appropriate dimensions such that (26) and ⎡ ˜ ⎤ ˜ 11 ˜ 12 h˜ T ˜ T H −T 0 ˜ T Q 1 2 4 0 ⎢ ˜ ⎥ ˜ T H −T 0 LT Q 22 hT3 D 0 1 0⎥ ⎢ ⎢ ˜ 33 ϒ3 := ⎢ 0 0 ⎥ ⎥ < 0 (28) ⎣ ˜ 44 0 ⎦ −I

˜ 0 , and ˜ i (i = 1, 2, 3, 4) are ˜ 12 , 22 , 3 , D ˜ 11 , ˜ 0, Q where ˜ 33 := R˜ 1 + R˜ 2 + R˜ 3 − 2Z, defined in Proposition 2; and ˜ 44 := 0 (0 − 2I). Moreover, the stabilizing controller (10) can be readily obtained, which is algebraically equivalent to x˙ˆ c (t) = Ac xˆ c (t) + Bc xˆ c (t − η(t)) + Cc y˜ (t) (29) u(t) = Dc xˆ c (t), rj ≤ t < rj+1 , j ∈ N ∪ {0} with

⎧ Ac = (C4 − YAX − YBC1 )(I − YX)−1 ⎪ ⎪ ⎨ Bc = (C3 − C2 CX)(I − YX)−1 ⎪ Cc = C2 ⎪ ⎩ Dc = C1 (I − YX)−1 .

(30)

−1 Proof: Since (I − 0 )−1 0 (I − ) ≥ 0 and (Z − R0 )R0 (Z − R0 ) ≥ 0, where R0 := R˜ 1 + R˜ 2 + R˜ 3 , it is clear that −−1 ≤ 0 − 2I −1 0 (31) ˜ −Z R1 + R˜ 2 + R˜ 3 Z ≤ R˜ 1 + R˜ 2 + R˜ 3 − 2Z.

By Schur complement, it is clear that if ϒ3 < 0, then ϒ2 < 0, where ϒ2 and ϒ3 are defined in (27) and (28), respectively. Therefore, it can be concluded from Propositions 1 and 2 that the closed-loop system (15) subject to (18) is asymptotically stable and strictly (Q0 , S0 , R0 )-dissipative if the LMIs in (26) and (28) are satisfied. Moreover, from the proof of Proposition 2, the controller gain matrices AK , BK , CK , and DK of the controller (10) can be obtained from (64) if the LMIs (26) and (28) are feasible. Performing an irreducible linear transformation xc (t) = N −1 xˆ c (t) on the state in (10) with (64) yields (29) with (30). The proof is completed. Since the controller in (10) with (64) is algebraically equivalent to the one in (29) with (30), hereafter, we refer to (AK , BK , CK , DK ) as (Ac , Bc , Cc , Dc ) defined in (30) for convenience if the condition of Proposition 3 is satisfied. However, if we design a memoryless controller, i.e., BK = Bc = 0, from Proposition 3, there is an extra constraint imposed on C2 , C3 , and X, that is, C3 = C2 CX. This implies that a memory controller can achieve better system performance than that using a memoryless controller because such a constrain is not required. Remark 6: For given threshold parameters σi (i = 1, 2, . . . , m), Proposition 3 presents an LMI-based approach to designing the dissipative controllers for the system (1) under the decentralized event-triggering scheme (3). More specifically: 1) if setting Q0 = −γ −1 I, S0 = 0, and R0 = γ I, Proposition 3 can be used to design a decentralized event-triggered H∞ controller. However, although the decentralized event-triggered H∞ control is also investigated in [10] by employing an impulsive system approach, the obtained results are not easy to be used to design suitable controllers. Moreover, the transmission delays, which are not avoidable in a communication channel, are not taken into account in [10]; 2) if setting Q0 = 0, S0 = I, and R0 = γ I, Proposition 3 can be used to design a decentralized event-triggered passive controller. Nevertheless, although event-triggered passive control is studied for input feed-forward output feedback passive (IF-OFP)


systems via IF-OFP controllers in [28], on the one hand, the controller design is not involved, and on the other hand, the computation of the minimum interevent time is not an easy task because a number of constrains are imposed on the systems under consideration (see [28, Proposition 1]), which certainly limits the application scope of the proposed method; 3) if taking Q0 = −γ −1 αI, S0 = (1 − α)I and R0 = γ I, α ∈ (0, 1), Proposition 3 delivers an LMI-based approach to the design of a suitable event-triggered mixed H∞ and passive controller. Remark 7: Proposition 3 depends on the sampling period h > 0. From (9), it is clear that both lower and upper bounds of the artificial time-varying delay function η(t) are closely dependent on h. Thus, Proposition 3 is a delay-dependent condition for the design of event-triggered DOFCs. For a larger h, it is more possible that Proposition 3 fails to draw any conclusion on the controller design. On the other hand, a smaller sampling period usually leads to some better performance of the resultant closed-loop system under study while introducing a larger number of network loads. Therefore, in the application of the proposed method, since the event-triggering scheme can greatly reduce network loads, one should choose a small sampling period h to design suitable event-triggered DOFCs such that the resultant closed-loop system can achieve the desired system performance, which can be seen from the simulation in the next section. Remark 8: It should be mentioned that compared with Proposition 1, Proposition 3 is conservative because of the estimation in (31) [45]. To reduce the conservatism, one can employ a cone complementary approach [46] to convert the corresponding nonconvex feasibility problem into a nonlinear minimization problem subject to a set of LMIs, which, however, the cost is much time-consuming. V. I LLUSTRATIVE E XAMPLES Example 1: Consider the plant ⎧ 0 1 0 ⎨ x˙ (t) = x(t) + u(t), x(0) = φ0 −2 3 1 ⎩ y(t) = [ − 1 4]x(t). The output-based ETC for this example is in [47] and [48], where the controller is given by ⎧ 0 1 0 ⎨ x˙ c = x + y˜ 0 −5 c 1 ⎩ u = [1 − 4]xc .

(32) studied

(33)

With the controller (33), it is shown in [47] and [48] that the system (1) with (32) is asymptotically stable under a certain event-triggering mechanism. The lower bounds on the interevent times are 11.4 ms in [47] and 0.1 ms in [48], respectively. To make a comparison, we turn to the method proposed in this paper. Since y(t) ∈ R, the number of the nodes m is one. Suppose that the sampling period h is 20 ms, and the transmission delay is less than h. The threshold parameter σ1 in the event-triggering condition (3) is set to be σ1 = 0.01.

Fig. 3.

State responses and the time intervals for Example 1.

Fig. 4.

Sketch of the satellite control system.

7

Under the proposed event-triggering scheme, it is found that the lower bound on the interevent times is 20 ms, which is much larger than the ones in [47] and [48]. Moreover, over the time period [0, 30 s], there are 1500 sampled packets in all, but only 535 packets are released, which means that the communication resources can be saved by 64.33%. Fig. 3 shows the state response of the system (32) with (33) under the initial condition x(0) = col{25/2, −25/2}, and the corresponding time intervals between two consecutive triggering instants are also shown in this figure. Example 2: Suppose that the plant in Fig. 1 is a satellite control system [30]. A sketch of the satellite control system is shown in Fig. 4, which consists of two rigid bodies, i.e., the main body and the instrumentation module. These bodies are joined by a flexible link (the “boom”). The boom is modeled as a spring with torque constant k and viscous damping constant d [31]. The dynamic equations can be given as

J1 θ¨1 + dθ˙1 − θ˙2 + k(θ1 − θ2 ) = Tc J2 θ¨2 + d θ˙2 − θ˙1 + k(θ2 − θ1 ) = 0

(34)

where Tc is the control torque; J1 and J2 are inertias; and θ1 and θ2 are the yaw angles for the main body and the instrumentation module, respectively. Let the state vector be x = col{θ2 , θ˙2 , θ1 , θ˙1 } and the control input be u = Tc . In order to deal with the decentralized event-triggered dissipative control, suppose that an external disturbance w(t) is imposed on the system. The state-space representation of the extended system of (34) is described by (1) with L = [0.6 0 0.3 0], D0 = col{0.1, 0.2, 0.1}, D1 = 0.02, D2 = 0.01,



B1 = col{0.01, 0, 0.01, 0}, B2 = col{0, 0, 0, 1/J1 }, and ⎡ ⎤ 0 1 0 0 ⎡ ⎤ k d ⎥ d ⎢ k 0 0 1 0 ⎢− ⎥ − ⎢ J2 J2 J2 ⎥ A = ⎢ J2 ⎥, C = ⎣0 1 0 1⎦. 0 0 1 ⎥ ⎢ 0 1 0 1 0 ⎣ k d k d⎦ − − J1 J1 J1 J1 In this paper, we set J1 = J2 = 1, k = 0.09, and d = 0.0219. Then the four eigenvalues of A are −0.0219 + 0.4237j, −0.0219−0.4237j, 0, and 0. Thus, the above system is unstable. For convenience of analysis, the external disturbance w(t) and the initial condition x0 are set to be w(t) = Fig. 5. System’s state responses. e−0.5t | sin t| and x0 = col{0.2, −0.3, 0.3, −0.2}, respectively. Let y = col{y1 , y2 , y3 }. Then, it is clear that y1 = θ1 , y2 = θ˙1 + θ˙2 , y3 = θ1 + θ2 . Thus, the entries y1 and y3 include information on the yaw angles of two bodies while the entry y2 includes information on the angular velocity. This means that the entries y1 and y3 are of the same physical properties but different from those of y2 . Now we employ the decentralized event-triggering scheme proposed in this paper to study the dissipative control of the satellite system. For this purpose, the measurement output y(t) is grouped into two nodes yˆ 1 and yˆ 2 , where yˆ 1 = col{y1 , y3 } and yˆ 2 = y2 . Then, 1 0the0 corresponding matrix H in (2) is obtained as H = 0 0 1 . Suppose that the sampling period 0 1 0 h is 0.1s and the parameter β in Proposition 3 is 0.9. A. Dissipative Control Set Q0 = −0.8147, S0 = 0.4583, R0 = 1.9572, σ1 = 0.02, and σ2 = 0.05. Applying Proposition 3, the controller gain matrices (AK , BK , CK , DK ) and i (i = 1, 2) can be obtained as 0.5211 −0.2987 1 = , 2 = 0.3805 −0.2987 1.3221 ⎡ ⎤ 0.0504 0.0025 0.0501 0.0021 ⎢ 0.1068 0.0051 0.1070 0.0046 ⎥ ⎥ × 103 Ak = ⎢ ⎣ 0.0116 0.0005 0.0115 0.0006 ⎦ −1.6388 −0.0782 −1.6274 −0.0707 ⎡ ⎤ −0.0729 −0.0029 −0.0724 −0.0029 ⎢−0.0706 −0.0029 −0.0703 −0.0030⎥ ⎥ Bk = ⎢ ⎣−0.0710 −0.0028 −0.0714 −0.0029⎦ −0.0703 −0.0029 −0.0699 −0.0029 ⎡ ⎤ 0.2481 0.0165 −0.8483 ⎢ 0.0483 −0.2176 −0.6283⎥ ⎥ Ck = ⎢ ⎣−0.1828 0.0226 −0.6011⎦ 0.0493 −0.2180 −0.6171

Dk = 0.6867 0.0328 0.6814 0.0296 . Under the above controller and decentralized eventtriggering scheme, the state responses of the closed-loop system are plotted in Fig. 5. The time intervals between two consecutive event-triggered instants for two nodes are also illustrated in this figure. On the time interval [0, 150 s], there are 3000 sampled data packets from two nodes, however, just 237 data packets triggered from nodes 1 and 67 data packets

Fig. 6.

System’s state responses and the release time intervals.

triggered from node 2. That is, the average transmission rate is calculated as 10.13% over the time interval, which means that the communication resources can be saved up to 89.87%. B. H∞ Control Set Q0 = −γ −1 I, S0 = 0, and R0 = γ I with γ = 0.8. By Proposition 3 with σ1 = 0.01 and σ2 = 0.03, the eventtriggered H∞ controller can be designed, under which, the state responses of the closed-loop system, and the time distance between two consecutive release time instants are plotted in Fig. 6, from which, the communication resources can be saved by 77.9% over the time interval [0, 150 s]. C. Passive Control Set Q0 = 0, S0 = I, and R0 = γ I with γ = 0.8. Applying Proposition 3 with σ1 = 0.01 and σ2 = 0.08, a suitable passive controller is designed. Under the passive controller, the state trajectories of the closed-loop system are illustrated in Fig. 7. Moreover, over the time interval [0, 150 s], the average transmission rate can be computed as 24.67%. Thus, the communication resources are saved by 75.33%. D. Mixed H∞ and Passive Control Set Q0 = −αγ −1 I, S0 = (1 − α)I, and R0 = γ I with γ = 0.8 and α = 0.5. Applying Proposition 3 with σ1 = 0.01 and σ2 = 0.05 yields a desired controller, under which the state responses are depicted in Fig. 8. The average transmission rate over the time interval [0, 150 s] comes to 22.5%, meaning that up to 77.5% communication resources are saved.


9

A PPENDIX A P ROOF OF P ROPOSITION 1 Choose a discontinuous Lyapunov–Krasovskii functional candidate as 4

Vj (t, x˜ t ), t ∈ Is ⊂ rj , rj+1 , j ∈ N ∪ {0} V(t, x˜ t ) = i=1

where V1 (t, x˜ t ) = x˜ T (t)P˜x(t) and t T V2 (t, x˜ t ) = x˜ (ω)Q1 x˜ (ω)dω + Fig. 7.


V3 (t, x˜ t ) = h

t−h 0

t

x˙˜ T (ω)R1 x˙˜ (ω)dωdθ

−2h

rj +(s−2)h t−h

Fig. 8.


x˜˙ T (ω)R2 x˙˜ (ω)dωdθ

t+θ

t

π2 − 4

x˜ T (s)Q2 x˜ (ω)dω

t−2h

−h t+θ −h t

+h 2 V4 (t, x˜ t ) = h

t−h

x˙˜ T (ω)R3 x˙˜ (ω)dω

rj +(s−2)h

ϕ T (ω)R3 ϕ(ω)dω

where x˜ t := x˜ (t + θ ), θ ∈ [−2h, 0]; ϕ(t) := x˜ (t) − x˜ (rj +(s−2)h), and P > 0, Q1 > 0, Q2 > 0, R1 > 0, R2 > 0, and R3 > 0 to be determined. Note that t x˙˜ T (ω)R3 x˙˜ (ω)dω + V41 (t, x˜ t ), t ∈ Is V4 (t, x˜ t ) = h2 t−h

where Remark 9: From Example 2, it is clear that the eventtriggered H∞ controller, passive controller, and mixed H∞ and passive controller obtained above work well for the satellite system. Under the decentralized event-triggering scheme, one can choose a proper control method to control the satellite system when there is a need. If a prescribed H∞ disturbance attenuation level γ > 0 is expected, then the H∞ control can be applied; if the controlled satellite system is expected to be passive, then the passive control can be employed; if both a prescribed H∞ level γ > 0 and passivity are expected, then the mixed H∞ and passive control can be applied. VI. C ONCLUSION The problem of decentralized event-triggered dissipative control for systems with the entries of the system outputs having different physical properties has been addressed. A decentralized PETC scheme has been introduced to transmit those necessary data packets for the design of suitable controllers. A DPP has been used to generate signals to actuate the controller at some certain time instants. Under the mechanism of the DPP, the closed-loop system has been modeled as a time delay system, and thus the Lyapunov functional method has been employed to formulate a sufficient condition to ensure the asymptotic stability and the strict (Q0 , S0 , R0 ) dissipativity. An LMI-based co-design algorithm has been presented to design desired controllers for given event-triggering threshold parameters. The effectiveness of the proposed method has been demonstrated through two examples.

V41 (t, x˜ t ) := h2

t−h

x˜˙ T (ω)R3 x˙˜ (ω)dω

rj +(s−2)h

−

π2 4

t−h rj +(s−2)h

ϕ T (ω)R3 ϕ(ω)dω.

(35)

Clearly, for t ∈ Is , one has ϕ(t) ˙ = x˙˜ (t). Note that ϕ(t)|t=rj +(s−2)h = 0. Use Lemma 1 to obtain t−h ϕ T (s)R3 ϕ(ω)dω rj +(s−2)h

≤

4h2 π2

t−h

rj +(s−2)h

x˙˜ T (ω)R3 x˙˜ (ω)dω, t ∈ Is

which leads to V41 (t, x˜ t ) ≥ 0. Moreover, V41 (t, x˜ t ) vanishes at t = rj + (s − 1)h. Thus, we have that V(t, x˜ t ) ≥ V(t, x˜ t )|t=rj +(s−1)h . (36) lim − t→(rj +(s−1)h) On the other hand, it is clear to see that there exist two scalars c1 > 0 and c2 > 0 such that c1 ˜x(t)22 ≤ V(t, x˜ t ) ≤ c2 ˜xt 2W .

(37)

Taking the time derivative along the trajectory of the system (15) yields ˙ x˜ t ) = V(t,

4

V˙ i (t, x˜ t ),

t ∈ Is

(38)

i=1

where

V˙ 1 (t, x˜ t ) = 2˜xT (t)P A0 x˜ (t) + A1 x˜ (t − η(t))

+ A2 e(t) + A3 w(t) ˜ (39)



V˙ 2 (t, x˜ t ) = x˜ T (t)Q1 x˜ (t) − x˜ T (t − 2h)Q2 x˜ (t − 2h) + x˜ T (t − h)(Q2 − Q1 )˜x(t − h) (40) V˙ 3 (t, x˜ t ) = x˙˜ T (t) h2 R1 + h2 R2 x˙˜ (t) t −h x˙˜ T (ω)R1 x˙˜ (ω)dω t−h t−h −h x˙˜ T (ω)R2 x˙˜ (ω)dω (41) t−2h

T π2

V˙ 4 (t, x˜ t ) = h2 x˙˜ T (t)R3 x˙˜ (t) − x˜ (t − h) − x˜ (t − η(t)) 4

× R3 x˜ (t − h) − x˜ (t − η(t)) . (42) For the first integral term in (41), use Jensen inequality to get t −h x˙˜ T (ω)R1 x˙˜ (ω)dω t−h

T

≤ − x˜ (t) − x˜ (t − h) R1 x˜ (t) − x˜ (t − h) . (43)

For the second integral term in (41), apply the reciprocally convex approach in [43] to obtain t−h −h x˙˜ T (ω)R2 x˙˜ (ω)dω t−2h

T

≤ − x˜ (t − η(t)) − x˜ (t − 2h) R2 x˜ (t − η(t)) − x˜ (t − 2h)

T

− x˜ (t − h) − x˜ (t − η(t)) R2 x˜ (t − h) − x˜ (t − η(t))

T

+ 2 x˜ (t − η(t)) − x˜ (t − 2h) S1 x˜ (t − h) − x˜ (t − η(t)) . (44) Substituting (39)–(42) into (38), and taking (43) and (44) into account, we have that ˙ x˜ t ) ≤ ξ T (t)11 ξ(t) + h2 x˙˜ T (t)(R1 + R2 + R3 )x˙˜ (t) V(t, + 2˜xT (t)PA3 w(t) ˜ + eT (t)H −T 0 H −1 e(t), t ∈ Is

which means that there exists a scalar > 0 such that ˙ x˜ t ) < −ξ T (t)ξ(t) ≤ −||˜x(t)||22 , t ∈ Is . V(t, Integrating both sides of (47) yields V(t, x˜ t ) − V(t, x˜ t )|t=rj +(s−1)h ≤ −

t rj +(s−1)h

ξ(t) := col{˜x(t), x˜ (t − η(t)), x˜ (t − h), x˜ (t − 2h), e(t)}. In what follows, we prove the conclusion from two aspects: 1) the system (20) subject to (18) is asymptotically stable and 2) under zero initial conditions, the closed-loop system (15) subject to (18) is strictly (Q0 , S0 , R0 )-dissipative. Proof of 1): Set w(t) ≡ 0. Then from (18), one has eT (t)H −T 0 H −1 e(t) ≤ ξ T (t)2T H −T 0 0 H −1 2 ξ(t) (46)

(48)

which, together with (37), leads to ˜ t ) ≤ c−1 ˜ t )|t=rj +(s−1)h . ||˜x(t)||22 ≤ c−1 1 V(t, x 1 V(t, x ≤ Taking (36) into account, one obtains ||˜x(t)||22 −1 − V(t, x ˜ )| ≤ c V(t, x ˜ )| ≤ c−1 t t=rj +(s−1)h t t=(rj +(s−1)h) 1 1 −1 −1 c1 V(t, x˜ t )|t=(rj +(s−2)h) ≤ · · · ≤ c1 V(t, x˜ t )|t=rj ≤ · · · ≤ c−1 ˜ t )|t=r0 ≤ c−1 xr0 ||2W , which means that the 1 V(t, x 1 c2 ||˜ system (20) is stable. Now we prove the asymptotic stability xr0 ||2W , of the system (20). First, since ||˜x(t)||22 ≤ c−1 1 c2 ||˜ from (20), it is clear that there exists a scalar c3 > 0 such that ||˙x(t)||2 ≤ c3 , which implies that x˜ (t) is uniformly continuous on [0, ∞). Second, from (48), it can be deduced that ∞

||˜x(t)||22 dt ≤ −1 V(t, x˜ t )|t=0 − V(t, x˜ t )|t=∞ 0

≤ −1 V(t, x˜ t )|t=0 < ∞

which means that ||˜x(t)||22 is integrable on [0, ∞). By Barbalat’s lemma [44], x˜ (t) → 0 as t → ∞. That is, the system (20) subject to (18) is asymptotically stable if (22) is satisfied. Proof of 2): Set w(t) = 0 and denote ζ (t) := col{ξ(t), w(t)}. ˜ Then from (18), one has

T ˜ 0 w(t) ˜ eT (t)H −T 0 H −1 e(t) ≤ 2 ξ(t) + D

˜ 0 w(t) × H −T 0 0 H −1 2 ξ(t) + D ˜ . (49)

(45) where 11 is defined in (23) and

||˜x(s)||22 ds

(47)

Then, (45) can be rewritten as, for t ∈ Is ⎧ ⎨ ˙ x˜ t ) ≤ ζ T (t) 11 3 V(t, 0 ⎩

T T 1T 1 (R + R + R ) 1 2 3 T A3 AT3 ! " ! "T ⎫ ⎬ T 2T −T −1 2 + H H ζ (t) (50) 0 0 ˜T ˜T ⎭ D D 0 0 + h2

where 2 is defined in (24). It follows from (45) that: ˙ x˜ t ) ≤ ξ T (t) 11 + h2 1T (R1 + R2 + R3 )1 V(t, + 2T H −T 0 0 H −1 2 ξ(t).

where 3 is defined in (25). On the other hand, if the matrix inequality (22) holds, then there exists a scalar θ > 0 such that

0 , 0, 0, 0 < 0. (51) ϒ1 + diag 0, θI 0 0

If (22) holds, then ⎡ 11 ⎣

Applying the Schur complement to (51), we have T T T 2 12 11 −T −1 2 ˜ ϒ1 := 0] + D ˜ T H 0 0 H ˜T 22 + [ θI D 0 0 0 0 T T T T T T ˜ 2 4 + h2 1T (−33 )−1 1T + T4 Q 0 such that (59) is satisfied. By Definition 1, the closed-loop system (15) subject to (18) is strictly (Q0 , S0 , R0 )dissipative. The proof is thus completed. A PPENDIX B

rj

+ ≥

(1 − β)hwT (rj + (s − 2)h)R0 w rj + (s − 2)h

s=1 rj+1

˙ x˜ t ) + θ wT (t)w(t) dt. V(t,

(56)

rj

For ∀T > 0, there exist integers j0 > 0 and s0 ≥ 1 such that T ∈ [rj0 + (s0 − 1)h, rj0 +1 + s0 h). Similar analysis to (56), we have that T

T z (t)Q0 z(t) + 2zT (t)S0 w(t) + βwT (t)R0 w(t) dt rj0 s0 −1

+

(1 − β)hwT rj0 + (i − 2)h R0 w rj0 + (i − 2)h

i=1

+ (1 − β) T − rj0 + (s0 − 1)h wT rj0 + (s0 − 2)h × R0 w rj0 + (s0 − 2)h T

˙ x˜ t ) + θ wT (t)w(t) dt. V(t, (57) ≥ rj0

Summing both sides of (56) with regard to j from 0 to j0 − 1, together with (57), yields T

T z (t)Q0 z(t) + 2zT (t)S0 w(t) + βwT (t)R0 w(t) dt 0

+ (1 − β)(w) T

˙ x˜ t ) + θ wT (t)w(t) dt V(t, ≥ 0

hwT rj0 + (i − 2)h R0 w rj0 + (i − 2)h

i=1

˜ = S0 w(t) and Note that S˜ 0 w(t)

ρj

11

T

= V(t, x˜ t )|t=T − V(t, x˜ t )|t=0 + θ 0

wT (t)w(t)dt

(58)

P ROOF OF P ROPOSITION 2 Sufficiency: Suppose that there exist real matrices P > 0, Q1 > 0, Q2 > 0, Rj > 0 ( j = 1, 2, 3), and S1 such that R2 RS12 ≥ 0 and (22) is satisfied. Now partition P as P = NYT YN1 , where Y, N, Y1 ∈ Rn×n . It is clear that Y > 0 and Y1 > 0. If N is singular, then there exists a sufficient ς0 > 0 such that N + ς0 I is nonsingular and 0 ς0 I (P + ς0 I 0 , Q1 , Q2 , R1 , R2 , R3 , S1 ) satisfies (22). Thus,

without loss generality, suppose that N is nonsingular. Introduce X > 0 such that Y1 = N T (Y − X −1 )−1 N. Clearly, Y − X −1 > 0 due to Y1 > 0 and N nonsingular. By the Schur complement, Y − X −1 > 0 ⇐⇒ Z > 0, where Z is defined in (26). Let X I I Y := . (60) , J J1 := 2 0 NT N −1 (I − YX) 0 It is clear that both J1 and J2 are nonsingular. Moreover, it can be verified that P = J2 J1−1 = J1−T J2T , Z = J2T J1 . Denote T ⎧ C ⎪ 1 ⎪ ⎨ C2 C ⎪ ⎪ ⎩ 3 C4

:= diag{J1 , J1 , J1 , J1 , I, I, J2 , I, I} := DK N −1 (I − YX) := NCK := C2 CX + NBK N −1 (I − YX) := YAX + YB2 C1 + NAK N −1 (I − YX).

(61) (62)



Performing a congruence transformation on ϒ1 defined in (22) by a nonsingular matrix T and after some simple algebraic manipulations, one obtains T T ϒ1 T = ϒ2

(63)

˜ 1 = J T Q1 J1 , Q ˜2 = (27), where Q 1 T ˜ = 1, 2, 3), and S1 = J1 S1 J1 . Thus

where ϒ2 is defined in J1T Q2 J1 , R˜ i = J1T Ri J1 (i the sufficiency of Proposition 2 is proved. Necessity: Suppose that there exist X > 0, Y > 0, ˜ 2 , R˜ i (i = 1, 2, 3), S˜ 1 , and Cj ( j = 1, . . . , 4) such ˜ 1, Q Q that (26) and (27) are satisfied. Then it is clear that the matrix I − YX is invertible due to Z > 0. Let N ∈ Rn×n be a nonsingular matrix and define two nonsingular matrices J1 and J2 given in (60). Denote P = J2 J1−1 , S1 = J1−T S˜ 1 J1−1 , Qi = ˜ i J −1 (i = 1, 2), Rj = J −T R˜ j J −1 ( j = 1, 2, 3), and J1−T Q 1 1 1 ⎧ −1 AK = N (C4 − YAX − YB2 C1 )(I − YX)−1 N ⎪ ⎪ ⎨ BK = N −1 (C3 − C2 CX)(I − YX)−1 N (64) C = N −1 C2 ⎪ ⎪ ⎩ K −1 DK = C1 (I − YX) N. Then, P > 0 and ϒ1 = T −T ϒ2 T −1 < 0. That is, there exist ˜ i J −1 (i = 1, 2), Rj = real matrices P = J2 J1−1 > 0, Qi = J1−T Q 1 −T ˜ −1 −T ˜ −1 J1 Rj J1 ( j = 1, 2, 3), and S1 = J1 S1 J1 such that (22) is satisfied. The proof is thus completed. R EFERENCES [1] E. Hendricks, M. Jensen, A. Chevalier, and T. Vesterholm, “Problems in event based engine control,” in Proc. Amer. Control Conf., vol. 2. Baltimore, MD, USA, 1994, pp. 1585–1587. [2] K. J. Åström and B. M. Bernhardsson, “Comparison of periodic and event based sampling for firstorder stochastic systems,” in Proc. IFAC World Congr., Beijing, China, 1999, pp. 301–306. [3] K.-E. Arzén, “A simple event-based PID controller,” in Proc. IFAC World Congr., vol. 18. Beijing, China, 1999, pp. 423–428. [4] X. Ge, F. Yang, and Q.-L. Han, “Distributed networked control systems: A brief overview,” Inf. Sci., Doi: 10.1016/j.ins.2015.07.047. [5] M. Velasco, J. M. Fuertes, and P. Marti, “The self triggered task model for real-time control systems,” in Proc. Real-Time Syst. Symp. (RTSS), Cancún, Mexico, 2003, pp. 67–70. [6] M. Lemmon, T. Chantem, X. Hu, and M. Zyskowski, “On self-triggered full-information H∞ controllers,” in Hybrid Systems: Computation and Control. Berlin, Germany: Springer, 2007 pp. 371–384. [7] X. Wang and M. Lemmon, “Self-triggered feedback control systems with finite-gain L2 stability,” IEEE Trans. Autom. Control, vol. 54, no. 3, pp. 452–467, Mar. 2009. [8] C. Fiter, L. Hetel, W. Perruquetti, and J.-P. Richard, “A state dependent sampling for linear state feedback,” Automatica, vol. 48, no. 8, pp. 1860–1867, 2012. [9] A. Anta and P. Tabuada, “To sample or not to sample: Self-triggered control for nonlinear systems,” IEEE Trans. Autom. Control, vol. 55, no. 9, pp. 2030–2042, Sep. 2010. [10] W. P. M. H. Heemels, M. C. F. Donkers, and A. R. Teel, “Periodic event-triggered control for linear systems,” IEEE Trans. Autom. Control, vol. 58, no. 4, pp. 847–861, Apr. 2013. [11] D. Yue, E. Tian, and Q.-L. Han, “A delay system method for designing event-triggered controllers of networked control systems,” IEEE Trans. Autom. Control, vol. 58, no. 2, pp. 475–481, Feb. 2013. [12] C. Peng and Q.-L. Han, “A novel event-triggered transmission scheme and L2 control co-design for sampled-data control systems,” IEEE Trans. Autom. Control, vol. 58, no. 10, pp. 2620–2626, Oct. 2013. [13] L. Zou, Z. Wang, H. Gao, and X. Liu, “Event-triggered state estimation for complex networks with mixed time delays via sampled data information: The continuous-time case,” IEEE Trans. Cybern., Doi: 10.1109/TCYB.2014.2386781. [14] C. Peng, Q.-L. Han, and D. Yue, “To transmit or not to transmit: A discrete event-triggered communication scheme for networked Takagi–Sugeno fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 21, no. 1, pp. 164–170, Feb. 2013.

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13

Xian-Ming Zhang received the M.S. degree in applied mathematics and the Ph.D. degree in control theory and engineering from Central South University, Changsha, China, in 1992 and 2006, respectively. In 1992, he joined Central South University, where he was an Associate Professor with the School of Mathematics and Statistics. From 2007 to 2013, he was a Senior Post-Doctoral Research Fellow with the Centre for Intelligent and Networked Systems, and a Lecturer with the School of Engineering and Technology, Central Queensland University, Rockhampton QLD, Australia. In 2014, he joined Griffith University, Gold Coast, QLD, Australia, where he is currently a Lecturer with the Griffith School of Engineering. His current research interests include H-infinity filtering, event-triggered control, networked control systems, neural networks, distributed systems, and timedelay systems. Dr. Zhang was a recipient of the First Hunan Provincial Natural Science Award in Hunan Province, China, in 2011, and the Second National Natural Science Award in China in 2013, both jointly with Profs. M. Wu and Y. He.

Qing-Long Han (SM’13) received the B.Sc. degree in mathematics from Shandong Normal University, Jinan, China, in 1983, and the M.Sc. and Ph.D. degrees in control engineering and electrical engineering from the East China University of Science and Technology, Shanghai, China, in 1992 and 1997, respectively. From 1997 to 1998, he was a Post-Doctoral Researcher Fellow with LAII-ESIP, Université de Poitiers, Poitiers, France. From 1999 to 2001, he was a Research Assistant Professor with the Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, USA. From 2001 to 2014, he was a Laureate Professor, the Associate Dean (Research and Innovation) with the Higher Education Division, and the Founding Director of the Centre for Intelligent and Networked Systems, Central Queensland University, Rockhampton, QLD, Australia. In 2014, he joined Griffith University, Gold Coast, QLD, Australia, where is currently the Deputy Dean (Research) with the Griffith Sciences, and a Professor with the Griffith School of Engineering. In 2010, he was appointed as the Chang Jiang (Yangtze River) Scholar Chair Professor by the Ministry of Education, Beijing, China. In 2011, he was appointed as the “100 Talents Program” Chair Professor by Shanxi Province of China. His current research interests include networked control systems, neural networks, timedelay systems, multiagent systems, and complex systems. Prof. Han is one of the World’s Most Influential Scientific Minds: 2014 and is a Highly Cited Researcher in the field of Engineering according to Thomson Reuters.