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Chen Peng and Qing-Long Han∗. Abstract—This paper is concerned with the event-triggered output feedback H∞ control for sampled-data control systems.

2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013

Output-based event-triggered H∞ control for sampled-data control systems with nonuniform sampling Chen Peng and Qing-Long Han∗ Abstract— This paper is concerned with the event-triggered output feedback H∞ control for sampled-data control systems with nonuniform sampling. Firstly, an output-based eventtriggered communication scheme is proposed, in which the state of the system is sampled with a nonuniform sampling period. Under this scheme, whether or not the sampled data should be transmitted is determined by a predetermined outputbased event-triggering condition. Secondly, an event-triggered networked control system is modeled as a time-delay system by taking into consideration the output-error between the output at the current sampling instant and the output at the last transmitted sampling instant. Thirdly, stability and stabilization criteria are derived to guarantee the uniform ultimate bounded stability and the desired performance while using less communication resources. Finally, two illustrative examples are used to show the effectiveness of the proposed method.

I. I NTRODUCTION A sampled-data control system is a control system where continuous-time plant is controlled with a digital device. In the development of sampled-data control systems, a general assumption is that the system takes samples at a constant rate and all the sampled-data by the sensors is transmitted over communication networks [1], [2], [3], [4], [5]. The advantage of periodic sampling procedures is that the well-developed theory on sampled-data control systems may be applied. However, from a resource allocation point of view, if all the sampled-data is transmitted over the communication networks, unnecessary utilization of the limited communication resources occur. For example, when no disturbances are acting on the system and the system approaching its operating point, periodic communication is clearly waste of communication resources [6]. To mitigate the over-provisioning of the computation and communication resources, as an alternative to traditional time-triggered control and communication schemes, event-triggered control and communication schemes are proposed, see, for example [6], [7], [8], [9], [10], [11]. This work was supported in part by the Australian Research Council Discovery Project under Grant DP1096780, the Research Advancement Awards Scheme Program (January 2010 - December 2012) at Central Queensland University, Australia; and the National Science Foundation of China under Grants 61074024 and 61273114. ∗ Corresponding author, Tel: +61 7 4930 9270; E-mail: [email protected] C. Peng is with the School of Mechatronic Engineering and Automation, Shanghai University, 200072, PR China; and also with the Centre for Intelligent and Networked Systems, Central Queensland University, Rockhampton QLD 4702, Australia (e-mail: [email protected]). Q.-L. Han is with the Centre for Intelligent and Networked Systems and the School of Engineering and Technology, Central Queensland University, Rockhampton QLD 4702, Australia.

978-1-4799-0178-4/$31.00 ©2013 AACC

In the implement of an event-triggered communication scheme, the signal is transmitted after the occurrence of an external event, generated by a pre-designed event-triggering condition, rather than the elapse of times as in conventional periodic communication. Generally, the existing eventtriggering conditions in the literature can be categorized into three types. The first type is current sampled-data-based method, where the current sampled data is transmitted only when the norm of the current sampled data is larger than a prescribed threshold, e.g. [8], [12]; The second type is absolute-difference-based method. For example, in [13], [14], the events are generated when the norm of the difference between the current sampled data and the previously transmitted one is larger than a certain threshold; The third type is relative-difference-based method, where the sampled data is transmitted when the ratio of the norm of the difference between the current state of plant and the previously transmitted one to the norm of the current sampled data violates a certain threshold, e.g. in [7], [9]. As an improvement, the eventtriggering mechanism used in [6] unifies two mechanisms in [7], [8]. In addition, another difference between the work aforementioned lies in the fact that in [6], [7], [8], [9], a hardware-dependent event detector must be deployed to generate an interrupt to release the transmisstion/control task, and a minimum inter-event time must be evaluated to avoid Zeno behavior [6], whereas in [10], [11], [15], since the system takes samples at a constant rate, it is clear that the minimum inter-event time is the constant sampling interval. Therefore, there is no Zeno behavior. In some applications, it is unreasonable or impractical to redeploy an existing system with such specialized hardware [16]. In particular, from the view point of resource scheduling, if there are multi systems to transmit signals over a shared network, time-varying sampling is more convenient to schedule than a constant sampling period. It is therefore also used in this study. In some practical applications, there are some states which are not available for feedback. Notice that most of the aforementioned results are in state-feedback cases. There are few results on output-feedback cases. Among the few results are Kofman and Braslavsky [17] on event-driven sampled-data scheme under data-rata constraints, Donders and Heemels [6] on output-based event-triggered control with decentralized event-triggering. However, one of common features in the aforementioned work is that they are ‘emulationbased approach’ in nature [6], i.e., to design a controller first without considering the communication load, and then to determine an event-triggering condition and/or network

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conditions to guarantee the stability and to maintain certain performance. Therefore, a disadvantage of above mentioned methods is that control analysis and design are more difficult when considering an event-triggering communication scheme. These issues motivate the current study. The purpose of this paper is to investigate the problem of output-based event-triggered H∞ control for sampled-data control systems with nonuniform sampling. The state of the system is taken sample with a nonuniform sampling period while whether or not the sampled data should be transmitted is determined by a predetermined well-designed output-based event-triggering condition. We propose an event-triggering mechanism that invokes transmissions of sampled-data when the difference between the current value of the output and its previously transmitted value becomes too large compared to the last transmitted value of the output, which represents a typical communication scheme for a sampled-data control system with nonuniform sampling. The sampling period is assumed to has nonzero lower and upper bounds, which is more general than the assumption of periodic sampling [10], [11], [15]. By making use of a time-delay modeling approach, a new output-error dependent time-delay model regarding the output-error between the output at the current sampling instant and the output at the last transmitted sampling instant is formulated. It is, therefore, convenient to consider the communication and control in a unified framework. In particular, a linear matrix inequality (LMI)based procedure is proposed for assessing H∞ stability of an NCS with time-varying transmission period, such that efficient verification is possible. Furthermore, different from some existing ones [6], [7], [9], [15], the output-feedback controller can be designed in this study when considering the communication and control simultaneously, for assuring the desired H∞ performance while to avoid the overprovisioning of limited transmission bandwidth. Finally, two illustrative examples are presented to show the effectiveness of the proposed event-triggered output-feedback H∞ control method. II. O UTPUT- BASED EVENT TRIGGERED

CONTROL

In this section, we will propose an output-based eventtriggered communication scheme to generate the transmission events by utilizing information about the system’s outputs at a time-varying sampling period and the output error between the output at the current sampling instant and the output at the last transmitted instant. A. Output-based event triggered communication scheme with nonuniform sampling Consider a linear time-invariant (LTI) plant given by  ˙ = Ax(t) + Bu(t) + B1 ω(t)  x(t) z(t) = Cx(t) + Dω(t) (1)  y(t) = Lx(t), t ≥ t0

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control input vector, z(t) ∈ Rq is the controlled output vector, and y(t) ∈ Rp is the measured output vector, ω(t) ∈ L2 [0, ∞)

is the exogenous disturbance, A, B, B1 , C, D are constant matrices with appropriate dimensions, and L is a full row rank. The initial condition of the system (1) is given by x(0) = x0 . Assume that before the transmitted signals arrive at the actuator, the control input is generated by a zero-order holder (ZOH) with the holding time t ∈ Ω , [tk , tk+1 ), where tk is the transmitted instant of the sensor. Consider an event-triggered static output-feedback control law of the form u(t) = Ky(tk ) = KLx(tk ), t ∈ Ω

(2)

where K is a constant matrix gain to be determined, which internally stabilizes the system (1) and guarantees that the H∞ norm of the transfer matrix from ω to z is less than a perdescribed level γ for x(t0 ) = 0 and all nonzero ω(t) ∈ L2 [t0 , ∞). In a conventional time-triggered control system, there are two generally assumptions, one is the sampling times are distributed equidistantly in time, i.e. tk+1 = tk +h for a constant sampling period h > 0; another is all of the sampling signals are transmitted over the communication network. However, the above mentioned assumptions are no longer needed in this paper. That is, the sampling period hl is allowed to be time-varying on a given interval, and whether the sampled signals should be transmitted or not is determined by an event-triggered communication mechanism, which will be introduced as follows. The conceptual framework of the proposed output-based event-triggered communication scheme is shown in Fig. 1, where the sensor takes samples at a time-varying sampling period hl and the event generator detects the output error between the output at the current sampling instant and the output at the last triggered sampling instant tk . When the difference between the current sampled output and their last transmitted sampled output becomes too large in an appropriate sense, we pursue an event-triggered communication mechanism that invokes transmissions of current sampled data. In other words, the above-mentioned output error will determine whether or not it is indeed necessary to transmit the sampled output to update the control law. For easy of presentation, at every transmission instant tk , we denote succedent sampling instants as h1 , h2 , ... until a new triggered transmission instant tk+1 . In particular, the eventtriggered communication mechanism proposed in this paper determines the next transmission instant tk+1 being designed as l−1 X hj +inf {hl (L+ e(il ))T ΦL+ e(il ) ≥ δY + ε3 } tk+1 = tk + j=0

l

(3) where e(il ) , y(il ) − y(tk ), Y = (L+ y(tk ))T ΦL+ y(tk ), tk is the latest transmitted instant, il is the current sampling l P instant and il = tk + hs , s is a nonnegative integer, s=0

l ∈ N+ , h0 = 0, hs is a time-varying sampling period, δ ∈ (0, 1) is a given scalar and Φ > 0 is a weighting matrix, ε ≥ 0 is a given scalar, L+ is the M-P inverse of L.

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From (3), it is clear that only part of the sampling data satisfying the predetermined threshold condition have the right to be transmitted over the communication networks. The parameters δ, Φ and ε determine how frequently the sampled data should be transmitted.

data should be transmitted or not, we divide the holding interval Ω of the ZOH into subsets Ωm = [im , im + hm+1 ), i.e., Ω = ∪Ωm , m = 0, 1, . . . , tk+1 − tk − 1, where im is defined just after (3). Define η(t) , t − im , t ∈ Ωm . Then, the control law (2) can be represented as u(t) = KLx(t − η(t)) − Ke(im ), t ∈ Ωm

Fig. 1. scheme

A conceptual framework of an event-triggered communication

Remark 1: As special cases, when δ → 0+ and ε = 0, which mean that all of sampling values should be transmitted, then the event-triggered communication scheme proposed in this paper is simplified to a time-triggered transmission scheme; when the system (1) is continuously sampled and Φ = I, ε = 0, the event-triggering condition in (3) can be seen as an extension of the event-triggering mechanism of [7] for output-based controllers. Remark 2: If the system sampling is equidistant in time and L is an identity matrix, that is, hl = h and L = I, then the event-triggering condition in (3) can be simplified as the discrete event-triggering condition of [10], [11] for state feedback controllers. Moreover, if the output of the system (1) can be continuously measured, and Φ is appropriately chosen to make (L+ )T ΦL+ = I, then the communication scheme (3) can be formulated as tk+1 = inf{t > tk eT (t)e(t) ≥ δy T (tk )y(tk ) + ε3 } (4)

which is in practice equivalent to the event-triggering condition in [6]. Thus, the event-triggering condition in [6] can be seen as a special case of the proposed event-triggering condition (3). However, there exist substantival differences between (3) and (4). One main difference is that the testing process based on (3) is a discrete time approach. It makes a distinction between sampling and transmission, and fully considers the non-uniform discrete sampling property of sampled-data control systems. Therefore, it is more practical than the event-triggered communication scheme in [6]. B. An output-error dependent time-delay system formulation In this section, we will reformulate the event-triggered control system as an output-error dependent time-delay system, which is convenient to consider the proposed communication and control schemes in a unified framework. Based on the event-triggered communication scheme (3), it is clear that there are some non-transmitted sampled-data between two transmitted sampled data. In order to introduce the proposed event-triggered communication scheme at every sampling instant to determine whether the current sampled

(5)

From the definition η(t), it is clear that η(t) is a piecewiselinear function satisfying η(t) ˙ = 1, 0 ≤ η(t) ≤ hm+1 ≤ η¯, for t ∈ Ωm and t 6= im (6) where η¯ = supm {hm+1 }. Combining (1) and (5) together, for t ∈ Ωm and m = 0, 1, 2, ..., we have the following output-error-dependent closed-loop system  ˙ = Ax(t) + BKLx(t − η(t)) − BKe(im ) + B1 ω(t)  x(t) z(t) = Cx(t) + Dω(t)  y(t) = Lx(t) (7) For the system (7), we supplement the initial condition of the state x(t) as [18] x(t0 + θ) = φ(θ), θ ∈ [−¯ η , 0] (8) where φ ∈ W with W denoting the Banach space of absolutely continuous functions [−¯ η, 0] → Rn with squareintegrable derivative andZ with the norm [19] Z kφk2W = kφ(0)k2 +

0

−¯ η

kφ(s)k2 ds +

0

−¯ η

2 ˙ kφ(θ)k dθ

where the vector norm k · k represents the Euclidean norm. Remark 3: Notice that the system (1) controlled with an output-based event-triggered control law has been modeled as an output-error-dependent system (7) with a sawtooth structured time-varying delay, which includes the timetriggered control model as a special case. For instance, if all the sampled data is transmitted, we have e(im ) = 0. Then the model (7) is simplified as the model in [20]. However, since the model (7) depends on x(t − η(t)) and e(im ) at the same time, and the communication condition (3) depends on e(im ). It is convenient to couple the proposed event-triggered communication scheme and use of the sawtooth structure characteristic of η(t) in the derivation of the results. To make the theoretical development easier, similar to [21], we introduce the following uniform ultimate boundedness definition and lemma to judge the state of the system (7) with ω(t) = 0 is uniformly ultimately bounded (UUB). Definition 1: For the system (7) with ω(t) = 0, if there exists a compact set U ∈ Rn such that for all x(t0 + θ) = η, 0], there exists an ε > 0 and a number xt0 ∈ U, θ ∈ [−¯ T (ε, xt0 ) such that kx(t)k < ε, ∀t ≥ t0 +T , it is said that the state of the system (7) with ω(t) = 0 is uniformly ultimately bounded. Lemma 1: Let V (t, xt ), where xt = x(t + θ), θ ∈ [−¯ η , 0], be a Lyapunov functional of the system (7) with ω(t) = 0 and satisfy ζ1 (kx(t)k) ≤ V (t, xt ) ≤ ζ2 (kxt kW ) (9) V˙ (t, xt ) ≤ −ζ3 (kx(t)k) + ζ3 (ε)

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where ε > 0 is a positive constant, ζ1 (·) and ζ2 (·) are continuous, strictly increasing functions, and ζ3 (·) is a continuous, nondecreasing function. If V˙ (t, xt ) < 0, for kx(t)k ≥ ε

(10)

then the state of the system (7) with ω(t) = 0 is UUB. Proof : Similar to the proof [21], it is omitted due to page limitation. Then under the proposed event-triggered communication framework and the UUB definition, the purpose of this paper is to pursue the stability analysis and controller synthesis for the system (1), such that i) The state of the system (7) with ω(t) = 0 is uniformly ultimately bounded; ii) Under the zero initial state condition, kz(t)k ≤ γ kω(t)k for any nonzero ω(t) ∈ L2 [0, ∞) and a prescribed γ > 0. III. H∞

where

Γ1 Γ2

= =

with Π11 Π21

Ξ11

=

Ξ41

=



Π11 Π21 Π31 0 B1T P1

     



Π + Γ1 + ΓT1 + ΓT2 Γ2 + (1 − i)Σ T (1 − i)¯ η QI2 + i¯ ηMi+2

∗ ∗ Π22 ∗ −δΦL+ L (δ − 1)Φ 0 0 0 0

∗ ∗ ∗ −P2 0

∗ ∗ ∗ ∗ −γ 2 I  −M1 0

M2 + M3 M1 − M2 − M3 0  C 0 0 0 D , Σ = 2¯ η I1 RI2

= P1 A + AT P1 + P2 − R + εI = LT K T B T P1 + R, Π31 = −LT K T B T P1

= δ(L+ L)T ΦL+ L − R  T I −I 0 0 0 I1 =  A BKL −BKL 0 I2 = Proof. See Appendix. Π22

where ˆ Π

STABILITY ANALYSIS AND CONTROL SYNTHESIS

In this section, under the proposed event-triggered communication scheme (3), we will first derive a stability criterion for the closed-loop system (7). Then based on the stability criterion, we will give a sufficient condition on the existence of an output feedback controller. We now state and establish the following stability criterion. Theorem 1: For some given constants γ > 0, δ ∈ (0, 1), η¯ > 0, and a matrix K, under the communication scheme (3), the state of system (7) is UUB with an H∞ norm bound γ, if there exist matrices P1 > 0, P2 > 0, Q > 0, Φ > 0, R > 0, S > 0, matrices M1 , M2 and M3 with appropriate dimensions such that   Ξ11 ∗ ∗ ∗  η¯SI2 −¯ ηS ∗ ∗   < 0, i = 0, 1  (11) T  η¯Mi+1 0 −¯ ηS ∗  Ξ41 0 0 −¯ ηQ

Π =

Based on Theorem 1, we are in a position to design the output feedback gain K to ensure that the output of the system (7) is UUB with an H∞ norm bound γ. The proof is similar to those in [10], thus it is omitted. Theorem 2: For some given constants γ > 0, δ ∈ (0, 1), η¯ > 0, under the communication scheme (3), the state of the system (7) is UUB with an H∞ norm bound γ and static output feedback gain K = Y X −1 L+ , if there exist matrices ˜ > 0, Φ ˜ > 0, R ˜ > 0, S˜ > 0, matrices M ˜ 1, X > 0, P˜2 > 0, Q ˜ ˜ M2 and M3 with appropriate dimensions such that  ˆ +Γ ˆ1 + Γ ˆ T1 Π ∗ < 0, i = 1, 2 (12) Ξi21 −Ξi22

B1



     

ˆ1 Γ ˆ2 Γ Ξ121 Ξ122 Ξ221 Ξ222

 ˆ Π11 ˆ  Π  21  ˆ =  Π31  0

 ∗ ∗ ∗ ∗ ˆ 22 Π ∗ ∗ ∗   + ˜ ˜ −δ ΦL L (δ − 1)Φ ∗ ∗   0 0 −P˜2 ∗  B1T 0 0 0 −γ 2 I   ˆ1 − M ˆ2 − M ˆ 3 0 −M ˆ1 0 ˆ2 + M ˆ3 M = M   CX 0 0 0 DX = √ ˜ T , η¯I˜2 , Γ ˆ 2 , η¯ RI ˜ T , I˜2 , εX} = col{¯ η I˜2 , η¯M 1 1 ˜ η¯X Q ˜ −1 X, I, Xλ−1 X, λI, I} = diag{¯ ηX S˜−1 X, η¯S, ˜ T , η¯M ˜ T,Γ ˆ 2 , X} = col{¯ η I˜2 , η¯M 2

3

˜ η¯Q, ˜ I, I} = diag{¯ ηX S˜−1 X, η¯S,

with ˆ 11 Π ˆ 31 Π Iˆ2

= = =

˜ Π ˆ 21 = Y T B T + R ˜ AX + XAT + P˜2 − R, ˆ 22 = δ(L+ L)T ΦL ˜ + L − R, ˜ −Y T B T , Π   AX BY −BY 0 B1

Remark 4: Notice that the matrix inequalities (12) are non-convex inequalities because of the nonlinear items ˜ −1 X and Xλ−1 X. Therefore, it can not be X S˜−1 X and X Q directly solved by the Matlab LMI Toolbox. Instead of the original non-convex minimization problem, one can consider a cone complementarity (CCL) algorithm [22], [23] to have a feasible solution. Remark 5: Compared with some existing results [6], [24] which require output/state feedback controller gain K being given a priori, the controller gain K can be obtained based on Theorem 2. Therefore, the requirement on controller gain K is no longer needed in this paper. IV. I LLUSTRATIVE EXAMPLES In this section, we use two numerical examples taken from [6], [7] to demonstrate the effectiveness of proposed method. In the following example, we take t0 = 0. Example 1: Consider the following   system:  0 1 0 x(t) ˙ = x(t) + u(t) (13) −2 3 1 Set L = I in (1). The output feedback control law (5) is simplified as a state feedback control law. For comparisons, set ε = 0.0001 and K = [1, −4] as those in [6], [7], and set the constant sampling period with zero communication delay in (3) and (6).

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In the case of δ = 0.003 and Φ = I, applying Theorem 1 in this paper, we obtain an inter-event time hm = 0.3029, which is larger than hm = 0.0318 obtained based on theorem IV.1 or its counterpart Theorem 5.1 in [25]. • In the case of δ = 0.0273 and Φ = I, applying Theorem V.1 in [6] and Theorem 1 in this paper yield hm =0.0840 and 0.1484, respectively. It is seen that a larger inter-event time is obtained based on the method proposed in this work. • In the case of δ = 0.588 and Φ > 0, we obtain hm = 0.1927 based on Theorem 1 in this paper, which is larger than hm = 0.1136 obtained from Theorem III.5 in [6]. In addition, the controller gain can be obtained based on the proposed research framework. For example, set δ = 0.3 and ε = 0.0001, we can obtained K = [−0.0043, −6.1324] and η¯ = 0.1041 based on Theorem 2. This is out of the field of the approaches in [6], [7]. Example 2: Consider the system (1) with the following parameters       0 1 0 0 A = ,B = , B1 = −2 3 1 1     1 0 C = , D = 0, L = 1 −4 (14) 0 1 Set γ = 200. Using the Matlab LMI Toolbox and applying Theorem 2, we obtain the corresponding η¯, Φ and K with the different given parameters δ and ε as listed in Table I. One can see that the obtained η¯ decreases as the given δ increases. Compared with those in [6], the condition of ε > 0 is no longer needed to guarantee that the minimum interevent time larger than zero. For the purpose of simulation, choose the corresponding parameters with δ = 0.4 listed in Table I, assume the constant sampling period h = 0.02 sec and a disturbance satisfying  sin 2t, t ∈ [5, 15] ω(t) = (15) 0.25 sin 2t, t ∈ [30, 40]

Figure 2 depicts the system output and communication conditions with the initial condition x(0) = [0.2, −0.2]T . One can see that the output of the plant indeed converges asymptotically to a ultimate bound, and the output of the plant has to be transmitted less often when the output fluctuates in the vicinity of the ultimate bound. For simulation time T = 50sec, 100sec and 150sec, respectively, only 81.8%, 71.8% and 68.5% of the sampled data needs to be transmitted. V. C ONCLUSION An output-based event-triggered communication scheme has been proposed for sampled-data control systems with nonuniform sampling. The proposed communication scheme only depends on the system output-error between the current sampled instant and the latest transmitted instant and an additional threshold, therefore, the requirements of continuous measurement and calculation in some existing ones have been no longer needed in this work. Based on the presented output-error-dependent model, the stability and stabilization criteria have been derived to guarantee the

1.5 output disturbance

1 0.5 0 −0.5

Transmission intervals

Disturbance and output



−1 0

10

20 30 Time (Second)

40

50

10

20 30 Time (Second)

40

50

0.6 0.4 0.2 0 0

Fig. 2. Trajectories of the system output z(t) and the disturbance ω(t) (top) and the communication intervals with a constant sampling period (bottom) for Example 2.

UUB stability and the desired performance. Compared with some existing event-triggering methods, the output feedback controller can be designed with respect to the given eventtriggering parameters. It has been shown through two examples that the proposed communication scheme has used less communication bandwidth while preserving the desired performance. A PPENDIX Proof of Theorem 1: Construct a Lyapunov-Krasovskii functional candidate as Z t V (t, xt ) = xT (t)P1 x(t) + xT (v)P2 x(v)dv t−¯ η

+

Z

t

t−¯ η

Z

t

x˙ T (v)S x(v)dvds ˙

s

+(¯ η − η(t))[˜ xT (t)R˜ x(t) Z t + x˙ T (v)Qx(v)dv] ˙

(16)

t−η(t)

with P1 > 0, P2 > 0, Q > 0, R > 0, S > 0 and x ˜(t) = x(t) − x(t − η(t)). From (3), it is clear that for im ∈ [tk , tk+1 ) T e (im )(L+ )T ΦL+ e(im ) < δ1 y T (tk )(L+ )T ΦL+ y(tk ) + ε3 (17) d Notice that dt x(t − η(t)) = [1 − η(t)] ˙ x(t ˙ − η(t)) = 0, for t ∈ Ωm and t 6= im . Taking the time derivative of V (t, xt ) along the trajectory of system (7) yields V˙ (t, xt ) =

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2xT (t)(P1 + P2 )x(t) ˙ − xT (t − η¯)P2 x(t − η¯) Z t T +¯ ηx˙ (t)S x(t) ˙ − x˙ T (s)S x(s)ds ˙ t−¯ η

+(¯ η − η(t)){2˜ xT (t)Rx(t) ˙ + x˙ T (t)Qx(t)} ˙ Z t − x˙ T (v)Qx(v)dv ˙ −x ˜T (t)T R˜ x(t)(18) t−η(t)

TABLE I C OMMUNICATION AND CONTROL PARAMETERS OBTAINED BASED ON DIFFERENT VALUES OF δ δ/ε K η¯ Φ

0.1/0.0001 1.3865 0.16 

0.2544 *

-0.9801 4.4176

0.2/0.0001 1.3266 0.12 



0.0323 *

0.3/0.01 1.4492 0.08

-0.2243 1.6113

Similar to [10], we use the Newton-Leibniz formula and introduce matrices M1 , M2 and M3 with appropriate dimensions to deal with integral items in (18). Combining (16)-(18) together, we have V˙ (t, xt ) < V˙ 1 (t, xt ) + V˙ 2 (t, xt ) − εx(t)T x(t) +δΨT ΦΨ + ε3 − (L+ e(im ))T ΦL+ e(im ) +[Cx(t) + Dω(t)]T [Cx(t) + Dω(t)]

+εx(t)T x(t) − z(t)T z(t) = ξ T (t)Ξ0 ξ(t) − z(t)T z(t) + γ 2 w(t)T w(t) +ε3 − εx(t)T x(t)

(19)

where Ψ = L+ Lx(t − η(t)) − L+ e(il ) and Ξ0

=

Ξ1 Ξ2

= =

Π + (¯ η − η(t))Ξ1 + η(t)Ξ2 +Γ1 + ΓT1 + ΓT2 Γ2 + η¯I2T SI2

M1 S −1 M1T + I2T QI2 + 2I1 RI2 M2 S −1 M2T + M3 Q−1 M3T

with Π, Γi , and Ii (i = 1, 2) being defined in Theorem 1. Since the condition of (11) ensures that Ξ0 < 0 in (19), then, it is clear that V˙ (t, xt ) ≤ γ 2 ω T (t)ω(t) − z T (t)z(t)

(20)

Integrating both sides of (20) from t0 to t, and letting m → ∞ and under the zero initial condition, we have kz(t)k ≤ γ kω(t)k. When kx(t)k ≥ ε, we have ε2 − x(t)T x(t) ≤ 0. Using the Lyapunov–Krasovskii functional (16), from (19) we have V˙ (t, xt ) < 0, for kx(t)k ≥ ε. Based on Lemma 1, it is readily derived that the state of the system (7) with ω(t) = 0 is UUB. Then based on Lyapunov stability theory and Lemma 1, we can conclude that the state of the system (7) is UUB with an H∞ norm γ. This completes the proof.  R EFERENCES [1] W. Heemels, A. Teel, N. van de Wouw, and D. Neˇsi´c, “Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance,” IEEE Trans. Autom. Control, vol. 55, no. 8, pp. 1781–1796, Aug. 2010. [2] X. Jiang and Q.-L. Han, “On designing fuzzy controllers for a class of nonlinear networked systems,” IEEE Trans. Fuzzy Syst., vol. 16, no. 4, pp. 1050-1060, Aug. 2008. [3] X.-M. Zhang and Q.-L. Han, “A delay decomposition approach to control of networked control systems,” European J. Control, vol. 15, no. 5, pp. 523-533, Sep.-Oct. 2009. [4] M. Donkers, W. Heemels, N. Van De Wouw, and L. Hetel, “Stability analysis of networked control systems using a switched linear systems approach,” IEEE Trans. Autom. Control, vol. 56, no. 9, pp. 2101–2115, 2011. [5] X.-M. Zhang and Q.-L. Han, “Network-based filtering using a logic jumping-like trigger,” Automatica, in press, doi:10.1016/j.automatica.2013.01.060





0.0302 *

-0.1931 1.2713

0.4/0.01 1.6263 0.05 



0.0212 *

-0.1236 0.7228



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