An improved memory-event-triggered control for networked control systems

Available online at www.sciencedirect.com

Journal of the Franklin Institute 356 (2019) 7210–7223 www.elsevier.com/locate/jfranklin

An improved memory-event-triggered control for networked control systems Engang Tian a,∗, Kunyu Wang c, Xia Zhao a,b, Shibin Shen c, Jinliang Liu d a School

of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, PR China b School of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 2111106, PR China c School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, China d College of Information Engineering, Nanjing University of Finance and Economics, Nanjing, Jiangsu 210023, PR China Received 20 January 2019; received in revised form 9 April 2019; accepted 25 June 2019 Available online 2 July 2019

Abstract In this paper, the H∞ control problem is investigated for a class of networked control systems with network-induced delay. A memory event-triggered scheme (METS) is proposed to reduce the redundant packet transmission in the network channel. Different from the normal event-triggered scheme (ETS), some recent released packets are stored at the event generator and controller sides, which are utilized for the first time to generate the triggered events and design the memory-based controller. The proposed METS has the following two merits. (1) The information of certain recent released signals are first utilized, which helps to improve the triggering instants at the crest or trough of the responses. (2) A state-dependent time-varying threshold parameter is designed, which can adjust the packet transmission rate according to the information of the state. Based on the proposed METS, a memory event-triggered controller is designed, the controller feedback gains and triggering parameters can be co-designed by solving a set of linear matrix inequalities. Finally, an example is given to illustrate the effectiveness of the proposed method. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

∗

Corresponding author. E-mail address: [email protected] (E. Tian).

https://doi.org/10.1016/j.jfranklin.2019.06.041 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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1. Introduction Networked control systems (NCSs) have been a hot research area in the recent two decades, and a rich body of outstanding research works have been reported, see [1–8] and the references therein. Nowadays, the research results of the NCSs have been applied in a great deal of practical systems, such as unmanned aerial vehicles, smart grid, industrial automation and mobile communications [9–18]. The main challenging issues in the NCSs are the network-induced phenomena caused by the data transmission and/or limited network bandwidth, such as transmission delay, packet loss, quantization and so on. In order to reduce the redundant sampling signals in the bandwidthlimited network, event-triggered scheme (ETS) has been proposed to guarantee desired system performance while reducing the signal transmission in the control loop [19–26]. This character is extremely important especially in wireless NCSs or sensor networks, wherein the batteries have limited energy, too much transmission will exhaust the power of the batteries. Compared with the classical time-triggered scheme, ETS can not only reduce the packet transmission, but also alleviate the computation burden of the controller. An early research work has been reported in [27], wherein both the periodic time-triggered scheme and ETS are considered and compared for a class of stochastic systems. After the publication of [27,28], researches on the ETS, event-triggered control and event-triggered filtering have gained ever-increasing attention. Generally speaking, the researches on ETS can be roughly classified into two categories, the first is called absolute ETS, for example, in [29], the absolute ETS is proposed as x(t ) − x(tk ) ≤ δ, where x(t) is the current measurement, x(tk ) is the most recent released measurement and δ is a constant threshold, which is independent with the state or output. An event is generated if the above inequality is violated. The second is called relative ETS [30,31], such as x(t ) − x(tk ) ≤ ρx(tk ), where the event is stimulated through a state-dependent threshold. The relative ETS can guarantee the asymptotical stability of the investigated systems, while the absolute ETS can only ensure the stable or uniform bounded of the system. It should be pointed out that the absolute ETSs are used in some early published works, which is easy to implement and can generate more triggering events at the transient process and is much sensitive to the variation of the state. Most of the recent works concerning ETS are relative ETS, for example, the continuous ETS [31], dynamic ETS [32,33], periodic ETS [34,35], discrete ETS [36,37], and so on. For example, in [34], a periodic ETS is proposed to strike a balance between conventional periodic sampled-data control and event-triggered control, which can naturally avoid the ZENO phenomenon. In [36], a discrete ETS is proposed, the triggering parameters and the controller feedback gains can be co-designed. Most of the proposed ETSs have a fixed threshold parameter, that is, the parameter remains the same regardless of the variation of the state and system dynamic. To overcome this problem, a dynamic ETS is proposed in [32,33], wherein a time-varying threshold parameter is proposed, which relies on some given variations or parameters. In [4,38], an adaptive ETS is proposed for a class of network-based T-S fuzzy control systems, under which the event is generated adaptively following the variation of both the nonlinear system and the reference model. In [39], another adaptive ETS is proposed for the load frequency control, where the threshold can be dynamically adjusted to save more limited network resources. From above description, it can be found that whether a new event is triggered by the ETS depends mainly on two factors. The first is the threshold parameter ρ, in fact, in many improved ETSs, such as the dynamic ETS [32], time-varying ETS [19] and

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Fig. 1. A simple example of response.

adaptive ETS [4,39], great efforts have been made to design a proper threshold parameter such that the ETS is much sensitive to the system dynamic. The second is the error between the current signal and the most recent released one. If the error is large, the new packet is more likely to be released. It seems natural that if the error is small, the new packet will not be transmitted to the controller, however, it is not always the case. For example, in Fig. 1, although the error between x(t3 h) and x(t4 h) and the error between x(t4 h) and x(t5 h) are much small, the values of these three packets are much larger than the other samples. In this situation, we usually hope the ETS to release them in order to shorten the transient process. That is, at the crest or trough of the state/output responses, we expect the ETS can transmit more packets to the controller although the error between them is small. Motivated by above discussion, a memory-based event-triggered scheme (METS) is proposed to design the H∞ controller for networked control systems. In order to increase the transmission instants at the crest or trough of the responses, two measures are taken into consideration, the first is to utilize the information of some recent released packets in the proposed METS. The second is to construct a time-varying threshold parameter, which will vary depending on the norm of the state values. Considering both the METS and networkinduced delay, new kind of networked control system model is built. And then a co-design strategy is provided to design the memory controller and the parameter in METS. A practical example is proposed to illustrate the effectiveness of the proposed design scheme and method. The main technical contributions of the current paper are summarized as follows. (1) A METS is, for the first time, proposed for the networked control systems, which utilizes some recent released packets and can improve the triggering events at the crest or trough responses. (2) A state-dependent dynamic threshold parameter is designed, which can change the transmission rate according to the norm of the state. (3) A memory event-triggered control method is given to co-design the controller and the trigger parameters. The rest of the paper is organized as follows. Section 2 formulates the METS and dynamic model of the NCSs. The main results are given in Section 3, two theorems and one corollary are given for the stability and controller design of the NCSs. A simulation example is given in Section 4 and this paper is concluded in Section 5.

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2. Dynamic model of memory event triggered NCSs Considering the following system x˙(t ) = Ax(t ) + Bu(t ) + B1 ω(t ) z(t ) = Cx(t )

(1)

where x(t ) ∈ Rn and z(t ) ∈ R p are, respectively, the state and controlled output. u(t ) ∈ Rq and ω(t ) ∈ Rv are the control input and disturbance, respectively. A, B, C and B1 are matrices with appropriate dimensions. Before introducing the METS, let us review the discrete ETS proposed in [36]   tk+1 h = tk h + min ih|etTk (t )etk (t ) > ρx T (tk h + ih)x(tk h + ih) , (2) i∈N

where etk (t ) = x(tk h + ih) − x(tk h), i ∈ N, {tk } are some integers belonging to the set {0, 1, 2, . . .}, tk h denotes the latest release instant by the ETS. h is the sampling interval of the sensors,  is a positive definite matrix and ρ is a given threshold parameter. x(t0 h) is assumed to be the first sampled and released signal. If the new sampled x(tk h + ih) satisfies the inequality in Eq. (2), tk+1 = tk h + ih and x(tk h + ih) is released by the ETS to the controller, else, new packet is sampled to verify in Eq. (2). For detailed explanation of the ETS in Eq. (2), please refer to [36]. In this paper, different from most of the ETSs, not only the error between the current and most recent released packet, but also the information of some recent released signals are utilized in the proposed METS, which is shown as   m  T T tk+1 h = tk h + min ih| εi etk−i+1 (t )etk−i+1 (t ) > ρ(t )x (tk h + ih)x(tk h + ih) (3) i∈N

i=1

where etk−i+1 (t ) = x(tk−i+1 h) − x(tk h + ih), i = 1, 2, . . . , m,

(4)

ρ(t ) = ρ0 + ρ1 e−λx(tk h+ih)2 ,

(5)

 x(tk−i+1 h) = x(t0 h) if k − i + 1 ≤ 0. εi ∈ [0, 1] are given weighting parameters and m i=1 εi = 1. ρ 0 , ρ 1 and λ are given positive constants. From Eq. (5), it can be found that ρ0 ≤ ρ(t ) ≤ ρ0 + ρ1  ρ. Remark 1. In the proposed METS Eq. (3), m is the number of the released packets used in the scheme, if m = 1, the METS reduces to the normal memoryless ETS. The parameters εi illustrate the weights of the released packets in the event condition. Usually, we regard that the new released packet is more important than the old ones. Therefore, ε1 is set much larger than the others and εi ≥ εi+1 (i = 2, . . . , m − 1). Remark 2. In Eq. (3), a state-dependent threshold ρ(t) is designed, the value of ρ(t) depends on the norm of current sampled data x(tk h + ih). A large x(tk h + ih)2 will result in a small ρ(t), thus more packets will be released to the controller in order to improve the transient performance. The parameter ρ 1 can adjust the proportion of the varying part in Eq. (5) and λ is introduced to change the varying rate of the state.

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Similar to the modeling method in [36], considering the network-induced delay, the interval [tk h + τk , tk+1 h + τk+1 )  I is divided into dk subintervals k I = ∪dl=1 Il

(6)

where τ k is the network-induced delay of the signal x(tk h), Il = [tk h + lh + ϑk , tk+1 h + lh + ϑk+1 ), dk = tk+1 − tk − 1 and ⎧ l = 0; ⎨τk , l = 1, . . . , dM − 1; ϑk = τ¯ , ⎩ τk+1 , l = dM . with τ¯ = max{τk }. Denoting η(t ) = t − tk h − lh for t ∈ Il , it can be found that η(t) is a k∈N

piecewise function satisfying 0 ≤ τk ≤ η(t ) ≤ h + τ¯  ηM , η(t ˙ ) = 1.

(7)

Corresponding to the METS, a memory feedback controller is designed as u(t ) =

m 

εi Ki x(tk−i+1 h),

(8)

i=1

where Ki are the controller feedback gains to be designed. Remark 3. To implement the METS and memory feedback controller, the past released packets are needed at both the event generator and controller sides. Therefore, two buffers are added and the length of the buffer is m. If we let m = 1 and ε1 = 1, the proposed METS reduces to the ETS in Eq. (2). Considering the proposed METS Eq. (3), memory feedback controller (8) and networkinduced delay, the system (1) can be rewritten as ⎧ m  ⎨x˙(t ) = Ax(t ) +  εi BKi x(t − η(t )) + etk−i+1 (t ) + B1 ω(t ), . (9) i=1 ⎩ z(t ) = L x (t ), t ∈ [tk h + τk , tk+1 h + τk+1 ) For system (9), when t ∈ [−ηM , 0], x(t ) = φ(t ) is the initial condition with φ(0) = x0 , where φ(t) is continuous on [−ηM , 0]. When k − i + 1 ≤ 0, etk−i+1 (t ) = x(t0 h) − x(tk h + lh). By using the proposed METS and memory feedback controller, the purpose of the current paper is to design the memory feedback controller gain and event parameters simultaneously, such that the following two requirements are satisfied. (1) The system (9) is asymptotically stable with ω(t ) = 0. (2) Under zero initial condition, for any nonzero ω(t ) ∈ L2 [0, ∞ ) and a prescribed index γ , the following inequality holds z(t )2 ≤ γ ω(t )2 . 3. Main results In this section, using the proposed METS Eq. (3), we will first derive the delay-dependent bounded real lemma for system (9) for given Ki , which is illustrated in Theorem 1. Second,

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based on the bounded real lemma, sufficient conditions for the memory event-triggered H∞ control are presented, see Theorem 2 and Corollary 1. Theorem 1. For given constants ηM > 0, ρ > 0, εi > 0 and matrices Ki (i = 1, . . . , m), if there exist P > 0, Q > 0, R > 0, S > 0,  > 0 and matrix W, such that



11 T21 <0 (10)

21 22

 R WT >0 (11) W R where

11

⎡

(1,1)

11

⎢ (2,1) ⎢ 11 ⎢ =⎢ ⎢ W ⎢ (4,1) ⎣ 11 B1T P

∗

∗

∗

(2,2)

11 R −W

∗ −Q − R

∗ ∗

0 0

0 0

(4,4)

11 0

⎤

∗

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∗ ⎦ −γ 2 I ∗ ∗

 2 (1,1) (2,1) π2 T T

11 = PA + AT P + Q − π4 S − R + C T C,

11 = m i=1 εi Ki B P + 4 S + R − W ,   2 (2,2) (4,1)

11 = − π4 S + ρ − 2R + W + W T , 11 =col −ε1 K1T BT P . . . − εm KmT BT P , (4, 4) = m diag{−ε1 , . . . , −εm }, 21 = col{ηM R S}, 22 = diag{−R, −S},  = [A i=1 εi BKi 0 ε1 BK1 . . . εm BKm B1 ], then system (9) is asymptotically stable with H∞ norm bound index γ . Proof. Construct the Lyapunov functional candidate as V (t ) = V1 (t ) + V2 (t ) where



V1 (t ) = x T (t )Px(t ) +  V2 (t ) =

(12) t

 x T (s)Qx(s)ds + ηM

t−ηM t

t −η(t )

x˙T (v)S x˙(v)dv −

π2 4



t

t−ηM t

t −η(t )



t

x˙T (v)Rx˙(v )dvds

s

(x(s) − x(s − η(s)) )T S (x(s) − x(s − η(s)) )ds

with P, Q, R, S are positive definite matrices. It is obvious that V1 (t) is positive symmetric definite, from the Wirtinger-type inequality discussed in [39], it can be seen that V2 (t) is also positive symmetric definite. Taking derivation on V(t), we obtain 2 T V˙ (t ) = 2x˙T (t )Px(t ) + x T (t )Qx(t ) − x T (t − ηM )Qx(t − ηM ) + ηM x˙ (t )Rx˙(t ) + x˙T (t )S x˙(t )  t π2 −ηM x˙T (v)Rx˙(v)dv − (13) (x(t ) − x(t − η(t )) )T S (x(t ) − x(t − η(t )) ) 4 t−ηM

By using the reciprocally convex approach [40], for Eq. (11) holds, we obtain ⎡ ⎤T ⎡ ⎤⎡ ⎤  t x(t ) −R R −WT WT x(t ) −ηM x˙T (v)Rx˙(v)dv ≤ −⎣x(t − η(t ))⎦ ⎣R − W −2R + 2W R − W T ⎦⎣x(t − η(t ))⎦ t−ηM x(t − ηM ) W R −W −R x(t − ηM ) (14)

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W is a free matrix satisfying (11). From Eqs. (3) and (13), (14), it can conclude that   V˙ (t ) + zT (t )z(t ) − γ 2 ωT (t )ω(t ) ≤ ζ T (t ) 11 − T21 −1 22 21 ζ (t ) where

 ζ T (t ) = x T (t )

x T (t − η(t ))

x T (t − ηM )

etTk (t )

...

etTk−m+1 (t )

(15)

 ωT (t )

By using Schur complement, it can be computed from Eq. (10) that 11 − T21 −1 22 21 < 0, then we obtain zT (t )z(t ) − γ 2 ωT (t )ω(t ) ≤ −V˙ (t )

(16)

Integrating both side of Eq. (16) from 0 to +∞ yields  +∞  T z (t )z(t ) − γ 2 ωT (t )ω(t ) ds ≤ V (0) − V (+∞ ) 0

Under zero initial condition, we obtain z(t )2 ≤ γ ω(t )2 . From the inequality (10), it can be concluded that when ω(t ) = 0, there exists a positive scalar μ > 0 such that V˙ (t ) ≤ −μx(t )22 from which we obtain that the system (9) is asymptotically stable with H∞ performance. This completes the proof.  Next, we will design the memory event-triggered controller for the system (9). ˜ = X X, W˜ = X W X and Yi = Ki X, Define X = P−1 , Q˜ = X QX, R˜ = X RX, S˜ = X SX,  pre- and post-multiplying both sides of (10) with diag{X , X , . . . , X , I , X , X }, pre- and postmultiplying both sides of (11) with diag{X, X} and using Schur complement, one can obtain ⎡ ⎤ ˜ 11

∗ ∗ ⎣ ˜ 21 ˜ 22 (17) ∗⎦<0 ˜ 31

0 −I

 R˜ W˜ T >0 (18) W˜ R˜ where

˜ 11

⎡ ˜ (1,1)

11 ⎢ ˜ (2,1) ⎢ 11 ⎢ =⎢ ⎢ W˜ ⎢ (4,1) ˜ ⎣ 11 B1T

∗

∗

∗

˜ (2,2)

11 R˜ − W˜

∗

∗

−Q˜ − R˜

0 0

0 0

∗ (4,4) ˜

11 0

∗

⎤

⎥ ∗ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ∗ ⎦

−γ 2 I m

˜ (1,1) = AX + X AT + Q˜ − π 2 S˜ − R˜ , ˜ (2,1) = i=1 εiYiT BT + π 2 S˜ + R˜ − W˜ , ˜ (2,2) = − π 2 S˜ +

11 11 11 4 4 4 ˜ ˜ (4,1) = col {−ε1Y1T BT · · · − εmYmT BT }, ˜ (4,4) = diag {−ε1 , ˜ . . . , −εm } ˜ , ρ −2 R˜ +W˜ +W˜ T , 11   11 ˜ 21 = ηM ˜ ˜ , ˜ 22 = diag {−X R˜ −1 X, −X S˜−1 X }, ˜ 31 = [CX 0 . . . 0], ˜ =

  m AX i=1 εi BYi 0 ε1 BY1 . . . εm BYm B1 .

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Theorem 2. For given constants ηM > 0, ρ > 0, δ 1 , δ 2 and εi > 0, if there exist X > 0, Q˜ > ˜ > 0 and matrix W˜ , Yi (i = 1, . . . , m) such that 0, R˜ > 0, S˜ > 0,  ⎡ ⎤ ˜ 11

∗ ∗ ⎣ ˜ 21 22 (19) ∗⎦<0 ˜

31 0 −I ˜ 11 , ˜ 21 and ˜ 31 are as defined and (18) hold, where 22 = diag{δ12 R˜ − 2δ1 X, δ22 S˜ − 2δ2 X }, as in (17). Then system (9) is asymptotically stable with an H∞ performance γ , the memory state feedback controller gain is Ki = Yi X −1 . Proof. For any constants δi , i = 1, 2, it holds that −X R˜ −1 X < δ12 R˜ − 2δ1 X, −X S˜−1 X < δ22 S˜ − 2δ2 X.

(20)

Therefore, Eq. (19) can be derived from Eq. (17).  For m = 1 and u(t ) = K x(tk h), a corollary can be obtained directly from Theorem 2. Corollary 1. For given constants ηM > 0, ρ > 0, δ 1 , δ 2 , if there exist X > 0, Q˜ > 0, R˜ > 0, S˜ > ˜ > 0 and matrix W˜ , Y such that 0,  ⎡ ⎤ ˆ 11

∗ ∗ ⎣ ˆ 21 22 (21) ∗⎦<0 ˜

31 0 −I ˜ 31 are as defined as in Eq.(17), and Eq.(16) hold, where 22 and ⎡ (1,1) ⎤ ˆ

∗ ∗ ∗ ∗ 11 ⎢ ˆ (2,1) ⎥ ˆ (2,2) ⎢ 11

∗ ∗ ∗ ⎥ 11 ⎢ ⎥ ˆ 11 = ⎢ W˜

R˜ − W˜ −Q˜ − R˜ ∗ ∗ ⎥ ⎢ ⎥ ⎢ T T ⎥ ˜ ⎣Y B 0 0 − ∗ ⎦ B1T 0 0 0 −γ 2 I ˆ (1,1) = AX + X AT + Q˜ − π 2 S˜ − R˜ , ˆ (2,1) = Y T BT + π 2 S˜ + R˜ − W˜ , ˆ (2,2) = − π 2 S˜ +

11 11 11 4 4 4     ˜ − 2R˜ + W˜ + W˜ T , ˆ 21 = ηM ˆ ηM ˆ , ˆ = AX BY 0 BY B1 . Then system ρ (9) is asymptotically stable with an H∞ performance γ , the state feedback controller gain is K = Y X −1 . 4. Simulation example Consider an satellite control system [41], the state space equation is described as the system (1) with the parameters ⎡ ⎤ ⎡ ⎤ 0 1 0 0 ⎡ ⎤ 0 k d d⎥ 0.01 ⎢ k ⎢− ⎢ 0 ⎥ − − ⎥ ⎢0.01⎥ ⎢ ⎢ ⎥ J2 J2 J2 ⎥ ⎥ A = ⎢ J2 (22) ⎥, B = ⎢ 0 ⎥, B 1 = ⎢ ⎣0.01⎦, 0 0 1 ⎥ ⎢ 0 ⎣ 1⎦ ⎣ k ⎦ d d d 0.01 − − − J1 J1 J1 J1 J1   C= 1 1 1 1,

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1.2 x1 (t)

1

x2 (t) x (t) 3

State Responses

0.8

x4 (t)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 0

5

10

15

20

25

t(s) Fig. 2. State responses by using Corollary 1 for time-varying ρ(t ).

Fig. 3. The variation of ρ(t ) and the release instants.

30

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Table 1 The average release instants for different group of εi (i = 1, 2, 3). ε1

0.998

0.9

0.7

0.5

ε2 ε3 AVI TPN

0.001 0.001 0.8571 35

0.07 0.03 0.6383 45

0.2 0.1 0.5357 56

0.3 0.2 0.4412 68

1.8 1.6 1.4

||x(t)||

1.2 1

0.2

0.8

0.1

0.6

0

0.4

-0.1 26

0.2

28

30

0 0

5

10

15

20

25

30

t(s) Fig. 4. State responses by using Corollary 1 for both time-varying and constant threshold parameter.

where J1 = J2 = 1 are two masses, k = 0.09 is the torque constant and d = 0.0219 is the viscous damping constant. The initial condition of the state is x0 =col{0.2 -0.3 0.3 0.2} and the external disturbance ω (t ) =sgn(sin t) when t ∈ [0, 10]; otherwise ω(t ) = 0. Firstly, for γ = 0.01, δ1 = 2.4 and δ2 = 0.3, by using Corollary 1, the maximum ηM is obtained as 0.26s, that is ηM = h + τ¯ = 260 ms, which shows a trade-off between the sampling interval h and maximum allowable transmission delay τ¯ , i.e., if we sample with a long sampling interval h, the system would only endure small transmission delay. If the system is delay-free, the maximum allowable sampling interval can be 260 ms for the periodic time-triggered scheme. In this example, we set h = 100 ms and τ¯ = 160 ms. The parameters in the triggering conditions ρ(t ) = ρ0 + ρ1 e−λx(tk h+ih) are given as ρ0 = 0.2, ρ1 = 0.2 and λ = 0.1, then we obtain ρ(t ) ≤ ρ0 + ρ1 = 0.4. By using Corollary 1, one

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2 ε1 = 0.9 ε1 = 0.7

1.8 1.6 1.8

1.4

||x(t)||

1.2 1.7 1 0.8

1.6

0.6 1.5

0.4

0

1

2

3

4

5

0.2 0 0

5

10

15

20

25

30

t(s) Fig. 5. State responses by using Theorem 2 for different group of εi .

can obtain  K = −4.1950 −19.0341 ⎡ 0.1219 −0.1607 ⎢−0.1607 0.2186 ˜ =⎢  ⎣ 1.4064 −1.9503 −0.3955 0.5515

−2.8408 1.4064 −1.9503 19.8790 −14.2690

 −3.4592 ,

⎤ −0.3955 0.5515 ⎥ ⎥. −14.2690⎦ 40.4544

The average release period is obtained as 0.8333 and only 36 packets are released to the controller. That is, only 12% sampled packets are released by the event-generator. The state responses are shown in Fig. 2, the release instants, time intervals and time-varying ρ(t) are shown in Fig. 3. To illustrate the effect of the time-varying threshold parameter ρ(t), we set ρ(t ) = ρ = 0.4 and run the simulation again. There are 35 packets released to the controller, the norm of the state responses with time-varying and constant ρ(t) are shown in Fig. 4. From Fig. 4, it can be found that by introducing the time-varying ρ(t), the state responses have a better dynamic performance with less sampling instants. Comparing the sampling instants of the above two cases, it can be concluded that by using the time-varying ρ(t), more packets are triggered at the initial instants and less packets are triggered when the system is steady, which can improve the transient performance of the system. In the following, we will consider the simulation results of the METS. Selecting m = 3, ε1 = 0.7, ε2 = 0.2 and ε3 = 0.1, the average release period is obtained as 0.5357 and 56 packets are released to the controller. For other groups of εi (i = 1, 2, 3), the average release

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periods (AVI) and transmitted packet numbers (TPN) are shown in Table 1. In order to show the effect of the METS, two groups of parameters εi (i = 1, 2, 3) = {0.9, 0.07, 0.03} and {0.7, 0.2, 0.1} are used for the simulation, the norm of x(tk h) under these two groups of parameters are shown in Fig. 5. It can be found from Fig. 5 that for εi (i = 1, 2, 3) = {0.7, 0.2, 0.1}, more packets are released at the transient process and the dynamic performance of the response is improved. From the simulation results, the following two conclusions can be obtained. Firstly, the METS proposed a method to improve the transient performance of the state responses by adjusting the parameters εi . Secondly, with the decrease of ε1 , more packets are released to improve the dynamic performance, these increased packets are mainly scattered in the transient process, especially at the crests and troughs period. 5. Conclusion A memory event-triggered scheme has been proposed for the networked control systems with network-induced delay. The information of some recent released packets has been firstly utilized in the triggering conditions, which will improve the triggering times at the transient process. Considering the METS scheme, new kind of networked control system model has been built. By using the proposed memory event-triggered control method, sufficient conditions for the H∞ control of the networked control systems have been obtained. One example has been given to show the effectiveness of the proposed METS and controller design technique. In the future researches, we will attempt to apply the METS in some practical systems, for example, the multi-agent systems and cyber-physical systems. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (No. 61773218), Natural Science Foundation of Jiangsu Province (BK20161561) and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning. References [1] Y. Shi, B. Yu, Output feedback stabilization of networked control systems with random delays modeled by Markov chains, IEEE Trans. Autom. Control 54 (7) (2009) 1668–1674. [2] J. Liu, T. Yin, M. Shen, X. Xie, J. Cao, State estimation for cyber-physical systems with limited communication resources, sensor saturation and denial-of-service attacks, ISA Trans. doi:10.1016/j.isatra.2018.12.032. [3] J. Liu, Y. Gu, L. Zha, Y. Liu, J. Cao, Event-triggered H∞ load frequency control for multi-area power systems under hybrid cyber attacks, IEEE Trans. Syst. Man Cybern. Syst. doi:10.1109/TSMC.2019.2895060. [4] Z. Gu, P. Shi, Y. Dong, Z. Ding, Decentralized adaptive event-triggered H-infinity filtering for a class of networked nonlinear interconnected systems, IEEE Trans. Cybern. 49 (5) (2019) 1570–1579. [5] L. Zou, Z. Wang, Q.L. Han, D. Zhou, Ultimate boundedness control for networked systems with try-once-discard protocol and uniform quantization effects, IEEE Trans. Autom. Control 62 (12) (2017) 6582–6588. [6] Y. Pan, G.H. Yang, Event-triggered fuzzy control for nonlinear networked control systems, Fuzzy Sets Syst. 329 (2017) 91–107. [7] X. Zhao, C.S. Liu, E. Tian, Finite-horizon tracking control for a class of stochastic systems subject to input constraints and hybrid cyber attacks, ISA Trans. (2019), doi:10.1016/j.isatra.2019.02.025. [8] X. Zhao, C.S. Liu, E. Tian, Probability-constrained tracking control for a class of time-varying nonlinear stochastic systems, J. Frankl. Inst. 355 (5) (2018) 2689–2702. [9] Y. Wang, Q. Han, Network-based modelling and dynamic output feedback control for unmanned marine vehicles in network environments, Automatica 91 (2018) 43–53.

7222

E. Tian, K. Wang and X. Zhao et al. / Journal of the Franklin Institute 356 (2019) 7210–7223

[10] M. Shen, S.K. Nguang, C.K. Ahn, Q.G. Wang, Robust H2 control of linear systems with mismatched quantization, IEEE Trans. Autom. Control 64 (4) (2018) 1702–1709. [11] M. Shen, J.H. Park, D. Ye, A separated approach to control of Markov jump nonlinear systems with general transition probabilities, IEEE Trans. Cybern. 46 (9) (2016) 2010–2018. [12] Y. Pan, G.H. Yang, Event-triggered fault detection filter design for nonlinear networked systems, IEEE Trans. Syst. Man Cybern. Syst. 48 (11) (2017) 1851–1862. [13] C. Peng, H. Sun, M. Yang, Y.L. Wang, A survey on security communication and control for smart grids under malicious cyber attacks, IEEE Trans. Syst. Man Cybern. Syst. (2019), doi:10.1109/TSMC.2018.2884952. [14] C. Dou, D. Yue, J.M. Guerrero, X. Xie, S. Hu, Multiagent system-based distributed coordinated control for radial DC microgrid considering transmission time delays, IEEE Trans. Smart Grid 8 (5) (2017) 2370–2381. [15] J. Liu, J. Xia, E. Tian, S. Fei, Hybrid-driven-based H∞ filter design for neural networks subject to deception attacks, Appl. Math. Comput. 320 (2018) 158–174. [16] X. Xiao, J.H. Park, L. Zhou, Event-triggered control of discrete-time switched linear systems with packet losses, Appl. Math. Comput. 333 (2018) 344–352. [17] X. Chen, J.H. Park, J. Cao, J. Qiu, Sliding mode synchronization of multiple chaotic systems with uncertainties and disturbances, Appl. Math. Comput. 308 (2017) 161–173. [18] X. Xie, Y. Dong, P. Chen, Relaxed real-time scheduling stabilization of discrete-time Takagi-Sugeno fuzzy systems via an alterable-weights-based ranking switching mechanism, IEEE Trans. Fuzzy Syst. 26 (6) (2018) 3808–3819. [19] E. Tian, Z. Wang, L. Zou, D. Yue, Probabilistic-constrained filtering for a class of nonlinear systems with improved static event-triggered communication, Int. J. Robust Nonlinear Control 29 (5) (2019) 1484–1498. [20] C. Peng, M. Wu, X. Xie, Y. Wang, Event-triggered predictive control for networked nonlinear systems with imperfect premise matching, IEEE Trans. Fuzzy Syst. 26 (5) (2018) 2797–2806. [21] Z. Gu, H. Zhao, D. Yue, F. Yang, Event-triggered dynamic output feedback control for networked control systems with probabilistic nonlinearities, Inf. Sci. 457–458 (2018) 99–112. [22] S. Hu, D. Yue, X. Xie, X. Yin, Resilient event-triggered controller synthesis of networked control systems under periodic DOS jamming attacks, IEEE Trans. Cybern. (2018), doi:10.1109/TCYB.2018.2861834. [23] S. Hu, D. Yue, X. Xie, X. Chen, X. Yin, Resilient event-triggered controller synthesis of networked control systems under periodic DoS jamming attacks, IEEE Trans. Cybern. (2019), doi:10.1109/TCYB.2018.2861834. [24] K. Mathiyalagan, H.P. Ju, R. Sakthivel, Finite-time boundedness and dissipativity analysis of networked cascade control systems, Nonlinear Dyn. 84 (4) (2016) 2149–2160. [25] R. Sakthivel, P. Selvaraj, Y. Lim, H.R. Karimi, Adaptive reliable output tracking of networked control systems against actuator faults, J. Frankl. Inst. 354 (9) (2017) 3813–3837. [26] R. Sakthivel, S. Santra, B. Kaviarasan, Resilient sampled-data control design for singular networked systems with random missing data, J. Frankl. Inst. 355 (3) (2018) 1040–1072. ˚ [27] K.J. Aström , B. Bernhardsson, Comparison of periodic and event based sampling for first-order stochastic systems, IFAC Proc. Vol. 32 (2) (1999) 5006–5011. ˚ [28] K.E. Aarzén , A simple event-based PID controller, IFAC Proc. Vol. 32 (2) (1999) 8687–8692. [29] J. Lunze, D. Lehmann, A state-feedback approach to event-based control, Automatica 46 (1) (2010) 211–215. [30] X. Wang, Event-Triggering in Cyber-Physical Systems, Department of Electrical Engineering, University of Notre Dame, Notre Dame, Indiana, USA, 2009 Ph.d thesis. [31] M. Mazo, P. Tabuada, Decentralized event-triggered control over wireless sensor/actuator networks, IEEE Trans. Autom. Control 56 (10) (2011) 2456–2461. [32] A. Girard, Dynamic triggering mechanisms for event-triggered control, IEEE Trans. Autom. Control 60 (7) (2013) 1992–1997. [33] V.S. Dolk, D.P. Borgers, W.P.M.H. Heemels, Output-based and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and zeno-freeness, IEEE Trans. Autom. Control 62 (1) (2016) 34–49. [34] W. Heemels, M. Donkers, A. Teel, Periodic event-triggered control for linear systems, IEEE Trans. Autom. Control 58 (2013) 847–861. [35] J. Liu, T. Yin, M. Shen, X. Xie, J. Cao, Distributed event-triggered state estimators design for networked sensor systems with deception attacks, IET Control Theory App. (2019), doi:10.1049/iet-cta.2018.5868. [36] D. Yue, E.G. Tian, Q. Han, A delay system method for designing event-triggered controllers of networked control systems, IEEE Trans. Autom. Control 58 (2013) 475–481. [37] S. Hu, D. Yue, X. Chen, Z. Cheng, X. Xie, Resilient H∞ filtering for event-triggered networked systems under nonperiodic DoS jamming attacks, IEEE Trans. Syst. Man Cybern. Syst. (2019), doi:10.1109/TSMC. 2019.2896249.

E. Tian, K. Wang and X. Zhao et al. / Journal of the Franklin Institute 356 (2019) 7210–7223

7223

[38] Z. Gu, T. Zhang, H. Zhao, F. Yang, M. Shen, A novel event-triggered mechanism for networked cascade control system with stochastic nonlinearities and actuator failures, J. Frankl. Inst. 356 (4) (2019) 1955–1974. [39] C. Peng, J. Zhang, H. Yan, Adaptive event-triggering H∞ load frequency control for network-based power systems, IEEE Trans. Ind. Electron. 99 (2017) 1685–1694. [40] P. Park, J.W. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica 47 (1) (2011) 235–238. [41] X. Zhang, Q. Han, Event-triggered dynamic output feedback control for networked control systems, IET Control Theory Appl. 8 (4) (2014) 226–234.