Guaranteed cost control of hybrid-triggered networked systems with stochastic cyber-attacks

ISA Transactions xxx (xxxx) xxx

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Research article

Guaranteed cost control of hybrid-triggered networked systems with stochastic cyber-attacks ∗

Jiancun Wu a , Chen Peng a , , Jin Zhang b , Mingjin Yang a , Bao-Lin Zhang c a

Department of Automation, School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, China School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel c College of Science, China Jiliang University, Hangzhou 310018, China b

highlights • A hybrid-triggered mechanism is presented to mitigate the pressure of network transmission. • Two stochastic variables are used to describe the switch process of two triggered schemes and the probability of cyber-attacks occurred. • The feedback guaranteed cost controller gain and parameters of triggered can be co-design.

article

info

Article history: Received 30 October 2018 Received in revised form 15 April 2019 Accepted 19 April 2019 Available online xxxx Keywords: Networked control systems Hybrid-triggered mechanism Cyber-attacks Guaranteed cost control

a b s t r a c t This paper is concerned with guaranteed cost control for a hybrid-triggered networked system subject to stochastic cyber-attacks. First, a hybrid-triggered mechanism including time-triggered mechanism and event-triggered mechanism is proposed to mitigate the pressure of network transmission, in which the switching between two mechanisms satisfies Bernoulli distribution. Second, the closedloop system subject to the hybrid communication scheme and stochastic cyber-attacks is modelled as a stochastic system with an interval time-varying delay. Then, based on the Lyapunov–Krasovskii functional approach, two theorems are presented for guaranteeing the mean-square stability of the studied system. Finally, the effectiveness of the proposed method is demonstrated through a numerical example. © 2019 Published by Elsevier Ltd on behalf of ISA.

1. Introduction Networked control systems (NCSs) is a special control system wherein the feedback control loops are connected via a shared communication network [1,2]. NCSs is playing an increasing critical role in different kinds of infrastructure fields for their superiorities such as easy wiring, easy operation and maintenance, low cost, high efficiency and flexibility [3–7]. Due to the limited bandwidth of communication network, some inevitable phenomena, e.g. network induced delays, data packet dropouts and disorder occur, which can deteriorate the performance of NCSs. Up to now, much research has been made to deal with those problem, for example, [8–11] and the references therein. In general, most of control strategies are executed based on time-triggered mechanism. How to save limited network communication resources while guaranteeing the performance of networked control systems is an urgent problem to be studied. Fortunately, a new triggering mechanism called event-triggered ∗ Corresponding author. E-mail address: [email protected] (C. Peng).

mechanism is proposed. The core idea of event-triggered mechanism is that the transmission of the current sampled data is determined by a pre-setting triggered condition [12]. On this basis of [12], some fruitful results have been obtained [13–15]. For example, a self event-triggering scheme for NCSs is presented in [16]. A fuzzy H∞ filter design approach is proposed for networked T–S fuzzy systems with an event-triggered control scheme [17]. An adaptive event-triggering communication scheme [18] is presented for power system, where the triggering threshold is dynamically adjusted to save more limited network resources. However, notice that the parameter of traditional event-triggered communication scheme is fixed, which ignores the variation of the network utilization. In practical systems, it is beneficial to take the variation of the network loads into consideration, therefore, it is desired to construct a more flexible communication scheme. By using the advantages of the timetriggering and event-triggering strategy [16], a hybrid-triggered mechanism is proposed in [19], where the stochastic switching between two triggered schemes is described by Bernoulli distributed. The reliable controller design for nonlinear systems with hybrid drive is discussed in [20], which takes the probabilistic actuator failure and probabilistic nonlinear perturbation

https://doi.org/10.1016/j.isatra.2019.04.017 0019-0578/© 2019 Published by Elsevier Ltd on behalf of ISA.

Please cite this article as: J. Wu, C. Peng, J. Zhang et al., Guaranteed cost control of hybrid-triggered networked systems with stochastic cyber-attacks. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.017.

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J. Wu, C. Peng, J. Zhang et al. / ISA Transactions xxx (xxxx) xxx

into account. The authors in [21] investigate the problem of filter design for T–S fuzzy systems with hybrid-driven scheme and quantization. The hybrid triggering mechanism proposed in aforementioned results are based on the difference between the current state and the latest release signal of the event generator. As an alternative method, the authors in [22] propose that a trigger condition that utilizes the latest network transmission signal, which is also employed in this paper. Note that the network security issue has not been considered in [22]. As a common phenomena in an unreliable communication network, the behaviour of cyber-attacks often exists, which can degrade or even deteriorate the performance of the system. The issues of the network security have obtained increasing attention [23–27]. For example, the attack scheduling problem for discrete-time networked systems is proposed in [28]. The authors in [29] investigate a security guaranteed filter for stochastic delay system with sensor saturations and deception attacks. A novel distributed recursive filter is proposed in [30] for stochastic system under cyber-attacks. To the best of our knowledge, with the hybrid-triggered mechanism proposed in [22], the results on network security problem for NCSs are scarce but important, this motivates this paper. The objective of this paper is to address a hybrid-triggered control for networked systems under stochastic cyber-attacks. The main contribution of this paper can be generalized as following: (1) a hybrid-triggered mechanism including time-triggered mechanism and traditional event-triggered mechanism is presented to mitigate the pressure of network transmission; (2) two stochastic variables satisfying Bernoulli distributed are used to describe the switching process of two triggered schemes and the probability of cyber-attacks occurred; (3) considering the network-induced delay, the closed-loop system subject to the stochastic cyber-attacks under a hybrid-triggered mechanism is modelled as an stochastic system with an interval time-varying delay, and the feedback guaranteed cost controller gain and triggering parameters can be obtained simultaneously. Notation. Rn stands for n-dimensional Euclidean space, denote block-diagonal matrix by diag{· · ·}, The superscript T stands for the transpose of a matrix or a vector, Z≥ 0 denotes the set of nonnegative integers and I is an identity matrix with an appropriate dimension. Prob{X } denotes the probability of event X to occur. Matrix X > 0 (respectively, X ≥ 0) means that the X is a positive definite (respectively, positive semi-definite) symmetric matrix. The symbol ∗ is the symmetric term in a matrix.

Fig. 1.

The structure of hybrid-triggered NCSs with cyber-attacks.

Assumption 1. The sensor and controller are connected via a communication network, which is suffering the threats of randomly occurring cyber-attacks. The sensor node is clock-driven with a constant h, while the controller and actuator with zero order holder (ZOH) are event-driven. Assumption 2. The network-induced delay at sampled instant kh is denoted by ηk . It is assumed that the time delay ηk has lower and upper bounds and η0 = 0 and all state variables of system (1) are measurable. Assumption 3. The hybrid-triggered mechanism is introduced which including time-triggered mechanism and event-triggered mechanism. The sampled data packets {x(0h), x(1h), x(2h), . . . , x(kh)} transmission over communication network is classified into two categories. Let Z≥1 0 and Z≥2 0 be the time stamp set of sampled data transmitted by time-triggered and event-triggered, ⋂ 2 Z≥0 = Ø. respectively. Z≥1 0 ∪ Z≥2 0 = Z≥0 , Z≥1 0 The hybrid-triggered mechanism is introduced to alleviate the transmission load and reduce the energy consumption of the network. In the following, the mathematical model for timetriggered and event-triggered schemes will be discussed, respectively. 2.1. Model under time-triggered scheme When k ∈ Z≥1 0 , the sampled data will be transmitted by timetriggered mechanism. Considering the time-delay induced by the network, a state feedback controller can be designed as 1 u1 (t) = Kx(kh), t ∈ [kh + ηk , kh + h + ηk+1 ), k ∈ Z≥ 0,

2. System framework and preliminaries

(3)

In this paper, the structure of networked control systems under stochastic cyber-attacks is shown in Fig. 1, where a hybridtriggered mechanism is introduced to mitigate the transmission frequency of NCSs. Consider the following linear system

where K ∈ Rn×m is the feedback gain matrix to be determined later. Then define τ (t) = t − kh for t ∈ [kh + ηk , kh + h + ηk+1 ), 1 k ∈ Z≥ 0 . It is clear that τ (t) is a linear piecewise function with ηk ≤ τ (t) ≤ h + ηk+1 . Denoting τ11 = mink {ηk |k ∈ Z≥1 0 }, τ21 = maxk {h + ηk+1 |k ∈ Z≥1 0 }, we obtain

x˙ (t) = Ax(t) + Bu(t),

u1 (t) = Kx(t − τ (t))

(1)

where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input, A and B are system matrices with appropriate dimensions. Associated with system (1), we consider the following cost function.

(4)

where

{

0 ≤ τ11 ≤ τ (t) ≤ τ21 ≤ +∞,

τ˙ (t) = 1, t ∈ [kh + ηk , kh + h + ηk+1 ), k ∈ Z≥1 0 .

(5)

+∞

∫

[xT (v )Mx(v ) + uT (v )Nu(v )]dv,

J =

(2)

0

where M and N are given positive definition symmetric matrices. The following assumptions are necessary for the further research.

Substituting (4) into system (1), the closed-loop system can be expressed by x˙ (t) = Ax(t) + BKx(t − τ (t)),

(6)

for t ∈ [kh + ηk , kh + h + ηk+1 ), k ∈ Z≥0 . 1

Please cite this article as: J. Wu, C. Peng, J. Zhang et al., Guaranteed cost control of hybrid-triggered networked systems with stochastic cyber-attacks. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.017.

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3

2.2. Model under event-triggered scheme

2.3. Modelling under hybrid-triggered scheme

When k ∈ Z≥2 0 , the event-triggered channel will be active. It can be seen that whether the sampled data x(kh) should be transmitted or not is determined by comparing the latest sampled data, which had been sent with the error between the current 2 sampled data. Inspired by [22], denote that tm h, tm ∈ Z≥ 0 is the mth release instant of the event generator. The condition of the event-triggered mechanism can be expressed as

In order to obtain the mathematical model of the NCSs under the hybrid driven scheme, a stochastic variable which obeys Bernoulli distributed is introduced to describe the switching process. Define a stochastic variable α (t), k ∈ Z≥0 taking values on either 0 or 1 with the following probability:

tm+1 h = min{kh|eT (kh)Φ1 e(kh) ≥ σ xT (kh)Φ1 x(kh)},

where Φ1 and Φ2 are positive weighting matrices, σ > 0 is a threshold parameter, e(kh) is the error between the current 2 sampled data x(kh), k ∈ Z≥ 0 and the latest released one x(sm h), 2 i.e., e(kh) = x(kh) − x(sm h), k ∈ Z≥ 0 with sm h = max{tm h, jm h},

Taking into account time-delay induced by network, the triggered data {x(t0 h), x(t1 h), . . .} arrive the actuator node at the instants {t0 h + τt0 , t1 h + τt1 , . . .}, respectively. Therefore, the control input of the plant is given by u2 (t) = Kx(sm h), t ∈ [sm h + ηsm , tm+1 h + ηtm+1 ),

(8)

In what follows, the interval [sm h + ηsm , tm+1 h + ηtm+1 ) is first divided into tm+1 − sm sub-intervals, i.e., tm+1 −1

[kh + ηk , kh + h + ηk+1 ), (9)

k=sm

where

{

ηsm , ηtm+1 ,

where 0 ≤ α¯ ≤ 1 is a constant, E {(α (t) − α¯ )2 } = α¯ (1 − α¯ ) = ρ1 is used to denote the mathematical variance of α (t). The state feedback controller can be described by the following form u(t) = α (t)u1 (t) + (1 − α (t))u2 (t), t ∈ [kh + ηk , kh + h + ηk+1 ), x(t) = φ (t), t ∈ [−τ2 , 0],

Remark 1. The hybrid-triggered mechanism used in this paper is different from the existing triggered scheme, such as [19,31]. It is worth noting that the time stamp of data packets used by the 2 trigger condition belongs to Z≥ 0 in [19,31]. However, the data packets used by the trigger condition proposed in this paper is the one latest sent x(sm h), that is, the time stamp may belong to 1 Z≥ 0 . Under this triggered mechanism, we can use the recently data packet x(sm h) to overcome this drawback and reduce the transmission frequency of data packets.

ηk =

k = sm , sm + 1, . . . , sm+1 − 1, k = tm+1 .

⋃

{

0 ≤ τ1 ≤ τ (t) ≤ τ2 ≤ +∞, τ˙ (t) = 1, t ∈ [kh + ηk , kh + h + ηk+1 ), k ∈ Z≥0 ,

where τ1 = min{τ11 , τ12 } and τ2 = max{τ21 , τ22 }. 2.4. Guaranteed cost control under hybrid-triggered scheme and cyber-attacks As shown in Fig. 1, both the time delay induced by the network and the stochastic cyber-attacks should be considered in the design. In this paper, deception attacks as a typical type of cyberattacks is considered, for deception attacks, the adversary intends to corrupted the data packets by inject a certain deception signal into the true signals of the control input u(t). Due to the random nature of the network conditions, the deception attacks launched by adversaries could be random or interval. In this paper, we characterize the phenomenon of the randomly occurring deception attacks by using a Bernoulli process. The hybrid-triggered networked system model subject to the randomly occurring deception attacks can be described by:

k ∈ Z≥0

{jm |jm ∈ Z≥1 0 }.

(10)

0 ≤ τ12 ≤ τ (t) ≤ τ22 ≤ +∞,

{

(11)

τ˙ (t) = 1, t ∈ [kh + ηk , kh + h + ηk+1 ), k ∈ Z≥2 0 ,

2 2 2 where τ12 = mink {ηk |k ∈ Z≥ 0 }, τ2 = maxk {h + ηk+1 |k ∈ Z≥0 }. Based on above analysis, we can arrive at

u2 (t) = Kx(t − τ (t)) − Ke(kh),

(12)

2 for t ∈ [kh + ηk , kh + h + ηk+1 ), k ∈ Z≥ 0 , where τ (t) satisfies (11) and e(kh) satisfies

e (kh)Φ1 e(kh) < σ x (t − τ (t))Φ1 x(t − τ (t)), k ∈ Z≥0 . T

(16)

u¯ (t) = u(t) + θ (t)[Kf (t − d(t)) − u(t)], t ∈ [kh + ηk , kh + h + ηk+1 ),

If k takes value on {jm |m ∈ Z≥0 }, the actual input of the plant at 1 is u(t) = Kx(jm h), [jm h + ηjm , jm h + h + ηjm +1 ), jm ∈ Z≥ 0 . Similar to the aforementioned operation, we can obtain that the state feedback controller can be expressed by (4). On the other hand, 2 when k ∈ Z≥ 0 , define τ (t) = t − kh for t ∈ [kh +ηk , kh + h +ηk+1 ), 2 k ∈ Z≥0 . It is easily obtained that

T

(15)

where φ (t) is the initial condition of x(t) and τ (t) in (15) satisfies

Notice that the variable k in (9) satisfies 2 k ∈ Z≥ 0

(14)

k ∈ Z≥0

1 jm = max{j|jh − kh < 0, j ∈ Z≥ 0 }.

⋃

Prob{α (t) = 1} = E {α (t)} ≜ α, ¯ Prob{α (t) = 0} = 1 − α¯ = α¯ 1 ,

(7)

k>sm

[sm h + ηsm , tm+1 h + ηtm+1 ) =

{

2

(13)

(17)

where f (t − d(t)) represents the function of deception attacks, d(t) ∈ [0, dm ] denotes the time delay of deception attacks. The stochastic variable θ (t), which characterizes the nature of the randomly occurring deception attacks, satisfies the Bernoulli distribution and has the following probability

{

¯ Prob{θ (t) = 1} = E {θ (t)} ≜ θ, Prob{θ (t) = 0} = 1 − θ¯ = θ¯1

(18)

where 0 ≤ θ¯ ≤ 1 is a constant, E {(θ (t) − θ¯ )2 } = θ¯ (1 − θ¯ ) = ρ2 is used to denote the mathematical variance of θ (t). Remark 2. The stochastic variable θ (t) is used to describe the nature property of the cyber-attacks. When θ (t) = 1, it means that the real input is replaced by the deception signal f (t − d(t)). Therefore, the closed-loop system subject to the deception attacks can be given by: x˙ (t) = Ax(t) + (1 − θ (t))BKx(t − τ (t))

− (1 − α (t))(1 − θ (t))BKe(kh) + θ (t)BKf (t − d(t))

(19)

Please cite this article as: J. Wu, C. Peng, J. Zhang et al., Guaranteed cost control of hybrid-triggered networked systems with stochastic cyber-attacks. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.017.

4

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Remark 3. It is assumed that Bernoulli variables α (t) and θ (t) are mutual independent. When θ (t) = 0, the closed-loop system in (19) is x˙ (t) = Ax(t)+BKx(t −τ (t))−(1−α (t))BKe(kh), which means no deception attacks in the transmission of the data packets. Assumption 4. The nonlinear function of the deception attacks f (t) is supposed to satisfy the following constraint condition

∥f (t)∥2 ≤ ∥Gx(t)∥2

(20)

where G is a constant matrix used to denote the upper bound of the deception attacks function. In what follows, we aim at designing a network-based control law such that the system under the hybrid-triggered scheme subject to stochastic deception attacks is mean square stable. For achieving this purpose, we first introduce the following crucial lemma. n×n Lemma 1[ ([32]). For and U ∈ Rn×n ] any matrices R ∈ R R ∗ satisfying > 0, τ (t) ∈ [τ1 , τ2 ], τ1 is a positive scalar, and U R vector function x˙ : [τ1 , τ2 ] → Rn , the following inequality holds:

− (τ2 − τ1 )

∫

t −τ1

x˙ T (s)Rx˙ (s)ds ≤ −ζ T (t)Λζ (t)

(21)

3.1. Mean square stability criterion In this section, we will analysis the mean square stability of the system (23) under the stochastic cyber-attacks. The following mean square stability criterion is derived via the Lyapunov– Krasovskii functional approach. Theorem 1. For given scalars 0 ≤ τ1 < τ2 , dm ≥ 0, 0 ≤ α¯ ≤ 1, 0 ≤ θ¯ ≤ 1 and matrices M > 0, N > 0, K , if there exist real matrices P > 0, Q1 > 0, Q2 > 0, Q3 > 0, R1 > 0, R2 > 0, R3 > 0, R4 > 0, Φ1 > 0 and U2 , U4 with appropriate dimensions such that the following inequalities hold

⎡ Ω11 ⎢ ∗ ⎢ ⎢ ∗ Ω=⎢ ⎢ ∗ ⎣ ∗ ∗ [

R2 U2

∗ R2

]

Ω12 Ω22 ∗ ∗ ∗ ∗

where

Ω11

In order to facilitate further development, the controller (17) and system (19) can be rewritten as the following form u¯ (t) = [D1 + (θ¯ − θ (t))D2 − (θ¯ − θ (t))(1 − α (t))D3 − θ¯1 (α¯ − α (t))D3 ]ξ (t)

(22) x˙ (t) = [F1 + (θ¯ −θ (t))F2 − (θ¯ −θ (t))(1 −α (t))F3 − θ¯1 (α−α ¯ (t))F3 ]ξ (t) (23)

Ω15

Ω16

0

0 0 0

0 0 0 0

Ω33 ∗ ∗ ∗

Ω34 Ω44 ∗ ∗

[

∗

R4 U4

≥ 0,

Γ11 ⎢ ∗ ⎢ ∗ ⎢ ⎢ ∗ =⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

⎡

3. Main results

0

Ω14

Ω55 ∗

]

R4

Ω66

Γ12 Γ22 ∗ ∗ ∗ ∗ ∗ ∗

Γ13 Γ23 Γ33 ∗ ∗ ∗ ∗ ∗

0

Γ24 Γ34 Γ44 ∗ ∗ ∗ ∗

Γ15

Γ16

Γ17

Γ18

⎤

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0 0

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

Γ55 ∗ ∗ ∗

Γ56 Γ66 ∗ ∗

Γ12 =R1 , Γ13 = θ¯1 PBK +

0

θ¯1 BK

F2 =

0

0

BK

F3 =

[

0

0

0

D1 =

[

0

0

θ¯1 K

D2 =

[

0

0

K

0

0

0

−K

D3 =

[

0

0

0

0

0

0

0

0 0

0 0

0

0 0

θ¯ BK

0 0

0 0

−BK 0

BK

0

−θ¯1 α¯ 1 BK ]

]

E {J } =

k=0

kh+h+ηk+1

Ψ (t , α, ¯ θ¯ )dt ,

4

R3 ,

R3 , Γ15 = R4 − U4 ,

π2 4

R3 + σ Φ1 ,

Γ55 = − 2R4 + U4 + U4T , Γ56 = R4 − U4 , Γ66 = − R4 − Q3 , Γ77 = −θ¯ I , Γ88 = −Φ1 ,

ρ3 =ρ2 α¯ 1 + ρ1 θ¯12

]

For the system (23), the expectation value of cost function (2) can be expressed as +∞ ∫ ∑

4

π2

Γ34 =R2 − U2 , Γ44 = −Q2 − R2 , τ = τ2 − τ1 ,

]

0

K

π2

Γ33 = − 2R2 + U2 + U2T −

]

−θ¯1 α¯ 1 K ]

0

Γ88

Γ16 =U4 , Γ17 = θ¯ PBK , Γ18 = −α¯ 1 θ¯1 PBK , Γ22 =Q2 − Q1 − R1 − R2 , Γ23 = R2 − U2 , Γ24 = U2 ,

and A

Γ77 ∗

Ω12 =[τ1 F1T R1 , τ F1T R2 , τ2 F1T R3 , dm F1T R4 ] √ Ω13 = ρ2 × [τ1 F2T R1 , τ F2T R2 , τ2 F2T R3 , dm F2T R4 ] √ Ω14 = ρ3 × [τ1 F3T R1 , τ F3T R2 , τ2 F3T R3 , dm F3T R4 ] [ ]T √ Ω15 = 0 0 0 0 θ¯ G 0 0 0 √ √ Ω16 =[DT1 N , ρ2 DT2 N , ρ3 DT3 N , M ] Ω22 =Ω33 = Ω44 = diag {−R1 , −R2 , −R3 , −R4 } Ω55 = − I , Ω66 = diag {−N , −N , −N , −M }

ξ (t) = col{x(t), x(t − τ1 ), x(t − τ (t)), x(t − τ2 ), x(t − d(t)), x(t − dm ), f (t − d(t)), e(kh)}

[

(25)

(26)

Γ11 =PA + AT P + Q1 + Q3 − R1 − R4 −

[

⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎦

≥ 0,

where

F1 =

⎤

where

t −τ2

ζ (t) = col[x(t − τ1 ), x(t − τ (t)), x(t − τ2 )] [ ] R ∗ ∗ Λ = U − R 2R − U − U ∗ −U U −R R

Ω13

(24)

kh+ηk

where Ψ (t , α, ¯ θ¯ ) = xT (t)Mx(t) + ξ T (t){DT1 ND1 + +θ¯ (1 − θ¯ )DT2 D2 + [θ¯ (1 − θ¯ )(1 − α¯ ) + α¯ (1 − θ¯ 2 )(1 − α¯ )]DT3 D3 }

then systems (23) under the stochastic deception attacks is asymptotically stable in the mean square, and the performance function E {J } is at least E {J } ≤ J ∗ =φ T (0)P φ (0) +

∫

0

φ T (v )Q1 φ (v )ds +

∫

−τ1

∫

0

φ T (v )Q3 φ (v )ds + τ1

+ −dm

−τ1

φ T (v )Q2 φ (v )ds

−τ2

∫

0

−τ1

∫

0

φ˙ T (v )R1 φ˙ (v )dv ds

s

Please cite this article as: J. Wu, C. Peng, J. Zhang et al., Guaranteed cost control of hybrid-triggered networked systems with stochastic cyber-attacks. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.017.

J. Wu, C. Peng, J. Zhang et al. / ISA Transactions xxx (xxxx) xxx

+ (τ2 − τ1 )

−τ1

∫

0

φ˙ T (v )R2 φ˙ (v )dv ds

0

∫

φ˙ T (v )R4 φ˙ (v )dv ds

+ dm

R2

(32)

[

t

∫

T

x˙ (s)R4 x˙ (s)ds ≤ −

− dm

V (t , xt ) = V1 (t , xt ) + V2 (t , xt ) + V3 (t , xt )

(28)

t −dm

⎡ where

t

x (v )Q1 x(v )dv + T

∫

t −τ1

∫

x˙ T (v )R2 x˙ (v )dv ds

Πx,e = σ xT (t − τ (t))Φ1 x(t − τ (t)) − eT (kh)Φ1 e(kh) ≥ 0

π2

s

t −τ2

∫

t

∫

4

[x(v ) − x(kh)]T R3 [x(v ) − x(kh)]dv kh ∫ t ∫ t ∫ t x˙ T (v )R3 x˙ (v )dv dm + τ22 x˙ T (v )R4 x˙ (v )dv ds ×

s

t −dm

kh

Applying the infinitesimal operator for Vi (t , xt )(i = 1, 2, 3) and taking expectation on it, on can get: E {LV1 (t , x(t))} =2xT (t)P [Ax(t) + θ¯1 BKx(t − τ (t))

− α¯ 1 θ¯1 BKe(kh) + θ¯ BKf (t − d(t))] E {LV2 (t , x(t))} =x (t)Q1 x(t) − xT (t − τ1 )Q1 x(t − τ1 ) + xT (t)Q3 x(t) T

− xT (t − dm )Q3 x(t − dm ) + xT (t − τ1 )Q2 x(t − τ1 ) − xT (t − τ2 )Q2 x(t − τ2 )

−

π2 4

4

xT (t)R3 x(t) +

x (kh)R3 x(kh) − τ1 T

∫

π2 2

xT (t)R3 x(kh)

t T

x˙ (s)R1 x˙ (s)ds t −τ1

− (τ2 − τ1 )

∫

t −τ1

x˙ T (s)R2 x˙ (s)ds

t −τ2

∫

t

− dm

E {LV (t , xt )} + Πx,f + Πx,e ≤ ξ T (t)Ω ξ (t)

x˙ T (s)R4 x˙ (s)ds

(29)

t −dm

(36)

By the Schur complement, we can obtained that Ω < 0, hence for t ∈ [kh + ηk , kh + h + ηk+1 ), k ∈ Z≥0 , one yields E {LV (t , xt )} ≤ −Πx,f − Πx,e < 0.

(37)

It is clear from Ω < 0 that for a sufficiently small ε > 0, E {LV (t , xt )} < −ε∥x(t)∥2 which implies that the mean-square asymptotical stability of system (23) is guaranteed. From (24), integrating both sides of (37) on t from 0 to +∞ yields E {V (+∞, x+∞ ) − V (0, x0 )}

≤−

+∞ ∫ ∑ k=0

E {LV3 (t , x(t))} =τ12 x˙ T (t)R1 x˙ (t) + (τ2 − τ1 )2 x˙ T (t)R2 x˙ (t) + τ22 x˙ T (t)R3 x˙ (t)

π2

(35)

Combining (28)–(35), we arrive at

t

+ d2m x˙ T (t)R4 x˙ (t) −

2R4 − U4 − U4T U4 − R4

Considering the condition of hybrid-triggered scheme (7), we can obtain that

x˙ T (v )R1 x˙ (v )dv ds

+ (τ2 − τ1 )

⎤ [ ] ∗ x(t) ∗ ⎦ × x(t − d(t)) x(t − dm ) R4

∗

Πx,f = θ¯ xT (t −τ (t))GT Gx(t −τ (t))−θ¯ f T (t −d(t))f (t −d(t)) ≥ 0 (34)

s

t −τ1

]T

From Assumption 4, which is the limited condition of the cyber-attacks in this paper, we can obtain

t

∫

∫

x(t) x(t − d(t)) x(t − dm )

(33)

xT (v )Q2 x(v )dv

xT (v )Q3 x(v )dv, t −dm t

V3 (t , xt ) = τ1

t −τ1

t −τ2

t −τ1 t

+

R4

× ⎣ U4 − R4 −U4

V1 (t , xt ) = xT (t)Px(t),

∫

R2

x(t − τ1 ) x(t − τ (t)) x(t − τ2 )

]

(27)

Proof. Construct the following Lyapunov–Krasovskii functional

V2 (t , xt ) =

∗ ∗ ⎦×

2R2 − U2 − U2T U2 − R2

[

s

−dm

∫

5

⎤

∗

× ⎣ U2 − R2 −U2

s

−τ2

∫

⎡

0

∫

kh+h+ηk+1

(Πx,f + Πx,e )dt = −E {J }.

(38)

kh+ηk

Therefore, one can obtain that E {J } ≤ E {V (0, x0 )−V (+∞, x+∞ )} ≤ E {V (0, x0 )} = J ∗ , this completes the proof. It is necessary to point out that the sufficient condition have been obtained which can guarantee the asymptotical stability in the mean-square of the system under cyber-attacks through Theorem 1. Owing to the existence of nonlinear terms such as θ¯1 BKP in Theorem 1, the controller gain cannot be directly obtained. To solve this problem, Theorem 2 will be introduced in the following subsection. 3.2. Guaranteed cost controller design

Notice that E {˙xT (t)Ri x˙ (t)} =ξ T (t){F1T Ri F1 + θ¯ (1 − θ¯ )F2T Ri F2 + [θ¯ (1 − θ¯ )

(30)

(1 − α¯ ) + α¯ (1 − θ¯ 2 )(1 − α¯ )]F3T Ri F3 }, i = 1, 2, 3 By utilizing the Jensen’s inequality and Lemma 1, it can be obtained that

− τ1

∫

t

x˙ T (s)R1 x˙ (s)ds ≤ −[xT (t) − xT (t − τ1 )]R1 [x(t) − (t − τ1 )] t −τ1

(31)

− (τ2 − τ1 )

∫

t −τ1

[ T

x˙ (s)R2 x˙ (s)ds ≤ − t −τ2

x(t − τ1 ) x(t − τ (t)) x(t − τ2 )

]T

In this section, we will solve the problem of guaranteed cost controller design. Based on Theorem 1, we can derived the following Theorem. Theorem 2. For given scalars 0 ≤ τ1 < τ2 , dm ≥ 0, 0 ≤ α¯ ≤ 1, 0 ≤ θ¯ ≤ 1 and matrices M > 0, N > 0, if there exist real matrices X > 0, Q¯ 1 > 0, Q¯ 2 > 0, Q¯ 3 > 0, R¯ 1 > 0, R¯ 2 > 0, R¯ 3 > 0, R¯ 4 > 0, ¯ 1 > 0 and U¯ 2 , U¯ 4 , K¯ with appropriate dimensions such that the Φ following inequalities hold

⎡¯ Ω11 ⎢ ∗ ⎢ ∗ ¯ =⎢ Ω ⎢ ⎢ ∗ ⎣ ∗ ∗

¯ 12 Ω ¯ 22 Ω ∗ ∗ ∗ ∗

¯ 13 Ω

¯ 14 Ω

¯ 15 Ω

0

0

¯ 33 Ω ∗ ∗ ∗

¯ 34 Ω ¯ 44 Ω ∗ ∗

0 0 0

¯ 55 Ω ∗

¯ 16 ⎤ Ω 0 ⎥ ⎥ 0 ⎥ ⎥ < 0, 0 ⎥ ⎦ 0 ¯ Ω66

(39)

Please cite this article as: J. Wu, C. Peng, J. Zhang et al., Guaranteed cost control of hybrid-triggered networked systems with stochastic cyber-attacks. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.017.

6

J. Wu, C. Peng, J. Zhang et al. / ISA Transactions xxx (xxxx) xxx

[

] ∗ ≥ 0, R¯ 2

R¯ 2 U¯ 2

[

] ∗ ≥ 0, R¯ 4

R¯ 4 U¯ 4

(40)

where

¯ 11 Ω

¯ 12 Ω ¯ 13 Ω ¯ 14 Ω ¯ 15 Ω ¯ 16 Ω ¯ 22 Ω ¯ 55 Ω

⎡¯ ⎤ Γ11 Γ¯ 12 Γ¯ 13 0 Γ¯ 15 Γ¯ 16 Γ¯ 17 Γ¯ 18 ⎢ ∗ Γ¯ 22 Γ¯ 23 Γ¯ 24 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ Γ¯ 33 Γ¯ 34 0 0 0 0 ⎥ ⎢ ⎥ ∗ ∗ Γ¯ 44 0 0 0 0 ⎥ ⎢ ∗ =⎢ ⎥, ∗ ∗ ∗ Γ¯ 55 Γ¯ 56 0 0 ⎥ ⎢ ∗ ⎢ ⎥ ¯ ∗ ∗ ∗ ∗ Γ66 0 0 ⎥ ⎢ ∗ ⎣ ∗ ¯ ∗ ∗ ∗ ∗ ∗ Γ77 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Γ¯ 88 =[τ1 F¯1T , τ F¯1T , τ2 F¯1T , dm F¯1T ] √ = ρ2 × [τ1 F¯2T , τ F¯2T , τ2 F¯2T , dm F¯2T ] √ = ρ3 × [τ1 F¯3T , τ F¯3T , τ2 F¯3T , dm F¯3T ] [ ]T √ = 0 0 0 0 θ¯ GX 0 0 0 √ √ =[D¯ T1 N , ρ2 D¯ T2 N , ρ3 D¯ T3 N , M ] ¯ 33 = Ω ¯ 44 = diag {R¯ 1 − 2X , R¯ 2 − 2X , R¯ 3 − 2X , R¯ 4 − 2X } =Ω ¯ 66 = diag {−N , −N , −N , −M } = − I, Ω

R3 , Γ¯ 15 = R¯ 4 − U¯ 4 ,

π2 ¯ ¯ 1, Γ¯ 33 = − 2R¯ 2 + U¯ 2 + U¯ 2T − R3 + σ Φ

4. An illustrative example

4

Γ¯ 34 =R¯ 2 − U¯ 2 , Γ¯ 44 = −Q¯ 2 − R2 , Γ¯ 55 = − 2R¯ 4 + U¯ 4 + U¯ 4T , Γ¯ 56 = R¯ 4 − U¯ 4 , ¯ 1 − 2X , Γ¯ 88 = −Φ ¯ 1, Γ¯ 66 = − R¯ 4 − Q¯ 3 , Γ¯ 77 = −θ/ [ ] F¯1 = AX 0 θ¯1 BK¯ 0 0 0 θ¯ BK¯ −θ¯1 α¯ 1 BK [ ] F¯2 = 0 0 BK¯ 0 0 0 −BK¯ 0 [ ] F¯3 = 0 0 0 0 0 0 0 BK¯ [ ] D¯1 = 0 0 θ¯1 K¯ 0 0 0 θ¯ K¯ −θ¯1 α¯ 1 K [ ] D¯2 = 0 0 K¯ 0 0 0 −K¯ 0 [ ] D¯3 = 0 0 0 0 0 0 0 K¯

Consider the system (1) with the following parameters given by:

[ A=

[ f (t) =

0

φ T (v )(Q¯ 1 − 2X )φ (v )ds

−τ1

∫

−τ1

φ T (v )(Q¯ 2 − 2X )φ (v )ds +

+

0

φ T (v )(Q¯ 3 − 2X )φ (v )ds

−dm

−τ2

+ τ1

∫

0

∫

0

∫

−τ1

φ˙ T (v )(R¯ 1 − 2X )φ˙ (v )dv ds

∫

−τ1

−τ2 0

∫

+ dm −dm

1 −2

]

,B =

[

0 1

]

−tanh(0.1x2 (t)) −tanh(0.05x1 (t))

]

,G =

[

s

K =

0

∫

φ˙ T (v )(R¯ 2 − 2X )φ˙ (v )dv ds s

0

φ˙ T (v )(R¯ 4 − 2X )φ˙ (v )dv ds

(41)

0.05 0

0 0.1

]

Supposing the initial condition is given as x0 = (−0.5, 0.5), our objective is to design the controller under the hybrid-triggered scheme such that the closed-loop system (23) is stable under the deception attacks. Then, we will give two cases to illustrate the effectiveness of the designed controller. Case 1: The system under the hybrid triggered control without cyber-attacks Set θ¯ = 0, which means the closed-loop system (23) under the hybrid triggered control without cyber-attacks can be expressed ˙ = Ax(t) + BKx(t − τ (t)) − (1 − α (t))BKe(kh). Applying as x(t) Theorem 2 with τ1 = 0, τ2 = dm = 0.5, α¯ = 0.4, σ = 0.2, the corresponding K and Φ1 can be derived as

s

+ (τ2 − τ1 )

∫

0 −1

The nonlinear function of the deception attacks in (22) is defined as f (t), and according to the constraint condition (20) of deception attacks, we can obtain:

then systems (23) under the stochastic deception attacks is asymptotically stable in the mean square. The controller gain matrix is given by K = K¯ X −1 and the performance function E {J } is at least

∫

P −1 = X , K¯ P −1 = K¯ , P −1 Ri P −1 = R¯i (i = 1, 2, 3),

1 −1 −1 where Σ1 = diag {X , R− i , Ri , Ri , I , I }(i = 1, 2, 3), Σ2 = diag {X , X }. Applying the inequalities, we can directly obtain results. The proof is completed.

Γ¯ 16 =U¯ 4 , Γ¯ 17 = θ¯ BK¯ , Γ¯ 18 = −α¯ 1 θ¯1 BK¯ , Γ¯ 22 =Q¯ 2 − Q¯ 1 − R¯ 1 − R¯ 2 , Γ¯ 23 = R¯ 2 − U¯ 2 , Γ¯ 24 = U¯ 2 ,

J ∗ =φ T (0)P φ (0) +

Proof. Left- and right-multiply (25), (26) by Σ1 , Σ2 and their transposes,respectively and setting

¯ i (i = 1, 2), P −1 Ui P −1 = U¯i (i = 2, 4) P −1 Φi P −1 = Φ

4

π2 ¯ 4

State responses of system (23) without cyber-attacks.

P −1 Qi P −1 = Q¯i (i = 1, 2, 3)

π2 ¯ Γ¯ 11 =AX + XAT + Q¯ 1 + Q¯ 3 − R¯ 1 − R¯ 4 − R3 , Γ¯ 12 =R¯ 1 , Γ¯ 13 = θ¯1 BK¯ +

Fig. 2.

[

0.12167

−0.23844

]

, Φ1 =

[

3.6554 0.67433

0.67433 4.5716

]

Under the obtained controller, the state trajectories of the system (23) without cyber-attacks are described in Fig. 2. The diagram of release intervals between two consecutive release instants is

Please cite this article as: J. Wu, C. Peng, J. Zhang et al., Guaranteed cost control of hybrid-triggered networked systems with stochastic cyber-attacks. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.017.

J. Wu, C. Peng, J. Zhang et al. / ISA Transactions xxx (xxxx) xxx

Fig. 3.

Fig. 4.

Fig. 5.

Release instants and release interval.

[

−0.35776

−0.48053

]

, Φ1 =

[

0.15724 0.11503

0.11503 0.1701

Release instants and release interval.

Fig. 6. State responses under sustained cyber-attacks.

State responses of system (23) under cyber-attacks.

shown in Fig. 3. One can see that the designed hybrid-triggered controller can stabilize the system without deception attacks. Case 2: System under the hybrid triggered control with cyberattacks Set θ¯ = 0.6, it means that the system are suffered from the random cyber-attacks with the probability of 60%, the closedloop system (23) under the hybrid triggered control subject to ˙ = Ax(t) + (1 − stochastic cyber-attacks can be expressed as x(t) θ (t))BKx(t − τ (t)) − (1 − α (t))(1 − θ (t))BKe(kh) + θ (t)BKf (t − d(t)). Applying Theorem 2 with τ1 = 0, τ2 = dm = 0.5, α¯ = 0.4, σ = 0.2, the corresponding K and Φ1 can be derived as K =

7

]

Under the obtained controller, Fig. 4 is the response curve of the system under stochastic cyber-attacks, and the release time intervals between any two consecutive release instants are described by Fig. 5. In fact, it is readily obtained that the proposed controller is capable of stabilizing the system with deception attacks and reducing the communication resources. Case 3: In order to further investigate the effectiveness of the proposed method, we set θ¯ = 1, which means the system with

sustained cyber-attacks. Fig. 6 plots the state responses of system, from which one can see that the hybrid-triggered controller proposed in this paper is still stabilize the system when the networked control system suffer a continuous attacks. In order to compare with the hybrid-trigger controller, we consider the situation of system with cyber-attacks under an event-triggered scheme. The state response of the studied system under stochastic cyber-attacks with traditional event-triggered mechanism is shown in Fig. 7. The diagram of release intervals between two consecutive release instants is shown in Fig. 8. Compared the above simulation results with the results in [19], one can obtain that the studied system can be stabilized under different situations. The transmission frequency of the network can be reduced by employing the event-triggered or the hybrid-triggered strategy. 5. Conclusion In this paper, a guarantee cost control for hybrid-triggered networked systems under stochastic cyber-attacks has been investigated. A novel hybrid-triggered mechanism has been introduced to alleviate the transmission load, and the phenomenon of the randomly occurring deception attacks has been characterized

Please cite this article as: J. Wu, C. Peng, J. Zhang et al., Guaranteed cost control of hybrid-triggered networked systems with stochastic cyber-attacks. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.017.

8

J. Wu, C. Peng, J. Zhang et al. / ISA Transactions xxx (xxxx) xxx

Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References

Fig. 7. State responses of system (23) under case 3.

Fig. 8. Release instants and release interval under case 3.

by using a Bernoulli process. By taking the above factors into consideration, the close-loop model of the studied system has been well constructed. The proposed method can ensure the stability of the system in the mean-square under stochastic cyberattacks. Finally, the effectiveness of the proposed method has been demonstrated through a numerical example. Following the method proposed in this paper, we will further study the H∞ filter and data dropout of NCSs under cyber-attacks in the future.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant 61833011, 61673255, 61273114 and 61773356, and the Outstanding Academic Leader Project of Shanghai Science and Technology Commission, China under Grant 18XD1401600, the Project 211, China under Grant D18003, the Key Project of Natural Science Foundation of Zhejiang Province of China under Grant LZ19F030001.

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Please cite this article as: J. Wu, C. Peng, J. Zhang et al., Guaranteed cost control of hybrid-triggered networked systems with stochastic cyber-attacks. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.04.017.