Improved event-triggered control for networked control systems under stochastic cyber-attacks

Neurocomputing 350 (2019) 33–43

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Brief papers

Improved event-triggered control for networked control systems under stochastic cyber-attacks Tao Li a, Xiaoling Tang a, Haitao Zhang b,∗, Shumin Fei c a

School of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China College of National Defense Engineering, The Army Engineering University of PLA, Nanjing 210007, PR China c School of Automation, Southeast University, Nanjing 210096, PR China b

a r t i c l e

i n f o

Article history: Received 9 December 2018 Revised 5 March 2019 Accepted 28 March 2019 Available online 19 April 2019 Communicated by Ma Lifeng Ma Keywords: Networked control systems Adaptive event-triggered control Stochastic cyber-attacks Asymptotical stability

a b s t r a c t This paper is concerned with an improved event-triggered control for a class of networked control systems (NCSs) under stochastic cyber-attacks with limited communication bandwidths. In order to reduce the unnecessary data transmissions, an adaptive event-triggered approach is employed to locally determine the sensor measurements whether to be transmitted or not, where the triggering thresholds can take much real-time information on the NCSs into consideration, such as the triggering error, last transmitted signal, and current sampled one. It is assumed that the network transmissions may be attacked by two types of randomly occurring cyber-attacks, then a novel mathematical model is achieved and three sufficient conditions are derived to guarantee the asymptotical stability of the closed-loop system. Especially, two augmented Lyapunov-Krasovskii functionals (LKFs) are constructed and an improved reciprocal convex technique is proposed to reduce the conservatism. Finally, two numerical examples are presented to show the effectiveness of the obtained results. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Normally, networked control systems (NCSs) can exchange data over a communication network with spatially separated sensors, actuators, and controllers. Together with rapid developments and widespread applications of computer science and communication network, the NCSs are playing an increasing role in the filed of smart grid, health care, water/gas distribution, and modern public transportation [1,2]. Obviously, networked-based control systems have many advantages such as great flexibility, low cost in installation, and easy maintenance. However, symbiotic with the advantages, the insertion of communication networks also results in some challenging problems such as limited network bandwidth, communication delay, packet loss and disorder, and cyber-attacks [3–7,12]. Traditionally, data transmission is determined purely on the elapse of a fixed or non-fixed time interval, which may produce a large number of redundant transmissions. Especially when there is little fluctuation of transmitted signals, the transmission will lead to unnecessary waste of communication resources [6]. Aiming to transmit less data and save energy efficiently, some communication methods were proposed, such as event-triggered control, ∗

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

https://doi.org/10.1016/j.neucom.2019.03.058 0925-2312/© 2019 Elsevier B.V. All rights reserved.

coding-decoding communication one, and so on [7,43,44]. Recently, the event-triggering technique has been widely utilized advocating that the data are transmitted only when some design function surpass a given threshold, which guarantees that only “necessary” signals can be sent out [8–15]. For instance, based on uniform sampling, some elegant designs on event-triggered controller was put forward [8–11] and in [12], a hybrid event scheme was proposed. In [13], a dynamic event-based controller was designed to study multi-agents and an overview was further presented [14]. Yet since the triggering thresholds were constants [8–14], then in [15], an adaptive event-triggered scheme was presented and the thresholds were time-variable. Meanwhile, owing to that data transmission plays an essential role in the NCSs, some event-triggered techniques was proposed and widely applied to realize the control targets [16–22]. Since an event-based idea was initially given under the networked circumstance [16], this method has been utilized to tackle the different control issues on the NCSs such as synchronization [17], quantization and packet loss [18], fuzzy controller [19], probabilistic nonlinearity [20], and actuator saturation [21]. Furthermore, in [22], a dynamic event-based scheme was proposed to study the state estimation in wireless sensor networks. However, based on the event results in [7–14,16–22], the thresholds are preselected constants in [0, 1] and cannot reflect the whole real-time information on controlled systems, which seem to be limited and need the further improvements.

34

T. Li, X. Tang and H. Zhang et al. / Neurocomputing 350 (2019) 33–43

Fig. 1. The structure of NCS under cyber-attacks.

Most recently, as for various systems including the NCSs, the cyber security has become a heated topic of the research and secure events around the world raise cyber security to a higher level [23–39]. The cyber-attacks aim to exploit the vulnerabilities in communication links and send wrong control signals to the operators [23–26]. For example, in electric power grids, an attacker can modify electric meters to inject bad measurements once they gain access to meters, which would possibly result in catastrophic consequences [27]. Some works showed that it was feasible to take cyber-attacks into discussions based on the event-triggered approach and resilient control [28–30]. Generally, the cyber-attacks basically include the Denial of Service (DoS) attacks and the deception ones. Specifically, sybil attack is a special case of DoS attacks that decreases the credibility of legitimate node by using its identity and accumulates important data or critical secret, seriously endangering the network services [31]. As one of most harmful DoS attacks in sensor network, the sybil ones serve as a gateway to other attacks such as the ones on routing and voting, data aggregation, fair resource allocation, misbehavior detection, and distributed storage [32,33]. Compared with the DoS ones, deception attacks are more difficult to be detected because the adversary can keep such attacks stealthy to the anomaly detector and it is worth mentioning that false data injection is a typical one of deception attack [34]. In [35], when bias injection attack is activated, the attacker can get access to sensor measurements and destroy the data delivered to the control unit when controller is an integral one. In order to mislead a dynamic detector, an attacker needs to inject a signal which consists with the network dynamics at each instant [36]. Especially, as for the NCSs, in [26,37], noting that the deception attacks were built as a nonlinear function satisfying an occurring Bernoulli distribution, the DoS attack was regarded as a kind of power-constraint periodic jamming signal which blocked the communication channel when the designed controller was not suitable for the cyber-attacks [38,39]. Overall, though the results in [12,23–39] are elegant, there still exist two issues to be addressed: first, as illustrated in [36], since the accumulated dynamic cyber-attack appeared, no works have taken this kind of attack into the NCSs; second, few results have appeared to study the relationship between communication delay and cyber-attacks’ delays when choosing the LKF, which remain important and challenging. Motivated by the discussions above, this paper focuses on an adaptive event-triggered control for the NCSs subject to two kinds of cyber-attacks. For each sensor, an adaptive event-based transmission scheme is designed to determine local transmitted signal, which can reduce the consumption of communication resources. Except for modeling a traditional randomly occurring cyber-attacks as a nonlinear function, we propose a new model of dynamic cyber-attack as an integral function satisfying a given restriction. Randomly switching two addressed attacks, the procedure satisfies a Bernoulli distribution. Considering all the factors above, a

closed-loop system model with multiple time-varying delays is established, enabling us to construct the mixed-delay-dependent augmented LKFs to derive less conservative results. These conditions are then applied to provide the algorithms on the co-design of suitable triggering parameters and controller gain by solving the derived linear matrix inequalities (LMIs). Finally, two numerical examples are presented to illustrate the feasibility of the proposed methods. Notations: Rn denotes the n-dimensional Euclidean space and Rn×m is the set of n × m real constant matrices. I is an appropriately dimensional identity matrix and 0 is an appropriately dimensional zero matrix. Prob{X} denotes the probability of occurring event X. E{ · } denotes the mathematical expectation. For a matrix   Z ∗ Y and two symmetric matrices Z and W, is a symmetric Y W ∗ matrix, where denotes the entries implied by symmetry. 2. Problem formulations In this paper, we consider the controlled plant described as

x˙ (t ) = Ax(t ) + Bu(t ),

(1)

(t )]T

where x(t ) = [x1 (t ) x2 (t ) . . . xn ∈ is the state vector, u(t ) ∈ Rm is the control input, and A, B are known matrices with appropriate dimensions. The diagram of the NCS is shown in Fig. 1, the sensors and collect and transmit information to the controller via a communication network with transmission delays and cyber-attacks. Considering the limited network bandwidth, the event-generators at each sensor side are employed to reduce the redundant network transmissions. If the sampled signals exceed the given event-triggered conditions, they will be sent out to the network. Then in our work, it is assumed that that the cyber-attacks occur randomly and make the transferred signals distrusted. Now an improved adaptive event-triggered scheme is introduced to determine whether the local sampled measurements of each sensor should be sent out to the communication network or not. The event-triggered condition of ith sensor is defined as the following:

eTi (tsi h )i 1 ei (tsi h ) ≤

Rn

αi (t )xTi (tki h )i 2 xi (tki h ) + βi (t )xTi (tsi h )i 2 xi (tsi h ), ∀ i ∈ {1, 2, . . . , n}, (2)

where tsi h = tki h + ji h with j ∈ {1, 2, . . . , δ} and δ = tk+1 − tk − 1, i 1 > 0, i 2 > 0 are the weighting matrix parameters, h is the sampling interval, ei (tsi h ) = xi (tki h ) − xi (tsi h ), tsi h is the currently sampled instants, and tki h is the last transmitted instant. Especially, the functions α i (t) and β i (t) satisfy

α˙ i (t ) =

β˙ i (t ) =

 1  − δ1 i eTi (tsi h )i 1 ei (tsi h ) αi (t )  − θ1 i xTi (tsi h )i 2 x(tsi h ) + θ1 i xTi (tki h )i 2 xi (tki h ) ,

(3)

 1  − δ2 i eTi (tsi h )i 1 ei (tsi h ) βi (t )  T i − θ2 i xi (tk h )i 2 xi (tki h ) + θ2 i xTi (tsi h )i 2 xi (tsi h ) ,

(4)

1 αi (t )

1 βi (t )

where δ1 i , δ2 i , θ1 i , θ2 i (i = 1, 2, . . . , n ) are given positive constants such that δ1 ≤ αi (t ) ≤ θ2 i , δ1 ≤ βi (t ) ≤ θ1 i . 1i

2i

Based on (2), the n channels’ event-triggering conditions can be summarized and described as

eT (t )1 e(t ) ≤ xT (tk h )α (t )2 x(tk h ) + xT (ts h )β (t )2 x(ts h ), (5)

T. Li, X. Tang and H. Zhang et al. / Neurocomputing 350 (2019) 33–43

where e(t ) = x(tk h ) − x(ts h ) is the diag{1l , 2l , . . . , nl } (l = 1, 2 ), and

triggering

error,

l =

35

the cyber-attacks should be considered when designing the controller.

α (t ) = diag{α1 (t ), α2 (t ), . . . , αn (t )}, β (t ) = diag{β1 (t ), β2 (t ), . . . , βn (t )}, δ1 = diag{δ11 , δ12 , . . . , δ1n };

(6)

δ2 = diag{δ21 , δ22 , . . . , δ2n }, θ1 = diag{θ11 , θ12 , . . . , θ1n }, θ2 = diag{θ21 , θ22 , . . . , θ2n }.

Remark 4. The functions f(x( · )) and g(x( · )) represent two types of cyber-attacks with different characteristics. The former is a function of conventional cyber-attacks and the latter is a function of accumulated dynamic ones when an attacker injects a signal which is consistent with the network dynamics at every instant of time [36].

(7)

Now combining (1) and (9), as for t ∈ [tk h + τtk , tk+1 h + τtk+1 ), we can obtain

Remark 1. Based on the event-triggered scheme in (5), we can check that, first, the scheme does not only include the information on each sensor’s triggering error, but also contains the sensor’s latest transmitted data and the current sampled one; second, the triggering parameters α (t), β (t) are not predetermined matrices but two time-varying ones where each element is determined by time-varying functions regulated by the adaptive laws in (3) and (4), in which the triggering error, latest transmitted data and current sampled one are fully discussed; third, when α (t) is zero matrix and β (t) is a constant one, our scheme can be converted to the one in [37]. Define τ (t ) = t − tk h − jh and consider the transmission delay τ k , then for t ∈ [tk h + τk , τk+1 h + τk+1 ), we can denote 0 ≤ τk ≤ τ (t ) ≤ τk+1 + h ≤ τM and the term (5) can be changed as

eT (t )1 e(t ) ≤ [eT (t ) + xT (t − τ (t ))]α (t )2 [e(t ) + x(t − τ (t ))] + x (t − τ (t ))β (t )2 x(t − τ (t )). T

(8)

Now assume that the communication network is vulnerable to an attacker who can easily alter the transmitted information. Considering that an attacker has a limited access to network resources and the malicious cyber-attacks can randomly modify controller’s input signals, then a new controller design concerning the eventtriggered scheme (5) and two kinds of cyber-attacks is proposed as

u(t ) =

ω (tk )Kx(tk h ) + ϕ (tk )K f (x(t − d (t ))) + γ (tk )K



t

t −υ (t )

x˙ (t ) = Ax(t ) + ω (tk )BKx(tk h ) + ϕ (tk )BK f (x(t − d (t )))  t + γ (tk )BK g(x(s ))ds. t −υ (t )

Recalling the definition on τ (t) and the characteristics of η(tk ), ρ (tk ), then (12) can be rewritten as

x˙ (t ) = Ax(t ) + ω ¯ BK[x(t − τ (t )) + e(t )] + ϕ¯ BK f (x(t − d (t )))  t + γ¯ BK g(x(s ))ds + [ω (tk ) − ω ¯ ]BK t −υ (t )

×[x(t − τ (t )) + e(t )] + [ϕ (tk ) − ϕ¯ ]BK f (x(t − d (t )))  t + [γ (tk ) − γ¯ ]BK g(x(s ))ds, (13) t −υ (t )

where t ∈ [tk h + τtk , tk+1 h + τtk+1 ) and ω ¯ = η¯ , ϕ¯ = (1 − η¯ )ρ¯ , γ¯ = (1 − η¯ )(1 − ρ¯ ). Assumption 1. The cyber-attacks f(x) and g(x) satisfy f (0 ) = 0, g(0 ) = 0, and

f i ( s1 ) − f i ( s2 ) gi ( s1 ) − gi ( s2 ) ≤ φ +f , φg−i ≤ i s1 − s2 s1 − s2 ≤ φg+i , ∀ s1 = s2 , i = 1, 2, . . . , n,

φ −fi ≤

(9)

where ω (tk ) = η (tk ), ϕ (tk ) = [1 − η (tk )]ρ (tk ), γ (tk ) = [1 − η (tk )][1 − ρ (tk )] with η(tk ) ∈ {0, 1}, ρ (tk ) ∈ {0, 1}, and K is controller gain to be designed. Here f (x(t − d (t ))) and t t−υ (s ) g(x (s ))ds represent the cyber-attacks of different characteristics and are individually presented as



T





Prob{η (tk ) = 1} = η¯ , Prob{η (tk ) = 0} = 1 − η¯ , Prob{ρ (tk ) = 1} = ρ¯ , Prob{ρ (tk ) = 0} = 1 − ρ¯ .

(11)

Remark 3. Notice that the communication channels are vulnerable to the cyber-attacks. The attacker can randomly release the attacks to modify the triggered transmitting measurements, which will unavoidably affect the controller to make wrong decisions. Therefore,

constants

denoting

both

i

Remark 5. It is worth noting that though Assumption 1 was utilized to describe the cyber-attacks [13,14], the restrictions on f(x), g(x) seem to be strong in some degree and will lead to the conservatism. Therefore, some less restrictive conditions can be further utilized in our future studies, such as the following ones





f (s1 ) − f (s2 ) − 1 (s1 − s2 )

≥ 0,

Remark 2. In the case of NCSs (1) with two types of cyberattacks which are modeled as a nonlinear function and a integral function with restrictions, thus the control input (9) may include three terms, which can be modified and switched randomly. In (9) η (tk ) = 1 means that the communication network is smooth and η (tk ) = 0 means that the cyber-attacks occur. Especially, ρ (tk ) = 1 means that the cyber attack f (x(t − d (t ))) oc t curs and ρ (tk ) = 0 means that the cyber attack t −υ (t ) g(x(s ))ds appears. The statistical properties are respectively given as

known

cyber-attacks in practical cases.

g( x ) = g 1 ( x 1 ) , g 2 ( x 2 ) , . . . , g n ( x n ) T ; (10)

are

(14)

the lower and upper bands of the cyber-attacks. Moreover, φ −f , φ +f , φg−i , φg+i can be obtained when we can estimate the

f ( x ) = f 1 ( x1 ), f 2 ( x2 ), . . . , f n ( xn ) , d (t ) ∈ (0, dM ], υ (t ) ∈ (0, υM ].

φ −f , φ +f , φg−i , φg+i i i

where

i

g(x(s ))ds,

(12)





 2 (s1 − s2 ) − f (s1 ) − f (s2 )

∀ s1 , s2 ∈ Rn ;

(15)



g( s 1 ) − g( s 2 ) − 1 ( s 1 − s 2 )

≥ 0,

T 

T 

 2 ( s 1 − s 2 ) − g( s 1 ) − g( s 2 )

∀ s1 , s2 ∈ Rn ,

(16)

where  1 ,  2 , 1 , 2 are the constant matrices of appropriate dimensions. Lemma 1 [40]. For any x(t), the nonlinear functions f(x(t)), g(x(t)) in Assumption 1 satisfy the following inequality for any diagonal matrices Vi ≥ 0 (i = 1, 2 )



T

x(t ) f (x(t ))



where

V1 φ +f −V1

−V2 φg− φg+V2

V2 φg+ −V2

T

x(t ) g(x(t ))



−V1 φ −f φ +f V1



x(t ) f (x(t ))

≥ 0,

x(t ) g(x(t ))

≥ 0,

  φ −f = diag φ −f1 (x1 ) φ +f1 (x1 ) , . . . , φ −fn (xn ) φ +fn (xn ) ,

36

T. Li, X. Tang and H. Zhang et al. / Neurocomputing 350 (2019) 33–43



φ −f1 (x1 ) + φ +f1 (x1 )

φ −fn (xn ) + φ +fn (xn )



2

2

Lemma 2 [41]. Assume that τ (t) ∈ [0, τ M ], for any constant matrices   R ∗ R, M ∈ Rn×n making ≥ 0, then M R

−τM



t

t−τM

 ξ T (t ) = xT (t )

where



x˙ T (s )Rx˙ (s )ds ≤

−R R−M M

∗ −2R + M + MT R−M

ξ T (t ) ξ (t ), xT (t − τ (t ))



xT (t − τM )

= M2 , 54 = G2 − M2 , 55 = −G2 − R2 + τd Q1 − dυ Q2 ,

51 61 71 77 84

φ = diag ,..., , 2 2   φg− = diag φg−1 (x1 ) φg+1 (x1 ) , . . . , φg−n (xn ) φg+n (xn ) ,  −  φg−n (xn ) + φg+n (xn ) φg1 (x1 ) + φg+1 (x1 ) + φg = diag ,..., . + f



and

=

∗ ∗ . −R

= G3 − M3 , 66 = −2G3 + M3 + M3T − U2 φg− , = M3 , 76 = G3 − M3 , = − G3 − R3 + dυ Q2 − υτ Q3 , 81 = ϕ¯ Y T BT , =

φ +f V1 , 88 = −V1 ,

91 = φg+U1 , 99 = −U1 + υM2 S4 , 10,6 = φg+U2 , 10,10 = −U2 , 11,1 = γ¯ Y T BT , 11,11 = − S4 , 12,1 = ω¯ Y T BT , 12,2 = (I + δ1 θ1 )2 , 12,12 = − (δ1 + δ2 )1 + (I + δ2 θ2 )2 . Proof. Firstly, we will construct an augmented Krasovskii functional for the system (13) as follows:

V (x(t )) =

3 

Vi (x(t )),

with

V1 (x(t )) = xT (t )P x(t ) + 0.5α T (t )α (t ) + 0.5β T (t )β (t )  t  t + xT (s )R1 x(s )ds + xT (s )R2 x(s )ds t−τM

 +

Based on the contents in last section, the following theorem will present a sufficient condition based on LMI forms, which ensure the asymptotical stability of the closed-loop system (13). Then as a conclusion, two corollaries will be further achieved and they can be regarded as the special cases of Theorem 1.

t−υM



Theorem 1. For given scalars τM , dM , υM , η¯ , ρ¯ , δ1 i , δ2 i , θ1 i , θ2 i (i = 1, 2, . . . , n ), r (r = 1, 2, 3 ), the closed-loop system (13) is asymptotically stable, if there exist definitely positive matrices P, R1 , R2 , R3 , Q1 , Q2 , Q3 , G1 , G2 , G3 , S4 , appropriately dimensioned matrices

Gl ∗ Ml (l = 1, 2, 3 ) making > 0, diagonal matrices 1 , 2 , Gl Ml V1 , U1 , U2 , and a constant matrix Y such that the LMI in (17) holds

∗

 0

∗ ∗





< 0,

(17)



V3 (x(t )) =

υM





t

t−dM





t−υM

t−dM







t

t−τM

t s

x˙ T (v )G1 x˙ (v )dvds

x˙ T (v )G2 x˙ (v )dvds x˙ T (v )G3 x˙ (v )dvds,

xT (s )Q1 x(s )ds + dυ

t−υM

− υM

t

s

t−τM

0

t

s

t

t−τM

+ υτ





t−υM

t−dM

xT (s )Q2 x(s )ds

xT (s )Q3 x(s )ds,

t

t+s

gT (x(v ))S4 g(x(v ))dvds,

where τd = τM − dM , dυ = dM − υM , υτ = υM − τM , and the definitely positive matrices P, R1 , R2 , R3 , Q1 , Q2 , Q3 , G1 , G2 , G3 , S4 wait to be determined. Now, taking the mathematical expectation of the derivative of V(x(t)) along the system (13), we can compute out

E{V˙ (x(t ))}



≤ E 2xT (t )P x˙ (t ) + α T (t )α˙ (t ) + β T (t )β˙ (t )

where

ϒi = [τM iT dM iT υM iT ]T (i = 1, 2 ),    = diag − 21 P + 12 G1 , −22 P + 22 G2 , −23 P + 32 G3 ,

(18)

  1 = PA ω¯ BY 0 0 0 0 0 ϕ¯ BY 0 0 γ¯ BY ω¯ BY T ,

(19)

  2 = 0 λ1 BY 0 0 0 0 0 λ2 BY 0 0 λ3 BY λ1 BY T .

(20)

Furthermore, except for zero terms, the elements of the matrix  = [i j ]12×12 can be listed as

11 = PA + AT P + R1 + R2 + R3 − G1 − G2 − G3 − U1 φg− , 21 22 31 33 44

+ υM

τd

x (s )R3 x(s )ds + τM T

+ dM

V2 (x(t )) =

t−dM

t

3. Main results

 ϒ1 ϒ2

(21)

i=1

Lemma 3 [42]. For a full column rank matrix B ∈ Rn×m , the singular decomposition is B = U V T with  = [0 0]T . Here U and V are orthogonal matrices,  ∈ Rm×n is a rectangular diagonal matrix with positive real numbers on the diagonal in decreasing order of magnitude, 0 ∈ Rm×m is an diagonal matrix with positive diagonal entries in decreasing order. Let P ∈ Rn×n and P = U diag{P1 , P2 }U T , then there must exist a constant matrix X ∈ Rm×m such that P B = BX and X = (0V T )−1 P1 0V T .



Lyapunov-

=ω ¯ Y T BT + G 1 − M1 , = − 2G1 + M1 + M1T + (2I + δ1 θ1 + δ2 θ2 )2 , = M1 , 32 = G1 − M1 , = −G1 − R1 − τd Q1 + υτ Q3 , 41 = G2 − M2 , = −2G2 + M2 + M2T − V1 φ −f ,

+ xT (t )(R1 + R2 + R3 )x(t ) − xT (t − τM )(R1 + τd Q1 − υτ Q3 )x(t − τM ) −xT (t − dM )(R2 − τd Q1 + dυ Q2 )x(t −dM ) 2 −xT (t − υM )(R3 − dυ Q2 + υτ Q3 )x(t − υM ) + x˙ T (t )(τM2 G1 + dM G2  t  t 2 +υM G3 )x˙ (t ) − τM x˙ T (s )G1 x˙ (s )ds − dM x˙ T (s )G2 x˙ (s )ds



− υM

 −

t−τM

t

t−υM t

t −υ (t )

t−dM

2 T x˙ T (s )G3 x˙ (s )ds + υM g (x(t ))S4 g(x(t ))

 g(x(s ))ds

 T

S4

t

t −υ (t )

 g(x(s ))ds

.

(22)

Based on the conditions in (2)–(4), we can estimate the term in (22) as follows

α T (t )α˙ (t ) + β T (t )β˙ (t ) ≤ −eT (t )(δ1 + δ2 )1 e(t ) + [x(t − τ (t ) + e(t ))]T (I + δ2 θ2 )2 [x(t − τ (t )) + e(t )] + xT (t − τ (t ))(I + δ1 θ1 )2 x(t − τ (t )).

(23)

T. Li, X. Tang and H. Zhang et al. / Neurocomputing 350 (2019) 33–43

Notice that



E x˙ (t )(τ T

2 M G1

+

2 dM G2

+υ

2 M G3



)x˙ (t ) = E



$T1

$1 +

$T2

+ ξ3T (t )3 ξ3 (t ) + $T1 $1 + $T2 $2



$2 , (24)

+

where

$1 = Ax(t ) + ω ¯ BK[x(t − τ (t )) + e(t )]  t + ϕ¯ BK f (x(t − d (t ))) + γ¯ BK

+

t −υ (t )

λ1 BK[x(t − τ (t )) + e(t )]

$2 =

+ λ2 BK f (x − d (t ))) + λ3 BK



g(x(s ))ds,

(25) +

 

t −υ (t )

g(x(s ))ds,

(26)

 2 2  = τM2 G1 + dM G2 + υM G3 , λ1 = ω ¯ (1 − ω ¯ ),   λ2 = ϕ¯ (1 − ϕ¯ ), λ3 = γ¯ (1 − γ¯ ).

 

x(t ) g(x(t ))





T

x(t − d (t )) f (x(t − d (t )))

T

x(t ) g(x(t ))



+

−V1 φ φ +f V1

−U1 φg− φg+U1

T

x(t − υ (t )) g(x(t − υ (t )))

V1 φ −V1

− f

U1 φg+ −U1

+ f





(27)

−τM

t

t−τM

−υM



x˙ T (s )G1 x˙ (s )ds − dM

t

t−υM



t

t−dM



−Gl Ml

x˙ T (s )G2 x˙ (s )ds

(30)

∗ −2Gl + Ml + MlT G l − Ml

∗ ∗ −Gl

( l = 1, 2, 3 ).

E{V˙ (x(t ))}



+ [x(t − τ (t ) + e(t ))]T (I + δ2 θ2 )2 [x(t − τ (t )) + e(t )] + xT (t − τ (t ))(I + δ1 θ1 )2 x(t − τ (t )) + xT (t )(R1 + R2 + R3 )x(t ) + xT (t − τM )(−R1 − τd Q1 + υτ Q3 )x(t − τM ) + xT (t − dM ) ×(−R2 + τd Q1 − dυ Q2 )x(t − dM ) + xT (t − υM )(−R3 + dυ Q2 − υτ Q3 )x(t − υM )

−



(x(t ))S4 g(x(t ))   g(x(s ))ds T S4

t −υ (t )

t

t −υ (t )

+ ξ (t )1 ξ1 (t ) + ξ (t )2 ξ2 (t ) T 1

T 2



By resorting to Lemma 3, there must exist a matrix X such that P BK = BXK. Then defining Y = XK and using Schur-complement, ¯ −1 ϒ ˜1 + ϒi (i = 1, 2 ) in (18), one can check that  + ϒ˜ 1  ¯ −1 ϒ ˜ 2 < 0 is equivalent to ϒ˜ 2 



 ϒ1 ϒ2

∗ ¯  0

∗ ∗ ¯ 



< 0.

(32)

g(x(s ))ds

−P G−1 P ≤ −2l P + l2 Gl . l

(33)

Remark 6. During proving Theorem 1, the novelties of the LKF can be illustrated as follows: (i) the terms in V1 (x(t)) have efficiently utilized the information on α (t), β (t) and τ (t ), d (t ), υ (t ); (ii) the terms in V2 (x(t)) have used the inter-relationship between the upper bounds of τ (t ), d (t ), υ (t ); (iii) the term V3 (x(t)) have studied the information of accumulated dynamic attacks g(x(t)), some of which have not appeared in present works. Based on the special case that f (x(· )) = g(x(· )), then we can derive the following corollary.

≤ E 2xT (t )P x˙ (t ) − eT (t )(δ1 + δ2 )1 e(t )

+υ

t −υ (t )

Then the LMI in (17) can guarantee the term (32) to be true, which ˜ 1 ¯ −1 ϒ ˜1 +ϒ ˜ 2 ¯ −1 ϒ ˜ 2 < 0, i.e., E{V˙ (x(t ))} < 0. Theremeans  + ϒ fore, the closed-loop system (13) is asymptotically stable and it completes the proof. 



T

Combining the terms from (22)–(30), then it can be deduced that

2 T Mg  t

(31)

On the other hand, there exists any l > 0 (l = 1, 2, 3 ) such that the following inequality must be true

T

l = G l − M l



x(t − υ (t )) g(x(t − υ (t )))

  ϒ˜ 1 = PA ω¯ PBK 0 0 0 0 0 ϕ¯ PBK 0 0 γ¯ PBK ω¯ BY T ,   ϒ˜ 2 = 0 λ1 PBK 0 0 0 0 0 λ2 PBK 0 0 λ3 PBK λ1 PBK T .

x(t − υ (t )) ≥ 0. (29) g(x(t − υ (t )))

x˙ T (s )G3 x˙ (s )ds

T



xT (t − dM ) xT (t − υ (t )) xT (t − υM ) f T (x(t − d (t )))

 t  gT (x(t )) gT (t − υ (t )) g(x(s ))ds T eT (t ) ,



ξ (t ) = x (t ) x (t − τ (t )) x (t − τM ) ,   ξ2T (t ) = xT (t ) xT (t − d (t )) xT (t − dM ) ,   ξ3T (t ) = xT (t ) xT (t − υ (t )) xT (t − υM ) ,   T 1

U2 φg+ −U2

¯ = diag − P G−1 P, −P G−1 P, −P G−1 P ,  1 2 3

≤ ξ1T (t )1 ξ1 (t ) + ξ2T (t )2 ξ2 (t ) + ξ3T (t )3 ξ3 (t ), where



x(t ) g(x(t ))

ξ T (t ) = xT (t ) xT (t − τ (t )) xT (t − τM ) xT (t − d (t ))

(28)

By using Lemma 2, the following inequality can

be obtained as Gl ∗ there exists Ml (l = 1, 2, 3 ) satisfying > 0: Gl Ml



−U2 φg− φg+U2



where

x(t − d (t )) ≥ 0, f (x(t − d (t )))



T

U1 φg+ −U1

V1 φ +f  x(t − d (t ))  −V1 f (x(t − d (t ))

 ξ T (t )ξ (t ) + $T1 $1 + $T2 $2     . ˜ 1 ¯ −1 ϒ ˜1 +ϒ ˜ 2 ¯ −1 ϒ ˜ 2 ξ (t ) , = E ξ T (t )  + ϒ

U2 φg+ −U2

 



x(t ) g(x(t ))

−U2 φg− φg+U2

−U1 φg− φg+U1

T

x(t − υ (t )) g(x(t − υ (t )))

From Lemma 1, it follows from the restrictions on f(x), g(x) that



 −V φ − 1 f T φ +f V1

x(t − d (t )) f (x(t − d (t ))

=E

t

37



Corollary 1. For given scalars τM , dM , υM , η¯ , ρ¯ , δ1 i , δ2 i , θ1 i , θ2 i (i = 1, 2, . . . , n ), r (r = 1, 2, 3 ), the closed-loop system (13) is asymptotically stable, if there exist definitely positive matrices P, R1 , R2 , R3 , Q1 , Q2 , Q3 , G1 , G2 , G3 , S4 , appropriate dimensioned matrices Ml (l = Gl ∗ 1, 2, 3 ) making > 0, diagonal matrices 1 , 2 , V1 , U1 , U2 Ml Gl and a constant matrix Y such that the LMI in (34) holds



 ϒ1 ϒ2

∗

 0

∗ ∗





< 0,

(34)

where ϒ 1 , ϒ 2 ,  are identical to the ones in Theorem 1. Most elements of the matrix  can be described as the corresponding ones in

38

T. Li, X. Tang and H. Zhang et al. / Neurocomputing 350 (2019) 33–43

+ ξ2T (t )ψ2 ξ2 (t ) + T1 1 1 + T2 1 2

Theorem 1 except for

11 = PA + A P + R1 + R2 + R3 − G1 − G2 − G3 − U1 φ T

− , f

66 = −2G3 + M3 + M3T − U2 φ −f , 91 = φ +f U1 , 10,6 = φ +f U2 . Proof. Based on the proof of Theorem 1 and replacing t t t −υ (t ) g(x (s ))ds as t −υ (t ) f (x (s ))ds, this corollary can be easily established and its detailed proof is omitted here. Once the accumulated dynamic attack g(x(t)) does not exist, then we can obtain the following corollary.  Corollary 2. For given scalars τM , dM , η¯ , ρ¯ , δ1 i , δ2 i , θ1 i , θ2 i (i = 1, 2, . . . , n ), r (r = 1, 2 ), the closed-loop system (13) is asymptotically stable, if there exist positive matrices P, R1 , R2 , Q1 , Q2 , G1 , G2 , Gl ∗ appropriate dimensioned matrices Ml (l = 1, 2 ) making > Gl Ml 0, diagonal matrices 1 , 2 , V1 and a matrix Y such that the LMI in (35) hold



 ϒ3 ϒ4

∗



∗ ∗

1

where ϒ3 = [τM 3T 21 P +



< 0,

1

0

12 G1 , −22 P

(35)

dM 3T ]T , ϒ4 = [τM 4T +

22 G2





dM 4T ]T , 1 = diag −

, and

 3 = PA ω¯ BY 0 0 0 ϕ¯ BY ω¯ BY T ,   4 = 0 λ1 BY 0 0 0 λ2 BY λ1 BY T . Except for the zero terms, other elements in the matrix  can be listed as

11 22 31 41

= PA + AT P + R1 + R2 − G1 − G2 , 21 = ω ¯ Y T BT + G 1 − M1 , = − 2G1 + M1 + M1T + (2I + δ1 θ1 + δ2 θ2 )2 , = M1 , 32 = G1 − M1 , 33 = −G1 − R1 − τd Q1 , = G2 − M2 , 44 = −2G2 + M2 + M2T − V1 φ −f , 51 = M2 ,

54 = G2 − M2 , 55 = −G2 − R2 + τd Q1 , 61 = ϕ¯ Y T BT , 64 = φ +f V1 , 66 = − V1 , 71 = ω¯ Y T BT , 72 = (I + δ1 θ1 )2 , 77 = − (δ1 + δ2 )1 + (I + δ2 θ2 )2 . Proof. Only consider the existence of cyber-attack f(x(t)), then ω (tk ) = η (tk ), ϕ (tk ) = 1 − η (tk ), ω¯ = η¯ , ϕ¯ = 1 − η¯ . The augmented LKF can be selected as

V (x(t )) = xT (t )P x(t ) + 0.5α T (t )α (t ) + 0.5β T (t )β (t )  t  t + xT (s )R1 x(s )ds + xT (s )R2 x(s )ds t−τM



+ τM

t

t−τM

 + dM + τd



t

t−dM t−dM

t−τM

t−dM



t

s



t s

x˙ (v )G1 x˙ (v )dvds T



=E

 −V φ − 1 f T φ +f V1

x(t − d (t )) f (x(t − d (t )))

V1 φ +f  x(t − d (t ))  −V1 f (x(t − d (t )))



   T (t )1 1 (t ) + T1 1 1 + T2 1 2 ,

where

  T (t ) = xT (t )

xT (t − τ (t ))

(36)

xT (t − τM )

xT (t −



d (t )) xT (t − dM ) xT (t − υ (t )) xT (t − υM ) f T (x(t − d (t ))) eT (t ) . Notice that









2 E x˙ T (t )(τM2 G1 + dM G2 )x˙ (t ) = E T1 1 1 + T2 1 2 ,

(37)

where

1 = Ax(t ) + ω¯ BK[x(t − τ (t )) + e(t )] + ϕ¯ BK f (x(t − d (t ))),   λ1 = ω¯ (1 − ω¯ ), λ2 = ϕ¯ (1 − ϕ¯ ), 2 = λ1 BK[x(t − τ (t )) + e(t )] + λ2 BK f (x − d (t ))), 2 1 = τM2 G1 + dM G2 . Then with the aid of the Theorem 1, the LMI in (35) can guarantee E{V˙ (x(t ))} < 0. Then the closed-loop system (13) is asymptotically stable and the proof is completed.  Remark 7. In this work, we do not only introduce a new kind of cyber-attack called the accumulated dynamic one but also initially utilize the inter-relationship between the upper bounds of timedelays τ (t ), d (t ), υ (t ) to construct the LKFs, which can enlarge the application area greatly. It is worth noting that, in recent years, many effective techniques have been proposed to tackle the issue on the reduced conservatism, such as Wirtinger-based integral inequalities, multiple integral Lyapunov method, extended reciprocal convex techniques, and dynamic Lyapunov approach, which also can be utilized in our work to extend the application area as large as possible. Remark 8. Though in our work, the adaptive event-triggering schemes can efficiently reduce the data transmissions, most recently, another important approach called coding-decoding communication protocol (CDCP) was utilized to tackle the control issue on discrete-time systems [43–45], which can send the symbolic data via communication networks and therefore occupy less resource when compared with the original data. Especially, since the feature of CDCP possesses a high communication reliability, it can prevent the transmitted data from being vulnerable to attacks launched by adversaries in order to meet the increasingly important security requirement. Then the adaptive event-based method in our work and the CDCP ones in [43–45] can be combined together to tackle the cyber-attacks in the NCSs, which will be reported in out future work.

x˙ T (v )G2 x˙ (v )dvds

xT (s )Q1 x(s )ds.

The mathematical expectation of V˙ (x(t )) can be obtained as

E{V˙ (x(t ))}



≤ E 2x (t )P x˙ (t ) − e (t )(δ1 + δ2 )1 e(t ) T

+

T

+ [x(t − τ (t ) + e(t ))]T (I + δ2 θ2 )2 [x(t − τ (t )) + e(t )] +xT (t − τ (t ))(I + δ1 θ1 )2 x(t − τ (t )) + xT (t )(R1 + R2 )x(t ) + xT (t − τM )(−R1 − τd Q1 )x(t − τM ) + xT (t − dM )(−R2 + τd Q1 )x(t − dM ) + ξ1T (t )ψ1 ξ1 (t )

4. Numerical examples In this section, two numerical examples will be presented to illustrate the proposed results in our work. Example 1. Consider the parameters of the system (1) and cyberattacks (9) as

⎡

−0.72 ⎢ 0.25 ⎢ 0 A=⎢ ⎢ 0 ⎣ 0 0

0.40 −0.56 0 0 0 0

0 0 −0.72 0.25 0 0

0 0 0.40 −0.56 0 0

0 0 0 0 −0.72 0.25

⎤

0 0 ⎥ 0 ⎥ ⎥, 0 ⎥ ⎦ 0.40 −0.56

T. Li, X. Tang and H. Zhang et al. / Neurocomputing 350 (2019) 33–43

⎡

0.1 ⎢0.5 ⎢0 B=⎢ ⎢0 ⎣ 0 0

39

Fig. 2. Event-triggered instants of sensor 1.

Fig. 4. Event-triggered instants of sensor 3.

Fig. 3. Event-triggered instants of sensor 2.

Fig. 5. State response of x(t).

0 0 0.1 0.5 0 0

⎤

0 0 ⎥ 0 ⎥ ⎥, 0 ⎥ ⎦ 0.1 0.5

⎡

−tanh(0.5x1 ) ⎢−tanh(0.01x1 ) ⎢ 0 f (x ) = ⎢ ⎢ 0 ⎣ 0 0

0 0 −tanh(0.5x2 ) −tanh(0.01x2 ) 0 0

⎤

0 0 ⎥ ⎥ 0 ⎥, ⎥ 0 ⎦ −tanh(0.5x3 ) −tanh(0.5x3 )

40

T. Li, X. Tang and H. Zhang et al. / Neurocomputing 350 (2019) 33–43

Fig. 6. The graph of attack rule η(t).

Fig. 8. Event-triggered instants of sensor 1. Table 1 The transmitted sensor measurements (Example 1). Sensor node

Sensor 1

Sensor 2

Sensor 3

Average transmission rate

Theorem 1 [37] Theorem 1

57 38

55 39

50 43

67.5% 50.0%

Table 2 The maximum allowable upper bounds of τ M for different dM (Example 1). Methods

0.3

0.6

0.9

1.2

Theorem 2 [26] Theorem 1 [37] Corollary 2

1.6405 1.6452 1.8879

1.5518 1.5366 1.8092

1.4161 1.3745 1.6672

1.1392 1.1174 1.4067

and



1 0 0

the −1 0 0

initial 0 1 0

0 −1 0

 φ −f = diag 0, 0,  φ +f = diag 0.25,  φg− = diag 0, 0,  φg+ = diag 0.05,

−tanh(0.1x1 ) ⎢−tanh(0.3x1 ) ⎢ 0 g( x ) = ⎢ ⎢ 0 ⎣ 0 0

0 0 −tanh(0.1x2 ) −tanh(0.3x2 ) 0 0

can

T

0 0 −1

be

selected

x (0 ) =

as

. It is easy to get that



0, 0, 0, 0 ,



0.005, 0.25, 0.005, 0.25, 0.005 ,



0, 0, 0, 0 ,



0.015, 0.05, 0.15, 0.05, 0.15 .

Choose the sampling period h = 0.5, η¯ = 0.5, ρ¯ = 0.5, τM = 0.8, dM = 0.6, υM = 0.3, 1 = 1, 2 = 1, 3 = 1, δ1 = 5I, δ2 = 5I, θ1 = 4I, θ2 = 4I, and employing MATLAB LMI Toolbox to solve the LMI in (17) of Theorem 1, we can obtain part feasible solutions as follows:

Fig. 7. The graph of switching attack rule ρ (t).

⎡

state 0 0 1

⎤

0 0 ⎥ ⎥ 0 ⎥, ⎥ 0 ⎦ −tanh(0.1x3 ) −tanh(0.3x3 )





W1 = diag 24.7627, 24.7627, 24.7627, 24.7627, 24.7627, 24.7627 ,





W2 = diag 0.6874, 0.6874, 0.6874, 0.6874, 0.6874, 0.6874 ,

⎡

⎤

−0.4117 −0.7918 0 0 0 0 ⎦. K =⎣ 0 0 −0.4303 −0.8276 0 0 0 0 0 0 −0.4454 −0.8565

T. Li, X. Tang and H. Zhang et al. / Neurocomputing 350 (2019) 33–43

41

Fig. 11. State response of x(t).

Fig. 9. Event-triggered instants of sensor 2.

the maximum allowable upper bounds (MAUBs) on τ M achieved by Corollary 2 of this work, Ref. [22], and Ref. [37] when only the cyber-attack f(x) is involved. Example 2. In this example, we consider the parameters of the system (1) and cyber-attacks (9) as









A = diag − 0.5, 0.1, −0.1 , B = 0.1 0.2 0.1 T ,









tanh(0.04x1 ) tanh(0.04x2 ) , tanh(0.04x3 )

f (x ) =

tanh(0.03x1 ) tanh(0.06x2 ) . tanh(0.03x3 )

g( x ) =

The initial state can be chosen as x(0 ) = [−1, 1, 2]T . Clearly, φ − = f

diag{0, 0, 0}, φ + = diag{0.02, 0.02, 0.02}, φg− = diag{0, 0, 0}, f

and φg+ = diag{0.015, 0.03, 0.015}. Choose the sampling period h = 0.1 and other scalars η¯ = 0.8, ρ¯ = 0.8, τM = 0.5, dM = 0.5, υM = 0.5, 1 = 1, 2 = 1, 3 = 1, δ1 = 5I, δ2 = 5I, θ1 = 4I, θ2 = 4I. Now by solving the LMI in Theorem 1, the corresponding controller gain matrix and triggering parameters can be respectively computed out as

K = Fig. 10. Event-triggered instants of sensor 3.

1 =





− 0.3748, −3.3408, −0.8723 ,



0 4.3771 0

0 0 , 4.3771

0.0427 0 0

0 0.0427 0

0 0 0.0427

 The event-triggered instants and corresponding releasing intervals of sensor nodes 1,2,3 are shown from Figs. 2–4, in which the transmitted sensor measurements are 38,39,43 on sensor nodes 1,2,3, respectively. The average transmission rate is 50% of total 240 sampled signal and Fig. 5 depicts the state response of the controlled system. Figs. 6 and 7 shows the attack rules η(t), ρ (t), respectively. Table 1 gives the transmitted sensor measurements when comparing with the methods in [37] since the Ref. [37] also uses this similar example. Under the identical situations, Table 2 presents

2 =



4.3771 0 0

 .

If we set sampling interval as h = 0.1, the transmitted sensor measurements of sensor nodes 1,2,3 are computed out as 84,81,32, respectively. The average transmission rate is obtained as 13.1% of total 1500 sampled measurements. The event-triggered instants and corresponding release intervals of sensor nodes 1,2,3 are respectively shown in Figs. 8–11 depicts the state response of the con-

42

T. Li, X. Tang and H. Zhang et al. / Neurocomputing 350 (2019) 33–43

unnecessary transmissions. The randomly occurring cyber-attacks which made the communication networks unreliable were respectively modeled as a nonlinear function and an integral of nonlinear function. By utilizing the whole information on time-delays and adaptive conditions, two augmented LKFs were constructed and some LMI results were presented to guarantee the asymptotical stability of the closed-loop system, in which the co-design of the controller and event-triggered schemes was given. Finally, the feasibility of the obtained methods have been illustrated by some simulation results and comparisons. Acknowledgments This work is supported by National Natural Science Foundation of China (Nos. 61873123, 61873127, 61573130) and Jiangsu Natural Science Foundation (Grant nos. BK20171419, BK20150888). References

Fig. 12. The graph of attack rule η(t).

Fig. 13. The graph of switching attack rule ρ (t).

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[45] L.F. Ma, Z.D. Wang, Q.L. Han, Y.R. Liu, Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays, Automatica 98 (2018) 358–362. Tao Li received his PHD degree in engineering from Southeast University in 2008 and was a postdoctoral research fellow at School of Instrument Science and Engineering of Southeast University during year 2008 and 2011, China. He has been a visiting scholar at Control System Center of The Manchester University from year 2016 to 2017. He is currently an associate professor at School of Automation Engineering, Nanjing University of Aeronautics and Astronautics in China. His current research interests include neural networks, time-delay systems, networked control systems, etc.

Xiaoling Tang received her bachelor’s degree in Engineering from Nanjing Normal University in 2017 China and currently, she is a master graduate student at Nanjing University of Aeronautics and Astronautics. Her research includes networked control systems with its application to the flight control.

Haitao Zhang received his bachelor’s and masters degrees in Engineering both from College of National Defense Engineering at The Army Engineering University of PLA of China in 2003 and 2006, respectively. Later he received his PHD degree from School of Automation at Southeast University of China in 2010. His current research includes time-delay systems and networked control systems with their applications to sensor networks.

Shumin Fei received the PHD degree from Beijing University of Aeronautics and Astronautics in 1995, China. Form year 1995 to 1997, he carried out his postdoctoral research at Research Institute of Automation of Southeast University, China. Presently, he is a professor and doctoral supervisor at School of Automation of Southeast University in China. He has published more than 100 journal papers and his current research interests include nonlinear systems, time-delay system, complex systems, and so on.