Event-triggered quantized H∞ control for networked control systems in the presence of denial-of-service jamming attacks

Nonlinear Analysis: Hybrid Systems 33 (2019) 265–281

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Event-triggered quantized H∞ control for networked control systems in the presence of denial-of-service jamming attacks✩ ∗

Xiaoli Chen a , Youguo Wang a , , Songlin Hu b a

School of Telecommunications and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China b Institute of Advanced Technology, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

article

info

Article history: Received 1 July 2018 Accepted 2 March 2019 Available online xxxx Keywords: Denial-of-service jamming attack Event-triggered control Networked control systems Piecewise Lyapunov functional Quantization

a b s t r a c t The problem of event-triggered quantized H∞ control of networked control systems (NCSs) under denial-of-service (DoS) jamming attacks is investigated in this paper. Firstly, a new resilient event-triggering transmission scheme is proposed to lighten the load of computing and communications while offsetting the DoS jamming attacks imposed by power-constrained Pulse-Width Modulated (PWM) jammers. Secondly, a new switched system model is established, which characterizes the effects of the eventtriggering scheme, quantization and DoS jamming attacks within a unified framework. Thirdly, linear matrix inequality (LMI)-based sufficient conditions for ensuring the exponential stability of the resulting switched system under the DoS jamming attacks are derived by using the piecewise Lyapunov functional method. Moreover, if the obtained LMIs are feasible, the co-design of the event-triggering parameter and the feedback gain matrix can be obtained. Finally, a satellite yaw angle control system is given to verify the effectiveness and feasibility of the developed theoretical results. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Recent advancement in wireless communications and electronics has enabled the development and application of low-cost, energy efficient and multi-function wireless networked control systems (WNCSs). As is known to all, wireless communications operate over a shared medium and are thus vulnerable to malicious attacks. Such malicious attacks include, but are not limited to, deception attacks and denial-of-service (DoS) attacks, see, e.g., [1–8] . The presence of those attacks in WNCSs (especially in security-critical infrastructures such as power grids, water distribution networks and transport systems [9] is often a source of instability and poor performance, and greatly increases the difficulty of stability analysis and control design as well. Therefore, stability analysis and controller synthesis of WNCSs subject to various cyber attacks are of theoretical and practical importance. Up to now, considerable attention has been paid to this topic, see, e.g., [10,11] and the reference therein. Among various cyber attack phenomena, DoS attack is regarded as the most probably type of cyber attack which has significant impact on the performance of the NCSs [12]. During the past few years, there are many contributions reported in the literature for stability analysis of NCSs under DoS attacks, see e.g., [13–15]. In addition, considering in WNCSs, ✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.nahs.2019.03.005. ∗ Corresponding author. E-mail address: [email protected] (Y. Wang). https://doi.org/10.1016/j.nahs.2019.03.005 1751-570X/© 2019 Elsevier Ltd. All rights reserved.

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where the energy consumption can be a critical issue, since the sensor nodes are in general feed by batteries. In fact, the energy of sensor nodes is mainly consumed in transmitting data [16] , and thus reducing the amount of transmitted data can extend the network lifetime. Further, given that event-triggered mechanism is useful for reducing the transmission of information without degrading the overall system performance [17], there has been a growing interest in using eventtriggered approach to studying security control problem of NCSs under DoS attacks in recent years. For example, in [12], the authors studied the stability analysis problem of event-triggered networked linear continuous-time systems under known/unknown Pulse-Width Modulated (PWM) DoS attacks for the first time. Inspired by this work, a more general DoS attack model was proposed in [18], where the DoS attack signal was only constrained by DoS frequency and DoS duration. The stability analysis of event-triggered networked linear continuous-time systems with external disturbance and such DoS attack was first investigated. Based on this idea, some extensions were also reported, see e.g., dynamic output-feedback controllers [19,20], nonlinear NCSs [21], and distributed NCSs [22]. Although significant progress has been made in the investigation of stability analysis of event-triggered NCSs under DoS attacks, most of the existing works are based on an emulation approach (i.e. the state feedback or observer gains are given in advance). The main limitation of the method is that it is difficult to obtain an optimal design given that one is restricted by the initial choice of the feedback control law [23]. To overcome this drawback, co-design method of security feedback control laws and event-triggering scheme in the presence of DoS attacks has been proposed in [24,25]. It is worth mentioning that the above mentioned works on co-design of event-triggering strategy and security controller do not consider the effects of signal quantization that always exists in NCSs. On the one hand, when the sampling signal is sent to the quantizer, it will reduce the size of the signal transmission, which can save the network resources occupied by the transmission signal. On the other hand, quantization errors may also have adverse effects on system’s performance. Unfortunately, by now, there is little co-design result available for event-triggered NCSs under quantization and DoS attacks except [26], where only periodic DoS attacks were considered and it is unrealistic to some extent. In addition, it is known that when a given system is with external disturbance, an effective control strategy is to design H∞ controllers such that the influence of disturbance can be attenuated. However, to the best of the authors’s knowledge, so far no attempt has been made towards solving the event-based H∞ control problem of NCSs under quantization and DoS attacks. It should be pointed out that DoS attacks lead to additional challenges: (1) How to guarantee the stability and Zeno-freeness of the closed-loop augmented quantized ETC system to be robust with respect to DoS attacks; (2) In the presence of DoS attacks, how to guarantee the L2 -gain from an external input to a controlled output to be a prescribed value. Based on the above analysis, the main purpose of this paper is to study the event-based H∞ quantized controller synthesis of WNCSs subject to DoS jamming attacks (When a user broadcasts a signal maliciously over a wireless medium to intentionally diminish the availability of the wireless channel, this is referred to as jamming, see [27] for more details). First, a new event-triggering mechanism is proposed to offset the effect of the nonperiodic DoS jamming attacks. Then a piecewise Lyapunov functional based method is developed to analyze the stability and L2 −gain of the considered event-triggered quantized NCSs under the DoS jamming attacks. Compared with the existing common Lyapunov function method and trajectory method adopted in [28], the proposed method can tackle the event-based H∞ controller synthesis problem of the NCSs under quantization and DoS jamming attacks. Some sufficient conditions in the form of linear matrix inequalities are derived to guarantee the switched system to be exponentially stable and to have a prescribed H∞ performance. If those conditions are solvable, a desired H∞ quantized state feedback controller gain matrix and the event-triggering parameters are obtained. Finally, the effectiveness and feasibility of the obtained results can be illustrated by using the satellite system. The preliminary results partially appeared in our conference paper [29]. Here we provide complete proofs not included in the conference version and make significant structural improvements. In particular, the present paper has additional results on the derivation of the weighted H∞ performance analysis. Notation: Rn represents the n-dimensional Euclidean space. In denotes the n × n identity matrix. Sym{x} denotes the expression X T + X . The X > 0 represents a real symmetric positive definite. In symmetric block matrices, “∗” is used as an ellipsis for terms caused by symmetry, diag {...} denotes the block-diagonal matrix. For a positive scalar τ , let C ([−τ , 0] ; Rn ) denote the space of bounded, continuous functions x : [−τ , 0] ↦ −→ Rn and for any x ∈ C ([−τ , 0] ; Rn ), we define ∥xt ∥τ = sup−τ ≤θ ≤0 {∥x (t + θ) ∥, ∥˙x (t + θ) ∥}. Throughout this paper, if not explicitly stated, matrices are assumed to have compatible dimensions. 2. Problem formulation and preliminaries Consider an event-triggered networked control system, shown in Fig. 1. Suppose the state space representation of the physical plant is given by:

{

x˙ (t) = Ax(t) + Bu(t) + E w (t ) y(t) = Cx(t) + Du(t) + F w (t )

(1)

where x(t) ∈ Rn is the state vector; u(t) ∈ Rp is the control input; w (t ) ∈ Rl is the disturbance input which belongs to L2 [0, ∞); y(t) ∈ Rq is the controlled output; and A, B, C , D, E , F are system matrices with appropriate dimensions. To

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267

Fig. 1. The structure of event-triggered NCSs under quantization and DoS attacks.

facilitate theoretical development, as in [24], we assume all state variables of the controlled plant are measurable, the signal in a network is transmitted with a signal packet, and the computation delay of the controller is negligible. In this paper, it is assumed that the sampled state is quantized by a logarithmic [ signal from sampler to the controller ]T quantizer and the quantizer is denoted as f (x) = f1 (x1 ) f2 (x2 ) · · · · · · fn (xn ) , which is symmetric, static, and time-invariant. For each fs (·) (s = 1, 2, . . . , n), the set of quantized levels is described by

{

}

(s)

Rs = ±ui , i = 0, ±1, ±2, . . . ∪ {0} .

According to [30], the set of quantized levels is defined as

{

(s)

(s)

Rs = ±ui , ui

} { } = ρsi u(0s) , i = ±1, ±2, . . . ∪ ±u(0s) ∪ {0} , u(0s) > 0, 0 < ρs < 1.

(s)

where u0 is the initial state of the quantizer and ρs is a parameter related to quantization density. In this setting, the associated quantizer f (x) is defined as

fs (ν) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(s)

ui , 0,

−fs (−υ) ,

if

(s) 1 u 1+δs i

υ>0 if υ = 0 if υ < 0

<υ≤

(s) 1 u 1−δs i

(2)

where δs = (1 − ρs ) /(1 + ρs ). Define ∆f = diag ∆f1 , ∆f2 , · · · , ∆fn , where ∆fs ∈ [−δs , δs ], then by the sector bound method proposed in [30], f (x) can be expressed as

{

}

f (x) = I + ∆f x.

(

)

(3)

Combining (2)–(3), and noting that the behavior of the ZOH, the real input signal u (t ) is given by u (t ) = K + K ∆f x (tk h) , t ∈ [tk h, tk+1 h)

(

)

(4)

where we have assumed that DoS attacks are absent in time interval [tk h, tk+1 h). Here, h is the constant sampling period, tk h denotes the last triggering instant, and tk+1 h is the next triggering instant. In the sequel, to investigate the effect of DoS attacks on the system performance, by referring to [31], we then introduce a type of power-constraint, nonperiodic DoS jamming attack signal, blocking the communication channels as follows: ZDoS (t ) =

{

0, t ∈ [hn , hn + ln ) 1, t ∈ [hn + ln , hn+1 ) .

(5)

For easy of exposition, we use a sequence of time intervals {Hn }n∈N where Hn represents the nth DoS interval and is given by Hn = [hn + ln , hn+1 ). Hence, hn + ln ∈ R≥0 denotes the time instant at which the (n+1)-th DoS interval commences and where hn+1 − hn − ln ∈ R≥0 denotes the length of the (n+1)-th DoS interval. The collection of sequences of DoS intervals {Hn }n∈N that satisfy 0 ≤ h0 ≤ h0 + l0 < h1 < h1 + l1 < h2 < · · · , is denoted by ZDoS . To establish our main results, it is necessary to make the following assumptions on the DoS attack signal (5): Assumption 1 ([18]). Let n (t ) denote the number of DoS off/on transitions occurring in the interval [0, t), i.e., n (t ) = card{n ∈ N | hn + ln ∈ [0, t)}, where card denotes the number of elements in the set. We say that the sequence of DoS

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attacks specified by {Hn }n∈N ∈ ZDoS satisfies the DoS frequency constraint for a given τD ∈ R>h , and a given υ1 ∈ R≥0 , if for all τD , t ∈ R≥0 n (t ) ≤ υ1 +

t

τD

.

Assumption 2 ([18]). There exists a uniform lower bound lmin on the lengths of the sleeping periods ln (∀n ∈ N) of the attacker, i.e., inf {ln } ≥ lmin .

n∈N

Similarly, there exists a uniform upper bound b max on the lengths of the active periods hn − hn−1 − ln−1 (∀n ∈ N) of the attacker, i.e., sup{hn − hn−1 − ln−1 } ≤ b max . n∈N

Remark 1. Assumptions 1 and 2 are reasonable for control design problem under unknown DoS jamming attacks. From a practical perspective, in order to guarantee the stability of the system, it is natural to require that communication is allowed to continue for some time after DoS jamming attacks stop. Also, DoS duration cannot last too long. This implies that the settings for the parameters ln and hn − hn−1 − ln−1 (∀n ∈ N) should ensure there is no overlap between a DoS jamming attack’s finish time and another one’s start time. According to [28], under the nonperiodic DoS jamming attacks (5), the real control input u (t ) (4) can be rewritten as

{

u (t ) =

{

}

where tk,n h

( ) [ ) (K + K ∆f ) x tk,n h , t ∈ tk,n h, tk+1,n h ∩ [hn−1 , hn−1 + ln−1 ) 0, t ∈ [hn−1 + ln−1 , hn ) , (

n∈N

denotes the set of successful control update instants

(6) ∆

t1,n h = hn−1

)

in the nth jammer action period, ∆

which are generated by the event-triggering mechanism to be designed later, k ∈ {1, . . . , k (n)} = ψ (n) with n ∈ N and k (n) = sup k ∈ N | tk,n h ≤ hn−1 + ln−1 .

{

}

∆

∆

∆

In to simplify representation, for n ∈ N, define O1,n = [hn , hn + ln ), O2,n = [hn + ln , hn+1 ), Lk,n = [ what follows, ) tk,n h, tk+1,n h , k ∈ ψ (n). Substituting (4) into (1), we can obtain the following closed-loop event-triggered control systems subject to nonperiodic DoS jamming attacks (5):

( ) ( ) ⎧ { Ax (t ) + E w (t ) + B K + K ∆f x tk,n h , t ∈ Lk,n ∩ O1,n−1 ⎪ ⎪ ⎪ ⎨ x˙ (t ) = Ax (t ) + E w (t ) , t ∈ O 2,n−1 , ) ( ) ( { ⎪ Cx (t ) + F w (t ) + D K + K ∆f x tk,n h , t ∈ Lk,n ∩ O1,n−1 ⎪ ⎪ ⎩ y (t ) = Cx (t ) + F w (t ) , t ∈ O2,n−1 .

(7)

Note that system (7) can be viewed as a switching one with two distinct modes. For t ∈ Lk,n ∩ O1,n−1 , the system is operating in closed loop mode; while for t ∈ O2,n−1 , the system is operating in open loop mode. When the uncontrolled subsystem is not stable, the stability of (7) will depend on the choices of event-triggering instants t1,n , t2,n , · · ·, tψ(n),n , the control parameter K . Next, we will introduce a new event-triggered communication scheme which is resilient towards the nonperiodic DoS jamming attacks. Recall that when we do not take the effect of the DoS jamming attacks into account, the following quadratic event-triggering condition is first proposed in [32] to investigate the stability and performance of the resulting event-triggered closed-loop control system:

Θ T V Θ > σ xT (tk h + ih) Vx (tk h + ih)

(8)

where Θ = x (tk h + ih) − x (tk h), σ ∈ (0, 1) is a design parameter, tk h denotes the last triggering instant, tk h + ih (k, i ∈ N) denote the subsequent sampling instant. Now, if the effect of the DoS jamming attacks is considered here, because the two network channels are intercepted over the jamming time intervals, the event-triggering scheme (8) cannot be used directly. In what follows, we will modify the above triggering scheme (8) to adapt to the nonperiodic DoS attacks (5). Definition 1. We define the event-triggering instant shown in (6) in the presence of DoS jamming attacks as follows:

{

}

tk,n h = tkj h satisfying (8) | tkj h ∈ O1,n−1 ∪ {hn }

(9)

where n, tkj , kj , j ∈ N, k denotes the number of triggering times occurring in nth jammer action period.

{

}

Remark 2. It can be seen from (9) that the time sequence tk,n h in n−th jammer action period lies in either time interval O1,n−1 or{right endpoint hn of each interval } [hn−1 , hn ). That is, if there is no event happening in the time interval O1,n−1 , namely, tkj h satisfying (8) | tkj h ∈ O1,n−1 = ∅, then the triggering instant only occurs at the right endpoint hn .

X. Chen, Y. Wang and S. Hu / Nonlinear Analysis: Hybrid Systems 33 (2019) 265–281

269

For technical convenience, similar to [26], define

{ } ∆ λk,n = sup m ∈ N | tk,n h + mh < tk+1,n h, m = 1, 2, . . . . Then the time interval Lk,n can be decomposed as λk,n +1

Lk,n = ∪m=1 ϖkm,n ,

(10)

where

{

) { } [ ϖkm,n = tk,n h + (m − 1) h, tk,n h + mh , m ∈ 1, 2, · · · , λk,n [ ) λ +1 ϖk,kn,n = tk,n h + λk,n h, tk+1,n h .

(11)

Note that k(n)

k(n)

O1,n−1 = ∪k=1 Lk,n ∩ O1,n−1 ⊆ ∪k=1 Lk,n .

{

}

(12)

Combining (10), (11) and (12), the interval O1,n−1 can be described as k(n)

λk,n +1

O1,n−1 = ∪k=1 ∪m=1

{ m } ϖk,n ∩ O1,n−1 .

Set

φkm,n = ϖkm,n ∩ O1,n−1 then k(n)

λk,n +1

O1,n−1 = ∪k=1 ∪m=1

φkm,n .

Now, for k ∈ ψ (n), n ∈ N, we define two piecewise functions as follows:

⎧ t − tk,n h, t ∈ φk1,n ⎪ ⎪ ⎪ ⎨ t − tk,n h − h, t ∈ φk2,n τk,n (t ) = .. ⎪ . ⎪ ⎪ ⎩ λk,n +1 t − tk,n h − λk,n h, t ∈ φk,n

(13)

⎧ 0,( t ∈ )φk1,n ( ⎪ ) ⎪ ⎪ ⎨ x tk,n h − x tk,n h + h , t ∈ φk2,n ek,n (t ) = .. ⎪ . ⎪ ⎪ ) ( ) ⎩ ( λk,n +1 x tk,n h − x tk,n h + λk,n h , t ∈ φk,n

(14)

and

Based on the above two definitions, it can be seen that

τk,n (t ) ∈ [0, h) , t ∈ Lk,n ∩ O1,n−1 . ( )

The event-triggered sampled state x tk,n h can be described as: x tk,n h = ek,n (t ) + x t − τk,n (t ) , t ∈ Lk,n ∩ O1,n−1 ,

(

)

(

)

with which the system (7) can be rewritten as:

( ) ( ) ( ) { ⎧ Ax (t ) + B K + K ∆f x t − τk,n (t ) + B K + K ∆f ek,n (t ) + E w (t ) , t ∈ Lk,n ∩ O1,n−1 ⎪ ⎪ ˙ x t = ( ) ⎪ ⎪ ⎪ Ax (t ) + E w (t ) , t ∈ O2,n−1 , ⎨ ( ) ( ) ( ) { Cx (t ) + D K + K ∆f x t − τk,n (t ) + D K + K ∆f ek,n (t ) + F w (t ) , t ∈ Lk,n ∩ O1,n−1 ⎪ y t = ( ) ⎪ ⎪ Cx (t ) + F w (t ) , t ∈ O2,n−1 , ⎪ ⎪ ⎩ x (t ) = ϕ (t ) , t ∈ [−h, 0] ,

(15)

where ϕ (t ) is initial function of x (t ), and the error vector ek,n (t ) satisfies eTk,n (t ) Vek,n (t ) ≤ σ xT t − τk,n (t ) Vx t − τk,n (t ) .

(

)

(

)

(16)

Now, the event-triggered H∞ control problem under DoS jamming attacks to be addressed in this paper can be formulated as follows: given a scalar γ > 0 and DoS parameters υ1 ≥ 0 and τD ≥ 0, find an event-triggered quantized state feedback controller (6), such that the following conditions are satisfied: (i) The switched system (15) is exponentially stable (ES), i.e., there exist two positive scalars α and β , such that ∀t ≥ 0, ∥x (t ) ∥ ≤ α∥ϕ0 ∥h e−β t for w (t ) = 0.

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(ii) The switched system (15) has L2 -gain less than γ , i.e., the controlled output y (t ) satisfies ∥y (t )∥2 ≤ γ ∥w (t )∥2 for

ϕ = 0 and all w (t ) ∈ L2 [0, +∞).

Before ending this section, let us give a technical lemma together with its corresponding proof. This lemma will be used to derive the main results of this paper in the next section. Lemma 1. Given fixed DoS parameters υ1 ≥ 0, τD ≥ 0, and the feedback gain matrix K . If for some prescribed scalars αi > 0, 1 > σ > 0, and h > 0, there exist symmetric positive definite matrices Pi , Qi , Ri , Zi , V , and matrices Mi , Ni , Si , i ∈ {1, 2}, such that the following linear matrix inequalities hold:

⎡ ⎢ ⎢ ⎢ Ξi = ⎢ ⎢ ⎣

i Ξ11 ∗ ∗ ∗ ∗ ∗

√

√ hNi

√

hSi 0

i Ξ22

i Ξ33 ∗ ∗ ∗

∗ ∗ ∗ ∗

√ hMi 0 0

i Ξ44 ∗ ∗

hATi 0 0 0

i Ξ55

√

hATi 0 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎦

(17)

i Ξ66

∗

i where Ξ11 = Ξ1i + Γi + ΓiT with

) ( P1 A + AT P1 + P B K + K ∆f ⎢ Q1 + 2α1 P1 1 ⎢ ∗ σV Ξ11 = ⎢ ⎡

∗ ∗

⎣

∗ ∗

0

P1 B K + K ∆f

0

0 0 −V

(

−e−2α1 h Q1 ∗

)⎤ ⎤ ⎡ P2 A + AT P2 + ⎥ 0 0 ⎥ ⎥ 2 ⎢ Q2 − 2α2 P2 ⎥ , Ξ1 = ⎣ ⎦ ∗ 0 0 ⎦ 2α2 h ∗ ∗ −e Q2

∆

∆

1 −1 i i i i i = −R− Ξ22 = Ξ33 = −e2(−1) αi λi h Ri , Ξ44 = −e2(−1) αi λi h Zi , λ1 = 1, λ2 = 0, Ξ55 i , Ξ66 = −Zi [ ] [ ] Γ1 = M1 + N1 −N1 + S1 −M1 − S1 0 , Γ2 = M2 + N2 −N2 + S2 −M2 − S2 ]T [ T ]T [ T ]T [ T T T T T T T T T T , S1 = S11 , N1 = N11 M1 = M11 S14 S13 S12 N14 N13 N12 M14 M13 M12 ]T [ T ]T [ T ]T [ T T T T T T T , S2 = S21 , N2 = N21 M2 = M21 S23 S22 N23 N22 M23 M22 ) ] [ ] ) ( [ ( , A2 = A 0 0 0 B K + K ∆f A1 = A B K + K ∆f ) [ hold, along the trajectories of the switched system (15), it follows that for t ∈ ti,n , t3−i,n+i−1 , i ∈ {1, 2}, n ∈ N, ( ) i Vi (t ) ≤ e2(−1) αi (t −ti,n ) Vi ti,n . i

i

where

{ ti,n =

hn−1 , i = 1 hn−1 + ln−1 , i = 2.

Proof. Construct a piecewise Lyapunov functional for switched system (15): Vβ(t ) (t ) = xT (t ) Pβ(t ) x (t ) +

0

∫

−h

∫

0

∫

x˙ T (s) exp (·) Rβ(t ) x˙ (s) dsdθ t +θ

t

x˙ T (s) exp (·) Zβ(t ) x˙ (s) dsdθ +

+ −h

t

∫

t +θ

∫

t

xT (s) exp (·) Qβ(t ) x (s) ds ∆

β(t ) α β(t ) (t −s) ,

where Pβ(t ) > 0, Qβ(t ) > 0, Rβ(t ) > 0, Zβ(t ) > 0, αβ(t ) > 0, exp (·) = e2(−1)

⎧ ⎨ 1, β (t ) = ⎩ 2,

(18)

t −h

t ∈ [−h, 0] ∪

(

∪ O1,n

and

)

n∈N

t ∈ ∪ O2,n . n∈N

Then, it follows the same arguments as for the proof of Lemma 1 in [26]. For reasons of space we omit these details here. Remark 3. Note that in deriving the exponential decay estimate of V2 (t ), we have assumed that the events cannot occur in the presence of DoS jamming attacks. This is reasonable. Since in the presence of DoS jamming attacks, data (from the quantizer and controller) can be neither sent nor received successfully. 3. Main results In this section, on the basis of Lemma 1, sufficient conditions are first presented to guarantee the exponential stability of the switched system (15) subjects to quantization and the nonperiodic DoS jamming attacks (5). Then based on the

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271

stability analysis result, H∞ performance analysis result and the co-design method of H∞ controller gain matrix and event-triggering parameter are obtained by solving a set of LMIs. 3.1. Stability analysis Theorem 1. Given fixed DoS parameters υ1 ≥ 0, τD ≥ 0 and the feedback gain matrix K , if for some prescribed scalars αi > 0, µi > 0 (µ1 µ2 > 1), 1 > σ > 0, and h > 0, there exist symmetric positive definite matrices Pi , Qi , Ri , Zi , V , and matrices Mi , Ni , Si , i ∈ {1, 2}, such that the LMIs (17), and the following conditions are satisfied: P1 ≤ µ2 P2

(19)

P2 ≤ µ1 e2(α1 +α2 )h P1

(20)

Qi ≤ µ3−i Q3−i

(21)

Ri ≤ µ3−i R3−i

(22)

Zi ≤ µ3−i Z3−i

(23)

0<λ=

2α1 lmin − 2 (α1 + α2 ) h − 2α2 bmax − ln µ1 µ2

(24)

τD

Then the switched system (15) under the nonperiodic DoS jamming attacks (5) is ES with decay rate ρ =

λ 2

.

Proof. Construct a piecewise Lyapunov functional candidate as in Lemma 1: V (t ) = Vβ(t ) (t ), β (t ) ∈ {1, 2}. From Lemma 1, we can get V (t ) ≤

e−2α1 (t −t1,n ) V1 t1,n , t ∈ t1,n , t2,n

)

(

{

[

) )

(25)

e2α2 (t −t2,n ) V2 t2,n , t ∈ t2,n , t1,n+1

)

(

[

According to (19)–(23), by simple calculations, it follows that

{

V1 (t1,n ) ≤ µ2 V2 t1−,n ( ) V2 t2,n ≤ µ1 e2(α1 +α2 )h V1 t2−,n .

(

)

(

)

(26)

For any given t ≥ 0, there exists a nonnegative integer n ∈ N such that t ∈ t1,n , t2,n or t ∈ t2,n , t1,n+1 . If t ∈ t1,n , t2,n , it follows from (25) and (26) that

)

[

)

[

[

)

V1 (t ) ≤ µ2 e−2α1 (t −t1,n ) V2 t1−,n ≤ en(t )×2(α1 +α2 )h+n(t ) ln(µ1 µ2 ) V1 (0)ed

(

)

with d = 2α2 (hn − hn−1 − ln−1 − ln−2 − · · · − l1 − l0 ) − 2α1 (ln−1 + ln−2 + · · · + l1 + l0 ), we have V1 (t ) ≤ e−2α1 (t −hn ) V1 ((hn )+ ) ≤ en(t )×2(α1 +α2 )h+n(t ) ln(µ1 µ2 ) V1 (0)ed ≤ eg1 (t) V1 (0) where g1 (t) =

(

υ1 +

t

)

τD

× 2(α1 + α2 )h + 2α2 bmax (υ1 +

t

τD

) − 2α1 lmin (υ1 +

t

τD

(

) + υ1 +

t

τD

)

ln(µ1 µ2 ). Here, we have

used Assumptions 1 and 2. It follows from (24) that V1 (t ) ≤ ec1 e−λt V1 (0)

(27)

where c1 = 2υ1 (α1 + α2 ) h + υ1 ln(µ1 µ2 ) + 2α2 bmax υ1 − 2α1 lmin υ1 . On the other hand, we can obtain V2 (t ) ≤

1

µ2

eg2 (t) V1 (0).

It follows from (24) that V2 (t ) ≤

V1 (0)

µ2

ec2 e−λt

(28)

where g2 (t) = (υ1 + τt + 1) × 2(α1 + α2 )h + 2α2 bmax (υ1 + D c2 = (υ1 + 1) (2 (α1 + α2 ) { h + ln(µ}1 µ2 ) + 2α2 bmax − 2α1 lmin ). Now, define M = max ec1 ,

ec2

µ2

t

τD

+ 1) − 2α1 lmin (υ1 +

t

τD

+ 1) + (υ1 +

, ϱ1 = min{λmin (Pi )}, and ϱ2 = λmax (P1 ) + hλmax (Q1 ) +

h2 2

t

τD

+ 1) ln(µ1 µ2 ),

λmax (R1 + Z1 ), and ρ = λ2 ,

from (27) and (28) , it yields that V (t) ≤ Me−λt V1 (0).

(29)

272

X. Chen, Y. Wang and S. Hu / Nonlinear Analysis: Hybrid Systems 33 (2019) 265–281

According to the definition of V (t), we have V (t) ≥ ϱ1 ∥x(t)∥2 , V1 (0) ≤ ϱ2 ∥ϕ0 ∥2h .

(30)

Substituting (30) into (29) yields

√ ∥x(t)∥ ≤

M ϱ2

ϱ1

e−ρ t ∥ϕ0 ∥h , ∀t ≥ 0

which implies that the switched system (15) is ES with decay rate ρ . Remark 4. It can be seen from (24) that the decay rate ρ is explicitly dependent on DoS parameters lmin , bmax , τD , the sampling period h, and the tuning parameters αi , µi > 0, i ∈ {1, 2}. How to choose the parameters h, αi and µi depends on Theorem 1. Moreover, it can be concluded that for fixed αi ∈ (0, +∞), µi ∈ (1, +∞), i ∈ {1, 2}, and h, given a bmax , if Theorem 1 holds for some l∗min , then it holds for any lmin ≥ l∗min . In fact, it follows from (24) that 0<

2α1 l∗min − 2 (α1 + α2 ) h − 2α2 bmax − ln µ1 µ2

τD

<

2α1 lmin − 2 (α1 + α2 ) h − 2α2 bmax − ln µ1 µ2

τD

.

Thus, the minimal sleeping time l∗min with given scalars bmax , αi ∈ (0, +∞), µi ∈ (1, +∞), i ∈ {1, 2}, and h can be estimated by l∗min = min {lmin | lmin satisfying (24) subjects to LMIs (17), (19)–(23)} .

(31) ∗

Similarly, given the minimal sleeping time lmin , if Theorem 1 holds for some bmax , then it holds for any bmax ≤ b∗max . Following the preceding analysis, it is easy to show that 0<

2α1 lmin − 2 (α1 + α2 ) h − 2α2 b∗max − ln µ1 µ2

τD

<

2α1 lmin − 2 (α1 + α2 ) h − 2α2 bmax − ln µ1 µ2

τD

.

Therefore, for given a lmin with fixed αi ∈ (0, +∞), µi ∈ (1, +∞), i ∈ {1, 2}, and h, the maximal DoS duration time b∗max can be calculated by b∗max = max {bmax | bmax satisfying (24) subjects to LMIs (17), (19)–(23)} .

(32)

3.2. H∞ Performance analysis Theorem 2. For given some scalars υ1 ≥ 0, τD ≥ 0, αi > 0, µi > 0 (µ1 µ2 > 1), 1 > σ > 0, lmin > 0, lmax > 0, b√ max > 0, γ > 0, and h > 0, the feedback gain matrix K , the switched system (15) is ES with an H∞ performance level γ¯ = ηηmax γ , min { } { l } max where ηmin = min µ1 , 1 , ηmax = max e µ , e2α2 bmax , if there exist symmetric positive definite matrices Pi , Qi , Ri , Zi , V , 2 2 and matrices Mi , Ni , Si , i ∈ {1, 2}, such that (19)–(24) and the following linear matrix inequalities hold: √ √ √ √ ⎡ i √ ⎤ Ψ11 hNi hSi hMi hRi ATi hZi ATi Γyi i ⎢ ∗ 0 0 0 0 0 ⎥ Ψ22 ⎢ ⎥ i ⎢ ∗ ∗ Ψ33 0 0 0 0 ⎥ ⎥ ⎢ i (33) Ψi = ⎢ ∗ ∗ ∗ Ψ44 0 0 0 ⎥<0 ⎢ ⎥ i ⎢ ∗ ∗ ∗ ∗ Ψ55 0 0 ⎥ ⎦ ⎣ i ∗ ∗ ∗ ∗ ∗ Ψ66 0 ∗ ∗ ∗ ∗ ∗ ∗ −In i where Ψ11 = Ψ1i + Γi + ΓiT with

⎡ ⎢ ⎢ ⎢ Ψ11 = ⎢ ⎢ ⎣ ⎡

P1 A + A T P1 + Q1 + 2α1 P1

∗ ∗ ∗ ∗

P1 B K + K ∆f

(

σV ∗ ∗ ∗

P2 A + AT P2 + Q2 − 2α2 P2

∗ ∗ ∗

⎢ Ψ12 = ⎣

)

0

P1 B K + K ∆f

0

0 0 −V

0 0 0

∗

−γ 2 In

(

−e−2α1 h Q1 ∗ ∗ 0 0

∗ ∗

0 0

−e2α2 h Q2 ∗

)

P1 E

P2 E 0 ⎥ ⎦ 0 −γ 2 In

⎤

∆

∆

i i i i Ψ22 = Ψ33 = −e2(−1) αi λi h Ri , Ψ44 = −e2(−1) αi λi h Zi , λ1 = 1, λ2 = 0, Ψ55 = −Ri i

i

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

X. Chen, Y. Wang and S. Hu / Nonlinear Analysis: Hybrid Systems 33 (2019) 265–281

]T

273

]T

i T T T T T T T T T , S2 = S21 = −Zi , S1 = S11 Ψ66 S22 S23 S24 S12 S13 S14 S15 ]T [ ( ) ( ) ] [ T T T T A 1 = A B K + K ∆f 0 B K + K ∆f E , N2 = N21 N22 N23 N24 [ ] [ ] Γ1 = M1 + N1 −N1 + S1 −M1 − S1 0 0 , A2 = A 0 0 E ]T [ ] [ T T T T Γ2 = M2 + N2 −N2 + S2 −M2 − S2 0 , M2 = M21 M22 M23 M24 ]T ]T [ T [ T T T T T T T T T , N1 = N11 M1 = M11 N12 N13 N14 N15 M12 M13 M14 M15 [ ( ) ( ) ]T [ ]T Γy1 = C D K + K ∆f 0 D K + K ∆f F , Γy2 = C 0 0 F

[

[

Proof. For any given k ∈ ψ (n + 1), n ∈ N, and for all w (t ) ∈ L2 [0, ∞), by a similar proof as that of Lemma 1, it follows from (33) that

{

V˙ 1 (t ) + 2α1 V1 (t ) + yT (t ) y (t ) − γ 2 w T (t ) w (t ) ≤ 0, t ∈ O1,n V˙ 2 (t ) − 2α2 V2 (t ) + yT (t ) y (t ) − γ 2 w T (t ) w (t ) ≤ 0, t ∈ O2,n .

(34)

In the sequel, for notation simplicity, let δ (t ) = γ 2 w T (t ) w (t ) − yT (t ) y (t ). Then for any given t ∈ [0, hn+1 ), n ∈ N, from (34), we can obtain n ∫ ∑

hk +lk

(

n ∫ ∑

hk +lk

(

δ (t ) dt +

hk+1

∫

e2α2 (hk+1 −t ) δ (t ) dt)

hk +lk

] e−2α1 (hk −t ) [ V˙ 1 (t) + 2α1 V1 (t) dt +

∫

µ2

hk

k=0

≥

µ2

hk

k=0

≥

e−2α1 (hk −t )

V1 (hn+1 ) − V1 (0)

µ2

hk+1

e2α2 (hk+1 −t ) [V˙ 2 (t) − 2α2 V2 (t)]dt)

hk +lk

+

n ∑

V1 (hk + lk )(

k=0

1

µ2

e2α1 lmin − µ1 e2α2 bmax +(2α1 +2α2 )h )

(35)

where we have used Assumption 2. From (24), one has µ1 e2α1 lmin − µ1 e2α2 bmax +(2α1 +2α2 )h > 0. Notice that V1 (hn+1 ) > 0, V1 (hk + lk ) > 0, k ∈ {0, 1, 2, . . . , n}, 2 n ∈ N, V1 (0) = 0, it can be inferred from (35) that n ∫ ∑

hk +lk

(

k=0

e−2α1 (hk −t )

µ2

hk

δ (t ) dt +

hk+1

∫

e2α2 (hk+1 −t ) δ (t ) dt) > 0.

(36)

hk +lk

Considering that DoS jamming attacks are intermittent, that is to say, ∀n ∈ N, ln < ∞. Therefore, there always exists a positive scalar lmax = supn∈N {ln }. Then, ∀t ∈ [hk , hk + lk ), k ∈ {0, 1, 2, . . . , n}, n ∈ N, 1 ≤ e2α1 (t −hk ) ≤ elk ≤ elmax .

(37)

Moreover, according to Assumption 2, ∀t ∈ [hk + lk , hk+1 ), we have 1 ≤ e2α2 (hk+1 −t ) ≤ e2α2 (hk+1 −hk −lk ) ≤ e2α2 bmax .

(38)

Combining (36)–(38), and noting the definitions of ηmin and ηmax in Theorem 2, one has n ∫ ∑ k=0

hk+1

ηmin y (t ) y (t ) dt ≤ T

hk

n (∫ ∑

∑ (∫ k=0 n

≤

µ2

hk +lk

e−2α1 (hk −t )

µ2

hk

∑∫ k=0

e−2α1 (hk −t )

hk

k=0 n

<

hk +lk

hk+1

y (t ) y (t ) dt + T

0

T

ηmax γ 2 wT (t ) w (t ) dt

hk

yT (t)y(t)dt <

0

that is ∥y (t )∥2 ≤ γ¯ ∥w (t )∥2 with γ¯ =

√

ηmax γ, ηmin

(

2α2 hk+1 −t

e

γ w (t ) w (t ) dt + 2

∫ ηmax hn+1 2 T γ ω (t)ω(t)dt . ηmin 0 0 When n → ∞ (hn+1 → ∞), it yields ∫ ∞ ∫ ∞ T y (t ) y (t ) dt ≤ γ¯ 2 wT (t ) w (t ) dt hn+1

hk+1

) yT (t ) y (t ) dt

)

hk +lk

which implies that

∫

∫

for w (t ) ∈ L2 [0, +∞).

∫

hk+1

e hk +lk

(

2α2 hk+1 −t

) γ 2 wT (t ) w (t ) dt

)

274

X. Chen, Y. Wang and S. Hu / Nonlinear Analysis: Hybrid Systems 33 (2019) 265–281

When w (t ) ≡ 0, it can be seen from ((34) ) that V˙ 1 (t ) + 2α1 V1 (t ) ≤ 0 for t ∈ O1,n and V˙ 2 (t ) − 2α2 V2 (t ) ≤ 0 for t ∈ O2,n . Then, based on Lemma 1 and Theorem 1 together with (33), it can be concluded that the switched system (15) is ES. This proof is completed. 3.3. H∞ Controller design Notice that the decay rate ρ of the augmented system ((15) ) is an important performance index for the control performance. In what follows, we aim to provide a co-design method of H∞ controller parameter K and the weighting matrix V in resilient event-triggering scheme (9) based on a give decay rate ρ . Theorem 3. For given scalars υ1 ≥ 0, τD ≥ 0, γ > 0, dj (j = 1, . . . , 5), h > 0, 1 > σ > 0, r > 0, ϵi > 0, αi > 0, µi > 0 (µ1 µ2 > 1), ζi > 0, ηi > 0, the system (15) is ES with an H∞ performance level γ¯ , if there exist symmetric positive ˜ i , N˜ i , S˜i (i = 1, 2), Y of appropriate definite matrices Xi > 0, P˜ i > 0, Q˜ i > 0, R˜ i > 0, Z˜i > 0, V > 0, Λ > 0, and matrices M dimensions and scalar ε1 > 0, such that the conditions (24), and the following matrix inequalities hold, [ ] −Λ Y T ≤0 (39) ∗ −rI

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𭟋i11

+ Φi + ΦiT ∗ ∗ ∗ ∗ ∗

𭟋i12

𭟋i13

𭟋i14

𭟋i22

0 𭟋i33

0 0 𭟋i44

∗ ∗ ∗ ∗

∗ ∗ ∗

Λ − 2d5 X1 + d25 I ≤ 0 ⎤ ς (i) 𭟋T15 ς (i) 𭟋T16 ⎥ ⎥ 0 0 ⎥ ⎥ 0 0 ⎥<0 ⎥ 0 0 ⎥ ⎦ i 𭟋 0

∗ ∗

55

∗ [

(40)

(41)

𭟋i66

]

X2T −X1

−µ2 X2 ∗

≤0

−µ1 e2(α1 +α2 )h X1 X1T ≤ ∗ −X2 ] [ −µ3−i Q˜ 3−i X3T−i ≤ ∗ ζi2 Q˜ i − 2ζi Xi ] [ −µ3−i R˜ 3−i X3T−i ≤ 2˜ ∗ ϵi Ri − 2ϵi Xi ] [ −µ3−i Z˜3−i X3T−i ≤ 2˜ ∗ ηi Zi − 2ηi Xi

(42)

]

[

0

(43)

0

(44)

0

(45)

0

(46)

where ς (1) = 1, ς (2) = 0, and

⎡

AX1 + X1 AT + 2α1 X1 + Q˜ 1

⎢ ⎢

∗ ∗ ∗ ∗

𭟋111 = ⎢ ⎢

⎣ ⎡ 𭟋211

∗ ∗ ∗

AX2 + X2 AT + Q˜ 2 − 2α2 X2

∗ ∗ ∗ −N˜ 1 + S˜1

⎢ =⎢ ⎣

Φ1 =

[

˜ 1 + N˜ 1 M

Φ2 =

[

˜ 2 + N˜ 2 M

˜1 = M

[

˜T M 11

˜T M 12

N˜ 1 =

[

T N˜ 11

T N˜ 12

A2 =

[

AX2

0

𭟋114 =

[

CX1

DY

BY σ V˜

0 0

BY 0 0 −V˜

−e−2α1 h Q˜ 1 ∗ ∗

0 0

0 0

⎥ ⎥ ⎥ ⎥ ⎦

−γ 2 I

∗ E 0 0

⎤

E 0 0 0

⎤

⎥ ⎥ ⎦ −e2α2 h Q˜ 2 ∗ −γ 2 I ] [ ˜ 1 − S˜1 0 0 , A1 = AX1 BY 0 BY −M ] [ √ √ ] ˜ 2 − S˜2 0 , 𭟋213 = AT2 hI hI −N˜ 2 + S˜2 −M √ [ ]T 2 ] T T T ˜ ˜ ˜ ˜ ˜ ˜ , 𭟋12 = h N2 S2 M2 M13 M14 M15 ]T [ ]T T T T , S˜1 = S˜ T S˜ T S˜ T S˜ T S˜ T N˜ N˜ N˜ 13

0

E 0

∗ ∗

14

]

15

, 𭟋112 = ]T

DY

F

11

√ [ h

N˜ 1

, 𭟋214 =

[

12

S˜1

˜1 M

CX2

0

13

14

]

15

√ ] , 𭟋113 = A1 hI hI ]T 0 F , 𭟋15 = ε1 H˜ m , 𭟋16 = r δf H˜ l

]

[ √ T

E

X. Chen, Y. Wang and S. Hu / Nonlinear Analysis: Hybrid Systems 33 (2019) 265–281

275

−R˜ 2 0 0 −e−2α1 h R˜ 1 0 0 2 −2α1 h ˜ ⎣ ⎦ , 𭟋22 = ∗ −R˜ 2 0 ⎦ ∗ −e R1 0 −2α1 h ˜ ∗ ∗ −Z˜2 ∗ ∗ −e Z1 [ 2 ] [ 2 ] d1 R˜ 1 − 2d1 X1 0 d3 R˜ 2 − 2d3 X2 0 1 2 𭟋33 = , 𭟋33 = ∗ d22 Z˜1 − 2d2 X1 ∗ d24 Z˜2 − 2d4 X2 ] [ ] [ T ˜ l = 0 X1 0 X1 0 0 0 0 0 0 0 , M ˜2 = M ˜ ˜T M ˜T M ˜T T H M 22 23 24 21

⎡

𭟋122

⎡

⎤

⎤

=⎣

𭟋144 = −I , 𭟋244 = −I , 𭟋155 = −ε1 I , 𭟋255 = 0, 𭟋166 = −ε1 rI , 𭟋266 = 0 S˜2 =

[

T S˜21

˜m = H

[

BT

T S˜22

0

T S˜23

0

]T

T S˜24

0

0

0

, N˜ 2 = 0

[

T N˜ 21

√

T N˜ 22

√

hBT

0

T N˜ 23

hBT

]T

T N˜ 24

DT

]

.

Moreover, if the above conditions are feasible, a desired controller gain matrix is given by K = YX1−1 . Proof. Notice that the inequalities (33) with i = 1 can be equivalently expressed as

{ } Ω + sym HmT KHf < 0

(47)

where

⎡ ⎢ ⎢ ⎢ ⎢ Ω=⎢ ⎢ ⎢ ⎣

√

Ω11 + Γ1 + Γ1T ∗ ∗ ∗ ∗ ∗ ∗

√ hN1

√

hS1 0

Ω22 ∗ ∗ ∗ ∗ ∗

√ hM1 0 0

Ω33 ∗ ∗ ∗ ∗

Ω44 ∗ ∗ ∗

hR1 AT1 0 0 0

√

Ω55 ∗ ∗

hZ1 AT1 0 0 0 0

Ω66 ∗

ΓyT

⎤

0 0 0 0 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Ω77

with P1 A + A T P1 +2α1 P1 + Q1

⎡ Ω11

⎢ ⎢ ⎢ =⎢ ⎢ ⎣

Γ1 =

[

∗ ∗ ∗ ∗ M1 + N1

P1 BK

0

P1 BK

P1 E

σV ∗ ∗ ∗

0

0 0 −V

0 0 0

∗ ]

−γ 2 I

−N1 + S1

−e−2α1 h Q1 ∗ ∗ −M1 − S1

0

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

, Ω22 = −e−2α1 h R1

, Ω33 = −e R1 , Ω44 = −e−2α1 h Z1 [ Ω55 = −R1 , Ω66 = −Z1 , Ω77 = −I , Hf = 0 ∆f 0 ∆f 0 0 0 0 0 0 [ T ]T [ T ]T T T T T T T T T M1 = M11 , S1 = S11 M12 M13 M14 M15 S12 S13 S14 S15 [ T ]T [ ] T T T T N1 = N11 , A1 = A BK 0 BK E N12 N13 N14 N15 [ ] √ T √ T Hm = BT P1 0 0 0 0 0 0 0 hB R1 hB Z1 DT Γy =

[

C

DK

0

DK

F

]

−2α1 h

0

]

Using Lemma 2 [33], it follows from (47) that there exist a scalar ε1 > 0 such that

Ω + ε1 HmT Hm + ε1−1 HfT K T KHf < 0.

(48)

From (39), we have r Λ − Y T Y ≥ 0. Notice that obtain

X1T X1

(49)

≥ 2d5 X1 −

d25 In

−1

by using Lemma 3 [33]. Moreover, since K = YX1 , then combining (40) and (49), we

K T K ≤ rIn .

(50)

From (48) and (50), we can conclude that if

Ω + ε1 HmT Hm + ε1−1 HfT rHf < 0.

(51)

then (48) holds. By the Schur complement, (51) is equivalent to

⎡

Ω ⎣ ∗ ∗

ε1 HmT −ε1 In ∗

⎤

r δf HIT ⎦<0 0 −ε1 rIn

(52)

276

X. Chen, Y. Wang and S. Hu / Nonlinear Analysis: Hybrid Systems 33 (2019) 265–281

with HI =

[

0

I

0

I

0

0

0

0

0

0

0

]

.

Define X1 = P1−1 , {J = diag}{J1 , J2 , J3 }, pre- and post-multiplying (52) with J, where J1 = diag {X1 , X1 , X1 , X1 , I }, J2 = 1 −1 {X1 , X1 , X1 }, J3 = R− . Define new matrix variables V˜ = X1 VX1 , Q˜ 1 = X1 Q1 X1 , Z˜1 = X1 Z1 X1 , R˜ 1 = X1 R1 X1 , 1 , Z1 ˜ 15 = X1 M15 and ˜ ˜ N1j = X1 N1j X1 , M1j = X1 M1j X1 , S˜1j = X1 S1j X1 (j = 1, 2, 3, 4), Y = KX1 , N˜ 15 = X1 N15 , S˜15 = X1 S15 , M 1 −1 2˜ 2˜ ˜ using the Lemma 3 [33], we can obtain −X1 R˜ − X ≤ d R − 2d X and − X Z X ≤ d Z − 2d X , then by the Schur 1 1 1 1 1 1 2 1 1 1 1 2 1 complement, (41) can be obtained. On the other hand, the inequalities (41) with i = 2 can be obtained by this method. Furthermore, pre- and post-multiplying (19) and (20) by X2 and X1 , respectively, and using the Schur complement, we obtain that (42) and (43) are equivalent to (19) and (20). Using the similar techniques, it is not difficult to derive that LMIs (44)–(46) ensure (21)–(23) hold. This completes the proof. Remark 5. It is worth mentioning that the major differences between the current paper and [19] are as follows: (1) a class of nonlinear continuous-time systems is considered in [19], while a sampled-data networked linear continuoustime system is discussed in this paper; (2) the description of the constraints on DoS attack behavior in [19] is based on the DoS duration and DoS frequency, while in this paper, the property of DoS attack behavior is constrained by minimal “sleeping” period, maximal “active” period of DoS attacks, and the DoS frequency; (3) output-based dynamic ETC strategies are proposed in [19], while a periodic sampled state-based static ETC scheme is developed in the present paper; (4) the common quadratic Lyapunov function is used in [19], while here, by introducing the conceptions of minimal “sleeping” period and maximal “active” period of DoS attacks, a piecewise Lyapunov–Krasovskii functional approach is proposed to analyze the exponential stability and H∞ performance; and (5) in the presence of exogenous disturbances and DoS attacks, the uniformly globally asymptotically stable and performance criteria in terms of induced normal-gain can be guaranteed in [19] by using emulation method, while an H∞ controller and resilient event-triggering scheme are obtained simultaneously to ensure the resultant switched system is ES and resilient to DoS jamming attacks by using a co-design method. Remark 6. This paper differs from our previous works [26,34] in several aspects. First, the considered problem is different. Specifically, the event-based robust state feedback controller design problem for uncertain networked control systems under quantization and denial-of-service attacks is investigated in [26], while here we deal with event-based quantized H∞ controller design for networked control systems in the presence of denial-of-service attacks. Moreover, notice that parameter uncertainties and both state and control input quantizations are considered in [26], while the present paper and [34] do not consider these issues (only state quantization is considered in this paper). Second, a different DoS attack model is considered. We focus on the nonperiodic DoS jamming attacks, and the description of the constraints on such DoS attack behavior is based on two parameters (see Assumption 2) and DoS frequency, while the works [26,34] consider periodic DoS jamming attacks, which do not need these requirements. In addition, the periodic DoS jamming attacks can be seen as a special case of nonperiodic DoS jamming attacks. Remark 7. Compared with [26,34], the main novelties of the current paper are as follows: (I) A new event-triggering mechanism is proposed to offset the effect of the nonperiodic DoS jamming attacks; (II) An explicit characterization of the relation among the sampling period, convergence rate, the minimal “sleeping” period and the maximal “active” period of DoS jamming attacks is obtained; and (III) A new weighted H∞ controller is developed to ensure that the resultant switched system is exponentially stable and resilient to nonperiodic DoS jamming attacks as well. The weighed H∞ performance analysis and controller design results are expressed by the feasible solutions of the obtained LMIs, which can be solved by the standard LMI Toolbox. 4. Illustrative example In this section, we present a simulation example to validate the effectiveness of the proposed controller design method for NCSs under quantization and unknown periodic DoS jamming attacks. We first co-design the event-triggering parameters (σ , V ) and the event-based quantized state-feedback control gain matrix K . Then the effect that different lmin has on bmax , and the decay rate ρ will be studied here. Additionally, the effect that the triggering parameter σ has on hmax , lmin , and H∞ performance γ¯ will be also discussed. Suppose the physical plant shown in Fig. 1 is a satellite system, which is taken from [35]. The satellite system consists of two rigid bodies joined by a flexible link. This link is modeled as a spring with torque constant k and viscous damping f . Denoting the yaw angles for the two bodies (the main body and the instrumentation module) by θ1 and θ2 , the control torque by u(t), and the moments of inertia of the two bodies by J1 and J2 , we see that the dynamic equations are given by J1 θ¨1 (t ) + k (θ2 (t ) − θ1 (t )) + f θ˙2 (t ) − θ˙1 (t ) = u(t)

(

)

( ) J2 θ¨2 (t ) + k (θ1 (t ) − θ2 (t )) + f θ˙1 (t ) − θ˙2 (t ) = 0.1w (t )

X. Chen, Y. Wang and S. Hu / Nonlinear Analysis: Hybrid Systems 33 (2019) 265–281

277

where w (t) denotes the external disturbance. Assume the output is the angular position θ2 (t ). Thus, the state-space representation of the above equation in the form of (1) is given by

⎡

1 ⎢ 0 ⎣ 0 0

+ [

y (t ) =

[ 0

0 1 0 0 0 1

θ˙1 (t ) 0 0 0 ⎥⎢ θ˙2 (t ) ⎥ ⎢ 0 ⎥ ⎢ =⎣ −k 0 ⎦ ⎣ θ¨1 (t ) ⎦ ¨θ2 (t ) k J2

1

⎤

⎤⎡

0 0 J1 0 0

0

0

0

][

]T

u (t ) +

θ1 (t )

[

0

θ2 (t )

⎡

0

0 0 k −k 0.1

1 0 −f f

θ1 (t ) 0 1 ⎥ ⎢ θ2 (t ) ⎥ f ⎦ ⎣ θ˙1 (t ) ⎦ −f θ˙2 (t )

⎤⎡

⎤

]T

w (t ) ]T . θ˙2 (t )

0

θ˙1 (t )

For this example, we choose J1 = J2 = 1, k = 0.09 and f = 0.04. Then the system matrices in (1) are given by

⎧ ⎡ ⎤ 0 0 1.0000 0 ⎪ ⎪ ⎪ ⎪ 0 0 1.0000 ⎥ ⎢ 0 ⎪ ⎪ ⎨ A = ⎣ −0.3 0.3 −0.0040 0.0040 ⎦ 0.3 −0.3 0.0040 −0.0040 ⎪ [ ]T [ ] ⎪ ⎪ ⎪ 0 0 1 0 B= ,C = 0 1 0 0 ⎪ ⎪ [ ] T ⎩ D = 0, E = 0 0 0 0.1 ,F = 0

(53)

By simple calculation, it is found that the eigenvalues of A are 0, 0, −0.04 + 0.4224j, −0.04 − 0.4224j, thus the considered system is not stable. For this example, our main objective is to design an event-based quantized state-feedback controller in the form of (6) such that the system (15) with (53) is ES with an H∞ disturbance attenuation level γ¯ . In the following, we will design the event-triggering parameters (σ , V ) and the event-based quantized state-feedback control gain matrix K simultaneously. To this end, we choose h = 0.02s, α1 = 0.05, α2 = 0.5, µ1 = µ2 = 1.01, lmin = 2s, and bmax = 0.15s satisfying (24). Then assume that the triggering parameter σ = 0.1, lmax = 3s, the parameters ρ1 = ρ2 = ρ3 = ρ4 = 0.818 for the quantizer f (·), the tuning parameters dj = 4(j = 1, 2, 3, 4), d5 = 0.1, ϵi = 10, ζi = 10, ηi = 10 (i = 1, 2), and r = 8, applying Theorem 3, we can obtain the feedback gain matrix K =

[

−3.6371

0.9762

−7.8759

−6.2011

]

(54)

⎤ −0.0361 −0.0269 ⎥ 0.0043 ⎦ 0.0144

(55)

the corresponding trigger matrix

⎡

0.1094

∗ ∗ ∗

V =⎣

⎢

0.0693 0.0725

∗ ∗

−0.0337 −0.0032 4.7595 ∗

and the obtained minimum guaranteed H∞ performance index γ¯min = 7.0363. We then show that under Assumptions 1 and 2, the switched system (15) with (53) is ES under the event-triggered controller (6) with (54) and (55) despite the presence of[ the nonperiodic DoS jamming attacks (5) with lmin = 2s and ]T bmax = 0.15s. The initial condition is assumed to be x0 = 0.1 −1 0.2 0.1 and simulation time is assumed to be T = 50s. On the one hand, the state responses of the system without DoS jamming attacks are depicted in Fig. 2 while the state responses in the presence of nonperiodic DoS jamming attacks are depicted in Fig. 3. By comparison, we can see that four state trajectories of the system (15) with (53) happen severe shocks at first and then converge to original point in Fig. 3. Further, the control input and the release time intervals between any two consecutive release instants under the nonperiodic DoS jamming signal are also depicted in Figs. 4–5, respectively. From Figs. 4–5, we can see that the above obtained event-triggering mechanism can reduce the amount of data transmission while offsetting the impact of the nonperiodic DoS jamming attacks. On the other hand, in that setup, the total duration(over which) the communication 3s is denied is obtained as N ∗ bmax = 20 ∗ 0.15s = 3s, thus a maximum duty cycle of 6% 50s ∗ 100% of a sustained DoS does not destroy the stability of the systems in the example. But in fact, this bound is conservative. Since from Figs. 3 and 4 (The gray stripes denote the DoS jamming attacks are active, while the white stripes denote the DoS attacks are sleeping), the systems can endure more DoS attacks while guaranteeing the stability of the system. The conservativeness of the results may stem from the treatment of the cross terms shown in Lemma 1. In this respect, the advanced matrix inequality techniques proposed in [36,37] can give a tighter bound, which may result in less conservative results. This point is left for our future study. Next, we illustrate the H∞ performance of the system (15) with (53). To this end, select a set of input signals as follows:

{ w (t ) =

1, if 5 ≤ t ≤ 10 −1, if 15 ≤ t ≤ 20 0, otherwise.

By calculation, ∥w∥2 = 3.4641, ∥y∥2 = 0.0315, which yields

∥y∥2 = 0.0091 < γ¯min = 7.0363 ∥w∥2 which shows the H∞ controller design for system ((15) ) with (53) is effective.

278

X. Chen, Y. Wang and S. Hu / Nonlinear Analysis: Hybrid Systems 33 (2019) 265–281

Fig. 2. State responses without DoS attacks.

Fig. 3. State responses under DoS attacks. Table 1 b∗max for different values of lmin . lmin

2

4

6

8

10

b∗max

0.15

0.35

0.55

0.75

0.95

Now, to illustrate the interaction effect of DoS attack parameters lmin and bmax , based on (32) in Remark 4, we solve the following optimization problem for different values of lmin in the time interval [0, 50s] (the parameters h, µi , αi (i = 1, 2), σ , and γ are chosen as the same before): b∗max = max{bmax |bmax satisfying (24) subjects to LMIs (39)–(46)}. Table 1 shows the maximum length b∗max of attack interval for which the exponential stability of the switched system ((15) ) with (53) is guaranteed for each lmin chosen. From Table 1, it can be seen that when the values of lmin increase, the values of b∗max also increase correspondingly. The reason for this phenomenon is that the parameter b∗max is increasing function of lmin . This also means that the system can relatively tolerate more malicious attacks (bmax ↑) with lmin ↑. We further show the relationship between lmin and the decay rate ρ . To this end, setting τD = 1 and the values of the other parameters remain unchanged. Some calculation results are listed in Table 2 . It is observed that the bigger the lmin ,

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279

Fig. 4. Control input.

Fig. 5. Release time intervals. Table 2 λ and ρ for different values of lmin with bmax = 0.35 and τD = 1. lmin

4

6

8

10

12

ρ

0.0040

0.1040

0.2040

0.3040

0.4040

the larger the ρ . This implies that in order to seek for larger value of ρ (i.e., guaranteeing the better stability performance), lmin should be chosen as large as possible while ensuring the relation (24) is satisfied. Similarly, we show the impact of the triggering parameter σ on the maximum allowable sampling period hmax and the corresponding lmin for the system (15) with ((53) ). For this purpose, we assume that µ1 = µ2 = 1.01, α1 = 0.05, α2 = 0.5, γmin = 1.57, lmin = 4, lmax = 5, b∗max = 0.35, ϵi = 10, ζi = 10, ηi = 10 (i = 1, 2), dj = 4 (j = 1, 2, 3, 4), and d5 = 0.1. The values of hmax and lmin for different σ are listed in Table 3. From Table 3, we find that with the increase of the triggering parameter σ (implying the reduction utilization in communication resources), the value of hmax is becoming smaller and, the value of lmin is decreasing correspondingly. This means that there is a tradeoff between system performance and the amount of data transmission. Finally, for a fixed h = 0.02, the minimum H∞ performance level γ¯min for different values of σ are given in Table 4 (The other parameters are chosen as the same before). It can be seen from Table 4 that the larger the σ , the larger the γ¯min , implying the worse the noise attenuation performance.

280

X. Chen, Y. Wang and S. Hu / Nonlinear Analysis: Hybrid Systems 33 (2019) 265–281 Table 3 hmax and l∗min for different values of σ .

σ

0.01

0.05

0.08

0.1

0.2

hmax l∗min

0.027 4.00

0.024 3.97

0.021 3.94

0.020 3.92

0.015 3.87

Table 4 The values of γ¯min for different σ .

σ

0.01

0.02

0.05

0.08

0.1

γ¯min

16.5682

16.8118

17.7864

18.6392

19.1265

Remark 8. In this paper, we have discussed the event-based H∞ stabilization problem of NCSs with quantization in the presence of nonperiodic DoS jamming attacks. The solution to the problem under the consideration is expressed in the form of the feasibility of a set of LMIs. The balance between the H∞ performance index, exponential stability criterion, event-triggering frequency and the tolerance of the nonperiodic DoS jamming attacks by taking full advantage of the design flexibility can be found by simulation study. 5. Conclusions This paper has studied the event-based H∞ stabilization problem of NCSs under quantization and DoS jamming attacks. A new switched control system model has been proposed to describe quantization and DoS jamming attacks in a unified framework. Then by using the piecewise Lyapunov functional, sufficient conditions for the exponential stability of the system under the nonperiodic DoS jamming attacks and quantization have been founded in the terms of LMIs. Finally, a practical example is used to demonstrate that the proposed event-triggered communication strategy can counteract nonperiodic DoS jamming attacks and significantly save valuable network resources while maintaining the required control performance. Our future works will be to further extend the proposed approach to nonlinear systems, stochastic systems, and power systems with the help of the existing works [38–42]. Acknowledgments This work was supported in part by the Open Foundation of National Engineering Research Center of Communications and Networking (No. GCZX001), the National Natural Science Foundation of China (Grant No. 61673223), the ‘‘Six Talent Peaks’’ Project of Jiangsu Province of China (Grant No. RLD201810); the QingLan Project of Jiangsu Province of China (Grant No. QL 04317006), the NUPTSF (Grant No. XJKY15001), and the China Postdoctoral Science Foundation (Grant No. 2015M571788). References [1] J. Liu, E. Tian, X. Xie, H. Lin, Distributed event-triggered control for networked control systems with stochastic cyber-attacks, J. Franklin Inst. B (2018) http://dx.doi.org/10.1016/j.jfranklin.2018.01.048. [2] J. Liu, L. Wei, E. Tian, S. Fei, J. Cao, Hybrid-driven-based H∞ filtering design for networked systems under stochastic cyber attacks, J. Franklin Inst. B 354 (2017) 8490–8512. [3] L. Zha, J.A. Fang, J. Liu, E. Tian, Reliable control for hybrid-driven T-S fuzzy systems with actuator faults and probabilistic nonlinear perturbations, J. Franklin Inst. B 354 (2017) 3267–3288. [4] A.Y. Lu, G.H. Yang, Secure state estimation for cyber-physical systems under sparse sensor attacks via a switched luenberger observer, Inform. Sci. 417 (2017) 454–464. [5] L. An, G.H. Yang, Secure state estimation against sparse sensor attacks with adaptive switching mechanism, IEEE Trans. Automat. Control 63 (2018) 2596–2603. [6] L. An, G.H. Yang, Decentralized adaptive fuzzy secure control for nonlinear uncertain interconnected systems against intermittent dos attacks, IEEE Trans. Cybern. (2018) http://dx.doi.org/10.1109/TCYB.2017.2787740. [7] L. An, G.H. Yang, L. An, G.H. Yang, Improved adaptive resilient control against sensor and actuator attacks, Inform. Sci. 423 (2017) 145–156. [8] D. Wang, Z. Wang, B. Shen, F.E. Alsaadi, T. Hayat, Recent advances on filtering and control for cyber-physical systems under security and resource constraints, J. Franklin Inst. B 353 (2016) 2451–2466. [9] J.Y. Keller, K. Chabir, D. Sauter, Input reconstruction for networked control systems subject to deception attacks and data losses on control signals, Internat. J. Systems Sci. 47 (2016) 814–820. [10] S. Amin, S.S. Sastry, Safe and secure networked control systems under denial-of-service attacks, in: International Conference on Hybrid Systems: Computation and Control, pp. 31–45. [11] P. Cheng, L. Shi, B. Sinopoli, Guest editorial special issue on secure control of cyber-physical systems, IEEE Trans. Control Netw. Syst. 4 (2017) 1–3. [12] H.S. Foroush, S. Martinez, On triggering control of single-input linear systems under pulse-width modulated dos signals, SIAM J. Control Optim. 54 (2016) 3084–3105. [13] D. Ding, Z. Wang, Q.L. Han, G. Wei, Security control for discrete-time stochastic nonlinear systems subject to deception attacks, IEEE Trans. Syst. Man Cybern. Syst. 48 (2018) 779–789. [14] A.Y. Lu, G.H. Yang, Input-to-state stabilizing control for cyber-physical systems with multiple transmission channels under denial-of-service, IEEE Trans. Automat. Control 63 (2018) 1813–1820.

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