Secure Synchronization Control for a Class of Cyber-Physical Systems With Unknown Dynamics

I. Introduction

CYBER-PHYSICAL systems (CPSs) combine physical processes, computational resources, with communication capability [1], which have been studied intensively for their various fields of applications, such as power grid systems in [2], multi-agent systems in [3]–[6] and smart networked systems in [7], [8]. However, CPSs embedded networked control technique are more vulnerable on account of cyber attacks. Therefore, the security of CPSs has recently attracted extensive concern in control field, such as secure state estimation under sparse attacks in [9], [10], secure control and distributed filtering under deception attacks in [11], [12], the resilient control for systems under replay attacks in [13] and the effects of denial-of-service (DoS) attacks on systems in [14]–[19].

DoS attacks, a kind of common attacks in CPSs, attempt to render some or all components of a control system inaccessible by preventing the information transmission. Some existing studies referred to DoS attacks have been published. For example, the stability analysis for a networked system in the presence of DoS attacks, which is on assumptions about the limited DoS attacks frequency and duration, is given in [14]. In [15]–[19], some resilient control methods for CPSs under DoS attacks, which is used to maximize frequency and duration of DoS attacks under the condition that the close-loop stability is not destroyed [15], are introduced. In order to improve the resilience ability of a system under DoS attacks, the state estimation problem is proposed in [20]–[22]. While, the above results are all for a single system.

Recently, based on the studies for a single system, the secure synchronization problem of complex networked systems, which results from that the synchronization behavior relying on the information of other nodes will be affected when DoS attacks successfully break down the information transmission channels, has triggered considerable attentions [23]–[26]. For example, for complex dynamical networks (CDNs) under recoverable attacks destroying the network topology, the authors in [24] introduced the attack frequency and the average recovering time to study the secure synchronization problem. In the presence of DoS attacks interrupting the communication channels of each agent in [25], [26], some sufficient conditions about achieving secure consensus of multi-agent systems are derived by using the Lyapunov function approach. However, it should be pointed that the exact system dynamic matrices are required to be known to analyze the stability of a networked system in [14]–[22] or to solve the secure synchronization control problem in [23]–[26].

In fact, in the absence of DoS attacks, the synchronization problem for the complex networked systems with unknown dynamics has been investigated. To solve optimal synchronization problems of multi-agent systems, a Q-learning method and a model-free off-policy reinforcement learning approach are developed in [27], [28], respectively. Besides, for CDNs of which the couplings are involved in node dynamic equations, the optimal synchronization control problem has been explored in our previous works [29], [30]. Nevertheless, the above results [27]–[30] about the complex networked systems with unknown dynamics were all based on the assumption that the networks were attack free. For the case that the CDNs are with unknown dynamics, how to design the secure synchronization control strategy and further to characterize the frequency and duration of DoS attacks under which the stability of synchronization errors is not destroyed, are challenging. Therefore, it is highly desired to study those problems, which motivates the current research.

In this paper, the secure synchronization control problem is considered for a class of CDNs with unknown dynamics in the presence of intermittent DoS attacks. The main contributions of this paper are summarized as follows.

1) For the attack cases, the stability analysis of synchronization errors cannot be directly carried out by using the existing Lyapunov function approach due to the presence of unknown system matrices. To overcome this difficulty, a matrix polynomial replacement method is proposed in the paper. More specifically, the unknown system matrices involved in the Lyapunov function are replaced with the known control gain matrices derived by iteratively solving an algebraic Riccati equation (ARE). And based on the replacement method, the decay rates about sleeping and active intervals of DoS attacks are further computed by solving a set of linear matrix inequalities.

2) According to the obtained decay rates and using the switching technique, the upper bounds of the frequency and duration of DoS attacks, under which the CDNs with unknown dynamics achieve secure synchronization, are given in terms of inequalities.

This paper is organized as follows. In Section II, some necessary concepts about the graph theory and the attack model are presented. An optimal synchronization control protocol is proposed in Section III. Stability analysis of secure synchronization is given in Section IV. In Section V, two examples of the CDNs are provided. Conclusion is given in Section VI.

Notations: In this paper, $\| \cdot \|$ is used to represent the Euclidean norm for vectors or matrices. For a matrix $P$, define $P^T$ as its transpose and $He(P) = P+P^T$. For a symmetric matrix $P$, $P>0$ means that $P$ is a positive definite matrix. Besides, given a real symmetric matrix $P$, $\lambda_{\min}(P)$ and $\lambda_{\max}(P)$ denote the smallest eigenvalue and the largest eigenvalue of $P$, respectively. $P = [p_{ij}]_{n\times n}$ denotes a matrix $P\in{\mathbb{R}}^{n\times n}$ with the $i$th row and $j$th column being $p_{ij}$. Define $vec(A)$ as an $mn$-vector for $A\in{\mathbb{R}}^{m\times n}$, i.e., $vec(A) = [\alpha_1^T,\alpha_2^T,\ldots,\alpha_n^T]$, where $\alpha_i\in{\mathbb{R}}^n$ is the $i$th column of $A$. Assume two sets $W$ and $N$ and $W\backslash N$ means that the elements belong to $W$, but not to $N$. Finally, let $\otimes$ represent the Kronecker product.

II. Preliminaries and Problem Formulation

A Graph Theory

A directed graph (digraph) ${\cal{G}} = (V, E)$ contains a set $V = 1, 2, \ldots, N$ of vertices and a set $E = 1, 2, \ldots, M$ of arcs $(i,j)$ leading from the initial vertex $i$ to the terminal vertex $j$. Each arc $(i, j) \in E$ is associated with a real-valued weight with $l_{ij} > 0$, while the $i$th node and the $j$th node have no connection with $l_{ij} = 0$. Assumed that there is no self loop in the graph. Furthermore, define the Laplacian matrix of a digraph as

More detailed concepts on the digraph were shown in [31].

B System Description

Consider the CDNs model with $N$ linear nodes given as

where $x_i\in{\mathbb R}^n$ and $u_i\in{\mathbb R}^m$ are the state and the controlled input, respectively. $A\in{\mathbb R}^{n\times n}$ is an unknown matrix and $B_i\in{\mathbb R}^{n\times n}$ is the constant invertible matrix. $l_{ij}$ are known coupling weights, and $\Gamma_i\in{\mathbb R}^m$ is the known inner connecting matrix of the $i$th node.

Assume the dynamic equation of the leader has the following form

where $z_0\in{\mathbb R}^n$ is the leader’s state. The leader can be regarded as a command generator, which generates the desired target trajectory.

Assumption 1: $(A,B_i)$ is controllable.

C Denial-of-Service Attack

In practice, the state information $x_i$ measured by using sensors is sent to controllers via networked channels. As a kind of common attacks in CPSs, DoS attacks compromise certain or all transmission channels of a control system (see Fig. 1), which leads to the unavailability of transmitted information.

Figure  1.  Framework of the CDNs under DoS attacks.

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Due to the energy constraint of adversaries [32], [33], DoS attacks need to terminate attack activities and shift to a sleep period to supply its energy for next attack. That is to say, one adversary launches DoS attacks discontinuously. As mentioned in [14] and [19], $\{z_s\}_{s\in{{{\mathbb{N}}}^+}}$ represents the time instant when the $s$th DoS attack is active. Then

means the time interval of the $s$th DoS attack with a length $d_s \in {\mathbb{R}}^+$. Thus, for given $t>\tau$, define that

which denote, during each interval $[\tau, t]$, the sets of time instants when the communication is denied or allowed, respectively. Moreover, $|\Xi(\tau,t)|$ and $|\Lambda(\tau,t)|$ represent the total lengths of DoS attacks being active and sleeping during the interval $[\tau, t]$.

Further, let $\{t_k\}_{k\in{{\mathbb{R}}_0}}$ with $t_0 = 0$ represent the sequence of time instants at which communication is attempted. Define the finite sampling rate as

for $k\in{\mathbb{N}}$. Thus, the state information received by controllers is considered as

with $i = 0,1,2,\ldots,N$. And the information received by actuators is given as

Similar to [14], there are two assumptions for DoS attacks.

Assumption 2 (DoS frequency): There exist constants $\eta\geq0$ and $\tau_D>0$ such that for all $t,\tau \in {\mathbb{R}}^+$

Assumption 3 (DoS Duration): There exists constant $T\geq1$ such that for all $t,\tau \in {\mathbb{R}}^+$

Assumption 2 is inspired by the concept of average dwell time [14] to specify the number of DoS attacks occurring on the interval $[\tau,t]$ and Assumption 3 is used to describe the length of the interval over which communication is interrupted. From [14] and [16], it is known that those two assumptions are common for reflecting the constraint energy of DoS attacks.

Now, the considered problem in this paper is presented as follows.

Problem 1: The main objective of this paper is to design a secure synchronization control law for the system (1) with unknown system matrices, such that the synchronization errors are asymptotically convergent for both attack-free and attack cases.

III. The Optimal Synchronization Controller Design Without Attacks

A The Optimal Synchronization Control Law

By using (1) and (2), define the synchronization error $e_i$ for the $i$th node as

Then the synchronization problem is described as

The dynamic equation of the synchronization error $e_i$ has the following form

where $\bar{A}_i = A-\sum_{j = 1,j\neq i}^Nl_{ij}\Gamma_i$, $\omega_i = \sum_{j = 1,j\neq i}^Nl_{ij}\Gamma_ie_j$ and $F_i$ satisfies $B_iF_i = I$.

Besides, similar to [30], define the performance function for the $i$th node (11), which involves the couplings information

where $Q_i\in \mathbb R^{n\times n}$ and $R_i\in{\mathbb R}^{n\times n}$ are symmetric positive weight matrices.

Theorem 1: For the error dynamics of CDNs (11) with the quadratic performance index (12), a control protocol is an optimal one if and only if it has the following form

where $P_i$ is a symmetric positive matrix and satisfies the following ARE

Proof: First, one gives the necessity proof of Theorem 1.

Based on (12), define the Hamiltonian function for (11) as

where $\lambda_i(t)$ is a multiplier to be determined. The necessary condition $\dfrac{\partial H_i}{\partial u_i} = 0$ on which a control protocol of system (11) is an optimal one results in

Since $\dfrac{{\partial}^2 H_i}{\partial u_i^2} = R_i>0$, the optimal control has the form of (16). Assume that $e_i$ and $\lambda_i$ satisfy the following linear equation [34]

where $P_i$ is obtained by solving the ARE (14). According to (17) and the canonical equation shown in [29], one gets the ARE

Using (16) and (17), it yields the control law (13). Therefore, the necessity is proven.

The proof process of sufficiency for Theorem 1, which is divided into two steps, is given as follows.

Step 1: It is necessary to prove that the control law (13) ensures the asymptotical stability of the synchronization errors. That is to say

Define the Lyapunov function of system (11) as $V(t) = e_i^TP_ie_i$, and its derivative is

According to ARE (14), (19) is rewritten as

Since $Q_i\geq0$, $R_i>0$, and $Q_i+P_iB_{i}R_i^{-1}B_{i}^TP_i>0$ for all states $e_i$, $\dot{V}_i(t)<0$, and the control law (13) ensures that the synchronization errors (9) asymptotically converge to zero.

Step 2: Consider the following equation:

Based on the ARE (14) and the condition (18), the performance index $J_i$ is rewritten as

with $u_i^* = -F_i\sum_{j = 1,j\neq i}^Nl_{ij}\Gamma_ie_j-R_i^{-1}B_{i}^TP_ie_i$. Due to $R_i>0$ and $X_i^T(t_0)P_iX_i(t_0)$ being a constant, the minimum value ${J_i}^* = \dfrac{1}{2}e_i^T(t_0)P_ie_i(t_0)$ of the performance function (22) can be obtained if and only if $u_i = {u_i}^*$.■

The optimal control law (13) is composed of the feedback control $-R_i^{-1}B_{i}^TP_ie_i$ and feedforward control $-F_i\sum\nolimits_{j = 1,j\neq i}^Nl_{ij}\Gamma_ie_j$ compensating the couplings in system dynamics. The details are also shown in [30].

B The Optimal Synchronization Controller Design With Unknown Dynamics

Consider the optimal controller with the following form

where

When all system dynamics are accurately known, the feedback gain $K_i$ is determined by solving the ARE (14). Since (14) is nonlinear in $P_i$, the solution $P_i$ cannot be obtained from (14) directly. To overcome this difficulty, an algorithm has been developed in [35] by solving (25) iteratively

with $K_i^{k+1} = R_i^{-1}B_i^TP_i^k$, where $K_i^0$ is a given feedback gain matrix stabilizing system (11) and $k$ represents the number of iterations.

However, when the system matrix $A$ is unavailable which means that the system matrices $\bar{A}_i$ are unknown, the above method is invalid. To obtain the feedback gain $K_i$, according to [29] and [30], an online iterative policy relying on the information of state, input and couplings is given as follows.

First, for a given stabilizing controller gain matrix $K_i^0$, the input signal for learning is chosen as $u_i = -K_i^0-F_i\omega_i+e$ with $e$ being the exploration noise. Based on (25) and $K_i^{k+1} = R_i^{-1}B_i^TP_i^k$, one has

with $\bar{A}_i^k = \bar{A}_i-B_iK_i^k$ and $Q_i^k = Q_i+{(K_i^k)}^TR_iK_i^k$. Note that, relying on (26), the requirement of the system matrix in (25) is replaced by the information of states $x_i$, inputs $u_i$ and couplings $F_i\omega_i$ measured online.

Further, for all $t = t_j,\; j = 1,2,\ldots,l$, satisfying $0\leq t_0\leq t_1\leq$$\cdots \leq t_l$ and $\delta t = t_l-t_0 = t_2-t_1 = \cdots = t_l-t_{l-1}$, (26) holds. Define matrices $\delta_{ixx}\in{\mathbb{R}}^{l\times\frac{1}{2}n(n+1)}$, $I_{ixx}\in{\mathbb{R}}^{l\times{n}^2}$, $I_{ixu}\in{\mathbb{R}}^{l\times n^2}$ and $I_{ixw}\in{\mathbb{R}}^{l\times n^2}$, so that

where $\hat{e}_i$ has been defined in [29]. For a matrix $P = [p_{ij}]_{n\times n}$, an operator is defined as

According to (26)–(28), one has the following equation:

where $\Theta_i^k\in{{\mathbb{R}}}^{l\times[\frac{1}{2}n(n+1)+n^2]}$ and $\Xi_i^k\in{{\mathbb{R}}}^{l}$ are given as

Furthermore, if $\Theta_i^k$ has full column rank ([36], Lemma 1), the unknown matrices $P_i^k$ and $K_i^{k+1}$ can be solved together by the least squares method as follows:

And the sequences of $P_i^k$ and $K_i^{k+1}$ have been proven to be convergent (that is, $\lim_{k\rightarrow\infty}K_i^{k} = K_i^*$, $\lim_{k\rightarrow\infty}P_i^{k-1} = P_i^*$) in [29].

Remark 1: The feedback gain $K_i$ is obtained under the attack free case, and the proposed iterative policy method requires that all the information of states $x_i$, control inputs $u_i$ and couplings $\omega_i$ are available.

IV. Synchronization Errors of Stability Analysis With Intermittent DoS Attacks

As shown in [14], there may be a time interval elapsing from the time $z_s+d_s$ to the time $z_s+d_s+v_s$ at which the state information is successfully sampled and transmitted. And the time interval $v_s$ (shown in Fig. 2) satisfies

Figure  2.  An example of DoS attacks.

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with $\Delta_*$ representing the upper bound on the inter-sampling rate in [14]. Thus the time interval $[\tau,t)$ is composed of the following two sub-intervals:

where $V_s = [z_s,z_s+d_s+v_s)$. Specially, for all $\tau,t \in {\mathbb{R}}_{\geq0}$,

Next, by using the Lyapunov function method and switching technique, the stability of the error systems (11) under DoS attacks will be discussed.

Note that, to analyze the stability of synchronization errors, the system dynamic matrices are required to be known in [23]–[26]. However, such a requirement cannot be satisfied in this paper, and then these existing methods are no longer applicable. To overcome this difficulty, a matrix polynomial replacement method is proposed based on the ARE (37) and the iterative learning results $P_{ia}$ and $K_i$. The details are shown as follows.

Divide the process of stability analysis into two steps.

Step 1: Consider the time interval $\tilde{\Lambda}(\tau,t)$ and define the Lyapunov function of the $i$th node

According to the ARE (14) and the condition $K_i = R_i^{-1}B_i^T\times P_{ia}$, one gets

Define $\varphi(\bar{A}_i,P_{ia}) = P_{ia}\bar{A}_{i}+\bar{A}_{i}^TP_{ia}-2P_{ia}B_{i}R_i^{-1}B_{i}^TP_{ia}$ and $\psi(K_i) = -(Q_i+K_i^TR_iK_i)$. Then the derivative of the Lyapunov function (36) is

By resorting to ARE (37) and the iterative learning results $P_{ia}$ and $K_i$ in (37), one has

As clarified in Theorem 1, the error systems (11) are asymptotically stable during the time interval $\tilde{\Lambda}(\tau,t)$. Thus suppose

with $\alpha_i>0$. That is to say,

Therefore, the convergence rate $\alpha_i$ of the system (11) can be computed by (41). Before applying the optimal control gain $K_i$ to system (1), the matrix $P_{ia}$ and the feedback gain matrix $K_i$ in ARE (37) have been obtained by iteratively solving (30).

Step 2: Consider the time interval $\tilde{\Xi}(\tau,t)$. The state information received by controllers is $x_i = 0$, i.e. $e_i = 0$. Thus controllers are out of action, i.e., $u_i = 0$. Define the Lyapunov function as

where its derivative satisfies

with the exponential divergence rates $\beta_i>0$. That is

Choose $P_{ib} = I_{n}$, and (44) becomes $\bar{A}_i^T+\bar{A}_i<\beta_i I_{nN}$. From (11), it is known that

Assumption 4: Assume $\lambda_{\max}(He(\bar{A}_i))$ is available.

Based on Assumption 4, one has

which yields the following inequality

Combining the above analysis results (41) and (47) of the piecewise Lyapunov functions (36) and (42), respectively, the stability of closed-loop system, which switches between stable and unstable subsystems, is established by using the switching technique. The conditions on the frequency and duration of DoS attacks, under which the secure synchronization can be guaranteed, are shown in Theorem 2.

Theorem 2: For system (11), suppose Assumptions 1–4 hold. Given any positive definite symmetric matrices $R_i$ and $Q_i$ with $i = 1,2,\ldots, N$, if there exist positive constants $\theta_i$ such that the following two inequalities hold

where $\alpha_i$ and $\beta_i$ satisfy (41) and (47), respectively, and $\mu_i = \max\{\lambda_{\max}(P_{ia})/\lambda_{\min}(P_{ib}), \lambda_{\max}(P_{ib})/\lambda_{\min}(P_{ia})\}$. Then, the optimal control laws (13) ensure that the synchronization errors are convergent under DoS attacks.

Proof: Consider any time interval $[z_s, z_{s+1})$ that $V_{ib}$ is activated in $[z_s, z_s+d_{s}+v_s)$ and $V_{ia}$ is activated in $[z_s+d_s+v_s, z_{s+1})$. By comparison lemma in [37], one gives

Similar to [26], there are the following two cases:

Case 1: If $t\in[z_s, z_s+d_{s})$, $n(t_0,t) = s$,

Case 2: If $t\in[z_s+d_s+v_s, z_{s+1})$, $n(t_0,t) = s$,

Thus, for $\forall t\geq t_0$,

According to Assumptions 2 and 3, and inequalities (34) and (35), it yields that

Let $\xi_i = {\mu}_i^{2\eta_i}e^{(\alpha_i+\beta_i)(1+\eta_i)\Delta_*}$ and $\delta_i = \alpha_i-\frac{2ln\mu_i+(\alpha_i+\beta_i)\Delta_*}{\tau_{D}}-\frac{\alpha_i+\beta_i}{T}$. Then, one has

From (48) and (49), it is known that $\delta_i>0$. Thus the synchronization errors $e_i$, $i = 1,2,\ldots,N$, are convergent exponentially, which implies that $\lim_{t\rightarrow\infty}e_i(t) = 0$.■

Remark 2: DoS attacks compromise certain or all transmission channels, which is also introduced in [15], [16], [19] and [26]. The case that all transmission channels are interrupted when DoS attacks occur is considered in Theorem 2. This case can also be found in [23] and [26].

Remark 3: The stability analysis of synchronization errors under DoS attacks in [23]–[26] is based on the Lyapunov function approach and the exact system dynamic matrices are required to be known. While, the system matrices are not available in this paper. To overcome this difficulty, the matrix polynomial $\varphi(\bar{A}_i,P_{ia})$ including the unknown matrices $\bar{A}_i$ and $P_{ia}$ is replaced by $\psi(K_i)$, which includes the known matrices $Q_i$, $R_i$ and the control gain matrix $K_i$ obtained by the iterative learning method.

V. Simulation Example

In this section, two complex dynamical network examples are taken to verify the effectiveness of above methods.

Example 1: Similar to [7], consider a spring-connected multi-vehicle system (shown in Fig. 3) composed of an isolated leader and four slave vehicles. The springs are regarded as the interconnection between two vehicles. In fact, the $i$th vehicle is modeled as

Figure  3.  Spring-connected multi-vehicle system with four slave vehicles.

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where $d_i$ is the position of $i$th vehicle from its equilibrium position $(i = 1,2,3,4)$, $m_i$ represents the weight of $i$th vehicle and $k$ means the stiffness coefficient of spring. The information of $d_i$ and the velocity $\dot{d}_i$ is transmitted to controllers through communication channels. From (56), the slave vehicle dynamics can be governed as

where $x_i = [d_i,\dot{d}_i]^T$,

Here, assume $m_1 = 1$, $m_2 = 2.5$, $m_3 = 1.5$, $m_4 = 0.5$ and $k = 1$. Moreover, the dynamic equation of leader is described as

The simulation process is divided into two steps:

Step 1: Solving process of the solutions ${K_i}$ and ${P}_{ia}$ satisfying (37).

Suppose the initial values $e_1 = [-9,-9]^T$, $e_2 = [-8,-8]^T$, $e_3 = [-7,-7]^T$ and $e_4 = [-6,-6]^T$. Then choose the weighting matrices ${Q_1} = 10I_2$, ${Q_2} = 20I_2$, ${Q_3} = 15I_2$, ${Q_4} = 18I_2$ and ${R_i} = I_2$ for $i = 1,2,3,4$. The initial feedback gain is assumed as

with $i = 1,2,3,4$. The exploration noise $e$, introduced in [36], is the sum of sinusoidal signals with different frequencies from $t = 0\;\rm s$ to $t = 4\;\rm s$ and it has the following form:

where $\omega_l$, with $l = 1,\ldots,50$, is randomly selected from $[-50,50]$. Besides, the internal $\sigma_k = 0.01\rm s$ is used to solve the optimal control gain ${K_i}$ based on policy iteration with the information of states $x_i$, inputs $u_i$ and couplings $\omega_i$. The matrices ${P}_{ia}$ and ${K_i}$, with $i = 1,2,3,4$, converge to their optimal values after 5 iterations, 4 iterations, 4 iterations, 4 iterations, respectively. Similar to [29], $P^k_{ia}$ and $K^k_i$ respectively converge to the corresponding optimal values in (59) and (60) when the stopping criterion $\parallel P_a^k-P_a^{k-1} \parallel\;\leq 0.005$ is met.

After applying the optimal control law (23) with obtained gains $K_{i}$ to this spring-connected multi-vehicle system (56), the state synchronization errors without any attacks are shown in Fig. 4. Clearly, the state synchronization errors of vehicles 1 – 4 are asymptotically convergent.

Figure  4.  Synchronization errors of vehicles 1–4 for the attack free case.

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Step 2: Stability analysis of secure synchronization.

Based on the analysis results (41) and (47), one gets

Assume the adversary launches DoS attacks with $\tau_D = 2$ and $T = 1.24$. Here $\Delta_*\geq\Delta_k = 0.001$. By computing (48) and (49), it is obtained that

From (62), it is known that the conditions in (48) and (49) of Theorem 2 are satisfied. Further, the synchronization errors with DoS attacks are plotted in Fig. 5 and it is known that the states of the spring-connected multi-vehicle system with unknown dynamics can track the leader’s if the DoS frequency and DoS duration satisfy $n(0,20)\leq\eta+10$ and $\mid\tilde{\Xi}(0,20)\mid \;\leq\;\;$16.1, respectively. While, the system performance shown in Fig. 5 degrades due to the effects of the DoS attacks. Besides, the control laws $u_i$, which are larger after each DoS attack ends, are plotted in Fig. 6.

Figure  5.  Synchronization errors of vehicles 1–4 with unknown system matrix $A$ for the DoS attacks case.

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Figure  6.  Control laws of vehicles 1–4 with unknown system matrix $A$ for the DoS attacks case.

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Moreover, under the assumption of availability on the system matrices $\bar{A}_i$, the solutions $P_{ia}^*$ of ARE (37) and the optimal controller gains $K_i^*$ are directly computed by (37) and (24). In particular, they are the same as the optimal values $P_{ia}$ and $K_{i}$ in (59) and (60), respectively ($i = 1,2,3,4$). Then, one gets the exponential convergence rates $\alpha_i^* = \alpha_i$ by solving (41) directly. Due to (44) is nonlinear in $P_{ib}$ and $\beta_i$, we search for solutions of (44) with $\beta_i^{k} = \beta_i^{k-1}+0.001$ ($\beta_i^0 = 0$) where $k$ represents the number of searches. Finally, the exponential divergence rates $\beta_i^* = 0.001$ and the following matrices $P_{ib}^*$ are obtained

According to $P_{ia}$ and $P_{ib}$, one has $\mu_i = 5.0233$ for $i = 1,2,3,4$. Assume the adversary launches DoS attacks with $\tau_D = 2$ and $T = 1.156$. By computing (48) and (49), it is obtained that

Obviously, the conditions in (48) and (49) of Theorem 2 are satisfied. The synchronization errors under DoS attacks with frequency satisfying $n(0,20)\leq\eta+10$ and the duration satisfying $\mid \tilde{\Xi}(0,20)\mid\; \leq17.3$ are plotted in Fig. 7. Besides, the corresponding control laws $u_i$ are plotted in Fig. 8. As shown in Figs. 7 and 8, the system performance degrades due to the effects of the DoS attacks. Further, after each DoS attack ends, much control effort is needed to drive the synchronization errors to be convergent.

Figure  7.  Synchronization errors of vehicles 1–4 with known system matrix $A$ for DoS attacks case.

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Figure  8.  Control laws of vehicles 1–4 with known system matrix $A$ for DoS attacks case.

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As shown above, the upper bounds of the DoS attack frequency and the DoS duration which are solved by the proposed method in this paper are slightly smaller than those obtained by the model based method.

To further prove the effectiveness of the proposed approaches in this paper, Example 2 with more complex graph structure is given as follows.

Example 2: Consider CDNs are with five nodes, and the correlative graph is shown in Fig. 9. The leader node’s dynamic is described as

Figure  9.  Graph structure for the CDNs.

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and the system dynamics of five nodes are shown as

with $B = I_2$, $\Gamma_1 = 0.9I_2$, $\Gamma_2 = 0.7I_2$, $\Gamma_3 = 0.9I_2$, $\Gamma_4 = I_2$, $\Gamma_5 = 2I_2$. Suppose the initial values $e_1 = [-6,-6]^T$, $e_2 = [-5,-5]^T$, $e_3 = [-4,-4]^T$, $e_4 = [-2,-6]^T$ and $e_5 = [-5,-3]^T$. Then choose the weighting matrices ${Q_5} = 12I_2$.

Similar to Example 1, the simulation process is divided into two steps.

Step 1: Use the policy iteration method to obtain the solutions $P_{ia}$ and $K_i$. The initial controller gain is chosen as

Besides, the other parameters are assumed to be the same as those in Example 1. By utilizing the PI method, the optimal solutions $P_{ia}$ and $K_i$ with $i = 1,2,3,4,5$ are obtained after 4 iterations. They are given as

Step 2: Stability analysis of secure synchronization

According to the analysis results (41) and (47), suppose that the DoS attacks with $\tau_D = 2$ and $T = 1.2$ occur during time interval $(0,20)$, which satisfying the conditions in (48) and (49) in Theorem 2.

The synchronization errors under the DoS attacks are given in Fig. 10. Besides, the corresponding control laws $u_i$ are plotted in Fig. 11. As shown in Fig. 10, the synchronization errors become large when the system is under DoS attacks. Finally, the system with unknown dynamics can track the leader, if the DoS frequency satisfies $n(0,20)\leq\eta+10$ and the DoS duration satisfies $\mid \tilde{\Xi}(0,20)\mid \;\leq16.6$.

Figure  10.  Synchronization errors of nodes 1–5 with unknown system matrix A for the DoS attacks case.

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Figure  11.  Control laws of nodes 1–5 with unknown system matrix A for the DoS attacks case.

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VI. Conclusion

This paper investigates the secure synchronization control problem for CPSs subject to intermittent DoS attacks. The considered CPSs are modeled as a class of CDNs with unknown dynamics. First, to deal with the state couplings, a distributed optimal controller is proposed based on our previous works [29]. And the optimal feedback gain matrix is learned by iteratively solving the ARE. Especially, based on the ARE and the iteratively learning results, the decay rates for each node about sleeping and active intervals of DoS attacks, are determined by solving a set of linear matrix inequalities. Moreover, by using the switching technique and the obtained decay rates, the upper bounds of the DoS attacks frequency and duration, under which the synchronization for all nodes is still achieved, have been proposed. Finally, the simulations of two examples verify the effectiveness of the proposed methods.