Secure Control of Uncertain Multi-agent Systems Under Cyber Attacks

Abstract

This paper considers the secure control problem for a class of uncertain nonlinear multi-agent systems. To deal with the uncertainties, the adaptive laws are designed in an iterative manner. Furthermore, the auxiliary variables and Nussbaum-type functions are introduced to handle the effects caused by the attacks. By developing a common Lyapunov function, a secure control method is designed to guarantee the asymptotical consensus of the considered systems against cyber attacks. Finally, we present an example to validate the effectiveness.

1 Introduction

In modern industrial process, physical components are connected with each other by network communication channels, such as transportation networks, UAVs, and formation of robots [1,2,3,4]. The connection by shared networks brings efficiency to control systems, but there are many vulnerabilities that could be maliciously exploited by hackers or attackers [5]. Therefor, it is paramountly important to pay more attention to the safety issue against malicious cyber attacks.

In the recent developments, denial-of-service (DoS) attacks and deception attacks are identified as two typical cyber attack and have been the subject of comprehensive research [6, 7]. To handle the complex cyber attacks, many control schemes have been developed. In [8, 9], the attack compensators and a dynamic surface-based resilient adaptive strategy were constructed to mitigate the effects of state-dependent actuator attacks. In [10], the Bernoulli distributed random model was introduced to characterize cyber attack, and a dynamic protocol was designed to guarantee the security against deception attacks. For a Lipschitz-type system, [11] used a pinning method and measured the attack through a stochastic variable to achieve the synchronization under deception attacks.

In practice, switched nonlinear multi-agent systems (SNMASs) provide an general framework for modeling the man-made process involving switching behaviors [12]. Considering the importance from both theoretical and practical points of view, many results on control synthesis for the NMASs have been proposed. In the existing results, a large amount of effort has been made to solve the consensus problem of SNMASs, such as [13,14,15]. In [16], an adaptive robust fault-tolerant consensus protocol was designed for nonlinear fractional-order SNMASs with general directed topology. In [17, 18], for MASs under fixed and switching topologies, the distributed event-triggered consensus was achieved when only the triggered information is available.

However, in the aforementioned results, the case that the control coefficients, and the constant parameters are required to be known. In addition, the time disturbances are ignored [19,20,21]. It is noted that many actual plants have uncertain natures, for example the automotive industry manufacturing process [22,23,24]. But, due to the difficulties caused by the unknown control coefficients, constant parameters, and time disturbances, the important secure control of uncertain NMASs in presence of cyber attacks has not been taken into account. This motivates the study.

2 Preliminaries

Consider the NMAS where the j-th agent is modeled as

$$\begin{aligned} \dot{\zeta }_{j,s} =\,\,&\zeta _{j,s+1} + f_{j,s}^{\sigma _j}(\zeta _{j,1},\cdots ,\zeta _{j,s},\theta _j),\ + r_{j,s}(t), \ s=1,\cdots ,n-1, \\\dot{\zeta }_{j,n} =\,\,&b_j u_j + f_{j,n}^{\sigma _j}(\zeta _j,\theta _j)+ r_{j,n}(t),\\z_i=\,\,&\zeta _{j,1}, \end{aligned}$$

(1)

where $\zeta _j=(\zeta _{j,1},\ldots ,\zeta _{j,n})^T\in R^{n}$, $z_i\in R$, $u_j\in R$, $j=1,2,\cdots ,N$, are the state, output and input of the j-th agent, respectively. $\sigma _j:[0,\infty )\rightarrow \{1, 2, \cdots , m\}$ is the switching signal. The control coefficient $b_j$, the constant parameter $\theta _j$ and the time disturbance $r_{j,s}$ are all unknown. Assume $|r_{j,s}| \le r_0$ with $r_0$ being a positive constant. The nonlinear term $f_{j,s+1}^{i}:R^{s}\rightarrow R$ with $f_{j,s}^{i}(0)=0$ is smooth. Consider the following sensor deception attack $\nu _j(\zeta _{j,s},t)$ on the data information,

$$\begin{aligned} \breve{\zeta }_{j,s} = \zeta _{j,s} + \nu _{j}(\zeta _{j,s},t), \quad s=1,2,\cdots ,n, \end{aligned}$$

(2)

where $\breve{\zeta }_{j,s}(t)$ is the obtained information of the j-th agent.

Assumption 1

The attacks $\nu _j(\zeta _{j,s},t)$, $j=1,2,\cdots ,n$, are modeled as $\nu _j(\zeta _{j,s},t)= \delta (t) \zeta _{j,s}$, where signals $\delta (t)$ are unknown. Denote $\varrho (t)=(1+ \delta (t))$ and $\zeta _{j,s}=\varrho (t)^{-1} \breve{\zeta }_{j,s}$. For $j=1,2,\cdots ,n$, there are uncertain constants $\varrho _0$, $\bar{\varrho }_1$ and $\bar{\varrho }_2$ such that $|\dot{\varrho }(t)\varrho ^{-1}(t)| \le \varrho _0$ and $0<\bar{\varrho }_1\le |\varrho (t)| \le \bar{\varrho }_2$.

Lemma 1

[25, 26]. For the vectors $\zeta$, y with suitable dimension, and the real-valued continuous function $f(y,\zeta )$, there are smooth functions $\psi (y)$ and $\bar{f}(\zeta )$ such that $|f(\zeta , y)| \le \psi (y)\bar{f}(\zeta )$.

By Lemma 1, there exist functions $\psi _{j,s}(\breve{\zeta }_{j,1},\cdots ,\breve{\zeta }_{j,s})$ and constant $\varpi$ such that

$$\begin{aligned} |f_{j,s}^i( \zeta _{j,1},\cdots , \zeta _{j,s}, \theta _j)| \le \varpi \psi _{j,s}(\breve{\zeta }_{j,1},\cdots ,\breve{\zeta }_{j,s}). \end{aligned}$$

(3)

3 Main Result

3.1 Design of Consensus Protocol

Step 1: For $j=1,\cdots ,N$, introduce the auxiliary variables as

$$\begin{aligned} \dot{\eta }_{j}&=-\sum _{k=1}^Nb_{j,k}(\breve{z}_i - \breve{z}_k), \\z_{j,1}&=\breve{\zeta }_{j,1} - \eta _{j}, \end{aligned}$$

(4)

where $\breve{z}_i = \breve{\zeta }_{j,1}$, $\eta _j(0)=\breve{\zeta }_j(0)$. Choose the Lyapunov function candidate as

$$\begin{aligned} V_{1} = \frac{1}{2}\sum _{j=1}^N z_{j,1}^2 + \frac{1}{2}\tilde{\varrho }^2 + \frac{1}{2}\tilde{\vartheta }^2, \end{aligned}$$

(5)

where $\tilde{\varrho } = \varrho - \hat{\varrho }$, $\tilde{\vartheta } = \vartheta - \hat{\vartheta }$, and $\hat{\varrho }$, $\hat{\vartheta }$ are the estimation of $\varrho _0$, $\vartheta$, respectively. By taking $\vartheta =\bar{\varrho }\varpi \max _{j=1,2,\cdots ,N}{|\theta _j|}$ and by using (3), one can calculate that

$$\begin{aligned} z_{j,1}\varrho f_{j,1}^{i}(\zeta _{j,1},\theta _j )\le \vartheta \pi + \vartheta \frac{(z_{j,1} \psi _{j,1})^2}{\sqrt{(z_{j,1} \psi _{j,1})^2 + \pi ^2}}. \end{aligned}$$

(6)

Construct the virtual protocol $\breve{\zeta }_{j,2}^*$ as

$$\begin{aligned} {\breve{\zeta }}_{j,2}^{*} =-\mu _{j,1}z_{j,1} + \dot{\eta }_j - \hat{\varrho } \bar{\omega }_{\varrho ,j,1} - \hat{\vartheta }\bar{\omega }_{\theta ,j,1}, \end{aligned}$$

(7)

where $\mu _{j,1}$ a positive constant, $\bar{\omega }_{\varrho ,j,1}=\frac{z_{j,1}\breve{\zeta }_{j,1}^2}{\sqrt{(z_{j,1}\breve{\zeta }_{j,1})^2 + \pi ^2}}, \bar{\omega }_{\theta ,j,1}=\frac{z_{j,1} \psi _{j,1}^2}{\sqrt{(z_{j,1} \psi _{j,1})^2 + \pi ^2}}.$ Then, it follows from (4)–(7) that

$$\begin{aligned} \dot{V}_{1}&\le -\sum _{j=1}^N\mu _{j,1} z_{j,1}^2 +\sum _{j=1}^N z_{j,1}(\breve{\zeta }_{j,2} - \breve{\zeta }_{j,2}^*) \\&\quad - \tilde{\varrho } (\dot{\hat{\varrho }}- \omega _{\varrho ,1}) - \tilde{\vartheta }(\dot{\hat{\vartheta }} - \omega _{\theta ,1}) + c_1, \end{aligned}$$

(8)

where $\omega _{\varrho ,1} = \sum _{j=1}^N z_{j,1} \bar{\omega }_{\varrho ,j,1}$, $\omega _{\theta ,1} =\sum _{j=1}^N z_{j,1}\bar{\omega }_{\theta ,j,1}$ and $c_1$ is a positive constant.

Step s $(2\le s \le n-1)$: Construct the auxiliary variable $z_{j,s}= \breve{\zeta }_{j,s} - \breve{\zeta }_{j,s}^*$ with $\breve{\zeta }_{j,s}^*$ being the virtual protocol. By (1), the following inequality holds,

$$\begin{aligned} \dot{z}_{j,s}= \dot{\varrho }\varrho ^{-1} \breve{\zeta }_{j,s} + \breve{\zeta }_{j,s+1} + \varrho f_{j,s}^i(\zeta _{j,1},\cdots ,\zeta _{j,s},\theta _j) - \dot{\breve{\zeta }}_{j,s}^*. \end{aligned}$$

(9)

Choose the Lyapunov function candidate as

$$\begin{aligned} V_s = V_{s-1} + \frac{1}{2}\sum _{j=1}^N z_{j,s}^2. \end{aligned}$$

(10)

Similar to step 1, one can design the virtual protocol $\breve{\zeta }_{j,s+1}^*$ such that

$$\begin{aligned} \dot{V}_{s}&\le -\sum _{j=1}^N\mu _{j,1} (z_{j,1}^2 + \cdots + z_{j,s}^2) +\sum _{j=1}^N z_{j,s}z_{j,s+1} \\&\quad - \tilde{\varrho } (\dot{\hat{\varrho }} - \omega _{\varrho ,s}) - \tilde{\vartheta } (\dot{\hat{\vartheta }} - \omega _{\theta ,s}) - \sum _{j=1}^N \sum _{l=2}^{s}z_{j,l}\frac{\partial \breve{\zeta }_{j,l}^*}{\partial \hat{\varrho }} (\dot{\hat{\varrho }}- \omega _{\varrho ,s}) \\&\quad - \sum _{j=1}^N \sum _{l=2}^{s}z_{j,l}\frac{\partial \breve{\zeta }_{j,l}^*}{\partial \hat{\vartheta }} (\dot{\hat{\vartheta }} - \omega _{\theta ,s}) + c_s, \end{aligned}$$

(11)

where $c_s$ is some a positive constant, $\omega _{\theta ,s}$ and $\omega _{\varrho ,s}$ are some functions.

Step n: At the finial step, the Lyapunov function is constructed as

$$\begin{aligned} V_{n} =V_{n-1} + \sum _{j=1}^N \frac{1}{2}z_{j,n}^2. \end{aligned}$$

(12)

Design the control protocol as

$$\begin{aligned} u_j= - \mathcal {C}_j(\xi _j) \breve{\zeta }_{j,n+1}^*, \ \ \dot{\xi }_j = \breve{\zeta }_{j,n+1}^*z_{j,n}, \end{aligned}$$

(13)

and the adaptive laws as

$$\begin{aligned} \dot{\hat{\varrho }}&= \omega _{\varrho ,n}, \ \ \dot{\hat{\vartheta }} = \omega _{\theta ,n}, \end{aligned}$$

(14)

where $\mathcal {C}_j(\xi _j) = \cosh (g_1\xi _j)\sin \big (\frac{\xi _j}{g_2^j}\big ), j=1,2,\cdots ,N,$ with $g_1>0$ and $g_2>0$, and the structures of $\breve{\zeta }_{j,n+1}^*$, $\omega _{\varrho ,n}$, $\omega _{\theta ,n}$ are same as that in previous steps. Then, after calculation, we can obtain that

$$\begin{aligned} \dot{V}_{n}&\le -\sum _{j=1}^N\mu _{j,1} (z_{j,1}^2 + \cdots + z_{j,n}^2) + \sum _{j=1}^N (\varrho b_j \mathcal {C}_j(\xi _j) -1 )\dot{\xi }_j + c_n, \end{aligned}$$

(15)

where $c_n>0$ is a constant.

3.2 Consensus Analysis

Theorem 1

For the SNMASs (1), consider the cyber attacks (2) with Assumption 1. The consensus protocol (13) and the adaptive laws (14), guarantee the asymptotical consensus of all outputs under arbitrary switching.

Proof: Integrating both sides of (15), it is deduced that

$$\begin{aligned} V_{n}(t)&\le \int _0^t\sum _{j=1}^N (\varrho b_j \mathcal {C}_j(\xi _j) -1 )\dot{\xi }_j \mathrm{d} \omega + c, \end{aligned}$$

(16)

where $c>0$ is a constant. By (16) and Barbalat’s Lemma, we have $\lim _{t\rightarrow \infty }z_{j,s}(t)=0$. Denote $\eta =(\eta _1,\cdots ,\eta _N)^T$, $z_1=(z_{1,1},\cdots ,z_{N,1})$, $\zeta _1=(\zeta _{1,1},\cdots ,\zeta _{N,1})^T$ and $\varsigma =P^{-1} \eta$. Then, we obtain that

$$\begin{aligned} \dot{\varsigma } = - J \varsigma - JP^{-1}z_1, \end{aligned}$$

(17)

where $P=[1_N;v_2;\cdots ;v_N]$ and $L_A= P J P^{-1}$. $J=\mathrm {diag}\{0,J_1\}$ is the Jordan canonical form. Let $\bar{\varsigma }= (\varsigma _2,\cdots ,\varsigma _N)$. By using Barbalat’s Lemma, it follows from (17) that $\lim _{t\rightarrow \infty }|\bar{\varsigma }|=0$. Furthermore, we can deduce that

$$\begin{aligned} \lim _{t\rightarrow \infty } (y_i(t)- z_i(t))=0, \end{aligned}$$

(18)

which means that the output of agents reach consensus asymptotically under arbitrary switching.

4 An Illustrative Example

To illustrate the effectiveness, this section presents a numerical simulation. Consider the following uncertain SNMAS whose communication graph is shown in Fig. 1.

$$\begin{aligned} \dot{\zeta }_{j,1} =\,\,&b_j u_j + f_{j,1}^{\sigma _j}(\zeta _{j,1},\theta _j) + r_{j,1}(t),\\z_i=\,\,&\zeta _{j,1}, \end{aligned}$$

(19)

where $j=1,2,3,4$, $\sigma _j:[0,\infty )\rightarrow M= \{1,2\}$. $f^i_{j,1}$, $f^i_{j,2}$, $f^i_{j,3}$ and $f^i_{j,4}$, $i\in {1,2}$, are selected as $f^1_{1,1} = \sin (\zeta _{1,1})\theta _1$, $f^2_{1,1} =\frac{\zeta _{1,1}\theta _1}{10 + \zeta _{1,1}^2}$, $f^1_{2,1} = \sin (\zeta _{2,1})\theta _2$, $f^2_{2,1} = \zeta _{2,1}\theta _2$, $f^1_{3,1} =\zeta _{3,1}\sin (\zeta _{3,1})$, $f^2_{3,1} =\zeta _{3,1}^2$, $f^1_{4,1} =\zeta _{4,1}$, $f^2_{4,1} =1-\cos (\zeta _{4,1}^2)$, $r_{1,1}=r_{2,1}=r_{3,1}=r_{4,1}=0.1\sin (t)$.

Fig. 1.

Communication graph.

According to the design method in Sect. 3, we design the consensus protocol and the adaptive laws as follows

$$\begin{aligned} u_j&= - \mathcal {C}_j(\xi _j) \breve{\zeta }_{j,2}^*, \quad \dot{\xi }_j = \breve{\zeta }_{j,2}^*z_{j,1},\\\dot{\hat{\varrho }}&= \omega _{\varrho ,1}, \quad \dot{\hat{\vartheta }} = \omega _{\theta ,1}, \end{aligned}$$

(20)

where $z_{j,1}=\breve{\zeta }_{j,1} - \eta _{j}$, $\breve{\zeta }_{j,2}^* =-\mu _{j,1}z_{j,1} + \dot{\eta }_j - \hat{\varrho } \bar{\omega }_{\varrho ,j,1} - \hat{\vartheta } \bar{\omega }_{\theta ,j,1}$, $\omega _{\varrho ,1} = \sum _{j=1}^N z_{j,1} \bar{\omega }_{\varrho ,j,1}$, $\omega _{\theta ,1} =\sum _{j=1}^N z_{j,1}\bar{\omega }_{\theta ,j,1}$, $\bar{\omega }_{\varrho ,j,1}=\frac{z_{j,1}\breve{\zeta }_{j,1}^2}{\sqrt{(z_{j,1}\breve{\zeta }_{j,1})^2 + \pi ^2}}$, $\bar{\omega }_{\theta ,j,1}=\frac{z_{j,1} \psi _{j,1}^2}{\sqrt{(z_{j,1} \psi _{j,1})^2 + \pi ^2}}$, $\psi _{1,1}= 2$, $\psi _{2,1}= |\breve{\zeta }_{2,1}|$, $\psi _{3,1}= 2 \breve{\zeta }_{3,1}^2 + 1$, $\psi _{4,1}= |\breve{\zeta }_{4,1}| + 1$.

For Simulation, the parameters are selected as $\theta _1=\theta _2=0.2$, $\theta _3=\theta _4=1$, $b_1=0.9$, $b_2=b_3=0.6$, $b_4=-0.5$, $\mu _{1,1}=2$, $\mu _{2,1}=5$, $\mu _{3,1}=2$, $\mu _{4,1}=4$, $g_1=20$, $g_2=0.26$, $a=1$, $\delta =0.5+0.2\sin (t)$, $(\zeta _{1,1}(0),\zeta _{2,1}(0),\zeta _{3,1}(0),\zeta _{4,1}(0))=(-0.2, 0.5, 0.1,0.2)$, and the other initial states are set as zero. The switching signal is described in Fig. 2. For convenience, we let $z_y=(y_1-y_2,y_2-y_3,y_3-y_4)$, and we will make a comparison to the control scheme in [24].

Fig. 2.

Switching law.

Figures 3 and 4 show the simulation results. Figure 3 shows the consensus error $\Vert z_y\Vert$ using the methods in the paper. It can be seen that the proposed control method can effectively deal with the cyber attacks of the uncertain MAS. Figure 4 show the adaptive laws of agents. The asymptotical consensus control objective is achieved under the proposed adaptive method.

Fig. 3.

Consensus error.

Fig. 4.

Adaptive laws.

5 Conclusion

This paper has established a consensus method for uncertain NMASs against cyber attacks. We introduce Nussbaum-type functions and auxiliary variables to handle the uncertainties and cyber attacks. How to extend the designed consensus protocol to the cases of more general cyber attacks is worthy of further study.