Secure consensus control for multi-agent systems under communication constraints via adaptive sliding mode technique

2.1. Graph theory

Graph theory is an important tool to study MASs, which is a graph composed of several nodes and edges connecting the node. Each agent can be represented as a node, and the information interaction between agents can be denoted as an edge in graph theory. A directed weighted graph is represented by $$ G=\left \{ \mathbb{V}, \mathbb{E} \right \} $$. For MASs with one leader and $$ N $$ agents, the node-set $$ \mathbb{V}=\left \{ {v}_{1}, {v}_{2}, \cdots, {v}_{N} \right \} $$ indicates the set of all points on the graph and $$ \mathbb{E}=\{(i, j), i, j \in \mathbb{V}, i \neq j\} $$ represents the set of all edges. $$ A=[a_{ij} ] \in \mathbb{R}^{N\times N} $$ is a non-negatively weighted adjacency matrix. If $$ a_{ij}>0 $$, it means that agent $$ i $$ can receive information from agent $$ j $$; conversely, if $$ a_{ij}=0 $$, agent $$ i $$ cannot receive information from agent $$ j $$. Define the matrix $$ B= $$ diag$$ (b_{1}, b_{2}, \cdots, b_{N} ) $$ to denote the communication between the leader and all followers, and the degree matrix $$ D= [d_{ii}] $$ with $$ d_{ii}= \sum_{j=1}^{N} a_{ij} $$. So, we can obtain the Laplace matrix $$ L= [l_{ij}] $$ as:

with

Lemma 1^[35] The matrix $$ L + B $$ is invertible if the directed graph $$ G $$ has a directed spanning tree.

Definition 1 Consider a multi-agent system with $$ N $$ agents and let $$ x_i(t) $$ represent the state of agent $$ i $$. If $$ lim_{t \to \infty}\| x_i(t)-x_j(t) \|=0 $$, for all $$ i, j=1, 2, \cdots, N $$, it is said that the multi-agent system can reach a consensus. Furthermore, if there exists a leader whose state is $$ x_0(t) $$, then $$ lim_{t \to \infty}\| x_i(t)-x_0(t) \|=0 $$, for all $$ i, j=1, 2, \cdots, N $$, means the tracking consensus is achieved.

2.2. System model

Consider a second-order MASs consisting of a leader labeled as node $$ 0 $$ and $$ N $$ followers indexed by $$ i\in\{1, 2, \cdots, N\} $$, and the $$ i $$th follower's dynamic is given as:

where $$ {x}_{i}(t)\in \mathbb{R}^{m} $$, $$ {v}_{i}(t)\in \mathbb{R}^{m} $$, $$ u_{i}(t)\in \mathbb{R}^{m} $$ represent the $$ i $$th follower's position, velocity and the control input, respectively. According to equation (3), it is obvious that we are focused on double integrators.

The leader's dynamic is of the following form:

with $$ {x}_{0}(t)\in \mathbb{R}^{m} $$, $$ {v}_{0}(t)\in \mathbb{R}^{m} $$ the leader's position and velocity, respectively, and $$ {u}_{0}(t)\in \mathbb{R}^{m} $$ representing the control input.

Define the $$ i $$th follower's consensus tracking errors as follows:

with $$ {e}_{1i}(t) $$ and $$ {e}_{2i}(t) $$ the tracking error variables of position and velocity, $$ {a}_{ij} $$ represents the element of $$ A $$, $$ {b}_{i} $$ determines whether there is information interaction between the leader and the followers, when $$ {b}_{i} > 0 $$, agent $$ i $$ can receive information from the leader, otherwise, $$ {b}_{i} $$ = 0.

The tracking errors can be rewritten in the compact form:

with $$ e_{1}(t)\triangleq [e_{11}^{T}(t), \cdots , e_{1 N}^{T}]^{T} $$, $$ e_{2}(t)\triangleq \left[e_{21}^{T}(t), \cdots , e_{2 N}^{T}(t)\right]^{T} $$, $$ x(t)\triangleq \left[x_{1}^{T}(t), \cdots , x_{N}^{T}(t)\right]^{T} $$, $$ v(t)\triangleq \left[v_{1}^{T}(t), \cdots , \right.\\ $$$$ v_{N}^{T}(t)]^{T} $$, $$ u(t)\triangleq \left[u_{1}^{T}(t), \cdots , u_{N}^{T}(t)\right]^{T} $$, $$ B\triangleq $$diag$$ \{b_{1}, b_{2}, \cdots, b_{N}\} $$.

From the above definition, one can obtain the tracking error system as:

Now, the consensus tracking problem of MASs (3)-(4) converts to the stabilization problem of the tracking error system (7). The objective of this work is to achieve leader-follower consistency.

2.3. Fading channel

As stated in the Introduction, the transmission between followers may be inevitably suffered from the channel fading phenomenon. In this work, the network channel is considered as a continuous one with time-varying channel gain, the transmitted data will be modeled as the actually received information with random attenuation. Hence, introduce the following memoryless multiplicative fading model:

where $$ {x}_{ij}(t) $$ and $$ {v}_{ij}(t) $$ are the fading position and speed signal of the $$ j $$th agent received by the $$ i $$th agent, and $$ {x}_{j}(t) $$ and $$ {v}_{j}(t) $$ are the signal and speed signal sent by the $$ j $$th agent, respectively. The random coefficient $$ \rho_{i j}(t) \in(0, 1] $$ are mutually independent random variables with mathematical expectation $$ \mathbb{E}\left(\rho_{ij}(t)\right)=\bar{\rho} $$.

Assuming that fading occurs only in the channel between followers, the special case of channel fading from the leader to the followers is not considered in this work. Hence, based on the fading information (8), the tracking errors (5) are rewritten as:

It can be seen that the tracking errors (9) involve the expectation of the random variable $$ \rho_{ij}(t) $$, which is introduced to compute the consistent tracking error variable among the agents more accurately.

Define $$ \bar{e}_{1}(t) $$$$ \triangleq $$$$ [\bar{e}_{11}^{T}(t), \cdots, \bar{e}_{1 N}^{T}(t)]^{T}, \bar{e}_{2}(t) $$$$ \triangleq $$$$ \left[\bar{e}_{21}^{T}(t)\right., \cdots, \bar{e}_{2 N}^{T}(t)]^{T} $$. Then, the compact form of tracking errors (9) is of the following form:

where $$ \alpha_{i} \triangleq \operatorname{diag} \{\delta_{i}^{1}, \delta_{i}^{2}, \ldots, \delta_{i}^{N}\} $$, $$ \delta_{i}^{k}(\cdot) $$ being the Kronecker delta function, which compares values of $$ i $$ and $$ k $$ and returns $$ 0 $$ when they are not equal; otherwise, it returns 1. $$ \Lambda_{1}(t)\triangleq\operatorname{diag}\{0, \rho_{1 2}(t), \cdots, \rho_{1N}(t)\} $$, $$ \cdots $$, $$ \Lambda_{N}(t) \triangleq \operatorname{diag}\{\rho_{N 1}(t), \rho_{N 2}(t), \cdots, 0\} $$. Considering the effect of channel fading, since only faded data is available, accurate neighbors' information cannot be used for the controller design. In the following section 3.1, the SMC law will be designed based on the consensus tracking errors $$ \bar{e}_1(t) $$ and $$ \bar{e}_2(t) $$.

Remark 1. There are two special cases considered in the channel fading model (8): when $$ \rho_{ij}(t)=0 $$, it means that there is no information interaction between agents and the communication channel is blocked, that is, the channel fading model is simplified to a packet loss model. In contrast, if $$ \rho_{ij}(t)=1 $$, it indicates that the data transmission between agents is complete and without any attenuation.

2.4. Deception attacks

Among various cyber-attacks, the deception attack on controllers is a common form and usually satisfies the following assumptions: the hackers can steal the state information or measurement output of the agents to generate false data, which can then be injected into the controller. As shown in Figure 1, the hackers can attack the controller of agent $$ i $$ by injecting false data. Thus, the actual data received by the actuator of agent $$ i $$ is as follows:

Figure 1. System over fading network subject to attacks on agent $$ i $$.

The compact form of expression (11) can be written as:

where $$ u(t) $$ is the designed control input and $$ W(t) \Psi_{a}(x(t), t) $$ is the false data. The matrix $$ W(t) $$ is an unknown and time-varying matrix that satisfies $$ \|W(t)\| \leq \mu(t) $$ with $$ \mu(t) $$ unknown and bounded, represents the injection patterns of the false data, for example, $$ W(t) $$ may be a matrix composed of elements $$ 0 $$ and $$ 1 $$, that is, sometimes false data is injected, sometimes not, to confuse users. Thus, the attack is difficult to be detected by users. $$ \Psi_{a}(x(t), t) $$ is a function of $$ x(t) $$, which means the false data generated via the state $$ x(t) $$, and satisfy $$ \left\|\Psi_{a}(x(t), t)\right\| \leq \psi(x(t), t) $$ with $$ \psi(x(t), t) $$ a known nonnegative function.

Remark 2. The deception attacks considered in this work focus on the controller, that is, $$ u(t) $$ may be suffered from the false data injection, such as the problem considered in some literature ^{[30, 36]} and so on. The deception attacks can also occur in the communication channel between agents, that is, $$ x(t) $$ may be affected by false data injection during transmission ^[34].

3.1. Adaptive SMC law

To cope with the impact of the deception attacks, the information about the attack is usually utilized to design the controller. For example, when the upper bounds $$ \mu(t) $$ and $$ \psi(x(t), t) $$ of the attack are known, the design of the controller is relatively easy to implement, but the fixed upper bounds will inevitably lead to larger conservativeness. To overcome this problem, an online estimation strategy will be employed to estimate the time-varying and unknown attacks, based on which, an adaptive sliding mode controller will be designed to attenuate the effect of the unknown attacks on MASs.

Design the sliding function as follows:

with $$ c>0 $$ the sliding gain, denoted $$ s(t)\triangleq\left[s_{1}^{T}(t), s_{2}^{T}(t), \cdots, s_{N}^{T}(t)\right]^{T} $$, the compact form of sliding function (13) can be written as:

From (7), we can obtain the derivative of the sliding function:

Under these constraints considered in this work, the $$ i $$th agent cannot receive accurate and complete information from neighbor agents, the switching function (13) under fading channel is rewritten as:

The compact form of expression (16) as:

Then, construct the sliding mode controller as follow:

where the robust term $$ u_{a}(t) $$ is designed as :

with $$ k_{1}>0 $$, and the adaptive term $$ u_{b}(t) $$ is designed as:

where $$ \hat{\mu}(t) $$ is the estimation of $$ \mu(t) $$ under the following adaptive law:

with $$ \theta $$ an adaptive parameter, and Proj the smooth projection ^[37] as:

where the continuous function $$ \varphi(\hat{\mu}(t)) $$ defined as:

with $$ \hat{\mu}_{max} $$ the given bound of projection, and scalar $$ 0<\delta<1 $$.

According to Imbedded Convex Sets Assumption ^[37], we obtain:

and

and these two conditions will be used in the following derivation.

Remark 3. In some existing reliable control methods, the known bounds of attacks are usually utilized, which may inevitably yield larger conservativeness. To overcome this shortcoming, the online estimation mechanism for unknown attacks/faults was proposed in some related works ^{[33, 38]}. Inspired by these works, the online estimation mechanism of the attack is integrated with the SMC technique in this work.

3.2. Consistence and Reachability

Theorem 1. Consider the MASs (3)-(4) with channel fading (8) and deception attacks (12), under the proposed SMC law (19)-(20), the reachability of the sliding surface $$ s(t)=0 $$ can be guaranteed in the sense of mean square.

Proof. Choose the Lyapunov function as follows:

where $$ \tilde{\mu}(t)=\hat{\mu}(t)-\mu(t) $$ is the estimated error with $$ \dot{\tilde{\mu}}(t)=\dot{\hat{\mu}}(t) $$.

Then, by the expressions (15) and (21), the derivative of $$ V_{1}(s, t) $$ can be given as:

Taking mathematical expectation to the above expression (27), one has:

It can be easily verified from expressions (5) and (9) that $$ \mathbb{E}\left(e_{1}(t)\right)=\mathbb{E}\left(\bar{e}_{1}(t)\right) $$, $$ \mathbb{E}\left(e_{2}(t)\right)=\mathbb{E}\left(\bar{e}_{2}(t)\right) $$. Meanwhile, it follows from (14) and (17) that $$ \mathbb{E}\left(s(t)\right)=\mathbb{E}\left(\bar{s}(t)\right) $$. Then, one can obtain:

By the conditions $$ \|W(t)\| \leq \mu(t) $$, $$ \left\|\Psi_{a}(x(t), t)\right\| \leq \psi(x(t), t) $$, one has:

Then, it follows from (25) that:

Hence, the reachability of the sliding surface $$ s(t)=0 $$ can be ensured in the sense of mean square.

□

Theorem 2. Considering the MASs (3)-(4) subject to deception attacks (12) and channel fading model (8), the consensus tracking for MASs (3)-(4) will be achieved under the proposed sliding surface (14) and the SMC law (19)-(20).

Proof. Select the Lyapunov function:

Its derivative is given as:

When the sliding surface $$ s(t)=0 $$, it follows from (14) and (7) that $$ e_{2}(t)=-ce_{1}(t) $$ and $$ \dot{e_{2}}(t)=-ce_{2}(t) $$, then we can obtain:

Combining the results of Theorem 1, the consensus tracking of MASs (3)-(4) can be ensured under the proposed sliding surface (14) and the SMC law (19)-(20). □

Acknowledgments

Special thanks to the School of Electronic and Electrical Engineering, Shanghai University of Engineering Science (China) for providing technical support for this research.

Authors' contributions

Methodology, software, validation, data curation, visual conceptualization, ritingization, writing- original draft: Ding M

Conceptualization, riting-reviewing and editing, investigation: Chen B

Availability of data and materials

Not applicable.

Financial support and sponsorship

This work is supported in part by the NNSF of China (62173222, 61803255), Shanghai Science and Technology Innovation Action Plan (22S31903700, 21S31904200), and the National Key R & D Program of China (Grant No. 2020AAA0109301).

Conflicts of interest

Both authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

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Copyright

© The Author(s) 2023.