Adaptive fuzzy fixed time formation control of state constrained nonlinear multi-agent systems against FDI attacks

1.   Introduction

Recently, with the continuous progress of information technology and the rapid development of artificial intelligence and swarm intelligence theory, the control technology of multi-agent system (MASs) has become a hot research direction^[1,2,3]. MASs are networks composed of multiple cooperating agents, which can jointly solve complex problems and provide innovative solutions. With the wide application of MASs in various fields, distributed control of MASs has attracted much attention, mainly including consensus control^[4], formation control^[5,6,7,8], containment control^[9,10], etc. Nevertheless, in some practical systems, especially in MASs, there are generally complex nonlinearities and uncertainties that can't be ignored, so the study of nonlinear MASs is of great essence. Because of the existence of nonlinear dynamics, the control problem of nonlinear MASs has become a challenge.

In the field of nonlinear multi-agent control, the key of control is to influence the behavior of the whole system by adjusting the interaction between each agent, and then achieve the control goal in a specific time scale. At present, finite-time control^[11,12,13], fast finite-time control^[14,15], fixed-time control^{[16,17,18,19]} and other methods have attracted the attention of many scholars, Cao et al. proposed finite-time control protocols in MASs consensus control in ^[11], and Du et al. designed a fixed-time consensus control protocol in heterogeneous MASs^[17]. Fixed-time control means that the system can reach the required state or goal within a predetermined fixed time. Compared with traditional control methods, fixed-time control could keep the stability and performance within a predetermined time and is not affected by the initial conditions and external disturbances of the system. It was applied in mechanical control, robot control, power system, etc. In addition, with the increasing requirements and limitations in the controlled system and the improvement of system modeling, scholars have proposed various control algorithms and methods, such as sliding mode control ^[20], adaptive control ^[21], reinforcement learning^[22,23], and iterative learning control ^[24], to cope with the challenges of complex and nonlinear systems.

Constraint problems in control systems have become one of the hot research directions^[25]. These constraints may involve state constraints^[25,26,27], input constraints^[28,29], output constraints^[30], etc. Due to the constraints in actual systems, many scholars hammered at formation control and constraint problems of MASs, and they certainly made progress in the theoretical research levels. In the research of MASs formation problem, similar to social networks^[31], the relationships between agents are complex. Agents need to be able to perceive the surrounding environment, including detecting the presence and position information of other neighbor agents, complete the coordination and control of complex tasks, and maintain predetermined geometric configurations. However, there are often more complex external conditions and human factors in practical applications.

At present, the security control problem of the control system under the complex network has been a concern, and the most typical attacks are the attacks in control system. Common malicious attacks include FDI attacks^{[10,32,33,34]}, replay attacks^[35], cyber attacks^[36], and denial of service (DOS) attacks^{[37,38,39,40]}. Among these, FDI attack is a kind of network attack method with strong damage ability to system stability where attackers inject false data into the network to change the state and destroy the stability of the systems. Miao et al. studied the control problem under attack and constraints in cyber-physical systems in ^[32], and Jiang et al. studied the tracking control problem under attack conditions based on MASs^[33].

Considering the universality and destructiveness in practical applications, many scholars are devoted to the study of FDI attacks. Nontheless, there are still few studies on the FDI attacks in the formation control of nonlinear MASs. Therefore, we realize the FTFC of nonlinear MASs under FDI attacks is required and it has more practical application reference value. This paper innovates from the following aspects:

1) Compared with previous control strategies, the FTFC for MASs with full state constraints proposed in this article takes FDI attacks into account. In dealing with the state constraint problem, a novel nonlinear transformation method is employed to transform the system state to an unconstrained state.

2) Integrating backstepping techniques with FTFC strategies, the designed controller can achieve stability during the attack period, and the attack coefficient can also be effectively compensated.

2.   System descriptions and preliminaries

2.1.   Graph theory

Define $\Xi = \left(\upsilon, \varepsilon, A \right)$ being a weighted-undirected graph in communication network, with a cluster of nodes $\upsilon = \left\{ {{v}_{1}}, {{v}_{2}}, \ldots, {{v}_{N}} \right\}$, a cluster of edges $\varepsilon \subseteq \left\{ \left({{v}_{i}}, {{v}_{j}} \right):{{v}_{i}}, {{v}_{j}}\subseteq \upsilon \right\}$, and $A = \left[{{a}_{ij}} \right]\in {{R}^{n\times n}}$ representing the adjacency matrix. In the undirected graph $\Xi$, an edge $\left({{v}_{i}}, {{v}_{j}} \right)$ is expressed as ${{v}_{ij}} = \left({{v}_{i}}, {{v}_{j}} \right), i\ne j$, and it means that the $j\text{th}$ agent is able to deliver the message to the $i\text{th}$ agent. The Laplacian matrix $L = \left[{{l}_{ij}} \right]\in {{R}^{n\times n}}$, associated with the undirected graph $\Xi$, is structured as

where $D = diag\left\{ {{d}_{1}}, \ldots, {{d}_{n}} \right\} = diag\left\{ \sum\nolimits_{j = 1}^{n}{{{a}_{1j}}}, \ldots, \sum\nolimits_{j = 1}^{n}{{{a}_{nj}}} \right\}$.

2.2.   Problem formulation

Consider the nonlinear MASs as

where ${{\bar{x}}_{i, m}} = {{\left[{{x}_{i, 1}}, \ldots, {{x}_{i, m}} \right]}^{T}}\in {{R}^{m}}$ and ${{\bar{x}}_{i, q}} = {{\left[{{x}_{i, 1}}, \ldots, {{x}_{i, q}} \right]}^{T}}\in {{R}^{q}}$ represent state vector of the $i\text{th}$ follower, $i = 1, \ldots, P$, $m = 1, \ldots, q-1$, ${{u}_{i}}$ and ${{y}_{i}}$ are the control input and output of the $i\text{th}$ follower, and ${{f}_{i, m}}\left({{{\bar{x}}}_{i, m}} \right)$ and ${{f}_{i, q}}\left({{{\bar{x}}}_{i, q}} \right)$ are both unknown smooth nonlinear functions.

Definition 1: Consider state-dependent parameterized FDI attacks model as

where ${{\overset{ \frown}{x}}_{i, r}}$ denotes the state destroyed by FDI attacks, ${{\zeta }_{s}}\left({{x}_{i, r}}, t \right)$ represents the FDI attacks into ${{x}_{i, r}}$, ${{\zeta }_{s}}\left({{x}_{i, r}}, t \right)$ is parameterized as ${{\zeta }_{s}}\left({{x}_{i, r}}, t \right) = {{\varsigma }_{s}}\left(t \right){{x}_{i, r}}$, and ${{\varsigma }_{s}}\left(t \right)$ represents the unknown and time-varying signals. The transformed state is denoted as ${{x}_{i, r}} = {{\kappa }_{s}}\left(t \right){{\overset{ \frown}{x}}_{i, r}}$, ${{\kappa }_{s}}\left(t \right) = {{\left(1+{{\varsigma }_{s}}\left(t \right) \right)}^{-1}}$.

The dynamic of the leader is scaled as

Consider the case of being attacked, which is modeled as

Remark 1: The FDI attacks in this paper targets MASs and occurs in both leader and follower agents.

Definition 2: Given any initial formation tracking errors $\left\| {{x}_{i}}\left({{t}_{0}} \right)-{{x}_{l}}\left({{t}_{0}} \right)-{{h}_{i}} \right\|\in R$, one has

where $T$ denotes the settling time for secure fixed-time formation tracking control, and ${{\hbar }_{i}} = {{[{{h}_{i, 1}}, \cdots, {{h}_{i, m}}]}^{T}}\in {{R}^{m}}$ represents the anticipated position vector between the $i\text{th}$ agent and the leader.

Remark 2: The time-varying structure ${{h}_{i}}$ of MASs in the paper is bounded, and its time derivative exists, that is, there exist ${{\hat{h}}_{i}}$ and ${{\bar{h}}_{i}}$ such that $\left| {{h}_{i}} \right| < {{\hat{h}}_{i}}$, $\left| {{{\dot{h}}}_{i}} \right| < {{\bar{h}}_{i}}$. Moreover, when ${{h}_{i}} = 0$, the formation control in this paper can be transformed into consensus tracking control.

Assumption 1 ^[32]: For the false data ${{\zeta }_{s}}\left({{x}_{i, r}}, t \right)$, there exists ${{\bar{\zeta }}_{s}}$, satisfying: ${{\zeta }_{s}}\left({{x}_{i, m}}, t \right) < {{\bar{\zeta }}_{i}}$, and the signals ${{\varsigma }_{s}}\left(t \right)$ are positive and meet the following conditions: ${{\underset{ -}{\varsigma }}_{s}} < {{\varsigma }_{s}}\left(t \right) < {{\bar{\varsigma }}_{s}}$, ${{\underset{ -}{\varsigma }}_{s}}$ and ${{\bar{\varsigma }}_{s}}$ stand by the unknown constants, the coefficient of attack ${{\kappa }_{s}}\left(t \right)$ is bounded, and $\left| {{\kappa }_{s}}\left(t \right) \right|\le {{\bar{\kappa }}_{s}}$.

Assumption 2: There exists an edge at least between each follower and the leader (root).

Lemma 1 ^[3]: The undirected graph $\Xi$ is connected if, and only if, the Laplacian matrix is irreducible.

Lemma 2 ^[18]: In continuously normalized and radiationally unconstrained functions $V\left(\mathbb{Z} \right)$, for any $\mathbb{Z}\left(t \right)\in R$, satisfy: $V\left(\mathbb{Z} \right)\le -\sigma {{V}^{\iota }}\left(\mathbb{Z} \right)-\upsilon {{V}^{\tau }}\left(\mathbb{Z} \right)$, where $\sigma > 0$, $\upsilon > 0$, $\iota > 1$, $0 < \tau < 1$ are constants and the system can achieve fixed-time stability within $T = \frac{1}{\upsilon \left(1-\tau \right)}+\frac{1}{\sigma \left(1-\iota \right)}$.

Lemma 3 ^[28]: For any variables $\bar{a}$, $\bar{b}$ and positive constants ${{\vartheta }_{1}}$, ${{\vartheta }_{2}}$, and ${{\vartheta }_{3}}$, one has

Lemma 4 ^[41]: For any ${{\lambda }_{i}}\in R$, the inequalities hold

2.3.   Fuzzy logic system

Lemma 5 ^[12]: $\omega \left(x \right)$ denotes a continuous but unknown function, and the domain is the compact set $\gamma$. For constant $\theta > 0$, the FLS $f\left(x \right) = {{W}^{T}}\Psi \left(x \right)$ satisfy the following:

where $x = {{\left({{x}_{1}}, {{x}_{2}}, \ldots, {{x}_{q}} \right)}^{T}}$, $\Psi \left(x \right) = {{\left[{{\Psi }_{1}}\left(x \right), {{\Psi }_{2}}\left(x \right), \ldots, {{\Psi }_{L}}\left(x \right) \right]}^{T}}$, ${{\Psi }_{\text{p}}}\left(x \right)$ is the fuzzy basis function, and $W = {{\left[{{W}_{1}}, \ldots, {{W}_{P}} \right]}^{T}}$ stands by the ideal weight vector.

Control objective: The principal objective of this article is to devise adaptive fuzzy FTFC controllers such that MASs with full state constraints satisfy the following:

$1)$ Each follower in MASs can achieve fixed-time formation control in the case of FDI attacks.

$2)$ The states of MASs are all bounded by the constraints and the closed-loop signals of the system are bounded.

3.   Main results

3.1.   Adaptive controller design

The coordinates are converted as

where ${{k}_{c}}$ denotes known smooth constants constraints on ${{x}_{i, m}}$, $\left| {{x}_{i, m}} \right| < {{k}_{c}}$ and define ${{\Phi }_{i, m}} = k_{c}^{2}-x_{i, m}^{2}$, ${{\mu }_{i, m}} = \left(k_{c}^{2}+x_{i, m}^{2} \right)/{{\left(k_{c}^{2}-x_{i, m}^{2} \right)}^{2}}$.

The formation tracking error is determined as

Step 1: The constrained system under attack can be constructed as

According to (3.1), take the derivative with respect to ${z}_{i, 1}$, and one has

where ${{\mu }_{i, 1}} = k_{c}^{2}+x_{i, 1}^{2}/{{\left(k_{c}^{2}-x_{i, 1}^{2} \right)}^{2}}$.

The formation tracking errors are defined as

then, one has

where $E = {{\left[{{e}_{i, 1}}, \cdots, {{e}_{P, 1}} \right]}^{T}}\in {{R}^{P}}$, ${{Z}_{f}} = {{\left[{{z}_{i, 1}}, \ldots, {{z}_{P, 1}} \right]}^{T}}$.

The unknown nonlinear function is defined as

From the FLSs, it follows that

where ${{\Psi }_{i, 1}}\left({{M}_{i, 1}} \right)\in {{R}^{{{\upsilon }_{m}}}}$ stands by fuzzy basis function vector, ${{\upsilon }_{m}}$ denotes the numbers of fuzzy rules, $\left\| {{W}_{i, m}} \right\|\le {{\bar{W}}_{i, m}}$, $\left| {{\varepsilon }_{i, m}}\left({{M}_{i, m}} \right) \right|\le {{\bar{\varepsilon }}_{i, m}}$, and $m = 1, 2, \ldots q$.

Define

where ${{\tilde{\omega }}_{i, m}} = {{\omega }_{i, m}}-{{\hat{\omega }}_{i, m}}$, ${{\hat{\omega }}_{i, m}}$ stands by the estimation of ${{\omega }_{i, m}}$, ${{\tilde{\kappa }}_{s}} = {{\kappa }_{s}}-{{\hat{\kappa }}_{s}}$, ${{\hat{\kappa }}_{s}}$ stands by the estimation of ${{\kappa }_{s}}$, ${{\upsilon }_{i, 1}} = {2{{\varphi }_{i, 1}}}/{\left(2{{\varphi }_{i, 1}}-1 \right)}\;$, and ${{\varpi }_{i, 1}} = {2{{\psi }_{i, 1}}}/{\left(2{{\psi }_{i, 1}}-1 \right)}\;$.

Take the derivative of ${{V}_{\text{1}}}$, and it obtains

According to Young's inequality,

where ${{d}_{i, j}}$ represents the positive design parameter.

The virtual controller is designed as

Substituing (3.13)–(3.16) into (3.12), one has

where ${{\Delta }_{i, 1}} = \frac{1}{2}d_{i, 1}^{2}+\bar{\varepsilon }_{i, 1}^{2}+\bar{h}_{i}^{2}$.

Step p$\left(2\le p\le q-1 \right)$: The derivative of ${{e}_{i, p}}$ is scaled as

The unknown nonlinear function is defined as

From the FLSs, it obtains

where ${{M}_{i, p}} = {{\left[{{x}_{i, 1}}, \cdots, {{x}_{i, p}}, {{{\hat{\omega }}}_{i, 1}}, \cdots, {{{\hat{\omega }}}_{i, p-1}} \right]}^{T}}$, ${{\Psi }_{i, p}}\left({{M}_{i, p}} \right)$ stands for fuzzy basis function, $W_{i, p}^{*T}$ denotes the optimal weight vector, and ${{\varepsilon }_{i, p}}\left({{M}_{i, p}} \right)$ denotes the error of approximation.

The Lyapunov function is chosen as

Take the derivative of (3.21), and it obtains

According to Young's inequality,

The virtual controller is

Step q: Derivation of ${{e}_{i, q}}$ is scaled as

The unknown nonlinear function is defined as

From the FLSs, it obtains

where ${{M}_{i, q}} = {{\left[{{x}_{i, 1}}, \cdots, {{x}_{i, q}}, {{{\hat{\omega }}}_{i, 1}}, \cdots, {{{\hat{\omega }}}_{i, q-1}} \right]}^{T}}$, ${{\Psi }_{i, q}}\left({{M}_{i, q}} \right)$ stands for fuzzy basis function, $W_{i, q}^{*T}$ denotes the optimal weight vector, and ${{\varepsilon }_{i, q}}\left({{M}_{i, q}} \right)$ denotes the error of approximation.

Choose the Lyapunov function

The actual controller is derived as

The adaptive law of design parameters is

where ${{\sigma }_{i, m}}$ and ${{r}_{i, m}}$ are design positive constants.

We then obtain

where $\Delta_{m} = \sum\limits_{m = 1}^{q}{\sum\limits_{i = 1}^{P}{{{\Delta }_{i, m}}}}$.

3.2.   Stability analysis

Theorem 1: For the high order MASs (2.1) with full state constraints under FDI attacks (2.2), with Assumptions 1–2, design virtual controllers (3.16) and (3.25), actual controllers (3.31), and choosing adaptation laws (3.32) and (3.33), boundedness of the MASs state can be achieved and the formation tracking error can fluctuate around zero with FTFC.

Proof of the Theorem 1:

Substituing (3.11), (3.21), and (3.30), we have

From Young's inequality,

then we get

where ${{\bar{\Delta }}_{i, q}} = \sum\limits_{m = 1}^{q}{\sum\limits_{i = 1}^{P}{\left(\frac{{{r}_{i, m}}{{\psi }_{i, m}}}{2}\kappa _{s}^{2}+\frac{{{\varphi }_{i, m}}{{\sigma }_{i, m}}}{2}\omega _{i, m}^{2} \right)}}$.

Thus, one obtains

From Lemma 3, we get

where ${{\Lambda }_{1}}\text{ = }\left(\frac{1-{{\gamma }_{\text{2}}}}{\text{2}} \right){{\left(\frac{\text{1+}{{\gamma }_{\text{2}}}}{\text{2}} \right)}^{\frac{\text{1+}{{\gamma }_{\text{2}}}}{1-{{\gamma }_{\text{2}}}}}}$ Define ${{c}_{r}} = \max \left\{ {{\lambda }_{\max }}\left[{{\left(L+B \right)}^{-1}} \right], \frac{1}{2}, \frac{1}{2{{\upsilon }_{i, m}}}, \frac{1}{2{{\varpi }_{i, m}}} \right\}$

From Lemma 4, one has

where ${{c}_{s}} = {{3}^{\frac{{{\gamma }_{1}}-1}{2}}}c_{r}^{\frac{1+{{\gamma }_{1}}}{2}}$, ${{\bar{c}}_{s}} = c_{r}^{\frac{1+{{\gamma }_{2}}}{2}}$.

From above, we get

where ${{\ell }_{1}} = \frac{a}{{{c}_{s}}} > 0$, ${{\ell }_{2}} = \frac{b}{{{{\bar{c}}}_{s}}} > 0$, $0 < \bar{\gamma } = \frac{1+{{\gamma }_{2}}}{2} < 1$, $\gamma = \frac{1+{{\gamma }_{1}}}{2} > 1$, ${{\Lambda }_{0}} = {{\bar{\Delta }}_{i, q}}+a{{\Lambda }_{1}}+b{{\Lambda }_{1}}$.

Suppose that there are unknown constants ${{\Gamma }_{i, m}}$ and ${{\Pi }_{i, m}}$, which satisfy: $\left| {{{\tilde{\omega }}}_{i, m}} \right|\le {{\Gamma }_{i, m}}$, $\left| {{{\tilde{\kappa }}}_{s}} \right|\le {{\Pi }_{i, m}}$

If ${{\Gamma }_{i, m}} < \sqrt{2{{\upsilon }_{i, m}}}$, one has

With ${{\Gamma }_{i, m}}\ge \sqrt{2{{\upsilon }_{i, m}}}$, one obtains

where ${{\Xi }_{1}} = \sum\limits_{m = 1}^{q}{\sum\limits_{i = 1}^{P}{{{\sigma }_{i, m}}{{\left(\frac{\Gamma _{i, m}^{2}}{2{{\upsilon }_{i, m}}} \right)}^{\frac{1+{{\gamma }_{1}}}{2}}}-\frac{{{\sigma }_{i, m}}}{2{{\upsilon }_{i, m}}}\Gamma _{i, m}^{2}}}$

If ${{\Pi }_{i, m}} < \sqrt{2{{\varpi }_{i, m}}}$, it gets

If ${{\Pi }_{i, m}}\ge \sqrt{2{{\varpi }_{i, m}}}$, one obtains

where ${{\Xi }_{2}} = \sum\limits_{m = 1}^{q}{\sum\limits_{i = 1}^{P}{{{r}_{i, m}}{{\left(\frac{\Pi _{i, m}^{2}}{2{{\varpi }_{i, m}}} \right)}^{\frac{1+{{\gamma }_{1}}}{2}}}-\frac{{{r}_{i, m}}}{2{{\varpi }_{i, m}}}\Pi _{i, m}^{2}}}$.

Through all above, it can be obtained that

To sum up, all the states of MASs are bounded within $T\le 1/{{\ell }_{\text{2}}}\left(1-\beta \right)\left(1-\bar{\gamma } \right)+1/{{\ell }_{1}}\left(\gamma -1 \right)$, and $\beta$ denotes a constant that satisfies: ${{\bar{\Lambda }}_{0}}\le {{\ell }_{\text{2}}}\beta V_{q}^{{\bar{\gamma }}}$.

Remark 3: The MASs in this paper are full state constrained, ${{x}_{i, m}}$ satisfying $-{{k}_{c}} < {{x}_{i, m}} < {{k}_{c}}$, and since ${{z}_{i, m}}$ is bounded, the states are all in a compact set.

4.   Simulations

In this section, MASs consist of the leader and 4 followers, and the adjacency matrix $A$ represents

The relationships between leader and followers are $B = diag\left\{ 1, 0, 0, 0 \right\}$, and the Laplacian matrix is in the form of

Figure 1 shows the communication topology of MASs.

The followers are modeled as

where the state of systems satisfies: $\left| {{x}_{i, 1}} \right| < {{k}_{c1}} = 0.7$, $\left| {{x}_{i, 2}} \right| < {{k}_{c2}} = 1.9$.

The leader trajectory is ${{x}_{l.1}} = 0.8\sin \left(2.5t \right)$, and the formation coefficient is ${{h}_{1}} = {{h}_{2}} = {{h}_{3}} = {{h}_{4}} = 2.2\cos \left(2.5t \right)$. The fuzzy membership function is set as $\mu _{{{F}_{i1}}}^{P} = \exp [-{{\left({{z}_{i, 1}}+0.25P \right)}^{2}}-{{\left({{z}_{i, 2}}+0.25P \right)}^{2}}]$, $\mu _{{{F}_{i2}}}^{P} = \exp [-{{\left({{z}_{i, 1}}+0.25P \right)}^{2}}-{{\left({{z}_{i, 2}}+0.25P \right)}^{2}}$-${{\left({{x}_{l, 1}}+0.25P \right)}^{2}}-{{\left({{{\dot{x}}}_{l, 1}}+0.25P \right)}^{2}}$-${{\left({{{\ddot{x}}}_{l, 1}}+0.25P \right)}^{2}}]$, selecting attack weight ${{\varsigma }_{s}}\left(t \right) = -0.18-0.42\cos \left(t \right)$ and the initial value is ${{x}_{i, 1}}\left(0 \right) = 0$, ${{x}_{i, 2}}\left(0 \right) = 0.45$, ${{\omega }_{i, j}}\left(0 \right) = 0$. ${{\kappa }_{i, j}}\left(0 \right) = 0$. The design parameters are set as ${{\gamma }_{1}} = 1.3$, ${{\gamma }_{2}} = 0.9$, ${{d}_{i, j}} = 1.5$, ${{\sigma }_{i, j}} = 2.5$, ${{r}_{i, j}} = 2$, ${{c}_{i, 1}} = 0.1$, ${{c}_{i, 2}} = 20$, ${{\bar{c}}_{1, 1}} = {{\bar{c}}_{2, 1}} = 5$, ${{\bar{c}}_{3, 1}} = {{\bar{c}}_{4, 1}} = 0.1$, ${{\bar{c}}_{i, 2}} = 0.1$, $i = 1, 2, 3, 4$, $j = 1, 2$.

The simulation results are shown in Figure 2 to Figure 7. The follower agents in Figure 2 can maintain the predetermined formation trajectory in the constrained state, and the tracking effects meet expectation and conform to the constraint range. The state output of the agents are shown in Figure 3, which conform to the constraint bound. Figures 4 and 5 illustrate the control inputs of the four groups of agents. Figure 6 illustrates the tracking error of the agents, and it can be seen that the tracking errors of the agents stably converge around zero. Figure 7 shows the state error of the formation process, and the convergence effect reaches the expected standard. During the simulation, the FDI attacks time is set within 5-15s, and the proposed method can still maintain the stability of MASs formation control after FDI attacks.

5.   Conclusions

In this paper, for the nonlinear MASs with full state constraints, the fixed-time formation strategy is proposed under FDI attacks conditions, the coordinate transformation is used to deal with the attacks state and the system state constraints, the nonlinear transformation method is used to remove the influence between the full state constraints and topological relations, and the FLS is used to approximate the uncertain function. An adaptive fuzzy fixed-time formation controller is designed based on backstepping method, and the effectiveness of the controller is verified by simulation examples. In the following work, we will work on MASs formation control under complex types of attacks.

Use of AI tools declaration

The authors stated that no artificial intelligence (AI) tools were used at the time of writing this article.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 62333009, in part by the Liaoning Revitalization Talents Program under Grant XLYC2202014, and in part by the Key Projects in Liaoning Province under Grant JYTZD2023085.

Conflict of interest

The authors declare that there is no competing financial interest or personal relationship that could have appeared to influence the work reported in this paper.