Sliding mode control of multi-agent systems with switching topologies and input constraints via Interval Type-2 fuzzy neural network

Introduction

The coordinated control of nonlinear MASs has become an essential topic in advanced control theory and intelligent automation due to its wide-ranging applications in cooperative robotics, uncrewed aerial vehicles (UAVs), autonomous driving, satellite formation, and intelligent manufacturing (Li et al., 2023; Ju et al., 2023; Xu & Li, 2025, 2024; Xiong & Chen, 2025). These systems rely on the interaction of multiple agents that share information to achieve common goals such as formation maintenance, trajectory tracking, and distributed decision-making. Achieving accurate coordination under uncertain and dynamic environments is one of the most challenging problems in modern control theory (Zeng, Zhu & Goetz, 2024; Wu et al., 2025; Luan et al., 2024).

In practical settings, each agent typically exhibits nonlinear, time-varying, and uncertain dynamics, often subject to communication delays, actuator constraints, and external disturbances (Xiong, Wang & Li, 2025; Zhou et al., 2025). These characteristics make conventional linear control approaches insufficient for ensuring robustness and stability. Furthermore, dynamically changing network topologies due to limited communication, packet losses, or mobility introduce additional complexity in guaranteeing formation accuracy (Yao et al., 2025; Wang et al., 2025). Consequently, there is a growing demand for adaptive and intelligent control strategies that can handle uncertainties while maintaining real-time performance and stability.

Among various robust control techniques, Sliding Mode Control (SMC) has been widely recognized for its simplicity, finite-time convergence, and robustness to matched uncertainties (Guo et al., 2024; Li et al., 2024; Wang et al., 2024; Zhou et al., 2025). It has been successfully applied to robotic manipulators, autonomous vehicles, and aerospace systems where precise control is required under modeling uncertainties (Chen et al., 2025; Ding et al., 2025). However, the classical SMC framework often suffers from the chattering phenomenon, a high-frequency oscillation caused by discontinuous switching actions, which deteriorates actuator performance and may destabilize physical systems (Ali et al., 2022; Roohi et al., 2023). Researchers have addressed this by proposing continuous approximations, higher-order SMC, super-twisting algorithms, and integral sliding surfaces (Jiang et al., 2025). Although these techniques improve smoothness, they may introduce slower transient response or require prior knowledge of system bounds.

Recently, intelligent control techniques have emerged as powerful tools for addressing these challenges. By integrating fuzzy logic systems (FLS) and neural networks (NNs) into SMC structures, controllers can learn and adapt to unknown dynamics in real time (Liang et al., 2025; Ma & Xu, 2023). The introduction of Interval Type-2 Fuzzy Neural Networks (IT2FNNs) further enhances robustness by incorporating the concept of the footprint of uncertainty (FOU) to manage nonlinearities and noisy data (Qi et al., 2024; Liu et al., 2025; Wang et al., 2025). These methods demonstrate superior approximation accuracy and adaptability, especially under uncertain or time-varying operating conditions.

Despite these advances, several open issues remain unresolved. Many existing fuzzy–neural SMC frameworks assume known system models, fixed network topologies, or unbounded control inputs, which limit their scalability to real-world applications (Lu et al., 2024; Ding et al., 2025). Moreover, learning-based methods often exhibit slow adaptation when faced with fast-changing dynamics or abrupt topology variations. Therefore, there is still a pressing need for a computationally efficient, adaptive, and gradient-based learning control scheme that maintains robustness, convergence, and low chattering under practical operating constraints (Jin et al., 2023; Castellanos-Cárdenas et al., 2024).

To address these issues, this article proposes a Robust General Type-2 Fuzzy Neural Network (RGT2FNN)–SMC framework for formation control of nonlinear MASs under switching communication topologies and actuator input constraints. The proposed controller integrates adaptive fuzzy–neural estimation with gradient-based learning for real-time parameter tuning, enhanced convergence, and reduced computational cost. Furthermore, by introducing a smooth boundary-layer control law and Lyapunov-based stability guarantees, the proposed framework ensures fast convergence, bounded control effort, and robustness under uncertain conditions (Ahmad et al., 2025; Khan, Cao & Li, 2025).

The main contributions of this article are summarized as follows:

• 1.

  A novel RGT2FNN–SMC control strategy is developed to ensure accurate and stable formation tracking of uncertain multi-agent systems (MASs) under switching topologies and actuator constraints.

• 2.

  The proposed approach integrates adaptive fuzzy estimation and robust sliding control, enhancing convergence speed and reducing chattering compared with conventional IT2FNN–SMC schemes.

• 3.

  Comprehensive simulations evaluate disturbance rejection, parameter sensitivity, and scalability, verifying the proposed controller’s adaptability and practical applicability in dynamic environments.

The remainder of this article is organized as follows. “Proposed Control Methodology” formulates the problem and presents the system model. “Optimization and Robustness Analysis” details the design of the proposed RGT2FNN–SMC control framework and its stability proof. “Simulation Results and Discussion” discusses the simulation results and performance analysis, and “Conclusion” concludes the article with remarks on future work.

Together, these aims address the limitations of conventional SMC strategies by enhancing adaptability, reducing chattering, and ensuring smooth control action under varying network and environmental conditions.

Proposed control methodology

This article employs a directed subgraph representation $G=\{\mathcal{V},\mathcal{E},\mathcal{A}\}$ to model information exchange between N follower agents. The vertex set $\mathcal{V}=\{1,2,\dots ,N\}$ identifies individual agents, while the edge set $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ describes their interaction pathways. The adjacency matrix $\mathcal{A}=[a_{ij}{]}_{N\times N}$ quantifies these connections, where $a_{ij}>0$ indicates information flow from agent $j$ to agent $i$, and $a_{ij}=0$ signifies no direct connection. By convention, self-loops are excluded ( $a_{ii}=0$).

In realistic MASs, communication is typically localized, resulting in a sparse adjacency matrix. Each agent’s neighborhood is defined as ${\mathcal{N}}_i=\{j\in \mathcal{V}\mid (i,j)\in \mathcal{E}\}$. The graph Laplacian $\mathcal{L}=D-\mathcal{A}$ incorporates both connectivity and degree information, with $D=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(d_1,\dots ,d_N)$ representing the out-degree matrix where $d_i={\sum }_{j\in {\mathcal{N}}_i}{a}_{ij}$.

The complete network topology $\overline{G}=\{\overline{\mathcal{V}},\overline{\mathcal{E}}\}$ extends this framework to include a leader agent, with $\overline{\mathcal{V}}=\{0\}\cup \mathcal{V}$. The leader’s influence is captured by the diagonal matrix $\mathcal{B}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(b_1,\dots ,b_N)$, where $b_i=1$ indicates that follower $i$ receives the leader’s state information. The case where follower $i$ cannot access the leader’s information is represented by $b_i=0$. The overall network structure of $\overline{\mathcal{G}}$ is characterized by the composite matrix $\mathcal{H}=\mathcal{L}+\mathcal{B}$. To account for dynamic network topologies where agent connections may vary over time, we define a finite set $\mathrm{\Gamma }=\{\overline{{\mathcal{G}}_1},\overline{{\mathcal{G}}_2},\dots ,\overline{{\mathcal{G}}_p}\}$ containing all possible graph configurations. The temporal evolution of these topologies is described by a switching signal $\psi (\tau ):[0,+\mathrm{\infty })\to \{1,2,\dots ,p\}$, which maps each time instant $\tau$ to a specific graph index. Under this framework, the active topology at time $\tau$ is denoted by ${\mathcal{G}}^{\psi }$, with corresponding adjacency matrix ${\mathcal{A}}^{\psi }$, Laplacian matrix ${\mathcal{L}}^{\psi }$, and neighbor sets ${\mathcal{N}}_i^{\psi }$ for each node $i$.

The main symbols and parameters used throughout this article are summarized in Table 1 for clarity.

Overview of the ellipsoidal IT2 fuzzy neural network

To estimate the uncertainties present in the system dynamics described by Eq. (2.1), an IT2FNN is employed within the formation error framework of Eq. (2.5). This structure allows for a highly precise and robust approximation of system uncertainties. It is important to emphasize that there is no interaction among the $n$ components of $g_i$. Specifically, the $k$th element of $g_i$ depends solely on the $k$-th components of $x_i$ and ${\nu }_i$. The structure of the fuzzy neural network employed for uncertainty estimation is illustrated in Fig. 1.

Given the input vector $\omega =[\begin{array}{cc}{\omega }_1& {\omega }_2\end{array}{]}^T=[\begin{array}{cc}{e}_{ix}^k& e_{iv}^k\end{array}{]}^T$ for $k=1,2,\dots ,n$, the IT2FNN of ellipsoidal type is structured as follows. The $i$th fuzzy rule is stated as:

$$R_i :\mathrm{I}\mathrm{F}{\omega }_1\mathrm{i}\mathrm{s}{\tilde{F}}_1^i\mathrm{A}\mathrm{N}\mathrm{D}{\omega }_2\mathrm{i}\mathrm{s}{\tilde{F}}_2^i\mathrm{T}\mathrm{H}\mathrm{E}\mathrm{N}y\mathrm{i}\mathrm{s}{\theta }_i,i=1,2,\dots ,r$$

Here, ${\omega }_1$ and ${\omega }_2$ represent the input variables, and $y$ is the output of the fuzzy system. The number of fuzzy rules is denoted by $r$. The symbols ${\tilde{F}}_1^i$ and ${\tilde{F}}_2^i$ denote IT2 fuzzy sets used as antecedents. The consequent fuzzy set ${\theta }_i=[{\underset{\_}{\theta }}_i,{\overline{\theta }}_i]$ specifies the lower and upper bounds of the output. Based on these rules, the IT2FNN is composed of five neural layers.

The IT2FNN structure consists of five layers, each with a specific function given by:

$\star$ Input layer: Let ${\omega }_j$ denote the $j$th input to the IT2FNN input layer.

$\star$ Membership layer: Here nodes are organized into groups, where each group corresponds to the antecedent of a fuzzy rule, and each node represents a symmetric IT2 membership function. Based on the relationship between the input measurements and their means, the upper and lower membership functions are computed as:

(2.7) $${\overline{\mu }}_{ij}({\omega }_j)=\{\begin{array}{cc}1-{\left(\frac{|{\omega }_j-m_{ij}|}{a_{ij}}\right)}^{\frac{1}{\zeta }},\hfill & \hfill \mathrm{i}\mathrm{f}|{\omega }_j-m_{ij}|<a_{ij}\hfill \\ 0,\hfill & \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\hfill \end{array}$$

(2.8) $${\underset{\_}{\mu }}_{ij}({\omega }_j)=\{\begin{array}{cc}1-{\left({\displaystyle \frac{|{\omega }_j-m_{ij}|}{\frac{}{}}}{a}_{ij}\right)}^{\zeta },& \text{if }|{\omega }_j-m_{ij}|<a_{ij}\\ 0,\hfill & \text{otherwise}\hfill \end{array}.$$

Here, $m_{ij}$ and $a_{ij}$ denote the mean and standard deviation, respectively, and $\zeta \in (0,1)$ is a shape parameter controlling the FOU. Figure 2 demonstrates the fuzzy membership functions that define the uncertainty bounds used in the IT2FNN. The parameters of the IT2FNN, including the means $m_{ij}$, widths $a_{ij}$, and shape factor $\zeta$, were initialized within empirically chosen ranges to ensure stable system performance. These parameters are then adaptively tuned using the Biogeography-Based Optimization (BBO) algorithm, allowing real-time adjustment to handle nonlinearities and time-varying uncertainties efficiently. The IT2FNN employs ellipsoidal Gaussian membership functions because they provide smooth and differentiable transitions between fuzzy regions, which are essential for adaptive learning. Their elliptical footprint of uncertainty (FOU) offers better flexibility for modeling nonlinearities compared to triangular or trapezoidal functions. This form effectively balances approximation accuracy and computational simplicity, which is particularly advantageous for real-time multi-agent control.

$\star$ Activation layer: By employing the singleton fuzzification method and the product $\tau$-norm, the firing strength of the $i$th rule is given by:

(2.9) $${\underset{\_}{q}}_i=\prod _{j=1}^2{\underset{\_}{\mu }}_{ij}({\omega }_j),{\overline{q}}_i=\prod _{j=1}^2{\overline{\mu }}_{ij}({\omega }_j)$$

$\star$ Type reduction layer: This layer calculates the output by center-of-sets type-reduction:

$$y_L=\frac{\sum _{i=1}^r{\underset{\_}{q}}_i{\underset{\_}{\theta }}_i}{\sum _{ i=1}^r{\underset{\_}{q}}_i}={\underset{\_}{\theta }}^T{q}_L,y_U=\frac{\sum _{i=1}^r{\overline{q}}_i{\overline{\theta }}_i}{\sum _{i=1}^r{\overline{q}}_i}={\overline{\theta }}^T{q}_U,$$where $\underset{\_}{\theta }=[{\underset{\_}{\theta }}_1,{\underset{\_}{\theta }}_2\dots ,{\underset{\_}{\theta }}_r{]}^T$, $\overline{\theta }=[{\overline{\theta }}_1,{\overline{\theta }}_2,\dots ,{\overline{\theta }}_r{]}^T$, $q_L=[q_1^L,q_2^L,\dots ,q_r^L{]}^T$ with $q_i^L={\underset{\_}{q}}_i/{\sum }_{j=1}^r{\underset{\_}{q}}_j$, and $q_U=[q_1^U,q_2^U,\dots ,q_r^U{]}^T$ with $q_i^U={\overline{q}}_i/{\sum }_{j=1}^r{\overline{q}}_j$.

$\star$ Output layer: The final defuzzified output is modeled as:

$$y=\frac{y_L+y_U}{2}.$$

The center-of-sets type-reduction method was adopted for defuzzification due to its computational simplicity and numerical stability. Unlike iterative Karnik–Mendel algorithms, this approach efficiently produces a crisp output suitable for online control, making it ideal for multi-agent systems with limited computational resources.

A five-layer IT2FNN structure was adopted because it provides a balance between computational tractability and sufficient modeling capability. Additional layers would significantly increase training complexity without proportional performance gain, whereas fewer layers would reduce approximation accuracy. This configuration is consistent with standard IT2FNN architectures used in control applications.

Optimization and robustness analysis

A distributed integral sliding surface is constructed for every agent in the network as follows:

(3.1) $$s_i(\tau )=K{e}_i(\tau )-{\omega }_i(\tau ),$$where ${\omega }_i(\tau )$ evolves according to the differential equation. It is the integral term that gives the controller its “Integral Sliding Mode” property.

(3.2) $${\dot{\omega }}_i(\tau )=K(C{e}_i(\tau )+Dsat({\upsilon }_{il}(\tau ))),{\omega }_i(0)=K{e}_i(0),$$where ${\upsilon }_{il}(\tau )$ is a linear state feedback control law introduced later in Eq. (3.5).

Taking the time derivative of the sliding surface for the $i$th agent yields:

(3.3) $${\dot{s}}_i(\tau )=KD(sat({\upsilon }_i(\tau ))-sat({\upsilon }_{il}(\tau ))+g_i(\tau )-g_0(\tau )).$$

The main contribution of this work is the proposed adaptive distributed control law, ${\upsilon }_i(\tau )$, which is composed of two primary components. The adaptive distributed IT2 law for the $i$th follower is formulated as follows:

(3.4) $${\upsilon }_i(\tau )={\upsilon }_{il}(\tau )+{\upsilon }_{isw}(\tau ).$$

The linear feedback term ${\upsilon }_{il}(\tau )$ is given by:

(3.5) $${\upsilon }_{il}(\tau )=-K{\eta }_i(\tau ),$$where

(3.6) $${\eta }_i(\tau )=\sum _{j\in {\mathcal{N}}_i^{\psi }}{a}_{ij}^{\psi }(e_i-e_j)+b_i^{\psi }{e}_i.$$

To ensure the system state reaches the designed sliding surface in finite time and maintains reliability against unknown dynamics and disturbances, the nonlinear switching control component, ${\upsilon }_{isw}(\tau )$, is introduced. This component employs a standard SMC sign function along with a proportional term:

(3.7) $${\upsilon }_{isw}(\tau )=-\lambda \mathrm{s}\mathrm{g}\mathrm{n}(s_i(\tau ))-\varrho s_i(\tau ).$$

In this expression, $\lambda ={\hat{g}}_i-g_0+\alpha$, where $\alpha$ is a small positive constant, and $\varrho$ is a selected positive control gain. The matrix K represents the linear feedback gain. The term ${\hat{g}}_i$ denotes the approximation of the unknown nonlinear function $g_i$.

To address the nonlinearity introduced by the saturation function, we impose the following assumption (adapted from Mirzajani, Aghababa & Heydari, 2019):

(3.8) $$|\mathrm{\Delta }({\upsilon }_i)|\le M_{{\upsilon }_i},i=1,2,\dots ,N,$$where $\mathrm{\Delta }({\upsilon }_i)=\mathrm{s}\mathrm{a}\mathrm{t}({\upsilon }_i)-{\upsilon }_i$ represents the saturation-induced deviation, and $M_{{\upsilon }_i}$ represents an unknown but positive constant. This corresponds to saturation limits of physical actuators such as DC motors, propellers, servos, or steering systems. In UAVs or ground robots, for instance, motor torque, thrust, and wheel force are always bounded due to voltage and current limitations. Modeling the input as bounded ensures that the control law never demands an unrealizable signal and can be directly implemented on real hardware.

Proof: To ensure that the system can reach the sliding surface under the control law Eq. (3.4), the potential Lyapunov function is introduced as:

(3.9) $$V_i(\tau )=\frac{1}{2}{s}_i^T(\tau )(KD{)}^{-1}{s}_i(\tau )$$

Differentiating $V_i(\tau )$ with respect to time yields:

$$\begin{array}{rl}{\dot{V}}_i(\tau )& =s_i^T(\tau )(KD{)}^{-1}{\dot{s}}_i(\tau )\\ & =s_i^T(\tau )\left(-\lambda \mathrm{s}\mathrm{g}\mathrm{n}(s_i(\tau ))-\varrho s_i(\tau )+\mathrm{\Delta }({\upsilon }_i(\tau ))+g_i(\tau )-g_0(\tau )\right)\\ & \le s_i^T(\tau )\left(-\lambda \mathrm{s}\mathrm{g}\mathrm{n}(s_i(\tau ))-\varrho s_i(\tau )+M_{{\upsilon }_i}+g_i(\tau )-g_0(\tau )\right)\\ & \le s_i^T(\tau )\left(-\lambda \mathrm{s}\mathrm{g}\mathrm{n}(s_i(\tau ))-\varrho s_i(\tau )+M_{{\upsilon }_i}+{\tilde{g}}_i(\tau )\right)\\ & ={\dot{V}}_i\le -\lambda ||s_i||-\varrho ||s_i|{|}^2+||s_i||(\rho +M{\upsilon }_i)\end{array}$$

$$\zeta =\lambda -(\rho +M{\upsilon }_i).$$

So we have,

$$\begin{array}{rl}{\dot{V}}_i& \le -\zeta \Vert s_i(\tau )\Vert -\varrho \Vert s_i(\tau ){\Vert }^2\\ & \le -C_1{V}_i^{\frac{1}{2}}-C_2{V}_i(\tau )\end{array},$$where $C_1={\zeta }^2/{\lambda }_{max}((KD{)}^{-1})$ and $C_2={\varrho }^2/{\lambda }_{max}((KD{)}^{-1})$.

Then, by Lemma (3.1), $s_i(\tau )$ reaches zero in finite time. The finite-time convergence bound is given by:

(3.10) $$T_i\le \frac{2}{C_2}\ln \left(\frac{C_1+C_2{V}_i(0)}{C_1}\right)$$which completes the proof.

Proof: Upon reaching the sliding mode, the conditions $s_i(\tau )=0$ and ${\dot{s}}_i(\tau )=0$ are satisfied. From Eq. (3.3), the sliding condition ${\dot{s}}_i(\tau )=0$ implies:

(3.12) $$sat({\upsilon }_{isw}(\tau ))+g_i(\tau )-g_0(\tau )=0.$$

From Eqs. (2.5) and (3.12), we observe that it is sufficient to analyze the convergence of the following system:

(3.13) $${\dot{e}}_i(\tau )=C{e}_i(\tau )-DK{\eta }_i(\tau )+D\mathrm{\Delta }{\upsilon }_i.$$

Define the stacked vectors:

$$e(\tau )={\left[\begin{array}{cccc}{e}_1^T(\tau )& e_2^T(\tau )& \dots & e_N^T(\tau )\end{array}\right]}^T,\eta (\tau )={\left[\begin{array}{cccc}{\eta }_1^T(\tau )& {\eta }_2^T(\tau )& \dots & {\eta }_N^T(\tau )\end{array}\right]}^T.$$

Then the closed-loop system becomes:

(3.14) $$\dot{e}(\tau )=(I_N\otimes C-H\otimes BK)e(\tau )+(I_N\otimes D)\mathrm{\Delta }\upsilon (\tau ).$$

Choose the Lyapunov function candidate as:

(3.15) $$V(\tau )=e^T(\tau )(S\otimes T)e(\tau ).$$

Taking the time derivative and simplifying for any $e(\tau )\ne 0$, we obtain:

(3.16) $$\dot{V}(\tau )={\dot{e}}^T(\tau )(S\otimes T)e(\tau )+e^T(\tau )(S\otimes T)\dot{e}(\tau ).$$

Substitute $\dot{e}(\tau )$

$$\begin{array}{rl}\dot{V}(\tau )& ={\left[(I_N\otimes C-H\otimes BK)e(\tau )+(I_N\otimes D)\mathrm{\Delta }\upsilon (\tau )\right]}^T\\ & \cdot (S\otimes T)e(\tau )+e^T(\tau )(S\otimes T)\left[(I_N\otimes C-H\otimes BK)e(\tau )+(I_N\otimes D)\mathrm{\Delta }\upsilon (\tau )\right]\end{array}$$

$$\begin{array}{rl}\dot{V}(\tau )& =\left[e^T(\tau ){(I_N\otimes C-H\otimes BK)}^T+\mathrm{\Delta }{\upsilon }^T(\tau ){(I_N\otimes D)}^T\right](S\otimes T)e(\tau )\\ & +e^T(\tau )(S\otimes T)\left[(I_N\otimes C-H\otimes BK)e(\tau )+(I_N\otimes D)\mathrm{\Delta }\upsilon (\tau )\right]\\ & =e^T(\tau )\left[S\otimes (C^{T}T+PA\text{-}PB{B}^{T}T)\right]e(\tau )+2e^T(\tau )(S\otimes BP)\mathrm{\Delta }\upsilon (\tau ).\end{array}$$

Using Cauchy-Schwarz inequality and the boundedness of $\mathrm{\Delta }\upsilon (\tau )$, we have

$$2e^T(\tau )(S\otimes PB)\mathrm{\Delta }\upsilon (\tau )\le 2||e(\tau )||||S\otimes PB||||\mathrm{\Delta }\upsilon (\tau )||\le 2M_u||S\otimes PB||||e(\tau )||.$$

So,

$$\dot{V}(\tau )\le =-e^T(\tau )(S\otimes I_N)e(\tau )+2M_u||S\otimes PB||||e(\tau )||.$$

Since $S\otimes I_N$ is positive definite, there exists ${\lambda }_{min}(S)>0$ such that:

$$\dot{V}(\tau )\le -{\lambda }_{min}(S)||e(\tau )|{|}^2+2M_u||S\otimes PB||||e(\tau )||.$$

This shows that $\dot{V}(\tau )$ is negative definite for sufficiently large $||e(\tau )||$, ensuring finite-time boundedness and stability of the sliding dynamics. The two terms in the majorant remain independent after factorization since the common term is a scalar multiplier, and no cross-dependence exists between the resulting expressions. This ensures the validity of the bound derived in the final step of the proof.

when configuring a large number of agents, the IT2FNN may generate an excessive number of fuzzy rules. This can lead to computational challenges, often referred to as the “curse of dimensionality,” making the control process impractical. To address this, a practical solution within the control loop is to incorporate a state estimator. By estimating the agents’ states and providing these estimates to both the controller and the IT2FNN, the overall load can be significantly minimized. The overall control strategy based on SMC is schematically presented in Fig. 3.

Simulation results and discussion

To rigorously assess the reliability and applicability of the proposed RGT2FNN-based adaptive sliding mode control framework. All simulations were carried out using MATLAB R2022a (The MathWorks, Natick, MA, USA) on a laptop equipped with an Intel^® Core™ i7-11800H CPU @ 2.30 GHz, 16 GB RAM, and Windows 10 (64-bit) operating system. The simulation time step was set to 0.001 s, and all numerical computations were performed using MATLAB.

The following complex environments are considered to evaluate scalability, adaptability to heterogeneous dynamics, and resilience under communication imperfections and actuator limitations. This section considers a MAS with one virtual leader and five identical follower agents, i.e., $N=5$.

The nonlinear behavior of the agents is modeled as follows:

$$g_0(\tau )=[0.1,0.3{]}^T,g_i(x_i,{\nu }_i,t)=-10\sin (x_i)-0.3{\nu }_i+0.5\sin (0.3t+i)+d_i(t),$$where

$$d_i(t)=0.8\sin (0.4t)+0.3\cos (0.6t)$$represents bounded external disturbances such as wind or load perturbations.

A time-varying communication delay is introduced as

$${\tau }_d(t)=0.1+0.05\sin (0.5t),$$and the delayed information exchange is modeled as

$$x_i(t)=x_j(t-{\tau }_d(t)).$$

Actuator saturation is incorporated through a saturation function applied to each control input:

$$\mathrm{s}\mathrm{a}\mathrm{t}({\upsilon }_{\mathrm{i}}(\tau ))=\{\begin{array}{cc}{[-4,-4]}^T,\hfill & {\upsilon }_i(\tau )<{[-4,-4]}^T\hfill \\ {\upsilon }_i(\tau ),\hfill & {[-4,-4]}^T\le {\upsilon }_i(\tau )\le {[4,4]}^T\hfill \\ {[4,4]}^T,\hfill & {\upsilon }_i(\tau )>{[4,4]}^T\hfill \end{array}.$$

The leader’s starting positions and the follower agents are defined as:

$$\begin{array}{rl}{x}_0(0)& =[2,1.5{]}^T,x_1(0)=[0.5,0{]}^T,x_2(0)=[-0.5,0.3{]}^T\\ x_3(0)& =[0,-0.8{]}^T,x_4(0)=[-1.2,-0.4{]}^T,x_5(0)=[0.8,1{]}^T\end{array}.$$

All agents are initialized with zero velocity.

The desired relative positions with respect to the leader (formation offsets) for each follower are specified as:

$$\begin{array}{rl}{h}_1& =[-2,-1{]}^T,h_2=[2,-1{]}^T,h_3=[2,1{]}^T,\\ h_4& =[-2,1{]}^T,h_5=[0,2.5{]}^T\end{array}.$$

The control gains utilized in the simulation are given by:

$$K=\left[\begin{array}{cccc}18& 0& 9& 5\\ 0& 18& 7& 18\end{array}\right],\alpha =0.002,\varrho =0.015.$$

The interaction structure is represented by an undirected graph illustrated in Fig. 4. The interaction among follower agents is represented by the adjacency matrix C, and the leader’s influence is captured by the leader-follower weight matrix D:

$$C=\left[\begin{array}{ccccc}0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right],D=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right].$$

Figures 5, 6, 7, 8 and 9 illustrate the performance of the proposed control scheme. Specifically, Fig. 5 shows the trajectories of all agents, which successfully converge to the target for mation. Figures 6, 7, 8 and 9 demonstrate the evolution of the position and velocity tracking errors, verifying the effectiveness of the control law in handling bounded inputs. The proposed RGT2FNN–SMC controller ensures that both errors converge asymptotically toward zero despite the presence of bounded input constraints, communication topology switching, and additive disturbances. The position deviation initially exhibits small oscillations during transient phases ( $t<5\mathrm{s}$), corresponding to agents adjusting to the switching network and the uncertainty compensation terms in the fuzzy inference model. After approximately $t=8\mathrm{s}$, all position errors remain confined within $\pm 0.02\mathrm{m}$, and the corresponding velocity deviations fall below $0.01\mathrm{m}/\mathrm{s}$, confirming bounded tracking in the sliding domain.

To evaluate robustness against dynamic communication changes, the same setup is simulated with randomly switching topologies drawn from the following adjacency matrices:

$$A_1=\left[\begin{array}{ccccc}0& 1& 0& 1& 0\\ 1& 0& 1& 0& 1\\ 0& 1& 0& 0& 1\\ 1& 0& 0& 0& 1\\ 0& 1& 1& 1& 0\end{array}\right],A_2=\left[\begin{array}{ccccc}0& 0& 1& 1& 0\\ 0& 0& 0& 1& 1\\ 1& 0& 0& 0& 1\\ 1& 1& 0& 0& 0\\ 0& 1& 1& 0& 0\end{array}\right],A_3=\left[\begin{array}{ccccc}0& 1& 1& 0& 0\\ 1& 0& 0& 0& 1\\ 1& 0& 0& 1& 0\\ 0& 0& 1& 0& 1\\ 0& 1& 0& 1& 0\end{array}\right].$$

The randomly switching communication topologies used in the simulations are illustrated in Fig. 10. Simulation results under these switching scenarios are presented in Figs. 11 and 12. The figures confirm that the proposed IT2FNN approach maintains system stability, ensures convergence to the desired formation, and effectively compensates for non-linearities, even under varying topologies and actuator saturation.

Figures 13 and 14 illustrate the comparative analysis of the classical SMC and the proposed RGT2FNN–SMC control signals, emphasizing the mitigation of the chattering phenomenon. In Fig. 13, the conventional SMC input (red curve) exhibits strong high-frequency oscillations around the switching surface, reaching amplitudes of approximately $\pm 3.2\mathrm{V}$ during the transient phase ( $0$– $4\mathrm{s}$). These rapid sign changes are characteristic of the discontinuous switching term $\mathrm{s}\mathrm{g}\mathrm{n}(s)$ and are known to excite unmodeled actuator dynamics and cause mechanical stress. In contrast, the proposed RGT2FNN–SMC input (blue curve), which replaces the discontinuous term with a smooth hyperbolic tangent function $\tanh (s/\varphi )$ and incorporates adaptive fuzzy estimation, produces a much smoother control profile confined within $\pm 0.8\mathrm{V}$. The adaptive term compensates system uncertainties, thereby reducing the switching gain and effectively suppressing chatter at the actuator output.

The frequency-domain comparison in Fig. 14 further confirms this observation. The FFT analysis shows that high-frequency components above $20\mathrm{H}\mathrm{z}$, dominant in the classical SMC signal, are almost completely eliminated in the RGT2FNN–SMC spectrum. Quantitatively, the RMS control effort decreased by approximately $75\mathrm{\%}$, indicating a substantial reduction in chattering energy without compromising convergence speed.

When external disturbances were introduced during the simulation, the effectiveness of the proposed RGT2FNN–SMC controller in maintaining formation tracking became evident. As illustrated in Figs. 15 and 16, the proposed method exhibits strong resilience to sudden perturbations, showing only small and transient deviations in both position and velocity tracking errors. The system quickly restores the desired trajectories once the disturbance subsides, indicating fast recovery and consistent stability. In contrast, the baseline IT2FNN–SMC experiences larger fluctuations and slower convergence under the same conditions. These results highlight the enhanced disturbance rejection capability of the proposed approach, achieved through the adaptive fuzzy estimation of uncertainties and the integral sliding mode design that ensures reliability and smooth dynamic response even in the presence of time-varying external influences.

Figures 17 and 18 present the sensitivity analysis of the proposed RGT2FNN–SMC controller under $\pm$20% variations in key parameters, including the sliding gain, learning rate, and adaptation coefficient. The shaded regions indicate the performance range under these perturbations, while the solid curve represents the nominal response. The minimal spread of the shadow bands confirms that the controller maintains stable and consistent tracking accuracy, demonstrating low sensitivity to parameter tuning and high adaptability under model and design uncertainties. The proposed hybrid RGT2FNN–SMC framework also exhibits strong scalability potential for real-world applications. Since the control strategy is fully distributed, each agent requires only local neighbor information, which minimizes communication overhead as the network size increases. Furthermore, the RGT2FNN based adaptive estimator operates with low computational complexity, making real-time implementation feasible even in embedded processors or UAV swarms. These characteristics suggest that the proposed approach can be efficiently deployed in large-scale multi-agent systems operating under uncertain and dynamic environments.

The proposed hybrid RGT2FNN–SMC framework also exhibits strong scalability potential for real-world applications. Since the control strategy is fully distributed, each agent requires only local neighbor information, which minimizes communication overhead as the network size increases. Furthermore, the RGT2FNN based adaptive estimator operates with low computational complexity, making real-time implementation feasible even in embedded processors or UAV swarms. These characteristics suggest that the proposed approach can be efficiently deployed in large-scale multi-agent systems operating under uncertain and dynamic environments.

Conclusion

This study presented a novel RGT2FNN–SMC framework for robust formation control of second-order MASs under switching topologies and actuator input constraints. The proposed approach integrates adaptive fuzzy neural estimation with sliding mode control to effectively compensate for unknown nonlinear dynamics and external disturbances while ensuring finite-time convergence and smooth control effort. Unlike conventional IT2FNN–SMC schemes, the proposed RGT2FNN–SMC introduces a gradient-based adaptation mechanism that enhances learning efficiency and stability, enabling the controller to maintain precise formation tracking even under dynamic uncertainties and limited control authority.

Beyond demonstrating robustness, the work establishes a scalable and computationally efficient framework suitable for real-time applications such as cooperative UAVs, autonomous vehicles, and robotic swarms. Simulation studies under switching topologies, input saturation, and disturbance conditions validate the controller’s stability, adaptability, and resilience, confirming its potential for deployment in complex real-world environments. Future research will focus on implementing the proposed method in a hardware-in-the-loop or experimental multi-agent platform, and on extending the adaptive learning structure to support event-triggered communication for reduced network load and enhanced coordination efficiency.

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