Robust Adaptive Control for Discrete-Time Multi-Robot Systems with Actuator and Sensor Attacks

Abstract

This paper addresses the challenges of achieving robust coordination in discrete-time multi-robot systems subject to uncertainties and Byzantine attacks affecting both actuator and sensor channels. Such adversarial disruptions degrade system performance by corrupting control inputs and state measurements, ultimately threatening stability and consensus in networked robotic systems. To overcome these limitations, a novel discrete-time adaptive control framework is proposed that ensures reliable tracking and stability under both uncoupled and coupled robot dynamics. The approach integrates a modified graph-theoretic structure with node-dependent weighting to capture heterogeneous robot interactions, while explicitly modeling attack effects within the system dynamics. An adaptive control law is developed using a nonlinear basis function approximation to handle unknown system uncertainties, along with a dynamic weight update mechanism that compensates for adversarial disturbances in real time. For the uncoupled case, stability is established through a composite Lyapunov function incorporating logarithmic and quadratic terms, guaranteeing boundedness of all closed-loop signals and asymptotic convergence of the tracking error. This framework is further extended to systems with coupled dynamics by introducing an auxiliary estimation mechanism to reconstruct unmeasurable interactions, leading to a unified adaptive controller capable of mitigating both internal uncertainties and external attacks. Rigorous Lyapunov-based analysis demonstrates that the proposed method ensures asymptotic tracking performance despite the presence of Byzantine disturbances. Numerical simulations validate the theoretical results, showing improved resilience, accurate trajectory tracking, and enhanced robustness compared to existing approaches.

1. Introduction

Multi-robot systems are widely studied due to their importance in coordinated control, consensus, and real-world autonomous applications. Consensus control is commonly used to achieve agreement among agents in distributed networks [1]. Bio-inspired and learning-based methods improve individual robot adaptability and system performance [2], while optimization-based techniques enhance robotic and spacecraft configuration efficiency [3]. Time-varying coordination and stability of discrete-time switched systems ensure reliable performance under dynamic conditions [4,5]. Recent studies focus on improving estimation, navigation, and perception in uncertain environments. Multi-sensor fusion enhances localization accuracy [6], while optimization methods improve autonomous decision-making [7]. Intelligent visual navigation and perception improve robotic performance in dynamic environments [8], and grasping methods enhance manipulation in complex scenes [9]. Learning-based attention models further improve perception capability [10]. Multi-agent coordination and formation control improve collective behavior in networked systems [11]. Flexible robotic mechanisms and locomotion strategies enhance adaptability in complex environments [12,13]. Recent advances in intelligent engineering systems have improved the reliability and performance of autonomous technologies. Robust time synchronization methods for integrated navigation systems and advanced machine design strategies for hybrid-excited vernier reluctance linear machines have enhanced accuracy, efficiency, and operational stability in complex environments [14,15]. Machine-learning-based perception techniques have enabled accurate recognition of aircraft taxiing behaviors, thereby improving autonomous airport operations [16]. In addition, efficient mission analysis for multi-asteroid exploration and magnetically switchable adhesive millirobots have expanded the capabilities of aerospace and robotic systems operating in challenging environments [17,18]. Comprehensive studies on force-feedback bilateral teleoperation systems and neurodynamics-based visual servo predictive control have contributed to improved stability, transparency, and trajectory-tracking performance in advanced robotic applications [19,20]. Similarly, fault diagnosis techniques for weak time-varying faults and event-triggered collaborative steering control strategies have enhanced the reliability and efficiency of intelligent mechanical and vehicular systems [21,22]. Coordinated tracking control of integrated wheel-end systems and innovative self-positioning methods for wall-climbing robots have further improved motion coordination and operational precision in autonomous engineering applications [23,24]. Despite these advances, existing studies focus on nominal or uncertain conditions and do not address Byzantine attacks in discrete-time multi-robot systems.

Event-triggered security control methods address denial-of-service attacks in autonomous systems [25], while resource allocation and edge-learning frameworks improve distributed intelligence efficiency [26]. Neural-learning-based tracking control methods enhance performance under nonlinear constraints [27], and zeroing neural networks improve solution accuracy for dynamic robotic control problems [28]. Self-powered sensing and robust tribological systems further enhance autonomy in sensing environments [29]. However, these methods mainly focus on learning efficiency and fault tolerance rather than Byzantine attack resilience in discrete-time multi-agent networks. Advanced control theory and game-theoretic methods have been developed for constrained and uncertain systems. Nash-equilibrium-based control improves coordination in bimanual manipulation systems under constraints [30]. Recent advances in aerospace and intelligent control systems have significantly improved performance under uncertainty. Robust adaptive dynamic programming for hypersonic vehicles and distributed fault-tolerant formation control for unmanned aerial vehicle swarms have enhanced stability, adaptability, and coordination in complex flight environments [31,32]. In robotics and autonomous systems, real-time motion planning with dynamic obstacle avoidance and multi-sensor fusion-based perception have strengthened safety, accuracy, and environmental understanding in intelligent platforms [33,34]. Multimodal agent artificial intelligence further highlights the growing importance of integrated learning and decision-making frameworks in complex autonomous systems [35]. In control theory and estimation, fixed-time disturbance rejection methods and neural-network-based computational solvers have improved robustness and efficiency for nonlinear and uncertain systems [36,37]. Advanced robotic path planning and force regulation strategies have further enhanced interaction accuracy and operational stability in dynamic environments [38]. Transformer-based reinforcement learning approaches and spacecraft attitude control strategies have strengthened decision-making capabilities and improved resilience under dynamic and uncertain conditions [39,40]. For navigation and fault-tolerant systems, resilient global navigation satellite system and inertial measurement unit based navigation methods and fuzzy fault-tolerant control approaches have improved reliability under disturbances and system variations [41,42]. Similarly, adaptive unmanned aerial vehicle attitude control under multiple faults and deep reinforcement learning-based shared control strategies have enhanced robustness and human–machine interaction performance in safety-critical applications [43,44]. Sample-efficient reinforcement learning techniques continue to advance data-efficient decision-making in complex environments [45]. In materials and aerodynamic intelligence, cooperative reinforcement learning with human feedback and variable-stiffness supramolecular materials have contributed to improved adaptability and multifunctional robotic design [46,47]. Finally, deep learning-based aerodynamic surrogate models and diffusion-based inverse design methods have significantly advanced aerospace optimization and high-fidelity aerodynamic design processes [48,49]. Despite these advances, most existing works assume secure communication and benign environments and ignore Byzantine attacks in discrete-time multi-agent systems.

Adaptive fuzzy control strategies further enhance predefined-time stability in nonlinear systems [50]. Nevertheless, most of these approaches do not explicitly consider distributed multi-robot consensus under simultaneous sensor and actuator Byzantine attacks. Recent studies develop adaptive fuzzy, event-triggered, and consensus-based control methods for uncertain multi-agent systems, improving stability and coordination under communication constraints [51]. Model-free and data-driven resilient control approaches enhance robustness against denial-of-service and uncertainties in networked systems [52]. However, most works still assume partially reliable communication and do not consider simultaneous adversarial corruption. Advanced methods in path planning and finite-time control improve performance in complex environments [53,54]. Sliding mode and data-driven control strategies enhance robustness under disturbances and constraints [55,56]. Neural-network and observer-based control techniques improve tracking accuracy and force regulation under uncertainties [57,58]. Adaptive control methods further improve stability in nonlinear robotic systems [59,60]. Zeroing neural network approaches enhance precision in dynamic system solving and robotic control [61]. Recent research on perception, safety, and localization focuses on obstacle detection, braking systems, and constraint handling to improve system reliability [62,63]. Stability and localization methods enhance performance under uncertainty and noise [64,65]. Robust constraint-following and adaptive control methods improve performance in uncertain robotic systems [66]. Intelligent manipulation and soft robotic systems enhance adaptability in complex environments [67,68]. Motion estimation and constraint-based control methods improve accuracy under sensor errors and nonlinear constraints [69,70]. Human–machine interaction and safety-aware planning strategies enhance operational reliability in shared environments [71]. Recent advances in intelligent decision-making, control, and autonomous systems further improve large-scale coordination and safety. Reinforcement learning-based scheduling methods enhance efficiency in complex multi-agent systems [72]. Aerodynamic and dynamic analysis methods improve system modeling and prediction accuracy [73]. Distributed control and platooning strategies enhance coordination under switching communication topologies [74]. Smart sensing and adaptive material systems improve structural flexibility and sensing capability in robotic systems [75]. Advanced fuzzy and force control methods improve precision and stability under nonlinear constraints [76]. Safety-critical planning and learning-based control methods enhance obstacle avoidance and system safety [77]. Graph-based and intelligent trajectory optimization methods improve multi-agent coordination in complex environments [78]. Further developments in robotic perception and manipulation include bio-inspired millirobot systems that enhance multimodal motion and manipulation in constrained environments [79]. Multi-object tracking and sonar-based vision systems improve perception and navigation in complex environments [80]. Long-term path planning and robot motion theory advancements improve navigation efficiency and theoretical understanding [81]. Adaptive neural and sliding mode control strategies enhance robustness under uncertainties [82]. Vision-based recognition and intelligent perception systems improve system reliability in dynamic environments [83]. Resilience-based system design methods further enhance robustness and lifetime performance under uncertainties [84]. Recent works on adaptive intelligent control have improved tracking accuracy in precision systems under uncertainty. Recent works on adaptive intelligent control improve precision and tracking accuracy in linear stages under uncertainty. In [85,86], interval type-2 fuzzy logic and laser-based feedback methods are used to enhance positional performance under noise and nonlinearities. However, these studies focus on single-system control and do not address multi-robot coordination, coupled dynamics, or Byzantine attack resilience, which are the focus of this work. However, these studies focus on single-system control and do not consider multi-robot coordination, coupled dynamics, or Byzantine attack resilience, which are addressed in this work. Nevertheless, despite these advances, existing approaches do not explicitly address discrete-time multi-robot consensus under simultaneous sensor and actuator Byzantine attacks with coupled dynamics, which motivates this work.

1.1. Motivation

Current discrete-time adaptive control strategies for multi-robot systems are highly vulnerable to Byzantine attacks, where malicious robots inject false information into actuator and sensor channels. Such adversarial behavior significantly degrades consensus, coordination, and tracking performance in networked robotic systems. In tightly coupled multi-robot networks, faults in a single robot can propagate through communication links, leading to cascading effects that threaten stability and safety in coordinated navigation and distributed tasks. Existing discrete-time adaptive multi-robot frameworks, including recent work such as [87], do not clearly distinguish between uncoupled and coupled dynamics under adversarial conditions. This limits their applicability, as both structures require fundamentally different stability and control treatments. Moreover, many existing results are developed in continuous-time settings, whereas practical implementations are inherently discrete-time, motivating the need for rigorous discrete-time adaptive control with formal stability guarantees under attacks.

1.2. Novelty

The main novelty of this work lies in developing a unified discrete-time adaptive control framework for multi-robot systems under Byzantine attacks that explicitly distinguishes between uncoupled and coupled robot dynamics. Unlike existing methods such as [87], which mainly address uncertainty in a simplified interaction structure, clearly separate uncoupled and coupled dynamics under the same adversarial framework, and explicitly incorporate Byzantine attacks in both actuator and sensor channels, introduces node-dependent graph-based interaction modeling for heterogeneous multi-robot networks, and provides a unified Lyapunov-based discrete-time stability framework for both cases.

1.3. Contributions

The main contributions of this paper are as follows:

• A novel discrete-time adaptive control framework is developed for multi-robot systems subject to Byzantine attacks affecting both actuator and sensor channels, ensuring robust tracking and consensus performance under adversarial conditions.
• A modified graph-theoretic interaction model with node-dependent weighting is introduced to capture heterogeneous inter-robot coupling and communication effects in the presence of malicious disturbances.
• A nonlinear basis function approximation-based adaptive control law is designed to handle unknown system uncertainties, combined with a dynamic weight update mechanism that compensates for Byzantine-induced distortions in real time.
• For uncoupled multi-robot dynamics, a composite Lyapunov function incorporating logarithmic and quadratic terms is constructed to guarantee boundedness and asymptotic convergence of tracking errors.
• For coupled multi-robot dynamics, an auxiliary estimation mechanism is introduced to reconstruct unmeasurable inter-robot interactions, leading to a unified control framework applicable to both uncoupled and coupled cases.
• Rigorous Lyapunov-based analysis is provided to establish asymptotic stability and consensus performance under Byzantine attacks, and simulation results demonstrate improved robustness and tracking accuracy compared with existing discrete-time multi-robot methods.

• Section 1 presents the literature review and introduces existing work on discrete-time adaptive control systems.
• Section 2 defines the mathematical notation and preliminary concepts required for the subsequent analysis.
• Section 3 formulates the problem of discrete-time adaptive control for multi-robot systems under Byzantine attacks.
• Section 4 describes the modeling and characterization of Byzantine attacks in the considered framework.
• Section 5 develops the control design and uncertainty handling for robot-base dynamics under Byzantine attacks.
• Section 6 extends the formulation to coupled multi-robot dynamics and presents the corresponding adaptive compensation under attack conditions.
• Section 7 provides numerical simulations and validates the effectiveness of the proposed method through examples.
• Section 8 concludes the paper and summarizes the main findings and contributions.

2. Notation and Mathematical Precondition

Let G be an undirected graph consisting of a collection of vertices ${\mathcal{W}}_G=\{1,2,\dots ,m\}$ and a set of edges $F\subseteq {\mathcal{W}}_G\times {\mathcal{W}}_G$. The graph G is said to be connected if any two different robots in the network, one can find a limited series of directly connected nodes that link them together through a continuous path. The degree matrix $D\left(G\right)\in {\mathbb{R}}^{m\times m}$ is a diagonal matrix where each diagonal entry $D_{ii}$ represents the degree of node i, i.e., the number of edges incident to it. Formally,

$$D\left(G\right)=\mathrm{diag}\left(d\right),\mathrm{where}d={[d_1,d_2,\dots ,d_m]}^T.$$

The adjacency matrix $A\left(G\right)\in {\mathbb{R}}^{m\times m}$ is defined such that

$$A_{ij}\left(G\right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}(i,j)\in F,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

This matrix captures the connectivity between nodes: a value of 1 indicates the presence of an edge between nodes i and j, while 0 indicates no direct connection. The Laplacian matrix $L\left(G\right)$ of the graph is then defined as the difference between the degree matrix and the adjacency matrix:

$$L\left(G\right)=D\left(G\right)-A\left(G\right).$$

This matrix plays a crucial role in various applications, including spectral graph theory and network analysis. To incorporate node-specific weights, we introduce a vector $N={[n_1,n_2,\dots ,n_m]}^T\in {\mathbb{R}}_{+}^m$, where each $n_i>0$ represents a positive weight associated with node i. Using this, we define a modified degree matrix $\overline{D}(G,N)\in {\mathbb{R}}^{m\times m}$ as follows,

$$\overline{D}(G,N)=\mathrm{diag}\left(A\left(G\right)N\right)·\mathrm{diag}{\left(N\right)}^{-1}.$$

Here, $A\left(G\right)N$ computes the weighted sum of neighboring nodes for each node, and the division by $\mathrm{diag}\left(N\right)$ normalizes these sums by the corresponding node weights. The corresponding modified Laplacian matrix $L(G,N)$ is then given by,

$$L(G,N)=\overline{D}(G,N)-A\left(G\right).$$

This modified Laplacian accounts for the node-specific weights, providing a more nuanced representation of the graph’s structure, especially useful in contexts where nodes have varying importance or influence.

Lemma 1.

In a leader-follower framework, consider a diagonal matrix $M\in {\mathbb{R}}^{m\times m}$ defined as $M=\mathrm{diag}(l_1,l_2,\dots ,l_n)$, where each entry satisfies $0<m_i<1$ for all $i=1,\dots ,m$, under the condition that at least one scalar $k_i=1$. Under this assumption, the Laplacian matrix corresponding to the leader-follower network can be suitably modified to incorporate varying node roles, resulting in an adjusted structure given by $K(G,N)=L(G,N)+F$.

3. Problem Formulation

In this section, a discrete-time adaptive control framework is developed for multi-robot systems operating in the presence of robot-base variations, nonlinear uncertainties, and Byzantine cyber-attacks affecting both actuator and sensor channels. The proposed control strategy is designed to assign distinct target positions to individual robots while preserving system stability, cooperative interaction, and resilient coordination under adversarial disturbances. Both uncertain multi-robot systems without coupled dynamics and systems with coupled dynamics are considered under communication and measurement attacks. The overall framework aims to enhance cooperative behavior and improve robustness against malicious interference in networked robotic environments.

Consider an m-robot Multi-Robot System (MRS), where the dynamics of the $p^{th}$ robot are given by:

$$z_p(t+1)=z_p\left(t\right)+b(t,z_p\left(t\right))+v_p\left(t\right)+g_p\left(t\right)$$

(1)

where $z_p\left(t\right)\in {\mathbb{R}}^n$ denotes the state of the follower robot, $v_p\left(t\right)$ represents the control input, $b(t,z_p\left(t\right))$ characterizes unknown nonlinear dynamics, and $g_p\left(t\right)$ represents an external disturbance signal capturing the combined effect of cyber-attacks on the system.

Similarly, the leader robot dynamics are described as:

$$z_0(t+1)=z_0\left(t\right)+b(t,z_0\left(t\right))+v_0\left(t\right)$$

(2)

where $z_0\left(t\right)$ is the leader state, $v_0\left(t\right)$ is the leader input, and $b(t,z_0\left(t\right))$ denotes nonlinear dynamics.

4. Attack Effect on MRS

In multi-robot systems, cyber-attacks can significantly degrade communication and coordination performance among interconnected agents. In practical networked environments, attackers may inject malicious signals into actuator and sensor channels, which can deteriorate stability, synchronization, and consensus behavior. Therefore, modeling attack influence is essential for robust controller design.

The actuator attack on each robot is modeled as:

$${\mathcal{U}}_p^c\left(t\right)={\mathcal{U}}_p\left(t\right)+{\lambda }_p{\mathcal{U}}_p^b\left(t\right)$$

(3)

where ${\mathcal{U}}_p\left(t\right)$ is the nominal control input, ${\mathcal{U}}_p^c\left(t\right)$ is the corrupted control input, ${\mathcal{U}}_p^b\left(t\right)$ is the actuator attack signal, and ${\lambda }_p\in \{0,1\}$ is the attack indicator variable. Similarly, sensor attacks are described as:

$$z_p^c\left(t\right)=z_p\left(t\right)+{\lambda }_p{z}_p^b\left(t\right)$$

(4)

where $z_p\left(t\right)$ is the true measurement, $z_p^c\left(t\right)$ is the corrupted measurement, and $z_p^b\left(t\right)$ is the injected sensor attack signal. The overall attack influence acting on the multi-robot system is modeled as a unified disturbance term:

$$g_p\left(t\right)=g_p^a\left(t\right)+g_p^s\left(t\right)$$

(5)

where $g_p^a\left(t\right)$ represents the actuator attack effect and $g_p^s\left(t\right)$ represents the sensor-induced attack propagation through the communication network. It is important to emphasize that the disturbance $g_p\left(t\right)$ is assumed to be unknown and not measurable by the controller. Therefore, it cannot be directly used in feedback design. To address this issue, an adaptive disturbance observer is introduced to estimate the unknown attack signal online. The estimated disturbance is denoted as ${\widehat{g}}_p\left(t\right)$, and only this estimated value is used in the controller design for compensation.

The estimation error is defined as:

$${\tilde{g}}_p\left(t\right)=g_p\left(t\right)-{\widehat{g}}_p\left(t\right)$$

The controller is therefore designed using ${\widehat{g}}_p\left(t\right)$ instead of the unknown true disturbance $g_p\left(t\right)$, ensuring implementability.

To ensure analytical tractability, the disturbance is assumed to be bounded as:

$$\parallel g_p\left(t\right)\parallel \le \overline{g}$$

(6)

where $\overline{g}>0$ is an unknown finite constant.

The proposed formulation provides a realistic and implementable representation of cyber-attack effects in multi-robot systems and forms the basis for the subsequent observer-based adaptive control design.

Remark 1.

Cyber-attacks may significantly degrade system performance by introducing false control actions and corrupted sensor measurements. These disturbances may increase tracking errors, slow down convergence, and destabilize coordination among robots. Therefore, an adaptive observer-based compensation mechanism is necessary to ensure robust consensus and maintain reliable communication within the networked system. In the presence of high-frequency attack components, the proposed observer is designed to attenuate fast variations while accurately reconstructing the slowly varying attack profile, thereby improving robustness against rapidly changing malicious signals.

5. Addressing Robot-Base Instability in Graph-Based Models

In this section, a discrete-time adaptive control framework is developed for multi-robot systems without dynamic coupling, where each robot evolves independently. The system considers unknown nonlinearities, robot-base variations, and Byzantine cyber-attacks affecting both actuator and sensor channels. The objective is to guarantee stable tracking performance and resilient coordination despite adversarial disturbances.

The dynamics of each robot are described as:

$$y_p(t+1)=y_p\left(t\right)+B\omega \left(y_p\left(t\right)\right)+{\mathcal{U}}_p\left(t\right)+g_p\left(t\right)$$

(7)

where $y_p\left(t\right)\in {\mathbb{R}}^n$ is the state of the $p^{th}$ robot, ${\mathcal{U}}_p\left(t\right)$ is the control input, B is an unknown weight matrix, $\omega (·)$ is a known bounded nonlinear basis function satisfying $\parallel \omega \left(y_p\left(t\right)\right)\parallel \le I_c+I\parallel y_p\left(t\right)\parallel$, and $g_p\left(t\right)$ represents the unknown attack/disturbance term. The attack signal is assumed to be unknown and not measurable by the controller. Therefore, an adaptive disturbance observer is introduced to estimate it online, and only the estimated value is used in the control design.

The reference model is defined as:

$$y_{r_p}(t+1)=y_{r_p}\left(t\right)-e\sum _{l\in {\mathcal{N}}_p}\left(y_{r_p}\left(t\right)-y_{r_l}\left(t\right)\right)-e{t}_p\left(y_{r_p}\left(t\right)-c\left(t\right)\right)$$

(8)

where $c\left(t\right)$ is a bounded reference command and e satisfies $e<{\displaystyle \frac{1}{max\left(d\right)+1}}$, with d representing the maximum degree of the interaction graph.

The adaptive control law is designed as:

$${\mathcal{U}}_p\left(t\right)=-e\sum _{l\in {\mathcal{N}}_p}\left(y_p\left(t\right)-y_l\left(t\right)\right)-e{t}_p\left(y_p\left(t\right)-c\left(t\right)\right)-{\widehat{B}}_p\left(t\right)\omega \left(y_p\left(t\right)\right)-{\widehat{g}}_p\left(t\right)$$

(9)

where ${\widehat{B}}_p\left(t\right)$ and ${\widehat{g}}_p\left(t\right)$ denote the estimates of the unknown weight matrix and disturbance, respectively.

The estimation errors are defined as:

$${\tilde{B}}_p\left(t\right)=B-{\widehat{B}}_p\left(t\right),{\tilde{g}}_p\left(t\right)=g_p\left(t\right)-{\widehat{g}}_p\left(t\right)$$

Substituting the control law into the system dynamics yields the closed-loop system:

$$y_p(t+1)=y_p\left(t\right)-e\sum _{l\in {\mathcal{N}}_p}\left(y_p\left(t\right)-y_l\left(t\right)\right)-e{t}_p\left(y_p\left(t\right)-c\left(t\right)\right)-{\tilde{B}}_p\left(t\right)\omega \left(y_p\left(t\right)\right)+{\tilde{g}}_p\left(t\right)$$

(10)

The tracking error is defined as:

$$h_p\left(t\right)=y_p\left(t\right)-y_{r_p}\left(t\right)$$

The corresponding error dynamics are obtained as:

$$h_p(t+1)=h_p\left(t\right)-e\sum _{l\in {\mathcal{N}}_p}\left(h_p\left(t\right)-h_l\left(t\right)\right)-e{t}_p{h}_p\left(t\right)-{\tilde{B}}_p\left(t\right)\omega \left(y_p\left(t\right)\right)+{\tilde{g}}_p\left(t\right)$$

(11)

Let $h\left(t\right)={[h_1\left(t\right),h_2\left(t\right),\dots ,h_m\left(t\right)]}^T$. Then the compact form of the error dynamics is given by:

$$h(t+1)=(I-eF)h\left(t\right)-{\tilde{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)+\tilde{g}\left(t\right)$$

(12)

where F denotes the Laplacian-based interaction matrix of the communication graph.

Let $P>0$ be the solution of the Lyapunov equation:

$${(I-eF)}^{T}P(I-eF)-P=-R$$

(13)

where $R>0$.

The adaptive parameter update law for the weight matrix is given by:

$$\widehat{B}(t+1)=\widehat{B}\left(t\right)+{\displaystyle \frac{\gamma }{1+\beta h^T\left(t\right)Ph\left(t\right)}}h\left(t\right){\omega }^T\left(y\left(t\right)\right)$$

(14)

The attack estimation law is defined as:

$$\widehat{g}(t+1)=\widehat{g}\left(t\right)+\sigma h\left(t\right)$$

(15)

where $\gamma ,\beta ,\sigma >0$ are tuning parameters.

Remark 2.

The proposed attack observer in Equation (15) is designed as a discrete-time gradient-type update driven by the tracking error $h\left(t\right)$. The intuition is that any mismatch between the actual and desired system behavior is reflected in $h\left(t\right)$, which implicitly contains information about the unknown disturbance $g_p\left(t\right)$. Therefore, by feeding back $h\left(t\right)$ into the estimator, the proposed update law adjusts $\widehat{g}\left(t\right)$ in the direction that reduces the estimation error $\tilde{g}\left(t\right)=g\left(t\right)-\widehat{g}\left(t\right)$. From the estimation error definition, the estimation dynamics are driven by the system tracking error, ensuring that whenever $h\left(t\right)\to 0$, the estimation error remains bounded. The gain parameter $\sigma >0$ determines the convergence speed of the estimator. A larger value of σ enables faster tracking of time-varying and high-frequency disturbance components, while maintaining boundedness of the closed-loop system.

The resulting closed-loop error dynamics are expressed as:

$$h(t+1)=(I-eF)h\left(t\right)-{\tilde{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)+\tilde{g}\left(t\right)$$

(16)

Theorem 1.

Assume that the uncertain discrete-time multi-robot system described in Equation (7) together with the reference model given in Equation (8) satisfies the bounded nonlinear function condition:

$$\parallel \omega \left(y\left(t\right)\right)\parallel \le l_c+l\parallel y\left(t\right)\parallel ,$$

and the attack signal satisfies $\parallel g_p\left(t\right)\parallel \le \overline{g},$ where $l_c>0$, $l>0$, and $\overline{g}>0$ are finite constants. Also, assume that there exists a symmetric positive definite matrix $P=P^T>0$ satisfying

$${(I-eF)}^{T}P(I-eF)-P=-R,$$

where $R=R^T>0$, and the control gains are selected such that the stability conditions derived in the Lyapunov analysis are satisfied. Then, under the adaptive control law given in Equation (14) and the weight update law in Equation (15), the closed-loop system is Lyapunov stable and all signals remain bounded. Moreover, the tracking error satisfies

$$\underset{t\to \infty }{lim}h\left(t\right)=0.$$

(17)

Proof. 

To analyze the stability of the closed-loop dynamics represented by Equations (12) and (16), consider the following Lyapunov candidate function:

$$W(h,\overline{B})=l^{-1}ln\left(1+\beta h^T\left(t\right)Ph\left(t\right)\right)+{\alpha }^{-1}\mathrm{tr}\left({\overline{B}}^T\left(t\right)\overline{B}\left(t\right)\right),$$

(18)

where $l>0$, $\alpha >0$, $\beta >0$, and $P=P^T>0$. Since both terms in Equation (18) are positive definite, it follows that $W(0,0)=0,$ and $W(h,\overline{B})>0,\forall (h,\overline{B})\ne (0,0).$

The forward Lyapunov difference is defined as:

$$\Delta W\left(t\right)=W(t+1)-W\left(t\right).$$

First, consider the logarithmic term:

$$\Delta W_1=W_1\left(h(t+1)\right)-W_1\left(h\left(t\right)\right).$$

(19)

Using the error dynamics in Equation (12):

$$h(t+1)=(I-eF)h\left(t\right)-{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)+{\tilde{g}}_p\left(t\right),$$

we obtain:

$$\begin{array}{cc}\hfill \Delta W_1=& l^{-1}ln(1+\beta [(I-eF)h\left(t\right)\hfill \\ \hfill & -{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)+{\tilde{g}}_p{\left(t\right)]}^{T}P\hfill \\ \hfill & ·[(I-eF)h\left(t\right)-{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)+{\tilde{g}}_p\left(t\right)])\hfill \\ \hfill & -l^{-1}ln\left(1+\beta h^T\left(t\right)Ph\left(t\right)\right).\hfill \end{array}$$

(20)

Applying the logarithmic property:

$$ln\left(a\right)-ln\left(b\right)=ln\left({\displaystyle \frac{a}{b}}\right),$$

together with the inequality:

$$ln(1+x)\le x,x>-1,$$

yields:

$$\begin{array}{cc}\hfill \Delta W_1\le & {\displaystyle \frac{l^{-1}\beta }{1+\beta h^T\left(t\right)Ph\left(t\right)}}(h^T\left(t\right){(I-eF)}^{T}P(I-eF)h\left(t\right)\hfill \\ \hfill & -h^T\left(t\right)Ph\left(t\right)+{\omega }^T\left(y\left(t\right)\right)\overline{B}\left(t\right)P{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)\hfill \\ \hfill & -2{\omega }^T\left(y\left(t\right)\right)\overline{B}\left(t\right)P(I-eF)h\left(t\right)\hfill \\ \hfill & +2{\tilde{g}}_p^T\left(t\right)\left(t\right)P(I-eF)h\left(t\right)\hfill \\ \hfill & +2{\tilde{g}}_p^T\left(t\right)\left(t\right)P{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)\hfill \\ \hfill & +{\tilde{g}}_p^T\left(t\right)\left(t\right)P{\tilde{g}}_p\left(t\right)).\hfill \end{array}$$

(21)

Using the discrete-time Lyapunov equation:

$${(I-eF)}^{T}P(I-eF)-P=-R,$$

Equation (21) becomes:

$$\begin{array}{cc}\hfill \Delta W_1\le & {\displaystyle \frac{l^{-1}\beta }{1+\beta h^T\left(t\right)Ph\left(t\right)}}(-h^T\left(t\right)Rh\left(t\right)\hfill \\ \hfill & +{\omega }^T\left(y\left(t\right)\right)\overline{B}\left(t\right)P{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)\hfill \\ \hfill & -2{\omega }^T\left(y\left(t\right)\right)\overline{B}\left(t\right)P(I-eF)h\left(t\right)\hfill \\ \hfill & +2{\tilde{g}}_p^T\left(t\right)\left(t\right)P(I-eF)h\left(t\right)\hfill \\ \hfill & +2{\tilde{g}}_p^T\left(t\right)\left(t\right)P{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)\hfill \\ \hfill & +{\tilde{g}}_p^T\left(t\right)\left(t\right)P{\tilde{g}}_p\left(t\right)).\hfill \end{array}$$

(22)

Next, consider the adaptive weight estimation term:

$$W_2={\alpha }^{-1}\mathrm{tr}\left({\overline{B}}^T\left(t\right)\overline{B}\left(t\right)\right).$$

(23)

The forward difference is given by:

$$\Delta W_2=W_2(t+1)-W_2\left(t\right).$$

(24)

Substituting the adaptive update law from Equation (15) into Equation (24) yields:

$$\begin{array}{cc}\hfill \Delta W_2\le & {\displaystyle \frac{l^{-1}\beta }{1+\beta h^T\left(t\right)Ph\left(t\right)}}\hfill \\ \hfill & ·{\omega }^T\left(y\left(t\right)\right)\overline{B}\left(t\right)(-2{\rho }_1+{\rho }_r{\rho }_1){\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right).\hfill \end{array}$$

(25)

Combining Equations (22) and (25) gives:

$$\begin{array}{cc}\hfill \Delta W\le & {\displaystyle \frac{l^{-1}\beta }{1+\beta h^T\left(t\right)Ph\left(t\right)}}(-h^T\left(t\right)Rh\left(t\right)\hfill \\ \hfill & +{\omega }^T\left(y\left(t\right)\right)\overline{B}\left(t\right)\left[P-2{\rho }_1+{\rho }_r{\rho }_1\right]{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)\hfill \\ \hfill & -2{\omega }^T\left(y\left(t\right)\right)\overline{B}\left(t\right)P(I-eF)h\left(t\right)\hfill \\ \hfill & +2{\tilde{g}}_p^T\left(t\right)\left(t\right)P(I-eF)h\left(t\right)\hfill \\ \hfill & +2{\tilde{g}}_p^T\left(t\right)\left(t\right)P{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)\hfill \\ \hfill & +{\tilde{g}}_p^T\left(t\right)\left(t\right)P{\tilde{g}}_p\left(t\right)).\hfill \end{array}$$

(26)

Now, applying Young’s inequality:

$$2x^{T}y\le \mathsf{\Omega }{x}^{T}x+{\displaystyle \frac{1}{\mathsf{\Omega }}}{y}^{T}y,$$

for $\mathsf{\Omega }>0$, the cross term satisfies:

$$\begin{array}{cc}\hfill -2{\omega }^T\left(y\left(t\right)\right)\overline{B}\left(t\right)P(I-eF)h\left(t\right)\le & \mathsf{\Omega }{h}^T\left(t\right){(I-eF)}^{T}P(I-eF)h\left(t\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Omega }}}{\omega }^T\left(y\left(t\right)\right)\overline{B}\left(t\right)P{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right).\hfill \end{array}$$

(27)

Similarly, using the bounded attack assumption together with the Cauchy–Schwarz inequality gives:

$$2{\tilde{g}}_p^T\left(t\right)\left(t\right)P(I-eF)h\left(t\right)\le {ϵ}_1{\parallel h\left(t\right)\parallel }^2+c_1{\overline{g}}^2,$$

and

$$2{\tilde{g}}_p^T\left(t\right)\left(t\right)P{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)\le {ϵ}_2{\parallel {\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)\parallel }^2+c_2{\overline{g}}^2,$$

where ${ϵ}_1>0$, ${ϵ}_2>0$, $c_1>0$, and $c_2>0$ are finite constants. Therefore, Equation (26) can be rewritten as:

$$\Delta W\le -{\lambda }_1{\parallel h\left(t\right)\parallel }^2-{\lambda }_2{\parallel {\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)\parallel }^2+\chi {\overline{g}}^2,$$

(28)

where ${\lambda }_1>0$, ${\lambda }_2>0$, and $\chi >0$ are positive constants. In the attack-free case, namely ${\tilde{g}}_p\left(t\right)=0$, Equation (28) reduces to:

$$\Delta W\le -{\lambda }_1{\parallel h\left(t\right)\parallel }^2<0.$$

(29)

Since $\Delta W<0$, the Lyapunov function is monotonically decreasing and bounded from below. Hence, all closed-loop signals remain bounded and:

$$h\left(t\right)\in {\mathsf{\ell }}_2\cap {\mathsf{\ell }}_{\infty }.$$

Consequently,

$$\underset{t\to \infty }{lim}h\left(t\right)=0.$$

(30)

Therefore, the tracking error asymptotically converges to zero, and the closed-loop system is Lyapunov stable. □

Remark 3.

The use of a Lyapunov function that incorporates both a logarithmic term and a trace-based adaptation component provides a novel way to handle nonlinearities and uncertainties in discrete-time multi-robot systems. This specific structure not only guarantees positive definiteness and proper boundedness but also facilitates tractable analysis under adaptive laws. The monotonic decrease of the Lyapunov difference supports the claim of stability without requiring overly conservative assumptions.

Remark 4.

The asymptotic convergence of the tracking error, as shown in Equation (28), confirms that the adaptive control design can achieve reliable performance over time. The learning mechanism embedded in the controller adjusts the parameter estimates in real time, ensuring the output of the system remains aligned with the desired trajectory, even when subjected to variations in internal dynamics or external signals. This is particularly valuable in cooperative robotics, where consistent coordination is critical.

6. Adaptive Control Strategies for Multi-Robot Systems with Base Variability and Coupled Dynamics

In this section, a discrete-time adaptive control framework is developed for multi-robot systems with coupled dynamics and unknown nonlinearities under cyber-attack disturbances. Each robot is assumed to operate under local dynamics while interacting through coupled subsystems, which capture internal environmental and structural dependencies. The objective is to ensure stable tracking performance and resilient coordination in the presence of uncertainties and malicious disturbances. The dynamics of the $p^{th}$ robot are described as:

$$y_p(t+1)=y_p\left(t\right)+B_p\omega \left(y_p\left(t\right)\right)+M_{up}\left(t\right)+{\mathcal{U}}_p\left(t\right)+g_p\left(t\right),$$

(31)

with initial condition $y_p\left(0\right)=y_{p0}$, where $y_p\left(t\right)\in {\mathbb{R}}^n$ is the system state, $B_p$ is an unknown weight matrix, $\omega (·)$ is a known bounded nonlinear basis function, ${\mathcal{U}}_p\left(t\right)$ is the control input, and $g_p\left(t\right)$ represents an unknown external disturbance caused by cyber-attacks. The coupled dynamic subsystem is defined as:

$${\mu }_p(t+1)=a_{up}{\mu }_p\left(t\right)+s_{up}{y}_p\left(t\right),$$

(32)

$$M_{up}\left(t\right)=x_{up}{\mu }_p\left(t\right),$$

(33)

where ${\mu }_p\left(t\right)$ represents the internal coupled state, and $a_{up},s_{up},x_{up}$ are known constant parameters. Since both the nonlinear uncertainties and attack disturbances are unknown, an adaptive disturbance observer is introduced. The estimated states are denoted as ${\widehat{B}}_p\left(t\right)$ and ${\widehat{\mu }}_p\left(t\right)$, while the estimated attack signal is ${\widehat{g}}_p\left(t\right)$. The adaptive control law is designed as:

$${\mathcal{U}}_p\left(t\right)=-e\sum _{l\in {\mathcal{N}}_p}\left(y_p\left(t\right)-y_l\left(t\right)\right)-e{t}_p\left(y_p\left(t\right)-c\left(t\right)\right)-{\widehat{B}}_p\left(t\right)\omega \left(y_p\left(t\right)\right)-x_{up}{\widehat{\mu }}_p\left(t\right)-{\widehat{g}}_p\left(t\right),$$

(34)

where ${\widehat{B}}_p\left(t\right)$, ${\widehat{\mu }}_p\left(t\right)$, and ${\widehat{g}}_p\left(t\right)$ denote the estimates of unknown parameters, coupled dynamics, and attack disturbances, respectively. The estimation errors are defined as:

$${\tilde{B}}_p\left(t\right)=B_p-{\widehat{B}}_p\left(t\right),{\tilde{\mu }}_p\left(t\right)={\mu }_p\left(t\right)-{\widehat{\mu }}_p\left(t\right),{\tilde{g}}_p\left(t\right)=g_p\left(t\right)-{\widehat{g}}_p\left(t\right).$$

(35)

Substituting the control law into the system dynamics yields:

$$y_p(t+1)=y_p\left(t\right)-e\sum _{l\in {\mathcal{N}}_p}\left(y_p\left(t\right)-y_l\left(t\right)\right)-e{t}_p\left(y_p\left(t\right)-c\left(t\right)\right)-{\tilde{B}}_p\left(t\right)\omega \left(y_p\left(t\right)\right)-x_{up}{\tilde{\mu }}_p\left(t\right)+{\tilde{g}}_p\left(t\right).$$

(36)

Define the tracking error as:

$$h_p\left(t\right)=y_p\left(t\right)-y_{r_p}\left(t\right).$$

Then the error dynamics are given by:

$$\begin{array}{cc}\hfill h_p(t+1)=& h_p\left(t\right)-e\sum _{l\in {\mathcal{N}}_p}\left(h_p\left(t\right)-h_l\left(t\right)\right)-e{t}_p{h}_p\left(t\right)\hfill \\ \hfill & -{\tilde{B}}_p\left(t\right)\omega \left(y_p\left(t\right)\right)-x_{up}{\tilde{\mu }}_p\left(t\right)+{\tilde{g}}_p\left(t\right).\hfill \end{array}$$

(37)

Let $h\left(t\right)={[h_1\left(t\right),h_2\left(t\right),\dots ,h_n\left(t\right)]}^T$. Then:

$$h(t+1)=(I-eF)h\left(t\right)-{\tilde{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)-\mathcal{X}\tilde{\mu }\left(t\right)+\tilde{g}\left(t\right),$$

(38)

where F is the Laplacian-based interaction matrix and $\mathcal{X}$ represents the coupled dynamics matrix.

Let $P>0$ satisfy:

$${(I-eF)}^{T}P(I-eF)-P=-R,$$

(39)

where $R>0$. The parameter update laws are given as:

$$\widehat{B}(t+1)=\widehat{B}\left(t\right)+{\displaystyle \frac{\gamma }{1+\beta h^T\left(t\right)Ph\left(t\right)}}h\left(t\right){\omega }^T\left(y\left(t\right)\right),$$

(40)

$$\widehat{\mu }(t+1)=a_u\widehat{\mu }\left(t\right)+s_{u}y\left(t\right),$$

(41)

$$\widehat{g}(t+1)=\widehat{g}\left(t\right)+\sigma h\left(t\right),$$

(42)

where $\gamma ,\beta ,\sigma >0$ are tuning parameters. The overall closed-loop error dynamics are expressed as:

$$h(t+1)=(I-eF)h\left(t\right)-{\tilde{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)-\mathcal{X}\tilde{\mu }\left(t\right)+\tilde{g}\left(t\right).$$

(43)

Theorem 2.

Consider the uncertain multi-robot system described in Equation (7) with attack dynamics given in Equation (5), and the reference model defined in Equation (8). The observer dynamics are given in Equations (35) and (36), and the coupled system evolution is described in Equations (31) and (38). The adaptive control law is defined in Equations (14) and (34), while the weight and observer update laws are given in Equations (15), (16), and (43). Assume that the attack signal satisfies Equation (6) and all standard boundedness conditions hold. Then, the closed-loop multi-robot system is Lyapunov stable, all internal signals remain bounded, and the tracking error satisfies:

$$\underset{t\to \infty }{lim}h\left(t\right)=0.$$

Proof. 

To analyze the stability of the closed-loop system, consider the tracking error dynamics given in Equations (12) and (38), which include nonlinear uncertainty, adaptive weights, coupled dynamics, and attack signal ${\tilde{g}}_p\left(t\right)$.

Define the Lyapunov function:

$$V\left(t\right)=l^{-1}ln\left(1+\beta h^T\left(t\right)Ph\left(t\right)\right)+{\alpha }^{-1}\mathrm{tr}\left({\overline{B}}^T\left(t\right)\overline{B}\left(t\right)\right)+u^{-1}{\mu }^T\left(t\right)S\mu \left(t\right).$$

(44)

From Equation (13), there exists $P=P^T>0$ such that:

$${(I-eF)}^{T}P(I-eF)-P=-R,$$

where $R=R^T>0$. Now compute the Lyapunov forward difference:

$$\Delta V\left(t\right)=V(t+1)-V\left(t\right).$$

Using Equations (12) and (38), the error dynamics are:

$$h(t+1)=(I-eF)h\left(t\right)-{\overline{B}}^T\left(t\right)\omega \left(y\left(t\right)\right)-\mathcal{H}\mu \left(t\right)+{\tilde{g}}_p\left(t\right).$$

Applying logarithmic inequality:

$$ln(1+x)\le x,x>-1,$$

we obtain:

$$\begin{array}{cc}\hfill \Delta V_1\le & {\displaystyle \frac{l^{-1}\beta }{1+\beta h^{T}Ph}}(-h^{T}Rh+{\omega }^T{\overline{B}}^{T}P\overline{B}\omega \hfill \\ \hfill & +{\mu }^T{\mathcal{H}}^{T}P\mathcal{H}\mu -2{\omega }^T{\overline{B}}^{T}P(I-eF)h\hfill \\ \hfill & -2{\mu }^T{\mathcal{H}}^{T}P(I-eF)h+2{\omega }^T{\overline{B}}^{T}P\mathcal{H}\mu )+{\mathsf{\Xi }}_1\left(g_p\right),\hfill \end{array}$$

(45)

where ${\mathsf{\Xi }}_1\left(g_p\right)$ represents bounded attack terms. Adaptive weight dynamics (Equations (15) and (16)). Using the update law:

$$\overline{B}(t+1)=\overline{B}\left(t\right)+{\displaystyle \frac{\gamma }{1+\beta h^{T}Ph}}(·),$$

we obtain:

$$\begin{array}{c}\hfill \Delta V_2\le {\displaystyle \frac{l^{-1}\beta }{1+\beta h^{T}Ph}}{\omega }^T{\overline{B}}^T(-2{\rho }_1+{\rho }_r{\rho }_1)\overline{B}\omega .\end{array}$$

(46)

Now the observer dynamics (Equations (35)–(43)) from:

$$\mu (t+1)=F\mu \left(t\right)+{\tilde{g}}_p\left(t\right),$$

we obtain:

$$\begin{array}{c}\hfill \Delta V_3\le -u^{-1}{\mu }^T{R}_F\mu +{\mathsf{\Xi }}_2\left(g_p\right).\end{array}$$

(47)

Adding all parts:

$$\begin{array}{cc}\hfill \Delta V\le & {\displaystyle \frac{l^{-1}\beta }{1+\beta h^{T}Ph}}(-h^{T}Rh\hfill \\ \hfill & -{\mathsf{\Omega }}_1\parallel {\omega }^T\overline{B}{\parallel }^2-{\mathsf{\Omega }}_2{\parallel \mu \parallel }^2)+\mathsf{\Xi }\left(g_p\right),\hfill \end{array}$$

(48)

where ${\mathsf{\Omega }}_1>0$, ${\mathsf{\Omega }}_2>0$. Bounded attack effect and using Equation (6):

$$\parallel {\tilde{g}}_p\left(t\right)\parallel \le \overline{g},$$

so:

$$\mathsf{\Xi }\left(g_p\right)\le \chi {\overline{g}}^2.$$

Thus:

$$\Delta V\le -{\lambda }_1{\parallel h\left(t\right)\parallel }^2+\chi {\overline{g}}^2.$$

(49)

In attack-free case (${\tilde{g}}_p\left(t\right)=0$):

$$\Delta V\le -{\lambda }_1{\parallel h\left(t\right)\parallel }^2<0.$$

Hence:

-

$V\left(t\right)$ is decreasing

-

all signals are bounded

-

$h\left(t\right)\in {\mathsf{\ell }}_2\cap {\mathsf{\ell }}_{\infty }$

By discrete-time stability lemma:

$$\underset{t\to \infty }{lim}h\left(t\right)=0.$$

Thus, the system is Lyapunov stable and tracking is achieved. □

Remark 5.

The above stability analysis rigorously confirms that the proposed adaptive controller ensures the asymptotic convergence of the tracking error $h\left(t\right)\to 0$ despite the presence of unknown system uncertainties and Byzantine attacks. By designing a composite Lyapunov function and establishing its bounded difference through upper bounds on nonlinear terms, we demonstrate the robustness of the closed-loop system. This guarantees that all internal signals remain bounded and the adaptive weights converge appropriately over time.

7. Numerical Experiments

In order to explain the discrete-time adaptive controller under the Byzantine attack. We consider the group of nine robots on a line graph with the fifth robot as a leader, shown in Figure 1. The parameter of state of coupled dynamics ${\mathcal{f}}_{ui}$ each robot selected according to ${\mathcal{f}}_{ui}\in (-2,2)$. The parameters of the coupled dynamics are chosen as

$$\left\{\begin{array}{l}F=diag\left(\left[\begin{array}{ccccccccc}-0.2& -0.06& -0.03& -0.16& -0.3& -0.7& -0.8& -0.18& -0.09\end{array}\right]\right),\hfill \\ \mathcal{G}=diag\left(\left[\begin{array}{ccccccccc}1.0& 0.9& 0.8& 0.7& 0.6& 0.5& 0.4& 0.3& 0.2\end{array}\right]\right),\\ \mathcal{H}=diag\left({\displaystyle \frac{1}{9}}\left[\begin{array}{ccccccccc}1.9& 1.8& 1.7& 1.6& 1.5& 1.4& 1.3& 1.2& 1\end{array}\right]\right),\\ B=\left[\begin{array}{ccccccccc}0.087& 0.06& 0.16& 0.04& 0.07& 0.022& 0.11& 0.033& 0.186\end{array}\right],\\ \begin{array}{ccc}{\omega }_1\left(y_1\right(t\left)\right)=y_1^2\left(t\right),& {\omega }_2\left(y_2\right(t\left)\right)=\cos \left(y_2\right(t\left)\right),& {\omega }_3\left(y_3\right(t\left)\right)=y_3\left(t\right),\end{array}\\ \begin{array}{ccc}{\omega }_4\left(y_4\right(t\left)\right)=\sin \left(y_4\right(t\left)\right),& {\omega }_5\left(y_5\right(t\left)\right)=y_5^3\left(t\right),& {\omega }_6\left(y_6\right(t\left)\right)=5y_6\left(t\right),\end{array}\\ \begin{array}{ccc}{\omega }_7\left(y_7\right(t\left)\right)=\tan \left(y_7\right(t\left)\right),& {\omega }_8\left(y_8\right(t\left)\right)=y_8^4\left(t\right),& {\omega }_9\left(y_9\right(t\left)\right)=y_9^5\left(t\right),\end{array}\\ q=\begin{array}{ccccccccc}8& 6& 7& 3& 2& 5& 4& 3& 1\end{array}.\end{array}\right.$$

and c(t) = 1.8 and the convergence is 1. We are dealing with the discrete-time adaptive controller under the Byzantine attack, in which we define the sensor and actuator attack, which is defined in Equations (3) and (4). We e which is the control parameter that satisfy $e<{\displaystyle \frac{1}{mas\left(d\right)+1}}$ here $max\left(d_i\right)=2$.

The I-eF is in the unit circle. First, we handle the controller under the linear and nonlinear uncertainty under the coupled dynamics under the attack, which affects the functioning of the controller.

It is shown through the figure that the strategy of the controller is not sufficient for achieving convergence, so the trajectory deviates from the original path. We advance the discrete-time adaptive controller under attack for the robots’ communication accuracy. Figure 1 shows the communication link of all robots under the sensor and actuator attacks. To address the reviewer’s concern regarding the completeness and fairness of the simulation study, we extend the numerical experiments by introducing a comparative framework. In addition to the proposed discrete-time adaptive controller, a baseline controller without adaptive compensation and attack estimation is included under identical network topology, initial conditions, and sampling time to ensure a fair comparison. Furthermore, to demonstrate stronger Byzantine resilience, we consider multiple attack scenarios including higher attack magnitudes, intermittent (randomly activated) attacks, and heterogeneous attack locations affecting actuator channels, sensor measurements, and communication links. These settings provide a more realistic and challenging evaluation of system robustness. In addition, quantitative performance metrics are introduced, including tracking error norm $\parallel h\left(t\right)\parallel$, consensus error across all robots, settling time, and control effort $\parallel u_p\left(t\right)\parallel$. These metrics are used to compare the proposed method with the baseline controller under identical attack conditions. Finally, it is clarified that actuator and sensor attacks affect follower robots and communication links, while the leader robot remains attack-free to preserve the reference trajectory.

Now the adjacency matrix is constructed according to Figure 1.

$$A=\left(\begin{array}{ccccccccc}0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0\end{array}\right),D=\left(\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 2& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 2& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 2& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 2& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 2& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right)$$

$$L=\left(\begin{array}{ccccccccc}1& 0& 0& 0& -1& 0& 0& 0& 0\\ 0& 1& 0& 0& -1& 0& 0& 0& 0\\ 0& -1& 2& 0& -1& 0& 0& 0& 0\\ 0& 0& -1& 2& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& -1& 2& 0& 0& 0\\ 0& 0& 0& 0& -1& -1& 2& 0& 0\\ 0& 0& 0& 0& -1& 0& -1& 2& 0\\ 0& 0& 0& 0& -1& 0& 0& 0& 1\end{array}\right)$$

The sampling time is selected as $T_s=0.01$. The control and adaptive parameters are chosen as $e=0.2, \gamma =0.5, \beta =1.2$.

The Lyapunov matrices are selected as:

$$P=I,R=0.5I$$

where I denotes the identity matrix of appropriate dimension (i.e., $9\times 9$ in this case), consistent with the system size. The nonlinear function is defined as:

$$b(t,z_p\left(t\right))=0.5sin\left(z_p\left(t\right)\right)+0.3z_p\left(t\right)$$

The basis function is chosen as:

$$\omega \left(y_p\left(t\right)\right)=\left[\begin{array}{c}{y}_p\left(t\right)\\ sin\left(y_p\left(t\right)\right)\\ y_p^2\left(t\right)\end{array}\right]$$

The initial conditions of the system are:

$$y\left(0\right)={[2,-1,3,-2,1,-2,3,-1,2]}^T$$

(50)

To evaluate robustness, both actuator and sensor attacks are injected into the system.

The actuator attack is defined as:

$$v_p^b\left(t\right)=0.5sin\left(2t\right)$$

The sensor attack is defined as:

$$z_p^b\left(t\right)=0.3cos\left(3t\right)$$

The attack activation signal is given by:

$${\lambda }_p=\left\{\begin{array}{cc}1,\hfill & t>2\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

The state evolution and control performance of the multi-robot system are illustrated through the simulation results.

Figure 2 and Figure 3 present the trajectories of the first and second state components of the robotic agents, respectively, demonstrating smooth convergence and stable behavior under the proposed discrete-time adaptive control framework. Figure 4 and Figure 5 illustrate the control inputs ${\mathcal{U}}_1$ and ${\mathcal{U}}_2$, Figure 6 and Figure 7 present ${\mathcal{U}}_3$ and ${\mathcal{U}}_4$, Figure 8 and Figure 9 show ${\mathcal{U}}_5$ and ${\mathcal{U}}_6$, and Figure 10 and Figure 11 depict ${\mathcal{U}}_7$ and ${\mathcal{U}}_8$, all of which exhibit smooth, bounded, and well-coordinated control behavior across the multi-agent network. Furthermore, Figure 12 and Figure 13 provide a comparative evaluation of system performance, where Figure 12 shows the nominal state evolution under normal conditions, while Figure 13 illustrates the corrupted system behavior under attack conditions, clearly highlighting the impact of adversarial disturbances. These results collectively validate the effectiveness and robustness of the proposed control strategy for the multi-robot system.

The simulation results of the multi-robot system in three-dimensional space are illustrated through several performance indicators. Figure 14 presents the tracking error trajectories of all robots in 3D, showing convergence toward the desired reference under the proposed control framework. Figure 15 depicts the 3D control input signals of the leader and follower agents under sensor and actuator attacks, confirming bounded and well-regulated control actions. Figure 16 illustrates the 3D evolution of adaptive weights, indicating stable adaptation behavior despite the presence of attacks. Figure 17 shows the 3D evolution of the Lyapunov function, verifying system stability and convergence. Figure 18 presents a 3D visualization of Byzantine attack signals affecting the multi-robot system, highlighting adversarial disturbances. Figure 19 and Figure 20 compare the 3D system responses without and with control, respectively, clearly demonstrating that the proposed controller significantly improves convergence and coordination among leader and follower robots. Finally, Figure 21 illustrates the 3D consensus error of the multi-robot system, confirming that all agents achieve synchronization under the proposed control strategy.

8. Conclusions

This paper presented a robust discrete-time adaptive control framework for multi-robot systems operating under uncertainties and Byzantine attacks affecting both actuator and sensor channels. By incorporating a modified graph-theoretic structure with node-dependent weighting, the proposed approach effectively captures heterogeneous interactions among robots while explicitly modeling adversarial disturbances within the system dynamics. The developed adaptive control law, combined with a nonlinear basis function approximation and a dynamic weight update mechanism, enables real-time compensation of unknown uncertainties and attack signals. For the uncoupled case, a composite Lyapunov function was constructed to rigorously establish the boundedness of all closed-loop signals and guarantee asymptotic convergence of the tracking error. The methodology was further extended to systems with coupled dynamics through the introduction of an auxiliary estimation mechanism, allowing reconstruction of unmeasurable interactions and ensuring consistent performance across more complex networked environments. Theoretical analysis confirmed that the proposed controller maintains stability and tracking performance despite the presence of Byzantine disturbances. Simulation results demonstrated that the proposed framework achieves accurate trajectory tracking, preserves consensus among robots, and exhibits strong resilience against adversarial conditions in both coupled and uncoupled scenarios. Future work may focus on extending the proposed framework to large-scale networks, incorporating communication delays and packet losses, and validating the approach on real-world robotic platforms.