Asymptotic Formation Tracking Control for Heterogeneous Multi-Agent Systems with External Disturbances and Time-Varying Actuator Faults

Abstract

This paper investigates the asymptotic formation tracking control problem for heterogeneous multi-agent systems (HMASs) subject to external disturbances and time-varying actuator faults. A comprehensive model incorporating these uncertainties is formulated to capture realistic engineering scenarios. To ensure robustness and adaptability, an integral sliding surface is constructed, followed by a rigorous analysis of the formation tracking error dynamics. A variable-gain disturbance observer is proposed to estimate the lumped disturbance and fault-induced input error using measured velocity and commanded input effectively. Leveraging information from neighboring agents and the leader, a distributed adaptive sliding mode fault-tolerant controller is synthesized. By utilizing the Lyapunov stability theorem, sufficient conditions for asymptotic tracking under smooth feedback are rigorously derived. Theoretical analysis demonstrates that under the proposed adaptive fault-tolerant control protocol, all follower agents asymptotically achieve the desired formation as $t\to \infty$. Numerical simulations are provided to validate the effectiveness and robustness of the proposed approach.

1. Introduction

Multi-agent systems (MASs) are distributed systems composed of multiple autonomous agents that collaborate with each other through communication networks. With the advancement of artificial intelligence and networking technologies, MASs have exhibited broad application prospects in fields such as robot formation [1], unmanned aerial vehicle (UAV) swarms [2], smart grids [3], and environmental monitoring [4,5]. Their core advantage lies in the ability to achieve global cooperative objectives through local information interaction, accomplishing complex tasks that are difficult for a single agent to handle, such as cooperative search [6], distributed perception [7], and collaborative control [8,9]. For a mobile team, changing loads or wind can act as disturbances, while reduced motor or drive effectiveness changes the applied control input. Maintaining coordination under these different mechanisms provides the practical motivation for this study. However, MASs encounter numerous challenges in practical applications: communication delays, parameter uncertainties, nonlinear dynamics, and cyber-attacks can all degrade system performance and even destabilize the system. To address these challenges, researchers have proposed various control strategies, including event-triggered control, sliding mode control, and fuzzy adaptive control, to improve the robustness, security, and resource efficiency of the system.

In recent years, the integration of intelligent optimization methods, such as adaptive dynamic programming (ADP), has further propelled research advancements in performance optimization and secure cooperation within MASs [10]. In particular, the study of HMASs holds significant importance in advancing the capabilities of MASs. Unlike homogeneous systems where all agents are identical, heterogeneous systems consist of agents with different dynamics, capabilities, or functions, which provide greater flexibility and adaptability in real-world applications. For instance, in robotic swarm applications [11], heterogeneous agents with different sensors and processing powers can handle different tasks simultaneously, improving the system’s overall efficiency and effectiveness. This heterogeneity enables the system to tackle more complex and diverse tasks, enhancing robustness and fault tolerance. Nevertheless, the inherent heterogeneity of agents in HMASs introduces additional challenges, such as increased coordination complexity and information inconsistency among heterogeneous agents [12]. Specifically, each agent must adapt to its unique dynamic constraints while ensuring harmonious operation with other agents to achieve the desired global cooperative objective. More critically, the presence of external disturbances (e.g., environmental noise or abrupt system changes) and actuator faults (e.g., motor-drive degradation or control input degradation) can significantly deteriorate the performance and undermine the stability of MASs. This issue becomes more prominent in heterogeneous settings, where different agents may exhibit distinct responses to such adverse factors [13]. Since external disturbances are inevitable in real-world operating environments, the design of advanced control strategies that can guarantee the stability and desired performance of MASs (especially HMASs) under both internal uncertainties (e.g., actuator faults) and external disturbances is of paramount importance for practical implementations. To surmount these aforementioned challenges, recent research efforts have focused on developing advanced fault-tolerant control (FTC) strategies and disturbance rejection mechanisms, including sliding mode control, adaptive control [14], and robust control techniques [15], which are capable of addressing both actuator faults and external disturbances simultaneously [16]. The primary objective of these approaches is to ensure that MASs can maintain their intended functionality and stable operation even under adverse operating conditions. Additionally, intelligent algorithms, such as ADP [17], are being extensively explored to enhance the performance optimization capability of MASs in the presence of disturbances, while concurrently improving the coordination efficiency and adaptive capacity of the system in dynamic and uncertain environments.

The consensus of MASs has become an important research topic, which has attracted great attention from researchers in many subjects such as control, mathematics and artificial intelligence, especially in swarm systems [18] and distributed sensor networks [19]. Various forms of distributed control protocols have been provided by robotics and control research teams. Saber and Murray [20] put forward a general hierarchy to solve the consensus problems of multi-agent systems for a single integrator. Ren and Beard [21] studied the consensus problem with a directed weighted graph and pointed out that under certain conditions, the system can achieve consensus. Li et al. [22] studied mean-square output consensus of heterogeneous multi-agent systems with multiplicative noises in dynamics and measurements. In the latest research, researchers have focused on switching directed networks [23,24], time delays [25,26], topologies [27], and consensus control with disturbance [28,29]. Various types of control theories are adopted in the control of the multi-agent system, such as distributed tracking control and leader-following control. Compared with the above control methods, sliding mode control is more stable under internal and external interferences. Additionally, it is superior in its application in dealing with the uncertainties of the system and its simplified realization [30]. In [31], predefined-time fault-tolerant consensus tracking was studied for multiple flexible robotic manipulators with prescribed performance. The present simulations concern single-input single-output (SISO) agents. The work in [32] studies finite-time formation control of second-order agents with relative-state constraints using a barrier-function sliding mode approach. In [33,34], consensus problems involving noises, unmodeled dynamics, or disturbances are investigated. Time delay and uncertainties are the most serious problems in the application of the MASs in fields like unmanned vehicle control and aircraft [35,36]. Many studies have focused on solving such problems. In [37], researchers employ the Lyapunov stability theory and linear matrix inequality (LMI) technique to deal with the uncertainties. In [38], the impulsive consensus of multi-agent nonlinear systems with control gain error was further studied. In [39], the sliding surface was designed to maximize the calculable set of admissible delays. In [40,41], delayed full-state feedback control was used to deal with the occasional time delay [42].

HMASs are more common in practical engineering scenarios, where agents usually differ in their dynamics, structures, and physical constraints [43]. In such systems, actuators and communication components often operate under harsh or uncertain environments, which can easily lead to performance degradation or partial failures due to factors such as component aging, external disturbances, overload, or mechanical wear [44]. As a result, the study of FTC for HMASs has attracted increasing attention, since ensuring reliable coordination despite these actuator faults is crucial for maintaining system safety and performance. Because the dynamics of agents are not identical, the fixed-time control methods developed for homogeneous MASs cannot be directly applied to HMASs [45]. To address this issue, the internal model principle has been introduced, and controllers for each agent have been designed based on global information [46]. For example, fixed-time output consensus of continuous-time linear HMASs has been achieved by tracking the output of an active leader, where the controllers are constructed using signum terms or power terms. However, this class of methods neglects the actual information exchanged among agents and their neighbors [47] and cannot estimate the settling time of each agent. These limitations significantly reduce their practicality in real distributed HMASs, especially under actuator faults and uncertainties.

Based on the above, the problem of formation tracking control with actuator failures and external disturbances in networked HMASs is considered, and an adaptive sliding mode fault-tolerant control protocol is designed based on a variable-gain disturbance observer. Compared to the fixed-time consensus controller designed in [46,47], the external disturbance and actuator faults are considered and the variable gain disturbance observer and adaptive sliding mode fault-tolerant formation tracking control strategy are designed. The resulting scheme is evaluated using matched no-observer and fixed-gain ablations. The main results in this paper can be summarized as follows:

• A variable-gain disturbance observer is proposed that uses a prescribed positive time-varying gain, a positive scheduled bandwidth, and a bounded output to provide implementable lumped-input compensation.
• An adaptive integral sliding mode control strategy is designed that incorporates an integral term to support asymptotic tracking and an adaptive law to adjust robustness bounds online, enabling effective compensation for both external disturbances and partial actuator failures.
• The established control framework is extended to HMASs with known unequal nominal input gains and a reference available to every follower. Lyapunov stability analysis is conducted, which proves that the smooth adaptive closed loop remains bounded and the physical formation errors tend to zero asymptotically.

This paper comprises six sections. The mathematical preliminaries of MASs and HMASs subject to actuator failure models are proposed in Section 2. In Section 3, the adaptive integral sliding surface is designed and the formation error is analyzed; meanwhile, a variable gain disturbance observer is constructed to estimate the external disturbances. Furthermore, the adaptive integral sliding fault-tolerant control laws are designed and the stability of the systems under the proposed protocol is also analyzed. To verify the effectiveness of the obtained theories, the numerical simulations are shown in Section 4. The parameter conditions, residual errors, and physical limitations are discussed in Section 5. Finally, the conclusions are condensed in Section 6.

2. Preliminaries and Model Formulation

This section introduces the communication graph, comparison results, and fault model used in the subsequent analysis.

2.1. Mathematical Preliminaries

In MASs, the communication topology among agents can be described by an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}=\{0,1,\dots ,n\}$ denotes the set of agents (0 is the leader, and $1,\dots ,n$ are followers), and $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ represents the communication edges. Define Laplacian matrix $L=\left[l_{ij}\right]\in {\mathbb{R}}^{n\times n}$, with elements satisfying $l_{ii}={\sum }_{j\ne i}{a}_{ij}$, $l_{ij}=-a_{ij}$ if $(j,i)\in \mathcal{E}$, otherwise $l_{ij}=0$; the weights satisfy $a_{ij}=a_{ji}\ge 0$; leader adjacency matrix $B=\mathrm{diag}\{b_1,\dots ,b_n\}$, where $b_i>0$ if follower i can directly obtain information from the leader, otherwise $b_i=0$; and composite matrix $H=L+B$, which is a positive definite matrix ($H>0$) for the fixed undirected, reference-pinned graph in Assumption 4. Indeed, $y^{T}Hy={\displaystyle \frac{1}{2}}{\sum }_{i,j}{a}_{ij}{(y_i-y_j)}^2+{\sum }_i{b}_i{y}_i^2>0$ for $y\ne 0$.

Lemma 1.

The differential inequality examined in this lemma is fundamental to the analysis of dynamical systems subject to persistent disturbances or modeling uncertainties. It formalizes the concept of exponential convergence towards a compact set, a property central to the framework of Input-to-State Stability (ISS). Consider a non-negative function $V\left(t\right)$ often serving as a Lyapunov function candidate. If its time derivative satisfies

$$\dot{V}\left(t\right)\le -\alpha V\left(t\right)+\beta ,(\alpha >0,\beta \ge 0)$$

(1)

then the trajectory of $V\left(t\right)$ is analytically bounded by

$$V\left(t\right)\le V\left(0\right)e^{-\alpha t}+{\displaystyle \frac{\beta }{\alpha }}(1-e^{-\alpha t}).$$

(2)

Proof. 

Multiplication of (1) by $e^{\alpha t}$ gives ${\displaystyle \frac{d}{dt}}\left(e^{\alpha t}V\right)\le \beta e^{\alpha t}$. Integrating from 0 to t gives (2).  □

Remark 1.

This result delineates a key trade-off: the convergence rate is governed by the scalar α, while the ultimate steady-state bound is determined by the ratio $\beta /\alpha$. It thus provides a quantitative description of how system dynamics dominate over external inputs, ensuring that the state remains ultimately bounded within a well-defined region.

Lemma 2.

While asymptotic stability guarantees convergence over an infinite time horizon, many control applications require system trajectories to reach equilibrium in finite time, a property that often enhances robustness and precision. This lemma provides a sufficient condition for such finite-time convergence, extending the classical Lyapunov stability theory. For the nonlinear system $\dot{x}=f(x,t)$, assume the zero set is invariant and solutions exist up to the stated hitting time. Suppose there exists a positive definite function $V(x,t)\ge 0$ such that its time derivative along the system trajectories satisfies

$$\dot{V}\le -c{V}^{\xi },(c>0,0<\xi <1)$$

(3)

then $x\left(t\right)$ converges to 0 in finite time, and the convergence time satisfies

$$T\left(x\left(0\right)\right)\le {\displaystyle \frac{V^{1-\xi }\left(x\left(0\right)\right)}{c(1-\xi )}}.$$

(4)

The settling time estimate is of paramount importance for controller design, as it allows engineers to a priori compute a worst-case convergence time based on initial conditions, thereby verifying the system’s transient performance specifications. Here, $\xi \in (0,1)$ is the exponent in the differential inequality.

Proof. 

While $V>0$, we have

$${\displaystyle \frac{d}{dt}}{V}^{1-\xi }\le -c(1-\xi ).$$

(5)

Integration it gives (4).  □

Lemma 3.

A uniformly continuous function $f:[0,\infty )\to {\mathbb{R}}^n$ satisfying ${\int }_0^{\infty }{\parallel f\left(t\right)\parallel }_{1}dt<\infty$ converges to zero.

Proof. 

Otherwise uniform continuity supplies disjoint intervals of a common positive length on which the norm exceeds a fixed positive value, contradicting integrability.  □

Assumption 1.

Define position compensation $h_i\left(t\right)\in {\mathbb{R}}^m$ and velocity compensation $c_i\left(t\right)\in {\mathbb{R}}^m$, which satisfy ${\dot{h}}_i\left(t\right)=c_i\left(t\right)$. The functions $h_i$ are twice continuously differentiable.

Assumption 2.

The derivative of the external disturbance $d_i\left(t\right)\in {\mathbb{R}}^m$ is bounded, i.e., $|{\dot{d}}_i\left(t\right)|\le \chi$ ($\chi >0$ is a constant). For the complete closed-loop result, additionally assume $|d_i\left(t\right)|\le D$ for some finite, possibly unknown D.

Remark 2.

This assumption on the boundedness of the disturbance variation rate is standard in robust and adaptive control theory. The derivative condition alone limits the rate of change without bounding the disturbance magnitude; the additional magnitude bound is imposed separately for the tracking theorem. This is a physically reasonable assumption for many practical systems, as it corresponds to disturbances that cannot change with infinite speed, thus ensuring that the control law can be designed to compensate for their effects within a finite bandwidth.

Assumption 3.

There exists an unknown constant $\rho \in (0,1]$ such that $0\le {\theta }_i\left(t\right)\le 1-\rho$ almost everywhere. Thus, each scalar channel retains positive control authority.

Assumption 4.

The follower graph is fixed and undirected, $b_i>0$, and each follower receives the leader reference. The signals $v_0,c_i,r_i:={\dot{c}}_i+{\dot{v}}_0$ are bounded and available. Measurements are noise-free and feedback is continuous-time. Hard actuator saturation is not included.

2.2. Formula and Modeling

The MASs with one leader and n followers are considered. The analysis is specialized to $m=1$, as in the simulations. The dynamics of the leader are given by

$${\dot{x}}_0\left(t\right)=v_0\left(t\right),$$

(6)

where $x_0\left(t\right)\in {\mathbb{R}}^m$ is the position of the leader, $v_0\left(t\right)\in {\mathbb{R}}^m$ is the velocity of the leader, and $v_0,c_i,r_i$ are bounded as specified above; a constant nonzero leader velocity does not imply bounded $x_0$. The dynamics of the followers are given by

$$\left\{\begin{array}{c}{\dot{x}}_i\left(t\right)=v_i\left(t\right),\hfill \\ {\dot{v}}_i\left(t\right)={\mu }_i{u}_i^F\left(t\right)+d_i\left(t\right),\hfill \end{array}\right.(i=1,2,\dots ,n)$$

(7)

here, ${\mu }_i=1/{\mathfrak{m}}_i>0$ is a known nominal input gain, with possibly unequal masses ${\mathfrak{m}}_i$. This is the specific heterogeneity included in the model. Here, $x_i\left(t\right)\in {\mathbb{R}}^m$, $v_i\left(t\right)\in {\mathbb{R}}^m$ denote the position and velocity of thei-th follower. The applied actuator input is

$$u_i^F\left(t\right)=(I_m-{\theta }_i\left(t\right))u_i\left(t\right),$$

(8)

where ${\theta }_i\left(t\right)=\mathrm{diag}\{{\theta }_{i1}\left(t\right),\dots ,{\theta }_{im}\left(t\right)\}$ is the actuator fault factor, satisfying $0\le \parallel {\theta }_i{\left(t\right)\parallel }_2<1$, $d_i\left(t\right)\in {\mathbb{R}}^m$ is the external disturbance, which satisfies Assumption 2. Only partial effectiveness loss is covered, complete loss, lock-in-place, and sign reversal are not claimed compensable. All followers may be partially degraded simultaneously if each retains the uniform positive authority.

Define position and velocity deviation vectors to describe the state difference between followers and the leader:

$${\overline{x}}_i\left(t\right)=x_i\left(t\right)-h_i\left(t\right)-x_0\left(t\right),{\overline{v}}_i\left(t\right)=v_i\left(t\right)-c_i\left(t\right)-v_0\left(t\right).$$

(9)

Based on the communication topology (Laplacian matrix L, leader adjacency matrix B), define composite consensus errors

$$e_x\left(t\right)=H\overline{x}\left(t\right),e_v\left(t\right)=H\overline{v}\left(t\right),$$

(10)

where $e_x\left(t\right)={[e_{x1}^T\left(t\right),\dots ,e_{xn}^T\left(t\right)]}^T,e_{xi}\left(t\right)={\sum }_{j\in {\mathcal{N}}_i}{a}_{ij}({\overline{x}}_i\left(t\right)-{\overline{x}}_j\left(t\right))+b_i{\overline{x}}_i\left(t\right)$. ${\mathcal{N}}_i$ is the neighbor set of the i-th follower. The form of $e_v\left(t\right)$ is consistent with that of $e_x\left(t\right)$, reflecting the consensus deviation at the velocity level.

3. Main Results

This section first analyzes the basic and integral sliding variables, then constructs an implementable observer and proves the stability of the corresponding smooth adaptive closed loop.

3.1. Sliding Surface Design and Convergence Analysis

To ensure the convergence of $e_{xi}\left(t\right)$ and $e_{vi}\left(t\right)$, a sliding mode surface is defined as follows

$$\widehat{S}\left(t\right)=k{e}_x\left(t\right)+e_v\left(t\right),$$

(11)

where $k>0$ is a constant gain used to adjust the error convergence rate.

Theorem 1.

If Assumption 1 holds and H is fixed and nonsingular, then for the sliding mode surface designed by (11), the consensus errors will converge to 0 when the sliding mode variable reaches and remains on the sliding mode surface $\widehat{S}\left(t\right)=0$.

Proof. 

Construct a Lyapunov function

$$V_1\left(t\right)={\displaystyle \frac{1}{2}}{e}_x^T\left(t\right)e_x\left(t\right),$$

(12)

for any $e_x\left(t\right)\ne 0$, $e_x^T\left(t\right)e_x\left(t\right)={\sum }_{i=1}^n{\parallel e_{xi}\left(t\right)\parallel }_2^2>0$ and $V_1\left(t\right)=0$ if and only if $e_x\left(t\right)=0$.

By differentiating $e_x\left(t\right)=H\overline{x}\left(t\right)$ using Assumption 1 and Equation (6), and substituting ${\dot{\overline{x}}}_i\left(t\right)={\overline{v}}_i\left(t\right)$, the following relation is obtained:

$${\dot{e}}_x\left(t\right)=H\overline{v}\left(t\right)=e_v\left(t\right).$$

(13)

When the system enters the sliding mode state ($\widehat{S}\left(t\right)=0$), the following relation is obtained from the definition of the sliding mode surface:

$$e_v\left(t\right)=-k{e}_x\left(t\right),$$

(14)

The closed-loop dynamics of the position error are obtained by substitution into ${\dot{e}}_x\left(t\right)=e_v\left(t\right)$:

$${\dot{e}}_x\left(t\right)=-k{e}_x\left(t\right),$$

(15)

the first-order linear system has an eigenvalue $-k<0$, showing a convergence trend. The following derivative is obtained by differentiating $V_1\left(t\right)$ and substituting (15):

$${\dot{V}}_1\left(t\right)=e_x^T\left(t\right){\dot{e}}_x\left(t\right)=-k{e}_x^T\left(t\right)e_x\left(t\right)=-2k{V}_1\left(t\right),$$

(16)

since $k>0$, ${\dot{V}}_1\left(t\right)\le 0$, and ${\dot{V}}_1\left(t\right)=0$ if and only if $e_x\left(t\right)=0$, which satisfies the asymptotic stability condition. As $t\to \infty$, $e_x\left(t\right)\to 0$.By combining this result with (14), the limit $e_v\left(t\right)\to 0$ is obtained. Thus, all consensus errors converge to 0 asymptotically, completing the proof. More explicitly, for $t\ge t_s$ on the invariant manifold, $e_x\left(t\right)=e_x\left(t_s\right)e^{-k(t-t_s)}$. Unless $e_x\left(t_s\right)=0$, this is not exact finite-time tracking.  □

3.2. Integral Sliding Surface Design and Error Convergence Analysis

To enhance the robustness and adaptability of the control system against uncertainties and disturbances, an integral sliding mode surface is constructed as follows:

$$S\left(t\right)=g_1\left(t\right)\widehat{S}\left(t\right)+{\int }_0^t{g}_2\left(\tau \right)\widehat{S}\left(\tau \right)\mathrm{d}\tau ,$$

(17)

where $S\left(t\right)\in {\mathbb{R}}^n$ denotes the proposed sliding vector. The basic sliding variable $\widehat{S}\left(t\right)$ is defined as

$$\widehat{S}\left(t\right)=k{e}_x\left(t\right)+e_v\left(t\right),$$

(18)

where $e_x\left(t\right)$ and $e_v\left(t\right)$ represent the state and velocity error vectors, respectively, and $k>0$ is a feedback gain.

The positive surface gains $g_1\left(t\right)$ and $g_2\left(t\right)$ are chosen as follows:

$$g_1\left(t\right)=1,g_2\left(t\right)=\lambda >0,$$

(19)

where $\lambda$ is a constant integral gain. Adaptation is assigned to the robust controller gains below, rather than to a surface gain that can vanish.

Theorem 2.

For the sliding mode surface described by (17), all consensus errors will converge to 0 provided $S\left(t\right)=0$ is maintained for every $t\ge t_s$.

Proof. 

Select the quadratic function of the basic sliding mode variable

$$V_2\left(t\right)={\displaystyle \frac{1}{2}}{\widehat{S}}^T\left(t\right)\widehat{S}\left(t\right),$$

(20)

then ${\widehat{S}}^T\left(t\right)\widehat{S}\left(t\right)={\sum }_{i=1}^n{\parallel {\widehat{S}}_i\left(t\right)\parallel }_2^2\ge 0$ and $V_2\left(t\right)=0$ if and only if $\widehat{S}\left(t\right)=0$. In the sliding mode state $S\left(t\right)=0$, the sliding mode surface is rewritten as

$$g_1\left(t\right)\widehat{S}\left(t\right)=-{\int }_0^t{g}_2\left(\tau \right)\widehat{S}\left(\tau \right)\mathrm{d}\tau .$$

(21)

Both sides are differentiated with respect to t, and the product rule is applied:

$${\displaystyle \frac{d}{dt}}\left[g_1\widehat{S}\right]={\dot{g}}_1\widehat{S}+g_1\dot{\widehat{S}},$$

(22)

and the following relation is obtained:

$${\dot{g}}_1\left(t\right)\widehat{S}\left(t\right)+g_1\left(t\right)\dot{\widehat{S}}\left(t\right)=-g_2\left(t\right)\widehat{S}\left(t\right),$$

(23)

since $g_1\left(t\right)>0$ (by (19)), the scalar inverse $g_1^{-1}\left(t\right)$ exists. Rearranging gives

$$\dot{\widehat{S}}\left(t\right)=-g_1^{-1}\left(t\right)({\dot{g}}_1\left(t\right)+g_2\left(t\right))\widehat{S}\left(t\right).$$

(24)

For the constant positive gains in (19), ${\dot{g}}_1=0$ and $({\dot{g}}_1+g_2)/g_1=\lambda >0$. Thus, the division is valid throughout the analysis. Let

$$\lambda =g_1^{-1}\left(t\right)({\dot{g}}_1\left(t\right)+g_2\left(t\right)),$$

(25)

then $\lambda >0$. The following derivative is obtained by differentiating $V_2\left(t\right)$ and substituting $\dot{\widehat{S}}\left(t\right)=-\lambda \widehat{S}\left(t\right)$:

$${\dot{V}}_2\left(t\right)={\widehat{S}}^T\left(t\right)\dot{\widehat{S}}\left(t\right)=-\lambda {\widehat{S}}^T\left(t\right)\widehat{S}\left(t\right)=-2\lambda V_2\left(t\right).$$

(26)

From Lemma 1, taking $\alpha =2\lambda$ and $\beta =0$, the following bound is obtained:

$$V_2\left(t\right)\le V_2\left(t_s\right)e^{-2\lambda (t-t_s)},$$

(27)

when $t\to \infty$, $V_2\left(t\right)\to 0$, that is, $\widehat{S}\left(t\right)\to 0$. By combining $\widehat{S}\left(t\right)=k{e}_x\left(t\right)+e_v\left(t\right)\to 0$ with (13), the relation $k{e}_x\left(t\right)+{\dot{e}}_x\left(t\right)\to 0$ is obtained. The complete solution is $e_x\left(t\right)=e^{-k(t-t_s)}{e}_x\left(t_s\right)+{\int }_{t_s}^t{e}^{-k(t-\sigma )}\widehat{S}\left(\sigma \right)d\sigma$. Since $\widehat{S}$ decays exponentially, the convolution tends to zero; when $k=\lambda$, it is proportional to $(t-t_s)e^{-k(t-t_s)}$. Finally, $e_v=\widehat{S}-k{e}_x\to 0$. This completes the proof.  □

3.3. Error Bound Analysis for Variable Gain Disturbance Observer

To accurately estimate and compensate for the lumped disturbance and fault-induced input error, a variable-gain disturbance observer is developed. The observer employs a filtered reconstruction with bounded output and a time-varying gain $\omega \left(t\right)$, which sets the filtering bandwidth. The structure is designed as follows

$$\begin{array}{cc}\hfill {\widehat{w}}_i^{\mathrm{raw}}& =p_i+\omega v_i,\hfill \\ \hfill {\dot{p}}_i& =-\omega (p_i+\omega v_i)-\omega {\mu }_i{u}_i-\dot{\omega }{v}_i,\hfill \\ \hfill {\widehat{w}}_i& =M\tanh ({\widehat{w}}_i^{\mathrm{raw}}/M),\hfill \end{array}$$

(28)

here, $p_i$ is the observer state, $M>0$ is the output bound, and $w_i=d_i-{\theta }_i{\mu }_i{u}_i$ is the lumped unknown input. Thus, ${\dot{v}}_i={\mu }_i{u}_i+w_i$. The observer uses measured velocity and commanded input, not the unknown applied input. Its gain is prescribed in time:

$$\omega \left(t\right)={\omega }_{\infty }+({\omega }_0-{\omega }_{\infty })e^{-\kappa t},$$

(29)

where the bandwidth parameters satisfy

$${\omega }_0>0,{\omega }_{\infty }>0,\kappa >0,M>0.$$

(30)

The raw output may be initialized at zero by choosing $p_i\left(0\right)=-{\omega }_0{v}_i\left(0\right)$. The controller-facing output satisfies $|{\widehat{w}}_i|\le M$. The gain is not described as error-adaptive, and no separate identification of $d_i$ and ${\theta }_i$ is claimed.

Theorem 3.

For the observer in (28), ${\dot{\widehat{w}}}_i^{\mathrm{raw}}=\omega (w_i-{\widehat{w}}_i^{\mathrm{raw}})$. The raw output is bounded whenever $w_i$ is bounded. If additionally $w_i$ is absolutely continuous and $|{\dot{w}}_i|\le L_w$, then, with ${\omega }_{\min }=\min ({\omega }_0,{\omega }_{\infty })$,

$$|{\widehat{w}}_i^{\mathrm{raw}}\left(t\right)-w_i\left(t\right)|\le e^{-{\omega }_{\min }t}|{\widehat{w}}_i^{\mathrm{raw}}\left(0\right)-w_i\left(0\right)|+{\displaystyle \frac{L_w}{{\omega }_{\min }}}(1-e^{-{\omega }_{\min }t}).$$

(31)

Proof. 

Differentiating $p_i+\omega v_i$ and substituting (28) gives

$${\dot{\widehat{w}}}_i^{\mathrm{raw}}=-\omega {\widehat{w}}_i^{\mathrm{raw}}-\omega {\mu }_i{u}_i+\omega {\dot{v}}_i=\omega (w_i-{\widehat{w}}_i^{\mathrm{raw}}).$$

(32)

This is a stable scalar filter. Subtracting ${\dot{w}}_i$ and using an integrating factor gives (31).  □

The derivative condition is needed only for the stated estimation-error bound. It is not an assumption required by the complete tracking theorem below. Filtering lag and bounded-output bias are permitted; exact or finite-time disturbance reconstruction is not claimed.

3.4. Design of Adaptive Sliding Mode Fault-Tolerant Controller

Due to the actuator faults in the system, an adaptive method is used to adjust a robustness bound. Combined with the above content, the following adaptive sliding mode fault-tolerant controller is designed:

$$u_i={\mu }_i^{-1}\left[-f_i-{\mathsf{\Gamma }}_i\tanh (S_i/{ϵ}_i)\right].$$

(33)

To specify the same integral surface as (17)–(19), define

$$z_i=k{\overline{x}}_i+{\overline{v}}_i,{\dot{J}}_i=z_i,J_i\left(0\right)=0,q_i=z_i+\lambda J_i,S=Hq.$$

(34)

Each node forms $S_i={\sum }_j{a}_{ij}(q_i-q_j)+b_i{q}_i$ from its own and neighboring q signals. No global graph inverse enters the control law. The nominal and adaptive gains are

$$f_i=k{\overline{v}}_i+\lambda z_i-r_i+{\widehat{w}}_i,A_i=\left|f_i\right|+1,{\mathsf{\Gamma }}_i=a_i{A}_i+c,$$

(35)

$${\dot{a}}_i=\tau A_i\left|S_i\right|,a_i\left(0\right)\ge 0,{ϵ}_i\left(t\right)={\displaystyle \frac{{ϵ}_0{e}^{-\nu t}}{1+{\mathsf{\Gamma }}_i\left(t\right)}},$$

(36)

here, $r_i={\dot{c}}_i+{\dot{v}}_0$, and $\tau ,c,{ϵ}_0,\nu$ are positive constants. The scalar $a_i$ adapts a conservative uncertainty-compensation gain; it is not the physical fault factor. The constant 1 is a unit acceleration scale in normalized variables. In dimensional notation, a fixed scale $a_{\mathrm{ref}}$ can be used in $A_i=\left|f_i\right|+a_{\mathrm{ref}}$ and $1+{\mathsf{\Gamma }}_i/a_{\mathrm{ref}}$; the numerical implementation uses $a_{\mathrm{ref}}=1$. No fixed floor on ${ϵ}_i$, adaptive-gain clipping, or actuator clipping is imposed.

Theorem 4.

For the heterogeneous multi-agent system, by designing the adaptive sliding mode fault-tolerant controller in (33), under the conditions that Assumptions 1–4 hold, this adaptive sliding mode fault-tolerant controller can ensure that the closed-loop errors, controller and observer states, and commanded inputs remain bounded, and the physical position and velocity errors converge to zero asymptotically.

Proof. 

Construct a composite Lyapunov function

$$V_4\left(t\right)={\displaystyle \frac{1}{2}}{S}^T\left(t\right)H^{-1}S\left(t\right)+{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{\rho {(a_{*}-a_i\left(t\right))}^2}{2\tau }},$$

(37)

where $a_{*}=[(1-\rho )+D+M]/\rho$ is a constant comparison value used only in the analysis. Neither D nor $\rho$ is required by the controller. Writing ${\tilde{a}}_i=a_{*}-a_i$, one has $(1-\rho )|f_i|+D+M\le \rho a_{*}{A}_i$.

The sliding mode term is ${\displaystyle \frac{1}{2}}{S}^T{H}^{-1}S$. Since $H^{-1}$ is a constant symmetric matrix, the following derivative is obtained:

$${\dot{V}}_{41}=S^T{H}^{-1}\dot{S}.$$

(38)

The exact error and surface derivatives are

$${\dot{\overline{v}}}_i={\mu }_i{R}_i{u}_i+d_i-r_i,$$

(39)

$$\dot{S}=H[\theta f+d-\widehat{w}-R\mathsf{\Gamma }\tanh (S/ϵ)],$$

(40)

where products involving diagonal actuator factors and the hyperbolic tangent are componentwise. No exchange of H with R is used. The derivative of the complete Lyapunov function is

$${\dot{V}}_4=S^T{H}^{-1}\dot{S}-\sum _i{\displaystyle \frac{\rho }{\tau }}{\tilde{a}}_i{\dot{a}}_i,$$

(41)

here, weight $\rho$ is constant, so there is no omitted time derivative of an instantaneous fault norm.

For a scalar s, $0\le \left|s\right|-s\tanh (s/ϵ)\le ϵ$. Moreover, ${\mathsf{\Gamma }}_i{ϵ}_i\le {ϵ}_0{e}^{-\nu t}$. Since $R_i\ge \rho$, $R_i\le 1$, and $|{\widehat{w}}_i|\le M$, each surface contribution is bounded by

$$\begin{array}{cc}\hfill S_i[{\theta }_i{f}_i+d_i-{\widehat{w}}_i-R_i{\mathsf{\Gamma }}_i\tanh (S_i/{ϵ}_i)]& \le \rho {\tilde{a}}_i{A}_i|S_i|-\rho c|S_i|+{ϵ}_0{e}^{-\nu t}.\hfill \end{array}$$

(42)

The update law (36) cancels the first term. Thus,

$${\dot{V}}_4\le -\rho c{\parallel S\parallel }_1+n{ϵ}_0{e}^{-\nu t},$$

(43)

$$V_4\left(t\right)+\rho c{\int }_0^t{\parallel S\left(\sigma \right)\parallel }_{1}d\sigma \le V_4\left(0\right)+{\displaystyle \frac{n{ϵ}_0}{\nu }}(1-e^{-\nu t}).$$

(44)

Consequently, S and $a_i$ are bounded and S is absolutely integrable. Set $q=H^{-1}S$. The identities

$$\dot{J}=-\lambda J+q,z=q-\lambda J,\dot{\overline{x}}=-k\overline{x}+z,\overline{v}=z-k\overline{x}$$

(45)

give bounded $J,z,\overline{x},\overline{v}$ through stable linear filters. Bounded r and $\widehat{w}$ then give bounded $f,A,\mathsf{\Gamma },u$. Therefore, $\dot{S}$ is bounded by (40), so S is uniformly continuous. Lemma 3 yields $S\to 0$. Applying the same stable filters in (45) to the input tending to zero gives $J,z,\overline{x},\overline{v}\to 0$.

Finally $w_i=d_i-{\theta }_i{\mu }_i{u}_i$ is bounded. The observer filter (32) gives bounded raw output, and $p_i={\widehat{w}}_i^{\mathrm{raw}}-\omega v_i$ is bounded because $v_i={\overline{v}}_i+c_i+v_0$ is bounded. These estimates prevent finite escape and establish forward completeness. Adaptive gains have finite limits, without requiring fault-parameter identification. The proof does not substitute an upper bound on $\parallel S\parallel$ into a negative inequality to claim finite-time convergence.  □

4. Simulation Results

To verify the effectiveness of the proposed adaptive integral sliding mode control scheme, numerical simulations are conducted on a second-order error system subject to external disturbances. The position error $e_x\left(t\right)$ and velocity error $e_v\left(t\right)$ are considered, and the basic sliding variable is defined as $\widehat{S}\left(t\right)=k{e}_x\left(t\right)+e_v\left(t\right)$. Based on this variable, the integral sliding surface is constructed according to (17), where the surface gains are $g_1=1$, $g_2=\lambda >0$, and the controller robustness gains $a_i$ are updated online.

The initial conditions are set as $e_x\left(0\right)=1.0$ and $e_v\left(0\right)=0$. The control parameters are chosen as $k=2.0$, $\lambda =0.7$, $\tau =0.15$, $a\left(0\right)=0.1$, and $c=0.25$. To alleviate numerical chattering, a hyperbolic tangent function is employed to approximate the discontinuous sign function, with the positive width in (36), ${ϵ}_0=0.02$, and $\nu =0.15$. The external disturbance is selected as $d\left(t\right)=0.5\sin \left(t\right)$. The total simulation time is $10 \mathrm{s}$ with an output interval of $0.005 \mathrm{s}$. The scalar example uses a stationary zero reference, unit nominal input gain, and $\theta \left(t\right)=0.25\psi \left(t\right)(1+\sin (0.5t\left)\right)/2$, with $\psi$ defined below. The observer parameters are the same as in the four-follower example.

To validate the proposed adaptive integral sliding mode control strategy with a time-varying gain disturbance observer, numerical simulations are conducted on a heterogeneous multi-agent system comprising one leader and four followers. The communication topology is defined by the Laplacian matrix L and leader adjacency matrix B:

$$L=\left[\begin{array}{cccc}1& -1& 0& 0\\ -1& 2& -1& 0\\ 0& -1& 2& -1\\ 0& 0& -1& 1\end{array}\right],B=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].$$

The composite matrix $H=L+B$ is positive definite, satisfying the connectivity requirement. Its eigenvalues are approximately 1, $1.5858$, 3, and $4.4142$. The known masses are $[1,1.2,0.8,1.5]$, so ${\mu }_i=1/{\mathfrak{m}}_i$ differs between followers. This is the modeled input-gain heterogeneity.

The first-order dynamics of the leader are given by ${\dot{x}}_0=v_0$ with $x_0\left(0\right)=0.0$ and $v_0=0.5$ (constant velocity). Followers are governed by second-order dynamics ${\dot{x}}_i=v_i$, ${\dot{v}}_i={\mu }_i(1-{\theta }_i)u_i+d_i$, where $x_i$ and $v_i$ denote position and velocity, respectively. Initial states of followers are set as $x_i\left(0\right)={[1.0,2.0,-1.0,-2.0]}^T$ and $v_i\left(0\right)={[0.8,1.0,-0.8,-1.0]}^T$. The desired offsets are $h_i\left(t\right)={\overline{h}}_i+0.05\sin \left(0.5t\right)$, with $\overline{h}={[0.8,1.8,-0.8,-1.8]}^T$, $c_i=0.025\cos \left(0.5t\right)$, and $r_i=-0.0125\sin \left(0.5t\right)$. These are distinct formation offsets with consistent derivatives.

External disturbances $d_i\left(t\right)$ are designed as $d_i\left(t\right)=0.1\sin (0.5t+i-1)$ to mimic environmental perturbations. Actuator faults are introduced for followers 2 and 3 using one-based indices, with

$$\begin{array}{cc}\hfill {\theta }_2\left(t\right)& =0.7{\theta }_{\max }\psi \left(t\right)(1+\sin \left(0.5t\right))/2,\hfill \\ \hfill {\theta }_3\left(t\right)& ={\theta }_{\max }\psi \left(t\right)(1+\cos \left(0.7t\right))/2,\hfill \\ \hfill \psi \left(t\right)& =\left\{\begin{array}{cc}0,\hfill & t\le 1,\hfill \\ 1-e^{-1.5{(t-1)}^2},\hfill & t>1,\hfill \end{array}\right.{\theta }_{\max }=0.45.\hfill \end{array}$$

(46)

The other fault factors are zero. The profiles are nonnegative and describe partial effectiveness loss. They satisfy $R_i\ge 0.55$ and $|d_i|\le 0.1$; these bounds are used for analysis, not supplied to the controller.

Control parameters are tuned as $k=2.0$, $\tau =0.15$, $\lambda =0.7$, $a_i\left(0\right)=0.1$, $c=0.25$, ${ϵ}_0=0.02$, $\nu =0.15$, $M=2$, ${\omega }_0=3$, ${\omega }_{\infty }=10$, and $\kappa =0.8$, satisfying the stability conditions. The simulation runs for 20 s with an output interval of $0.01 \mathrm{s}$.

Both programs implement (28)–(36) and the same reference derivatives. Integration uses Radau with relative tolerance ${10}^{-8}$, absolute tolerance ${10}^{-10}$, and maximum internal step $0.02$ s. The output interval is not a digital sample-and-hold period. A refined run uses tolerances ${10}^{-10}$ and ${10}^{-12}$ and maximum step $0.01$ s. There is no hard actuator clipping or adaptive-gain clipping in this model.

Figure 1 depicts the evolution of the adaptive integral sliding variable $\left|S\right(t\left)\right|$. The scalar program uses positive constant integral coefficients and an adaptive robustness gain; the latter is not a physical fault estimate. Its position RMS over 8-10 s is 0.005568 m. The final gain sample is evaluated by the same equations, rather than left at an initialized zero.

Figure 1. The integral variable, adaptive robustness gain, and errors of the scalar example.

Figure 2 depicts the position trajectories of the agents, from which it can be observed that all followers approach their distinct desired formation positions in spite of initial position deviations. Figure 3 illustrates velocity tracking relative to $v_0+c_i\left(t\right)$, together with commanded forces. As shown in Figure 4, the physical position tracking errors converge to a neighborhood near zero, which is quantified using finite-window RMS rather than described as exact finite-time elimination. Figure 5 presents a comparison between the true lumped unknown input and its estimated value for followers 1–4, using the actual target $w_i=d_i-{\theta }_i{\mu }_i{u}_i$ only for offline evaluation.

Figure 2. Position tracking: leader-follower formation with distinct desired offsets under actuator faults and disturbances. Upper: full position histories. Lower: offsets relative to the leader over 0–4 s, with dotted desired offsets.

Figure 3. Velocity tracking: actual and reference velocities, tracking errors, and commanded input forces.

Figure 4. Position error convergence: integral sliding mode surface performance. The upper panel shows composite $e_{xi}$; the lower panel shows physical ${\overline{x}}_i$, with a 16–20 s inset.

Figure 5. Lumped-input estimation: time-varying gain observer tracking with nonzero filtering lag and possible bounded-output bias.

For 16–20 s, the per-agent physical position RMS values are $[4.474,3.265,4.216,7.132]\times {10}^{-6}$ m in Figure 6, and the aggregate $E_x=\sqrt{mean\parallel \overline{x}{\parallel }_2^2}$ is $9.966\times {10}^{-6}$ m in Table 1. The raw lumped-observer RMS values are approximately $[0.003969,0.005211,0.005683,0.003787]$ $\mathrm{m} {\mathrm{s}}^{-2}$. Peak commanded force is 7.781 N, so this run is not a demonstration under a 5 N limit. Tightening the integration settings changes physical position-error curves by at most $2.051\times {10}^{-9}$ m at common output times. The maximum transient physical position-error magnitude is 0.473 m, compared with an initial magnitude of 0.2 m. All followers enter and remain within ±0.01 m at subsequent output samples from 6.82 s through 20 s; this is a finite-horizon sampled statistic, not a theoretical settling-time guarantee.

Figure 6. Same-time pairwise distances, including the leader, and the partial-loss histories. This is a sampled diagnostic, not a global collision-avoidance guarantee.

Table 1. Matched ablations: aggregate physical RMS over 16–20 s.

Case	${\mathit{E}}_{\mathit{x}}$ (m)	${\mathit{E}}_{\mathit{v}}$ (m/s)	Peak Force (N)
Adaptive with observer	$9.966\times {10}^{-6}$	$6.137\times {10}^{-6}$	7.781
Adaptive without observer	$4.304\times {10}^{-5}$	$2.409\times {10}^{-5}$	7.781
Fixed gain with observer	$7.162\times {10}^{-5}$	$4.739\times {10}^{-5}$	7.781
Adaptive, no fault	$9.945\times {10}^{-6}$	$6.107\times {10}^{-6}$	7.781

Matched ablations retain the same graph, plant, reference, initial states, faults, disturbances, and solver. The no-observer case sets the control-facing estimate to zero; the fixed-gain case keeps $a_i=0.1$; the no-fault case sets ${\theta }_i=0$, as shown in Figure 7.

Figure 7. Matched no-observer and fixed-gain ablations on the same plant.

The adaptive observer-assisted run reduces aggregate final-window position RMS relative to the first two ablations in this example, without uniform improvement for every follower or every metric. Additional fault-level runs, without retuning, are reported as Table 2.

Table 2. Fault-level runs with the same controller settings. The limit is a profile parameter, not a maximum compensability boundary.

${\mathit{\theta }}_{\mathbf{max}}$	Guaranteed ${\mathit{R}}_{\mathit{i}}\mathbf{\ge }$	${\mathit{E}}_{\mathit{x}}$ (m)	Sampled Minimum Distance (m)
0.30	0.70	$9.956\times {10}^{-6}$	0.6478
0.45	0.55	$9.966\times {10}^{-6}$	0.6478
0.60	0.40	$9.986\times {10}^{-6}$	0.6478
0.85	0.15	$1.016\times {10}^{-5}$	0.6478

These are tested partial-loss profiles, not a demonstrated maximum fault boundary.

5. Discussion

The chosen graph satisfies the symmetric positive-definiteness condition. Positive k and $\lambda$ make the error filters stable; positive $\tau$ and c give adaptation and the negative term in (43). Positive ${ϵ}_0,\nu$ give an integrable smoothing budget, and the output bound M protects the controller without assuming exact observer convergence. The complete tracking theorem uses a disturbance magnitude bound; a disturbance derivative bound alone is insufficient. The gain schedule is used in the implemented observer at every solver evaluation.

Finite-window residual errors and observer lag must not be confused with exact asymptotic limits. The raw observer is a filter of the combined disturbance and actuator-induced input error. Bounded output can add bias. A fixed smoothing floor, measurement noise, actuator saturation, or digital sample-and-hold would require an extension of the current ideal continuous-feedback result. The present solver output spacing is not a deployment sampling-rate guarantee.

The position plots are one-dimensional position versus time, not planar paths. All ten simultaneous pairwise distances, including the leader, were checked in the new example. The minimum sampled separation is approximately $0.6478$ m. Distinct desired offsets replace the common-target example, but no collision-avoidance constraint or arbitrary-initial-condition safety theorem is claimed. Agent size and physical input limits must be included before interpreting this example as a safe hardware deployment.

The model permits simultaneous partial degradation of every follower provided each channel retains uniformly positive effectiveness. It does not include complete failure, lock-in-place, or sign reversal. The fault tests at ${\theta }_{\max }=0.30,0.60,0.85$ are not a proof of a largest tolerable fault. The heterogeneity is specifically known nominal input-gain differences; arbitrary heterogeneous nonlinear dynamics and unknown masses are not included. Every follower receives the reference, as stated in Assumption 4.

6. Conclusions

This paper focuses on addressing the formation control problem of HMASs subject to time-varying external disturbances and actuator faults. To accurately estimate the lumped disturbance and fault-induced input error affecting the HMAS, the variable gain disturbance observer is designed. Furthermore, to achieve robust formation tracking under actuator faults, an adaptive integral sliding mode fault-tolerant control strategy is proposed. Under the proposed control framework, the HMASs asymptotically achieve the desired formation configuration. Simulation results demonstrate that the proposed control strategy reduces the aggregate final-window tracking RMS relative to the specified no-observer and fixed-gain ablations in the tested example.