Human-in-the-Loop Time-Varying Formation Tracking of Networked UAV Systems with Compound Actuator Faults

Highlights

What are the main findings?

• A novel two-layer human-in-the-loop control framework is developed for time-varying formation tracking of networked UAVs under compound actuator faults and external disturbances.
• A distributed observer and an adaptive fault-tolerant controller jointly ensure bounded convergence of formation tracking errors using only local information exchange.

What are the implications of the main findings?

• The proposed approach enhances the reliability and robustness of cooperative UAV missions involving human operators in fault-prone environments.
• The results provide a scalable and practical control solution for real-world multi-UAV applications such as surveillance, encirclement, and search-and-rescue operations.

Abstract

Time-varying formation tracking of networked unmanned aerial vehicle (UAV) systems plays a crucial role in cooperative missions such as encirclement, cooperative surveillance, and search-and-rescue operations, where human operators are often involved and system reliability is challenged by actuator faults and external disturbances. Motivated by these practical considerations, this paper investigates a human-in-the-loop time-varying formation tracking problem for networked UAV systems subject to compound actuator faults and external disturbances. To address this problem, a novel two-layer control architecture is developed, comprising a distributed observer and a fault-tolerant controller. The distributed observer enables each UAV to estimate the states of the human-in-the-loop leader using only local information exchange, while the fault-tolerant controller is designed to preserve formation tracking performance in the presence of compound actuator faults. By incorporating dynamic iteration regulation and adaptive laws, the proposed control scheme ensures that the formation tracking errors converge to a bounded neighborhood of the origin. Rigorous Lyapunov-based analysis is conducted to establish the stability, convergence, and robustness of the resulting closed-loop system. Numerical simulations further demonstrate the effectiveness of the proposed method in achieving practical time-varying formation tracking under complex fault scenarios.

1. Introduction

With the rapid advancement of artificial intelligence and automation technologies [1,2,3], unmanned aerial vehicle (UAV) swarms have attracted significant research attention and are increasingly applied across diverse domains, including cooperative surveillance, large-scale mapping, precision agriculture, and payload transportation [4,5,6]. Compared to individual UAVs, networked UAV systems offer distinct advantages, such as enhanced operational efficiency, increased redundancy, expanded spatial coverage, and improved system robustness, all of which stem from their ability to perform collaborative tasks [7,8,9]. However, the realization of these benefits is contingent upon the development of effective coordination and control strategies that ensure the swarm operates as a unified, cohesive entity [10,11].

In most existing literature on cooperative tracking control for networked systems, the reference trajectory for the swarm is typically generated by an autonomous system, often modeled as an autonomous leader or a pre-defined command generator [12,13]. While full autonomy remains a long-term goal, current artificial intelligence and autonomous decision-making technologies still face significant challenges in addressing the highly dynamic, uncertain, and complex real-world environments. Unforeseen events, intricate obstacle fields, or ambiguous mission objectives can easily surpass the capabilities of fully autonomous systems, potentially leading to mission failure or unsafe conditions. To mitigate these limitations, the human-in-the-loop control paradigm has emerged as a promising solution. By incorporating human cognitive abilities, experiential knowledge, and high-level decision-making into the control loop, human-in-the-loop systems can substantially enhance the reliability, adaptability, and safety of UAV swarms operating in complex scenarios [14,15]. Human operators can provide strategic guidance, address unexpected situations, and override autonomous decisions when necessary, thus fostering a synergistic human–machine team. For instance, a fuzzy iterative learning control approach was proposed in [14] to address the human-in-the-loop consensus problem in unknown or mixed-order nonlinear networked systems. Additionally, in [16], the human-in-the-loop time-varying formation-containment tracking control problem for general linear networked systems was tackled using a control scheme combined with a local observer and an unknown input reconstruction (UIR) mechanism.

In practical missions, UAV actuators are exposed to prolonged operational loads, harsh environmental conditions, and potential external disturbances. As a result, compound actuator faults, such as partial loss of effectiveness and bias faults, are inevitable over time [17,18]. If not appropriately addressed, these faults can severely degrade system performance, lead to unstable swarm behavior, or even cause catastrophic failures. Therefore, the development of fault-tolerant control strategies is not merely an enhancement but a critical necessity for ensuring the safe and reliable operation of UAV swarms [19]. Fault-tolerant control schemes are designed to maintain acceptable system performance and stability despite the presence of specified actuator faults, thus enhancing the resilience and operational availability of the swarm. For instance, an adaptive neural network-based fault-tolerant control method was proposed in [20] to solve the optimized consensus tracking control problem for nonlinear networked systems. In [21], a specified-time adaptive control algorithm was introduced to address the fault-tolerant consensus tracking control problem for high-order nonlinear networked systems with actuator faults.

Furthermore, a common limitation in many existing studies on swarm formation tracking control is the focus on predominantly fixed or static formations, where the relative offsets between agents remain constant over time [22,23]. While suitable for simple environments, fixed formations have significant limitations in complex operational scenarios, such as those involving obstacles, threats, or challenging terrains like urban canyons and dense forests. The inherent rigidity of fixed formations prevents the swarm from navigating narrow passages or adaptively reconfiguring to avoid threats. In contrast, time-varying formation control offers a more flexible and powerful alternative. A time-varying formation allows the swarm to dynamically adjust its collective shape in real time, enabling it to efficiently circumvent obstacles, distribute sensing resources from various angles and baselines to improve situational awareness, and enhance survivability in contested environments [24,25]. Time-varying formations inherently contribute to system resilience; if a swarm member develops a fault or is lost, the remaining agents can dynamically reassign roles and reposition themselves to fill the gap, thereby maintaining the overall formation’s integrity and mission capability [26,27].

In summary, to address the combined challenges of environmental adaptation, human oversight integration, and system fault tolerance, this paper investigates the problem of fault-tolerant, time-varying formation tracking control for networked UAV systems, based on a human-in-the-loop architecture specifically designed to handle compound actuator faults. The main contributions of this work are as follows:

(1) Unlike conventional control strategies that depend entirely on autonomous leaders [12,13], the proposed framework introduces a human operator into the control loop for reference trajectory generation. This human-in-the-loop design leverages human cognitive capabilities for high-level decision-making in complex and uncertain environments, substantially improving the overall reliability and safety of the UAV swarm system.

(2) In contrast to control methods developed solely for fault-free operation, the proposed controller integrates an adaptive fault-compensation mechanism to address actuator impairments [8,12,14,24,26,27]. This approach ensures robust tracking performance and maintains system stability even under partial actuator failures or performance degradation, thereby significantly enhancing operational adaptability in post-fault scenarios.

(3) The proposed scheme supports dynamic, real-time reconfiguration of the swarm’s geometric formation [22,23,28], offering enhanced flexibility and resilience compared to static formation strategies. This capability allows the swarm to navigate through cluttered environments by adaptively adjusting its shape, which improves overall survivability. Moreover, the time-varying formation inherently provides fault tolerance, enabling the swarm to reconfigure and compensate for the loss or impairment of individual members, thus preserving collective mission objectives.

The remainder of this paper is organized as follows. Section 2 presents the preliminaries and problem formulation, including the necessary graph theory and the problem description. Section 3 introduces the main results, detailing the design of the distributed observer and the fault-tolerant controller. In Section 4, simulation results are provided to validate the proposed methods. Finally, Section 5 concludes the paper and discusses potential directions for future research.

2. Preliminaries and Problem Formulation

2.1. Graph Theory

Consider a networked UAV system comprising N followers. The interactions among N agents is represented by a structurally balanced signed $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$ in this paper, where $\mathcal{V}=\{v_1,\dots ,v_N\}$ denotes the node set, $\mathcal{E}=\{{\varpi }_{ij}=(v_i,v_j)\}\subset \mathcal{V}\times \mathcal{V}$ means the edge set, and $\mathcal{A}={\left[a_{ij}\right]}_{N\times N}$ represents the adjacent matrix. If the agent i has access to the information of the agent j, then $a_{ij}\ne 0$; otherwise, $a_{ij}=0$. $a_{ij}>0$ and $a_{ij}<0$ represent the cooperative and antagonistic relationship between agent i and j, respectively. Then, the Laplacian matrix is defined as $L=D-\mathcal{A}$, where D is represented as $D={\mathbf{diag}}_i^N\left[{\zeta }_i\right],$ with ${\zeta }_i={\displaystyle \sum _{j=1}^N}\left|a_{ij}\right|$. This paper postulates the presence of the “human-in-the-loop system” with the index number 0. We define $B={\mathbf{diag}}_i^N\left[a_{i0}\right]$. If the ith agent is informed about the leader’s information, then $a_{i0}>0$; otherwise, $a_{i0}=0$.

2.2. Problem Description

To model the dynamics of the ith follower UAV, we employ the following second-order system, which describes the position and velocity of each UAV in the network:

$$\begin{array}{cc}\hfill {\dot{p}}_i\left(t\right)& =q_i\left(t\right),\hfill \\ \hfill {\dot{q}}_i\left(t\right)& =g_i{u}_i\left(t\right)+{\overline{F}}_i,i=1,2,\dots ,n,\hfill \end{array}$$

(1)

where $p_i\left(t\right)$ and $q_i\left(t\right)$ represent the position and velocity of the ith UAV, respectively. The control input $u_i\left(t\right)$ corresponds to the control signal for the ith UAV, while $g_i$ denotes the unknown health index of the actuator. Additionally, ${\overline{F}}_i$ represents the uncertain fault term that models external disturbances and internal system uncertainties.

Moreover, to better align with practical applications, Equation (2) can be transformed as follows:

$$\begin{array}{cc}\hfill M_i\left(p_i\right){\ddot{p}}_i+C_i(p_i,{\dot{p}}_i){\dot{p}}_i+G_i\left(p_i\right)& =g_i{u}_i+f_i\left(t\right),i=1,2,\dots ,N,\hfill \end{array}$$

(2)

where $p_i$, ${\dot{p}}_i$, and ${\ddot{p}}_i$ represent the generalized coordinate, velocity, and acceleration of the ith UAV, respectively, with each belonging to ${\mathbb{R}}^n$. The control input $u_i\in {\mathbb{R}}^n$, the Coriolis and centrifugal terms $C_i(p_i,{\dot{p}}_i)\in {\mathbb{R}}^{n\times n}$, the inertia matrix $M_i\left(p_i\right)\in {\mathbb{R}}^{n\times n}$, and the gravitational force vector $G_i\left(p_i\right)\in {\mathbb{R}}^n$ are associated with the ith UAV. The term $f_i\left(t\right)\in {\mathbb{R}}^n$ represents the unknown fault function, and $g_i$ denotes the unknown health index of the actuator, where $0<g_i\le 1$. Specifically, if $g_i=1$, the ith actuator is fully operational, while if $0<g_i<1$, the actuator is subject to faults due to external disturbances.

Property 1.

$M_i\left(q_i\right)$ is a symmetric, positive definite matrix.

Property 2.

${\dot{M}}_i\left(q_i\right)-2C_i(q_i,{\dot{q}}_i)$ is skew-symmetric, i.e., for all $x\in {\mathbb{R}}^n$, one gets that $x^T[{\dot{M}}_i\left(q_i\right)-2C_i(q_i,{\dot{q}}_i)]x=0$.

Furthermore, the following human-in-the-loop system is considered to generate the reference trajectory in this paper:

$$\begin{array}{cc}\hfill {\dot{\eta }}_0\left(t\right)& =A_0{\eta }_0\left(t\right)+B_0{u}_0\left(t\right),\hfill \\ \hfill y_0\left(t\right)& =C_0{\eta }_0\left(t\right),\hfill \end{array}$$

(3)

where ${\eta }_0\left(t\right)\in {\mathbb{R}}^m$ and $y_0\left(t\right)\in {\mathbb{R}}^n$ represent the state and output, respectively. The matrices $A_0\in {\mathbb{R}}^{m\times m}$, $B_0\in {\mathbb{R}}^{m\times p}$, and $C_0\in {\mathbb{R}}^{n\times m}$ denote the state matrix, the input matrix, and the output matrix, respectively. The input $u_0\left(t\right)\in {\mathbb{R}}^p$ of the human-in-the-loop system is unknown to all followers and bounded, i.e., there exists a positive constant $\alpha$ such that $\parallel u_0\left(t\right)\parallel <\alpha$.

Assumption 1.

At least one agent can access the information of the human-in-the-loop system. Furthermore, a structurally balanced connected graph can describe the interaction relationships among agents.

Assumption 2.

The ranks of matrices $A_0$, $B_0$, and $C_0$ satisfy the following conditions:

(1) $\mathrm{rank}\left(C_0{B}_0\right)=\mathrm{rank}\left(B_0\right)=p$;

(2) $\mathrm{rank}\left[\begin{array}{cc}\epsilon I_m-A_0& B_0\\ C_0& 0\end{array}\right]=m+p$, where ε is a constant.

Assumption 3.

The false data injected into the follower model (1) is bounded, that is, $∥f_i\left(t\right)∥<d_i$, where $d_i$ is an unknown positive constant.

Lemma 1

([29]). If the Lyapunov function $V\left(x\right)$ with respect to x satisfies the following inequality

$$\begin{array}{c}\hfill \dot{V}\left(x\right)+\lambda V\left(x\right)\le \omega ,\end{array}$$

(4)

where λ and ω are positive constants, then $V\left(x\right)$ will converge to the bound defined as $V\left(x\right)\le {\displaystyle \frac{\omega }{\lambda (1-\delta )}}$ within the finite time $t_f$. There exists an upper bound of $t_f$, that is, $t_f\le {\displaystyle \frac{1}{\lambda \delta }}ln{\displaystyle \frac{\lambda (1-\delta )V\left(0\right)}{\omega }}$, where δ is a constant satisfying that $0<\delta <1$, and $V\left(0\right)$ is the initial value of $V\left(x\right)$.

Lemma 2

([30]). Define a vector $\gamma ={[\gamma ,\dot{\gamma },\dots ,{\gamma }^{(n-1)}]}^T$ and a function as follows:

$$\begin{array}{c}\hfill {\gamma }_f(\gamma ,t)={({\displaystyle \frac{d}{dt}}+\theta )}^{(n-1)}\gamma ,\end{array}$$

(5)

where θ is a positive constant. If ${\gamma }_f(\gamma ,t)$ is bounded, that is, $∥{\gamma }_f(\gamma ,t)∥\le \overline{\gamma }$, then γ will be asymptotically bounded by $\left|{\gamma }^{\left(k\right)}\right|\le {\left(2\theta \right)}^k{\displaystyle \frac{\overline{\gamma }}{{\theta }^{n-1}}},k=0,1,\dots ,n-1$.

Lemma 3

([31]). Under Assumption 1, we can select a transformation matrix $H={\mathit{diag}}_i^N\left[h_i\right]$ such that $\overline{L}=HLH$, where $h_i=1$. Hence, we can get $Z={\mathit{diag}}_i^N\left[z_i\right]$ by solving ${\mathit{col}}_i^N\left[z_i\right]={\left(\overline{L}\right)}^{-T}{1}_N$ such that ${\overline{L}}^{T}Z+Z\overline{L}>0$. Define $w={\lambda }_{min}(Z^{-1}{\overline{L}}^{T}Z+\overline{Z})$. Then, under Assumption 1, we have $w>0$.

Definition 1.

Consider the networked systems (3), (1). If the below condition holds true

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}(p_i\left(t\right)-y_0\left(t\right)-b_i\left(t\right))\le \chi ,\end{array}$$

(6)

where $\chi >0$ denotes the tracking error bound and $b_i\left(t\right)$ represents the time-varying formation parameter, then the networked systems (3) and (1) achieve practical time-varying formation tracking.

3. Main Results

In this section, we present the control scheme, which consists of a distributed observer and a fault-tolerant controller. Specifically, the distributed observer is designed for each follower to estimate the dynamic states of the human-in-the-loop system (3). Based on these estimates, the fault-tolerant controller is developed to drive the networked Euler–Lagrange system, achieving practical bipartite consensus tracking and mitigating the impact of actuator faults on the system.

3.1. Distributed Observer Design

A distributed observer is designed in this subsection to estimate the state of the human-in-the-loop system (3).

At first, the estimation error is defined as

$$e_{\eta i}\left(t\right)=\sum _{j=1}^N{a}_{ij}({\eta }_i\left(t\right)-{\eta }_j\left(t\right))+a_{i0}({\eta }_i\left(t\right)-{\eta }_0\left(t\right)),$$

where ${\eta }_i$ denotes the $i_{th}$ follower’s estimation of the state of the human-in-the-loop system. Let ${\widehat{e}}_{\eta i}\left(t\right)$ be the estimation of $e_{\eta i}\left(t\right)$. The estimation system of the estimation error is designed as

$$\begin{array}{cc}\hfill {\dot{\mu }}_i\left(t\right)& =[\Phi A_0-\Theta C_0]{\mu }_i\left(t\right)+[\Theta (I+C_0\Gamma )-\Phi A_0\Gamma ]{\phi }_i\left(t\right),\hfill \\ \hfill {\widehat{e}}_{\eta i}\left(t\right)& ={\mu }_i\left(t\right)-\Gamma {\phi }_i\left(t\right),\hfill \end{array}$$

(7)

where ${\phi }_i\left(t\right)={\displaystyle \sum _{j=1}^N}{a}_{ij}(C_0{\eta }_i-C_0{\eta }_j)+a_{i0}(C_0{\eta }_i-y_0)$, $\Gamma =-B_0{\left[{\left(C_0{B}_0\right)}^T\left(C_0{B}_0\right)\right]}^{-1}{\left(C_0{B}_0\right)}^T$, and $\Phi =I+\Gamma C_0$. $\Theta$ is a matrix which can make $\Phi A_0-\Theta C_0$ satisfy the Hurwitz property. The estimation error vector is defined as ${\tilde{e}}_{\eta i}={\widehat{e}}_{\eta i}-e_{\eta i}$. Then we have

$$\begin{array}{c}\hfill {\dot{\tilde{e}}}_{\eta i}=(\Phi A_0-\Theta C_0){\tilde{e}}_{\eta i}.\end{array}$$

(8)

It can be observed that system (8) is asymptotically stable. Consequently, it follows that

$$\underset{t\to \infty }{lim}{\tilde{e}}_{\eta i}={\widehat{e}}_{\eta i}-e_{\eta i}=0,$$

which implies that the estimated signal ${\widehat{e}}_{\eta i}\left(t\right)$ asymptotically converges to the true signal $e_{\eta i}\left(t\right)$ and thus provides an accurate estimation.

Subsequently, based on the estimation dynamics in (7), a distributed observer is designed as follows:

$$\begin{array}{cc}\hfill {\dot{\varphi }}_i\left(t\right)& =-({\alpha }_{1i}+{\overline{\alpha }}_{1i}){\varphi }_i\left(t\right),\hfill \\ \hfill {\dot{\eta }}_i\left(t\right)& =A_0{\eta }_i\left(t\right)+B_0(-{\lambda }_{1}Q{\widehat{e}}_{\eta i}\left(t\right)-{\lambda }_2{\displaystyle \frac{Q{\widehat{e}}_{\eta i}\left(t\right)}{∥Q{\widehat{e}}_{\eta i}\left(t\right)∥+{\varphi }_i\left(t\right)}}),\hfill \end{array}$$

(9)

where the initial state of ${\varphi }_i\left(t\right)$ satisfies that ${\varphi }_i\left(0\right)>0$, ${\alpha }_{1i}$ and ${\overline{\alpha }}_{1i}$ are positive constants, ${\lambda }_1$, ${\lambda }_2$, and Q are control parameters. The estimation error is defined as ${\overline{e}}_{\eta i}\left(t\right)={\eta }_i\left(t\right)-h_i{x}_0\left(t\right)$. According to (3), (7), and (9), we can get that

$$\begin{array}{c}\hfill {\dot{\overline{e}}}_{\eta i}\left(t\right)=A_0{\overline{e}}_{\eta i}\left(t\right)-{\lambda }_1{B}_{0}Q{\widehat{e}}_{\eta i}\left(t\right)-{\lambda }_2{B}_{0}Q\chi \left(t\right)-B_0{u}_0\left(t\right),\end{array}$$

(10)

where $\chi \left(t\right)={\displaystyle \frac{Q{\widehat{e}}_{\eta i}\left(t\right)}{∥Q{\widehat{e}}_{\eta i}\left(t\right)∥+{\varphi }_i\left(t\right)}}$.

Define ${\overline{e}}_{\eta }\left(t\right)={\mathbf{col}}_i^N\left[{\overline{e}}_{\eta i}\left(t\right)\right]$ and ${\widehat{e}}_{\eta }\left(t\right)={\mathbf{col}}_i^N\left[{\widehat{e}}_{\eta i}\left(t\right)\right]$. Thus, we obtain that $e_{\eta }\left(t\right)=(\overline{L}\otimes {\mathbf{1}}_m){\overline{e}}_{\eta }\left(t\right)$. It can further yield that

$$\begin{array}{c}\hfill {\dot{e}}_{\eta }\left(t\right)=(I_N\otimes A_0)e_{\eta }\left(t\right)-{\lambda }_1(L\otimes B_{0}Q){\widehat{e}}_{\eta }\left(t\right)-{\lambda }_1(L\otimes B_{0}Q)\chi \left(t\right)-(L{\mathbf{1}}_N\otimes B_0)u_0\left(t\right),\end{array}$$

(11)

where $e_{\eta }\left(t\right)={\mathbf{col}}_i^N\left[e_{\eta i}\left(t\right)\right]$. Define ${\tilde{e}}_{\eta }\left(t\right)={\mathbf{col}}_i^N\left[{\tilde{e}}_{\eta i}\left(t\right)\right]$. From (8), one gets that

$$\begin{array}{c}\hfill {\dot{\tilde{e}}}_{\eta }\left(t\right)=(I_N\otimes (\Phi A_0-\Theta C_0)){\tilde{e}}_{\eta }.\end{array}$$

(12)

From (10) and (11), we have

$$\begin{array}{cc}\hfill {\dot{\widehat{e}}}_{\eta }\left(t\right)=& (I_N\otimes A_0){\widehat{e}}_{\eta }\left(t\right)-{\lambda }_1(L\otimes B_{0}Q){\widehat{e}}_{\eta }\left(t\right)-{\lambda }_1(L\otimes B_{0}Q)\chi \left(t\right)-(L{1}_N\otimes B_0)u_0\left(t\right)\hfill \\ & +(I_N\otimes (\Phi A_0-\Theta C_0-A_0)){\tilde{e}}_{\eta }\left(t\right).\hfill \end{array}$$

(13)

Define ${\epsilon }_{\eta }\left(t\right){\mathbf{col}}_i^N\left[{\epsilon }_{\eta i}\left(t\right)\right]$ and $\overline{\chi }\left(t\right)={\mathrm{col}}_i^N\left[{\overline{\chi }}_i\left(t\right)\right]$, where ${\overline{\chi }}_i\left(t\right)=h_i{\chi }_i\left(t\right)={\displaystyle \frac{{\epsilon }_{\eta i}\left(t\right)}{∥Q{\epsilon }_{\eta i}\left(t\right)∥+{\varphi }_i\left(t\right)}}$. From (12) and $HH=I_N$, one gets that

$$\begin{array}{cc}\hfill {\dot{\epsilon }}_{\eta i}\left(t\right)=& (I_N\otimes A_0){\epsilon }_{\eta i}\left(t\right)-{\lambda }_1(\overline{L}\otimes B_{0}Q){\epsilon }_{\eta i}\left(t\right)-{\lambda }_1(\overline{L}\otimes B_{0}Q)\overline{\chi }\left(t\right)-(\overline{L}{\mathbf{1}}_N\otimes B_0)u_0\left(t\right)\hfill \\ & +(H\otimes (\Phi A_0-\Theta C_0-A_0)){\tilde{e}}_{\eta i}\left(t\right).\hfill \end{array}$$

(14)

Theorem 1.

Assuming both Assumptions 1 and 2 hold, if the parameters of the distributed observer satisfy that ${\lambda }_1>{\displaystyle \frac{{\overline{\lambda }}_1}{w}}$, ${\lambda }_2>\alpha$, and $Q=B_0^T{T}^{-1}$, where ${\overline{\lambda }}_1>0$, and T is the solution of the following inequality:

$$\begin{array}{c}\hfill A_0^{T}T+T{A}_0-{\overline{\lambda }}_1{B}_0{B}_0^T+{\delta }_{1}T<0,\end{array}$$

(15)

where ${\delta }_1>0$ and ${\delta }_2$ satisfies the following inequality:

$$\begin{array}{c}\hfill {(\Phi A_0-\Theta C_0)}^T\Omega +\Omega (\Phi A_0-\Theta C_0)+{\delta }_2\Omega <0.\end{array}$$

(16)

Then, the distributed observer (9) can ensure that the estimation error converges, that is, $\underset{t\to \infty }{lim}{e}_{\eta i}=0$.

Proof. 

Similarly, construct a Lyapunov function candidate as

$$\begin{array}{cc}\hfill V_1\left(t\right)=& {\epsilon }_{\eta }^T\left(t\right)\left(Z\otimes T^{-1}\right){\epsilon }_{\eta }\left(t\right)+2{\lambda }_2\sum _{i=1}^N{\overline{l}}_{ii}{z}_i{\displaystyle \frac{{\varphi }_i\left(t\right)}{{\overline{\alpha }}_{1i}}}+k_1{\tilde{e}}_{\eta }^T\left(t\right)\left(I_N\otimes \Omega \right){\tilde{e}}_{\eta }\left(t\right)\hfill \\ \hfill =& V_a\left(t\right)+2{\lambda }_2{V}_b\left(t\right)+k_1{V}_c\left(t\right),\hfill \end{array}$$

(17)

where $k_1$ is a positive constant, and ${\overline{l}}_{11},\dots ,{\overline{l}}_{NN}$ denote the diagonal elements of the matrix $\overline{L}$. It is straightforward to verify that $V_1\left(t\right)$ is positive definite.

Define $\overline{T}=T^{-1}{B}_0{B}_0^T{T}^{-1}$. From (13) and the relation $Q=B_0^T{T}^{-1}$, one can obtain that

$$\begin{array}{cc}\hfill {\dot{V}}_a\left(t\right)=& {\epsilon }_{\eta }^T\left(t\right)(Z\otimes (A_0^T{T}^{-1}+T^{-1}{A}_0)){\epsilon }_{\eta }\left(t\right)-{\lambda }_1{\epsilon }_{\eta }^T\left(t\right)(({\overline{L}}^{T}Z+Z\overline{L})\otimes \overline{T}){\epsilon }_{\eta }\left(t\right)\hfill \\ & -2{\lambda }_2{\epsilon }_{\eta }^T\left(t\right)(Z\overline{L}\otimes \overline{T}){\overline{\chi }}_i\left(t\right)-2{\epsilon }_{\eta }^T\left(t\right)(ZHL{1}_N\otimes T^{-1}{B}_0)u_0\left(t\right)\hfill \\ & +2{\epsilon }_{\eta }^T\left(t\right)(ZH\otimes T^{-1}(\Phi A_0-\Theta C_0-A_0)){\epsilon }_{\eta }\left(t\right).\hfill \end{array}$$

(18)

Due to ${\overline{L}}^{T}Z+Z\overline{L}\ge wZ$, ${\lambda }_1>{\displaystyle \frac{{\overline{\lambda }}_1}{w}}$, and (14), it yields

$$\begin{array}{cc}\hfill & {\epsilon }_{\eta }^T\left(t\right)(Z\otimes (A_0^T{T}^{-1}+T^{-1}{A}_0)){\epsilon }_{\eta }\left(t\right)-{\lambda }_1{\epsilon }_{\eta }^T\left(t\right)(({\overline{L}}^{T}Z+Z\overline{L})\otimes \overline{T}){\epsilon }_{\eta }\left(t\right)\hfill \\ & \le -({\delta }_1+{\widehat{\delta }}_1){\epsilon }_{\eta }^T\left(t\right)(Z\otimes T^{-1}){\epsilon }_{\eta }\left(t\right),\hfill \end{array}$$

(19)

where $0<{\widehat{\delta }}_1<{\delta }_1$. Define ${\overline{\delta }}_i\left(t\right)=B_0^T{T}^{-1}{\epsilon }_{\eta i}\left(t\right)$. One goes to

$$\begin{array}{cc}\hfill -{\epsilon }_{\eta }^T\left(t\right)(Z\overline{L}\otimes \overline{T})\overline{\chi }\left(t\right)=& -\sum _{i=1}^N{z}_i{\epsilon }_{\eta i}^T\left(t\right)\overline{T}(\sum _{j=1}^N\left|a_{ij}\right|({\overline{\chi }}_i\left(t\right)-{\overline{\chi }}_j\left(t\right)))\hfill \\ \hfill & -\sum _{i=1}^N{z}_i{a}_{0i}{\epsilon }_{\eta i}^T\left(t\right)\overline{T}{\overline{\chi }}_i\left(t\right)\hfill \\ \hfill =& -\sum _{i=1}^N{z}_i(\sum _{j=1}^N\left|a_{ij}\right|({\displaystyle \frac{{∥{\overline{\delta }}_i\left(t\right)∥}^2}{∥{\overline{\delta }}_i\left(t\right)∥+{\varphi }_i\left(t\right)}}-{\displaystyle \frac{{\overline{\delta }}_i^T\left(t\right){\overline{\delta }}_j\left(t\right)}{∥{\overline{\delta }}_j\left(t\right)∥+{\varphi }_j\left(t\right)}}))\hfill \\ \hfill & -\sum _{i=1}^N{z}_i{a}_{ij}{\displaystyle \frac{{∥{\overline{\delta }}_i\left(t\right)∥}^2}{∥{\overline{\delta }}_i\left(t\right)∥+{\varphi }_i\left(t\right)}}.\hfill \end{array}$$

(20)

Then, we can obtain that

$$\begin{array}{cc}\hfill {\epsilon }_{\eta }^T\left(t\right)(ZHL{\mathbf{1}}_N\otimes T^{-1}{B}_0)u_0\left(t\right)& =-\sum _{i=1}^N{z}_i{a}_{i0}{h}_i{\epsilon }_{\eta i}^T\left(t\right)U^{-1}{B}_0{u}_0\left(t\right)\hfill \\ & =-\sum _{i=1}^N{z}_i{a}_{i0}{h}_i{\epsilon }_{\eta i}^T\left(t\right)U^{-1}{B}_0{u}_0\left(t\right)\hfill \\ & =-\sum _{i=1}^N{z}_i{a}_{i0}{\overline{\delta }}_i^T\left(t\right)u_0\left(t\right)\hfill \\ & =\alpha \sum _{i=1}^N{z}_i{a}_{i0}∥{\overline{\delta }}_i\left(t\right)∥.\hfill \end{array}$$

(21)

According to ${\overline{l}}_{ii}={\displaystyle \sum _{i=1}^N}\left|a_{ij}\right|+a_{i0}$, we have

$$\begin{array}{c}\hfill {\dot{V}}_b\left(t\right)=-\sum _{i=1}^N{\alpha }_{1i}{\overline{l}}_{ii}{z}_i{\displaystyle \frac{{\varphi }_i\left(t\right)}{{\overline{\alpha }}_{1i}}}-\sum _{i=1}^N{z}_i(\sum _{j=1}^N\left|a_{ij}\right|{\varphi }_i\left(t\right))-\sum _{i=1}^N{z}_i{a}_{i0}{\varphi }_i\left(t\right).\end{array}$$

(22)

Due to ${\lambda }_2>\alpha$, we can deduce that

$$\begin{array}{cc}\hfill & -\sum _{i=1}^N{z}_i{a}_{i0}({\lambda }_2{\displaystyle \frac{{∥{\overline{\delta }}_i\left(t\right)∥}^2}{∥{\overline{\delta }}_i\left(t\right)∥+{\varphi }_i\left(t\right)}}-\alpha ∥{\overline{\delta }}_i\left(t\right)∥+{\lambda }_2{\varphi }_i\left(t\right))\hfill \\ & =-\sum _{i=1}^N{z}_i{a}_{i0}{\displaystyle \frac{({\lambda }_2-\alpha )({∥{\overline{\delta }}_i\left(t\right)∥}^2+{\varphi }_i\left(t\right)∥{\overline{\delta }}_i\left(t\right)∥)+{\lambda }_2{\varphi }_i^2\left(t\right)}{∥{\overline{\delta }}_i\left(t\right)∥+{\varphi }_i\left(t\right)}}\le 0,\hfill \end{array}$$

(23)

and

$$\begin{array}{cc}\hfill -& \sum _{j=1}^N\left|a_{ij}\right|({\displaystyle \frac{{∥{\overline{\delta }}_i\left(t\right)∥}^2}{∥{\overline{\delta }}_i\left(t\right)∥+{\varphi }_i\left(t\right)}}-{\displaystyle \frac{{\overline{\delta }}_i^T\left(t\right){\overline{\delta }}_j\left(t\right)}{∥{\overline{\delta }}_j\left(t\right)∥+{\varphi }_j\left(t\right)}}+{\varphi }_i\left(t\right))\hfill \\ \hfill =& -\sum _{j=1}^N\left|a_{ij}\right|{\displaystyle \frac{∥{\overline{\delta }}_i\left(t\right)∥{\varphi }_j\left(t\right)(∥{\overline{\delta }}_i\left(t\right)∥+∥{\overline{\delta }}_i\left(t\right)∥)}{(∥{\overline{\delta }}_i\left(t\right)∥+{\varphi }_i\left(t\right))(∥{\overline{\delta }}_j\left(t\right)∥+{\varphi }_j\left(t\right))}}\hfill \\ \hfill & -\sum _{j=1}^N\left|a_{ij}\right|{\displaystyle \frac{{\varphi }_i^2\left(t\right)(∥{\overline{\delta }}_j\left(t\right)∥+{\varphi }_j\left(t\right))}{(∥{\overline{\delta }}_i\left(t\right)∥+{\varphi }_i\left(t\right))(∥{\overline{\delta }}_j\left(t\right)∥+{\varphi }_j\left(t\right))}}\le 0.\hfill \end{array}$$

(24)

From (18)–(24), it yields that

$$\begin{array}{cc}\hfill {\dot{V}}_a\left(t\right)+2{\lambda }_2{\dot{V}}_b\left(t\right)\le & -({\delta }_1+{\widehat{\delta }}_1){\epsilon }_{\eta }^T\left(t\right)(Z\otimes T^{-1}){\epsilon }_{\eta }\left(t\right)-2{\lambda }_2\sum _{i=1}^N{\alpha }_{1i}{\overline{l}}_{ii}{z}_i{\displaystyle \frac{{\varphi }_i\left(t\right)}{{\overline{\alpha }}_{1i}}}\hfill \\ & +2{\epsilon }_{\eta }^T\left(t\right)(ZH\otimes T^{-1}(\Phi A_0-\Theta C_0-A_0)){\tilde{e}}_{\eta }\left(t\right).\hfill \end{array}$$

(25)

According to (12) and (16), it follows that

$$\begin{array}{cc}\hfill {\dot{V}}_c\left(t\right)=& {\tilde{e}}_{\eta }^T\left(t\right)[I_N\otimes {(\Phi A_0-\Theta C_0)}^T\Omega +\Omega (\Phi A_0-\Theta C_0)]{\tilde{e}}_{\eta }\left(t\right)\hfill \\ \hfill \le & -({\delta }_2+{\widehat{\delta }}_2){\tilde{e}}_{\eta }^T\left(t\right)(I_N\otimes \Omega ){\tilde{e}}_{\eta }\left(t\right),\hfill \end{array}$$

(26)

where $0<{\widehat{\delta }}_2<{\delta }_2$. Define ${\tilde{\alpha }}_1=min\{{\alpha }_{11},\dots ,{\alpha }_{1N}\}$. We can deduce that

$$\begin{array}{c}\hfill {\dot{V}}_1\left(t\right)=-{\delta }_1{V}_a\left(t\right)-{\tilde{\alpha }}_{1}2{\lambda }_2{V}_b\left(t\right)-{\delta }_2{V}_c\left(t\right)+[{\epsilon }_{\eta }^T\left(t\right),{\tilde{e}}_{\eta }^T\left(t\right)]E{[{\epsilon }_{\eta }^T\left(t\right),{\tilde{e}}_{\eta }^T\left(t\right)]}^T,\end{array}$$

(27)

where $E=\left[\begin{array}{cc}-{\widehat{\delta }}_1(Z\otimes T^{-1})& \overline{E}\\ {\overline{E}}^T& -k_1{\widehat{\delta }}_2(I_N\otimes \Omega )\end{array}\right]$, $\overline{E}=ZH\otimes T^{-1}(\Phi A_0-\Theta C_0-A_0)$. Due to $-{\widehat{\delta }}_1(Z\otimes T^{-1})<0$, according to the Schur complement lemma, as $-k_1{\widehat{\delta }}_2(I_N\otimes \Omega )+{\displaystyle \frac{1}{{\widehat{\delta }}_1}}{\overline{E}}^T(Z^{-1}\otimes T)\overline{E}<0$, then $E<0$. Thus we choose $k_1>{\displaystyle \frac{1}{{\widehat{\delta }}_1{\widehat{\delta }}_2}}{\lambda }_{max}\left((I_N\otimes {\Omega }^{-1}){\overline{E}}^T(Z^{-1}\otimes T)\overline{E}\right)$; then it can ensure that $E<0$. Define $\tilde{\delta }=min\{{\delta }_1,{\tilde{\alpha }}_1,{\delta }_2\}>0$. According to (27), it yields ${\dot{V}}_1\left(t\right)\le -\tilde{\delta }{V}_1\left(t\right)$, i.e., ${\dot{V}}_1\left(t\right)$ is negative definite. Therefore, we can obtain that $\underset{t\to \infty }{lim}{\epsilon }_{\eta i}\left(t\right)=\underset{t\to \infty }{lim}{\tilde{e}}_{\eta i}\left(t\right)=0$, that is, $\underset{t\to \infty }{lim}{\widehat{e}}_{\eta i}=\underset{t\to \infty }{lim}{e}_{\eta i}=0$.

The proof is complete. □

3.2. Fault-Tolerant Controller Design

In this subsection, a fault-tolerant controller is designed to ensure that the tracking error converges into a neighborhood around the origin.

At first, the tracking error vector and auxiliary variables are defined as follows:

$$\begin{array}{c}\hfill e_{ti}\left(t\right)=p_i\left(t\right)-C_0{\eta }_i\left(t\right)-b_i\left(t\right),{\chi }_{si}\left(t\right)={\dot{e}}_{ti}\left(t\right)+Q_{ti}{e}_{ti}\left(t\right),\end{array}$$

(28)

where $Q_{ti}={\mathbf{diag}}_r^n\left[{\lambda }_{ir}\right]$, ${\lambda }_{ir}>0$ are the control parameters to be designed, and $b_i\left(t\right)$ is the time-varying formation parameter.

Then, the fault-tolerant controller is designed as

$$\begin{array}{cc}\hfill u_i=& {\widehat{g}}_i(C_i{\dot{q}}_i+G_i+M_i(C_0{\ddot{\eta }}_i-Q_{ti}{\dot{e}}_{ti})-C_i{\chi }_{si}-{\widehat{J}}_i\mathrm{sgn}\left({\chi }_{si}\right)-{\Phi }_{ti}{\chi }_{si})\hfill \\ \hfill =& {\widehat{g}}_i{\Psi }_i,\hfill \end{array}$$

(29)

where ${\widehat{J}}_i={\mathbf{diag}}_r^n\left[{\widehat{\theta }}_{ir}\right]$,${\Phi }_{ti}={\mathbf{diag}}_r^n\left[k_{1ir}\right]$, and $k_{1ir}$ is a positive constant. ${\widehat{g}}_i$ is the estimation of ${\overline{g}}_i$, and ${\overline{g}}_i=1/g_i$. The estimation error of ${\overline{g}}_i$ is represented by ${\tilde{g}}_i={\widehat{g}}_i-{\overline{g}}_i$.

The adaptive laws are proposed as

$$\begin{array}{c}\hfill {\dot{\widehat{\theta }}}_i=\left|{\chi }_{si}\right|-k_{2i}{\widehat{\theta }}_i,{\dot{\tilde{g}}}_i=-{\chi }_{si}^T{M}_i{\Psi }_i,\end{array}$$

(30)

where $k_{2i}$ is a positive constant.

As illustrated in Figure 1, this paper presents a two-layer control architecture. The human-in-the-loop system generates the reference trajectories. The upper layer consists of a distributed observer designed to estimate the dynamic states of the human-in-the-loop system. The lower layer implements a fault-tolerant controller for each follower agent, enabling trajectory tracking of the human-in-the-loop reference while mitigating the effects of system uncertainties and actuator faults. Through the integrated operation of these two layers, the networked UAV system achieves time-varying formation tracking and enhances overall robustness.

Theorem 2.

Assuming Assumptions 1, 2, and 3 hold, considering the networked system (1)–(3), under the action of the observer (9) and the controller (29), the auxiliary variables and the tracking error will converge to the following regions:

$$\begin{array}{c}\hfill \left|{\chi }_{si}\right|\le {\overline{\chi }}_{si},e_{ti,r}\le {\displaystyle \frac{{\overline{\chi }}_{si}}{{\lambda }_{ir}}},\end{array}$$

(31)

where ${\overline{\chi }}_{si}\le \sqrt{{\displaystyle \frac{2{\vartheta }_1}{{\lambda }_{min}\left(M\right){\vartheta }_2(1-\rho )}}}$, $0<\rho <1$, and ${\vartheta }_1$ and ${\vartheta }_2$ are positive constants related to the system inertia matrix $M_i$, control parameters and false data upper bound $d_i$.

Proof. 

Select a Lyapunov function as

$$\begin{array}{c}\hfill V_{2i}={\displaystyle \frac{1}{2}}{\chi }_{si}^T{M}_i{\chi }_{si}+{\displaystyle \frac{1}{2}}{({\widehat{\theta }}_i-d_i)}^T({\widehat{\theta }}_i-d_i)+{\displaystyle \frac{1}{2}}{g}_i{\left({\tilde{g}}_i\right)}^2.\end{array}$$

(32)

Then we have

$$\begin{array}{cc}\hfill {\dot{V}}_{2i}=& {\displaystyle \frac{1}{2}}{\chi }_{si}^T{\dot{M}}_i{\chi }_{si}+{\chi }_{si}^T{M}_i{\dot{\chi }}_{si}+{({\widehat{\theta }}_i-d_i)}^T{\dot{\widehat{\theta }}}_i+g_i{\tilde{g}}_i{\dot{\tilde{g}}}_i\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{\chi }_{si}^T({\dot{M}}_i-2C_i){\chi }_{si}+{\chi }_{si}^T{C}_i{\chi }_{si}+{\chi }_{si}^T{M}_i{\dot{\chi }}_{si}+{({\widehat{\theta }}_i-d_i)}^T{\dot{\widehat{\theta }}}_i+g_i{\tilde{g}}_i{\dot{\tilde{g}}}_i.\hfill \end{array}$$

(33)

According to Property 2, we have

$$\begin{array}{cc}\hfill {\dot{V}}_{2i}=& {\chi }_{si}^T{C}_i{\chi }_{si}+{\chi }_{si}^T{M}_i{\dot{\chi }}_{si}+{({\widehat{\theta }}_i-d_i)}^T{\dot{\widehat{\theta }}}_i+g_i{\tilde{g}}_i{\dot{\tilde{g}}}_i\hfill \\ \hfill =& {\chi }_{si}^T{C}_i{\chi }_{si}+{\chi }_{si}^T{M}_i(({\ddot{q}}_i-C_0{\ddot{\eta }}_i)+Q_{ti}{\dot{e}}_{ti})+{({\widehat{\theta }}_i-d_i)}^T{\dot{\widehat{\theta }}}_i+g_i{\tilde{g}}_i{\dot{\tilde{g}}}_i.\hfill \end{array}$$

(34)

From (1), it follows that

$$\begin{array}{cc}\hfill {\dot{V}}_{2i}=& {\chi }_{si}^T{M}_i(g_i{u}_i+f_i-C_i{\dot{q}}_i-G_i)-{\chi }_{si}^T{M}_i{C}_0{\ddot{\eta }}_i+{\chi }_{si}^T{M}_i{Q}_{ti}{\dot{e}}_{ti}+{({\widehat{\theta }}_i-d_i)}^T{\dot{\widehat{\theta }}}_i+g_i{\tilde{g}}_i{\dot{\tilde{g}}}_i.\hfill \end{array}$$

(35)

Substituting (29) and (30) into (35), we can get that

$$\begin{array}{cc}\hfill {\dot{V}}_{2i}\le & -{\widehat{\theta }}_i^T\left|{\chi }_{si}\right|-{\chi }_{si}^T{\Phi }_{ti}{\chi }_{si}+d_i^T\left|{\chi }_{si}\right|+{({\widehat{\theta }}_i-d_i)}^T(\left|{\chi }_{si}\right|-k_{2i}{\widehat{\theta }}_i)\hfill \\ \hfill =& -{\chi }_{si}^T{\Phi }_{ti}{\chi }_{si}-{({\widehat{\theta }}_i-d_i)}^T{k}_{2i}{\widehat{\theta }}_i.\hfill \end{array}$$

(36)

Noting that

$$\begin{array}{c}\hfill -{({\widehat{\theta }}_i-d_i)}^T{k}_{2i}{\widehat{\theta }}_i\le -{\displaystyle \frac{1}{2}}{({\widehat{\theta }}_i-d_i)}^T{k}_{2i}({\widehat{\theta }}_i-d_i)+{\displaystyle \frac{1}{2}}{d}_i^T{k}_{2i}{d}_i.\end{array}$$

(37)

From (36) and (37), we can get that

$$\begin{array}{cc}\hfill {\dot{V}}_{2i}\le & -{\chi }_{si}^T{\Phi }_{ti}{\chi }_{si}-{\displaystyle \frac{1}{2}}{({\widehat{\theta }}_i-d_i)}^T{k}_{2i}({\widehat{\theta }}_i-d_i)+{\displaystyle \frac{1}{2}}{d}_i^T{k}_{2i}{d}_i\hfill \\ \hfill \le & -{\displaystyle \frac{{\lambda }_{min}\left({\Phi }_{ti}\right)}{{\lambda }_{max}\left(M_i\right)}}{\chi }_{si}^T{M}_i{\chi }_{si}-{\displaystyle \frac{1}{2}}{\lambda }_{min}\left(k_{2i}\right){({\widehat{\theta }}_i-d_i)}^T({\widehat{\theta }}_i-d_i)+{\displaystyle \frac{1}{2}}{d}_i^T{k}_{2i}{d}_i\hfill \\ \hfill \le & -min\{{\displaystyle \frac{2{\lambda }_{min}\left({\Phi }_{ti}\right)}{{\lambda }_{max}\left(M_i\right)}},{\lambda }_{min}\left(k_{2i}\right)\}{V}_{2i}+{\displaystyle \frac{1}{2}}{d}_i^T{k}_{2i}{d}_i\hfill \\ \hfill =& -{\Omega }_{v2i}{V}_{2i}+{\chi }_{v2i},\hfill \end{array}$$

(38)

where ${\Omega }_{v2i}=min\{{\displaystyle \frac{2{\lambda }_{min}\left({\Phi }_{ti}\right)}{{\lambda }_{max}\left(M_i\right)}},{\lambda }_{min}\left(k_{2i}\right)\},{\chi }_{v2i}={\displaystyle \frac{1}{2}}{d}_i^T{k}_{2i}{d}_i.$ According to Lemma 1, it follows that the auxiliary variable can converge to the following boundary:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\lambda }_{min}\left(M_i\right){∥{\chi }_{si}∥}^2\le {\displaystyle \frac{1}{2}}{\chi }_{si}^T{M}_i{\chi }_{si}\le {\displaystyle \frac{{\chi }_{v2i}}{{\Omega }_{v2i}(1-{\delta }_i)}}.\end{array}$$

(39)

Then we can obtain that

$$\begin{array}{c}\hfill ∥{\chi }_{si}∥\le \sqrt{{\displaystyle \frac{2{\chi }_{v2i}}{{\lambda }_{min}\left(M_i\right){\Omega }_{v2i}(1-{\delta }_i)}}}={\overline{\chi }}_{si}.\end{array}$$

(40)

From (28) and Lemma 2, it yields that $e_{ti}$ can converge to the following boundary:

$$\begin{array}{c}\hfill \left|e_{ti,r}\right|\le {\displaystyle \frac{{\overline{\chi }}_{si}}{{\lambda }_{ir}}}.\end{array}$$

(41)

The proof is complete. Based on the above discussion, it can be concluded that the proposed control scheme can achieve the time-varying formation tracking of networked system and enhance the robustness of the system. □

In this section, a two-layer control architecture, consisting of a distributed observer and a fault-tolerant controller, is proposed to achieve human-in-the-loop resilient bipartite consensus tracking of networked Euler–Lagrange systems under actuator faults, thereby enhancing the system’s reliability and robustness. The effectiveness of the control scheme is rigorously validated using Lyapunov stability theory.

Remark 1.

For practical implementation, the gain matrices Θ, T, and Ω are obtained offline by solving the linear matrix inequalities (LMIs) (15) and (16) using standard LMI solvers (e.g., MATLAB 2022a LMI Toolbox). Feasibility is assured under Assumptions 1–2, which guarantee the stabilizability and detectability of the leader system. Specifically, Θ is chosen to place the eigenvalues of $\Phi A_0-\Theta C_0$ in the left-half plane, while T and Ω are derived as positive definite solutions to their respective LMIs. The control gains (${\lambda }_1$, ${\lambda }_2$, Q, $k_{1ir}$, $k_{2i}$) are then selected according to the conditions stated in Theorem 1 and Theorem 2. This offline design process ensures that all inequalities hold before deployment, and the online algorithm reduces to the distributed observer update (9) and the fault-tolerant controller update (29) and (30), which rely only on local communication and measurements.

Remark 2.

It is worth noting that cooperative and competitive interactions may coexist in practical multi-agent systems, which cannot be fully captured by purely cooperative communication graphs [32,33,34,35]. For instance, antagonistic couplings may naturally arise due to conflicting objectives, resource competition, or adversarial behaviors among agents. Although the present work focuses on cooperative network structures, the proposed analytical framework provides a potential foundation for further extensions. In particular, future research may investigate the considered problem over signed graphs, where both positive and negative interactions are explicitly modeled, thereby enabling a more comprehensive characterization of cooperation–competition coexistence in complex networked systems.

4. Simulation Results

In this section, the validity of the proposed control scheme will be demonstrated by a simulation example. The communication topology among agents is shown in Figure 2.

The system matrices are shown in

Table 1. Notation and key parameters.

Symbol	Dimension	Description
$A_0$	${\mathbb{R}}^{m\times m}$	State matrix of the human-in-the-loop leader
$B_0$	${\mathbb{R}}^{m\times p}$	Input matrix of the leader
$C_0$	${\mathbb{R}}^{n\times m}$	Output matrix of the leader
Q	${\mathbb{R}}^{p\times m}$	Observer gain matrix, $Q=B_0^T{T}^{-1}$
$\Phi$	${\mathbb{R}}^{m\times m}$	$\Phi =I+\Gamma C_0$
$\Theta$	${\mathbb{R}}^{m\times n}$	Observer gain matrix
$\Gamma$	${\mathbb{R}}^{m\times n}$	$\Gamma =-B_0{\left[{\left(C_0{B}_0\right)}^T\left(C_0{B}_0\right)\right]}^{-1}{\left(C_0{B}_0\right)}^T$
L	${\mathbb{R}}^{N\times N}$	Laplacian matrix of the communication graph
Z	${\mathbb{R}}^{N\times N}$	Positive definite matrix satisfying $L^{T}Z+ZL>0$
T	${\mathbb{R}}^{m\times m}$	Solution of the LMI in Equation (15)
$\Omega$	${\mathbb{R}}^{m\times m}$	Positive definite matrix in Equation (16)

.

The initial conditions and control parameters are shown in

Table 2. Simulation settings and parameters.

Category	Parameter	Value/Expression
Human-in-the-loop System	$A_0$	$\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$
	$B_0$	$\left[\begin{array}{c}0\\ 1\end{array}\right]$
	$C_0$	$\left[\begin{array}{cc}1& 1\\ -2& 2\end{array}\right]$
	$u_0\left(t\right)$	$2cos\left(t\right)$
Initial States	$p_1\left(0\right)$	${[20,15]}^T$
	$p_2\left(0\right)$	${[-20,20]}^T$
	$p_3\left(0\right)$	${[20,-15]}^T$
	$p_4\left(0\right)$	${[23,14]}^T$
	$p_5\left(0\right)$	${[-14,21]}^T$
	$x_0\left(0\right)$	${[0,0]}^T$
Disturbances	$f_1\left(t\right)$	$4sin\left(2t\right)+1$
	$f_2\left(t\right)$	$3cos\left(t\right)+e^{-2t}$
	$f_3\left(t\right)$	$2sin\left(t\right)$
	$f_4\left(t\right)$	$4cos\left(t\right)$
Actuator Health Indices	$g_i$	$0.95$ (for all $i=1,\dots ,5$)
Observer Gains	${\lambda }_1$	$1.9$
	${\lambda }_2$	2
	Q	${[35.4,14.9]}^T$
Controller Gains	${\lambda }_{ir}$	40
	$k_{2i}$	1
	$k_{1ir}$	20
Formation Parameters	$b_1^0$	${[-4,0]}^T$
	$b_2^0$	${[-1,-4]}^T$
	$b_3^0$	${[3,-2]}^T$
	$b_4^0$	${[-2,4]}^T$
	$b_5^0$	${[3,2]}^T$
	Rotation $R\left({\tau }_i\left(t\right)\right)$	$\left[\begin{array}{cc}cos\left({\tau }_i\left(t\right)\right)& -sin\left({\tau }_i\left(t\right)\right)\\ sin\left({\tau }_i\left(t\right)\right)& cos\left({\tau }_i\left(t\right)\right)\end{array}\right]$

.

The follower model (1) can be established as

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{m}_{i11}& m_{i12}\\ m_{i21}& m_{i22}\end{array}\right]{\ddot{p}}_i+\left[\begin{array}{cc}-c_i{\dot{p}}_i\left(2\right)& -c_i({\dot{p}}_i\left(1\right)+{\dot{p}}_i\left(2\right))\\ c_i{\dot{p}}_i\left(1\right)& 0\end{array}\right]{\dot{p}}_i+\left[\begin{array}{c}{\lambda }_{i1}g\\ {\lambda }_{i2}g\end{array}\right]=u_i+f_i,\end{array}$$

where

$$\begin{array}{cc}\hfill m_{i11}=& (m_{i1}+m_{i2})r_{i1}^2+m_{i2}{r}_{i2}^2+2m_{i2}{r}_{i1}{r}_{i2}cos\left(q_i\left(2\right)\right)+J_{i1},\hfill \\ \hfill m_{i12}=& m_{i21}=m_{i2}{r}_{i2}^2+m_{i2}{r}_{i1}{r}_{i2}cos\left(q_i\left(2\right)\right),\hfill \\ \hfill m_{i22}=& m_{i2}{r}_{i2}^2+J_{i2},\hfill \\ \hfill c_i=& m_{i2}{r}_{i1}{r}_{i2}sin\left(q_i\left(2\right)\right),\hfill \\ \hfill {\lambda }_{i1}=& (m_{i1}+m_{i2})r_{i1}cos\left(q_i\left(1\right)\right)+m_{i2}{r}_{i2}cos(q_i\left(1\right)+q_i\left(2\right)),\hfill \\ \hfill {\lambda }_{i2}=& m_{i2}{r}_{i2}cos(q_i\left(1\right)+q_i\left(2\right)).\hfill \end{array}$$

The model parameters of each follower are selected as $m_{i1}=0.5\mathrm{kg}$, $m_{i2}=1.5\mathrm{kg}$, $r_{i1}=1\mathrm{m}$, $r_{i2}=0.8\mathrm{m}$, $J_{i1}=J_{i2}=5\mathrm{kg}·\mathrm{m}$, and $g=9.81\mathrm{m}/{\mathrm{s}}^2$. The time-varying formation parameters are defined as $b_i\left(t\right)=R\left({\tau }_i\left(t\right)\right)b_i^0$, $i=1,\dots ,N$, where $b_i^0$ denotes the constant formation offset and $R\left({\tau }_i\left(t\right)\right)=\left[\begin{array}{cc}cos\left({\tau }_i\left(t\right)\right)& -sin\left({\tau }_i\left(t\right)\right)\\ sin\left({\tau }_i\left(t\right)\right)& cos\left({\tau }_i\left(t\right)\right)\end{array}\right]$ is the rotation matrix. Here, ${\tau }_i\left(t\right)=arctan2({\dot{\eta }}_{i2}\left(t\right),{\dot{\eta }}_{i1}\left(t\right))$. The constant formation offsets are chosen as $b_1^0={[-4,0]}^T$, $b_2^0={[-1,-4]}^T$, $b_3^0={[3,-2]}^T$, $b_4^0={[-2,4]}^T$, and $b_5^0={[3,2]}^T$.

Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 illustrate the process of the networked UAV system performing time-varying formation tracking control. As shown, the networked system (1), (3) successfully achieves practical time-varying formation tracking control under the proposed control scheme. From Figure 11, it can be concluded that the distributed observer (9) ensures the leader’s state estimation error converges to zero. Based on Figure 12, it can be inferred that, under the influence of the fault-tolerant controller (29), the tracking error of the networked UAV system, despite external disturbances, converges to a bounded region near zero.

To further validate the effectiveness of the fault-tolerant controller proposed in this paper, the fault-tolerant factor $g_i$ in the original controller (29) was removed, that is,

$$\begin{array}{c}\hfill u_i=C_i{\dot{q}}_i+G_i+M_i(C_0{\ddot{\eta }}_i-Q_{ti}{\dot{e}}_{ti})-C_i{\chi }_{si}-{\widehat{J}}_i\mathrm{sgn}\left({\chi }_{si}\right)-{\Phi }_{ti}{\chi }_{si}.\end{array}$$

(42)

The resulting tracking errors of the system are illustrated in the figure.

Figure 13 illustrates that the controller lacking the fault-tolerant factor achieves the final tracking performance but requires more time to converge compared with the fault-tolerant controller (29), validating the effectiveness of the fault-tolerant design presented herein.

5. Conclusions

This paper has presented a fault-tolerant control framework for human-in-the-loop time-varying formation tracking in networked UAV systems, particularly addressing the challenges posed by compound actuator faults. A distributed observer has been developed to accurately estimate the leader’s state, while a fault-tolerant controller has been designed to mitigate the effects of actuator faults, ensuring robust performance. The proposed two-layer control strategy has been shown to guarantee that both tracking and estimation errors converge to small bounded regions, with the stability and convergence rigorously verified through Lyapunov stability theory. The effectiveness of the proposed approach has been demonstrated through extensive simulation results, which show its capability to achieve practical formation tracking even in the presence of actuator faults and external disturbances.

In future work, the focus will be on extending this framework to handle more complex communication topologies, such as directed and time-varying communication networks, and on real-world experimental validation to further evaluate the approach’s performance under realistic conditions. Furthermore, investigating the integration of obstacle avoidance algorithms into the control scheme would enhance the applicability of the control framework proposed in this work.