Observer-Based Adaptive Event-Triggered Fault-Tolerant Control for Bidirectional Consensus of MASs with Sensor Faults

Abstract

The adaptive event-triggered fault-tolerant control problem for bidirectional consensus of multi-agent systems (MASs) subject to sensor faults and external disturbances is investigated. A hierarchical algorithm is first introduced to eliminate the dependence on Laplacian matrix information, thereby reducing computational complexity. Subsequently, a disturbance observer (DO) and a compensation signal were constructed to accommodate external disturbances, filtering errors, and approximation errors introduced by the radial basis function neural network (RBFNN). Compared with the absence of a disturbance observer, the tracking performance was improved by $15.2\%$. In addition, a switching event-triggered mechanism is considered, in which the advantages of fixed-time triggering and relative triggering are integrated to balance communication frequency and tracking performance. Finally, the boundedness of all signals under the proposed fault-tolerant control (FTC) scheme is established. It has been clearly demonstrated by the simulation results that the proposed mechanism achieves a $39.8\%$ reduction in triggering frequency relative to the FT scheme, while simultaneously yielding a $5.0\%$ enhancement in tracking performance compared with the RT scheme, thereby highlighting its superior efficiency and effectiveness.

1. Introduction

With the advancement of modern industrial machinery, the consensus problem in MASs has become one of the central research topics in cooperative control, with broad applications in robotic networks, unmanned aerial vehicle formations [1], sensor networks [2], and related areas [3,4]. In practical engineering scenarios, communication topology often involves asymmetric weights and adversarial interaction patterns, thereby giving rise to bidirectional consensus challenges [5,6]. For instance, ref. [7] proposed resilient consensus algorithms capable of mitigating the influence of mobile malicious agents on directed graphs, effectively ensuring convergence despite non-symmetric and adversarial interactions. Ref. [8] developed dynamic event-triggered bidirectional consensus protocols for nonlinear MASs over signed networks, demonstrating that efficient convergence can be achieved while reducing communication overhead. Furthermore, ref. [9] developed a differentially private bidirectional consensus algorithm over signed networks by injecting time-varying noise and carefully tuning step-sizes, showing that MASs can achieve mean-square and almost-sure convergence while preserving a predefined differential-privacy level. Unbalanced and switching topologies may further introduce more complex cooperative dynamics and stability challenges. Therefore, the development of a bidirectional consensus control framework under asymmetric topology with switching behavior possesses significant theoretical importance as well as broad practical value.

Meanwhile, from an engineering implementation perspective, the system is often subjected to external disturbances and unknown faults during the dynamic modeling process. In recent years, DOs have been widely employed in adaptive and robust control to enable real-time estimation of unknown disturbances [10,11]. A finite-time DO is designed to estimate unknown external disturbances in real time, and this estimation is integrated with a control law to achieve rapid and robust attitude tracking of a constrained second-order UAV system [12]. A fixed-time disturbance-observer-based sliding mode control strategy was proposed for robotic manipulators, in which convergence and tracking performance are guaranteed even in the presence of unknown external disturbances and model uncertainties [13]. However, the observer itself introduces additional tracking errors due to filtering, estimation inaccuracies, and numerical implementation. Therefore, an auxiliary compensation signal was introduced, through which the observation errors were integrated with the system dynamics, ensuring the effective attenuation of both the tracking and consensus errors.

On the other hand, system faults represent an inevitable adverse influence in practical operations and can generally be categorized as sensor faults and actuator faults [14,15,16]. These faults degrade system performance and hinder the achievement of the desired control objectives. A distributed robust FTC protocol has been designed for uncertain MASs in the presence of actuator faults, thereby overcoming the practical adverse effects of such faults on system performance [17]. For the tracking control problem of nonlinear systems subject to sensor faults, an event-triggered FTC method based on bounded estimation has been proposed, in which several auxiliary filters and auxiliary functions are incorporated to overcome the difficulties induced by network delays and unknown control directions [18]. However, the aforementioned results involve only a single type of fault. Consequently, the quality control problem in which sensor faults and actuator faults occur simultaneously remains a challenging issue and warrants further investigation.

To conserve communication resources and enhance overall system efficiency, event-triggered control has emerged as a highly popular strategy. In contrast to periodic control schemes, event-triggered mechanisms update the control input or initiate communication only when specified triggering conditions are satisfied, thereby significantly reducing the communication burden and computational overhead [19,20,21]. In recent years, various event-triggered control strategies have been proposed, including fixed-threshold (FT) [22] and relative-threshold (RT) [23,24]. FT triggering can effectively prevent Zeno behavior, while its adaptability to continuously varying system conditions remains limited [25]. To accommodate both communication efficiency and performance requirements, a switching-threshold mechanism that integrates FT and RT triggering strategies has been proposed by researchers [26,27,28]. In [29], a switching-threshold event-triggering mechanism was proposed and integrated with an adaptive controller, by which the triggering frequency was significantly reduced while the tracking error of nonlinear systems with dynamic uncertainties was ensured to converge a tunable small neighborhood. Nevertheless, the aforementioned studies have been primarily applied to single nonlinear systems; therefore, rational application of the switching-threshold event-triggered mechanism to MASs subject to external disturbances and sensor faults is regarded as a worthwhile objective.

Building upon the aforementioned results, a disturbance-observer-based adaptive switching event-triggered FTC scheme is proposed in this study to achieve bidirectional consensus tracking for MASs subject to external disturbances and sensor faults. The main contributions of the present research are summarized as follows.

• A hierarchical algorithm is proposed to address unbalanced topology induced by negative weighting effects, in contrast to prevalent bidirectional consensus control designs under balanced communication topology [6]. Tracking error for each agent is defined solely based on its immediate parent node, thereby simplifying analysis of local errors and eliminating dependence on Laplacian matrices.
• DO and compensation signals are constructed to address external disturbances and command filtering errors in MASs [13]. This design enables overall FTC strategy to simultaneously account for both external disturbances and sensor or actuator faults, thereby broadening its applicability. The DO reduces the steady-state tracking error by $15.2\%$ compared to no-observer designs.
• Integrating the advantage of fewer triggering instances inherent in FT with the enhanced flexibility offered by RT [25], a novel switching threshold-based time-triggered mechanism is developed. This approach effectively balances communication frequency and tracking performance in MASs, thereby further mitigating the zero phenomenon while ensuring boundedness of signals is rigorously demonstrated. It has been demonstrated by the simulation results that the triggering frequency is reduced by $39.8\%$ compared with the FT scheme, while the tracking performance is improved by $5.0\%$ relative to the RT scheme.

The remainder of this manuscript is organized as follows. In Section 2, the system model, preliminary results, and the disturbance-observer formulation are presented. Section 3 provides a detailed description of the switching event-triggered mechanism and the backstepping-based controller design, followed by a comprehensive stability analysis. In Section 4, simulation studies of the proposed switching event-triggered fault-tolerant control scheme are reported, and comparative simulations are conducted to verify its effectiveness. Finally, Section 5 summarizes the main conclusions of this work.

2. Problem Description and Preliminaries

A system model problem formulation is presented, and sensor fault and actuator fault models are analyzed. A hierarchical algorithm is employed to address the imbalance in communication topology caused by negative weight effects, thereby enabling bidirectional consensus under switching topology scenarios. A disturbance observer is constructed to estimate external disturbances and to compensate for external perturbations and approximation errors generated by RBFNNs. The relevant notation definitions are given in

Table 1. Notation list.

Symbol	Meaning
$a_{ij}$	Signed adjacency weight between agents i and j
${\overline{\iota }}_{i,k}$	State vector
${\widehat{\theta }}_{i,k}$	Estimated weight vector
${\varsigma }_{f_i}\left({\overline{\iota }}_{i,k}\right)$	Approximation error
$\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)$	Basis function vector
$d_{i,k}$	Unknown disturbances
${\widehat{D}}_{i,k}$	Estimated disturbance
${\widehat{Q}}_{i,k}$	Estimated extended disturbance state
${\overline{\varpi }}_1$	Baseline constant threshold
${\overline{\varpi }}_2$	Fixed triggering threshold
H	Switching threshold boundary for ETC
$w\left(t\right)$	Continuous control law
$v\left(t\right)$	Actual control input
$|·|$	Euclidean norm for vectors; absolute value for scalars

.

2.1. Problem Statement

Consider the dynamics of an ith follower agent with the following inputs saturation:

$$\left\{\begin{array}{c}{\dot{\iota }}_{i,k}={\iota }_{i,k+1}+f_{i,k}\left({\overline{\iota }}_{i,k}\right)+d_{i,k}\left(t\right),i=1,\dots ,M\hfill \\ {\dot{\iota }}_{i,n}=u\left(v_i\right)+f_{i,n}\left({\overline{\iota }}_{i,n}\right)+d_{i,n}\left(t\right),k=1,\dots ,n-1\hfill \\ y_i={\iota }_{i,1},\hfill \end{array}\right.$$

(1)

where ${\overline{\iota }}_{i,k}={[{\iota }_{i,1},\dots ,{\iota }_{i,k}]}^T\in R^k$. $u\left(v_i\right)\in R$ and $y_i\in R$ denote the control input subject to actuator faults and the output of the $ith$ follower.

$f_{i,k}(·)(k=1,2,\dots ,n)$ denotes the unknown smooth nonlinear function, which is approximated by RBFNNs and formulated as

$$f_{i,k}\left({\overline{\iota }}_{i,k}\right)={\theta }_{i,k}^{*T}\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)+{\varsigma }_{f_i}\left({\overline{\iota }}_{i,k}\right),\left|{\varsigma }_{f_i}\left({\overline{\iota }}_{i,k}\right)\right|⩽{\varsigma }_{f_i,m},$$

(2)

with ${\varsigma }_{f_i,m}>0$ being a constant, and ${\theta }_{i,k}^{*}$ denoting the ideal weight vector. The form of the Gaussian functions are shown as

$$\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)=-exp\left({\displaystyle \frac{{({\overline{\iota }}_{i,k}-{\varsigma }_{i,k})}^T({\overline{\iota }}_{i,k}-{\varsigma }_{i,k})}{{{\zeta }_{i,k}}^2}}\right),$$

(3)

where ${\varsigma }_{i,k}={[{\varsigma }_{i,1},{\varsigma }_{i,2},\dots ,{\varsigma }_{i,n}]}^T$ and ${\zeta }_{i,k}(k=1,2,\dots ,n)$ represent the center and width respectively.

The system output and input of an ith follower agent are, respectively, affected by sensor faults and actuator faults of the following forms:

$$\left\{\begin{array}{c}{y}_i^f=\mathrm{\Gamma }{\iota }_{i,1}+\varphi \left(t\right),\mathrm{\Gamma }\in \left(0,1\right]\hfill \\ u\left(v_i\right)=\eta v_i\left(t\right)+\varpi \left(t\right),\eta \in \left(0,1\right]\hfill \end{array}\right.$$

(4)

where $\mathrm{\Gamma }$ and $\eta$ are known constants, and $\varphi \left(t\right)$ and $\varpi \left(t\right)$ denote the sensor bias and actuator bias, respectively. Two unknown scalar quantities, ${\varphi }^{*}$ and ${\varpi }^{*}$, are assumed to exist such that $|\varphi \left(t\right)|⩽{\varphi }^{*}$ and $\left|\varpi \left(t\right)\right|⩽{\varpi }^{*}$.

Control objective: The primary aim of this work is to propose an adaptive switching event-triggered fault-tolerant control framework based on disturbance-observer techniques so that a bidirectional consensus can be achieved under switching-topology scenarios in the presence of external disturbances and sensor faults. The proposed control scheme is designed to accomplish two key objectives.

• Bidirectional consensus tracking with guaranteed communication efficiency and tracking performance;
• Boundedness of all closed-loop signals regardless of the presence or absence of actuator and sensor faults.

Lemma 1 

([28]). For $\forall c\in R$, one has

$$0⩽\left|c\right|-ctanh\left({\displaystyle \frac{c}{\tau }}\right)<\kappa \tau ,\kappa =0.2785,$$

(5)

where τ is a positive constant.

Lemma 2 

([19]). Given that the Lyapunov function $V\left(x\right)$ is a continuous and positive definite function, and its initial value $V\left(0\right)$ lies within a finite range, it can be inferred that $V\left(x\right)$ remains bounded and satisfies the following condition:

$$\dot{V}\left(x\right)\le aV\left(x\right)+b,$$

(6)

where a and b are positive constants, then ${lim}_{t\to \infty }supV\le {\displaystyle \frac{b}{a}},{lim}_{t\to \infty }\dot{V}=0$.

Lemma 3. 

The unknown continuous nonlinear function $F\left(S\right)$ can be approximated by RBFNNs with the approximated error $\varsigma \left(S\right):{sup}_{x\in \mathrm{\Omega }}|F\left(S\right)-{\widehat{\theta }}^T\mathrm{\Phi }\left(s\right)|\le \varsigma \left(S\right)$, where $\varsigma \left(S\right)\le \overline{\varsigma }$ and $\overline{\varsigma }>0$ is an unknown constant.

Assumption 1 

([3]). There is at least one path between the leader and the followers. $\mathcal{G}$ has a fixed spanning tree if the leader is the root node.

Assumption 2 

([6]). The reference signal $y_d$ satisfies $y_d^2+{\dot{y}}_d^2\le \overline{y}$, with $\overline{y}>0$ being a positive constant.

Assumption 3 

([27]). There exist unknown constants $\overline{d}$ and ${\overline{d}}^{*}>0$ such that $|d_{i,k}\left(t\right)|⩽\overline{d}$ and $|{\dot{d}}_{i,k}\left(t\right)|⩽{\overline{d}}^{*}$.

2.2. Preliminary Work

The communication topology of MASs to represent the interactions between agents, which is described by a signed digraph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$, where $\mathcal{X}=\{x_1,x_2,\dots ,x_M\}$ is a node set, $\mathcal{E}\subseteq \mathcal{X}\times \mathcal{X}=\left\{(x_j,x_i)\right|x_j,x_i\in \mathcal{V}\}$ is an edge set, and $\mathcal{A}=\left[a_{ij}\right]\in {\mathbb{R}}^{M\times M}$ is the adjacency matrix. There are three relationships between agent $x_i$ and $x_j$ for the directions of $a_{ij}$ as follows: (1) the cooperative interaction relationship: $a_{ij}>0$; (2) the antagonistic relationship: $a_{ij}<0$; (3) otherwise: $a_{ij}=0$.

The treatment of imbalanced topology induced by negative-weight effects is generally not amenable to classical consensus analysis since the error dynamics involve cross-coupling terms originating from multiple neighbors, thereby complicating the computation. To address this issue, a hierarchical algorithm is designed in which the original imbalanced topology is analyzed through a spanning tree. Under this framework, each agent interacts only with its unique parent node, allowing the error analysis to be simplified to a local level. Design hierarchical matrix ${\mathcal{A}}^{\prime }=\left[a_i^{\prime }\right]\in {\mathbb{R}}^{1\times M}$, where $a_i^{\prime }=sign\left(a_{i,j}\right)max\left\{\right|a_{i,j}\left|\right\}$. To provide a more intuitive illustration of the hierarchical algorithm, an example is presented in Figure 1. Agents $x_1$ and $x_2$ can directly receive the information from $y_d$, so $y_d$ is in the layer 0 and $x_1$ and $x_2$ are in the layer 1. For the agents $x_3$ and $x_4$, they can receive the information from a unique agent in the layer 1, so they are in the layer 2. Then, interactions at each layer are restricted to nodes in the upper layer. For $x_3$ and $x_4$, the corresponding $a_3^{\prime }=sign\left(a_{i,j}\right)max\left\{\right|a_{3,1}|,|a_{3,2}\left|\right\}=a_{3,2}=-3$, $a_4^{\prime }=sign\left(a_{i,j}\right)max\left\{\right|a_{4,1}|,|a_{4,2}\left|\right\}=a_{4,1}=3$. Similarly, it can be concluded that $a_1^{\prime }=1$ and $a_2^{\prime }=1$. There are two shortest routes in the layered directed graph as follows: $\left(i\right)$ $R_1>0$: $y_R\to x_1\to x_4$; $\left(ii\right)$ $R_2<0$: $y_R\to x_2\to x_3$. We determine that if $R_i>0,x_i\in {\mathcal{X}}_1$; otherwise, $x_i\in {\mathcal{X}}_2$ holds. In Figure 1, $y_d,x_1,x_2,x_4\in {\mathcal{X}}_1$, and $x_3\in {\mathcal{X}}_2$. The hierarchical matrix after the topology switching is designed as ${\mathcal{A}}^{\prime \prime }$. As shown in Figure 1, prior to the switching, one has ${\mathcal{A}}^{\prime }={[1,1,-3,3]}^T$. After the topology switching occurs, the corresponding $a_{ij}$ values are altered, leading to ${\mathcal{A}}^{″}={[1,-3,1,3]}^T$.

2.3. Design of DO

Due to complexity and diversity in practical operating conditions, influence of external disturbances on controlled systems is often unavoidable, which may lead to degradation of system performance. Hence, suppression or compensation of such disturbances is regarded as an indispensable component in controller design. In this work, a DO has been proposed, through which external disturbances and approximation errors generated by RBFNNs can be effectively compensated. For convenience in subsequent design, the following auxiliary variables are introduced.

$$Q_{i,k}=D_{i,k}-{\nu }_{i,k}{\iota }_{i,k},k=1,\dots ,n,$$

(7)

where $D_{i,k}=d_{i,k}+{\delta }_{d_i,M}$ denotes unknown lumped disturbance, with ${\nu }_{i,k}>0$ denoting the design parameters.

Considering (1), (2), and (7), one has

$$\begin{array}{cc}\hfill {\dot{Q}}_{i,k}& ={\dot{D}}_{i,k}-{\nu }_{i,k}\left({\iota }_{i,k+1}+{\theta }_{i,k}^{*T}\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)+{\delta }_{f_i}\left({\overline{\iota }}_{i,k}\right)+d_{i,k}\right)\hfill \\ \hfill & ={\dot{D}}_{i,k}-{\nu }_{i,k}\left({\iota }_{i,k+1}+{\theta }_{i,k}^{*T}\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)+Q_{i,k}+{\nu }_{i,k}{\iota }_{i,k}\right),\hfill \end{array}$$

(8)

where ${\iota }_{i,n+1}=u$, ${\nu }_{i,k}$ is a known constant and denotes observer gain. The DO is designed in the following form:

$$\left\{\begin{array}{c}{\widehat{D}}_{i,k}={\widehat{Q}}_{i,k}+{\nu }_{i,k}{\iota }_{i,k}\hfill \\ {\dot{\widehat{Q}}}_{i,k}=-{\nu }_{i,k}\left({\iota }_{i,k+1}+{\theta }_{i,k}^{*T}\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)+{\nu }_{i,k}{\iota }_{i,k}+{\widehat{Q}}_{i,k}\right)\hfill \\ {\dot{\tilde{D}}}_{i,k}={\dot{\tilde{Q}}}_{i,k}={\dot{D}}_{i,k}-{\nu }_{i,k}{\tilde{\theta }}_{i,k}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)-{\nu }_{i,k}{\tilde{D}}_{i,k},\hfill \end{array}\right.$$

(9)

where ${\tilde{D}}_{i,k}=D_{i,k}-{\widehat{D}}_{i,k}$ and ${\tilde{Q}}_{i,k}=Q_{i,k}-{\widehat{Q}}_{i,k}$, representing the disturbance estimation error and extended disturbance estimation error.

3. Main Results

In this section, a detailed controller design procedure is presented, as illustrated in Figure 2. An adaptive event-triggered FTC scheme based on a switching-threshold mechanism is proposed, and stability of the system is verified.

3.1. Event-Triggered Mechanism

To balance the utilization of communication resources and the tracking performance of the controlled system [27], an event-triggered mechanism with switching thresholds is introduced.

$$\left\{\begin{array}{c}v\left(t\right)=w\left(t_l\right),\forall t\in [t_l,t_{l+1})\hfill \\ t_{l+1}=\left\{\begin{array}{cc}inf\{t>t_l\mid |\sigma \left(t\right)|\ge \Delta \left(t\right)|v\left(t\right)|+{\overline{\varpi }}_1\},\hfill & \left|v\right(t\left)\right|<H\hfill \\ inf\{t>t_l\mid |\sigma \left(t\right)|\ge {\overline{\varpi }}_2\},\hfill & \left|v\right(t\left)\right|\ge H\hfill \end{array}\right.\hfill \\ \dot{\Delta }\left(t\right)=-g\Delta \left(t\right){\sigma }^2\left(t\right),\hfill \end{array}\right.$$

(10)

where $w\left(t\right)$ denotes the continuous control law, which will be provided later. The term $\sigma \left(t\right)=w\left(t\right)-v\left(t\right)$ represents the deviation between the continuous control law $w\left(t\right)$ and the actual control input $v\left(t\right)$. The quantities ${\overline{\varpi }}_1,{\overline{\varpi }}_2,g$, and H are known constants, and the initial value of the dynamic threshold parameter $\Delta \left(t\right)$ is assumed to satisfy $\Delta \left(0\right)={\Delta }_0,{\Delta }_0\in (0,1)$.

A continuous control law $w\left(t\right)$ and an actual control law $\nu \left(t\right)$ are constructed:

$$w\left(t\right)=-{\displaystyle \frac{\overline{\Delta }+1}{\eta }}\left({\alpha }_{i,n}tanh\left({\displaystyle \frac{z_{i,n}{\alpha }_{i,n}}{\overline{\rho }}}\right)+\sigma tanh\left({\displaystyle \frac{z_{i,n}\overline{\varpi }}{\overline{\rho }}}\right)\right),$$

(11)

$$v\left(t\right)={\displaystyle \frac{w\left(t\right)-\overline{\varpi }{\beta }_2\left(t\right)}{\overline{\Delta }{\beta }_1\left(t\right)+1}},$$

(12)

where ${\alpha }_{i,n}$ denotes the virtual control law, which will be provided in the controller design section, $\overline{\rho }$ represents smoothing coefficient, and ${\beta }_1\left(t\right)$ and ${\beta }_2\left(t\right)$, designed as time-varying auxiliary functions introduced to ensure the boundedness of the continuous control law, satisfy $|{\beta }_1\left(t\right)|⩽1$ and $|{\beta }_2\left(t\right)|⩽1$. The parameters $\overline{\Delta }$ and $\sigma$ are defined as follows:

$$\overline{\Delta }=\left\{\begin{array}{cc}{\Delta }_0,\hfill & \left|v\right(t\left)\right|<H\hfill \\ 0,\hfill & \left|v\right(t\left)\right|\ge H,\hfill \end{array}\right.\overline{\varpi }=\left\{\begin{array}{cc}{\overline{\varpi }}_1,\hfill & \left|v\right(t\left)\right|<H\hfill \\ {\overline{\varpi }}_2,\hfill & \left|v\right(t\left)\right|\ge H.\hfill \end{array}\right.$$

(13)

Remark 1. 

The event-triggered control scheme with switching thresholds is proposed such that the advantages of both the FT and RT mechanisms are incorporated. Through this integration, an effective balance between communication resource utilization and tracking performance is achieved. Since ${\alpha }_{i,n}$, $\widehat{D}$, $\widehat{\theta }$, and $w\left(t\right)$ are all bounded, $\dot{w}\left(t\right)$ is also bounded, and $|\dot{w}\left(t\right)|⩽{\overline{L}}_w$. From the switching event-triggered mechanism, it can be observed that the minimum inter-event time satisfies $t_{l+1}-t_l\ge {\displaystyle \frac{{\overline{\varpi }}_1}{L_w}}$, ${\overline{\varpi }}_1>0$ and $L_w<\infty$. Inter-event time is strictly positive. Hence, Zeno behavior is excluded.

3.2. Controller Design

First, in order to prevent the occurrence of complexity explosion, the following command filter, together with its corresponding compensation signal, is introduced for the $ith$ agent

$$\left\{\begin{array}{c}{\dot{\rho }}_{i,j}^{\prime }=a_1{\rho }_{i,j}^{″}\hfill \\ {\dot{\rho }}_{i,j}^{″}=-2a_2{a}_1{\rho }_{i,j}^{″}+a_1{e}_{i,j}\hfill \\ e_{i,j}={\alpha }_{i,j}-{\rho }_{i,j}^{\prime },j=1,\dots ,n-1,\hfill \end{array}\right.$$

(14)

$$\left\{\begin{array}{c}{\delta }_{i,1}=-c_1{\delta }_{i,1}+\widehat{\lambda }{\delta }_{i,2}-\widehat{\lambda }{e}_{i,1}\hfill \\ {\dot{\delta }}_{i,k}=-c_k{\delta }_{i,k}+{\delta }_{i,k+1}-e_{i,k}-{\delta }_{i,k-1}\hfill \\ {\dot{\delta }}_{i,n}=-c_n{\delta }_{i,n}-{\delta }_{i,n-1},k=2,\dots ,n-1,\hfill \end{array}\right.$$

(15)

where ${\rho }_{i,j}^{\prime }\left(0\right)={\alpha }_{i,j}\left(0\right)$, ${\rho }_{i,j}^{\prime \prime }\left(0\right)=0$, constants $a_1>0$, $a_2\in (0,1]$, $c_{i,j}>0$, and $e_{i,j}$ are filtering errors, and ${\alpha }_{i,j}$ are virtual controllers, which will be given later.

The following coordinate transformation and auxiliary error signals are defined:

$$\left\{\begin{array}{c}{\tau }_{i,k}={\pi }_{i,k}-{\rho }_{i,k-1}^{\prime }\hfill \\ z_{i,k}={\tau }_{i,k}-{\delta }_{i,k},k=1,\dots ,n,\hfill \end{array}\right.$$

(16)

where ${\pi }_{1,1}=y_i^f$, ${\pi }_{i,j}={\iota }_{i,j}$, $i=2,\dots ,M$ and $j=2,\dots ,n$, ${\rho }_{1,0}^{\prime }=y_d\left(t\right)$, ${\rho }_{i,0}^{\prime }=a_i^{\prime }{\pi }_{q,1}$, ${\pi }_{q,1}$ is the state of the node $x_q$ in the upper layer, which transmits information to the node $x_i$ in the adjacent lower layer.

The virtual controller ${\alpha }_{i,k}$ is constructed as

$$\begin{array}{cc}\hfill {\alpha }_{i,k}& =-{\displaystyle \frac{1}{{\Lambda }_{i,k}}}({\Lambda }_{i,k}{\widehat{\theta }}_{i,k}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)+{\displaystyle \frac{{\mu }_{i,k}}{2}}{z}_{i,k}+{\Lambda }_{i,k}{\widehat{D}}_{i,k}-{\dot{\rho }}_{i,k-1}^{\prime }\left(t\right)\hfill \\ \hfill & +c_{i,k}{\tau }_{i,k}+{\widehat{\mathrm{\Gamma }}}_{i,k}{\tau }_{i,k-1}+{\displaystyle \frac{m_{i,k}}{2}}{z}_{i,k}{\widehat{\varphi }}_h+{\displaystyle \frac{b_{i,k}}{2}}{z}_{i,k}\widehat{r}),i=1,\dots ,n,\hfill \end{array}$$

(17)

where ${\nu }_i={\mathrm{\Gamma }}^2$ if $k=1$; otherwise, ${\nu }_i=2$. ${\mathrm{\Gamma }}_i=0$ if $k=1$; otherwise, ${\mathrm{\Gamma }}_i=\mathrm{\Gamma }$. $m_i=1$ if $k=1$; otherwise, $m_i=0$. $b_i=0$ if $k=1$; otherwise $b_i=1$.

In addition, the adaptive laws are designed as

$${\dot{\widehat{\theta }}}_{i,k}={\gamma }_{{\theta }_{i,k}}{\Lambda }_{i,k}{z}_{i,k}\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)-h_{i,k}{\widehat{\theta }}_{i,k},$$

(18)

$${\dot{\widehat{\varphi }}}_h={\displaystyle \frac{{\gamma }_h}{2}}{z}_{i,1}^2-h_{\varphi }{\widehat{\varphi }}_h,$$

(19)

$$\dot{\widehat{r}}={\displaystyle \frac{{\gamma }_r}{2}}{z}_{i,n}^2-h_r\widehat{r},$$

(20)

where ${\gamma }_{{\theta }_{i,k}}>0$, ${\gamma }_h>0$, ${\gamma }_r>0$, $h_{i,k}>0$, $h_{\varphi }>0$, and $h_r>0$ are design parameters. ${\varphi }_h>{\parallel {\theta }^{*}\parallel }^2$ and $r>\parallel {\theta }^{*}{\parallel }^2$ to guarantee the negativity of the Lyapunov derivative.

Based on the proposed controller and adaptive laws, the adaptive bidirectional consensus switching event-triggered FTC scheme is formally established in Theorem 1. The detailed proof of Theorem 1 is provided in the subsequent stability analysis section.

3.3. Stability Analysis

Theorem 1. 

Under Assumptions 1–3, for the multi-agent system described by Equation (1) and operating in the presence of sensor faults (4), the switching-threshold event-triggered mechanism (10)–(12), the command filter (14) together with its associated compensation signal (15), the virtual controller (17), and the adaptive laws (18)–(20) collectively ensure the realization of bidirectional consensus with a balanced trade off between communication efficiency and tracking performance. Furthermore, the boundedness of all closed-loop signals is guaranteed, irrespective of whether actuator or sensor faults occur.

Proof. 

$\mathrm{Step}\mathrm{k}(1⩽\mathrm{k}⩽\mathrm{n}-1):$ From Equations (1), (4), (9), and (14)–(16), the following can be deduced:

$${\dot{z}}_{i,k}={\dot{\tau }}_{i,k}-{\dot{\delta }}_{i,k}={\iota }_{i,k+1}+f_{i,k}\left({\overline{\iota }}_{i,k}\right)+d_{i,k}\left(t\right)-{\dot{\rho }}_{i,k-1}^{\prime }\left(t\right)-{\dot{\delta }}_{i,k},$$

(21)

where $f_{i,k}\left({\overline{\iota }}_{i,k}\right)$ is the unknown function, which can be approximated by RBFNNs, and one can obtain the following:

$${\dot{z}}_{i,k}={\iota }_{k+1}+{\tilde{\theta }}_{i,k}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)+{\widehat{\theta }}_{i,k}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)+D_{i,k}-{\dot{\rho }}_{i,k-1}^{\prime }\left(t\right)-{\dot{\delta }}_{i,k}.$$

(22)

The Lyapunov function $V_{i,k}$ is chosen as

$$V_{i,k}=V_{i,k-1}+{\displaystyle \frac{1}{2}}{z}_{i,k}^2+{\displaystyle \frac{1}{2{\gamma }_{{\theta }_{i,k}}}}{\tilde{\theta }}_{i,k}^T{\tilde{\theta }}_{i,k}+{\displaystyle \frac{1}{2}}{\tilde{D}}_{i,k}^2+{\displaystyle \frac{{\lambda }_{i,k}}{2{\gamma }_h}}{\tilde{\varphi }}_h^2,$$

(23)

where $V_{i,0}=0$, ${\lambda }_{i,k}=1$ if $k=1$; otherwise, ${\lambda }_{i,k}=0$.

By computing Equation (23), the following result is obtained

$$\begin{array}{cc}\hfill {\dot{V}}_{i,k}& ={\dot{V}}_{i,k-1}+z_{i,k}{\dot{z}}_{i,k}-{\displaystyle \frac{1}{{\gamma }_{{\theta }_{i,k}}}}{\tilde{\theta }}_{i,k}^T{\dot{\widehat{\theta }}}_{i,k}+{\tilde{D}}_{i,k}{\dot{\tilde{D}}}_{i,k}-{\displaystyle \frac{{\lambda }_{i,k}}{{\gamma }_h}}{\tilde{\varphi }}_h{\dot{\widehat{\varphi }}}_h\hfill \\ & ={\dot{V}}_{i,k-1}+z_{i,k}({\Lambda }_{i,k}{\varsigma }_{i,k+1}+{\Lambda }_{i,k}{\tilde{\theta }}_{i,k}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)+{\Lambda }_{i,k}{\widehat{\theta }}_{i,k}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)+{\Lambda }_{i,k}{\tilde{D}}_{i,k}+{\Lambda }_{i,k}{\widehat{D}}_k+{\lambda }_{i,k}\dot{\varphi }\left(t\right)\hfill \\ & -{\dot{\rho }}_{i,k-1}^{\prime }\left(t\right)-{\dot{\delta }}_{i,k})-{\displaystyle \frac{1}{{\gamma }_{{\theta }_{i,k}}}}{\tilde{\theta }}_{i,k}^T{\dot{\widehat{\theta }}}_{i,k}+{\tilde{D}}_{i,k}\left({\dot{D}}_{i,k}-{\nu }_{i,k}{\tilde{\theta }}_{i,k}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)-{\nu }_{i,k}{\tilde{D}}_{i,k}\right)\hfill \\ & -{\displaystyle \frac{{\lambda }_{i,k}}{{\gamma }_h}}{\tilde{\varphi }}_h{\dot{\widehat{\varphi }}}_h,\hfill \end{array}$$

(24)

where ${\Lambda }_{i,k}=\mathrm{\Gamma }$ if $k=1$; otherwise, ${\Lambda }_{i,k}=1$.

Based on Young’s inequality, we have

$$\begin{array}{cc}& z_{i,k}{\Lambda }_{i,k}{\tilde{D}}_{i,k}⩽{\displaystyle \frac{{\Lambda }_{i,k}^2}{2}}{z}_{i,k}^2+{\displaystyle \frac{1}{2}}{\tilde{D}}_{i,k}^2,{\tilde{D}}_{i,k}{\dot{D}}_{i,k}⩽{\displaystyle \frac{1}{2}}{\tilde{D}}_{i,k}^2+{\displaystyle \frac{1}{2}}{D}_{i,k}^{\prime 2},\hfill \\ & z_{i,k}{\lambda }_{i,k}\dot{\varphi }\left(t\right)⩽{\displaystyle \frac{1}{2}}+{\displaystyle \frac{{\lambda }_{i,k}^2}{2}}{z}_{i,k}^2{\varphi }^{{*}^2},-{\nu }_{i,k}{\tilde{D}}_{i,k}{\tilde{\theta }}_{i,k}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)⩽{\displaystyle \frac{1}{2}}{\tilde{D}}_{i,k}^2+{\displaystyle \frac{{\nu }_{i,k}^2{M}^{\prime }}{2}}{\tilde{\theta }}_{i,k}^T{\tilde{\theta }}_{i,k},\hfill \end{array}$$

(25)

where $D_{i,k}^{\prime }={\varsigma }_{f_k,m}+d_{k,m}$. $M^{\prime }$ is RBFNNs node number.

Combining Equations (17)–(19) and Equations (24) and (25) gives

$$\begin{array}{cc}\hfill {\dot{V}}_{i,k}& ⩽{\dot{V}}_{i,k-1}+z_{i,k}\left({\Lambda }_{i,k}{z}_{k+1}+{\Lambda }_{i,k}{\delta }_{k+1}-{\Lambda }_{i,k}{e}_{i,k}-c_{i,k}{\tau }_{i,k}-{\dot{\delta }}_{i,k}-{\widehat{\mathrm{\Gamma }}}_{i,k}{\tau }_{i,k-1}\right)\hfill \\ & +{\displaystyle \frac{h_{i,k}}{{\gamma }_{{\theta }_{i,k}}}}{\tilde{\theta }}_{i,k}^T{\widehat{\theta }}_{i,k}-\left({\nu }_{i,k}-{\displaystyle \frac{3}{2}}\right){\tilde{D}}_{i,k}^2+{\displaystyle \frac{M^{\prime }{\nu }_{i,k}^2}{2}}{\tilde{\theta }}_{i,k}^T{\tilde{\theta }}_{i,k}+{\displaystyle \frac{1}{2}}{D}_{i,k}^{\prime 2}+{\displaystyle \frac{{\lambda }_{i,k}}{2}}+{\displaystyle \frac{{\lambda }_{i,k}{h}_{\varphi }}{{\gamma }_h}}{\tilde{\varphi }}_h{\widehat{\varphi }}_h,\hfill \end{array}$$

(26)

where ${\tau }_0=0$.

By applying Young’s inequality, we have

$${\displaystyle \frac{h_{i,k}}{{\gamma }_{{\theta }_{i,k}}}}{\tilde{\theta }}_{i,k}^T{\widehat{\theta }}_{i,k}⩽-{\displaystyle \frac{h_{i,k}}{2{\gamma }_{{\theta }_{i,k}}}}{\tilde{\theta }}_{i,k}^T{\widehat{\theta }}_{i,k}+{\displaystyle \frac{h_{i,k}}{2{\gamma }_{\theta }}}{\theta }_{i,k}^T{\theta }_{i,k},$$

(27)

$${\displaystyle \frac{{\lambda }_{i,k}{h}_{\varphi }}{{\gamma }_h}}{\tilde{\varphi }}_h{\widehat{\varphi }}_h⩽-{\displaystyle \frac{{\lambda }_{i,k}{h}_{\varphi }}{2{\gamma }_h}}{\tilde{\varphi }}_h^2+{\displaystyle \frac{{\lambda }_{i,k}{h}_{\varphi }}{2{\gamma }_h}}{\varphi }_h^2.$$

(28)

According to Equations (15), (27), and (28), ${\dot{V}}_{i,k}$ can be derived

$$\begin{array}{c}{\dot{V}}_{i,k}⩽-{\displaystyle \sum _{j=1}^k}\left(c_{i,j}{z}_{i,j}^2+\left({\nu }_{i,j}-{\displaystyle \frac{3}{2}}\right){\tilde{D}}_{i,j}^2+\left({\displaystyle \frac{h_{i,j}}{2{\gamma }_{{\theta }_{i,k}}}}-{\displaystyle \frac{M^{\prime }{\nu }_{i,j}^2}{2}}\right){\tilde{\theta }}_{i,j}^T{\tilde{\theta }}_{i,j}\right)-{\displaystyle \frac{h_{\varphi }}{2{\gamma }_h}}{\tilde{\varphi }}_h^2\\ +z_{i,k}{z}_{i,k+1}+{\displaystyle \sum _{j=1}^k}\left({\displaystyle \frac{1}{2}}{D}_{i,j}^{\prime 2}+{\displaystyle \frac{h_{i,j}}{2{\gamma }_{{\theta }_{i,k}}}}{\theta }_{i,j}^T{\tilde{\theta }}_{i,j}+{\displaystyle \frac{{\lambda }_{i,j}}{2}}\right)+{\displaystyle \frac{h_{\varphi }}{2{\gamma }_h}}{\varphi }_h^2.\end{array}$$

(29)

$\mathrm{Step}\mathrm{n}:$ From Equations (1), (4), (12), and (14)–(16), the following can be deduced:

$$\begin{array}{cc}\hfill {\dot{z}}_{i,n}& ={\dot{\tau }}_{i,n}-{\dot{\iota }}_{i,n}={\dot{\iota }}_{i,n}-{\dot{\rho }}_{i,n-1}^{\prime }\left(t\right)-{\dot{\delta }}_{i,n}\hfill \\ & ={\displaystyle \frac{\eta w\left(t\right)}{\overline{\Delta }{\beta }_1\left(t\right)+1}}-{\displaystyle \frac{\eta \overline{\varpi }{\beta }_2\left(t\right)}{\overline{\Delta }{\beta }_1\left(t\right)+1}}+\varpi \left(t\right)+f_{i,n}\left({\overline{\iota }}_{i,n}\right)+d_{i,n}\left(t\right)-{\dot{\rho }}_{i,n-1}^{\prime }\left(t\right)-{\dot{\delta }}_{i,n},\hfill \end{array}$$

(30)

where $f_{i,n}\left({\overline{\iota }}_{i,n}\right)$ is the unknown function, which can be approximated by RBFNNs, and one can obtain

$$\begin{array}{cc}\hfill {\dot{z}}_{i,n}& ={\displaystyle \frac{\eta w\left(t\right)}{\overline{\Delta }{\beta }_1\left(t\right)+1}}-{\displaystyle \frac{\eta \overline{\varpi }{\beta }_2\left(t\right)}{\overline{\Delta }{\beta }_1\left(t\right)+1}}+\varpi \left(t\right)+{\tilde{\theta }}_{i,n}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,n}\right)+{\widehat{\theta }}_{i,n}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,n}\right)\hfill \\ \hfill & +D_{i,n}-{\dot{\rho }}_{i,n-1}^{\prime }\left(t\right)-{\dot{\delta }}_{i,n}.\hfill \end{array}$$

(31)

Choosing the Lyapunov function $V_{i,n}$ as

$$V_{i,n}=V_{i,n-1}+{\displaystyle \frac{1}{2}}{z}_{i,n}^2+{\displaystyle \frac{1}{2{\gamma }_{{\theta }_{i,n}}}}{\tilde{\theta }}_{i,n}^T{\tilde{\theta }}_{i,n}+{\displaystyle \frac{1}{2}}{\tilde{D}}_{i,n}^2+{\displaystyle \frac{1}{2{\gamma }_r}}{\tilde{r}}^2.$$

(32)

By computing Equation (32), the following result is obtained

$$\begin{array}{cc}\hfill {\dot{V}}_{i,n}& ={\dot{V}}_{i,n-1}+z_{i,k}{\dot{z}}_{i,n}-{\displaystyle \frac{1}{{\gamma }_{{\theta }_{i,n}}}}{\tilde{\theta }}_{i,n}^T{\dot{\widehat{\theta }}}_{i,n}+{\tilde{D}}_{i,n}{\dot{\tilde{D}}}_{i,n}-{\displaystyle \frac{1}{{\gamma }_r}}\tilde{r}\dot{\widehat{r}}\hfill \\ & ={\dot{V}}_{i,n-1}+z_{i,n}\left({\displaystyle \frac{\eta w\left(t\right)}{\overline{\Delta }{\beta }_1\left(t\right)+1}}-{\displaystyle \frac{\eta \overline{\varpi }{\beta }_2\left(t\right)}{\overline{\Delta }{\beta }_1\left(t\right)+1}}+\varpi \left(t\right)+{\tilde{\theta }}_{i,n}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,n}\right)+{\widehat{\theta }}_{i,n}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,n}\right)\right)\hfill \\ & +{\tilde{D}}_{i,n}+{\widehat{D}}_{i,n}-{\dot{\rho }}_{i,n-1}^{\prime }\left(t\right)-{\dot{\delta }}_{i,n})-{\displaystyle \frac{1}{{\gamma }_{{\theta }_{i,n}}}}{\tilde{\theta }}_{i,n}^T{\dot{\widehat{\theta }}}_{i,n}-{\displaystyle \frac{1}{{\gamma }_r}}\tilde{\widehat{r}}\dot{\widehat{r}}\hfill \\ & +{\tilde{D}}_{i,n}\left({\dot{D}}_{i,n}-{\nu }_{i,n}{\tilde{\theta }}_{i,n}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,n}\right)-{\nu }_{i,n}{\tilde{D}}_{i,n}\right).\hfill \end{array}$$

(33)

Based on Young’s inequality, we have

$$\begin{array}{cc}& z_{i,n}{\tilde{D}}_{i,n}⩽{\displaystyle \frac{1}{2}}{z}_{i,n}^2+{\displaystyle \frac{1}{2}}{\tilde{D}}_{i,n}^2,{\tilde{D}}_{i,n}{\dot{D}}_{i,n}⩽{\displaystyle \frac{1}{2}}{\tilde{D}}_{i,n}^2+{\displaystyle \frac{1}{2}}{D}_{i,n}^{\prime 2},\hfill \\ & -{\nu }_{i,n}{\tilde{D}}_{i,n}{\tilde{\theta }}_{i,n}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,n}\right)⩽{\displaystyle \frac{1}{2}}{\tilde{D}}_{i,n}^2+{\displaystyle \frac{{\nu }_{i,n}^2{M}^{\prime }}{2}}{\tilde{\theta }}_{i,n}^T{\tilde{\theta }}_{i,n},z_{i,n}\varpi \left(t\right)⩽{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}{z}_{i,n}^2{\varpi }^{*2},\hfill \\ & {\displaystyle \frac{z_{i,n}\eta w\left(t\right)}{\overline{\Delta }{\beta }_1\left(t\right)+1}}⩽{\displaystyle \frac{z_{i,n}\eta w\left(t\right)}{\overline{\Delta }+1}},{\displaystyle \frac{z_{i,n}\eta w\left(t\right)}{\overline{\Delta }{\beta }_1\left(t\right)+1}}⩽{\displaystyle \frac{z_{i,n}\eta w\left(t\right)}{\overline{\Delta }+1}},\hfill \end{array}$$

(34)

where $D_{i,n}^{\prime }={\varsigma }_{f_n,m}+d_{n,m}$.

Substituting Equations (11), (18), (20), and (34) into (33) leads to

$$\begin{array}{cc}\hfill {\dot{V}}_{i,n}& ⩽{\dot{V}}_{i,n-1}+z_{i,n}(-{\alpha }_{i,n}tanh\left({\displaystyle \frac{z_{i,n}{\alpha }_{i,n}}{\overline{\rho }}}\right)+{\widehat{\theta }}_{i,n}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,n}\right)+{\displaystyle \frac{1}{2}}{z}_{i,n}+{\widehat{D}}_{i,n}-{\dot{\rho }}_{i,n-1}^{\prime }\left(t\right)\hfill \\ & -{\dot{\delta }}_{i,n}+{\displaystyle \frac{1}{2}}{z}_{i,n}\widehat{r})+{\displaystyle \frac{h_{i,n}}{{\gamma }_{{\theta }_{i,n}}}}{\tilde{\theta }}_{i,n}^T{\widehat{\theta }}_{i,n}-\left({\nu }_{i,n}-{\displaystyle \frac{3}{2}}\right){\tilde{D}}_{i,n}^2+{\displaystyle \frac{M^{\prime }{\nu }_{i,n}^2}{2}}{\tilde{\theta }}_{i,n}^T{\tilde{\theta }}_{i,n}+{\displaystyle \frac{1}{2}}{D}_{i,n}^{\prime 2}+{\displaystyle \frac{1}{2}}\hfill \\ & +{\displaystyle \frac{h_r}{{\gamma }_r}}\tilde{r}\widehat{r}+\left|z_{i,n}{\displaystyle \frac{\eta \overline{\varpi }}{1-\overline{\Delta }}}\right|-\overline{\varpi }tanh\left({\displaystyle \frac{z_{i,n}\overline{\varpi }}{\overline{\rho }}}\right).\hfill \end{array}$$

(35)

Similar to Equation (26), we have

$$\begin{array}{cc}\hfill {\dot{V}}_{i,n}& ⩽{\dot{V}}_{i,n-1}+z_{i,n}(-{\alpha }_{i,n}tanh\left({\displaystyle \frac{z_{i,n}{\alpha }_{i,n}}{\overline{\rho }}}\right)+{\widehat{\theta }}_{i,n}^T\mathrm{\Phi }\left({\overline{\iota }}_{i,n}\right)+{\displaystyle \frac{1}{2}}{z}_{i,n}+{\widehat{D}}_{i,n}-{\dot{\rho }}_{i,n-1}^{\prime }\left(t\right)\hfill \\ & -{\dot{\delta }}_{i,n}+{\displaystyle \frac{1}{2}}{z}_{i,n}\widehat{r})-\left({\displaystyle \frac{h_{i,n}}{2{\gamma }_{{\theta }_{i,n}}}}-{\displaystyle \frac{{\eta }_{i,n}^2}{2}}\right){\tilde{\theta }}_{i,n}^T{\tilde{\theta }}_{i,n}-\left({\eta }_{i,n}-{\displaystyle \frac{3}{2}}\right){\tilde{D}}_{i,n}^2-{\displaystyle \frac{h_r}{2{\gamma }_r}}{\tilde{r}}^2+{\displaystyle \frac{h_r}{2{\gamma }_r}}{r}^2\hfill \\ & +{\displaystyle \frac{h_{i,n}}{2{\gamma }_{{\theta }_{i,n}}}}{\theta }_{i,n}^T{\theta }_{i,n}+{\displaystyle \frac{1}{2}}{D}_{i,n}^{\prime 2}+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\eta \overline{\varpi }}{1-\overline{\Delta }}}\right)}^2-\overline{\varpi }tanh\left({\displaystyle \frac{z_{i,n}\overline{\varpi }}{\overline{\rho }}}\right).\hfill \end{array}$$

(36)

We define $V={\sum }_{i=1}^M{\sum }_{j=1}^n{V}_{i,j}$. Considering Equations (15) and (17) and applying Lemma 1, the following inequality can be obtained

$$\begin{array}{cc}\hfill \dot{V}& ⩽-\sum _{j=1}^n\left(c_{i,j}{z}_j^2+\left({\nu }_{i,j}-{\displaystyle \frac{3}{2}}\right){\tilde{D}}_{i,j}^2+\left({\displaystyle \frac{h_{i,j}}{2{\gamma }_{{\theta }_{i,j}}}}-{\displaystyle \frac{{\nu }_{i,j}^2}{2}}\right){\tilde{\theta }}_{i,n}^T{\tilde{\theta }}_{i,n}\right)-{\displaystyle \frac{h_{\varphi }}{2{\gamma }_{\varphi }}}{\tilde{\varphi }}_h^2-{\displaystyle \frac{h_r}{2{\gamma }_r}}{\tilde{r}}^2\hfill \\ & +\sum _{j=1}^n\left({\displaystyle \frac{1}{2}}{D}_{i,j}^{\prime 2}+{\displaystyle \frac{h_{i,j}}{2{\gamma }_{{\theta }_{i,j}}}}{\theta }_{i,n}^T{\theta }_{i,n}\right)+{\displaystyle \frac{h_{\varphi }}{2{\gamma }_h}}{\varphi }_h^2+{\displaystyle \frac{h_r}{2{\gamma }_r}}{r}^2+1+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\eta \overline{\varpi }}{1-\overline{\Delta }}}\right)}^2\hfill \\ & +0.557\overline{\rho }\hfill \\ & ⩽-\overline{a}V+\overline{b},\hfill \end{array}$$

(37)

where $\overline{a}=min\left\{2c_{i,j},h_{i,j}-{\gamma }_{{\theta }_{i,j}}{\nu }_{i,j}^2,2{\nu }_{i,j}-3,h_{\varphi },h_r\right\}$ and $\overline{b}={\sum }_{j=1}^n\left({\displaystyle \frac{1}{2}}{D}_{i,j}^{\prime 2}+{\displaystyle \frac{h_{i,j}}{2{\gamma }_{{\theta }_{i,j}}}}{\theta }_{i,j}^T{\theta }_{i,j}\right)+{\displaystyle \frac{h_{\varphi }}{2{\gamma }_h}}{\varphi }_h^2+{\displaystyle \frac{h_r}{2{\gamma }_r}}{r}^2+1+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\eta \overline{\varpi }}{1-\overline{\Delta }}}\right)}^2+0.557\overline{\rho }$.

By Lemma 2, integrating Equation (37) over [0,t], one has

$$\left\{\begin{array}{c}V\left(t\right)⩽\left(V\left(t_0\right)-{\displaystyle \frac{\overline{b}}{\overline{a}}}\right)e^{-\overline{a}\left(t-t_0\right)}+{\displaystyle \frac{\overline{b}}{\overline{a}}}\hfill \\ V\left(+\infty \right)⩽{\displaystyle \frac{\overline{b}}{\overline{a}}}.\hfill \end{array}\right.$$

(38)

Based on the definition of $V\left(t\right)$ and the coordinate transformation in Equation (16), it can be ensured that all closed-loop system signals remain bounded. □

Remark 2. 

The proposed ETC scheme with a switching threshold integrates the advantages of both FT and RT mechanisms, thereby achieving an effective balance between communication resources and tracking performance. While ensuring flexibility in the triggering threshold, the occurrence of Zeno behavior is simultaneously avoided. From inequality (38), it can be inferred that achieving a faster convergence rate while reducing the convergence error requires either a larger value of $\overline{a}$ or a smaller value of $\overline{b}$. In practical scenarios, multiple control objectives—such as convergence rate, tracking performance, and communication frequency—must be taken into account. Therefore, the selection of these parameters should be conducted with the aim of balancing the overall performance requirements.

4. Simulation Examples

In this section, the effectiveness of the proposed adaptive bidirectional consensus FTC scheme, employing a switching event-triggered mechanism, is demonstrated through simulation examples. In addition, comparative simulations are conducted to further verify the superiority of the designed controller.

4.1. Main Simulations

The communication graph is illustrated in Figure 3, where the multi-agent system consists of four agents, and $y_d$ denotes the leader agent. The communication topology is directed and connected, indicating that the proposed algorithm is applicable to a broad class of systems. The dynamics of each agent follow the reference model in [28] can be described as

$$\left\{\begin{array}{c}{\dot{\iota }}_{i,1}={\iota }_{i,2}+d_{i,1}\left(t\right),\hfill \\ {\dot{\iota }}_{i,2}=u_i-{\iota }_{i,1}{e}^{-{\iota }_{i,1}}-0.5{\iota }_{i,2}+d_{i,2}\left(t\right),\hfill \end{array}\right.$$

(39)

where $d_{i,1}\left(t\right)=0.8sin\left(2t\right)+0.1sin\left(5t\right)+0.05\mathcal{N}(0,1)$, $d_{i,2}\left(t\right)=0.4cos\left(1.2t\right)+0.1cos\left(4t\right)+0.05\mathcal{N}(0,1)$, and random Gaussian noise is incorporated into the disturbance. Then, unknown function $f_{i,k}\left({\overline{\iota }}_{i,k}\right)$ in the system is approximated by RBFNNs, for which the basis functions are selected as

$$\mathrm{\Phi }\left({\overline{\iota }}_{i,k}\right)=-exp\left({\displaystyle \frac{{({\overline{\iota }}_{i,k}-3+\varsigma )}^T({\overline{\iota }}_{i,k}-3+\varsigma )}{{\zeta }^2}}\right),$$

where $\zeta =2$, $\varsigma =1,\dots ,7$, $k=1,2$.

The leader’s dynamic is described as $y_d=0.8sin\left(t\right)+0.2sin\left(3t\right)$. The communication graph is shown in Figure 3. The agents are divided into two different groups based on the hierarchical algorithm. The shortest routes for agents $x_1-x_4$ are $R_1>0:y_R\to x_1\to x_2$, $R_2<0:y_R\to x_1\to x_3$, and $R_3<0:y_R\to x_1\to x_4$. Thus, according to the shortest routes, ${\mathcal{X}}_1=\{y_d,x_1,x_2\}$ and ${\mathcal{X}}_2=\{x_3,x_4\}$ can be given. As shown in Figure 3, prior to the switching, one has ${\mathcal{A}}^{\prime }={[1,1,-1,-1]}^T$. Topology switching occurs at $t=19$ s; the corresponding $a_{ij}$ values are altered, leading to ${\mathcal{A}}^{″}={[1,-1,1,-1]}^T$.

Considering that follower 1 is affected by a sensor fault, while followers are all subjected to actuator faults, the corresponding sensor-fault and actuator-fault models are formulated

$$y^f=\left\{\begin{array}{cc}{\iota }_{1,1}\left(t\right),\hfill & t\le 15\mathrm{s},\hfill \\ 0.34{\iota }_{1,1}\left(t\right)+0.28sin\left(t\right),\hfill & t>15\mathrm{s},\hfill \end{array}\right.$$

(40)

$$u_i\left(v\right)=\left\{\begin{array}{cc}{v}_i\left(t\right),\hfill & t\le 10\mathrm{s},\hfill \\ 0.5v_i\left(t\right)+0.5sin\left(t\right),\hfill & t>10\mathrm{s}.\hfill \end{array}\right.$$

(41)

In this simulation, the initial conditions are specified as follows: ${\iota }_1\left(0\right)={[0.2,0.2]}^T$, ${\iota }_2\left(0\right)={[0.15,0.2]}^T,{\iota }_3\left(0\right)={[0.1,0.3]}^T,{\iota }_4\left(0\right)={[0.15,0.3]}^T,Q_1\left(0\right)=Q_2\left(0\right)=Q_3\left(0\right)=Q_4\left(0\right)={[0,0]}^T,{\widehat{\theta }}_1\left(0\right)={[0.05,0.05]}^T,{\widehat{\varphi }}_h\left(0\right)=0.1$, ${\widehat{r}}_1\left(0\right)={\widehat{r}}_2\left(0\right)={\widehat{r}}_3\left(0\right)={\widehat{r}}_4\left(0\right)=0.1$, ${\delta }_1\left(0\right)={\delta }_2\left(0\right)={\delta }_3\left(0\right)={\delta }_4\left(0\right)={[0,0]}^T,{\Delta }_1\left(0\right)={\Delta }_2\left(0\right)={\Delta }_3\left(0\right)={\Delta }_4\left(0\right)=0.5$; and the initial values of the design parameter are assigned as follows: $c_{11}=c_{21}=c_{31}=c_{41}=15,c_{12}=c_{22}=c_{32}=c_{42}=20,a_{11}=a_{21}=a_{31}=a_{41}=100,a_{12}=a_{22}=a_{32}=a_{42}=1$, ${\nu }_{i,1}={\nu }_{i,2}=15,(i=1,\dots ,4),\overline{\rho }=0.05,h_{1,1}=h_{2,1}=h_{3,1}=h_{4,1}=5,h_{1,2}=h_{2,2}=h_{3,2}=h_{4,2}=8,h_{\varphi }=12,h_r=4,{\gamma }_{{\theta }_{1,1}}={\gamma }_{{\theta }_{2,1}}={\gamma }_{{\theta }_{3,1}}=0.15,{\gamma }_{{\theta }_{1,2}}={\gamma }_{{\theta }_{2,2}}={\gamma }_{{\theta }_{3,2}}={\gamma }_{{\theta }_{4,2}}=0.08$, ${\gamma }_{r_1}={\gamma }_{r_2}={\gamma }_{r_3}={\gamma }_{r_4}=0.008,{\overline{\varpi }}_{1,1}={\overline{\varpi }}_{2,1}={\overline{\varpi }}_{3,1}={\overline{\varpi }}_{4,1}=0.08,{\overline{\varpi }}_{1,2}={\overline{\varpi }}_{2,2}={\overline{\varpi }}_{3,2}={\overline{\varpi }}_{4,2}=0.15,H=0.6,g=0.005$.

Based on the aforementioned selection of control parameters, the simulation results under random noise disturbances, sensor faults, and switching topology are presented in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. As illustrated in Figure 4, after the sensor fault occurring at $t=15$ s and the topology switching at $t=19$ s, the tracking performance is rapidly recovered due to the designed controller, and all agents are still able to maintain stable trajectory tracking. These observations indicate that the proposed control strategy can accommodate topology switching scenarios and effectively compensate for sensor faults. Figure 5 shows the tracking errors of all agents under the switching event-triggered mechanism. In comparison with Figure 4, it further demonstrates that the designed controller ensures stable and effective tracking performance. Figure 6 depicts the trajectories of the disturbance observer subjected to random Gaussian noise, from which it can be observed that the designed observer accurately tracks the system disturbance. Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 display the trajectories of continuous controllers $w\left(t\right)$ and actual controllers $v\left(t\right)$ for each agent under the proposed switching event-triggered mechanism. Figure 11, Figure 12, Figure 13 and Figure 14 present the triggering intervals of all agents under the designed scheme. Each curve represents the updates of the controller, and its amplitude corresponds to the triggering interval. It can be observed that the proposed triggering mechanism is free of Zeno behavior.

4.2. Comparison Studies

Under the condition that all other scenarios remain identical, comparative simulations are first conducted using the FT and RT mechanisms, and the results are contrasted with those obtained under the switching event-triggered mechanism by taking agent 1 as an illustrative example. Figure 15 presents the controller triggering intervals under the RT mechanism, while Figure 16 shows those under the FT mechanism. Figure 17 illustrates the differences in tracking performance of agent 1 under the three triggering mechanisms, simultaneously demonstrating that all mechanisms are capable of achieving the desired tracking performance. To more clearly highlight the superiority of the proposed switching event-triggered mechanism,

Table 2. Update times and average tracking errors of follower 1 under three different triggering controllers.

	Update Times	Mean Absolute Error 1
ETC with FT	1295	0.012613
ETC with RT	2339	0.012056
ETC with ST	1407	0.011986

¹ ${\mathrm{MAE}}_i={\displaystyle \frac{1}{N}}{\sum }_{t_s=1}^N\left|{\tau }_i\left(t_s\right)\right|$. $t_s$ denotes the data sampling instant. N represents the total number of sampling steps.

summarizes the triggering counts and average tracking errors of agent 1 under the three mechanisms. The data in Table 2 indicate that compared with the RT mechanism, the proposed switching mechanism reduces the triggering count by $39.8\%$, and compared with the FT mechanism, it decreases the average tracking error by $5.0\%$. These results verify that the proposed event-triggered mechanism achieves a favorable balance between tracking performance and communication frequency.

On the other hand, while keeping the control mechanism unchanged, additional simulations are performed by modifying the disturbance-handling component. Taking agent 1 as an example once again, Figure 18 illustrates the tracking performance differences under different disturbance-handling designs. Without the disturbance observer, the average tracking error is $0.014141$. Under the switching event-triggered mechanism, the incorporation of the disturbance observer reduces the average tracking error by $15.2\%$, thereby significantly enhancing the tracking performance.

5. Conclusions

A switching event-triggered FTC scheme based on a disturbance observer has been proposed for MASs subject to sensor faults, achieving bidirectional consensus under switching topology. A hierarchical algorithm is adopted to address communication topology imbalance induced by negative weights; each agent interacts with only one parent node in the upper layer, which simplifies local error analysis and avoids complexity associated with asymmetric global Laplacian operators. A disturbance observer and its associated compensation signal are constructed to accommodate external disturbances and filtering errors. A switching event-triggered mechanism is developed, combining advantages of FT and ST mechanisms and enabling a balanced trade-off between communication frequency and tracking performance. Numerical simulations confirm superiority and boundedness of the proposed scheme. Future research will focus on more complex settings, such as heterogeneous MASs.