Adaptive state observer event-triggered consensus control for multi-agent systems with actuator failures

1.   Introduction

Currently, many engineering problems are complex and dynamic, posing challenges for traditional single-agent systems. Multi-agent systems (MASs) address these challenges through distributed problem-solving via collaboration, competition, and division of labor. Consequently, the coherent cooperative control of MASs has become a key research focus ^{[1,2,3,4,5,6,7]}. In ^[1], the authors studied the coherent adaptive cooperative control of uncertain nonlinear MASs from the three uncertainties of dead zone, disturbances, and uncertain time-varying control directions. Similarly, ^[3] extends the investigation to include uncertainties such as unknown control directions, input unmodeled dynamics, sensor failures, and prescribed performance. In ^[5], a distributed control protocol is formulated through the integration of adaptive control techniques and matrix theory. The design incorporates Fourier series expansion and neural networks (NNs) to approximate uncertain nonlinearities with unmeasured periodic time-varying perturbations. A variety of industrial field agents are further reorganized into new heterogeneous MASs for cooperative control, and a review and outlook of the latest results are presented in ^[6]. However, all the above control methods presuppose the assumption that all states throughout the system can be measured.

The acquisition of state information is limited during the operational phase due to equipment constraints, various other variables, and environmental conditions ^{[8,9,10,11,12,13,14]}. These limitations hinder the system's tracking performance and stability. This constraint ultimately hinders the system's tracking performance and stability. In ^[8], heterogeneous MASs comprising numerous unmanned ground vehicles (UGVs) and unmanned aerial vehicles (UAVs) are investigated to design a fixed-time observer aimed at estimating the mismatch interference and set total uncertainty.

In ^[9], the aid of a fuzzy observer for estimating the unavailable states is accompanied by utilizing the characteristics of fuzzy logic systems (FLSs) to address mismatches among agents and uncertainties within the MASs. In ^[12], the combination of the designed NNs state observer with the introduced adaptive control algorithm addresses the system's steady-state control and transient performance. To deal with the problems arising from unknown perturbations, a low-coupling, simple structure and easy-to-implement nonlinear perturbation observer is designed in ^[14].

In addition, nonlinearities widely exist in engineering control systems, for instance, actuator failures are unavoidable problems that may negatively affect the stability, performance, safety, and lifetime of the system. Therefore, it is especially critical for systems that require a high degree of controllability and robustness. To cope with this problem, many scholars have proposed a multitude of innovative control schemes with adaptive backstepping control techniques as the research background ^{[15,16,17,18,19,20,21]}. In ^[15], an adaptive fuzzy fixed-time fault-tolerant controller was designed by utilizing the inverse step control technique and the fixed-time stabilization theory. In ^[16], an adaptive output feedback compensation method is proposed by considering both actuator and sensor failure. Compared with ^[16] Authors in ^[17] relaxes the assumption requirement on the nonlinear functions, and proposes a new compensation mechanism to compensate for the impact of actuator failures by employing the approach of cubic absolute value Lyapunov functions and the novel $(\sigma, \sigma^f)$-modification.

On the contrary, the aforementioned research accomplishments rely on a traditional time-triggered mechanism (TTM). The fixed sample frequency and limitations in communication bandwidth associated with TTM lead to data redundancy in network signal transmission, thus occupying communication resources. Consequently, there has been a growing interest among scholars in exploring the event-triggered mechanism (ETM)^{[22,23,24,25,26,27,28,29,30]}. In ^[22], an event-triggered output feedback control method incorporating adaptive dynamic programming (ADP) with a state observer is proposed to reduce the signal transmissions on network channels, and thus control signals are updated only at specific moments where the triggering conditions are violated. In^[25], researchers construct an adaptive progressive tracking control strategy with a temporal control mechanism and an event triggering mechanism with variable thresholds, and comparative experiments indicate better robustness with the latter. In ^[27], the combination of command filtering backstepping and adaptive event-triggered communication (e.g., when a certain threshold is reached) solves the virtual controller complexity problem while effectively avoiding the waste of communication resources. While ^[29] is opposite to ^[27,29], the preset triggered condition is established by utilizing the negative semi-definiteness of the Lyapunov function's derivative, aiming to alleviate the communication burden and conserve computational resources.

In summary, with regard to a category of uncertain nonlinear MASs with actuator failures, an adaptive NNs event-triggered consensus control strategy incorporating the state observer is proposed to reduce the impact of the system by actuator failures. In comparison to existing findings, the main innovations of the proposed control method include:

(1) Different from the existing algorithms, such as ^[15,16,27] an event-triggered adaptive states observation consistency control strategy is proposed for a class of uncertain nonlinear MASs with actuator failures.

(2) With regard to the actuator failures that are common in the actual industrial production system, the existing compensation programs, such as ^{[17,18,19,20]}, need to occupy a large amount of communication resources. Thus, the design of an adaptive ETM to achieve dynamic compensation for actuator failures while reducing the update frequency of control signals, which reduces the occupation of communication resources, is more widespread in practical engineering.

(3) Due to the actual systems, the states of the system are often unmeasurable and the system is characterized by nonlinearities. Therefore, an adaptive NNs control method incorporating a state observer is proposed, which utilizes radial basis function neural networks (RBFNNs) to approximate MASs' unknown nonlinear functions, and the state observer is formulated for estimating the states of MASs.

The rest is presented in the following sections. Section 2 contains the problem description and preliminaries. Section 3 designs an adaptive NNs event-triggered consensus control scheme. Section 4 gives two simulation examples to verify the effectiveness of the designed scheme. Section 5 is the conclusion.

2.   Problem description and preliminaries

2.1.   Problem description

Consider the following category of uncertain nonlinear MASs ^[31]

where $X_{k, i} = [x_{k, 1}, x_{k, 2}, \ldots, x_{k, i}]^T$ and $X_{k, n} = [x_{k, 1}, x_{k, 2}, \ldots, x_{k, n}]^T$ are the state vectors, which are not measurable; $f_{k, i}(\cdot)$ is the unknown smooth nonlinear function; $u_{k}$ represents the control input, while $y_{k} \in [y_{k, 1}, y_{k, 2}, \ldots, y_{k, n}]^T$ denotes system output. The system consists of a leader labeled $G$ and $k(k = 1, 2, \ldots, N)$ followers.

Assume each agent is equipped with $b = 1, 2, \ldots, Z, (Z > 1)$ actuators, where $u_{k} = \sum_{b = 1}^{Z} l_{k, b} u_{k, b}$ and $l_{k, b}$ denotes the known constant control gain. During actual operation, agents will inevitably suffer from actuator failures. The actuator failure model of $b-th$ $(1, 2, \ldots, Z)$ is given as follows:

where the health factor is $\rho_{k, b} \in[0, 1]$; $c_{k, b}$ and $t_{bw}$ are unknown constants, $t_{bw}$ indicates the moment of the $b$th actual failure. The crucial working states of the actuator operation are categorized into three situations:

1) $\rho_{k, b} = 0$, $u_{k, b} = c_{k, b}$. The actuator is in a state of complete failure.

2) $\rho_{k, b}\in (0, 1)$, $u_{k, b} = \rho_{k, b} \bar u_{k, b}$. The partial failure will occur during actuator working.

3) $\rho_{k, b} = 1$, $u_{k, b} = \bar u_{k, b}$. The actuator works properly.

To facilitate the subsequent analysis, the work situations of the three actuators mentioned above are further summarized. Thus, two sets $L_{m1}$ and $L_{m2}$ are set and $L_{m1} \cup L_{m2} = \{1, 2, \ldots, Z\}$. Here, $L_{m1}$ indicates that the actuator is in a set of partial failure and normal operating states; $L_{m2}$ denotes the set in which the actuator is in a complete failure state.

Then, the MASs (2.1) is transformed into the following equation:

where

where $A_e$ is a strict Hurwitz matrix. Given a positive definite matrix $P = P^T > 0$, there exists a positive definite matrix $O = O^T > 0$, which satisfies $A_e ^T O +OA_e = -P$.

Assumption 1 (^[32]). The desired trajectory $y_{s} = \left[y_{s1}, y_{s2}, \ldots, y_{sn}\right] ^T \in R^n$ has the derivative of the $(n+1)-th$ order. There are constants $s_0 > 0$ and $s_1 > 0$ such that $s_1 > s_0$ and $\left| y_{s} \right| \leq s_0 \leq s_{1}$.

Assumption 2 (^[33]). During the operation, the system can stand $Z-1$ actuator failures at the same time to the greatest extent.

Lemma 1 (^[34]). For any $\upsilon$, $\varrho \in \mathcal{R}$, the following inequality holds for

2.2.   Graph theory

Through a directed graph $\mathcal{Q} (\mathcal{F}, \mathfrak{D})$, the communication topology among agents including leaders and followers is presented, in which $\mathcal{F} = \left\{G, 1, 2, \ldots, N\right\}$ represents the set of agents and $\mathfrak{D} \subseteq \mathcal{F} \times \mathcal{F}$ denotes the set of edges. $\mathbf{A} = [a_{kj}]_{N \times N}$ signifies the adjacency matrix of the graph $\mathcal{Q}$ between agents. If there is an edge $(j, k) \subseteq \mathfrak{D}$ (from agent $j$ to agent $k$), which signifies that agent $k$ can receive information from agent $j$, then $a_{kj} > 0$; contrarily, $a_{kj} = 0$. In addition, a matrix $\mathbf{D} = diag \{s_1, s_2, \ldotp, s_N\} \in R^{N \times N}$ is obtained, in which $s_i = \sum_{j = 1}^{N} a_{kj}$, $i = 1, 2, \ldots, N$, and a Laplacian matrix is $\mathbf{L} = \mathbf{D} - \mathbf{A} \in R^{N \times N}$.

2.3.   RBFNNs

By using the characteristic that NNs can arbitrarily approximate the unknown nonlinear function, the unknown nonlinear function of $Q(\theta)$ is processed, and there are

where $W(\theta) = [W_1(\theta), W_2(\theta), \ldots, W_n(\theta)]^T$ is expressed as the basis function vector and $\psi = [\psi _1, \psi _2, \ldots, \psi _n]^T$ signifies the ideal weight vector, in which $n > 0$ is the number of neural nodes; there exists a positive constant $\bar \varepsilon$ denoted as the approximation error such that $\left\lvert \varepsilon (\theta) \right\rvert \leq \bar \varepsilon$. Generally, the following basis function is selected as

where $i = 1, 2, \ldots, n$; $\kappa_i$ signifies the width, while $p_i = [p_{i1}, p_{i2}, \ldots, p_{in}]^T$ denotes the center of the Gaussian function.

Remark 1. $\psi \in R^\iota$ encompasses $\iota$ unknown constants, represented by an unknown constant value $\zeta$ for $|| \psi ||^2 = \zeta$. By updating the norm of $\psi$ instead of directly estimating it, thus, only the parameter $\zeta$ is estimated. This is done to minimize the estimation error $\tilde \zeta = \zeta - \hat \zeta$, thereby reducing computational load and simplifying controller design.

3.   Adaptive NNs event-triggered consensus control design

3.1.   State observer design

Since the states of the MASs are unmeasurable, the following state observer is designed for its estimation and its expression is

where $\hat X_{k, i} = \left[\hat x_{k, 1}, \hat x_{k, 2}, \cdots, \hat x_{k, i}\right]^T$ is the estimated value of $X_{k, i}$ and $\hat X_{k, n} = \left[\hat x_{k, 1}, \hat x_{k, 2}, \cdots, \hat x_{k, n}\right]^T$ is the estimated value of $X_{k, n}$. The observer error is $e_k = X_{k, n} - \hat X_{k, n}$, and its derivative is obtained as

Select the following Lyapunov function:

By taking the derivative of $V_{k, 0}$, one has

According to Young's inequality and $\sum_{i = 1}^{n} W_{k, 0}^T W_{k, 0} \leq 1$. Additionally, $\zeta$ is defined as $\zeta = max\left\{ n\left\lVert \psi _{0}\right\rVert^2, \right. \left. \left\lVert \psi _{i}\right\rVert^2, i = 1, \ldots, n \right\}$ and $n$ is the number of neural nodes. Therefore, there will be $\left\lVert \psi _{0}\right\rVert^2 \leq \zeta /n$, and $\left\lvert \varepsilon _{k, 0}\right\rvert \leq \bar \varepsilon _{k, 0}$ represents the approximate error, $\bar \varepsilon _{k, 0} > 0$.

Then, substitute (3.5) into (3.4) to get

where $\Delta = -2\left(\lambda _{max} (O_k)\right) ^2 + \lambda _{min}(P_k)$.

3.2.   Control design

The construction of virtual control laws within the backstepping control framework follows. Initially, the error system is expressed as

where $z_{k, 1}$ and $z_{k, n}$ represent a synchronization error of agent $k$ and a virtual error, respectively; $y_{s}$ is the leader's output signal; $\upsilon _k \geq 0$ denotes the weight vector associated with the edge from agent $k$ to leader $G$. In particular, when $\upsilon _k = 0$, it means that there is no direct information exchange between agent $k$ and leader $G$. Moreover, at least one agent will receive the synchronization information from leader $G$.

Step 1: The first Lyapunov function is constructed as follows:

where $r_{k, 1} > 0$ is a constant. $\tilde{\zeta} _{k, 1} = \zeta_{k, 1} - \hat \zeta_{k, 1}$ signifies the estimation error, and $\hat \zeta_{k, 1}$ represents the estimation of the uncertain parameter $\zeta_{k, 1}$. Then, from (2.1) and (3.7), the derivation of $z_{k, 1}$ is

According to (3.9), the $\dot V_{k, 1}$ has

Utilizing Young's inequality yields

Let $Q_{k, 1}(\theta_{k, 1}) = (\upsilon _k + s_k)f_{k, 1} + \frac{1}{4}z_{k, 1}(\upsilon _k + s_k)^2- \dot y_{s} \upsilon _k - \sum_{j = 1}^{N} a_{kj}(x_{j, 2}+ f_{j, 1})$, the unknown nonlinear function $Q_{k, 1}(\theta_{k, 1})$ is approximated by RBFNN, and it obtained that

where $\theta_{k, 1} = [\hat x_{k, 1}, \hat x_{j, 1}, \hat x_{j, 2}, \dot y_{s}]^T$. Utilizing Young's inequality yields

where $\zeta _{k, 1} = \left\lVert \psi _{k, 1} \right\rVert^2$, $\sigma_{k, 1}$ is a positive constant to be designed, and $\left\lvert \varepsilon _{k, 1}\right\rvert \leq \bar \varepsilon _{k, 1}$ represents the approximate error, $\bar \varepsilon _{k, 1} > 0$.

By incorporating (3.11)–(3.13) into (3.10) yields

$\alpha_{k, 1}$ and $\dot {\hat\zeta}_{k, 1}$ are designed as

where $\beta _{k, 1} > 0$ and $\xi _{k, 1} > 0$ are constants.

By incorporating (3.15)-(3.16) into (3.14) yields

where $\Delta_1 = \Delta-1$ and $M_1 = \left\lVert O_k \right\rVert^2 \left(\zeta_k + \bar \varepsilon ^2_{k, 0}\right)$.

Step i $(i = 2, 3, \ldots, n-1)$: The Lyapunov function is constructed as

where $\tilde{\zeta} _{k, i} = \zeta_{k, i} - \hat \zeta_{k, i}$ signifies the estimation error, and $\hat \zeta_{k, i}$ represents the estimation of the uncertain parameter $\zeta_{k, i}$. According to (3.7), one has

where

According to (3.19)-(3.20), one has

Utilizing the Young's inequality yields

If $i = 2$, let $Q_{k, i}(\theta_{k, i}) = h_i(y_k - \hat x_{k, 1}) - \sum_{q = 1}^{i-1} \frac{\partial \alpha_{k, i-1}}{\partial \hat x_{k, q}} (\hat x_{k, q+1} + h_q e_{k, 1})$ - $\sum_{q = 1}^{i-1} \frac{\partial \alpha_{k, i-1}}{\partial \hat \zeta _{k, q}} \dot {\hat \zeta} _{k, q} - \frac{\partial \alpha_{k, i-1}}{\partial x_{k, 1}} (\hat{x}_{k, 2} + f_{k, 1})$ - $\sum_{q = 1}^{i-1} \sum_{j = 1}^{N} \frac{\partial \alpha_{k, i-1}}{\partial \hat x_{j, q}} \dot x_{j, q}$ - $\sum_{q = 1}^{i} \frac{\partial \alpha_{k, i-1}}{\partial y_{s}^{(q-1)}} \dot y_{s}^{(q)}$ + $\frac{1}{4} z_{k, i}^2 \left(\frac{\partial \alpha_{k, i-1}}{\partial _{k, i}} \right)^2 + \frac{z_{k, i-1}^2}{2} + (\upsilon _k + s_k)z_{k, i-1}$. If $i \geq 3$, let $Q_{k, i}(\theta_{k, i}) = h_i(y_k - \hat x_{k, 1}) - \sum_{q = 1}^{i-1} \frac{\partial \alpha_{k, i-1}}{\partial \hat x_{k, q}}$ $(\hat x_{k, q+1} + h_q e_{k, 1}) - \sum_{q = 1}^{i-1} \frac{\partial \alpha_{k, i-1}}{\partial \hat \zeta _{k, q}} \dot {\hat \zeta} _{k, q} -$ $\frac{\partial \alpha_{k, i-1}}{\partial x_{k, 1}} (\hat{x}_{k, 2} + f_{k, 1}) - \sum_{q = 1}^{i-1}$ $\sum_{j = 1}^{N} \frac{\partial \alpha_{k, i-1}}{\partial \hat x_{j, q}} \dot x_{j, q} -$ $\sum_{q = 1}^{i} \frac{\partial \alpha_{k, i-1}}{\partial y_{s}^{(q-1)}} \dot y_{s}^{(q)} + \frac{1}{4} z_{k, i}^2 \left(\frac{\partial \alpha_{k, i-1}}{\partial _{k, i}} \right)^2 + \frac{z_{k, i-1}^2}{2} + z_{k, i-1}$. Then, the unknown nonlinear function $Q_{k, i}(\theta_{k, i})$ is approximated by an RBFNN, and it is obtained that

where $\theta_{k, i} = [\hat x_{k, i}^T, \hat x_{j, i}^T, \dot y_{s}]^T$. Applying Young's inequality yields

where $\zeta _{k, i} = \left\lVert \psi _{k, i} \right\rVert^2$, $\sigma_{k, i}$ is a positive constant to be designed, and $\left\lvert \varepsilon _{k, i}\right\rvert \leq \bar \varepsilon _{k, i}$ represents the approximate error, $\bar \varepsilon _{k, i} > 0$.

By incorporating (3.22)–(3.24) into (3.21) yields

$\alpha_{k, i}$ and $\dot {\hat\zeta}_{k, i}$ are designed as

where $\beta _{k, i}$ and $\xi _{k, i}$ are positive constants to be designed.

Further, by incorporating (3.26)-(3.27) into (3.25) yields

Step n: Construct the Lyapunov function as

where $r_{k, n}$ is a positive constant to be designed. $\tilde{\zeta} _{k, n} = \zeta_{k, n} - \hat \zeta_{k, n}$ signifies the estimation error, and $\hat \zeta_{k, n}$ represents the estimation of the uncertain parameter $\zeta_{k, n}$. $\Gamma _{k, b}^{-1}$ is the inverse matrix of $\Gamma _{k, b}$, which moreover denotes a positive definite matrix.

Since the $\dot z_{k, n} = \sum_{b = 1}^{Z} l_{k, b} \left(\bar u_{k, b} \rho_{k, b} + c_{k, b} \right) + h_n (y_k - \hat x_{k, 1}) - \dot \alpha_{k, n-1}$, it is the same as Step i.

The expression for $V_{k, n}$ after derivation is

Utilizing Young's inequality yields

Let $Q_{k, n}(\theta_{k, n}) = h_n(y_k - \hat x_{k, 1}) - \sum_{q = 1}^{n-1} \frac{\partial \alpha_{k, n-1}}{\partial \hat x_{k, q}} (\hat x_{k, q+1} + h_q e_{k, 1}) -$ $\sum_{q = 1}^{n-1} \frac{\partial \alpha_{k, n-1}}{\partial \hat \zeta _{k, q}} \dot {\hat \zeta} _{k, q} -$ $\frac{\partial \alpha_{k, n-1}}{\partial x_{k, 1}} (\hat{x}_{k, 2} + f_{k, 1}) - \sum_{q = 1}^{n-1} \sum_{j = 1}^{N} \frac{\partial \alpha_{k, n-1}}{\partial \hat x_{j, q}}$ $\dot x_{j, q} - \sum_{q = 1}^{n} \frac{\partial \alpha_{k, n-1}}{\partial y_{s}^{(q-1)}} \dot y_{s}^{(q)}$ $+ \frac{1}{4} z_{k, n}^2 \left(\frac{\partial \alpha_{k, n-1}}{\partial _{k, n}} \right)^2 + \frac{z_{k, n-1}^2}{2} + z_{k, n-1}$. Then, the unknown nonlinear function $Q_{k, n}(\theta_{k, n})$ is approximated by an RBFNN, and it is obtained that

where $\theta_{k, n} = [\hat x_{k, n}^T, \hat x_{j, n}^T, \dot y_{s}]^T$. Applying Young's inequality, one has

where $\zeta _{k, n} = \left\lVert \psi _{k, n} \right\rVert^2$, $\sigma_{k, n}$ is a positive constant to be designed, and $\left\lvert \varepsilon _{k, n}\right\rvert \leq \bar \varepsilon _{k, n}$ represents the approximate error, $\bar \varepsilon _{k, n} > 0$.

By incorporating (3.32)–(3.34) into (3.31) yields

The ETM of relative threshold is designed as follows

where $0 < \eta < 1$, $\lambda > 0$, and $\gamma > 0$ are design constants; $E_{k, b}(t) = \bar u_{k, b}(t)- \varpi_{k, b}(t)$, in which $\bar u_{k, b}(t)$ denotes the control signal and $\varpi_{k, b}(t)$ is the event-triggered control input; and $\gamma > \eta_1 / (1-\eta)$ and $m \in Z^+$. When the trigger condition $\left| E_{k, b}(t)\right| \geq \eta \left\lvert \bar u_{k, b}(t)\right\rvert + \eta_{1}$ holds true, $\bar u_{k, b} (t) = \varpi_{k, b} \left(t_{k, F}\right)$ is updated and keeps this value until the next event is triggered. $t_{k, F}$ denotes the moment when the event is triggered, $t_{k, F} > 0$, and $F \in Z^+$. The control law $\tilde{u}_{k, b}$ will be designed later.

For any $t \in\left[t_{k, F}, t_{k, F+1}\right)$, $\varpi_{k, b}(t) = \left(1+ \eta \phi _{1}(t)\right)\bar u_{k, b}(t) + \eta_1 \phi _{2}(t)$, in which $|\phi _{1}(t)| \le 1$ and $|\phi _{2}(t)| \le 1$. Thereby, it can be further articulated as

Based on the uncertain actuator model and ETM, the control law $\tilde{u}_{k, b}$ is constructed as follows

where $K_{k, b} = \left[K_{k, b11}, K_{k, b21}, \ldots, K_{k, b2Z} \right]^T$, $K_{k, b11} = \frac{1}{\sum_{b = 1, b \in L_{m1}}^{Z} |l_{k, b}|\rho_{k, b} }$, and $H = \left[\alpha_{k, n}, 1, \ldots, 1 \right]^T$. Especially, if $b \in L_{m2}$, one has $K_{k, b2Z} = - \frac{|l_{k, b}|c_{k, b}}{\sum_{b = 1, b \in L_{m1}}^{Z} |l_{k, b}|\rho_{k, b} }$. In contrast, if $b \in L_{m1}$, one has $K_{k, b2Z} = 0$.

Further, it can be inferred that

Remark 2. It is noteworthy that, from Equations (3.39)–(3.40), it can be observed that $\rho_{k, b}$ and $c_{k, b}$ are all unknown constants. Therefore, the value of $K_{k, b}$ cannot be directly measured. To achieve the feasibility of the intermediate control law $\tilde{u}_{k, b}$, the estimation $\hat K_{k, b}$ is utilized to estimate the value of $K_{k, b}$, resulting in an estimation error $\tilde{K}_{k, b} = K _{k, b}- \hat{K}_{k, b}$.

Then, $\tilde{u}_{k, b}$ is re-expressed as

According to (3.36), (3.38), and Lemma 1, it can be obtained that

By incorporating (3.38)–(3.42) into (3.35) yields

$\dot {\hat{\zeta}} _{k, n}$, $\dot {\hat K}_{k, b}$, and $\alpha_{k, n}$ are designed as

Substituting (3.44)–(3.46), one has

where $\Delta_{n} = \Delta_{n-1}-1$.

Utilizing Young's inequality yields

Further, combining (3.47)–(3.49) yields

where $\Omega = M_1 + \sum_{q = 1}^{n} \frac{1}{2r_{k, q}} \xi_{k, q} \zeta_{k, q}^2 +$ $\sum_{b = 1, b \in L_{m1}}^{Z} \frac{1}{2} |l_{k, b}| \rho_{k, b} g_{k, b} K_{k, b}^T K_{k, b}$ $+ 0.557 \sum_{b = 1, b \in L_{m1}}^{Z} |l_{k, b}| \rho_{k, b} \lambda + \sum_{q = 1}^{n}$ $\left(\frac{ \sigma_{k, q}^2 }{2} + \frac{ \bar \varepsilon _{k, q}^2 }{2} \right)$.

Ultimately, it can be given as follows

where $\mu_1 = \min \left\{\frac{2 \Delta_n}{\lambda_{max}(O_k)}\right\}$, $\mu_2 = \min \left\{ 2 \beta_{k, q} \right\}$, $\mu_3 = \min \left\{ 2 \beta_{k, n} -1 \right\}$, $\mu_4 = \min\left\{ \xi_{k, q} \right\}$, and $\mu_5 = \min \left\{\frac{g_{k, b}}{\lambda_{max} (\Gamma_{k, b}^{-1})}\right\}$. $\lambda_{max}(O_k^{-1})$ and $\lambda_{max} (\Gamma_{k, b}^{-1})$ are the maximum eigenvalues of $O_k^{-1}$ and $\Gamma_{k, b}^{-1}$, respectively.

Hence, it could be obtained that

where $\mu_6 = \min \left\{ \mu_1, \mu_2, \mu_3, \mu_4, \mu_5\right\}$. The proposed control method can be showed by Figure 1.

3.3.   Stability analysis

Theorem 1. Under the Lemma 1, Assumption 1 and Assumption 2, combining the MASs (2.1) with virtual control laws (3.15), (3.26), (3.46) and adaptive laws (3.16), (3.27), (3.44), (3.45) designed based on the ETM (3.36), (3.37) and the state observer (3.1), the following conditions are satisfied:

1) All signals within the closed-loop system remain bounded, with each agents' output following the trajectory of the virtual leader;

2) The occurrence of Zeno's phenomenon can be successfully prevented.

Proof: Multiplying both sides of Equation (3.52) by $e^{\mu_6 t}$ simultaneously yields

Then, solving the differential equation gives

Thus, $z_{k, 1}^2$ converges exponentially to the tight set $\Xi = \left\{ z_{k, 1} | z_{k, 1}^2 \leq \frac{2\Omega}{\mu_6} \right\}$ at the rate of $\mu_6$ and can be tuned by adjusting the design parameters for $\Xi$.

According to the tight set $\Xi$ it follows that the error signals are all bounded, and $y_s$ is bounded, and $\zeta_{k, 1}$ is a constant so that $x_{k, 1}$, $\tilde \zeta _{k, 1}$, and $e_{k}$ are also bounded. Further, the bound of (3.7), (3.15), (3.26), and (3.46) yields that $\alpha_{l, 1}$, and $x_{k, 2}$ are all bounded. Similarly, $x_{k, i}$, $\hat \zeta _{k, 1}$, $\hat x_{k, i}$ and $\alpha_{k, i}$ are all bounded. Moreover, because $\tilde K_{k, b} = K_{k, b}- \hat K_{k, b}$ and $\tilde K_{k, b}$ are bounded, $\hat K_{k, b}$ is also bounded. Thus, all signals within the closed-loop system are semi-global consistent and eventually bounded.

Based on the literature ^[35], defining the synchronization error vector $\Theta _{k} = [z_{1, 1}, z_{2, 1}, ..., z_{k, 1}]^T$, one has:

where $\bar \Theta_{k} = [\bar z_{1, 1}, \bar z_{2, 1}, ..., \bar z_{k, 1}]^T = Y_{k} - y_{s}$, $Y_{k} = [y_{1, 1}, y_{2, 1}, ..., y_{k, 1}]^T$, $y_{s} = [y_{s1}, y_{s2}, ..., y_{sn}]^T$, $\lambda _{max} \left(L+A \right)$ denotes $(L+A)$ the maximum eigenvalue. Thus, the tracking error of the MASs converges to the following set:

(3.28), as well as $t \in [t_{k, F}, t_{k, F+1})$, yields

Expressed by (3.38), $\dot \varpi _{k, b}$ remains continuously bounded with the existence of a constant $\bar \varpi _{k, b} > 0$ ensuring, $|\dot \varpi _{k, b}| < \bar \varpi _{k, b}$. Furthermore, it can be found that $lim_{t_{k, m} \rightarrow t_{k, m+1}} E_{k, b} (t) = \eta |\bar u_{k, b} (t)|+ \eta_1$. According to the Lagrange mean value theorem, it can be obtained that:

Since \(t_{k, F+1} - t_{k, F} \geq t^\diamond\), \(t^\diamond\) should be guaranteed to satisfy \(t^\diamond \geq \frac{\eta |\bar u_{k, b} (t)|+ \eta_1}{\bar \varpi _{k, b}}\). Obviously, the considered system can effectively avoid the Zeno behavior.

Remark 3. Zeno's phenomenon involves an infinite number of events occurring in a finite period of time, resulting in a system requiring an infinite update. This may aggravate the computational weight of the system, making real-time control unattainable, further leading to communication overload, and possibly even causing system crashes or performance degradation. Therefore, designing event-triggered control systems that exclude the Zeno phenomenon ensures that the system maintains computational feasibility, communication efficiency, and utility in long-term operation while maintaining the desired performance and reliability.

Remark 4. Further analysis reveals that $|z_{k, 1}| \leq \max \left\{ \left(2V_{k, n}(0)\right)^{\frac{1}{2}}, \left(\frac{2\Omega}{\mu_6}\right)^{\frac{1}{2}}\right\}$. It is evident that the set can be adjusted by selecting the parameters to be designed, such as $\beta_{k, q}$, $\beta_{k, n}$, $\xi_{k, q}$, $g_{k, b}$, and so on. Additionally, from Eqs (3.50)–(3.54), it can be observed that when $\rho_{k, b}$ remains constant, the residual set varies with the changes in $c_{k, b}$. Similarly, when $c_{k, b}$ remains constant, $\Omega$ increases with the rising of $\rho_{k, b}$, resulting in an expansion of the residual set of tracking errors. □

4.   Simulation example and analysis

This section aims to validate the control method's efficacy by conducting a comprehensive analysis of numerical and practical examples.

4.1.   Numerical example

Consider a MAS with actuator failures comprising a virtual leader and three follower agents. The communication topology is illustrated in Figure 2, and the model for the follower agents are detailed as

where $k = 1, 2, 3$ and selecting the nonlinear functions as $f_{k, 1} = 0.1(1+sin^2(x_{k, 1}))x_{k, 1}$ and $f_{k, 2} = 2.5x_{k, 2}+x_{k, 1}x^2_{k, 2}$; $u_{k, b} = \rho_{k, b} \bar u_{k, b} + c_{k, b}$, $l_{k, b} = 1(b = 1, 2)$ unknown constants and $u_{k, 1}$ and $u_{k, 2}$ are represent system input signals. $x_{k, 1}(0) = x_{k, 2}(0) = 0.1$ are initial states and the initial values of the state observer estimated are $\hat x_{k, 1}(0) = \hat x_{k, 2}(0) = 0.2$. The desired tracking signal is $y_s = sin(2t)$.

The basis function for RBFNNs is designed as follows:

where $i = 1, 2, \dots, 16$; the Gaussian function centered at ${\kappa_i}$ has a distribution interval in $[-1, 1]$.

Other correlated parameters to be designed as $\eta = 0.2$, $\lambda = 0.5$, $\eta_1 = 0.2$, $\gamma = \eta_1 /(1-\eta) + 0.01$, $\sigma_{k, 1} = \sigma_{k, 2} = r_{k, 1} = r_{k, 2} = \xi_{k, 1} = \xi_{k, 2} = 0.1$, $h_{k, 1} = 120$, $h_{k, 2} = 60^2$, $\beta_{k, 1} = 37$, $\beta_{k, 2} = 18$, $g_{k, b} = 0.3$, $\Gamma_{k, b} = [1, 0; 0, 1]$, $\hat K_{k, b}(0) = [0, 0, 0]^T$, and $\hat \zeta_{k, 1}(0) = \hat \zeta_{k, 2}(0) = 0$.

Consider the cases of two actuator failures, which are shown as follows. Case 1: Actuator 1 of each agent keeps operating normally, and Actuator 2 of each agent loses $30\%$ of performance after $t = 8 s$. Case 2: Each agent's Actuator 1 fails $20\%$ of the faults throughout the run and each agent's Actuator 2 fails completely.

The simulation results from Figure 3 to Figure 8 indicate that all signals are bounded. From Figures 3(a) and 4(a), it can be observed that when facing actuator failures, every agent's output can effectively track the output signal of the virtual leader, enabling MASs to achieve signal synchronization. Figures 4 and 7 show the values of observer state and real state for two cases, with small error values between them. In Figures 5 and 8, it is evident that whether the actuator partially fails or a complete failure occurs after 8 seconds of normal operation, the updated control signal will be delivered to the system upon satisfying the event-triggered condition. This serves as compensation for the actuator failures, demonstrating the better effectiveness of the proposed control method on the system. By observing Figure 3(c), it significantly exceeds the simulation time step of 0.01s, indicating the absence of the Zeno phenomenon. The values in Table 1 elucidate that the two cases conserve a maximum of $72.60\%$ and $77.60\%$ of communication resources, respectively. This illustrates that, in contrast to traditional TTM, ETM offers significant conservation of communication resources.

The designed control method is compared with the method in the literature ^[24] to compare its tracking performance in the uncertain nonlinear MASs with actuator failures. From Figures 3 and 4, it can be observed that the designed state observer is effective in estimating the system state and the tracking error of each agent converges quickly to the range of the $\pm 0.05$ error band within $0.2s$. The tracking performance is better than that of the literature ^[24], and there is no need to assume in advance that the system state is measurable at the time. Therefore, the designed control method is more general.

4.2.   Practical example

To assess the proposed control approach in the actual system effectiveness, reference ^[35] for the multi-single-link robotic arm system ($k = 1, 2, 3$)

where $f_{k, 2} = -\frac{1}{J}(B x_{k, 2} + mgl sin(x_{k, 1}))$ denotes the unknown smooth nonlinear functions; $u_{k, b} = \rho_{k, b} \bar u_{k, b} + c_{k, b}$, $l_{k, b} = 1(b = 1, 2)$ are unknown constants; and $u_{k, 1}$ and $u_{k, 2}$ represent system input signals. $x_{k, 1}(0) = x_{k, 2}(0) = 0.1$ are initial states, representing joint angle and angular velocity, respectively and the initial values of the state observer's estimated state are $\hat x_{k, 1}(0) = \hat x_{k, 2}(0) = 0.2$. The robotic arm system parameters are specified as follows: $J = 0.8$, $B = 1$ and $mgl = 10$. The desired tracking signal is $y_s = sin(2t)$.

The communication topology graph and the selection of the basis functions for the cases of actuator failures are the same as for the numerical example. The other relevant parameters to be designed are as follows: $\eta = 0.1$, $\lambda = 3$, $\eta_1 = 0.5$, $\gamma = \eta_1 /(1-\eta) + 0.001$, $\sigma_{k, 1} = \sigma_{k, 2} = r_{k, 1} = r_{k, 2} = 0.1$, $\xi_{k, 1} = \xi_{k, 2} = 0.5$, $h_{k, 1} = 112$, $h_{k, 2} = 56^2$, $\beta_{k, 1} = 44$, $\beta_{k, 2} = 16$, $g_{k, b} = 0.01$, $\Gamma_{k, b} = [1, 0; 0, 1]$, $\hat K_{k, b}(0) = [0, 0, 0]^T$ and $\hat \zeta_{k, 1}(0) = \hat \zeta_{k, 2}(0) = 0$.

Figure 9 to Figure 14 and Table 1 show the simulation results of the multi-single-link robotic arm system (4.3), which is analyzed similarly to the numerical simulation. By utilizing the proposed control method, the multi-single-link robotic arm system achieves the desired tracking performance and signal bound while synchronizing the signals in the case of actuator failures subjected to the two cases respectively. Furthermore, the nonexistence of the Zeno phenomenon is guaranteed.

5.   Conclusions

This research is targeted at a class of uncertain nonlinear MASs with actuator failures, whose control objective is to design an adaptive NNs event-triggered consensus control method with state observers to estimate unmeasurable states of the MASs, in order to realize the dynamic compensation of actuator failures while reducing the communication resources among the agents and avoiding Zeno phenomenon, and to make the all follower synchronization. Finally, the simulation results show the effectiveness of the control method. In future work, we will utilize the proposed method in combination with a fixed-time disturbance observer, speeding up the convergence of the system as well as directly measuring the disturbance signals, which is a valuable research work.

Author contributions

Kairui Chen: Conceptualization, methodology, investigation, writing - review and editing, validation, project administration; Yongping Du: methodology, Methodology, software, formal analysis, writing-review and editing, methodology; Shuyan Xia: Supervision, project administration, writing - review and editing, validation. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (62103115, 52075108), in part by the Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJZD-K202301203), in part by the Scientific and Technological Research Program of Wanzhou District(wzstc-20230309), in part by the Multi-dimensional Data Sensing and Intelligent Information Processing Key Laboratory Open Foundation (DWSJ2306), and in part by the Science and Technology Research Program of Guangzhou (2024404T9895, SL2023A04J00530).

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.