Neighbor event-triggered adaptive distributed control for multiagent systems with dead-zone inputs

1.   Introduction

The event-triggered approach has attracted considerable attention because it provides an alternative solution for updates or operations based on specific events or conditions, rather than relying on fixed time intervals. Originally, the event-triggered strategy was regarded as an optimal control problem ^[1], and there were studies investigating the input constraints problem of unknown nonlinear systems ^[2] or switched nonlinear systems ^[3]. These investigated problems involve the implementation of a nominal system and a discounted cost function, but ignore how to guarantee system performance simultaneously. In recent years, event triggering has emerged as a pivotal concept in the realm of control systems. The fundamental idea behind control system based on event triggering was to update control input only when specific triggering conditions were met, thereby optimizing the use of system resources ^[4]. The result performed how to keep balance between communication wastage and system performance, and this index has been noticed by more researchers ^[5]. For example, an adaptive neural network control scheme based on event-triggering ^[6] was developed for nonlinear systems characterized by sensor faults, input saturation, unknown control directions, and external disturbances. The study ^[7] proposed a method to set up periodically updated event-triggered conditions, leading the controller and parameter estimator to trigger events simultaneously. This research assures the asymptotic convergence of the calming error, effectively avoiding Zeno behavior. Separately, another study ^[8] introduced an adaptive event-triggered tracking approach based on a backstepping algorithm for a class of non-strict feedback nonlinear systems. This method combined intelligent technology for modeling unknown nonlinearities, and the event-triggering mechanism conserved communication resources through the transmission of discrete signals. Particularly, the results are further expanded to deal with multi-agent systems carrying with triggering control input ^[9]. For the multi-agent system with directed graphs, the research ^[10] introduced a new adaptive event-based protocol to tackle the consensus problem in linear quality. Furthermore, a novel dynamic control approach based on event triggering was introduced to address the consensus problem ^[11]. Further research ^[12] designed a new event-triggering mechanism suitable for the leader and an asynchronous edge event-triggering mechanism applicable to all edges. Combining with backstepping techniques, another article ^[13] proposed an adaptive scheme for fuzzy-based event-triggered containment control. The mentioned scheme is designed to update conditionally at specific sampling times for random nonlinear multi-agent systems with unidentified Bouc-Wen hysteresis inputs. All the approaches aim at conserving limited resources and reducing communication burdens. However, it is worth noting that nearly all the results have not accounted for bandwidth constraints associated with signal transmission between neighbor agents, let alone consider non-smooth actuator constraints additionally.

In the study of multi-agent systems, nonlinearities emerge as pivotal elements that profoundly influence system performance. Particularly, the occurrence of actuator dead zones are a pretty common phenomenon that cannot be overlooked. A dead zone is delineated by a specific region of input signal values where the actuator's output response is either zero or approaches zero. This phenomenon can lead to instability in the system, reducing control precision and production efficiency ^{[14,15,16,17]}. Therefore, research on actuator dead zones is essential for ensuring the stable operation and high performance of multi-agent systems. Some compensated methods have been used in the fields of industrial application. For example, the result in ^[18] proposed a new adaptive Fault Tolerant Control (FTC) method, in which the adaptive law estimated the effective gain and dead zone breakpoint of the actuator, effectively solving the dead zone phenomena in the robotic manipulator. Besides, the authors designed a compensator to mitigate the detrimental effects of dead zones on the control signal, successfully transforming the dead zone function into a linear form ^[19]. Additionally, another approach ^[20] proposed an innovative adaptive active FTC scheme specifically for wheeled robot systems. This scheme effectively compensated for actuator faults and dead zones, significantly enhancing the system's resistance to stability threats caused by dead zone phenomena. A novel distributed control strategy was advanced to effectively resolve complex game problems in the presence of actuator dead zones ^[21]. For intelligent control purpose, a fuzzy logic-based compensation method in ^[22] was set out to overcome the nonlinear challenges of actuator dead zones. This strategy avoids reliance on traditional dead zone inverse functions, instead using fuzzy rules to introduce offsets into the control input. The fuzzy-based scheme was further applied to handle the problem posed by input dead zones in nonlinear systems ^[23], or addressed the nonlinear estimation and compensation issues caused by quantization and dead zones ^[24]. Another intelligent approach, named the Radial Basis Function Neural Networks, was utilized in ^[25], where an adaptive consistency control strategy was introduced, accomplishing comprehensive compensation for both input dead zones and actuator faults. Actually, the challenges posed by dead zone nonlinearites and network communication resources have been investigated in ^[26]. This study proposed an efficient designated tracking performance control method for Steer-by-Wire systems. This method not only compensates for the impacts of actuator faults and dead zones, but also achieves resource savings in network communication between actuators. However, despite the insights gained from the review of the aforementioned references, it is observed that there is a notable absence of research that concurrently considers dead zone phenomena of non-smooth actuators and neighbor communication constraints for multi-agent systems.

To fill this gap, this paper concentrates on the distributed control problem for multi-agent systems, taking into account the constraints on neighbor communication bandwidth and generalized dead zone inputs. The main contributions of this study are outlined as follows.

1. The existing literature has scarcely addressed the constraints imposed on neighbor communication bandwidth in multi-agent systems. In this context, we propose a more nuanced approach to construct neighbor event-triggered mechanisms, which feature a more adaptable structure for relative thresholds.

2. A novel compensatory strategy is presented to address the coupling model posed by generalized dead zone inputs and event-triggered coefficients. Moreover, the potential Zeno behavior can be avoided.

3. The majority of studies focus on event-triggered strategies to make the sacrifice of system performance achieve higher communication rates, but in this paper, stable performance in terms of consensus tracking errors are guaranteed to asymptotically converge to zero, while transient performance is also established.

The outline of this paper is as follows. Section 2 introduces the prerequisite knowledge and systems modeling. Controller designed procedure and stability analysis are given in section 3. Simulation study is shown to verify the effectiveness of the proposed method in section 4. Finally, section 5 gives a conclusion of this paper.

2.   System modeling and background

2.1.   Graph theory

By combining the principles of algebraic graph theory, we can abstract the communication networks of multi-agent systems. In this abstraction, the agents are represented as nodes, and the communication links are represented as edges. Depending on the nature of communication, the graph can be classified into different types. If the communication between any two nodes is mutual, it is referred to as a bidirectional graph. On the other hand, if the communication is unidirectional, the graph is called a directed graph, or simply an undirected graph. For a directed communication graph with $M$ nodes, the set of all nodes is denoted as: $V = {v_{1}, v_{2}, ..., v_{M}}$. Each edge connecting two nodes $v_{i}$ and $v_{j}$ can be represented as $e_{i, j} = (v_{i}, v_{j})$. The collection of all edges is denoted as $E$. If node $v_{i}$ passes information to node $v_{j}$, we consider $v_{j}$ as a neighbor of $v_{i}$. Additionally, there may be information incoming from other nodes, forming the neighborhood set of $v_{i}$, which is also known as the in-degree of $v_{i}$. To account for the amplification or attenuation of information transmission, a weight coefficient $a_{ij}$ is assigned to each edge. The weight coefficients of all edges form an adjacency matrix, where if $(v_{i}, v_{j})\in E$, then $a_{ij} > 0$; otherwise, $a_{ij} = 0$ equals 0. A directed path from node $i$ to node $j$ is a sequence of nodes starting from $i$ and ending at $j$ such that each consecutive pair of nodes in the sequence corresponds to a directed edge in the graph.

2.2.   Multi-agents model

In this study, we introduce a class of nonlinear multi-agent systems that are preceded by dead zone inputs.

In this representation, the term $i = 1, 2, ..., m$ denotes the $i$th subagent, and $j = 1, 2, ..., n$ specifies the sequential order of each agent. The vector $X_{i, j} = [x_{i, 1}, ..., x_{i, j}]^{T}$, which belongs to $R^{j}$, is defined as the variable vector, while the output of the $i$th agent is stated as $y_{i}$. The nonlinear functions $\varphi_{i, 0}$ and $\varphi_{i, j}$ are known to be elements of $R$ and $R^{n}$, respectively, and are characterized by their smoothness and continuity. Conversely, the parameter vector $\theta_{i}$, also an element of $R^{n}$, remains unidentified and inaccessible. The input of the normal system corresponding to the $i$th subagent is signified by $u_{i}$, which is constrained by the dead zone nonlinearity expressed as $D_{i}(u_{i})$. As illustrated in Figure 1, it is important to note that the model under examination includes $m$th dead zone models, and the dead zone constraints are encapsulated within a generalized structure detailed as follows:

Here, $\rho_{i1}$ and $\rho_{i2}$ represent bounded functions defined by the derivatives of the outputs of the $i$th dead zone model at specific instants, where $\rho_{i1} = {\partial{d_{i, r}(u_{i})}\over\partial{u_{i}}}\big|_{u_{i}(t) = u_{i}(t_{i, r})}$ and $\rho_{i2} = {\partial{d_{i, l}(u_{i})}\over\partial{u_{i}}}\big|_{u_{i}(t) = u_{i}(t_{i, l})}$. For the purpose of designing the controller, it is requisite to transform the dead zone model into a decomposed form:

where control coefficients are defined by $\rho_{i}(u_{i})$ and bounded disturbance-like terms by $\epsilon_{i}(u_{i})$ as indicated below:

for these two terms, which satisfy that

in which the limits are yet-undetermined positive constants. The control objectives of the present study are delineated as follows:

(a) The objective is to ensure that the outputs from all agents, denoted by $y_{i}$, can synchronously track the output signal of the leader, $y_{r}$.

(b) Another goal is to guarantee that all closed-loop signals remain cooperatively semi-globally uniformly ultimately bounded (CSUUB). In order to achieve these objectives, we establish the following assumptions:

Assumption 1. The information transmission graph $G$ is fixed and connected. The output signal of the leader $y_{r}$ is sufficiently smooth, bounded, and continuous, while its first $n$th derivative is available for its neighbor agents.

Assumption 2. The functions $d_{i, r}(u_{i}), d_{i, l}(u_{i})$ are smooth, and there exist unknown positive constants $b_{i, r}, b_{i, l}$, such that $0 < \rho_{i, r}(t_{i, r}) < +\infty, \ \forall u_{i}\in[b_{i, r}, +\infty)$ and $0 < \rho_{i, l}(t_{i, l}) < +\infty, \ \forall u_{i}\in(-\infty, b_{i, l}]$.

2.3.   Neighbor event-triggered mechanism

In the domain of event-triggered control, two predominant methodologies are employed for designing triggering mechanisms: the fixed threshold strategy and the relative threshold strategy.

The fixed threshold strategy is predicated on the establishment of a constant threshold. The criterion for event initiation is the attainment or breach of this threshold by the state change of the system under control. Typically, this threshold is predetermined and remains invariant to fluctuations in the state of the system. Nonetheless, this strategy may be susceptible to detection lapses and false alarms when processing voluminous datasets. Such pitfalls occur particularly when there are substantial shifts in data patterns, rendering the preset thresholds unsuitable and leading to erroneous triggerings.

To mitigate these dilemmas, the relative threshold strategy is conceptualized. Distinct from its fixed counterpart, the relative threshold strategy adapts the triggering conditions dynamically in response to variations in the system state. Its core tenet is that if the system's state diverges from a set threshold by a specific margin for a given duration, an event trigger is deemed to have occurred, necessitating subsequent action. The usage of relative thresholds fortifies the system's stability and curtails superfluous energy expenditure and potential system volatility due to excessive event triggers.

Within the scope of this article, the relative threshold strategy is articulated through its application to inter-agent signal relay, particularly when the state variation surpasses a designated boundary. The neighbor control signal for the $i^{th}$ agent from its $j^{th}$ neighbor at time $t$ is defined as $\breve{u}_{j}(t)$, which complies with the following equation:

where $0 < \delta < 1$ and $m_{j} > 0$ are predetermined constants. The term $e_{t}(t)$ represents the error threshold between $u_{j}(t)$ and $\breve{u}_{j}(t)$. When $e_{t}(t)$ surpasses the relative threshold as defined by $\delta|u_{j}(t)|+m$, the control signal $\breve{u}_{j}(t)$ is propagated to the neighboring agent.This nuanced approach is instrumental in achieving a balance between sensitivity and stability in event-triggered control systems.

Remark 1. Contrary to the existing literature by constructing event-triggered mechanisms for single agents, this paper works on solving the challenge of neighbor communication bandwidth constraints, and develops a more flexible relative threshold structure, where the system model is shown by Figure 2.

3.   Distributed adaptive control design

In this section, we endeavor to amalgamate an event-triggering strategy with an adaptive control framework, refined by backstepping techniques, to fulfill our control objectives. The design process of the controller necessitates attention not only to the imperative of sustaining system stability but also to the judicious use of bandwidth during communication between adjacent agents.

3.1.   Controllers and adaptive laws design

The controller design unfolds through a recursive procedure, comprehensively described from step $1$ through step $n$. As a preliminary to the design process, we introduce the definition of consensus errors. These errors will serve as pivotal variables in the development of our control laws and will inform the calculation of feedback signals necessary to drive the system toward the desired consensus state.

where the connection coefficients $b_{i}, d_{i}$ and $a_{i, j}$ are all known, and $j\in N_{i}$ represents the neighbor agent of the $i$th agent. The variables are further given as

Step 1: Following (2.1), $\dot{z}_{i, 1}$ is shown as

where the first virtual controller $\alpha_{i, 1}$ is set by

where $\omega_{i, 1} = \varphi_{i, 1}^{T}(X_{i, 1})$ and $\omega_{j, 1} = \varphi_{j, 1}^{T}(X_{j, 1})$. The including coefficient $c_{i, 1}$ is a positive user-defined parameter, while $\hat{\theta}_{i}$ and $\hat{\theta}_{j}$ are the estimators of $\theta_{i}$ and $\theta_{j}$, respectively.

Set a Lyapunov candidate as

whose derivative is computed as

where the tuning functions $\tau_{i, 1}$ and $\tau_{ij, 1}$ are expressed by

Step 2: From the definition of (2.1) and (3.3), $\dot{z}_{2}$ can be expressed as

where

Set the virtual controller $\alpha_{i, 2}$ by

where $\Gamma_{1}$ and $\Gamma_{2}$ are user-defined symmetric positive matrices, and the tuning functions are given as

accompanied by $\omega_{i, 2} = \varphi_{i, 2}^{T}-{\partial{\alpha_{i, 1}}\over\partial{x_{i, 1}}}\varphi_{i, 1}^{T}$ and $\omega_{j, 2} = \sum\limits^{2}_{l = 1}{\partial{\alpha_{i, 1}}\over\partial{x_{j, l}}}\varphi_{j, l}^{T}$.

Choose the Lyapunov candidate $V_{i, 2}$ as

With the previous definitions, one has

Step l (l = 3, ..., n-1): The following steps can be summarized as

where

The $l$th virtual controller is designed as

where

with $\omega_{i, l} = \varphi_{i, l}^{T}-\sum\limits_{k = 1}^{l-1}{\partial{\alpha_{i, l-1}}\over\partial{x_{i, k}}}\varphi_{i, k}^{T}$ and $\omega_{j, l} = \sum\limits^{l}_{k = 1}{\partial{\alpha_{i, l-1}}\over\partial{x_{j, k}}}\varphi_{j, k}^{T}$.

The Lyapunov candidate is set as

then its derivative is expressed by

Step $n$: In this final step, the event-triggered mechanisms, consensus controllers and adaptive laws will be designed, respectively.

The neighbor control signals $\breve{u}_{j}$ are transmitted if the designed events are triggered. As discussed in the result ^[27], $\breve{u}_{j}(t)$ is able to be rewritten as

where $\lambda_{1}(t), \lambda_{2}(t)$ are continuous parameters, satisfying $\lambda_{*k}(t) = 0, \lambda_{*k+1}(t) = \pm1$ and $|\lambda_{*}(t)|\leq 1$, and $t_{k}(k\in Z^{+})$ is the controller update time. The neighbor control signals contain the control coefficient terms and the unknown conventional constant terms. The virtual controller $\alpha_{i, n}$ is designed as

where

with $\omega_{i, n} = \varphi_{i, n}^{T}-\sum\limits_{k = 1}^{n-1}{\partial{\alpha_{i, n-1}}\over\partial{x_{i, k}}}\varphi_{i, k}^{T}$ and $\omega_{j, n} = \sum\limits^{n}_{k = 1}{\partial{\alpha_{i, n-1}}\over\partial{x_{j, k}}}\varphi_{j, k}^{T}$.

The final controller $u_{i}$ is designed as

where $\hat{\varrho}_{i} = [\hat{\varrho}_{i1}, \hat{\varrho}_{i2}, \hat{\varrho}_{i3}, \hat{\varrho}_{i4}]^{T}$ is the estimate of $\varrho_{i} = [\frac{1}{\underline{\rho}_{i}}, \frac{\bar{\epsilon}_{i}}{\underline{\rho}_{i}}, \frac{1}{\underline{\rho}_{i}(1-\delta)}, \frac{1}{\underline{\rho}_{i}(1-\delta)}]^{T}$, accompanied by the update laws

where $\Gamma_{\varrho}$ is a user defined symmetric positive matrix. It is noted that $\dot{\hat{\varrho}}_{i}(t)\geq0$ can be held. The term $\nu_{i} = [\nu_{i1}, \nu_{i2}, \nu_{i3}, \nu_{i4}]^{T}$ is defined as

It is worthy to point out that $z_{i, n}\hat{\varrho}_{i}^{T}\nu_{i}\leq0$ can be always held if the initial values are chosen by $\hat{\varrho}_{i}(0) > 0$ for $i = 1, 2, 3, 4$. The update laws for $\hat{\theta}_{i}$ and $\hat{\theta}_{j}$ are given as

3.2.   Stability analysis

Theorem 1. Consider the nonlinear MASs (2.1) with generated dead zone inputs (2.3), consisting of neighbor event-triggered mechanisms (2.8), consensus controllers (3.24) and adaptive update laws (3.27)–(3.28), under Assumptions 1–2. With any initial condition $V(0) < C$, the considered systems satisfy these characteristics:

a. All the closed-loop signals of the MASs are CSUUB.

b. The consensus errors asymptotically converge to zero.

c. Zeno behavior is able to be avoided.

Proof. Set the final Lyapunov function as

Via derivation operation, $\dot{V}_{i, n}$ can be expressed as

Applying (3.21) and (3.24) to the above equation, the including terms can be expressed as

Remark 2. As shown in Eq (3.31), the coupling model, which emerges from neighbor event-triggered mechanisms combined with decomposition dead zone models, can be classified into three components: the control coefficients term, the bounded disturbance-like term and the neighbor control signal term. To effectively handle this coupling model, the controller design necessitates the incorporation of multiple polynomial compensators, i.e., $u_{i} = \hat{\varrho}_{i1}\nu_{i1}+\hat{\varrho}_{i2}\nu_{i2}+\hat{\varrho}_{i3}\nu_{i3}+\hat{\varrho}_{i4}\nu_{i4}$.

With the property of $0\leq |z|-\frac{z^{2}}{\sqrt{z^{2}+\varepsilon^{2}}}\leq \varepsilon$, $\dot{V}_{i, n}$ is rewritten by

With the aid of update laws in (3.27)–(3.28), the derivative of final Lyapunov function $V$ is expressed by

where $\Delta = m(\varepsilon_{1}+\bar{\epsilon}_{i}\varepsilon_{2}+\frac{\varepsilon_{3}}{1-\delta}+\frac{\varepsilon_{4}}{1-\delta})$. Via integration operation, the above inequation can be computed as

This implies that $V(0)$ and $V(t)$ are bounded, accompanied by $z_{i, k}, \tilde{\theta}_{i}, \tilde{\theta}_{j}, \tilde{\varrho}_{i}\in L_{\infty}$. Based on the predefinition, $y_{r}, \theta_{i}, \theta_{j}, \varrho_{i}$ are ensured bounded, then it is easy to obtain that $x_{i, 1}, x_{j, 1}, \hat{\theta}_{i}, \hat{\theta}_{j}, \hat{\varrho}_{i}\in L_{\infty}$. Since the continuous function $\varphi_{i, j}$ and the virtual controller $\alpha_{i, 1}$ are bounded defined, $x_{i, 2}$ is also bounded. Subsequently, all the variables $x_{ij}$ can be proved bounded and the neighbor control signals $\breve{u}_{j}$ can be ensured bounded.

Based on the Barbalat's Lemma ^[28], the consensus tracking errors can be guaranteed to asymptotically converge to zero with the conditions of $z_{i, 1}(t)\in\mathrm{£}_{2}$ and $\dot{z}_{i, 1}(t)\in\mathrm{£}_{\infty}$, i.e., $\lim_{t\longrightarrow\infty}z_{i, 1}(t) = 0$. The bound for the $L_{2}$-norm of the tracking error can be expressed as

The Zeno phenomenon, will lead to physical unrealization, i.e., the preset events may infinitely be triggered in finite time. To prove that the established event-triggered mechanisms are able to avoid this issue, a lower bound of the inter-execution time $t_{*} > 0$ satisfying the condition of $\{t_{h+1}-t_{h}\}$ is required. Actually, $e_{t, k}(t) = u_{j, k}(t)-\breve{u}_{j, k}(t), \forall t \in [t_{h}, t_{h+1})$ is bounded defined, so

The function $\dot{u}_{i, 1}$ contains the variable of $\alpha_{i, 1}$, therefore $\dot{u}_{i, 1}\in L_{2}$, i.e., $|\dot{u}_{i, 1}|\leq \zeta$, where $\zeta$ is a positive constant. Give $e_{t}(t_{h}) = 0$, which yields that $\lim\limits_{t\to t_{h+1}}e_{t}(t) = |\breve{u}_{j, k}(t)|+b$. The following inequation is finally given as

This means that the lower bound of the inter-execution time is successfully found, i.e., $t^{*}\geq (|\breve{u}_{j}(t)|+b)/\zeta$. That is to say, all the events can avoid triggering infinite times within a finite time interval.

Remark 3. Most studies on event-triggered strategies sacrifice system performance to increase communication rates. However, based on Theorem 1, it is observed that our method uniquely maintains high communication rates while ensuring stable system performance, accompanied with consensus tracking errors converging to zero asymptotically. Additionally, with the guidance of the established transient performance inequation (3.35), a controlled system can quickly and efficiently adapt to changes.

4.   Simulation study

The section works on verifying feasibility of the proposed method. As discussed in ^[29], the four continuous torsional pendulum models are employed for our studies, which can be expressed as

where $\beta_{i}$ and $\omega_{i}$ are treated as the angle and angular velocity for $i$th link, respectively. $M_{i}, l_{i}$ and $J_{i}$ represent the mass, the length and the rotary inertia of $i$th pendulum, while the coefficients $g_{i}, f_{d, i}$ are the acceleration of gravity and the frictional factor, respectively. All these parameters are unknown and unavailable. The motor torques of the systems contain dead zone nonlinearity, i.e., $D_{i}(u_{i})$. By giving the definition of $x_{i, 1} = \beta_{i}$ and $x_{i, 2} = \omega_{i}$, one has the following parametric strict-feedback form.

where $\theta_{i} = [\frac{M_{i}g_{i}l_{i}}{J_{i}}, \frac{f_{d, i}}{J_{i}}]^{T}$ is remaining to estimate. The physical system model and the communication topology can be graphically represented in Figure 3. From the connection graph of 4 subsystems, the values are preset as $b_{i} = 1$ and $a_{i, j} = 1$ only when two neighbor agents are connected, or $b_{i} = 0$ and $a_{i, j} = 0$. The leader's signal is set by $y_{r} = \sin(t)$ and the dead zone parameters are given as $\rho_{i, r} = 2, \rho_{i, l} = 2, b_{ir} = 0.1, b_{il} = -0.2$. The event-triggered neighbor control signals $\breve{u}_{j}(t)$ are defined as (2.8), where the relative threshold is set as $\delta|u_{j}(t)|+m_{j}, \delta = 0.1, m_{j} = 0.01$. Via the operator of initialization with $x_{1, 1}(0) = 0.1, x_{1, 2}(0) = 0.5, x_{2, 1}(0) = x_{3, 1}(0) = x_{4, 1}(0) = 0.8, x_{2, 2}(0) = x_{3, 2}(0) = x_{4, 2}(0) = 0, y_{r}(0) = 0,$ $u_{i}(0) = 0.1, \hat{\theta}_{i}(0) = [0, 0]^{T}, \hat{\varrho}_{i}(0) = [0,0,0,0]^{T}$, and the operator of selecting proper user-defined parameters as $c_{i, j} = 2, \Gamma_{1} = \Gamma_{2} = 5.2, \varepsilon_{i} = 0.001$, for $i = 1, 2, 3, 4, j = 1, 2$, the simulation results of this method can be seen from Figure 4.

The results presented in Figure 4 illustrate the following points:

● The outputs of all the subagents, represented by $y_{i}$, can successfully track the leader's trajectory, denoted as $y_{r}$, despite the presence of dead zone nonlinearities in the motor actuators.

● The consensus errors, $e_{i}$, are guaranteed to asymptotically converge to zero.

● The neighbor control signals, $\breve{u}_{i}$, are discretized using the jointly designed event-triggered mechanisms.

● Each of the triggering mechanisms is configured to avoid Zeno behavior.

Additionally, Figure 5 demonstrates the implementation of the neighbor event-triggered mechanisms that utilize relative thresholds.

5.   Conclusions

The distributed control problem for multi-agent systems with neighboring communication bandwidth constraints and dead zone inputs is addressed in this paper. The aforementioned bandwidth constraints are managed using the proposed event-triggered mechanisms, which are applicable to the control signals exchanged among neighbors and feature more adaptable relative threshold configurations. Moreover, the decomposition of the dead zone nonlinearity, when coupled with event-triggered coefficients, is effectively neutralized by the specially designed consensus controllers. The proposed methodology ensures closed-loop signal ultimate boundedness for all closed-loop signals, guarantees that consensus errors asymptotically approach zero, and precludes the possibility of Zeno behavior. Simulation results corroborate the effectiveness of our approach.

Use of AI tools declaration

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

Acknowledgments

This research was funded by the National Natural Science Foundation of China [No.621002153], and the Guangzhou Municipal Science and Technology Project [No.202201010381].

Conflict of interest

The authors declare that they have no conflict of interest.