Fixed-time event-triggered control strategy for multi-group multi-agent average consensus

Abstract

This paper investigates distributed multi-group average consensus in a leader–follower framework using a robust fixed-time event-triggered control (FxTETC) approach. Unlike conventional consensus problems, agents are partitioned into multiple groups, achieving intra-group synchronization while tracking distinct inter-group trajectories. A complex number-based communication topology is introduced to facilitate intra-group coordination while preserving inter-group formation differences. To achieve fixed-time consensus under model uncertainties and external disturbances, fully distributed event-triggered sliding mode controllers (SMC) are designed. Distributed sliding manifolds establish consensus tracking, and a fixed-time event-triggering mechanism reduces unnecessary communications. The proposed distributed fixed-time control (FxTC) law requires no global network information, ensuring scalability and robustness. It guarantees convergence of all agent states to their group average consensus within a predetermined fixed time and provides a strictly positive lower bound on inter-event intervals to eliminate Zeno behavior. Numerical simulations validate the efficiency of the approach in achieving both intra-group synchronization and inter-group trajectory tracking over directed network topologies.

Introduction

There has been growing interest in cooperative control of networked multi-agent systems(MAS), driven by its applicability to various domains including spacecraft, robotic teams, vehicular networks, and smart power grids.^1–4 A key challenge in such systems is to achieve consensus, ensuring that all agents reach an agreement on certain states or trajectories. ⁵ To address this, extensive research has explored distributed consensus protocols for both linear and nonlinear multi-agent systems (MASs), leading to the development of adaptive and robust distributed control strategies. ⁶

Among these, fixed-time (FxT) control has become a strong framework because it guarantees convergence within a settling time that does not depend on initial conditions. This is different from standard finite-time methods, where settling time relies on initial states.^7–9 Researchers have studied FxT consensus in various MAS configurations, including high-order and heterogeneous nonlinear systems. Recently, bipartite and multipartite consensus models have been introduced. These models expand on classical consensus by allowing group agreement under opposing or structured interactions.^10–12 Bipartite consensus makes sure that two groups converge to equal but opposite values. Multipartite consensus generalizes this idea, enabling multiple groups to reach distinct yet related values. These models have proven helpful in clustering, formation control, and cooperative behaviors.^3,10–13

At the same time, disturbances and uncertainties are unavoidable in real-world MAS applications. This drives research on consensus strategies that can handle external disruptions. ¹⁴ Despite these advances, much of the existing literature relies on time-triggered control, which needs regular communication and often creates significant communication overhead.^15,16 To address this, event-triggered control (ETC) has emerged as a promising alternative. In ETC, communication occurs only when it’s necessary.^17,18 Various ETC schemes have been proposed, including fully distributed adaptive designs,^19,20 leader–follower consensus approaches, ¹⁷ dynamic and self-triggered strategies, ²¹ and event-based broadcasting protocols. ²² These strategies effectively reduce unnecessary transmissions by adapting triggering intervals to system dynamics. However, many earlier works relied on global information—such as the smallest positive eigenvalue of the Laplacian matrix—thereby limiting their fully distributed nature. ²³ Recent studies have further explored robust and scalable ETC designs for MASs, including cooperative output regulation, ²⁴ heterogeneous and homogeneous agent tracking, ²⁵ and FxT convergence protocols. ²⁶ Leveraging these recent developments, we propose a distributed, FxTETC framework designed for multi-group MAS. This framework achieves both intra-group synchronization and inter-group trajectory tracking, all while minimizing communication load and ensuring robustness. This study presents unique ETC distributed multi-group consensus algorithms to overcome the drawbacks of conventional time-triggered control and to fully utilize the advantages of FxT and ETC.^27,28 These tactics employ a leader–follower architecture to synchronize agents both within and across groups with the least amount of communication overhead. Some recent studies have explored event-triggered strategies to enhance communication efficiency and system performance in MAS. In, ²⁶ consensus convergence is optimized while minimizing data transmissions. The work in ²⁹ addresses predictor design for nonlinear MIMO systems with large communication delays by proposing a periodic event-triggered mechanism with a cascade predictor to ensure exponential error convergence. The work in ³⁰ presents an observer-based event-triggered formation control scheme specifically designed for linear MAS experiencing distributed infinite delays, achieving time-varying formation with reduced communication costs and guaranteed Zeno-free behavior.

In addition, SMC has proven to be a reliable strategy for achieving robust control of MAS when subject to external disturbances. ³¹ Several studies have explored SMC-based consensus control, addressing challenges such as robustness to uncertainties and disturbances. ²⁷ Recently, FxT control has also emerged as a novel approach to address stability and convergence issues in MASs. ³² Unlike traditional finite-time control,^33,34 FxT control offers stability guarantees irrespective of initial conditions, prompting research into FxT consensus across various MAS configurations and dynamics.^32,35,36

To mitigate the challenges arising from fixed-interval (time-triggered) control mechanisms while harnessing the strengths of FxT and ETC in multi-group MASs, this paper proposes event-triggered distributed multi-party consensus strategies. These strategies utilize a leader–follower architecture and FxT control to synchronize agents within and across different parties while minimizing communication overhead. The proposed approaches aim to strike a balance between reducing data transmission frequency and ensuring FxT convergence for nonlinear agent dynamics.

The primary contributions of this work are summarized as follows:

1. We propose FxTETC strategy for multi-group MAS, extending cooperative control beyond prior works such as,^12,37 which study multi-group consensus without FxTETC mechanisms. The proposed approach ensures FxT convergence while significantly reducing communication load.

2. Unlike, ²⁷ which considers single-group consensus using SMC protocols, this work introduces a FxTETC framework for multi-group leader–follower consensus. The approach ensures intra-group synchronization while preserving inter-group separation, thereby enabling structured formation and coordination.

3. The proposed strategy exhibits resilience to uncertainties in the model and exogenous disturbances, and it scales effectively through a fully distributed design. These features collectively mitigate critical challenges of communication overhead, resilience, and network scalability in distributed consensus control.

The effectiveness of the proposed FxTETC scheme is established through rigorous theoretical analysis and validated via comprehensive numerical simulations.

Preliminary mathematical concepts and notations

• Notation: The set complex numbers is denoted by $\mathbb{C}$ , and

$${\mathbb{C}}_{+}=\{x\in \mathbb{C}\mid \mathrm{Re}\left(x\right)>0\}.$$

For any vector $x\in {\mathbb{C}}^n$ (or ${\mathbb{R}}^n$ ) and scalar a > 0, the generalized power operator is defined as

$$\lfloor x{\rceil }^a:={\left|x\right|}^a\mathrm{sign}\left(x\right)$$

• Lemma 1. (Adapted from ⁸ ) Consider the nonlinear system

$$\stackrel{·}{z}=g\left(z\right),$$

where the vector field g(z) exhibits homogeneity in the bi-limit. Specifically, its behavior near the origin and at infinity is described by the homogeneity triples $(r_a,{\lambda }_a,g_a)$ and $(r_b,{\lambda }_b,g_b)$ , respectively. Assume that the origin is globally asymptotically stable (GAS) for the system $\stackrel{·}{z}=g\left(z\right)$ as well as for its associated limiting systems $\stackrel{·}{z}=g_a\left(z\right)$ and $\stackrel{·}{z}=g_b\left(z\right)$ . Under these conditions, the following results hold:

1. If the degree of homogeneity at infinity λ _b is positive and the degree near the origin λ _a is negative (i.e. ${\lambda }_b>0>{\lambda }_a$ ), then the equilibrium at the origin is FxT stable.

2. Suppose there exist constants δ _a and δ _b such that ${\delta }_a>\mathrm{ma}{x}_{1\le j\le n}{r}_{a,j}$ and ${\delta }_b>\mathrm{ma}{x}_{1\le j\le n}{r}_{b,j}$ . Then there exists a Lyapunov function W(z), its partial derivatives $\frac{\partial W}{\partial z_j}$ are homogeneous in the bi-limit, associated with the homogeneity triples $(r_a,{\delta }_a-r_{a,j},\frac{\partial W_a}{\partial z_j})$ and $(r_b,{\delta }_b-r_{b,j},\frac{\partial W_b}{\partial z_j})$ , respectively. Additionally, the Lie derivative $\frac{\partial W}{\partial z}g\left(z\right)$ is negative definite.

Graph theory

Let a directed graph denoted as $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{L})$ , where:

• $\mathcal{V}$ represents the set of nodes (or agents) in the network, with cardinality $\mathcal{S}$ ,

• $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ denotes the set of directed edges that define the information exchange among agents, and

• $\mathcal{L}=[{\mathcal{L}}_{i,j}]\in {\mathbb{C}}^{\mathcal{S}\times \mathcal{S}}$ is the adjacency (or weighted connection) matrix, whose complex-valued elements quantify the communication strength between nodes.

For each node $v_i\in \mathcal{V}$ , the set of its incoming neighbors is defined as

$${\mathcal{S}}_i=\{v_j\in \mathcal{V}\mid (v_j,v_i)\in \mathcal{E}\},$$

which includes all nodes v _j that have directed edges pointing to v _i .

Multi party consensus

Definition (Multi-Partite Consensus): Let $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{L})$ denote the original communication topology, where $\mathcal{V}$ is the set of nodes (agents), $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is the set of directed edges, and $\mathcal{L}$ is the associated complex-weighted adjacency matrix.

An augmented graph $\overline{\mathcal{G}}=(\overline{\mathcal{V}},\overline{\mathcal{E}},\overline{\mathcal{L}})$ is constructed by adding a leader node, denoted as ${\mathcal{V}}_{\mathcal{S}+1}$ , to the original graph. Thus, the set of nodes in the augmented graph becomes

$$\overline{\mathcal{V}}=\mathcal{V}\cup \{{\mathcal{V}}_{\mathcal{S}+1}\}.$$

The nodes in $\overline{\mathcal{V}}$ are partitioned into T disjoint groups G _f such that:

$$\bigcup _{f=1}^T{G}_f=\overline{\mathcal{V}},G_f\cap G_l=\varnothing \mathrm{for}f\ne l.$$

The collection of group partitions is denoted by

$$G:=\{G_f\subseteq \overline{\mathcal{V}}|f=1,\dots ,T\},$$

where each $G_f\in G$ represents a distinct group of agents in the augmented graph.

A sequence of alternating nodes $i_m\in \mathcal{V}$ , for $m=1,\dots ,n$ , such that either $(i_m,i_{m+1})\in \mathcal{E}$ or $(i_{m+1},i_m)\in \mathcal{E}$ , is called a weak path $i_1,i_2,\dots ,i_{n+1}$ . If $(i_m,i_{m+1})\in \mathcal{E}$ for all $m=1,\dots ,n$ , the path is said to be directed.

A weak path becomes a weak cycle if $i_1=i_{n+1}$ ; similarly, a directed cycle requires all edges to follow the direction consistently. A directed graph $\mathcal{G}$ is called strongly connected if, for any two distinct vertices $v_i,v_j\in \mathcal{V}$ , there is at least one directed path leading from v _i to v _j . A cycle formed by a sequence of nodes $i_1,i_2,\dots ,i_n,i_1$ is said to be consistent if the following condition holds:

$$\underset{m=1}{\overset{n}{\Pi }}{w}_m=1,$$

where

$$w_m=\{\begin{array}{c}{\mathcal{L}}_{i_{m+1},i_m},\mathrm{if}(i_m,i_{m+1})\in \mathcal{E},\hfill \\ \multicolumn{1}{c}{{\mathcal{L}}_{i_m,i_{m+1}}^{-1},\mathrm{if}(i_{m+1},i_m)\in \mathcal{E}.}\end{array}$$

If every weak cycle in the graph is consistent, the graph is said to be consistent; otherwise, it is called inconsistent.^11,12

A spanning tree of the graph is selected, and each node is assigned a unique label in ascending numerical order. For example, assign the root as node 1, and label its child nodes as 2, 3, etc. Define:

$${\mathcal{P}}_1=1.$$

Let $i\in \{2,\dots ,\mathcal{S}\}$ , and denote its parent (ancestor) in the tree as $î$ . Then define:

$${\mathcal{P}}_i={\mathcal{P}}_{\hat{i}}{\mathcal{L}}_{i,\hat{i}}^{-1}.$$

This is equivalent to the following identity:

$${\mathcal{P}}_i{\mathcal{L}}_{i,\hat{i}}{\mathcal{P}}_{\hat{i}}^{-1}=1.$$

(1)

This construction ensures that all edges in the spanning tree satisfy the consistency condition. Due to the consistency of the graph, this also extends to any other arcs not in the spanning tree.

Consider now two nodes m₁ and m₂ whose common ancestor is node n, as shown in Figure 1. Assume that there are $n_1+1$ nodes along the path from n to m₁, labeled $n,n+1,\dots ,n+n_1$ , and $n_2+1$ nodes along the path from n to m₂, labeled $n+n_1+1,\dots ,n+n_1+n_2$ . Then:

$$\begin{array}{c}{\mathcal{P}}_{m_1}={\mathcal{P}}_n\underset{i=n}{\overset{n+n_1}{\Pi }}{\mathcal{L}}_{i+1,i}^{-1},\hfill \\ \multicolumn{1}{c}{{\mathcal{P}}_{m_2}={\mathcal{P}}_n{\mathcal{L}}_{n+n_1+1,n}^{-1}\underset{j=n+n_1+1}{\overset{n+n_1+n_2}{\Pi }}{\mathcal{L}}_{j+1,j}^{-1}.}\end{array}$$

(2)

Figure 1.

Tree-based construction of ${\mathcal{P}}_i$ and consistency verification via cross-edge $(m_2\to m_1)$ .

Assume there exists an edge from m₂ to m₁. Then:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathcal{P}}_{m_1}{\mathcal{L}}_{m_1,m_2}{\mathcal{P}}_{m_2}^{-1}={\mathcal{P}}_n\left(\underset{i=n}{\overset{n+n_1}{\Pi }}{\mathcal{L}}_{i+1,i}^{-1}\right){\mathcal{L}}_{m_1,m_2}{\mathcal{P}}_n^{-1}{\mathcal{L}}_{n+n_1}}\\ \multicolumn{1}{c}{+1,n\left(\underset{j=n+n_1+1}{\overset{n+n_1+n_2}{\Pi }}{\mathcal{L}}_{j+1,j}\right).}\end{array}$$

(3)

Since the graph is consistent, we have:

$$\left(\underset{i=n}{\overset{n+n_1}{\Pi }}{\mathcal{L}}_{i+1,i}^{-1}\right){\mathcal{L}}_{m_1,m_2}{\mathcal{L}}_{n+n_1+1,n}\left(\underset{j=n+n_1+1}{\overset{n+n_1+n_2}{\Pi }}{\mathcal{L}}_{j+1,j}\right)=1,$$

(4)

which, together with the previous identity, implies:

$${\mathcal{P}}_{m_1}{\mathcal{L}}_{m_1,m_2}{\mathcal{P}}_{m_2}^{-1}=1.$$

(5)

This completes the proof.^3,12

Problem formulation

This section addresses the problem of multi group multi agent consensus. Consider a set of $\mathcal{S}$ follower agents and a single leader (administrator), resulting in a total of $\mathcal{S}+1$ agents. The leader has access to the position information of all $\mathcal{S}$ follower agents. In line with the multi-party consensus objective, each follower agent updates its state based on the leader’s influence and coordination with neighboring agents.

The motion of each follower agent is described by

$$\begin{array}{c}{\stackrel{·}{p}}_i\left(t\right)=q_i\left(t\right),\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{q}}_i\left(t\right)=f_i(p_i\left(t\right),q_i\left(t\right),t)+u_i\left(t\right)+w_i\left(t\right),i\in \mathcal{S},}\end{array}$$

(6)

where

• $p_i\left(t\right)\in {\mathbb{C}}^n$ and $q_i\left(t\right)\in {\mathbb{C}}^n$ denote the position and velocity vectors of agent i, respectively,

• $f_i(p_i\left(t\right),q_i\left(t\right),t)$ captures the agent’s inherent nonlinear dynamics,

• $u_i\left(t\right)\in {\mathbb{C}}^n$ is the control input, and

• $w_i\left(t\right)\in {\mathbb{C}}^n$ represents an unknown but bounded external disturbance affecting agent i.

Assumption 1 ^2,38: The disturbance $w_i\left(t\right)$ is assumed to be norm-bounded as

$$\left|w_i\left(t\right)\right|\le {\delta }_i,\forall t\ge 0,$$

for some known constant ${\delta }_i>0$ .

• Assumption 2 ² : The nonlinear function

$$f_i(p_i\left(t\right),q_i\left(t\right),t):{\mathbb{C}}^{n_p}\times {\mathbb{C}}^{n_q}\times {\mathbb{R}}_{\ge 0}\to {\mathbb{C}}^m$$

is assumed to be bounded, that is,

$$\left|f_i(p_i\left(t\right),q_i\left(t\right),t)\right|\le \overline{f},\forall t\ge 0,\forall i\in \{1,\dots ,\mathcal{S}\},$$

for some constant $\overline{f}>0$ .

The leader (agent $\mathcal{S}+1$ ) evolves according to the following dynamics:

$$\begin{array}{c}{\stackrel{·}{p}}_{\mathcal{S}+1}\left(t\right)=q_{\mathcal{S}+1}\left(t\right),\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{\mathcal{S}+1}\left(t\right)=f_{\mathcal{S}+1}(p_{\mathcal{S}+1}\left(t\right),q_{\mathcal{S}+1}\left(t\right),t),}\end{array}$$

(7)

where $p_{\mathcal{S}+1}\left(t\right)\in {\mathbb{C}}^n$ and $q_{\mathcal{S}+1}\left(t\right)\in {\mathbb{C}}^n$ represent the leader’s position and velocity, while $f_{\mathcal{S}+1}(·)$ characterizes the nonlinear dynamic behavior of the leader. The overall leader state is represented by the vector:

$${\left[\begin{array}{cc}{p}_{\mathcal{S}+1}\left(t\right)\hfill & q_{\mathcal{S}+1}\left(t\right)\hfill \end{array}\right]}^{\top }.$$

Let $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{L})$ denote the communication graph among the $\mathcal{S}$ follower agents. The augmented graph $\overline{\mathcal{G}}=(\overline{\mathcal{V}},\overline{\mathcal{E}},\overline{\mathcal{L}})$ is obtained by incorporating the leader node $v_{\mathcal{S}+1}$ into the original graph $\mathcal{G}$ .

Each agent $i\in \{1,\dots ,\mathcal{S}+1\}$ shares its state $(p_i\left(t\right),q_i\left(t\right))$ over the augmented graph $\overline{\mathcal{G}}$ , and is assigned to a group ${\mathcal{P}}_i$ , representing a cooperative subset of agents. Specifically, for any agents i and j, if $\overline{i}=\overline{j}$ , then the corresponding interaction weight satisfies:

$${\mathcal{L}}_{i,j}\in \{I,0\},$$

indicating either identity coupling within the group or no interaction.

Based on the multi-group FxTETC framework, the sliding surface(SS) of the system in (6) can be computed as follows:

$${\sigma }_i\left(t\right)=q_i\left(t\right)+{\xi }_i\left(t\right)$$

(8)

with

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\xi }}_i\left(t\right)=f_i(p_i\left(t\right),q_i\left(t\right),t)+l_1\sum _{j\in \mathcal{S}}{\lceil (p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right))\rfloor }^{{\varpi }_1}}\\ \multicolumn{1}{c}{+l_1^{″}\sum _{j\in \mathcal{S}}{\lceil (p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right))\rfloor }^{{\varpi }_1^{\prime }}}\\ \multicolumn{1}{c}{+l_2\sum _{j\in \mathcal{S}}{\lceil (q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right))\rfloor }^{{\varpi }_2}}\\ \multicolumn{1}{c}{+l_2^{″}\sum _{j\in \mathcal{S}}{\lceil (q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right))\rfloor }^{{\varpi }_2^{\prime }}}\end{array}$$

(9)

where the parameters are chosen such that the parameters ϖ _i and ${\varpi }_i^{\prime }$ for $i=1,2$ are defined as follows:

$${\varpi }_1=\frac{\varpi }{2-\varpi },{\varpi }_2=\varpi ,{\varpi }_1^{\prime }=\frac{4-3\varpi }{2-\varpi },{\varpi }_2^{\prime }=\frac{4-3\varpi }{3-2\varpi }$$

(10)

and the integers $l_1,l_2,l_1^{\prime \prime },l_2^{\prime \prime }$ are all positive.

Based on equation (9), define a variable as

$$\begin{array}{c}\multicolumn{1}{c}{{\tilde{z}}_i\left(t\right)=-f_i(p_i\left(t\right),q_i\left(t\right),t)-l_1\sum _{j\in \mathcal{S}}{\lceil (p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right))\rfloor }^{{\varpi }_1}}\\ \multicolumn{1}{c}{-l_1^{″}\sum _{j\in \mathcal{S}}{\lceil (p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right))\rfloor }^{{\varpi }_1^{\prime }}}\\ \multicolumn{1}{c}{-l_2\sum _{j\in \mathcal{S}}{\lceil (q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right))\rfloor }^{{\varpi }_2}}\\ \multicolumn{1}{c}{-l_2^{″}\sum _{j\in \mathcal{S}}{\lceil (q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right))\rfloor }^{{\varpi }_2^{\prime }}}\end{array}$$

(11)

for ith agent, $t\in [t_k^i,t_{k+1}^i)$ , a decentralized FxTETC and the sliding surface are constructed as follows:

$$\begin{array}{c}\multicolumn{1}{c}{u_i\left(t\right)={\tilde{z}}_i\left(t_k^i\right)-K_{1i}\mathrm{sign}\left({\sigma }_i\left(t_k^i\right)\right)-K_{2i}\lceil {\sigma }_i\left(t_k^i\right){\rfloor }^{\overline{\alpha }}}\\ \multicolumn{1}{c}{-K_{3i}\lceil {\sigma }_i\left(t_k^i\right){\rfloor }^{\overline{\beta }}}\end{array}$$

(12)

$${\sigma }_i\left(t_k^i\right)=q_i\left(t_k^i\right)-{\int }_{t_k^i}^t{\tilde{z}}_i\left(\tau \right)d\tau$$

(13)

We introduce the following error variable to reach the average multi party synchronization of the nonlinear MAS in equation (6):

$$\begin{array}{c}\multicolumn{1}{c}{e_i\left(t\right)={\tilde{z}}_i\left(t\right)-K_{1i}\mathrm{sign}\left({\sigma }_i\left(t\right)\right)-K_{2i}{\lceil {\sigma }_i\left(t\right)\rfloor }^{\overline{\alpha }}-K_{3i}{\lceil {\sigma }_i\left(t\right)\rfloor }^{\overline{\beta }}}\\ \multicolumn{1}{c}{-{\tilde{z}}_i\left(t_k^i\right)+K_{1i}\mathrm{sign}\left({\sigma }_i\left(t_k^i\right)\right)+K_{2i}{\lceil {\sigma }_i\left(t_k^i\right)\rfloor }^{\overline{\alpha }}}\\ \multicolumn{1}{c}{+K_{3i}{\lceil {\sigma }_i\left(t_k^i\right)\rfloor }^{\overline{\beta }}}\end{array}$$

(14)

Where $K_{2i}>0$ , $K_{3i}>0$ , $0<\overline{\alpha }<1$ , and $\overline{\beta }>1$ . The event-triggering condition is defined as

$$t_{k+1}^i=\inf \{t>t_k^i|\left|e_i\left(t\right)\right|\ge {\pi }_i\},$$

(15)

where the triggering threshold ${\pi }_i>0$ is selected individually for each agent. Once the condition in (15) is satisfied, the values ${\tilde{z}}_i\left(t_k^i\right)$ and $q_i\left(t_k^i\right)$ are transmitted to update $u_i\left(t\right)$ and ${\sigma }_i\left(t_k^i\right)$ as defined in (12), with $K_{1i}>0$ . Moreover, there exists a constant ${\overline{\rho }}_i>0$ such that the following inequality is satisfied:

$${\overline{\rho }}_i+{\delta }_i+{\pi }_i\le K_{1i}.$$

(16)

• Theorem 1: The fixed-time (FxT) distributed multi-party consensus control protocol (12) and its associated triggering condition (15), when applied with (6), collectively force the closed-loop MAS trajectories onto the sliding surface within a fixed time.

Proof. Consider the Lyapunov function

$$V_1=\frac{1}{2}\sum _{i\in \mathcal{S}}{\sigma }_i{\left(t\right)}^2.$$

(17)

By differentiating (17) and applying (8) together with the triggering condition (15), we obtain

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1=\sum _{i\in \mathcal{S}}{\sigma }_i\left(t\right)(e_i\left(t\right)-K_{1i}\mathrm{sign}\left({\sigma }_i\left(t\right)\right)}\\ \multicolumn{1}{c}{-K_{2i}{\lfloor {\sigma }_i\left(t\right)\rceil }^{\overline{\alpha }}-K_{3i}{\lfloor {\sigma }_i\left(t\right)\rceil }^{\overline{\beta }}+w_i\left(t\right)).}\end{array}$$

(18)

Using the bounds $\left|e_i\left(t\right)\right|\le {\pi }_i$ and $\left|w_i\left(t\right)\right|\le {\delta }_i$ , we obtain

$${\stackrel{·}{V}}_1\le \sum _{i\in \mathcal{S}}\left|{\sigma }_i\left(t\right)\right|({\pi }_i-K_{1i}+{\delta }_i)-K_{2i}{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\alpha }+1}-K_{3i}{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\beta }+1}.$$

(19)

Rewriting,

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1\le \sum _{i\in \mathcal{S}}(-\left|{\sigma }_i\left(t\right)\right|(K_{1i}-{\pi }_i-{\delta }_i)}\\ \multicolumn{1}{c}{-K_{2i}{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\alpha }+1}-K_{3i}{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\beta }+1}).}\end{array}$$

(20)

With the constraint $K_{1i}\ge {\overline{\rho }}_i+{\pi }_i+{\delta }_i$ , (20) becomes

$${\stackrel{·}{V}}_1\le \sum _{i\in \mathcal{S}}(-{\overline{\rho }}_i\left|{\sigma }_i\left(t\right)\right|-K_{2i}{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\alpha }+1}-K_{3i}{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\beta }+1}).$$

Define the minimum gains

$$K_1=\underset{i\in \mathcal{S}}{\min }(K_{1i}-{\pi }_i-{\delta }_i),K_2=\underset{i\in \mathcal{S}}{\min }{K}_{2i},K_3=\underset{i\in \mathcal{S}}{\min }{K}_{3i}.$$

Since $\left|{\sigma }_i\left(t\right)\right|\le \sqrt{2V_1}$ , we get

$${\stackrel{·}{V}}_1\le -a_1{V}_1^{1/2}-a_2{V}_1^{(\overline{\alpha }+1)/2}-a_3{V}_1^{(\overline{\beta }+1)/2},$$

where

$$a_1=K_1\sqrt{2\left|\mathcal{S}\right|},a_2=K_2\left|\mathcal{S}\right|2^{(\overline{\alpha }+1)/2},a_3=K_3\left|\mathcal{S}\right|2^{(\overline{\beta }+1)/2}.$$

Let

$$p=\min \{\frac{1}{2},\frac{\overline{\alpha }+1}{2},\frac{\overline{\beta }+1}{2}\},q=\max \{\frac{1}{2},\frac{\overline{\alpha }+1}{2},\frac{\overline{\beta }+1}{2}\},$$

and

$$a=\min (a_1,a_2,a_3),b=\max (a_1,a_2,a_3).$$

Then the FxT convergence time satisfies

$$T^s\le \frac{1}{a(1-p)}+\frac{1}{b(q-1)}.$$

Thus, all ${\sigma }_i\left(t\right)$ converge to zero within a fixed time. □

Homogeneity-based fixed-time stability analysis

To establish FxT stability,^39,40 the proof is carried out in two main steps. In the first step, it is demonstrated that the system described by equation (22) achieves GAS under the control law specified in equation (12). In the second step, it is shown that the corresponding limiting systems also exhibit GAS, as they characterize the system dynamics near the origin and at infinity. The proof is finally completed by applying Lemma 1 and utilizing the bi-limit homogeneity property.

The system dynamics, when constrained to evolve on the sliding manifold defined in equation (8), exhibit the following behavior under the control law (12):

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{p}}_i\left(t\right)=q_i\left(t\right)}\\ \multicolumn{1}{c}{{\stackrel{·}{q}}_i\left(t\right)=-l_1\sum _{j\in \mathcal{S}}{\lceil (p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right))\rfloor }^{{\varpi }_1}}\\ \multicolumn{1}{c}{-l_1^{″}\sum _{j\in \mathcal{S}}{\lceil (p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right))\rfloor }^{{\varpi }_1^{\prime }}}\\ \multicolumn{1}{c}{-l_2\sum _{j\in \mathcal{S}}{\lceil (q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right))\rfloor }^{{\varpi }_2}}\\ \multicolumn{1}{c}{-l_2^{″}\sum _{j\in \mathcal{S}}{\lceil (q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right))\rfloor }^{{\varpi }_2^{\prime }}}\end{array}$$

(21)

By defining $e_{\mathrm{pi}}\left(t\right)=p_i\left(t\right)-\frac{1}{S}\sum _{j\in \mathcal{S}}{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)$ and $e_{\mathrm{qi}}\left(t\right)=q_i\left(t\right)-\frac{1}{S}\sum _{j\in \mathcal{S}}{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)$ , the resulting dynamics are given by:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{e}}_{\mathrm{pi}}\left(t\right)=e_{\mathrm{qi}}\left(t\right)}\\ \multicolumn{1}{c}{{\stackrel{·}{e}}_{\mathrm{qi}}\left(t\right)=-l_1\sum _{j\in \mathcal{S}}{\lceil (e_{\mathrm{pi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{pj}}\left(t\right))\rfloor }^{{\varpi }_1}}\\ \multicolumn{1}{c}{-l_1^{″}\sum _{j\in \mathcal{S}}{\lceil (e_{\mathrm{pi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{pj}}\left(t\right))\rfloor }^{{\varpi }_1^{\prime }}}\\ \multicolumn{1}{c}{-l_2\sum _{j\in \mathcal{S}}\lceil {(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right)\rfloor }^{{\varpi }_2}}\\ \multicolumn{1}{c}{-l_2^{″}\sum _{j\in \mathcal{S}}{\lceil (e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))\rfloor }^{{\varpi }_2^{\prime }}}\end{array}$$

(22)

To assess GAS, consider the Lyapunov function:

$$\begin{array}{c}\multicolumn{1}{c}{V_2=\frac{1}{2}\sum _{i\in \mathcal{S}}\sum _{j\in \mathcal{S}}{\int }_0^{(e_{\mathrm{pi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{pj}}\left(t\right))}{l}_1\mathrm{sig}{\left(v\right)}^{{\varpi }_1}\mathrm{dv}}\\ \multicolumn{1}{c}{+\frac{1}{2}\sum _{i\in \mathcal{S}}\sum _{j\in \mathcal{S}}{\int }_0^{(e_{\mathrm{pi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{pj}}\left(t\right))}{l}_1^{″}\mathrm{sig}{\left(v\right)}^{{\varpi }_1^{\prime }}\mathrm{dv}}\\ \multicolumn{1}{c}{+\frac{1}{2}\sum _{i\in \mathcal{S}}{e}_{\mathrm{qi}}^2\left(t\right)}\end{array}$$

(23)

Differentiating the Lyapunov function in equation (23) and substituting from equation (22) yields:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2=\frac{1}{2}\sum _{i\in \mathcal{S}}\sum _{j\in \mathcal{S}}{l}_1{\mathcal{L}}_{i,j}{\mathrm{sig}}^{{\varpi }_1}(e_{\mathrm{pi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{pj}}\left(t\right))}\\ \multicolumn{1}{c}{(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}\\ \multicolumn{1}{c}{+\frac{1}{2}\sum _{i\in \mathcal{S}}\sum _{j\in \mathcal{S}}{l}_1^{″}{\mathcal{L}}_{i,j}{\mathrm{sig}}^{{\varpi }_1^{\prime }}(e_{\mathrm{pi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{pj}}\left(t\right))}\\ \multicolumn{1}{c}{(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}\\ \multicolumn{1}{c}{+\sum _{i\in \mathcal{S}}{e}_{\mathrm{qi}}{\left(t\right)}^2.}\end{array}$$

(24)

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2=-\frac{l_2}{2}\sum _{i\in \mathcal{S}}\sum _{j\in \mathcal{S}}(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}\\ \multicolumn{1}{c}{{\mathrm{sig}}^{{\varpi }_2}(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}\\ \multicolumn{1}{c}{-\frac{l_2^{\prime \prime }}{2}\sum _{i\in \mathcal{S}}\sum _{j\in \mathcal{S}}(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}\\ \multicolumn{1}{c}{{\mathrm{sig}}^{{\varpi }_2^{\prime }}(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right)).}\end{array}$$

(25)

Because

$$\begin{array}{c}\multicolumn{1}{c}{(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right)){\mathrm{sig}}^{{\varpi }_2}(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}\\ \multicolumn{1}{c}{={|e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right)|}^{1+{\varpi }_2},}\end{array}$$

and

$$\begin{array}{c}\multicolumn{1}{c}{(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right)){\mathrm{sig}}^{{\varpi }_2^{\prime }}(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}\\ \multicolumn{1}{c}{={|e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right)|}^{1+{\varpi }_2^{\prime }},}\end{array}$$

one can conclude that ${\stackrel{·}{V}}_2<0$ . Consequently, consensus is achieved by the states of the multi-group MASs.

Furthermore, to verify the FxT stability using the bi-limit homogeneity method, ⁸ consider the error dynamics in (22). Let ${\mathcal{F}}^i$ denote the vector field. Then, its zero-limit and infinity-limit vector fields can be expressed as:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathcal{F}}_0^i=[e_{\mathrm{qi}}\left(t\right),-l_1\sum _{j\in \mathcal{S}}{\lceil e_{\mathrm{pi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{pj}}\left(t\right)\rfloor }^{{\varpi }_1}}\\ \multicolumn{1}{c}{-l_2\sum _{j\in \mathcal{S}}{\lceil e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qi}}\left(t\right)\rfloor }^{{\varpi }_2}{]}^T}\\ \multicolumn{1}{c}{{\mathcal{F}}_{\infty }^i=[e_{\mathrm{qi}}\left(t\right),-l_1^{\prime \prime }\sum _{j\in \mathcal{S}}{\lceil e_{\mathrm{pi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{pj}}\left(t\right)\rfloor }^{{\varpi }_1^{\prime }}}\\ \multicolumn{1}{c}{{-l_2^{″}\sum _{j\in \mathcal{S}}{\lceil e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qi}}\left(t\right)\rfloor }^{{\varpi }_2^{″}}]}^T}\end{array}$$

(26)

From (26) and (10), it follows that the condition $0<{\varpi }_l<1<{\varpi }_l^{\prime \prime },(l=1,2)$ holds for any $\varpi \in (0,1)$ , where the integers $l_1,l_2,l_1^{\prime \prime },l_2^{\prime \prime }$ are strictly positive.

Moreover, the vector fields ${\mathcal{F}}_0^i$ and ${\mathcal{F}}_{\infty }^i$ can be interpreted as approximations of ${\mathcal{F}}^i$ in the limits 0 and $\infty$ , respectively. Specifically, the 0-limit of the approximated vector field ${\mathcal{F}}_0^i$ is r₀-homogeneous of degree $k_0=-1$ , which can be verified using

$$r_0=\left[\begin{array}{c}\frac{2-\varpi }{1-\varpi }\hfill \\ \multicolumn{1}{c}{\frac{1}{1-\varpi }}\end{array}\right].$$

(27)

Hence, in the 0-limit, ${\mathcal{F}}^i$ serves as an approximation of ${\mathcal{F}}_0^i$ .

Similarly, in the $\infty$ -limit, ${\mathcal{F}}_{\infty }^i$ is $r_{\infty }$ -homogeneous of degree $k_{\infty }=1$ , with

$$r_{\infty }=\left[\begin{array}{c}\frac{2-\varpi }{1-\varpi }\hfill \\ \multicolumn{1}{c}{\frac{3-2\varpi }{1-\varpi }}\end{array}\right].$$

(28)

Thus, the system is bilimit homogeneous within the framework of Lemma 1.

Furthermore, the vector field ${\mathcal{F}}_0^i$ is homogeneous of degree ϖ− 1, while ${\mathcal{F}}_{\infty }^i$ is homogeneous of degree ${\varpi }_1^{\prime }-1$ , as shown in (27). ³⁹ To examine the GAS of (31), consider the Lyapunov function:

$$\begin{array}{c}\multicolumn{1}{c}{V_2^0=\frac{1}{2}\sum _{i\in \mathcal{S}}\sum _{j\in \mathcal{S}}{\int }_0^{(e_{\mathrm{pi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{pj}}\left(t\right))}{l}_1{\mathrm{sig}}^{{\varpi }_1}\left(v\right)\mathrm{dv}}\\ \multicolumn{1}{c}{+\frac{1}{2}\sum _{i\in \mathcal{S}}{e}_{\mathrm{qi}}{\left(t\right)}^2.}\end{array}$$

(29)

By differentiating the Lyapunov function and utilizing equation (22), one can obtain:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2^0=\frac{1}{2}\sum _{i\in \mathcal{S}}\sum _{j\in \mathcal{S}}{l}_1{\mathcal{L}}_{i,j}{\mathrm{sig}}^{{\varpi }_1}(e_{\mathrm{pi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{pj}}\left(t\right))}\\ \multicolumn{1}{c}{·(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))+\sum _{i\in \mathcal{S}}{e}_{\mathrm{qi}}\left(t\right){\stackrel{·}{e}}_{q_i}\left(t\right).}\end{array}$$

(30)

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2^0=-\frac{l_2}{2}\sum _{i\in \mathcal{S}}\sum _{j\in \mathcal{S}}(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}\\ \multicolumn{1}{c}{\mathrm{sig}{(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}^{{\varpi }_2}.}\end{array}$$

(31)

Because

$$\begin{array}{c}(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))\mathrm{sig}{(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}^{{\varpi }_2}\hfill \\ \multicolumn{1}{c}{={|e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right)|}^{1+{\varpi }_2},}\end{array}$$

(32)

one can conclude that

$${\stackrel{·}{V}}_2^0<0.$$

Consequently, the states of the multi-group MAS reach average consensus.

It can be observed ${\mathcal{F}}_0^i$ is homogeneous of degree ϖ− 1, while ${\mathcal{F}}_{\infty }^i$ is homogeneous of degree ${\varpi }_1^{\prime }-1$ , as demonstrated in (27).^8,39 To find the GAS of (31), the following Lyapunov function is considered:

$$\begin{array}{c}\multicolumn{1}{c}{V_2^{\infty }=\frac{1}{2}\sum _{i\in \mathcal{S}}\sum _{j\in \mathcal{S}}{\int }_0^{(e_{\mathrm{pi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{pj}}\left(t\right))}{l}_1^{″}\mathrm{sig}{\left(v\right)}^{{\varpi }_1^{\prime }}\mathrm{dv}}\\ \multicolumn{1}{c}{+\frac{1}{2}\sum _{i\in \mathcal{S}}{e}_{\mathrm{qi}}{\left(t\right)}^2}\end{array}$$

(33)

By taking the time derivative of $V_2^{\infty }\left(t\right)$ , one obtains:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2^{\infty }=-\frac{l_2^{\prime \prime }}{2}\sum _{i\in \mathcal{S}}\sum _{j\in \mathcal{S}}(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}\\ \multicolumn{1}{c}{\mathrm{sig}{(e_{\mathrm{qi}}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{\mathrm{qj}}\left(t\right))}^{{\varpi }_2^{\prime }}.}\end{array}$$

(34)

Because

$$\begin{array}{l}\left(e_{qi}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{qj}\left(t\right)\right)\mathrm{sig}{\left(e_{qi}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{qj}\left(t\right)\right)}^{{{\varpi }^{\prime }}_2}\\ ={\left|e_{qi}\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{e}_{qj}\left(t\right)\right|}^{1+{{\varpi }^{\prime }}_2},\end{array}$$

(35)

it follows that

$${\stackrel{·}{V}}_2^{\infty }<0.$$

Based on the principles of homogeneity theory,^9,40 the GAS of both the zero-limit and infinity-limit systems under bi-limit homogeneity ensures that the original nonlinear system possesses FxT stability.

This completes the FxT stability analysis grounded in the homogeneous structure of the system. As previously established, the system exhibits GAS, ³¹ and the conditions for homogeneity are satisfied. Therefore, by using Lemma 1, it follows that the system described by equation (33) is FxT stable.

Justification of triggering condition

let a sequentially progressing time instants $0=t_{i0}<t_{i1}\dots <t_k^i<t_{k+1}^i<\dots$ represent the instants, where $t_{k+1}^i-t_k^i={\varphi }_k^i$ . The difference ${\varphi }_k^i$ between the consecutively activated instants $t_i^k$ and $t_i^{k+1}$ is then calculated to determine the minimal time.

• Theorem 2: When the distributed FxTC in equation (12) and the event-trigger rule in equation (15) are in place, the system does not display Zeno behavior.

Proof. By defining

$$\begin{array}{c}\multicolumn{1}{c}{{\tilde{z}}_i\left(t\right)=-f_i(p_i\left(t\right),q_i\left(t\right),t)-l_1\sum _{j\in \mathcal{S}}{\lceil p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)\rfloor }^{{\varpi }_1}}\\ \multicolumn{1}{c}{-l_1^{″}\sum _{j\in \mathcal{S}}{\lceil p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)\rfloor }^{{\varpi }_1^{\prime }}}\\ \multicolumn{1}{c}{-l_2\sum _{j\in \mathcal{S}}{\lceil q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)\rfloor }^{{\varpi }_2}}\\ \multicolumn{1}{c}{-l_2^{″}\sum _{j\in \mathcal{S}}{\lceil q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)\rfloor }^{{\varpi }_2^{\prime }}.}\end{array}$$

(36)

Under Assumption 1 and 2, the derivative of the absolute error (14) $\left|e_i\left(t\right)\right|$ can be bounded as

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d\left|e_i\left(t\right)\right|}{\mathrm{dt}}\le l_1{\varpi }_1\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1-1}\sum _{j\in \mathcal{S}}|q_i\left(t\right)}\\ \multicolumn{1}{c}{-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|+l_1^{″}{\varpi }_1^{\prime }\sum _{j\in \mathcal{S}}|p_i\left(t\right)}\\ \multicolumn{1}{c}{-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right){|}^{{\varpi }_1^{\prime }-1}\sum _{j\in \mathcal{S}}|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}\\ +l_2{\varpi }_2\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2-1}\sum _{j\in \mathcal{S}}|(u_i\left(t\right)\hfill \\ \multicolumn{1}{c}{+w_i\left(t\right))-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}(u_j\left(t\right)+w_j\left(t\right))|}\\ +l_2^{″}{\varpi }_2^{\prime }\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2^{\prime }-1}\sum _{j\in \mathcal{S}}|(u_i\left(t\right)\hfill \\ \multicolumn{1}{c}{+w_i\left(t\right))-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}(u_j\left(t\right)+w_j\left(t\right))|}\\ +K_{1i}|1-{\tanh }^2\left(\mathrm{\Gamma }{\sigma }_i\left(t\right)\right)|·\left|\mathrm{\Gamma }{\stackrel{·}{\sigma }}_i\left(t\right)\right|\hfill \\ +K_{2i}\overline{\alpha }{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\alpha }-1}\left|{\stackrel{·}{\sigma }}_i\left(t\right)\right|+K_{3i}\overline{\beta }{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\beta }-1}\left|{\stackrel{·}{\sigma }}_i\left(t\right)\right|.\hfill \end{array}$$

(37)

where $\mathrm{\Gamma }>>1$ is a constant. Utilizing the result in, ² the inequality can be simplified as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d\left|e_i\left(t\right)\right|}{\mathrm{dt}}\le l_1{\varpi }_1\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1-1}\sum _{j\in \mathcal{S}}|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}\\ +l_1^{″}{\varpi }_1^{\prime }\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1^{\prime }-1}\sum _{j\in \mathcal{S}}|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|\hfill \\ +l_2{\varpi }_2\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2-1}\hfill \\ \times (\sum _{j\in \mathcal{S}}|u_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{u}_j\left(t\right)|+\left|w_i\left(t\right)\right|+\sum _{j\in \mathcal{S}}\left|w_j\left(t\right)\right|)\hfill \\ +l_2^{″}{\varpi }_2^{\prime }\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2^{\prime }-1}\hfill \\ \times (\sum _{j\in \mathcal{S}}|u_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{u}_j\left(t\right)|+\left|w_i\left(t\right)\right|+\sum _{j\in \mathcal{S}}\left|w_j\left(t\right)\right|)\hfill \\ +(K_{1i}\mathrm{\Gamma }+K_{2i}\overline{\alpha }{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\alpha }-1}+K_{3i}\overline{\beta }{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\beta }-1})\left|{\stackrel{·}{\sigma }}_i\left(t\right)\right|.\hfill \end{array}$$

(38)

Let us assume that ${\sigma }_i\left(t\right)$ is bounded; hence we obtain

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d\left(\left|e_i\left(t\right)\right|\right)}{\mathrm{dt}}\le l_1{\varpi }_1\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1-1}\sum _{j\in \mathcal{S}}|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}\\ +l_1^{″}{\varpi }_1^{\prime }\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1^{\prime }-1}\sum _{j\in \mathcal{S}}|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|\hfill \\ +l_2{\varpi }_2\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2-1}(\sum _{j\in \mathcal{S}}|u_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{u}_j\left(t\right)|+\mathcal{S}\left|w_i\right|+\sum _{j\in \mathcal{S}}\left|w_j\right|)\hfill \\ +l_2^{″}{\varpi }_2^{\prime }\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2^{\prime }-1}(\sum _{j\in \mathcal{S}}|u_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{u}_j\left(t\right)|+\mathcal{S}\left|w_i\right|+\sum _{j\in \mathcal{S}}\left|w_j\right|)\hfill \\ +(K_{1i}\mathrm{\Gamma }+K_{2i}\overline{\alpha }{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\alpha }-1}+K_{3i}\overline{\beta }{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\beta }-1})[\left|u_i\left(t\right)\right|+\left|w_i\right|\hfill \\ +l_1\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1}+l_1^{″}\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1^{\prime }}\hfill \\ +l_2\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2}+l_2^{″}\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2^{\prime }}].\hfill \end{array}$$

(39)

The event is commonly understood not to occur until $\left|e_i\left(t\right)\right|$ reaches π _i as per the criterion (15). Taking (39) into further account, we find:

$$\begin{array}{c}\multicolumn{1}{c}{{\pi }_i\le \left|e_i\left(t\right)\right|\le {\varphi }_k^i[l_1{\varpi }_1\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1-1}\sum _{j\in \mathcal{S}}|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}\\ +l_1^{″}{\varpi }_1^{\prime }\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1^{\prime }-1}\sum _{j\in \mathcal{S}}|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|\hfill \\ +l_2{\varpi }_2\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2-1}(\sum _{j\in \mathcal{S}}|u_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{u}_j\left(t\right)|+\mathcal{S}\left|w_i\right|+\sum _{j\in \mathcal{S}}\left|w_j\right|)\hfill \\ +l_2^{″}{\varpi }_2^{\prime }\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2^{\prime }-1}(\sum _{j\in \mathcal{S}}|u_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{u}_j\left(t\right)|+\mathcal{S}\left|w_i\right|+\sum _{j\in \mathcal{S}}\left|w_j\right|)\hfill \\ +(K_{1i}\mathrm{\Gamma }+K_{2i}\overline{\alpha }{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\alpha }-1}+K_{3i}\overline{\beta }{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\beta }-1})[\left|u_i\left(t\right)\right|+\left|w_i\right|\hfill \\ +l_1\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1}+l_1^{″}\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1^{\prime }}\hfill \\ +l_2\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2}+l_2^{″}\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2^{\prime }}]]\hfill \end{array}$$

(40)

Therefore, the inter-event interval ϕ _ik is calculated as

$${\varphi }_{\mathrm{ik}}\ge \frac{{\pi }_i}{{\epsilon }_i}>0$$

With

$$\begin{array}{c}\multicolumn{1}{c}{{\epsilon }_i=[l_1{\varpi }_1\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1-1}\sum _{j\in \mathcal{S}}|q_i\left(t\right)}\\ \multicolumn{1}{c}{-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|+l_1^{″}{\varpi }_1^{\prime }\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1^{\prime }-1}}\\ \multicolumn{1}{c}{\sum _{j\in \mathcal{S}}|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|+l_2{\varpi }_2\sum _{j\in \mathcal{S}}|q_i\left(t\right)}\\ \multicolumn{1}{c}{-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right){|}^{{\varpi }_2-1}}\\ \multicolumn{1}{c}{(\sum _{j\in \mathcal{S}}|u_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{u}_j\left(t\right)|+\mathcal{S}{\delta }_i+\sum _{j\in \mathcal{S}}{\delta }_j)}\\ +l_2^{″}{\varpi }_2^{\prime }\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2^{\prime }-1}\hfill \\ \multicolumn{1}{c}{(\sum _{j\in \mathcal{S}}|u_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{u}_j\left(t\right)|+\mathcal{S}{\delta }_i+\sum _{j\in \mathcal{S}}{\delta }_j)}\\ +(K_{1i}\mathrm{\Gamma }+K_{2i}\overline{\alpha }{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\alpha }-1}+K_{3i}\overline{\beta }{\left|{\sigma }_i\left(t\right)\right|}^{\overline{\beta }-1})\hfill \\ \multicolumn{1}{c}{·[\left|u_i\left(t\right)\right|+{\delta }_i+l_1\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1}}\\ +l_1^{″}\sum _{j\in \mathcal{S}}{|p_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{p}_j\left(t\right)|}^{{\varpi }_1^{\prime }}\hfill \\ +l_2\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{\overline{q}}_j\left(t\right)|}^{{\varpi }_2}\hfill \\ +l_2^{″}\sum _{j\in \mathcal{S}}{|q_i\left(t\right)-{\mathcal{L}}_{i,j}\angle {\theta }_{i,j}{q}_j\left(t\right)|}^{{\varpi }_2^{\prime }}]].\hfill \end{array}$$

(41)

It demonstrates that the Zeno behavior is excludable. This completes the proof. □

Simulation results

Consider a multi-agent system (MAS) comprising ten agents, including one leader node (denoted as node L) and nine follower agents. The directed communication topology among these agents is illustrated in Figure 2.

Figure 2.

Communication topology showing intra-group and inter-group weighted links.

In the communication topology, Agents 1, 2, and 3 constitute Group 1; Agents 4, 5, and 6 form Group 2; and Agents 7, 8, and 9 comprise Group 3.

The multi-agent system (MAS) dynamics are described by equation (6). The associated nonlinear functions and external disturbances are defined as:

$$\begin{array}{c}\multicolumn{1}{c}{f(p_i\left(t\right),q_i\left(t\right),t)=-0.1\sin \left(p_i\left(t\right)\right)-0.2\cos \left(q_i\left(t\right)\right),}\\ \multicolumn{1}{c}{w_i(p_i\left(t\right),q_i\left(t\right),t)=0.1\sin (0.1p_i^2\left(t\right))}\\ \multicolumn{1}{c}{+0.2\cos (0.5q_i\left(t\right)),i\in \mathcal{S}.}\end{array}$$

(42)

The initial positions of the agents are set as:

$$\begin{array}{c}\multicolumn{1}{c}{p_0=[p_1\left(0\right),\dots ,p_9\left(0\right){]}^T}\\ \multicolumn{1}{c}{={[\left[\begin{array}{c}\sqrt{2}{e}^{i\frac{\pi }{4}}\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}\sqrt{2}{e}^{i\frac{\pi }{4}}\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}\sqrt{2}{e}^{i\frac{\pi }{4}}\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right]]}^T.}\end{array}$$

The initial velocities are set as: $q_0=[q_1\left(0\right),\dots ,q_9\left(0\right){]}^T={[\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right]\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \end{array}\right]]}^T$ . A total of nine agents, organized into three groups, is illustrated in Figure 2. Each group is led by a designated leader agent. In this simulation, agent 1 is the leader of group 1, with agents 2 and 3 following. For group 2, agent 4 serves as the leader, followed by agents 5 and 6. Similarly, in group 3, agent 7 leads, with agents 8 and 9 following.

The objective of this controller is to design a multi-formation configuration for the agents while maintaining the communication graph structure depicted in Figure 1. For group 1, the reference trajectory is chosen as $p_{\mathcal{S}+1}=0.1t+j0.1t$ . Group 2 is oriented at 45 ° relative to this reference trajectory, and group 3 is similarly oriented at an additional 45 ° with respect to group 2.

Based on equation (6), the interaction weights among agents within the same group are assigned a value of 1. For inter-group communication, the weight between Group 1 and Group 2 is specified as ${\mathcal{L}}_{1,4}\angle {\theta }_{1,4}=1\angle \frac{\pi }{4}$ , and the weight between Group 2 and Group 3 is set as ${\mathcal{L}}_{4,7}\angle {\theta }_{4,7}=1\angle \frac{\pi }{4}$ . Applying equation (5), the resulting weight from Group 3 to Group 1 is determined as:

$${\mathcal{L}}_{7,1}\angle {\theta }_{7,1}=1\angle \left(\frac{3\pi }{2}\right).$$

Using controller (12) and the event-triggering function (15) with the following parameters:

$l_1=-7$ , $l_1^{″}=-2$ , $l_2=-15$ , and $l_2^{″}=-5$ . Additionally, ${\varpi }_1=0.43$ , ${\varpi }_2=0.6$ , ${\varpi }_1^{\prime }=1.57$ , and ${\varpi }_2^{\prime }=1.22$ .

The parameters for designing the sliding surface are chosen as $K_{1i}=0.6$ , $K_{2i}=1.5,K_{3i}=0.1$ , $\overline{\alpha }=0.35$ , and $\overline{\beta }=1.75,{\overline{\rho }}_i=0.1,{\delta }_i=.2,{\pi }_i=0.5$ .

Time invariant formation control

The initial positions and corresponding controller parameters of the MAS are specified above. Figure 3 shows that the followers are able to track the trajectory of the leader, it shows that in a multi group consensus the each agent in a group 1 synchronize with each other. Similarly for Group 2 and Group 3 the inter group synchronization is shown by Figures 4 and 5 respectively.

Figure 3.

Group 1 positions and velocities.

Figure 4.

Group 2 positions and velocities.

Figure 5.

Group 3 positions and velocities.

The Figure 6 illustrates the sliding surface trajectories associated with the leader of each group. Each curve corresponds to one group leader, showing how the designed sliding surface converges toward zero over time. The decreasing trend confirms that the control law effectively drives the leader dynamics onto the sliding surface, thereby ensuring stability and coordination within each group. The convergence of all leader surfaces highlights the robustness of the proposed strategy in handling inter-group synchronization.

Figure 6.

Evolution of the sliding surface corresponding to the leader within each group.

Figures 7 to 9 depict the time courses of updates triggered by events and the respective time intervals under the condition of event triggering for agents 1, 4, and 7, respectively. These figures demonstrate when each agent executes the control updates under the event-triggered event. Importantly, for all of these agents, the time between two subsequent triggering events is guaranteed to be constant forwards from zero. This observation implies that there are not infinitely many events in a finite length of time, a behavior that is known in the literature as Zeno behavior. The event time consistent across all of the selected agents demonstrates that the proposed event-triggered strategy is effective for eliminating Zeno execution in the multi-group consensus problem.

Figure 7.

Inter-event time plot of agent 1 for ramp function.

Figure 8.

Inter-event time plot of agent 4 for ramp function.

Figure 9.

Inter-event time plot of agent 7 for ramp function.

Figure 10 displays the three-dimensional trajectories of all agents. The results indicate that the followers within each group successfully follow the trajectory of their respective leaders. Additionally, the plots confirm that consensus is maintained not only within individual groups but also across different groups, indicating proper inter-group coordination.

Figure 10.

3D position trajectories of different groups for ramp function.

To quantitatively evaluate the benefits of the multi-group consensus framework, mean squared error (MSE) formation error is introduced. For a system of $\mathcal{S}$ agents, the MSE of the i-th agent at time t is defined as

$${\mathrm{MSE}}_i\left(t\right)={(p_i\left(t\right)-p_{\mathrm{avg}}\left(t\right))}^2,$$

where $p_i\left(t\right)$ is the position of the i-th agent and

$$p_{\mathrm{avg}}\left(t\right)=\frac{1}{\mathcal{S}}\sum _{j=1}^{\mathcal{S}}{p}_j\left(t\right)$$

is the instantaneous group average. This definition provides a time-varying measure of how far each agent is from the consensus state. To summarize the long-term behavior, the average MSE of agent i over the entire simulation interval of length T is given by

$${\mathrm{MSE}}_i=\frac{1}{T}\sum _{k=1}^T{(p_i\left(k\right)-p_{\mathrm{avg}}\left(k\right))}^2.$$

Figure 11 depicts the MSE of the agents. When consensus is achieved, MSE _i for each agent goes to a very small value, thereby confirming that the proposed FxTETC strategy ensures convergence of all agents to the desired group average.

Figure 11.

MSE error in multi group consensus.

For the MAS problem, in the absence of ETC, a conventional time-triggered strategy must be employed, which incurs a communication cost of 5000 sampling or data transmission instants for multi-group consensus tracking of a ramp function, given a step size of 0.01. In contrast, when ETC is applied, each agent triggers its communication at different rates, as summarized in Table 1. This clearly demonstrates the significant reduction in communication load achieved with ETC, which is approximately $48.8\%$ compared to the time-triggered strategy.

Table 1.

Number of triggering instants and savings under event-triggered control (ETC).

Agent	No. of triggers (ETC)	Saving in transmissions	Saving (%)
1	2345	2655	53.1
2	1955	3045	60.9
3	2907	2093	41.9
4	2515	2485	49.7
5	2476	2524	50.5
6	3093	1907	38.1
7	2813	2187	43.7
8	2030	2970	59.4
9	2897	2103	42.1

Time variant formation control

For a time variant tracking the leader reference signal is chosen as $x_0=\sin (0.1t)+\mathrm{jcos}(0.1t)$ With the initial conditions for the followers and their controllers chosen at random. All agents’ 3-D trajectories are displayed in Figure 12. With sinusoidal formation the followers of each group follow the leader’s trajectory effectively, whereas the they have a definite phase separation based on the complex polar weights chosen among the different parties.

Figure 12.

3D position trajectories of different groups for time variant formation control.

Figures 13 to 15 present the event-triggered time intervals for Agents 1, 4, and 7, representing three different groups. Due to space limitations, only one representative agent from each group is shown. These plots validate the efficiency of the proposed ETC strategy in minimizing both energy usage and the frequency of controller updates. Despite the occurrence of multiple triggering events, the presence of nonzero intervals between them confirms that Zeno behavior is successfully avoided.

Figure 13.

Inter-event time plot of agent 1 for time variant formation control.

Figure 14.

Inter-event time plot of agent 4 for time variant formation control.

Figure 15.

Inter-event time plot of agent 7 for time variant formation control.

Conclusion

In this paper, a distributed multi-group average consensus problem is presented, employing a leader-follower architecture with fixed-time event-triggered control. The proposed method ensures that agents, divided into distinct parties, achieve synchronization within their groups while maintaining distinct target trajectories across different parties. Using distributed sliding manifolds and integral functions, the consensus problem in disturbed multi-agent systems (MASs) was introduced under a directed graph setting. The event-triggering mechanisms proposed improved the network resource usages by minimizing unnecessary communication, while guaranteeing fixed-time convergence and avoiding Zeno behavior. The theoretically derived sufficient conditions for average multi-agent consensus in nonlinear MASs were validated through numerous numerical simulations and empirical comparisons demonstrating the applicability and performance of the proposed methodology.

Scope for Future Work: Future studies can focus on improving the suggested fixed-time distributed event-triggered control scheme for multi-group average consensus for network control with respect to cyber-physical challenges. A constructive future direction will be expanding the framework to model various forms of cyber-attacks (data injection, denial-of-service (DoS) attacks, replay attacks, deception-based, etc.) that may corrupt the integrity of the network. The development of resilient detection and mitigation strategies and a feasible adaptive security layer will be important for effective consensus performance under these adversarial conditions. Moreover, the implementation of secure communication protocols and intelligent triggering mechanisms based on network vulnerabilities will enhance the robustness and scalability of the system substantially. Another opportunity is to validate the proposed method experimentally in a networked cumbersome platform such as cooperative robotic systems, smart grids, and IoT-enabled multi-agent networks in order to assess performance under with real uncertainties and communication delays.