Fixed-Time Bipartite Formation of Multi-Agent Systems Using Dynamic Event-Triggered Scheme

1  Introduction

Due to their flexibility and scalability, multi-agent systems have been widely applied across fields such as spacecraft attitude coordination [1,2], smart grids [3,4], and intelligent transportation [5,6]. Formation control [7–10] is a fundamental problem in the coordination of multi-agent systems, aiming to design distributed control protocols that enable multiple agents to form a desired geometric shape to accomplish complex tasks in challenging environments, such as limited communication bandwidth [11], unknown disturbances [12], and cyber-attacks [13]. Convergence rate is an important performance metric in formation control and remains a topic of active research. In [14–16], asymptotic convergence algorithms were proposed, while in [17–19], finite-time convergence algorithms were presented. Among these, finite-time algorithms converge faster than asymptotic ones. However, the upper bound of the convergence time for finite-time convergence algorithms depends on the system’s initial state. To address the upper bound on convergence time in finite-time consensus algorithms, researchers have begun studying fixed-time convergence algorithms [20]. In [21], a fixed-time formation control algorithm for heterogeneous multi-agent systems with disturbances was proposed. In [22], an adaptive optimal fixed-time output feedback formation algorithm for time-varying scenarios was introduced. In [23], a fixed-time formation control algorithm for uncertain nonlinear multi-agent systems with actuator faults was proposed. In addition, some scholars further developed prescribed-time control approaches [24–26]. Although research on fixed-time formation control algorithms exists, these studies primarily focus on cooperative relationships among agents. Therefore, further research on the competitive relationships among multi-agent systems and on implementing fixed-time formation control for such systems is a meaningful endeavor.

In nature, agents exhibit both cooperative and competitive relationships, as seen in wolf pack hunting. Addressing the cooperation and competition in multi-agent systems, reference [27] first introduced the concept of bipartite consensus. In recent years, researchers have made significant progress in studying bipartite formation in multi-agent systems. In [28], a bipartite formation algorithm for second-order nonlinear multi-agent systems with hybrid pulses was proposed. In [29], a bipartite time-varying formation algorithm for nonlinear multi-agent systems based on disturbance observers was developed. In [30], a fixed-time bipartite time-varying formation tracking algorithm for networked Euler-Lagrange systems was presented. It is noted that although numerous studies have examined bipartite formation control algorithms, few have addressed communication constraints in bipartite formation control for multi-agent systems. Compared with standard consensus problems, bipartite formation issues are more complex because they involve antagonistic relationships and negative weights, which necessitate greater communication and computational resources. Therefore, the main motivation of this paper is to design a fixed-time bipartite formation control algorithm that further reduces communication energy consumption.

In practical applications, continuous controller updates can waste significant communication resources and increase wear on actuators. To overcome these drawbacks, researchers have proposed event-triggered control mechanisms, in which the controller is updated only when event-triggered conditions are met [31,32]. In [33], a data-driven event-triggered bipartite formation control method was designed for nonlinear multi-agent systems with unknown dynamics. In [34], an event-triggered bipartite time-varying formation control method was achieved for linear multi-agent systems with uncertain dynamics. Additionally, to further reduce the number of event triggers, researchers proposed a dynamic event-triggering strategy that introduces a dynamic variable for the static event-triggered strategy, allowing the triggering condition to change over time. In [35], a dynamic event-triggered strategy was proposed for the fixed-time consensus problem, reducing the number of event triggers in the system. In [36,37], bipartite consensus problems for multi-agent systems under dynamic event-triggered control were studied. However, ref. [35] did not account for competitive relationships among agents, and refs. [36,37] achieved only asymptotic consensus. Therefore, research on dynamic event-triggered fixed-time bipartite formation control for multi-agent systems remains highly important.

On the other hand, most control algorithms in [34–37] require continuous information measurement and transmission from the system. However, due to limited onboard communication and the actual hardware’s driving capabilities, achieving real-time sampling and communication is difficult. To address these issues, researchers have combined periodic sampling [38–40] with event-triggered mechanisms, in which the core idea is to sample from continuous systems at specific times, with event detection occurring only on the sampled data. This approach accounts for the practical situation of periodic sensor sampling. It ensures that the minimum event-triggered interval is at least as long as the sampling interval, thereby further eliminating Zeno behavior. Currently, there is limited research on event-triggered consensus in sampled-data multi-agent systems. Reference [41] studied an event-triggered consensus problem for a class of nonlinear multi-agent systems based on sampled data. Reference [42] investigated a finite-time tracking consensus method for second-order multi-agent systems based on sampled data with an event-triggered scheme. Reference [43] formulated an asynchronous control problem of continuous-time positive Markov jump systems with a dynamic event-triggered scheme based on sampled data. However, the designs in [41,42] employ static event-triggered mechanisms, which can result in unnecessary triggering. Although ref. [43] proposed a dynamic event-triggered communication protocol, it did not account for competitive relationships among agents and did not achieve fixed-time consensus. Therefore, researching the dynamic event-triggered fixed-time convergence problem based on sampled data is a meaningful and challenging task.

Based on the above analysis, this paper addresses sampling periods, communication constraints, and fixed-time bipartite formation issues in multi-agent systems by proposing a dynamic event-triggered fixed-time bipartite consensus control algorithm using sampled data. The main contributions are summarized as follows:

(1) Design a periodic sampling mechanism. Compared to the existing algorithm in [36], it can reduce the system’s communication frequency.

(2) Develop a dynamic event-triggered control algorithm based on sampled data to further reduce the communication burden compared with the existing algorithm in [42].

(3) Design a distributed fixed-time bipartite formation control method. Compared with the existing method in [43], the proposed method further accounts for competitive relationships among agents and converges faster.

Notations: R, R+, RN, RN×n, and RN×N represent the set of real numbers, positive real numbers, N-dimensional column vectors, N×n-dimensional matrices, and N-dimensional square matrices, respectively.

2  Mathematical Preliminaries and Problem Statements

2.1 Signed Graph Theory

Consider an undirected graph 𝒢=(𝒱,ℰ,𝒜) consisting of N nodes, where 𝒱={v1,v2,…,vN} is the set of nodes of 𝒢. Each node vi(i=1,2,…,N) represents an agent i, and ℰ⊆𝒱×𝒱 represents the set of edges. If (vi,vj)∈ℰ, it indicates that agent i can interact with neighbor agent j. 𝒜=[aij]∈RN×N is the adjacency matrix of the graph 𝒢. If (vi,vj)∈ℰ and i≠j then aij≠0, otherwise, aij=0. If there is a path between any pair of nodes in graph 𝒢, then the graph is said to be connected. The node set 𝒱 of the signed graph 𝒢 can be partitioned into {𝒱1,𝒱2}, such that 𝒱1∪𝒱2=𝒱 and 𝒱1∩𝒱2=∅. If all edges within 𝒱1 or 𝒱2 are positively weighted and all edges between 𝒱1 and 𝒱2 are negatively weighted, then the signed graph is structurally balanced. If ∀vi and vj∈𝒱m(m∈{1,2}), then aij≥0. If vi∈𝒱m, vj∈𝒱n, and m≠n(m,n∈{1,2}), then aij≤0. The Laplacian matrix of the graph 𝒢 is ℒ=[lij]∈RN×N, where lii=∑i=1N|aij| and lij=−aij(i≠j). The 𝒜 and ℒ are both symmetric matrices.

2.2 Relevant Lemmas

Lemma 1 ([44]): If there exists a continuous positive definite and radially unbounded function Γ:Rn→R+∪{0} such that:

(1) Γ(x)=0⇔x=0;

(2) For parameters α, β>0, p=1−12r, q=1+12r, and r>1, if any solution x(t) satisfies the inequality

D∗Γ(x(t))≤−αΓp(x(t))−βΓq(x(t))(1)

the system’s origin is globally fixed-time stable, and the settling time is obtained as

T≤Tmax=πrαβ.(2)

Lemma 2 ([27]): If the signed graph 𝒢 is structurally balanced, then there exists a diagonal matrix D=diag{d1,d2,…,dN}, with di={±1}, such that D𝒜D is nonnegative.

Lemma 3 ([27]): If the signed graph 𝒢 is undirected and connected, then according to Lemma 2, it is obtained that the Laplacian matrix ℒ of 𝒢 is positive semi-definite, and its eigenvalues satisfy 0<λ2≤…≤λN. The eigenvector corresponding to the eigenvalue 0 is 1=[1,1,…,1]T∈RN. Furthermore, 𝒢 is a signed graph and structurally balanced. If 1TDx=0, where x=[x1,x2,…,xN]T, then λ2xTx≤xTℒx≤λNxTx, where xTℒx=(1/2)∑i=1N∑j=1N|aij|(xi−sign(aij)xj)2.

Lemma 4 ([35]): For ℓ1,ℓ2,…,ℓm≥0, 0<p≤1, and 1<q<∞, we have

m1−p(∑i=1mℓi)p≥∑i=1mℓip≥(∑i=1mℓi)p,m1−q(∑i=1mℓi)q≤∑i=1mℓiq≤(∑i=1mℓi)q.(3)

Remark 1: Lemmas 1-4 are used to analyze the stability of the system controlled by our designed method, which is presented later. Moreover, it should be noted that Lemma 2 is a basic property for the signed graph 𝒢 when it satisfies the structurally balanced condition, which ensures that the gauge transformation can be conducted and the Laplacian matrix ℒ of the signed graph 𝒢 is positive semi-definite. Further discussion can be found in [27,34], and [36].

2.3 System Dynamics Descriptions

Consider a system composed of N agents, where i agent is described as

{x˙i(t)=ui(t),x(0)=x0(4)

where xi(t) represents the state of agent i, and ui(t) represents the control input of agent i.

Definition 1: Let f=[f1T,f2T,⋯,fNT]T∈RN×n be the desired formation vector, where fi=[fi1, fi2,⋯,fin]T. The fixed-time bipartite formation of the multi-agent systems (4) implies there exists a prescribed time T such that limt→T⁡‖xi(t)−fi−didj(xj(t)−fj)‖=0 and for t≥T, it satisfies ‖xi(t)−fi‖=‖xj(t)−fj‖. The prescribed time T is bounded, and for any initial state of the system, ∃Tmax>0 such that T≤Tmax.

Assumption 1: The communication topology 𝒢 is undirected, connected, and structurally balanced.

Remark 2: A structurally balanced graph of the control systems’ communication topology is a basic requirement of the controlled systems to realize bipartite formation control. As given in Section 2.1, this requirement is that all agents can be assigned to two distinct groups. The relationships within the same group are cooperative, but those between different groups are antagonistic. More deities are discussed in [27].

2.4 Problem Statements

This study is focused on a class of continuous-time multi-agent systems to implement bipartite formation control tasks, where the main challenges are all listed as

(1) How to realize sampling control for the continuous system. Most existing methods are based on continuous systems and assume an infinite communication resource. In fact, when controlling the physical systems, we also need to detect the system and set a sampling time.

(2) How to realize dynamic event-triggered communication for the controlled system. Most existing methods are time-triggered or strictly event-triggered communication strategies, in which the event-triggered conditions are constant. Designing a dynamic event-triggered condition can further reduce the communication frequencies.

(3) How to realize fixed-time bipartite formation control for agents with a competitive relationship. Most existing formation control approaches consider only the cooperative relationships among agents. However, cooperative and competitive relationships coexist. Only considering one of the relationships is insufficient.

The objective of this study is to realize a sampling-based fixed-time dynamic event-triggered bipartite formation control method for a class of continuous-time multi-agent systems with cooperative and competitive relationships.

3  Main Results

For the multi-agent systems (4), the control law for the agent i is given as

ui(t)=−a1Υiμ(tki)−a2Υiυ(tki)−a3Υi(tki),t∈[tki,tk+1i),(5)

where a1, a2, and a3 are all positive constants, μ>1, 0<υ<1, μ, and υ are ratios of two positive odd numbers, tki represents the triggering time of agent i, and tk+1i represents the next triggering time of agent i. Moreover, k=1,2,… represents the specific number of triggers. Υi(t) is defined as Υi(t)=∑j=1N|aij|(x^i(t)−sign(aij)x^j(t)), where x^i(t)=xi(t)−fi,i=1,2,…,N. Then, we have Υ(t)=[Υ1(t),Υ2(t),…,ΥN(t)]T=ℒx^, where x^=[x^1,x^2,…,x^N]T.

To reduce the real-time computation burden of the system, a combined measurement error equation based on periodic sampling is designed as

Δi(tki+mh)=a1Υiμ(tki)+a2Υiυ(tki)+a3Υi(tki)−a1Υiμ(tki+mh)−a2Υiυ(tki+mh)−a3Υi(tki+mh),(6)

where m=1,2,…, and h>0 is the synchronous sampling period of the agents. Combining Eqs. (6) with (5), we get that

ui(t)=−Δi(tki+mh)−a1Υiμ(tki+mh)−a2Υiυ(tki+mh)−a3Υi(tki+mh).(7)

The compact form of Eq. (7) for t∈[kh,kh+h) is ui(t)=−Δi(kh)−a1Υiμ(kh)−a2Υiυ(kh)−a3Υi(kh). Hence, it follows that

x^˙(t)=−Δ(kh)−a1Υμ(kh)−a2Υυ(kh)−a3Υ(kh),(8)

where Δ=[Δ1,Δ2,…,ΔN]T.

To further reduce the number of event triggers, the dynamic event-triggered condition for agent i is designed as

τ(Δi2(kh)−b1Υi2μ(kh)−b2Υi2υ(kh))≥Φiμ+12(kh)+Φiυ+12(kh),(9)

where τ, b1, and b2 are positive design parameters. Moreover, Φi is the internal dynamic variable, and we have

Φ˙i(t)=−c1Φiμ+12(kh)−c2Φiυ+12(kh)+σ(b1Υi2μ(kh)+b2Υi2υ(kh)−Δi2(kh)),(10)

where Φi(0)>0 and t∈[kh,kh+h). Moreover, c1 and c2 are positive constants, and σ∈(0,1) is a design parameter.

From Eqs. (9) and (10), we get that

Φ˙i(t)≥−(c1+στ)Φiμ+12(kh)−(c2+στ)Φiυ+12(kh).(11)

According to reference [35], we consider a function as

Φ˙i(t)≥−(c1+στ)Φiμ+12(t)−(c2+στ)Φiυ+12(t).(12)

when t≥0, we have Φi(t)≥e∫0tφ(s)dsΦi(0)>0, with φ(t)=−(c1+στ)Φiμ−12(t)−(c2+στ)Φiυ−12(t). Hence, the sample of Φi(t) results in Φi(kh)>0.

When the dynamic event-triggered condition (9) is satisfied, the controller updates its value using the neighbor agent’s state at the triggering time. When the triggering condition is not satisfied, the system employs a zero-order holder (ZOH) to maintain the controller’s output at the control value from the previous trigger time.

In summary, a sampling-based dynamic event-triggered fixed-time bipartite formation algorithm is schematically outlined in Fig. 1.

Figure 1: Diagram of the designed control method.

Remark 3: From Eqs. (9) and (10), it is found that the left items τ(Δi2(kh)−b1Υi2μ(kh)−b2Υi2υ(kh)) is similar to the right items σ(b1Υi2μ(kh)+b2Υi2υ(kh)−Δi2(kh)) in Eq. (9). However, Eq. (10) includes extra negative items −c1Φiμ+12(kh)−c2Φiυ+12(kh), which can further ensure that the condition (9) holds at early states. When the controlled systems achieve stability, the left-hand side of Eq. (9) converges to the origin, but the condition (9) is not satisfied. In other words, at the early stage, communication is more frequent to facilitate rapid convergence, whereas during the transition to stability, it is reduced to prevent unnecessary triggering.

Remark 4: Reference [36] uses continuous state information xi(t) of agent i under ideal conditions, assuming that agents can obtain and transmit information at any moment, without considering that actual embedded systems can only sample and communicate based on a fixed period dictated by the clock. This paper examines the impact of the sensor sampling period on the system, using sampled data xi(kh) that more closely reflect real-world conditions and align with the operational mechanisms of actual systems. Each agent’s state is sampled at a fixed sampling period h and transmitted to neighboring agents and the dynamic event-triggered detector, where the range of the sampling period is provided in the following sections. Since the event detection is based on sampled data, xi(tki) is a subset of xi(kh) and {t0i,t1i,t2i,…}⊆{0,h,2h,…}. The minimum value of tk+1i−tki is one sampling period h, which excludes Zeno behavior.

Theorem 1: When the following conditions are satisfied:

h(a32+a3+a1a3+a2a3)λN<a3−14,(13)

h(a1+a2+a3+1)λN<g0τ+σ−1,(14)

g1(2λN)μ<a1N1−μ2(2λ2)μ+12,(15)

g2N1−υ(2λN)υ<a2(2λ2)υ+12,(16)

where g0=min{c1,c2}, g1=h(b1+a1b1+a2b1+a3b1+a1+a12+a1a2+a1a3)λN+b1, and g2=h(b2+a1b2+a2b2+a3b2+a2+a22+a1a2+a2a3)λN+b2, using the above control and triggering conditions, the multi-agent system (4) can achieve fixed-time bipartite formation within a specific sampling period range.

Proof: Consider a Lyapunov function as

V(t)=12x^T(t)ℒx^(t)+∑i=1NΦi(t),t∈[kh,kh+h).(17)

Let V1(t)=12x^T(t)ℒx^(t) and V2(t)=∑i=1NΦi(t), then V(t)=V1(t)+V2(t). From the above, we know V1(t)≥0 and V2(t)>0, so V(t)≥0. For convenience, let kh=κ. The derivative of V(t) is

V˙(t)=x^T(t)ℒx^˙(t)+∑i=1NΦ˙i(t)=(x^(κ)+(t−κ)(−Δ(κ)−a1Υμ(κ)−a2Υυ(κ)−a3Υ(κ)))Tℒ×(−Δ(κ)−a1Υμ(κ)−a2Υυ(κ)−a1Υ(κ))−c1∑i=1NΦiμ+12(κ)−c2∑i=1NΦiυ+12(κ)+σ(b1Υμ(κ)TΥμ(κ)+b2Υυ(κ)TΥυ(κ)−ΔT(κ)Δ(κ))≤−ΥT(κ)Δ(κ)−a1ΥT(κ)Υμ(κ)−a2ΥT(κ)Υυ(κ)−a3ΥT(κ)Υ(κ)+h(ΔT(κ)ℒΔ(κ)+a12Υμ(κ)TℒΥμ(κ)+a22Υυ(κ)TℒΥυ(κ)+a32ΥT(κ)ℒΥ(κ)+2a1ΔT(κ)ℒΥμ(κ)+2a2ΔT(κ)ℒΥυ(κ)+2a3ΔT(κ)ℒΥ(κ)+2a1a2Υμ(κ)TℒΥυ(κ)+2a1a3Υμ(κ)TℒΥ(κ)+2a2a3Υυ(κ)TℒΥ(κ))−c1∑i=1NΦiμ+12(κ)−c2∑i=1NΦiυ+12(κ)+σ(b1Υμ(κ)TΥμ(κ)+b2Υυ(κ)TΥυ(κ)−ΔT(κ)Δ(κ)).(18)

Combine Lemma 3 and the following inequality

aTℒb≤12aTℒa+12bTℒb,∀a,b∈Rn.(19)

Then, we obtain that

V˙(t)≤|ΥT(κ)||Δ(κ)|−a1ΥT(κ)Υμ(κ)−a2ΥT(κ)Υυ(κ)−a3ΥT(κ)Υ(κ)+h(1+a1+a2+a3)λNΔT(κ)Δ(κ)+h(a1+a12+a1a2+a1a3)λNΥμ(κ)TΥμ(κ)+h(a2+a22+a1a2+a2a3)λNΥυ(κ)TΥυ(κ)+h(a3+a32+a1a3+a2a3)λNΥT(κ)Υ(κ)−c1∑i=1NΦiμ+12(κ)−c2∑i=1NΦiυ+12(κ)+σ(b1Υμ(κ)TΥμ(κ)+b2Υυ(κ)TΥυ(κ)−ΔT(κ)Δ(κ))≤(14−a3+h(a3+a32+a1a3+a2a3)λN)ΥT(κ)Υ(κ)−a1ΥT(κ)Υμ(κ)−a2ΥT(κ)Υυ(κ)+ΔT(κ)Δ(κ)+h(1+a1+a2+a3)λNΔT(κ)Δ(κ)+h(a1+a12+a1a2+a1a3)λNΥμ(κ)TΥμ(κ)+h(a2+a22+a1a2+a2a3)λNΥυ(κ)TΥυ(κ)−c1∑i=1NΦiμ+12(κ)−c2∑i=1NΦiυ+12(κ)+σ(b1Υμ(κ)TΥμ(κ)+b2Υυ(κ)TΥυ(κ)−ΔT(κ)Δ(κ)).(20)

when conditions (13)–(16) are met, we get that

V˙(t)≤−a1ΥT(κ)Υμ(κ)−a2ΥT(κ)Υυ(κ)+ΔT(κ)Δ(κ)+h(1+a1+a2+a3)λNΔT(κ)Δ(κ)+h(a1+a12+a1a2+a1a3)λNΥμ(κ)TΥμ(κ)+h(a2+a22+a1a2+a2a3)λNΥυ(κ)TΥυ(κ)−c1∑i=1NΦiμ+12(κ)−c2∑i=1NΦiυ+12(κ)+σ(b1Υμ(κ)TΥμ(κ)+b2Υυ(κ)TΥυ(κ)−ΔT(κ)Δ(κ))≤−a1ΥT(κ)Υμ(κ)−a2ΥT(κ)Υυ(κ)+h(1+a1+a2+a3)λNΔT(κ)Δ(κ)+(h(a1+a12+a1a2+a1a3)λN+b1)Υμ(κ)TΥμ(κ)+(h(a2+a22+a1a2+a2a3)λN+b2)Υυ(κ)TΥυ(κ)−c1∑i=1NΦiμ+12(κ)−c2∑i=1NΦiυ+12(κ)+(σ−1)(b1Υμ(κ)TΥμ(κ)+b2Υυ(κ)TΥυ(κ)−ΔT(κ)Δ(κ)).(21)

Combining the dynamic event-triggered condition (9), we obtain that

V˙(t)≤−a1ΥT(κ)Υμ(κ)−a2ΥT(κ)Υυ(κ)+g1Υμ(κ)TΥμ(κ)+g2Υυ(κ)TΥυ(κ)−(c1−1−σ+h(1+a1+a2+a3)λNτ)∑i=1NΦiμ+12(κ)−(c2−1−σ+h(1+a1+a2+a3)λNτ)∑i=1NΦiυ+12(κ).(22)

From Lemma 4, we derive that

ΥT(κ)Υμ(κ)=∑i=1NΥiμ+1(κ)=∑i=1N(Υi2(κ))μ+12≥N1−μ2(∑i=1NΥi2(κ)).(23)

and

∑i=1NΥi2(κ)=ΥT(κ)Υ(κ)=(ℒx^)T(ℒx^)=x^Tℒ2x^=(ℒ12x^)Tℒ(ℒ12x^)≥λ2x^Tℒx^=2λ2V1.(24)

By substituting Eq. (24) into Eq. (23), we get that

ΥT(κ)Υμ(κ)≥N1−μ2(2λ2V1)μ+12.(25)

Similarly, we obtain that

ΥT(κ)Υυ(κ)≥(2λ2V1)υ+12Υμ(κ)TΥμ(κ)≤(2λNV1)μΥυ(κ)TΥυ(κ)≤N1−υ(2λNV1).(26)

Substituting Eqs. (25) and (26) into Eq. (22), and combining them with Lemma 4, we get that

V˙(t)≤−a1N1−μ2(2λ2)μ+12V1μ+12−a2(2λ2)υ+12V1υ+12+(g1(2λN)μV1μ2)2+(g2N1−υ(2λN)υV1υ2)2−N1−μ2(c1−1−σ+h(1+a1+a2+a3)λNτ)V2μ+12−(c2−1−σ+h(1+a1+a2+a3)λNτ)V2υ+12.(27)

when conditions (15) and (16) are satisfied, we get that

V˙(t)≤−g3(V1μ+12+V2μ+12)−g4(V1υ+12+V2υ+12)≤−g321−μ2Vμ+12−g4Vυ+12.(28)

where g3=min{a1N1−μ2(2λ2)μ+12,N1−μ2(c1−1−σ+h(1+a1+a2+a3)λNτ)}, and g4=min{a2(2λ2)υ+12,c2−1−δ+h(1+a1+a2+a3)λNτ}.

From Eq. (28) and Lemma 1, we conclude that the multi-agent systems (4) can achieve fixed-time bipartite formation, with the settling time as

T≤Tmax=π(μ−1)21−μ2g3g4(29)

Since the agent’s event detection occurs within the sampling period h, the minimum interval for triggering an event is h, thereby fundamentally eliminating the Zeno phenomenon.□

4  Simulation Analysis

4.1 Example 1

Consider a system composed of seven agents, with the communication topology shown in Fig. 2, where solid lines represent cooperative relationships and dashed lines indicate competitive relationships. From Fig. 2, it is obtained that the second smallest eigenvalue of ℒ is λ2=0.75, and the largest eigenvalue is λN=3.80. The initial state and initial dynamic variable values of the system are x(0)=[−16−25−10062011]T and Φi(0)=40. The parameters in Eqs. (5), (9) and (10) are designed as a1=3, a2=1, a3=1, μ=11/9, υ=7/9, f1=−1, f2=−2, f3=1, f4=2, f5=3, f6=4, f7=−3, b1=0.09, b2=0.01, c1=1, c2=1, σ=0.01, and τ=10. Using these design parameters and Eq. (2), the settling time is T≤Tmax=18.6 s, where Tmax is the maximum settling time, providing an upper bound estimate. From Theorem 1, the system’s sampling period can be ensured to be within h≤0.033 s.

Figure 2: Communication topology of the controlled multi-agent systems.

When h=0.01 s, under the same parameters, the state evolution and triggering instants of the system using different sampled-data-based event-triggered mechanisms are illustrated in Fig. 3. According to reference [42], the static event triggering condition is designed as Δi2(kh)−b1Υi2μ(kh)−b2Υi2υ(kh)≥0. From Fig. 3, it can be observed that under nearly identical convergence rates, the number of triggers for agents 1–7 under dynamic event-triggered strategy are 20, 23, 23, 22, 18, 12, and 20, respectively, whereas under static event-triggered strategy, the triggers are 76, 77, 81, 79, 72, 72, and 71, respectively. Compared to the static event-triggered strategy in [42], the dynamic event-triggered strategy proposed in this paper can further reduce the number of triggers by 73.8%, thereby reducing the system’s communication resources.

Figure 3: Fixed-time bipartite formation using different event-triggered mechanisms.

Furthermore, this paper conducts simulations with varying sampling periods, as shown in Fig. 4. The effectiveness is characterized by V1(t)=0.5xT(t)ℒx(t). Fig. 4 shows that: (1) With a smaller sampling period, event detection frequency is higher, and the controller updates more frequently, allowing timely adjustment of the system state. Thus, the collection process is smoother with less fluctuation. However, shorter sampling periods increase computational requirements, making real-time implementation more difficult. (2) With a larger sampling period, event detection frequency is lower, and the trigger count decreases. However, as the sampling period approaches a stable value, the controller cannot adjust the system state in time, leading to poor system effectiveness because the event triggering condition is often not satisfied, thereby increasing the trigger count. Therefore, selecting an appropriate sampling period is critical to the stability of real systems and to the number of triggers, which is one of the research objectives of this paper.

Figure 4: The impact of different sampling periods on the controlled multi-agent systems.

Finally, to verify the superiority of this algorithm, a comparative experiment with the existing method in [43] was conducted, and the results are shown in Fig. 5. Figs. 3a and 5a are the same. Fig. 5c changes the initial state to x′(0)=[−180−290−110080230120]T and the initial dynamic variable value to Φi(0)=500 and σ=0.001, with other parameters as in Fig. 5a. Comparing Fig. 5b and d, the collection time of the algorithm in reference [43] increases with the increase in the initial state. In contrast, the upper bound on the algorithm’s collection time in this paper is independent of the initial state. Comparing Fig. 5a and b, with the same initial state, the actual states coincide at t=0.9 s and t=7 s, indicating that the algorithm in this paper has a faster convergence speed compared to the existing algorithm in reference [43].

Figure 5: The results of different methods for the controlled multi-agent systems.

4.2 Example 2

Consider a system composed of five unicycle-like nonholonomic robots [45]. The system’s communication topology is shown in Fig. 6a, and the schematic diagram of each unicycle-like nonholonomic robot is shown in Fig. 6b. The dynamic model of each robot is given as

{p˙i=vicos⁡ϑi,q˙i=visin⁡ϑi,ϑ˙i=ϖi.(30)

where i=1,2,…,5, Θi=[pi,qi]T represents the position of robot i in the global coordinate system, ϑi represents the orientation of robot i in the coordinate system, vi is the linear velocity input, and ϖi is the angular velocity input. Let the position-like output as Θ⌢i=[p⌢i,q⌢i]T, with

{p⌢i=pi+rcos⁡ϑi,q⌢i=qi+rsin⁡ϑi.(31)

where r is a small positive constant.

Figure 6: Unicycle-like nonholonomic robots.

From Eqs. (30) and (31), we have

{p⌢˙i=v⌢i,q⌢˙i=ϖ⌢i,ϑ˙i=1r(−v⌢isin⁡ϑi+ϖ⌢icos⁡ϑi).(32)

where v⌢i=vicos⁡ϑi−rϖisin⁡ϑi and ϖ⌢i=visin⁡ϑi+rϖicos⁡ϑi are the virtual inputs for robot i. Since the dynamics of p⌢i and q⌢i have already been transformed into a double-integrator model, we can apply the control methods similar to those in Section 3.

The controller is analogous to Eq. (5).

v⌢i(t)=−a1Piμ(tki)−a2Piυ(tki)−a3Pi(tki),t∈[tki,tk+1i),ϖ⌢i(t)=−a1Qiμ(tki)−a2Qiυ(tki)−a3Qi(tki),t∈[tki,tk+1i).

where Pi(t)=∑j=1N|aij|(p⌢i(t)−fi1−sign(aij)(p⌢j(t)−fj1)) and Qi(t)=∑j=1N|aij|(q⌢i(t)−fi2−sign(aij)(q⌢j(t)−fi2)). The dynamic event-triggered conditions are given by Eq. (9).

The initial states of the system are (p1,q1,ϑ1)=(2,3,0), (p2,q2,ϑ2)=(−1,2,π), (p3,q3,ϑ3)=(−3,−1,0), (p4,q4,ϑ4)=(4,4,π/2), (p5,q5,ϑ5)=(0,−1,0), and Φi(0)=4. The desired formation vectors are f1=[1,2]T, f2=[−1,−2]T, f3=[−2,−3]T, f4=[2,3]T, and f5=[3,2]T. Other parameters are set as h=0.01, r=0.5, a1=2, a2=0.5, a3=0.5, μ=11/9, υ=7/9, b1=0.3, b2=0.1, c1=2, c2=1, σ=0.01, and τ=10. The position-like output is shown in Fig. 7a, where p⌢i and q⌢i converge within 0.2 s, exhibiting a fast convergence rate. The dynamic event-triggered intervals are shown in Fig. 7b, which indicates that the designed dynamic event-triggered strategy has a relatively small number of triggers.