1. Introduction

Recently, the cooperative control of MASs has been extensively studied across numerous scientific fields, including urban transportation scheduling, distributed power generation, and unmanned aerial vehicles formation control, etc.[1–8]. As a primary problem in the cooperative control of MASs, consensus control has achieved many significant results [9–15].

Lag consensus, as an unusual case of consensus, has recently attracted considerable attention for its utilisation in preventing congestion [16–19]. Assuming to be a lag time, lag consensus denotes that the followers’ states at time are same as the leader’s states at time. The asymptotic stability of lag consensus has been studied for MASs [16–19]. However, the asymptotic lag consensus cannot satisfy the requirements of practical engineering applications. Finite-time lag consensus (FNTLC) control has recently gained attention for enhancing the convergence speed of the MASs [20–22]. However, FNTLC control is subject to initial states of MASs. For resolving the problem, fixed-time lag consensus (FXTLC) control has been presented [23–26]. But FXTLC control depends upon the control protocol parameters. To address the issue, prescribed-time lag consensus (PTLC) control has been proposed. The prescribed-time lag consensus control protocol makes the arbitrary assignment of convergence time possible, which is independent of both the controller parameters and the initial states of the agents. Obviously, PTLC has important academic value and practical significance.

The problem of prescribed-time partial component lag consensus of MASs subject to both disturbances and switching topologies is investigated [27]. A PTLC control scheme with adaptive gains for lag consensus is introduced [28]. It should be noted that the above works, including [27,28], all assume that the communications among agents are continuous. However, continuous communication results in excessive resource consumption and may compromise the system’s stability due to the limited computational capabilities and storage capacity in applications. Obviously, this research of discontinuous communication control protocols is important.

To reduce resource consumption, a control scheme for successive lag consensus of nonlinear MASs is presented based on event-triggered mechanism with both constant and time-varying lag consensus scenarios [29,30]. By a novel static event-triggered mechanism, reference [31] designs a control protocol to research the lag consensus of MASs. Reference [1] develops a dynamic memory event-triggered mechanism to research the lag consensus control of multi-unmanned aerial vehicle systems under hybrid attacks. The aforementioned studies have effectively addressed the resource constraint of agents in practical scenarios using event-triggered mechanisms, but they rarely consider the convergence time at the same time. It is more challenging to create a control scheme for reducing resource consumption while setting the prescribed convergence time at will for the lag consensus control problem.

Notably, the aforementioned works about lag consensus have primarily considered cooperative relationships while neglecting competitive relationships within agents. However, competitive relationship is widely present in natural systems, social and engineering systems. In MASs, cooperation and competition often coexist. Motivated by competitive relationships and lag consensus, the problem of lag bipartite consensus is presented [32–35]. Reference [34] investigates the LBC control problem of linear MASs under saturating input, but utilizes continuous communication control protocol. Using the event-triggered mechanism, reference [35] discusses the LBC control of nonlinear MASs with external disturbances, but just considers the asymptotic convergence.

Significantly, the aforementioned studies on LBC assume that the leader’s states are available. Obviously, it is unrealistic in numerous practical applications. So, to devise an observer is important for estimating the states of leader. In light of the discussions above, the paper comprehensively studies the PTLBC of nonlinear MASs via the PTDO and event-triggered mechanism. The main contributions of this paper are summarized as follows:

1. 1). Different from the system investigated in [36], this paper studies a second-order integrator system with nonlinear components, which extends the research scope of relevant works.
2. 2). Different from the method adopted in [28], this paper proposes a novel event-triggered mechanism by combining the prescribed-time time-varying function with the measurement error.
3. 3). Considering that follower agents cannot directly access the state of the leader agent, a prescribed-time dynamic observer is designed to estimate the leader’s state information.
4. 4). To address the problem of communication congestion, a prescribed-time lag-bipartite consensus control protocol based on the event-triggered mechanism is designed by integrating prescribed-time control with the event-triggered mechanism.

This paper is organized into the following chapters: The problem statement and preliminaries are shown in Section 2. Section 3 proposes a novel PTDO, studies an innovative event-triggered mechanism, develops a PTLBC control protocol to realize LBC within the arbitrarily prescribed time, and discusses the Zeno behaviour in detail. The theoretical results are validated through a numerical example in Section 4. Section 5 gives the conclusions.

2. Preliminaries and problem statement

2.2. Some assumptions and lemmas

The section lists the useful assumptions and lemmas.

Assumption 1 ([26]). Let the graph be structurally balanced. There is a spanning tree rooted at the leader in the graph .

Assumption 2 ([37,38]). There always exist positive real numbers and satisfying the below equation, where vectors .

Lemma 1 ([14]). If there is a matrix , where , and all elements of are greater than or equal to zero, where , then the structural balance of the directed signed graph holds.

Lemma 2 ([19]). The below inequality is true

where , are any given vectors with proper dimensions and is a positive constant.

Lemma 3 ([26]). If there is a positive define matrix , then is positive define,where the vectors , and .

Lemma 4 ([26]). Analyze the following systems

(1)

where is the state, is the time-bounded vector and indicates the system initial state. The time-varying function below is first introduced [36].

(2)

and , , in which , is a prescribed time, and is the initial time. is defined as

(3)

For system, if there is a Lyapunov function which satisfies

(4)

where is defined in, and , then the system is prescribed-time stable with the prescribed time . It can get the solution below.

(5)

Lemma 5 ([38]). For system，if there exist a continuously differentiable and positive-definite function , two positive constants , , and a scalar satisfying , such that

(6)

then the system can realize the prescribed-time stability. It gets the solution as

(7)

2.3. Problem statement

Analyze the nonlinear MASs system of followers and one leader. The leader’s dynamics is modelled below.

(8)

where represents the position, represents the velocity, represents nonlinear functions, and represents the acceleration or the control input.

The i-th follower’s nonlinear dynamics is modelled below.

(9)

where represents the position, represents the velocity, indicates nonlinear functions satisfying Lipschitz condition, and indicates the control input.

Definition 1. For the lag time and the arbitrarily prescribed time , if the solution of and satisfies

(10)

and

(11)

then the PTLBC is said to be achieved for the nonlinear MASs and, where is defined in Lemma 1.

3. Main results

3.1. The design of PTDO

Suppose Assumption 1 holds. For the followers, the PTDO below is designed to estimate the states information of the leader.

(12)

where

Here, the estimate of the leader state and are denoted as and , respectively. is the observer gains, is as follows

(13)

where , and . Set three times , , and , where and represent the prescribed convergence time of the position states and velocity states for the state observer, respectively; denotes the prescribed convergence time of the closed-loop nonlinear system under the proposed controller.

Theorem 1. For the PTDO, estimation errors are categorised into velocity errors and position errors . If , , and , then and will converge to zero within the prescribed time and , respectively.

Proof. The convergence analysis of prescribed time dynamic observer will be implemented by the following two steps.

1. (1). For convenience, let be , and be . Considering the observer error , by differentiating the first equation in, we obtain . Selecting the following Lyapunov function

(14)

the derivative of is as follows

(15)

From (14), it can be obtained that . It concludes that . Then, substituting these inequalities into, it obtains

(16)

where , i.e., . Especially, the following solution can be obtained

In other words, within the arbitrarily prescribed time , converges to zero according to Lemma 4.

1. (2). Similar to the derivation processes of Equations (14) and (15) and, it obtains that within the arbitrarily prescribed time , will converge to zero. That is to say, the PTDO (12) is effective.

3.2. Event-triggered mechanism

The innovative event-triggered mechanism is devised in the section. For simplicity, let be , be , be , and be . Define the lag bipartite consensus error as and , respectively. Let two auxiliary states be

(17)

Then, let , where , . Let for , and for . Define the state measured error . Denote . Choose the event-triggered function as follows

(18)

where , , are positive constants, and defined in (2). If , the position and velocity are update to and , and transmitted to the MASs, where

3.3. PTLBC control protocol

Via the above event-triggered mechanism and PTDO, the PTLBC control protocol is designed. For , we design the i-th follower’s PTLBC control protocol as follows

(19)

where and corresponds to the case when in (3). Besides, represents the triggering time of the k-th event for the follower , where . For , and record the transmission states of follower at . When , the and transition to and in the control protocol (19). Then, it obtains

(20)

Taking the derivative of , one gets

(21)

Consequently, we obtain the feedback control system of (8) and (9) together with (19) and (21) as

(22)

The system (22) has the concise form below,

(23)

where , , , , , and .

Substituting state measurement error into (23), it has

(24)

Theorem 2. Assume that Assumptions 1 and 2 hold. Consider the MASs (8) and (9) with the PTLBC control protocol (19). If the following conditions can be satisfied,

(25)

where,, and definition . Then the lag bipartite consensus errors and converge to 0 within arbitrarily prescribed time , i.e., using the proposed control protocol (19), the PTLBC of the MASs in (8) and (9) can realized.

Proof. Let . It can choose the Lyapunov function below,

(26)

where

Here, and are explained in Lemma 3. It can get that is positive define based on Lemma 2. Using the first inequality of (25), it gets . Then

(27)

Based on Assumption 2, Lemma 3 and , one gets

(28)

From the trigger condition (18), one gets

(29)

where

(30)

Applying Lemma 3, it gets

(31)

The can be simplified as follows

(32)

Combining (28), (29), (30) and (32), it gets

(33)

Let , then

(34)

and

(34)

Then, by Lemma 5, the system is prescribed-time stable. The following solution can be obtained

That is, the lag bipartite consensus error and converges to zero within . Then, the PTLBC of the MASs and can be achieved.

Theorem 3. Considering the MASs (8) and (9) with the PTLBC control protocol, the Zeno behaviour can be excluded.

Proof. Taking the derivative of , it has

Let , and then

According to (34), it gets

(36)

Then, according to (26), it obtains

(37)

Hence

(38)

Obviously

(39)

And then

(40)

Considering , it obtains

(41)

At trigger instant , it follows

(42)

According to (18), at trigger instant

(43)

combining with (42) and (43), it follows

(44)

Next, using reduction to absurdity, it will be proved that the trigger interval . Assume , it can be deduced from that , which leads to a contradiction. Hence, the assumption does not hold, that is, the proposed event-triggered mechanism excludes Zeno behaviour.

4. Numerical results

The simulation results of nonlinear MASs (8) and (9) under the PTLBC control protocol (19) are given in the section.

In S1 Fig, the nonlinear MASs with one leader and six followers is presented. The solid line indicates cooperation relationships, and the dashed line indicates the competitive relationships of the followers. The followers are partitioned into two subsets and , which means that . The Laplacian matrix is below.

Obviously, . For the leader, the control input is expressed as , and the nonlinear part is expressed as . The nonlinear parts of the followers are expressed as Obviously, we can verify that Assumption 2 is satisfied with and . Set , and . The conditions (25) of Theorem 2 are satisfied when we choose , . The initial states values are given by , , and respectively. Setting the parameters, , and in the event-triggering function, the function is given as follows

The initial values and of the PTDO are set to and , respectively. S2 Fig shows that converges to 0 within the arbitrary prescribed time , and converges to 0 within the arbitrary prescribed time , with the lag time . This demonstrates that the PTDO is effective.

S3 Fig A and S3 Fig B provide the detailed variation curves of the position and velocity for all agents. It indicates that LBC is realized within the prescribed time , with the lag time .

S4 Fig presents the event intervals for follower, which confirms that Zeno behaviour does not occur.

5. Conclusion

The paper has researched the PTLBC control problem of nonlinear MASs. The PTDO has been devised for followers to acquire the states of the leader, and the event-triggered mechanism has been devised to save communication consumption. Via the PTDO and the event-triggered mechanism, the PTLBC control protocol has been designed. Utilizing Lyapunov stability analysis method, it has been rigorously proven that the nonlinear MASs (8) and (9) can achieve PTLBC. Furthermore, the Zeno behaviour of the controllers has been excluded. And then, the theoretical results were verified through the simulation example. We will further explore the problem subject to cyber attacks in our future work.

Supporting information