Dynamic Event-Triggered Consensus Control for Input Constrained Multi-Agent Systems With a Designable Minimum Inter-Event Time

Meilin Li , Yue Long ,,, Tieshan Li , Senior,,Hongjing Liang ,,, and C.L.Philip Chen ,,

Abstract—This paper investigates the consensus control of multi-agent systems (MASs) with constrained input using the dynamic event-triggered mechanism (ETM).Consider the MASs with small-scale networks where a centralized dynamic ETM with global information of the MASs is first designed.Then, a distributed dynamic ETM which only uses local information is developed for the MASs with large-scale networks.It is shown that the semi-global consensus of the MASs can be achieved by the designed bounded control protocol where the Zeno phenomenon is eliminated by a designable minimum inter-event time.In addition, it is easier to find a trade-off between the convergence rate and the minimum inter-event time by an adjustable parameter.Furthermore, the results are extended to regional consensus of the MASs with the bounded control protocol.Numerical simulations show the effectiveness of the proposed approach.

I.INTRODUCTION

IN recent years, research on multi-agent systems (MASs)has covered a wide range of scientific fields, including intelligent transportation system, mobile robot and so on[1]-[3].It has a far-reaching and significant impact on the development of engineering technology.A typical problem of MASs is consensus control, i.e., making each agent converge to a common state driven by the controller.Up until now, a large number of research works have contributed to determining how to handle the consensus control problem of the MASs with different types of system dynamics [4]-[6].

As is known to all, input constraints are common in engineering systems, which is inevitable with practical implementation.At present, there are some related results to solving input constrained MASs consensus control.In [7], the secure consensus control of MASs subject to different saturation levels and multiple denial-of-service attacks was studied.The control protocols were designed by adopting a low-gain feedback method, and the sufficient conditions for secure consensus control were established.In [8], for the MASs with parameter uncertainties and input saturation, a novel low-gain feedback law was established to reach robust semi-global consensus.In [9], an improved low-and-high gain feedback method was proposed to address consensus control of leader-following MASs with actuator faults and input constraints, and the finite-time stability of the MASs was realized.In [10], a fully distributed control protocol with an expected consensus result was proposed for the second-order MASs constrained by both velocity and input.In [11], an observer-based consensus algorithm was presented for input constrained MASs under switching communication topology, and the leader-following consensus was reached.In the above results of consensus control of the input constrained MASs, the control input of each agent needs to be updated continuously, which undoubtedly increases the computing cost, processor power consumption,and actuator loss of the MASs.This is because each agent is usually composed of some energy limited modules, such as embedded microprocessor, battery-powered module, and multi-functional sensor.Therefore, the design of energy-saving control strategy can reduce the continuous update of control input among agents and is in line with the actual demand[12].

Fortunately, the event-triggered mechanism (ETM) is an effective method to solve this problem [13]-[16].At present,the consensus control of MASs based on ETM has many related results [17]-[19].However, these results focus on the research of traditional ETMs, which is known as static ETM.In [20], a new ETM called a dynamic ETM was proposed.It is worth mentioning that, the triggered event under the dynamic ETM can appropriately and significantly prolong the average inter-event time (IET) in most cases, so as to further reduce the controller update number, compared with the static ETM[21], [22].Thus, research on a dynamic ETM was extended to the coordination control of MASs.In [12], an adaptive consensus protocol based on distributed dynamic ETM was given to solve the leaderless and leader-following consensus control of MASs.In [23], a decentralized dynamic ETM was proposed based on output feedback which can make the state of each agent achieve consensus without continuous communication.In [24], a novel dynamic ETM with an adjustable minimum IET was presented for the linear MASs, which can ensure the consensus of MASs without the Zeno phenomenon.For heterogeneous MASs with nonuniform communication delay, a dynamic periodic ETM was proposed in [25], which effectively reduced the communication among agents and achieved the output consensus of the MASs.In order to realize the consensus control of MASs with faults, an innovative framework of fault estimation and consensus control based on dynamic ETM was proposed in [26], which can compensate for the performance loss caused by faults and save communication resources simultaneously.In [27] and [28], effective consensus protocols based on dynamic ETMs were designed for the MASs with input constraints, respectively.However,these two results only focus on consensus control with the undirected network topology, rather than the directed network topology.Moreover, the minimum IET has a complex relationship with system matrices and can not be adjusted according to parameters, which is not beneficial for engineering implementation.Accordingly, it is necessary to design effective control methods to remedy the shortcomings of the above results.

Inspired by the above discussions, this paper aims to solve the consensus control problem of MASs with constrained input under directed communication topology.Inevitably, we will encounter the following difficulties and challenges.

1) From the perspective of further reducing energy use and cost of the control protocol updating, determining how to design a more effective ETM for different scale network structures is the first problem to be considered.

2) The second problem to be solved is determining how to design a bounded control protocol based on novel ETM for the input constrained MASs under the directed graph, so that the consensus control of MASs can be achieved with the conditions of energy saving and avoiding actuator saturation.

3) Some existing results such as [29], [30] show that the relationship between the expression of the minimum IET and the system matrix is complex.Hence, determining how to establish an expression of the minimum IET that can not only avoid the complex relationship with the system matrix, but also can be adjusted by the parameter, is the third problem to be addressed.

To address the problems above, a centralized dynamic ETM and a distributed dynamic ETM are proposed, respectively.Then, the dynamic event-triggered-based bounded control protocol is designed.Four main contributions are detailed as follows.

1) Two novel dynamic ETMs known as the centralized dynamic ETM and distributed dynamic ETM are proposed,respectively.It is shown that the proposed centralized dynamic ETM is more suitable for small-scale MASs and the distributed dynamic ETM is more suitable for the large-scale MASs.Different from the ETM in [31], the dynamic ETMs proposed in this paper can ensure a larger IET, which can further reduce the energy use and cost of the control protocol updating.

2) The bounded event-triggered control protocols based on two novel dynamic ETMs under the directed graph are proposed.Moreover, compared with [32], which only focuses on the control of general linear MASs, in this paper, the linear MASs with constrained input is considered, which is more common in practical engineering systems.Then, by designing the bounded control protocols based on dynamic ETMs the semi-global consensus is achieved.

3) With the designed centralized dynamic ETM and the distributed dynamic ETM, the expressions of the minimum IET are established which avoids the complex relationship with the system matrix.Finally, a trade-off between minimum IET and convergence rate can be easily found by adjusting the parameter.In particular, a strict positive lower bound of the minimum IET can be found with the centralized dynamic ETM which is more meaningful to the application of ETC in engineering practice.

4) Furthermore, by relaxing the constraint on the eigenvalue of matrixA, the results can be extended to handle the regional consensus control of the input constrained MASs by the designed dynamic ETMs.

The rest of this paper is structured as below.The preliminaries and problem statement are stated in Section II.The bounded control protocol based on dynamic ETM is investigated in Section III.In Section IV, the effectiveness of the control scheme is verified by numerical simulation.The conclusion is given in Section V.

Notations: Denote Rnas an-dimensional Euclidean real space.Let matrixXbe a positive (or semi positive) definite matrix; thenXTis the transpose ofX.The maximum eigenvalue ofXis denoted by λmax(X) and λmin(X) is the minimum eigenvalue.tr(X) is the trace ofX.INis theN×Nidentity matrix.For matricesX∈Rm×nandY∈Rp×q,X?Y∈Rmp×nqrepresents their Kronecker product.The notation ‖X‖ is the induced two-norm ofX.

II.PRELIMINARIES AND PROBLEM STATEMENT

A. Graph Theory

a node can be reached by any other nodes.

B. Problem Statement

Consider there areN(N≥2) agents forming the MASs, in which each agent is subjected to actuator saturation.Fori=1,...,N, the dynamic of theith agent is shown as follows:

whereAandBare the system matrices;xi∈Rnandui∈Rmare the state vector and the control input, respectively.The symbol sat(ui) is the saturation function andsat(ui)=sign(ui)min{Δ,|ui|} , where Δ ＞0 is the saturation threshold.

The main purpose of this paper is to design bounded dynamic event-triggered-based control protocols for MASs with small-scale and large-scale network structures such that semi-global consensus can be achieved.To accomplish this mission, the following problems must be solved.

Problem 1(Semi-global consensus by centralized dynamic ETM with small-scale network structure): For input constrained MASs (1), a centralized dynamic ETM which utilizes global information of the MASs and a bounded control inputui(t)are designed, so that for any given bounded set Ωc?RnN+1,the semi-global consensus can be achieved as limt→∞‖xi(t)-xj(t)‖=0,?i,j∈N as long as for any (xi(0),ζ(0))∈Ωc.Thus,a designable minimum IET with strict positive lower bound should be guaranteed.

Problem 2(Semi-global consensus by distributed dynamic ETM with large-scale network structure): For input constrained MASs (1), a distributed dynamic ETM which utilizes local information of the MASs and a bounded control inputui(t) are designed, so that for any given bounded set Ωd?RN(n+1),the semi-global consensus can be achieved as limt→∞‖xi(t)-xj(t)‖=0,?i,j∈N as long as for any (xi(0),ζi(0))∈Ωd.Thus, a designable minimum IET with lower bound greater than zero should be guaranteed.

To solve Problems 1 and 2, the following assumptions and lemmas are introduced.

Assumption 1: The pair (A,B) is controllable.

Assumption 2: All the eigenvalues of matrixAare on the imaginary axis.

It is noteworthy that, as pointed out in [33], Assumption 2 is necessary for the input constrained MASs to realise semiglobal consensus.

Assumption 3: The graph G is directed and strongly connected.

Lemma 1([33],[34]): Under Assumptions 1 and 2, consider there is a constant γ ＞0; then a parametric Lyapunov equation is given as follows:

whereP=P(γ) is a unique positive definite matrix andδis a positive design parameter.Note that there is a Lyapunov equation as shown below:

Remark 1: The parametric Lyapunov equation in Lemma 1 is a special form of the general algebraic Riccati equation,which is designed by Zhouet al.[33], and has many unique advantages compared with the general algebraic Riccati equation.More information regarding the proof of the above properties and some other problems of the parametric Lyapunov equation can be seen in Appendix A of [34].

where the meaning of variablesυ,Υ,Lˉ are given in Lemma 2 andυi=1.For the strongly connected graph G, θ(L) is called the general algebraic connectivity.

III.BOUNDED CONTROL PROTOCOL DESIGN BASED ON DYNAMIC ETM

In this section, the bounded control protocols based on centralized dynamic ETM and distributed dynamic ETM are proposed for the input constrained MASs (1) to realise semiglobal consensus.Then, the expressions of the minimum IET with an adjustable parameter of the dynamic ETMs are established.

A. Solution to Problem 1 by Centralized Dynamic ETM

Considering the MASs (1) with small-scale network structure, a bounded control protocol based on centralized dynamic ETM is designed in this part.

First of all, definezi(t) as a combined measurement variable [37] whose expression is given as follows:

Noting that in the centralized case, all the agents share the same event-triggered instants.Hence, the triggering instants of the MASs are defined ast0,t1,...,tk,..., wheret0=0.Then,the measurement error of agentiis given as follows:

For the sake of designing the novel centralized dynamic ETM, based on the adjustable parameterγin the parametric Lyapunov equation (2), the differential equation is given as below:

where ζ=ζ(t) is the internal dynamic variable and the initial value of ζ(t) satisfies ζ(0)＞0.Then, the centralized dynamic ETM is proposed by the rule as follows:

The dynamics ofg(t) ont∈[tk,tk+1) are given as follows:

Recalling (23), one has

Similarly, we can get the inequality as shown below:

According to (7), (25) and (26), one has that

Next, define a differential equation as below:

where ω (t) is a variable with ω (tk)=g(tk)=0.From (28), one has that ω(t) is an increasing function.Hence, one has thatg(t)≤ω(t).Define ω(tk+τ)=and recalling (28), we can get that

If we further relax the constraint on the eigenvalues of matrixA, by borrowing some ideas in [34] and using similar technical methods in Theorem 1, one can get a corollary as shown below.

Remark 2: In fact, for event-triggered sampling, any non-Zeno behavior is mathematically acceptable, whereas the sampling rate is always limited in real-time control systems [39].Even if the Zeno behavior is avoided and the minimum IET of all agents are greater than zero; unfortunately, it is still not enough to ensure that the solution can be realized by physical devices.Because technically, although the minimum IET is positive, sometimes they may become too small such that no physical hardware can keep up with the action speed required by the event-triggered algorithm [40].Therefore, in practical applications, a computable minimum IET can better determine whether the limitation of the sampling rate is satisfied than a positive minimum IET without an exact value [41].This further illustrates that the centralized dynamic ETM proposed in this paper, which can obtain an adjustable minimum IET with strict positive lower bound, is very meaningful to the application of the ETC method in engineering practice.

B. Solution to Problem 2 by Distributed Dynamic ETM

In this section, considering the consensus control of MASs(1) with large-scale network structure, a distributed dynamic ETM is proposed.It should be noted that in the distributed case, each agent has its own event-triggered instants.Therefore, some intermittent instantsare denoted as the triggering times of agenti.Then, define the measurement error as follows:

Next, we will prove that with the designed distributed dynamic ETM (33), there is a minimum IET with adjustable parameter which greater than zero.Recalling (31), the time derivative ofei(t) is shown as follows:

Remark 4: Inspired by [31], a centralized dynamic ETM and a distributed dynamic ETM are proposed in this paper, respectively.In the centralized case, only one event-triggered detector is required, so the detector cost is very low.However, the detector must receive the information transmitted from all agents to judge whether the triggering condition is violated,which may result in information congestion or information overload, particularly in large-scale networks.Therefore, the centralized dynamic ETM is more suitable for the MASs with small-scale networks structure.In the distributed case, every agent is equipped with an event-triggered detector, so the cost of verifying event-triggered conditions may be quite high.However, the advantage is that each agent can independently determine its own event-triggered time by receiving information only from its neighbors.Hence, the distributed dynamic ETM is more suitable for the MASs with large-scale networks structure.In order to better show the characteristics of these two ETMs, Table I is given.

TABLE I CHARACTERISTICS OF CENTRALIZED AND DISTRIBUTED DYNAMIC ETMS

Remark 5: In this paper, the dynamic ETMs are proposed based on internal dynamic variablesζand ζi.In addition, the only adjustable parameterγof the parametric Lyapunov equation (2) is introduced into the design of the internal dynamic variables (7) and (32).In this way, the expressions of the minimum IET which avoid the complex relationship with the system matrix are established.In addition, a trade-off between minimum IET and convergence rate can be easily found by adjusting the parameterγ.It is worth noting that this kind of dynamic ETM aims to minimize the use of communication resources.In [44], a flexible dynamic ETM based on dynamic threshold parameter was proposed to respond different system stability and performance requirements.The dynamic ETM designed in [44] is based on online adjustable dynamic threshold parameters.This dynamic ETM alleviates bandwidth consumption in an intelligent and flexible way.Although this method has some good merits, our designed methods also have some unique advantages such as the clear and simple expression of minimum IET and easy stability analysis of the closed-loop system.

IV.SIMULATIONS

In this part, two simulation examples are carried out to test the validity and superiority of the method proposed in this paper.

Consider that there are 6 agents in the MASs and the system matrices of each agent is chosen as

Example 1(Semi-global consensus with centralized dynamic ETM): Through Figs.2-6, one can determine that the semi-global consensus is guaranteed by the proposed centralized dynamic ETM (8) and the bounded control protocol (11)of the input constrained MASs (1).From Figs.2-4, it can be seen that the states of each agent converge to some common values, which means that the consensus is achieved.Further,one can determine that with the increase ofγ, the convergence rate of the system state is accelerated.Fig.5 illustrates that the control input is updated only at the event-triggered instants determined by the centralized dynamic ETM (8) and the magnitudes of the control inputs are limited below the saturation threshold Δ=8.The time evolution of the internal dynamic variableζis given in Fig.6.

In order to illustrate the impact of changing the value ofγon the centralized dynamic ETM, Table II is given.From Table II, one can determine that the minimum IET as well as the average IET of the MASs decrease, and the event-triggered numbers increase with the increases ofγ.Since the centralized dynamic ETM (8) is adopted in Example 1, every agent is triggered at the same time, which can also be seen from Table II.Note that the minimum IET is larger than the sampling interval, which means that no Zeno behavior is happened.

Fig.5.The control inputs of the MASs by the centralized dynamic ETM.

Fig.6.The internal dynamic variable ζ of the MASs.

TABLE II THE MINIMUM IET/AVERAGE IET/EVENT-TRIGGERED NUMBERS(ETNS) UNDER CENTRALIZED ETM BY DIFFERENT γ

Fig.7.The consensus errors xi1-x j1,i,j=1,...,6 by the distributed dynamic ETM.

Fig.8.The consensus errors xi2-x j2,i,j=1,...,6 by the distributed dynamic ETM.

Fig.9.The consensus errors xi3-x j3,i,j=1,...,6 by the distributed dynamic ETM.

Example 2(Semi-global consensus with distributed dynamic ETM): Through Figs.7-11, one can determine that the semiglobal consensus is guaranteed by the distributed dynamic ETM (33) and the proposed event-triggered-based bounded control protocol (36) of the input constrained MASs (1).By observing Figs.7-9, one can determine that the states of the MASs tend to approach consensus asymptotically.Moreover,the time it takes for the MASs converge to consensus is shorter with the increase ofγ.Under the control of the eventtriggered-based bounded control protocol (36), the control inputs only update at some discrete event-triggered time and the amplitude of the control inputs do not exceed the saturation threshold which can be seen from Fig.10.As can be seen from Fig.11, the internal dynamic variables ζi＞0 are always maintained.

Fig.10.The control inputs of the MASs by the distributed dynamic ETM.

Fig.11.The internal dynamic variable ζi of the MASs.

TABLE III THE MINIMUM IET/AVERAGE IET/ETNS UNDER DISTRIBUTED ETM BY DIFFERENT γ

Next, we will analyze the impact of changing the value ofγon the distributed dynamic ETM through Table III.When the value ofγincreases, the minimum IET as well as the average IET of the MASs decrease; in the meantime, the event-triggered numbers increase.In this case, each agent in the MASs has its own event-triggered times which is decided by the distributed dynamic ETM (33).Since the minimum IET is larger than the sampling interval, it is demonstrated that the Zeno behavior does not happen.

It can be seen from the above simulation results that the control algorithm proposed in this paper is effective.

V.CONCLUSION

The consensus control of multi-agent systems (MASs) with constrained input is studied in this paper.For the sake of reducing the energy as well as the cost of control protocol update, a centralized dynamic event-triggered mechanism is first designed for the MASs with small-scale structure.Then,for the MASs with large-scale structure, a distributed dynamic event-triggered mechanism is given.Moreover, the semiglobal consensus of the MASs is realised and the exclusion of Zeno behavior is guaranteed in both cases.In addition, the trade-off between the convergence rate and the minimum inter-event time is established.Furthermore, the regional consensus of the MASs is achieved with a bounded control protocol by relaxing the restriction on the eigenvalues of system matrix.The simulation results show that the feasibility of the proposed algorithm is verified.In the future, the authors will focus on solving the problem of consensus control of input constrained MASs with faults under general connected graphs.