Ruizhuo SONG，Shi XING，Zhen XU

1School of Automation and Electrical Engineering，University of Science and Technology Beijing，Beijing 100083，China

2Beijing Engineering Research Center of Industrial Spectrum Imaging，Beijing 100083，China

3Research Institute of Urbanization and Urban Safety，School of Civil and Resource Engineering，University of Science and Technology Beijing，Beijing 100083，China

Abstract: In this study，we solve the finite-time leader-follower consensus problem of discrete-time second-order multi-agent systems (MASs) under the constraints of external disturbances.First，a novel consensus scheme is designed using a novel adaptive sliding mode control theory.Our adaptive controller is designed using the traditional sliding mode reaching law，and its advantages are chatter reduction and invariance to disturbances.In addition，the finite-time stability is demonstrated by presenting a discrete Lyapunov function.Finally，simulation results are presented to prove the validity of our theoretical results.

Key words: Finite-time;Leader-follower consensus;Adaptive sliding mode control;Multi-agent systems

1 Introduction

In the last few years，distributed cooperative control has been an important research direction within the area of multi-agent systems (MASs) because of its high efficiency and robustness compared to control of the classical single agent (Olfati-Saber and Murray，2004;Olfati-Saber et al.，2007;Ren，2008;Ren and Beard，2008).The distributed cooperative control method is applied to both theoretical and practical areas.The theoretical research includes consensus(Li ZK et al.，2009;Xie DS et al.，2016;Zhang HG et al.，2020b，2021b)，formation control (Oh et al.，2015)，coverage control(At?n? et al.，2014)，rendezvous(Li P et al.，2018)，flocking(Olfati-Saber，2006)，reinforcement learning (Li Q et al.，2021;Xia LN et al.，2022a)，and attack (Li Q et al.，2019b;Xia LN et al.，2022b).At the same time，the MAS has been used in many practical areas，such as design of sensor networks(Cruz-Piris et al.，2018)and unmanned aerial quadrotors(Han et al.，2020).

Among the numerous distributed cooperative MAS control methods，the consensus problem is a basic and significant research topic.It is the key to solving other cooperative control problems.Normally，consensus can be classified into three types:leaderless consensus (Cui Q et al.，2020)，leaderfollower consensus(one leader or a virtual leader)(Li Q et al.，2020;Zhang J et al.，2021)，bipartite consensus (Li Q et al.，2019a;Zhang HG et al.，2021a)，and containment control(multiple leaders)(Cui GZ et al.，2018;Wang W et al.，2020;Zhang HG et al.，2020a).The consensus problem states that，with the evolution of time，the global or partial MAS states tend to be asymptotically stable.Specifically，in the case of leader-follower consensus，it requires that the states of all followers drive into the trajectory of a single leader，which is widely used in the field of formation control as well.

An important performance index of the consensus problem is the convergence rate.In many articles(Chowdhury et al.，2018;Deng et al.，2020;Wang QL and Sun，2020)，the MAS consensus is asymptotic;that is to say，the consensus situation can be reached only when the time tends towards infinity.However，we need to achieve finite-time consensus (FTC) in real applications，which means that the trajectory of the system can converge to the designed equilibrium state within a finite time horizon and remain there since then.Therefore，various kinds of controllers have been constructed to meet the demand of FTC stabilization problems in MAS research.In Zou et al.(2020)，second-order FTC was discussed in a switched MAS，and in Shi et al.(2019) it was extended to the FTC of high-order MASs.Ning and Han (2019) and Lu et al.(2021) studied the prescribed and bipartite FTC，respectively.Liu XY et al.(2019)compared the FTC and fixed-time consensus (FxTC)，in which the MAS can reach consensus within a fixed time.At the same time，the importance of FTC research was demonstrated because FTC has been used in a wider range of control areas (Min et al.，2018) and extended to practical applications such as the financial sector (Wang YL et al.，2021).

Significantly，most finite-time controllers are discussed based on the continuous models in the MAS field (Tong et al.，2018).In fact，the useful information may not be transmitted continuously in practice due to unreliable communication channels or the limited perceptual capabilities of agents.In addition，in a real-time computer control system，the controller demand must be discrete.Therefore，we focus on the study of discrete-time MASs，in which the agents can obtain only the state information at discrete times from their neighbors.There have been some recent research findings about the MAS discrete-time consensus.Leader-follower and highorder switched discrete MASs have been reported(Liu JW and Huang，2021;Liu YF and Su，2021;Zhang YY et al.，2021)，and the time-varying situation has been discussed in Zhang JL et al.(2021).Wang B and Tian(2021)and Zhang WL et al.(2021)gave the discrete consensus with multiplicative uncertainties delays and unconfined cyber-attacks in MAS，respectively.Wang QS et al.(2021) presented a new method to achieve the optimal consensus of discrete MAS and Zhou et al.(2021)extended the discrete-time research in the area of financial systems.

Another crucial problem concerns disturbances.Most disturbances originate from uncertainty and the impact of external disturbances，and severely affect the dynamic performance of the MAS.At the same time，MAS control performance and accuracy will be reduced.Research on anti-disturbance control has been carried out using multiple traditional control methods，such as robust control (Wang JY et al.，2021)，adaptive control (Liang et al.，2021)，back stepping control(Wang XY et al.，2018)，fuzzy control (Shao and Ye，2021)，and sliding mode control (SMC) (Yu and Long，2015;Sinha and Mishra，2020).

SMC has significant advantages against the uncertainty of the system and the influence of external disturbances，especially for closed-loop systems(Young et al.，1999).The traditional SMC method is essentially a variable structure nonlinear control(Utkin，1977).It can drive a decentralized controller to reach the sliding surface，which is constructed as a state function of the system in finite time and maintained on the sliding surface thereafter.The programming of the sliding surface is irrelevant to the system parameter of the system and the external disturbances.Many researchers have noticed the excellent performance of SMC，such as robustness and expandability，and have conducted extensive research，especially in the field of MASs (Sun et al.，2018;Wang GD et al.，2018;Qin et al.，2019;Zhang Z et al.，2019;Sinha and Mishra，2020;Yao et al.，2020).Furthermore，Liu HY et al.(2020)extended the research in fractional-order MASs，F(xiàn)ei et al.(2020)combined the sliding mode method with the neural network，and Chen et al.(2020)introduced SMC in the application of a non-holonomic spherical robot.

However，the SMC algorithm can cause highfrequency chattering of the control system.Because of the discontinuous control structure，the control trajectory may not reach the sliding surface without inertia，and a zigzag trajectory may be superimposed in the practical sliding mode，which causes devastating consequences in the controller.To solve the chattering problem，Gao et al.(1995)first proposed the reaching law theory of discrete-time variable structure control to eliminate the high-frequency chattering of SMC.Recent applications to avoid chattering include the synthetic trigger function(Liu JH et al.，2019)and the adaptive event-triggered controller strategy(Xu and Wu，2021).

Motivated by the above observations，the main task of this paper is to analyze the FTC of a discretetime，leader-follower MAS under the controller of discrete-time SMC.The main contributions are summarized below:

1.Aiming at the second-order MAS over a directed graph，based on the traditional SMC method，we propose a new adaptive discrete sliding mode controller that leads the followers to achieve finite-time tracking control of the single leader’s position and velocity information in a discrete-time MAS.

2.The existence and reachability of discrete sliding mode are discussed in our research.Using the method of finite-time stability by constructing a discrete Lyapunov function(Hamrah et al.，2019)in a discrete system，it is demonstrated that the presented control algorithm can ensure the stability of the discrete MAS in finite time.Furthermore，the stability is not affected by variation of the discrete sampling time.

3.Compared with the existing discrete SMC method，the adaptive discrete sliding mode controller can adjust the time-varying parameterεto decrease the inherent chattering of traditional SMC.Meanwhile，the speed at which the sliding mode switching surface is reached is not affected.Using the saturation function sat(·)，which is widely used in the ideal SMC，and substituting the signum function sgn(·)，the advantage of the proposed controller in our research is used to the greatest extent to create robustness and invariance to internal system perturbation and external disturbances.

2 Preliminaries and formulation of problems

2.1 Graph theory

A weighted graphG=(V，E，A)is defined as the MAS communication topology withnagents.For the definition of the graph，we setV={1，2，···，n}as the node set andE ?V×Vas the edge set.Then we set the weighted adjacency matrixA=[aij]∈Rn×nas ann-order real matrix.An edge ofGfrom nodeito nodejis expressed by(i，j)，and on behalf of that，nodejcan consume information from nodei.If(i，j)∈E，thenaij ＞0，if not，aij=0，and we setaii=0，which means that the graph has no self-loop.We call(j，i)∈Ebecause nodejis the neighborhood of nodei.So，we defineNi={j |j ∈V，(j，i)∈E}as the neighbor set ofi.The directed spanning tree exists when and only whenVhas a root node，and the root node can lead to a directed path to all other nodes.If (j，i)∈E ?(i，j)∈E，then we defineGas an undirected graph.Another crucial definition is the Laplacian matrixL=[lij]∈Rn×n，which is based on the adjacency matrix presented aboveA:

From the definition of the Laplacian matrix，we haveL·1n=0，where 1n=[1，1，...，1]Tis a column vector.For a leader-follower MAS，we designate node 0 as the leader and the series 1，2，...，nas the followers.Another graph，=(，，)，represents thenfollowers and a single leader in leader-follower MASs.Specifically，=V ∪{0}，andandare the corresponding edge set and adjacency matrix，respectively.We set the Laplacian matrix of the leader-follower directed graph as ˉL，and denote another column vectorB=[b1，b2，...，bn]T，wherebi=1 represents the follower agent withibeing a neighborhood of the leader，andbi=0 otherwise.The matrix=diag(b1，b2，···，bn).We assume that the followers receive information only from the leader，and that the followers cannot provide information to the leader.We call the leader globally reachable if each follower has a directed path from the leader.

2.2 Problem formation

In this subsection，we propose the issue of leader-follower consensus by considering secondorder discrete linear dynamics.The dynamic function includes one single leader and the remainingnagents are followers.The leader is labeled 0，while the followers are labeled{1，2，···，n}.First，we suppose that the dynamics of follower agents is as follows:

wherei=1，2，...，n，andui(k)，xi(k)，vi(k)∈R are the control input，position，and velocity of theithfollower agent at sampling timek，respectively.di(k)represents the matched slowly varying disturbance acting on the follower agents at timek.Tkindicates the sampling period andk(ork+1)is the abbreviation forkTk(or (k+1)Tk).Similarly，the leader’s dynamics is presented by another second-order discrete linear system:

Similar to the above definition，x0(k)，v0(k)∈R indicate the position and velocity of the leader，respectively.

We assumeξi(k)=[xi(k)，vi(k)]T，and the leader’s column vector is denoted byξ0(k)=[x0(k)，v0(k)]T.Then，from Eqs.(1) and (2)，we obtain

To explain the process of consensus controller design and stability analysis，we introduce some necessary assumptions and lemmas.

Assumption 1The disturbancedi(k)is bounded and meets the condition|di(k)whereis a positive number，i=1，2，...，n.

Assumption 2The communication topology graph for the above leader-follower MAS has at least one single directed spanning tree，which can be expressed as／=0.

Lemma 1(Tsai et al.，2018) To avoid the SMC chattering phenomenon，we can use the saturation function to replace the traditional sign function.The saturation function is defined as

whereδiis an arbitrary small positive constant.

Lemma 2We set=L+B ∈Rn×nas the Laplacian matrix of graph.It is positive definite if Assumption 2 holds.

Lemma 3(Hamrah et al.，2019) Define a discretetime system output variableyk ∈Rn.Then，consider a positive definite Lyapunov functionVk=V(yk)，V:Rn →R.Define the real numbersα，h ∈(0，1)，andγk=γ(Vk) is also a positive definite function.We have

if the following inequality is satisfied:

We can say that the discrete system achieves Lyapunov stability.The output variableykconverges toy=0 ifk ≥N，in which the finite positive integerN ∈N.

ProofBased on the definition of Lyapunov stability，the difference of the Lyapunov function ΔVkis negative definite along the trajectories of the above discrete-time system，while the right-hand side of inequality(4)tends to 0 when，and only when，Vk=0.

Using the definition ofγk，the equality form of inequality(4)is given as follows:

We set the initial valueV0of the above Lyapunov function as

Then，substituting the value ofV0in inequality(5)，we have

Thus，the value ofV1is defined as

The value range of parameterg0should beg0＞1.We have noticed that ifg0≤1，Eq.(8) means thatg1≤0.This leads to a paradox unlessg1=0，because the variableV1must be non-negative based on the definition of the Lyapunov function.So，the value of the Lyapunov function has converged to 0 at the first iteration step，which meansk=1.To avoid this case，supposeg0＞1;then，substituting the variableV1in inequality (5)，we can have a similar iteration expression forV2:

If we continue iterating in this method，we can obtain the following expression forVk+1and the recurrence relation involvinggkforak:

We suppose thatVkis in the scope of inequality(5);then，based on Eq.(11) and inequality (5)，we have

BecauseVk+1is positive definite，gk+1should be greater than or equal to 0 according to Eq.(11).Furthermore，ifVkis within the range given in inequality(5)，we have

in which parameterβis arbitrarily small，especially forβg0＜1.Rearrange the right-hand side of inequality(13)as follows:

Whengk →βg0＜1 falls in the interval ofVk，a finite positive integerk=N -1 can meet condition(14).For example，gN-1≤(1-h)11-α，sogN ≤0.In addition，gN ≥0 becauseVN ≥0 andγ0＞0.Hence，gN=0 based on the contradictory conditions.Therefore，we use Eq.(11) again，and it can be concluded thatgj=0 andVj=0 underj ≥N.Consequently，yjcan converge to 0 underj ≥N.We now have finite-time stability of the discrete system.

Definition 1The consensus control for the discrete-time second-order MAS (1) and (2) can be reached in finite time，if for any initial states，there is a point of timeT ∈[0，+∞)，such that the following conditions are satisfied:

and

3 Main results

In this section，we aim to achieve leader-follower consensus by designing a distributed discrete sliding mode controller.Using the control scheme，the followers can track the position and velocity trajectory of the leader in finite time.We assume that all agents in the designed discrete MAS can update their own position and velocity information at sampling timeTk，and that the position and velocity information can be observed in real time.

First，let us define the position and velocity tracking errors:

Then we define the local neighborhood consensus error for our discrete second-order MAS (1) and(2)as

wherei=1，2，...，n.

Based on Eqs.(3)and(20)，we obtain

We defineliias the main diagonal element of the graph Laplacian matrixL.Our aim is to make the consensus error vectore(ξi(k))tend toward0in finite time.

To ensure the convergence of the consensus error in finite time，we construct a new adaptive discrete SMC scheme.First，the sliding mode surface function has been designed as

whereei(k) is short for the consensus error vectore(ξi(k)).

Similarly，we haveSi(k+1)=Ceei(k+1)，where the sliding mode parameterCeshould satisfy the pole-placement approach and the second-order Hurwitz condition such thatCe=[c1，1]，c1＞0 andc1∈N.

Our aim is to design a stable discrete time sliding mode controller based on the classical exponential reaching law.Discrete movement from an arbitrary state，which is driven by our novel control law in Theorem 1，can reach the sliding surface in a finite number of steps，and the system can remain on the sliding surface without zigzagging，that is，the so-called ideal quasi-sliding mode.This can reduce the occurrence of chattering.

The sliding surface function that satisfies the ideal quasi-sliding mode condition is expressed asSi(k)=Ceei(k)=0.The sliding mode parameterCeis a constant，so when the above equation holds，the consensus errorei(k)=0.

Theorem 1Using the following control law:

ProofFirst，we design a new discrete adaptive sliding mode controller according to the traditional exponential reaching law (Gao et al.，1995).We define the traditional exponential reaching law as follows:

whereDi(k)=[d1(k)，d2(k)，...，di(k)]，i=1，2，...，n，and parameterεcan influence the chattering phenomenon performance and the speed at which the sliding surface is reached.Our ideal parameterεshould change over time.At the beginning of the sliding mode movement，εshould be larger，and with the increase of time，εneeds to decline exponentially to eliminate the chattering.

We useε=|Si(k)|/2 as the adaptive parameter instead of a fixed parameter，and replace the signum function sgn(·) with the saturation function sat(·).The improved adaptive sliding mode reaching law is defined as

Then we construct a positive definite Lyapunov function of consensus error vector.Thus，the iteration of the above Lyapunov function can be calculated as

Inserting Eq.(29)into Eq.(28)，we have

Then，inserting Eq.(30) into Eq.(27)，we can update ΔVkas

Therefore，we obtain

under the following circumstances:

to ensure the monotonic decrease of ΔVk ＜0.

Based on Lemma 3，and from inequality(5)，we can haveVk(Si(k))，and the consensus error vectorei(k)converges to0in a finite number of steps of the discrete-time domain.So，the local neighborhood consensus error tends to0in finite time.This means that the discrete MAS achieves consensus in finite time.

Based on the sliding mode function(22)and the improved adaptive sliding mode reaching law (26)，we can use the iteration of the sliding mode function ΔS=Si(k+1)-Si(k)to construct the new adaptive sliding mode controller.

Using the definition of the local neighborhood consensus error vectore(ξi(k)) and the iteration form of the error vectore(ξi(k+1))，the adaptive sliding mode reaching law (26) can be updated using the following iterative form of the sliding mode surface functionSi(k+1)=Ceei(k+1):

Then，inserting Eq.(35)in Eq.(26)and reorganizing the function，we have

Finally，we arrive at the adaptive sliding mode controller:

Remark 1To guarantee the stability of the designed control scheme，the existence and reachability of our novel adaptive discrete sliding mode controller are discussed.We set another Lyapunov functionVk1=(k)/2.Once the following condition

is met based on the Lyapunov stability theorem，Si(k)=0 is a globally asymptotically stable switched sliding surface，which means that any initial position of the state (velocity) will tend toward the switched sliding surface.Taking the reachability condition as

when the sampling timeTkis small enough，the existence and reachability of the discrete sliding mode can be described as

Substituting function(25)with the adaptive parameterε=|Si(k)|/2 and the adaptive sliding mode reaching law

This completes the proof of existence and reachability of our novel adaptive discrete sliding mode.

Remark 2As a typical controller design method of variable structure control sliding mode，the reaching law method has the advantage of causing the motion to get close to the sliding surface monotonically.Using the reaching law method，we can solve the SMC high-frequency chattering problem very well.

Remark 3Based on the proof of Theorem 1，we can obtain the time-variant adaptive parameterεin the process of system movement.To guarantee the convergence performance of sliding mode surface functionSi(k)，we first analyze the traditional exponential reaching law:

We can have|p|＜1，which means|Si(k+1)|＜|Si(k)|，and|Si(k)|is decreasing;thus，the convergence of sliding functionSi(k) can be guaranteed.

Based on the decreasing condition|Si(k)| ＞εT/(2-qT)，T q+εT/|Si(k)|＜2 is required.Then the adaptive parameter is reorganized as

Thus，the sampling timeTkmeets the condition

andSi(k) can monotonically approach the sliding mode surface.

4 Simulations

In this section，the efficacy of the proposed sliding mode consensus is verified in a discrete-time MAS through simulations.

We consider the directed communication topology graph with five followers and one leader.The graph is shown in Fig.1.We assume that the leader node is labeled 0，and that the follower nodes are labeled 1–5.

Fig.1 Communication topology graph of the secondorder discrete-time multi-agent system

Based on Fig.1，we can define the adjacency matrixA，pinning gain matrixB，and Laplacian matrixLas follows:

Let the discrete sampling interval beT=0.01.The external disturbance was applied to each follower asdi(k)=0.002cos(0.86k).The system matricesAandBwere set as

The useful parameters were defined as follows:sliding gainCe=[2.5，1]，reaching law gainq=1.1，and the Lyapunov gainτ=10.

Then we set the initial position and velocity state of each follower agent as[1，1]，[2，2]，[3，3]，[4，4]，and[5，5].

From Figs.2 and 3，we can see that all the follower position and velocity trajectories can reach finite-time consensus under the proposed control scheme.Furthermore，the consensus time was less than 10 s.

Fig.4 shows the trajectories of the control inputui(k) of each follower agent.Compared with previous research results，the use of the saturation function to replace the sign function and the time varying parameterεcan greatly reduce the chattering of the traditional control law.

Fig.2 Position state of finite-time leader-follower consensus with disturbances

Fig.3 Velocity state of finite-time leader-follower consensus with disturbances

Fig.4 Control input of finite-time leader-follower consensus with disturbances

Fig.5 presents the sliding surface function of each follower agent.The sliding surface reached and remained at a quasi-sliding mode band within(–0.1，0.1)in finite time.

Fig.5 Sliding surface function of finite-time leaderfollower consensus with disturbances

We performed a comparative experiment to show the effectiveness of our controller.First，we replaced the sign function as our saturation function，as defined in Lemma 1.The contrast effect of the trajectories in sliding surfaceSi(k)with the sign function is shown in Fig.6.

Fig.6 Sliding surface function of finite-time leaderfollower consensus with disturbances by the sign function

From Fig.6，the sliding surfaceSi(k) reached the steady-state region with a delay of more than 2 s compared to using the saturation function.

Second，we considered parameterε.This is the main parameter for the SMC system to overcome external disturbances.Using our proposed controller，the time varying parameter was updated to the adaptive form:ε=|Si(k)|/2.Compared with the fixed value，the adaptive parameterεwas larger at the beginning to accelerate toward stability，and then decreased to reduce the chattering.The contrast effect of the trajectories in sliding surfaceSi(k)with a fixedεis shown in Fig.7.

Fig.7 Sliding surface function of finite-time leaderfollower consensus with disturbances by a fixed ε

The chattering phenomenon was severe and may have a bad effect on the performance of the system.Our aim is to avoid the damage and extend the life of the system.

Based on the above comparison of Figs.6 and 7 with Fig.5，we conclude that the use of the saturation function and adaptive sliding mode parameter in the proposed controller is helpful and effective.

5 Conclusions

In this study，we proposed a new control algorithm to solve the finite-time leader-follower consensus problem of second-order discrete-time multi-agent systems with external disturbances.We built a distributed adaptive sliding mode controller to realize finite-time consensus.Specifically，a timevarying sliding mode parameterεwas adopted and a saturation function was designed for use in the novel sliding mode function to eliminate the influence of chattering.The results of the numerical simulations showed the efficiency and effectiveness of our control protocol.

Contributors

Ruizhuo SONG and Shi XING designed the research.Shi XING processed the data.Ruizhuo SONG，Shi XING，and Zhen XU drafted the paper.Ruizhuo SONG and Shi XING revised and finalized the paper.

Compliance with ethics guidelines

Ruizhuo SONG，Shi XING，and Zhen XU declare that they have no conflict of interest.