# Finite-Time Consensus of Networked Multiagent Systems with Time-Varying Linear Control Protocols

Authors: Jiayan Wen; Chen Wang; Wenguang Luo; Guangming Xie
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Finite-time consensus problems for networked multiagent systems with first-order/second-order dynamics are investigated in this paper. The goal of this paper is to design local information based control protocols such that the systems achieve consensus at any preset time. In order to realize this objective, a class of linear feedback control protocols with time-varying gains is introduced. We prove that the multiagent systems under such kinds of time-varying control protocols can achieve consensus at the preset time if the undirected communication graph is connected. Numerical simulations are presented to illustrate the effectiveness of the obtained theoretic results.


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## Body

## 1. Introduction

Multiagent systems have extensive potential applications, ranging from multiple spacecraft alignment, formation control of multiple robots, and heading direction in flocking behavior to group average in distributed computation and rendezvous of multiple vehicles. Among these, achieving consensus in networked multiagent systems has been increasingly attracting more attention in recent years, which is a comprehensive interdisciplinary research field, including control theory, mathematics, biology, physics, computer science, robot, and artificial intelligence. Great efforts have been made on the consensus problems of multiagent systems [1, 2].From the viewpoint of system and control theory, the study of consensus algorithms is mainly impelled by the particles swarm model introduced by Vicsek et al. [3]. This discrete model of finite autonomous agents assumes that all agents move in a plane with equal speed but with different headings, while each agent’s heading is updated using the so-called nearest neighbor rule based on the average of its own heading plus the heading of its neighbor. Numerical simulations have been provided to demonstrate that, under their proposed rule, all agents eventually move in the same direction without the centralized coordination. Later, Jadbabaie et al. [4] gave a strict theoretical explanation of the consensus behavior of Vicsek’s model and derived convergence results for several similarly inspired models. They have proven that Vicsek’s model can still be valid under switching topology, but for it there does not exist a common quadratic Lyapunov function. From then on, plenty of researches have been performed on the consensus problem. Olfati-Saber and Murray [5] have introduced a systematical framework of consensus problem in networks of dynamic agents with fixed/switching topology and communication time-delays. Ren and Beard [6] have investigated a more comprehensive discrete-time consensus scheme which includes Jadbabaie’s result as a special case and have presented some more relaxable conditions for consensus of information under dynamically changing interaction topologies. In [7, 8], Moreau and Lin have separately considered the more general discrete-time consensus model and continuous-time consensus model. Meanwhile, consensus problems with switching topologies and time-varying delays have been considered [9–12]. In [13–16], consensus of multiagent systems with second/higher-order dynamics has been considered. Part or all of the agents update their states according to second-order or higher-order dynamics.In the study of consensus problem, convergence rate is an important performance index of the proposed consensus protocols. It has been shown in [5] that the second smallest eigenvalue of interaction graph Laplacian, called algebraic connectivity of graph, quantifies the speed of convergence of the consensus algorithm. In [17], Xiao and Boyd have considered the problem of the weight design via semidefinite convex programming so that the algebraic connectivity can be increased. Although maximizing the second smallest eigenvalue of interaction graph Laplacian allows for a better convergent rate of the linear protocols, the state consensus can never occur in finite time. In some practical situations, however, it may be required that agreement has to be reached in finite time.The idea of finite-time convergence has been introduced to finite-time consensus for multiagent systems in [18–22]. Reference [18] has introduced the normalized and signed gradient dynamical systems associated with a differentiable function and has identified conditions that guarantee finite-time convergence. In [19], finite-time consensus tracking of multiagent systems has been reached on the terminal sliding-mode surface. Under both the global information and the local information, [20] has developed a new finite-time formation control framework for multiagent systems with a large population of members. Reference [21] has investigated finite-time consensus problems for multiagent systems and has presented a framework for constructing effective distributed protocols. In [22], weighted average consensus with respect to a monotonic function has been studied for a group of kinematic agents with time-varying topology. In the existing results, their protocols are generally discontinuous and nonlinear which, however, may not be suitable for real applications. Another shortage is the fact that only the upper bound of the convergence time is given and accurate convergence time can not be preset.Motivated by these analyses, in this paper, we try to design a control protocol such that the consensus can be achieved at any preset time. In order to reach this goal, preset time dependent time-varying but linear feedback control protocols are presented. We find that, under the same communication conditions as those in asymptotical consensus, our control protocols work well; that is, the terminal time dependent time-varying control protocol can solve a consensus problem at any present time if the undirected communication tropology is connected.The remaining part of this paper is organized as follows. Section2 gives the preliminary knowledge about graph theory. Sections 3 and 4 discuss the first-order case and second-order case, respectively; both finite-time consensus control protocols are obtained. Section 5 gives the simulation results. Finally, Section 6 concludes the whole paper.Notations. Let R denote the set of all real numbers. 1N represents the all 1 vector with dimension N. Notation diag{a1,…,aN} represents the diagonal matrix(1)a10⋱0aN.


## 2. Preliminaries on Algebraic Graph Theory

In this section, we present some definitions and properties about algebraic graph theory that will be used in this paper. For more details, we refer to [5, 23].Graph will be used to describe the communication topology among agents. LetG=(V,E,A) be an undirected graph with the set of vertices V={1,2,…,N}, the set of edges E⊆V×V, and a weighted adjacency matrix A=[aij] with nonnegative adjacency elements aij. An edge of G is denoted by eij=(j,i). The adjacency elements associated with the edges are positive; that is, eij∈E⇔aij>0. Moreover, we assume aii=0 for all i∈V. The set ofneighbors of node i is denoted by Ni={j∈V:(j,i)∈E}. Since the graph is undirected, it means that once eij is an edge of G, eji is an edge of G as well. As a result, the adjacency matrix A is a symmetric nonnegative matrix.The degree of nodei is the number of its neighbors Ni and is denoted by deg(i). Thedegree of node i is given by(2)deg⁡i=∑j=1Naij.Thedegree matrix is defined as Δ=diag{deg(1),deg(2),…,deg(N)}. Then theLaplacian of graph G is defined by(3)L=Δ-A.An important fact ofL is that all the row sums of L are zero and thus 1N=[1,1,…,1]T∈RN is an eigenvector of L associated with the zero eigenvalue.Apath between distinct vertices i and j means a finite ordered sequence of distinct edges of G in the form i,k1,k1,k2,…,(kl,j). A graph is calledconnected if there exists a path between any two distinct vertices of the graph.Lemma 1 (see [23]).
An undirected graphG is connected if and only if the rank of its Laplacian matrix L is N-1.By Lemma1, for a connected graph, there is only one zero eigenvalue of L; all the other ones are positive and real.


## 3. First-Order Dynamics

Consider a multiagent system consists ofN identical agents with the first-order dynamics(4)x˙it=uit,i=1,2,…,N,where xi(t)∈R and ui(t)∈R are the state and the control input of the agent i, respectively.We propose a time-varying linear feedback control protocol for system (4): (5)uit=kt∑j∈Niαijxjt-xit,where k(t)∈R is a time-varying feedback gain to be designed and the weights αij, for i,j=1,2,…,N, are assumed to be given by the interaction topology G.Using the notationx(t)=[x1T(t),…,xNT(t)]T, system (4) with protocol (5) can be written into a matrix form:(6)x˙t=-ktLGxt,where LG is the Laplacian matrix of the graph G.For system (4), the objective of finite-time consensus is to achieve the following requirement.Given any finite timetf∈(0,+∞), system (6) satisfies that, for any initial state x(0), xi(t)-xj(t)→0 as t→tf-, ∀i,j=1,2,…,N.If the above requirement is achieved, we say that control protocol (5)solves the finite-time consensus problem at timetf for system (4).In what follows, we try to find suitable control protocol (5) to solve the finite-time consensus problem for system (6).Consider the communication topology described by an undirected graphG; we assume it is connected. Then there exists a nonsingular matrix P∈RN×N such that P-1LGP=diag{0,λ2,…,λN}.Letη=P-1x; we have(7)η˙t=-ktdiag0,λ2,…,λNηt.Lemma 2.
Assuming the communication topology graphG is undirected and connected, then control protocol (5) solves the finite-time consensus problem at time tf if ηi(t)→0 as t→tf-, i=2,3,…,N.Proof.
Without loss of generality, we assume that the first column vector of the matrixP is 1N. Since(8)limt→tf-⁡xt=Plimt→tf-⁡ηt,it follows that(9)limt→tf-⁡xt=Pη10,0,…,0T=η101N.It means that(10)xit⟶x∗=η10ast⟶tf-,i=1,2,…,N.Proposition 3.
Suppose that the communication topology graphG is undirected and connected. Given any finite time tf, the time-varying feedback control protocol(11)uit=ctf-t∑j∈Niαijxjt-xitsolves the finite-time consensus problem at time tf for system (4), where c is a positive constant scalar.Proof.
From (7), we have(12)η˙it=-cλitf-tηit,t∈0,tf.It follows that(13)ηit=tf-ttfcλiηi0.Since c and λi are positive, we have(14)ηit⟶0,ast⟶tf-.By Lemma 8, we know that control protocol (11) solves the finite-time consensus problem at time tf.Remark 4.
In Proposition3, assuming cλi>1, from (13), we have(15)η˙it=cλitf-tcλi-1tfcλiηi0.This implies that η˙i(t) is bounded, ∀t∈[0,tf). It is easy to verify that ui is bounded, ∀t∈[0,tf). This means that if we select c such that(16)c>maxi=2,…,N⁡1λi,control protocol (11) is always bounded.Proposition 5.
Assuming the communication topology graphG is undirected and connected, given any finite time tf, the time-varying feedback control protocol(17)uit=ctf-tm∑j∈Niαijxjt-xitsolves the finite-time consensus problem at time tf for system (4), where c is a positive constant scalar and m≥1 is a positive integer.Proof.
Ifm=1, we come back to Proposition 3. If m>1, we have(18)η˙it=-cλitf-tmηit,(19)ηit=exp⁡cλi1-mtf-tm-1-cλi1-mtfm-1ηi0.It is obvious that(20)cλi1-mtf-tm-1⟶-∞ast⟶tf-.It means that(21)ηit⟶0,ast⟶tf-.Remark 6.
In Proposition5, assuming m>1, from (19), we have(22)η˙it=cλitf-tmexp⁡cλi1-mtf-tm-1-cλi1-mtfm-1ηi0.Consider(23)limt→tf-⁡1tf-tmexp⁡cλi1-mtf-tm-1=lims→0+⁡1smexp⁡cλi1-msm-1=limp→+∞⁡pmexp⁡cλipm-11-m=0.This implies that η˙i is bounded, ∀t∈[0,tf). It follows that ui is bounded, ∀t∈[0,tf). This means that once we select c>0, control protocol (17) is always bounded.In conclusion, we present the following theorem.Theorem 7.
Assuming the communication topology graphG is undirected and connected, given any finite time tf, the time-varying feedback control protocol(24)uit=K˙t∑j∈Niαijxjt-xitsolves the finite-time consensus problem at time tf for system (4), where function K(t) satisfies that(i)
K(t) is differentiable, t∈[0,tf);(ii)
K(t)→+∞, as t→tf-.Moreover, if function K(t) satisfies that (iii)
K˙texp⁡(-K(t)) is bounded, t∈[0,tf),control protocol (24) is always bounded.Proof.
The conclusion is obvious since it is easy to verify that(25)ηit=exp⁡λiK0-Ktηi0.


## 4. Second-Order Dynamics

Consider a multiagent system consists ofN identical agents with the second-order dynamics(26)x˙i=viv˙i=uii=1,2,…,N,where xi(t)∈R, vi(t)∈R, and ui(t)∈R are the state, the velocity, and the control input of the agent i, respectively.The control law studied in this section is a time-varying feedback protocol(27)uit=∑j∈Niαijk2txj-xi+k1tvj-vi,where k1(t),k2(t)∈R are time-varying feedback gains to be designed and the weights aij, for i,j=1,2,…,N, are assumed to be given by the interaction topology G.Denotex(t)=[x1(t),x2(t),…,xN(t)]T, v(t)=[v1(t),v2(t),…,vN(t)]T, and(28)zt=xt⊗10T+vt⊗01T.Moreover, let(29)A=0100,B=01,Kt=k1tk2t;system (26) with protocol (27) can be rewritten in a matrix form:(30)z˙t=IN⊗A-LG⊗BKtzt,where LG is the Laplacian matrix of the graph G.The objective of finite-time consensus is to achieve the following requirement.Given any finite timetf∈(0,+∞), system (30) satisfies that, for any initial state and initial speed, xi(t)-xj(t)→0 andvi(t)-vj(t)→0, ∀i,j=1,2,…,N, as t→tf-.In what follows, we try to find suitable control protocol (27) to solve the finite-time consensus problem for system (30).Similar to the first-order case, consider the communication topology described by an undirected graphG, we assume it is connected, and there exists a nonsingular matrix P∈RN×N such that P-1LGP=diag{0,λ2,…,λN}. Without loss of generality, we assume that the first column vector of the matrix P is 1N.Letη(t)=(P-1⊗I2)z(t); then we can obtain(31)η˙t=P-1⊗I2z˙t=IN⊗A-diag0,λ2,…,λN⊗BKtηt=diagA,A-λ2BKt,…,A-λNBKtηt.It follows that(32)limt→tf-⁡zt=P⊗I2limt→tf-⁡ηt.Lemma 8.
Suppose that the undirected communication topology graphG is connected; then control protocol (27) solves the finite-time consensus problem at time tf if ηi(t)→0 as t→tf-, i=3,…,2N.Proof.
From the assumption thatηi(t)→0 as t→tf-, i=3,…,2N, we have(33)limt→tf-⁡zt=P⊗I2η10+η20tfη200⋮0=1N⊗η10+η20tfη20.It means that(34)xit⟶η10+η20tf,vit⟶η10,as t→tf-, i=1,2,…,N. Thus, consensus is achieved at the preset time tf.Lemma 9 (see [24]).
Consider the linear second-order differential equation(35)y¨t+pty˙t+qtyt=0,t≥0,where p(t) and q(t) are given smooth functions of t. When y1(t) is one fundamental solution, then the other solution y2(t) is given by(36)y2t=y1t∫0tWsy12sds,where(37)Wt=exp⁡-∫0tpsds.Moreover, the general solution is given by(38)yt=C1y1t+C2y2t,where C1 and C2 are constants.Theorem 10.
Assuming the undirected communication topology graphG is connected, for any given finite time tf, control protocol (27) with time-varying feedback gains(39)k1t=2ctf-t3,k2t=ctf-t2solves the finite-time consensus problem at time tf for system (26), where c is a positive constant scalar.Proof.
According to Lemma8, we only need to show that ηi(t)→0 as t→tf-, i=3,…,2N. Now, let us consider the dynamics of ηi(t), i=3,…,2N. It is noted that(40)η˙2i-1tη˙2it=01-2cλitf-t3-cλitf-t2η2i-1tη2it.
It is easy to verify thate-cλi/tf-t is one of the fundamental solutions for the second equation in (40). By Lemma 8, the general solution of (40) is(41)η2i-1t=e-cλi/tf-tC1+C2∫0tecλi/tf-s+cλi/tfds,η2it=-cλitf-t2e-cλi/tf-tC1+C2∫0tecλi/tf-s+cλi/tfds+C2ecλi/tf,where C1 and C2 are constants.
It follows that(42)limt→tf⁡η2i-1t=C1+C2∫0tf-ecλi/tf-s+cλi/tfdslimt→tf⁡e-cλi/tf-t=0,limt→tf⁡η2it=C2ecλi/tf-limt→tf⁡cλi/tf-t2C1+C2∫0tecλi/tf-s+cλi/tfdsecλi/tf-t=C2ecλi/tf-limt→tf⁡cλi/tf-t2C2ecλi/tf-t+cλi/tfcλi/tf-t2ecλi/tf-t-limt→tf⁡2cλi/tf-t3C1+C2∫0tecλi/tf-s+cλi/tfdscλi/tf-t2ecλi/tf-t=-limt→tf⁡2cλiC1+C2∫0tecλi/tf-s+cλi/tfdstf-tecλi/tf-t=0.This completes the proof.Now, we extend the specific time-varying gain functions to a general form.Suppose that a functionf(t) satisfies(1)
f(t)∈C2[0,tf), where C2[0,tf) represents a second-order continuously differentiable function on [0,tf);(2)
f(t)→+∞ as t→tf-;(3)
exp(-f(t))f¨(t)/f˙(t)→0 as t→tf-.We consider the following time-varying gains with a general form:(43)k1t=f¨t,k2t=f˙t.Then protocol (27) with (43) covers a wide range of algorithms including the specific form in Theorem 7.Theorem 11.
Assume that the undirected communication topology graphG is connected. For system (26), control protocol (27) with time-varying gains (43) solves the finite-time consensus problem at any preset finite time tf.Proof.
Similarly, we have(44)η˙2i-1tη˙2it=01-λif¨t-λif˙tη2i-1tη2it,i=2,…,N.
It is easy to verify thate-λif(t) is one of the fundamental solutions for the second equation in (44). By Lemma 8, the general solution of (44) is(45)η2i-1t=e-λiftC1+C2∫0teλifs+λif0ds,η2it=-λif˙te-λiftC1+C2∫0teλifs+λif0ds+C2eλif0,where C1 and C2 are constants.
It follows that(46)limt→tf⁡η2i-1t=limt→tf⁡e-λiftC1+C2∫0tfeλifs+λif0ds=0,limt→tf⁡η2it=limt→tf⁡-λif˙te-λiftC1+limt→tf⁡-λif˙te-λiftC2∫0teλifs+λif0ds+C2eλif0=-C1limt→tf⁡λif˙teλift-C2limt→tf⁡λif˙t∫0teλifs+λif0dseλift+C2eλif0=-C1limt→tf⁡f¨tf˙teλift-C2limt→tf⁡f¨t∫0teλifs+λif0ds+f˙teλift+λif0f˙teft+C2eλif0=-C2∫0tfeλifs+λif0dslimt→tf⁡f¨tf˙teft=0.This completes the proof.


## 5. Numerical Simulations

To demonstrate our theoretical results in the previous two sections, we carry out numerical simulations in this section. Considering a multiagent system consisting of eight agents, the communication graphG is given in Figure 1.Figure 1
Communication graph.The Laplacian matrix ofG is given by (47)LG=5-1-100-1-1-1-12-100000-1-15-10-1-1000-120-10000001-100-10-1-1-1400-10-10003-1-100000-12.First, we consider the dynamics of the agent with the first order. Control protocol (11) is applied as c=2. We run the simulations with tf=10 and tf=1, respectively, and show the results in Figure 2. In the simulations, the initial states of the agents are generated randomly. For ease of comparison, we use the same initial states of the eight agents in both simulations. The simulation results have shown that the groups of agents can reach consensus as t→tf-. By simple calculation, we know (16) is held; thus the inputs of all the agents are bounded. In addition, comparing the cases of tf=10 and tf=1, one can find that the dynamics of the agents’ states adapt the preset finite time tf while the inputs of the agents increase linearly with the decreasing of tf.Figure 2
Simulation results of control protocol (11) for the single-integrator dynamics case when c=2. ((a) and (b)) The case with tf=10; ((c) and (d)) the case with tf=1; ((a) and (c)) the states of all the eight agents; ((b) and (d)) the inputs of all the eight agents.
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1Next, we consider the second-order dynamics case. Control protocol (27) with (39) is applied as c=2. We run the simulations with tf=10 and tf=1, respectively, and show the results in Figure 3. In the simulations, the initial states and velocities are generated randomly. Similarly, we use the same initial states and velocities in both simulations. The simulation results have shown that the groups of agents can reach consensus as t→tf-. In addition, one can check that the smaller tf will not affect the effectiveness and the performance of our control protocols.Figure 3
Simulation results of control protocol (27) with (39) for the double-integrator dynamics case when c=2. ((a) and (b)) The case with tf=10; ((c) and (d)) the case with tf=1; ((a) and (c)) the states of all the eight agents; ((b) and (d)) the velocities of all the eight agents.
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## 6. Conclusion

Finite-time consensus problems for multiagent systems have been investigated in this paper. We have considered both of the first-order and second-order cases. Time-varying linear feedback control protocols have been established under which the systems achieve consensus at any preset time. The condition on the communication topology has been proven to be the same as those in the asymptotical consensus case. The future work includes finite time with switching topology or time-delay.



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