# Fixed-Time Connectivity Preserving Tracking Consensus of Multiagent Systems with Disturbances

Authors: Fenglan Sun; Peiyong Liu; Jürgen Kurths; Wei Zhu
Journal: Complexity
 (2020)

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## Abstract

This text studies the fixed-time tracking consensus for nonlinear multiagent systems with disturbances. To make the fixed-time tracking consensus, the distributed control protocol based on the integral sliding mode control is proposed; meanwhile, the adjacent followers can be maintained in a limited sensing range. By using the nonsmooth analysis method, sufficient conditions for the fixed-time consensus together with the upper and lower bounds of convergence time are obtained. An example is given to illustrate the potential correctness of the main results.


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## Body

## 1. Introduction

Consensus of multiagent systems (MASs) is a very active research topic in control field [1–15]. The key to consensus problem is through designing an appropriate control protocol, such that all the agents can achieve a certain value. The convergence velocity is an important performance index of a control protocol, and the fast convergence velocity is very important for improving the performance and robustness of a system [4, 7–15]. Olfati-Saber and Murray [9] proved that the algebraic connectivity of the interaction graph determines the convergence velocity. The larger the second small eigenvalue of the Laplace matrix, the faster the convergence velocity. Zhou and Wang [4] characterized the convergence velocity for the distributed discrete-time consensus algorithm over a variety of random networks with arbitrary weights. They proposed the asymptotic and per-step convergence factors as measures of the convergence velocity and derived the exact value for the per-step convergence fact. Draief and Vojnović [11] derived an exact relation between the expected convergence time and the voting margin for some of these graphs, which revealed how the expected convergence time tends to infinity as the voting margin approaches zero. By maximizing the algebraic connectivity of the interaction graph, one can increase the convergence velocity with respect to the linear protocol, but the consensus can never be reached in finite time [4, 9–15]. That is, the convergence time is infinite or inestimable. However, in practice, it is often required the consensus to be reached in finite time. Comparing with the asymptotic consensus, the finite-time consensus has strong antiinterference, antiuncertainty, robustness, and fast convergence velocity [16–19]. Though the finite-time consensus can ensure the system converge in finite time, the convergence time depends on the initial states. However, due to the complexity of environment, the initial states are usually not available in advance [20–28]. In this case, the convergence time cannot be estimated. Correspondingly, the practical application of the multiagents systems will be restricted. To overcome this shortcoming, the fixed-time stability is proposed and analyzed in [20–23]. Based on the fixed-time stability concept, Parsegov et al. [22] first proposed the fixed-time consensus (FdTC), which means that the convergence time is limited and the upper bound of convergence time is independent of the initial states. Zuo and Tie [24] studied the FdTC of first-order MASs over undirected topology. It is shown that the convergence time of the proposed general framework is upper bounded for any initial states. Fu and Wang [25] studied the fixed-time tracking problem for second-order systems with bounded input uncertainties. Based on a fixed-time distributed observer and sliding mode control method, the fixed-time tracking consensus (FdTTC) for high-order MASs with unknown bounded external disturbances is studied in [26]. A new cascade control structure is developed to achieve the FdTTC. Wang et al. [27] studied the FdTC of the second-order nonlinear MASs with time delay over undirected topology. Shang and Ye studied the leader-follower fixed-time group consensus control of MASs over directed topology [28]. A display estimation of convergence time is given. Though there are many results on FdTC, the preserving connectivity between agents communicating with each other is seldom taken into account. However, in practice, the communication ability of multiple agents is usually limited, so it is very important to ensure that the agents communicate with each other within a limited communication radius [29–32]. In addition, for some complex systems, the Lipschitz condition is hard to satisfy. Hence, it is very important to study the system with the nonlinear term regardless of the Lipschitz conditions [29, 30]. By the nonsmooth analysis method, Cao et al. [29] studied the finite-time consensus of MASs with unknown Lipschitz terms and ensured that the communication between agents are always within a limited communication radius. Sun et al. [30] studied the finite-time connectivity-preserving consensus of second-order MASs with nonlinear terms which did not satisfy the Lipschitz conditions. In [30], by using the energy function idea and the technique of arc method, it is proved that the designed control protocol can ensure that the neighbor agents are always in a limited communication radius. Hong et al. [31] studied the finite-time consensus problem for second-order nonlinear MASs under communication constraints, and the interaction patterns can be preserved for the system with the objection of disturbance rejection, but the display estimation of convergence time is not given. Hong et al. [32] studied fixed-time connectivity-preserving distributed average tracking for first-order MASs without considering the interference and the nonlinear terms. Zuo et al. [33] studied the FdTC for MASs over directed and switching interaction topology. They addressed that the explicit bounds of the convergence time for both protocols are independent of the initial condition. However, as far as we know, there are few results related to the FdTTC of MASs with the disturbance and general nonlinearity under the limited communication radius between neighbor agents. Hence, inspired by [27–34], this work studies the FdTTC of the first-order leader following MASs with disturbances and general nonlinearities. Based on the integral sliding surface, a new control protocol is designed. The main contributions of this paper are as follows: (a) the FdTTC of MASs with disturbances and nonlinear term is studied; (b) the agents communicating with each other are always maintaining in a limited communication radius; (c) the upper and lower bounds of convergence time that are independent of initial conditions are given; (d) the unknown disturbances and nonlinear terms are not required to satisfy the Lipschitz condition.The remainder of this paper is as follows. Notations and preliminaries are described in Section2. Problem statement and some necessary lemmas are given in Section 3. Main results are given in Section 4. A simulation example is presented in Section 5 to illustrate the correctness of the obtained theoretical results. And the conclusion is concluded in Section 6.


## 2. Notations and Preliminaries

In this paper,R is the real set, Rn is the n-dimensional real space, E⊗F is the Kronecker product of matrices E and F, ∙ is the 2-norm of a matrix, ⋅1 is the 1-norm of a matrix, 1n=1,…,nT is the n-dimensional column vector with all elements being 1, and ⋅T is the transposition of a matrix or a vector. Let signX=signx1,…,signxnT, where X=x1,…,xnT, and sign⋅ is the sign function, andsig⋅a=⋅asign⋅.GraphG=ε,φ with the edge set ε⊆φ×φ describes the communication connection between followers, and φ=φii=1,2,…,N is the node set. The neighbor set of agent φi is denoted as Ni=φjφj,φi∈ε. The adjacency matrix between followers is A=aijN×N∈RN×N, where aij=0 if node φj does not belong to Ni or the relative distance between nodes φi and φj is larger than r, and r>0 is the communication radius of the agent; otherwise aij>0. In the context, assume that aii=0,∀i=1,2,…,N. If for any k∈2,…,s, there is a sequence of edges i1,i2,i2,i3,…,is−1,is with ik−1,ik∈ε, it is called there is a path from agent i1 to agent is. If there is a path between any two different nodes of G, graph G is a connected graph. The degree matrix of G is D=diagd1,d2,…,dN, where di=∑j=1Naij for i=1,2,…,N. Let ℒ=lij=D−A represent the Laplacian matrix of G. Denote H=ℒ+ℬ, where ℬ=diagb1,…,bN, and bi>0 if the ith follower can get information from the leader, otherwise bi=0.


## 3. Problem Formulation

This section studies the FdTTC for MASs withN agents. The dynamics of the i-th follower is(1)p˙it=uit+fipit,t+dipit,t,i=1,…,N,where pit∈R is the position, uit∈R is the control input, fipit,t∈R is the nonlinear item, and dipit,t∈R is the interference item of the ith agent, respectively.The dynamics of the leader, that is, agent 0, is described as(2)p˙0t=f0p0t,t,where p0t∈R is the position state and f0p0t,t∈R is the nonlinear item.Lemma 1 (see [21]).
If there is a continuous unbounded positive definite functionVx, such that for some d1>0,d2>0,d3>1, and 0≤d4<1, there is(3)V∙x≤−d1Vd3x−d2Vd4x.
Then, the origin of systems (1) and (2) is globally fixed-time stable, and the convergence time T satisfies(4)1d1+d21−d4≕Tmin≤T≤Tmax≔1d1d3−1+1d21−d4.Lemma 2.
(i)
(see [33]) Let σ1,σ2,…,σN≥0 and w>1, then there is
(5)
σ
1
+
⋯
+
σ
N
w
≥
σ
1
w
+
⋯
+
σ
N
w
≥
N
1
−
w
σ
1
+
⋯
+
σ
N
w
.
(ii)
(see [7]) Let σ1,σ2,…,σN≥0 and 0<w≤1, then there is
(6)
N
1
−
w
σ
1
+
⋯
+
σ
N
w
≥
σ
1
w
+
⋯
+
σ
N
w
≥
σ
1
+
⋯
+
σ
N
w
.
(iii)
Sequence inequality
Letε1≤ε2≤⋯≤εn and μ1≤μ2≤⋯≤μn be two sequences of the real number. Then, there is(7)ε1μn+ε2μn−1+⋯+εnμ1≤ε1ρ1+ε2ρ2+⋯+εnρn≤ε1μ1+ε2μ2+⋯+εnμn,where ρ1,…,ρn is an arrangement of μ1,…,μn.(iv)
Letm1,m2,…,mn,l be the positive real numbers. Then, according to (iii) there is(8)m1+m2+⋯+mnm1l+m2l+⋯+mnl≤nm1l+1+m2l+1+⋯+mnl+1.Lemma 3 (see [18]).
Assume that functionφ:R2⟶R+ satisfies φyi,yj=−φyj,yi,i,j=1,…,N,i≠j. Then, for any symmetric matrix B=bijN×N and real numbers ς1,ς2,…,ςN, the following equality holds:(9)∑i=1N∑j=1Nbijςiφyj,yi=−12∑i=1N∑j=1Nbijςj−ςiφyj,yi.Lemma 4 (see [9]).
Laplace matrixℒ of the undirected connected graph G satisfies the following:(i)
For any N-dimensional column vectorsx=x1,…,xNT∈RN, there is(10)xTℒx=∑i=1N∑j=1Naijxi−xj2.(ii)
The eigenvalues of matrixℒ are recorded as 0<λ2≤⋯≤λN−1, and xTℒx≥λ2xTx.Assumption 1.
GraphG is connected, and there is at least one agent that can access the leader’s information, and at the initial time, the distance between any follower and its neighbors is less than r.Assumption 2.
There is a constantc>0, such that(11)fipit,t+dipit,t−f0p0t,t≤c,i=1,…,N.Definition 1.
For any initial state, there is(12)limt⟶Tpit−p0t=0,pit−p0t=0,∀t≥T,i=1,2,…,N,where T is the settling time and the bound of T is independent of the initial states; then, systems (1) and (2) are said to achieve the FdTTC.Definition 2.
If for anyt>0, graph G is connected, and the distance between any agent and its neighbor agents is less than the communication radius r and the states of systems (1) and (2) satisfy Definition 1; then, systems (1) and (2) are said to achieve the fixed-time connectivity-preserving tracking consensus.
For systems (1) and (2), design the following sliding mode control protocol:(13)uit=ui1t+ui2t,with(14)ui1=−k1∑j=1Naijsigα2pit−pjt−k2∑j=1Naijsigα3pit−pjt,and(15)ui2t=−k5signsi−k6sigα1si,where k1, k2,k5, and k6 are the positive constants; α1>1, 0<α2<1, α3>1, and the sliding surface sit∈R, i=1,…,N, is designed as(16)si=∑j=1Naijp¯i−p¯j+bip¯i−p¯0−∫0t∑j=1Naijui1τ−uj1τ+biui1τdτ,where p¯i=pit−pi0 with pi0 being the initial position state of agent i,i=1,…,N. Throughout this paper, we denote a^min=minaijaij≠0,∀i,j∈1,…,N,c0=maxN^1,N^2,…,N^N, where N^i is the number of elements in the set Ni,H¯=H⊗In,st=s1t,…,sNtT,p¯=p¯1,…,p¯NT,u1=u11,…,uN1T,u2=u12,…,uN2T,f=f1,…,fNT and dt=d1t,…,dNtT. Then, (16) can be rewritten as(17)s=H¯p¯−1n⊗p¯0−∫0tH¯u1τdτ.


## 4. Main Results

Theorem 1.
Suppose Assumptions 1 and 2 hold, and for anyt∈0,t1, the distance between followers and their neighbors is less than r. Then, under the control protocol (13), each agent of systems (1) and (2) can maintain its dynamic behavior on the sliding surface in a limited time. That is, sit=0 and t∈0,t1, ∀i=1,…,N, provided that the control gain α1>1 and k5>c and the convergence time T1 satisfies(18)2λmax1/2H¯−1k5+k6−c≕Tmin≤T1≤Tmax≔2λmax1/2H¯−11k5−cα1−1+1k6.Proof.
LetV1s,t=1/2sTH¯−1s. Then, the derivative of V1s,t along system (17) is(19)V∙1=sTH¯−1s∙=sT−k5signs+f+dt−1n⊗f0p0t,t−k6sigα1s≤−k5−cs1−k6s21+α1≤−k5−c2λmaxH¯−1V11/2−k62λmaxH¯−1V11+α1/2.
From Lemma1, one can get the convergence time which satisfies(20)22/λmaxH¯−1k5+k6−c≕Tmin≤T1≤Tmax≔12/λmaxH¯−11k5−cα1−1/2+2k6,which implies that (18) holds. The proof is completed.Theorem 2.
For systems (1) and (2) under the control protocol (13), if k1>0, k2>0, k5>c, k6>0, m≥2c1c0, and E0<c2r2a^min, where E0 is the initial value of an energy function, then the distance between all followers and their neighbor agents is less than r.Proof.
Similar to the argument in [30], assume that there exists a time t0>0 and φj∈Ni,i=1,…,N, such that pit0−−pjt0−<r and pit0−pjt0≥r. According to Theorem 1, sit=0 for t∈0,t0i=1,…,N. Denoting ei1=pit−p0t, then one can get(21)e∙i1=−k1∑j=1Naijsigα2ei1−ej1−k2∑j=1Naijsigα3ei1−ej1.
Define an energy functionEe on 0,t0 as(22)Ee=∑i=1N∑j=1Naij∫0∣ei1−ej1∣hsds+m∑i=1Nei12,where hpi−pj satisfies the following conditions:(i)
hpi−pj is continuous for pi−pj;(ii)
c2pi−pj≤hpi−pj≤c1pi−pj,c1>c2>0.
The derivative ofEe along (21) is(23)E˙=2∑i=1N∑j=1Naijhei1−ej1−∑j=1Nk1aijsigα2ei1−ej1−k2∑j=1Naijsigα3ei1−ej1+2m∑i=1Nei1−k1∑j=1Naijsigα2ei1−ej1−k2∑j=1Naijsigα3ei1−ej1,that is,(24)E˙≤2c1∑i=1N∑j=1Naijei1−ej1∑j=1Nk1aijei1−ej1α2+k2∑j=1Naijei1−ej1α3−2m∑i=1Nei1k1∑j=1Naijsigα2ei1−ej1+k2∑j=1Naijsigα3ei1−ej1.
From (iv) in Lemma2, one can get(25)E˙≤2c1c0∑i=1N∑j=1Nk1aijei1−ej1α2+1+k2aijei1−ej1α3+1−2m∑i=1Nei1k1∑j=1Naijsigα2ei1−ej1+k2∑j=1Naijsigα3ei1−ej1≤2c1c0−m∑i=1N∑j=1Nk1aijei1−ej1α2+1+k2aijei1−ej1α3+1≤0,which implies that Et≤E0<c1r2a^min. Then, there is limt⟶t0−ei1−ej1=r. Since ∫0rhsds>c1r2/2 and Et0−>c2r2a^min, which contradicts Et≤E0<c1r2a^min. Hence, the assumption is wrong. The proof is completed.
Denotek3=k1/221+α2/2λ21+α2/2ℒ1 and k4=k2/2N1−α321+α3/2λ21+α3/2ℒ2, where ℒ1 and ℒ2 are the Laplacian matrix of matrix A1=aij2/1+α2N×N and A2=aij2/1+α3N×N and λ2ℒ1 and λ2ℒ2 are the second small eigenvalue of matrix ℒ1 and ℒ2, respectively.Theorem 3.
Let Assumptions 1 and 2 hold. For systems (1) and (2) under the control protocol (13) with (14)–(16), if k1>0,k2>0, and 0<α2<1,α3>1, then for any i∈1,…,N,ei1 could converge to zero with the convergence time T2 satisfying(26)2k3+k41−α2≤T2≤2k4α3−1+2k31−α2.Proof.
According to Theorem1, sit=0, i=1,…,N. Then, based on the sliding manifold (17), there is(27)e∙i1=−k1∑j=1Naijsigα2ei1−ej1−k2∑j=1Naijsigα3ei1−ej1.
Choose the candidate Lyapunov function(28)V2=12∑i=1Nei12.
The derivative ofV2 along (27) is(29)V2∙=∑i=1Nei1e∙i1=−k1∑i=1N∑j=1Naijei1sigα2ei1−ej1−k2∑i=1N∑j=1Naijei1Tsigα3ei1−ej1.
From Lemma3, one can obtain(30)V2∙≤−k12∑i=1N∑j=1Naijei1−ej11+α2−k22∑i=1N∑j=1Naijei1−ej11+α3≤−k12∑i=1N∑j=1Naij2/1+α2ei1−ej121+α2/2−k22∑i=1N∑j=1Naij2/1+α3ei1−ej121+α3/2.
According to (i) and (ii) in Lemma2 and 4, one can get(31)V2∙≤−k12∑i=1N∑j=1Naij2/1+α2ei1−ej121+α2/2−N1−α3k22∑i=1N∑j=1Naij2/1+α3ei1−ej121+α3/2≤−k1221+α2/2λ21+α2/2ℒ1V21+α2/2−k22N1−α321+α3/2λ21+α3/2ℒ2V21+α3/2≤−k3V21+α2/2−k4V21+α3/2.
From Lemma1, one can get for any i∈1,…,N,ei1 converges to zero in a finite time and the settling time T2 satisfies(32)2k3+k41−α2≕Tmin≤T2≤Tmax≔2k4α3−1+2k31−α2.
The proof is completed.From Theorems1–3, one can find that systems (1) and (2) can achieve the fixed-time connectivity-preserving tracking consensus and the convergence time T=T1+T2 satisfies(33)2λmax1/2H¯−1k5+k6−c+2k3+k41−α2≤T≤2λmax1/2H¯−11k5−cα1−1+1k6+2k4α3−1+2k31−α2.


## 5. Examples

Example 1.
Consider MASs (1) and (2) consisting of four followers i=1,2,3,and4 and one leader i=5. Under the consensus protocol (13) over the communication network as shown in Figure 1, choose α1=2, α2=0.3, α3=3, r=20, k1=k2=1, k5=15, k6=1, fipit,t=−pitcost, dipit,t=2sin0.5it+iπ/6, f0p0t,t=2sint. Let d12=p1−p2, d13=p1−p3, and d34=p3−p4. The initial state is e1=e11,e21,e31,e41T=6,2,3,5T, and the adjacency matrix(34)A=aij4×4=0110100010010010,ℬ=diag0,0,1,1.
By calculation, one can getTmax=28.465, Tmin=4.785, and T=7.690, where T is the convergence time and Tmin andTmax are the lower and upper bound of T. Hence, the convergence time is limited. Trajectories of the position error between followers and the leader are given in Figure 2, which shows that the position error between followers and the leader tends to zero. The distances between neighbor agents are shown in Figure 3. The maximum distance dmax and minimum distance dmin are given in Table 1. Figure 3 and Table 1 show that the distance between followers and their neighbors is always within a given limited range. The simulations illustrate that under the consensus protocol (13), systems (1) and (2) can achieve the FdTTC, which verifies the correctness of the main results in the work.Figure 1
Network graph.Figure 2
Trajectories of the position error between followers and the leader.Figure 3
Distances between neighbor agents of systems (1) and (2).Table 1
The maximum and minimum distance between neighbor agents.
d
12
d
13
d
34
d
max
4.000
3.000
2.000
d
min
0
0
0


## 6. Conclusion

The FdTTC of first-order MASs with general nonlinearity and disturbances is studied in this paper. Firstly, the integral sliding surface is designed to solve the nonlinear term and disturbance in the system; then, the Lyapunov function theory and principle of fixed-time stability are adopted to prove that the integral sliding surface can switch to the surface in a finite time, and the upper and lower bounds independent of the initial conditions of the time to the sliding surface are obtained. Secondly, it is proved that the agents communicating with each other are always keeping within a limited communication radius by using the idea of arc method and designing the energy function. Thirdly, it is proved that the system can achieve FdTTC and the upper and lower bounds of convergence time that are independent of the initial conditions are given. Finally, a simulation example is presented, which shows that all the states of the followers can converge to that of the leader, and the convergence time is between the given lower and upper bounds. Table1 gives that the agents communicating with each other are always within the given limited communication radius. Hence, the presented example verifies the correctness of theoretical results and the validity of the adopting methods. The FdTTC for the nonlinear MASs with time delay or noise over the switching network is the future work to be done.



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