# On Multitracking of First-Order MASs with Adaptive Coupling Strength

Authors: Da Huang; Xiaolin Fan; Zhiyong Yu; Haijun Jiang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

The problem of cluster consensus with multiple leaders is called multitracking. In this article, a sort of multitracking of first-order multiagent systems with adaptive coupling strength is studied by the application of adaptive strategy, and the delayed relation between various leaders and clusters is considered. To reach the clustered multitracking goal, a novel pinning-like control protocol with adaptive approach is designed according to the properties of network topology. In addition, the structure of the networked system is a weakly connected digraph. Some conditions are derived to ensure that the nodes in the same cluster reach the consensus via tracking their leader, while leaders will keep a delayed relation with the settled leader node as time goes on to form the required delay consensus.


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## Body

## 1. Introduction

A multiagent system (MAS) is a networked system consisted of multiple interconnected computerized agents. Consensus is a sort of distributed coordination problem of MASs, and one basic aim for the research is to design reasonable control strategies so that all agents can achieve a common status value. In the system model related with consensus, the agents are required to communicate with each other based on the graph of the networked system so that they can cooperate effectively to finish some desired tasks.As one of the interdisciplinary research topics related to the field of applied mathematics, control theory, and computer science, consensus problem has attracted much attention of both researchers and engineers thanks to its wide application in the coordination problem of aerial vehicles, satellite clusters, sensor networks, etc. In the past decades, consensus problems have been explored from various aspects in the existing research works [1–14], and during this period, synchronization problems of complex networks, which share similar control strategies with consensus problems, have also been widely investigated [15–34]. The study methods for the two coordination problems are essentially the same.Cluster consensus, as a sort of extended consensus problems, means that all agents in one cluster achieve the same target status while there exists no consensus for agents among different clusters. As the scales and complexities of complex systems increase, cluster consensus problem exhibits more flexibility in practical applications, and it has a more general concept than the original global consensus. In fact, many multitasking issues need to be solved by partitioning all the nodes into different clusters. It is familiar that the clustered consensus phenomenon is very common in practical circumstances, such as formation of group opinions and multispecies foraging. The case of cluster consensus with multileaders is usually called as multitracking, for which the main aim is to make the reasonable controller design so that the position or velocity status of nodes in each cluster will be consistent with that of the leader.Over the past decade, the study of coordination control related to clustered structure has caused wide attention, and an amount of research works on clustered networks have been done from various aspects [2, 4–7, 9, 10, 13–15, 17–22, 31–33].In [15], some simple and useful criteria are derived by constructing an effective control scheme and adjusting automatically the adaptive coupling strength. In [17], the article investigated the cluster synchronization with time-varying coupling strengths and delayed dynamical systems by applying pinning scheme. Ma et al. [9] investigated the second-order group consensus through pinning control and leader-following approach, and the pinning scheme is established by the structural properties of the graph.Based on the rationality of the existence of time delays in the coordination system, many study works on clustered networked systems with time delays have been done [2, 15, 17, 22]. Ma et al. [2] have proposed the notion of cluster-delay consensus. In [2], the authors find a method to deal with problem on the delayed relations between different clusters through the tracking process of the leader nodes. In real situation, the notion of cluster-delay consensus may imply that the nodes in different clusters should arrive at the same position at different times, which means the possible congestion can be avoided in the MASs.As we know, the studies on cluster consensus models which considered the possible existence of communication links among the multiple leader nodes are relatively rare.In real systems, the graph structure may be very complicated and has the group pattern because of the accomplishment of multitasks which have some exact delayed relations of final states. For instance, in the traffic system, to avoid congestion and make roads orderly, time delays can be designed among different sorts of vehicles.It is familiar that an adaptive controller is a controller that modifies its properties to adapt to changes in the dynamics properties of disturbances, and the adaptive control ([15, 16, 18, 35, 36]) strategy which acts on the coupling strengths of networks is very effective in enhancing the stability of consensus problem. In the study of clustered coordinate control problem, many researchers use adaptive approach to synchronize the complex network or to make the MAS reach consensus.Through the above observations and the advantages of adaptive control, an issue arises naturally: How can this sort of multitracking problem be solved if the adaptive couplings and the adaptive control method are involved, and what sort of control strategy can be suitably designed for reaching the consensus? For solving this problem, we establish multitracking models with adaptive coupling strength, and this sort of cluster consensus will be realized by designing suitable adaptive control strategies.Specifically, the novelties of this research are listed as follows:(i)
This article studies a novel multitracking model with adaptive coupling strength, and the delayed relation among the states of agents in different clusters is embodied through the tracking process for the leading subsystem(ii)
Under the clustered digraph, a new pinning-like control strategy combined with adaptive approach is designed(iii)
Compared with similar problems on cluster consensus, two kinds of time delays and two sorts of adaptive coupling strengths are considered in this paperIn this paper, the intrinsic dynamics of all followers and leader nodes satisfy a Lipschitz-like condition. Lyapunov method, matrix theory, and graph theory are used for deriving the results which can solve the multitracking problem.The rest part is organized as follows. Model description and some preliminaries are given in Section2. Section 3 includes the main results. Section 4 gives the numerical simulations to verify the theoretical results, and Section 5 has made the final conclusions.Notation: throughout this article,R represents the set of real numbers. Let Rn denotes the n-dimensional Euclidean space and RM×N denotes the M×N real matrices. ON×N denotes the zero matrix, and In is the n-dimensional identity matrix. For a real matrix A∈RN×N, let AT be its transpose and denote its symmetric matrix as As=A+AT/2, and λ2A denotes the maximum eigenvalue of A. The norm of a vector is denoted by x=xTx1/2, for x∈Rn. For a real symmetric matrix F, denote F<0F>0, if F is negative (positive) definite. For any two nonempty sets X and Y, X∖Y denotes the complementary set of Y respect to X. ⊗ is the Kronecker product.


## 2. Model Description

In this research, the graph structure of a MAS is denoted by a digraphB=V,E,A, where the vertex set V=v1,v2,…,vN denotes the agents, E⊆V×V is the edge set, and A=aijN is the weighted adjacency matrix which denotes the linking structure of the MAS. A directed edge of B denoted by vi,vj means that there is a directed information link from vi to vj, which can be understood as vj can receive information from vi. The elements of A are defined as follows: aij>0, if vj,vi∈E; aij=0, if vj,vi∉E.The digraph is supposed to be simple [37] and weakly connected. The in-degree of node vi is defined as deginvi=∑j=1,j≠iNaij. The Laplacian matrix of B is denoted by L=lij∈RN×N and is defined as lij=−aij, i≠j. lii=∑j=1,j≠iNaij, which ensures that ∑j=1Nlij=0.Let us consider a first-order multiagent network consisting ofN followers and m leader nodes. The dynamics of the follower with time-varying delay is modeled by(1)x˙it=fxit,xit−ςt+ct∑j≠iaijxjt−xit+uit,i=1,2,…,N,where xit∈Rn is the state of the ith node, uit∈Rn is the control input, and fxit,xit−ςt∈Rn is the intrinsic nonlinear dynamics of the ith agent. ct>0 is the time-varying coupling strength. ςt>0 is the inherent time-varying delay.Suppose the networked system hasm clusters with 2≤m<N, and each cluster has one leader. Let Vr be the node set of the rth cluster, thus V=v1,v2,…,vN=V1∪V2∪…∪Vm and Vr∩Vs=∅ (r,s=1,2,…,m;r≠s). Let i^ be the subscript of the cluster that the ith vertex belongs, i.e., vi∈Vi^. Let the leader set be V∗=v1∗,v2∗,…,vm∗. The followers of the rth leader are the nodes in Vr, r∈1,2,…,m. Let Vr˜⊆Vr be subset of nodes which can receive information from other clusters, i.e., for any vertex vi∈Vr˜, there exists at least one node vj∈V∖Vp such that aij≠0.The leaders of clusters for system (1) are described by the following equation:(2)h˙wt=fhwt,hwt−ςt−γwhwt−h1t−ςw,w=1,2,…,m,where hwt is the state of vk∗, γw>0 is the coupling coefficient between vw∗ and v1∗, and ςw is the time delay between the vw∗ and v1∗, ς1=0 and ςw>0, w=2,…,m.Remark 1.
We can deduce from (2) that the linking structure of the term −γwhwt−h1t−ςw implies a star coupled digraph, which has v1∗ as its center vertex and vw∗ as its leaf vertex. The multitracking model can be interpreted as adding pinning communication links between agents of the star subnetwork and the pinned nodes of the clustered subnetwork (see Figure 1)
The main goal of the research is to impose suitable control effectsuit on subsystem (1) such that it can keep pace with the subsystems (2) with pinning-like adaptive approach. In the meantime, subnetwork (2) is designed to reach the desired delayed consensus to be defined later. For getting the main conclusions, the following definition and assumptions are necessary.Figure 1
The graph of a multitracking model. Thekth leader is denoted by triangle dot. The i th agent is labelled with i and denoted by circle dot, i=1,2,…,12; k=1,2,3.Definition 1.
The MAS with delayed inherent dynamics (1) and (2) is said to reach the delayed cluster consensus, if the solutions of (1) and (2) satisfy limt⟶∞xit−hi^t=0 and limt⟶∞hwt−h1t−ςw=0.Remark 2.
One can see from the dynamics that the movement ofv1∗ is determined by its own dynamic behavior. In fact, the MAS with (1) and (2) can be interpreted as an entire system consisting of two subsystems. The entire network is designed to achieve the delayed cluster consensus via applying adaptive control strategies.Assumption 1.
There exist two positive constantsθ and η, such that (3)ϕt−φtTfϕt,ϕt−ςt−fφt,φt−ςt≤θϕt−φtTϕt−φt+ηϕt−ςt−φt−ςtTϕt−ςt−φt−ςt,for any φt,ϕt∈Rn, where ςt is the time-varying delay of intrinsic dynamic.Assumption 2.
The inequalities0<ςt≤ς and 0<ς˙t≤ϵ<1 hold for the time-varying delay ςt, where ς and ϵ are constants.Lemma 1 (see [9]).
Based on the notations of Section2, if node vi∈Vi^∖V˜i^, then ∑j=1Nlijhj^t−ςj^=0.Lemma 2 (see [38]).
For matricesA,B,C, and D with suitable dimensions, the following equations hold:(i)
A+B⊗C=A⊗C+B⊗C(ii)
A⊗BT=AT⊗BTLemma 3 (see [11]).
Letϑ·: t0−ς,+∞⟶0,+∞ be a continuous function and ϑ.t≤−cϑt+dϑ¯t holds for t≥t0, where ϑ¯t=sup−ς≤ζ≤0ϑt+ζ. If c>d>0, then one has(4)ϑt≤ϑ¯t0exp−ϵt−t0,t≥t0,where ϵ>0 satisfies the equation: ϵ−c+dexpϵς=0.


## 3. Main Results

Sincelij=−aij, i≠j, and lii=∑j=1,j≠iNaij, (1) can be rewritten by(5)x˙it=fxit,xit−ςt−ct∑vj∈Vlijxjt+uit,vi∈V.The consensus error in each cluster is as follows:(6)ξit=xit−hi^t.Then the error system with (1) and (2) can be obtained: (7)ξ˙it=fxit,xit−ςt−fhi^t,hi^t−ςt−ct∑j=1Nlijξjt−ct∑j=1Nlijhj^t+γi^hi^t−h1t−ςi^+uit.By the network structure and Lemma1, the control input uit can be constructed as follows:(8)uit=−γi^hi^t−h1t−ςi^+ct∑j=1Nlijhj^t−ctdiξit,vi∈V˜i^,−γi^hi^t−h1t−ςi^−ctdiξit,vi∈Vi^∖V˜i^∧degviin=0,−γi^hi^t−h1t−ςi^,otherwise,where ct>0 is the adaptive coupling strength, di>0 is the feedback control gain, and the adaptive updating law is as follows: (9)c˙t=α∑i=1NξiTtξit,where α>0 is the adaptive gain.For simplicity, the nodes inVi^ can be classified as V1i=V˜i^, V2i=vi|vi∈Vi^∖V˜i^,degviin=0, and V3i=Vi^∖V1i∪V2i. Let D=diagd1,d2,…,dN with di>0 for vi∈V1i∪V2i and di=0 for vi∈V3i.Remark 3.
By the form of controller (8), one can see that in the kth cluster (k=2,3,…,m), each of the three types of nodes has been controlled by individual scheme, i.e., to the nodes vi∈V1i∪V2i, the first term is used for counteracting the interaction between the leaders vk∗ and v1∗, while the third term −ctdiξit is a feedback term exert on the ith node, and it is applied to make the nodes reach the consensus intra one cluster. The second term applied to the nodes in vi∈V˜i^ is used to balance the interactions among clusters.
By (7) and (8), the error system with adaptive coupling strength has the following description: (10)ξ˙it=fxit,xit−ςt−fhi^t,hi^t−ςt−ct∑j=1Nlijξjt−ctdiξit,vi∈Vi^.
Thus one can acquire the following theorem.Theorem 1.
Based on Assumptions1 and 2, if the following conditions hold:(i)
θ−βc∗+η/1−ε<0 and βIN−D−L is negative definite(ii)
λ˜<−a2Then systems (1) and (2) with protocol (8) and adaptive updating law (9) can solve the multitracking problem, i.e., the delayed cluster consensus can be achieved.Proof.
Consider the following Lyapunov functional candidate:(11)Vt=12∑i=1NξiTtξit+η1−ϵ∑i=1N∫t−ςttξiTsξisds+βct−c∗22α,where β and c∗ are positive constants.
Denoteξt=ξ1Tt,ξ2Tt,…,ξNTtT, then under Assumptions 1 and 2, by Lemma 1, the time derivative of Vt along the trajectories of (10) can be derived as follows:(12)V.t=∑i=1NξiTtfxit,xit−ςt−fhi^t,hi^t−ςt−ct∑i=1N∑j=1NlijξiTtξjt−ct∑vi∈V˜i^diξiTtξit+η1−ϵ∑i=1NξiTtξit−1−ς˙tη1−ϵ∑i=1NξiTt−ςtξit−ςt+βct−c∗c¯.t≤θ∑i=1NξiTtξit+η∑i=1NξiTt−ςtξit−ςt−ctξTtL⊗Inξt−ct∑vi∈V˜i^diξiTtξit+η1−ϵ∑i=1NξiTtξit−1−ς˙tη1−ϵ∑i=1NξiTt−ςtξit−ςt+βct−c∗∑i=1NdiξiTtξit=θξTtξt+ηξTt−ςtξt−ςt−ctξTtL⊗Inξt+η1−ϵξTtξt−1−ς˙tη1−ϵξTt−ςtξt−ςt+βctξTtD⊗Inξt−βc∗ξTtIN⊗Inξt−ctξTtD⊗Inξt≤θξTtξt−ctξTtL⊗Inξt+η1−εξTtξt−βc∗ξTtIN⊗Inξt+βctξTtIN⊗Inξt−ctξTtD⊗Inξt≤ξTtθ−βc∗+η1−ϵIN⊗In+ctβIN−D−L⊗Inξt.
One can acquire thatβIN−D−Q is negative definite for an appropriately small β and a sufficiently large c∗ can ensure θ−βc∗+η/1−ϵ<0 holds, therefore, V.t≤0, which further implies that limt⟶∞xit−hi^t=0, (i=1,2,…,N) holds for the solutions of (1) and (2) with any initial values.
On the other aspect, for achieving the delayed consensus among leader nodes, the error for system (2) is denoted by(13)ϱ¯wt=hwt−h1t−ςw.
Then the error system is denoted by(14)ϱ¯.wt=fhwt,hwt−ςt−fh1t−ςw,h1t−ςw−ςt−γwϱ¯wt,w=2,…,m.
The Lyapunov function candidate is established as follows:(15)V∗t=12∑w=2mϱ¯wTtϱ¯wt.
Denoteϱ∗t=ϱ2Tt,…,ϱmTtT; then system (2) can reach the LDC by the following derivation:(16)V∗.t=∑w=2mϱwTtfhwt,hwt−ςt−fh1t−ςw,h1t−ςt−ςw−∑w=2mγwϱwTtϱwt≤a1∑w=2mϱkTtϱkt−∑w=2mγwϱwTtϱwt+a2∑w=2mϱwt−ςtTϱwt−ςt≤ϱ∗Tta1IN−Θ⊗Inϱ∗t+2a2sup0≤ςt≤ςV∗t−ςt≤2λ˜V∗t+2a2sup0≤ςt≤ςV∗t−ςt,where λ˜=λmaxa1IN−Θ and Θ=diagα2,…,αm. Therefore, by Lemma 3 and the condition λ˜<−a2, we have(17)V∗t≤sup0≤ς0≤ςV∗−ς0exp−ϵt,whereϵsatisfiesϵ+2λ˜+2a2expϵς=0andϵ>0.
Then one can derive that the delayed consensus among leaders can be achieved in this situation, and therefore, the delayed cluster consensus can be reached under controller (8); furthermore, the uncertain parameters ct adapt itself to some certain values, i.e., ct⟶c∗; thus, the multitracking problem under the adaptive approach is solved, and this completes the proof.Remark 4.
In view of subsystem (2), if γw is time varying and it is described by the form γwt=pwqt, in which qt is the coupling strength and pw is the corresponding coupling weights. Set the adaptive updating law as the form: q˙t=δ∑w=2mpwϱwTtϱwt, where δ>0 is the adaptive gain, and thus the second theorem can be obtained.Theorem 2.
Suppose Assumptions1 and 2 hold, based on Remark 4, if qt satisfies the following updating law:(18)q˙t=δ∑w=2mpwϱwTtϱwt,and if the following conditions hold:(i)
θ−βc∗+η/1−ϵ<0 and βIN−D−L is negative definite(ii)
a1+ρ/1−εIm−1−q∗Ξ is negative definiteThen systems (1) and (2) under controller (8) with adaptive updating law (9) can solve the multitracking problem.Proof.
The proof of the first part is similar to that of Theorem1, and to the leading system, we condiser the following Lyapunov functional candidate:(19)V∗t=12∑w=2mϱwTtϱwt+12δqt−q∗2+ρ1−ε∑i=2m∫t−τttϱiTsϱisds,where the constant q∗ is positive. Denote ϱ∗t=ϱ2Tt,…,ϱmTtT,Ξ=diagq2,q3,…,qm, and choose ρ≥a2, and then we have(20)V∗.t≤a1∑w=2mϱwTtϱwt+a2∑w=2mϱwt−ςtTϱwt−ςt−∑w=2mpwqtϱwTtϱwt+∑w=2mpwqtϱwTtϱwt−q∗∑w=2mpwϱwTtϱwt+ρ1−ε∑w=2mϱwTtϱwt−1−ς˙tρ1−ε,∑w=2mϱwt−ςtTϱwt−ςt≤a1∑w=2mϱwTtϱwt−q∗∑w=2mpwϱwTtϱwt+ρ1−ε∑w=2mϱwTtϱwt=ϱw∗Tta1+ρ1−εIm−1−q∗Ξ⊗Inϱw∗t.
Some sufficiently largeq∗ can be chosen so that a1+ρ/1−εIm−1−q∗Ξ is negative definite, then V∗.t≤0, therefore, the delayed consensus among leaders can be achieved, and therefore, the multitracking problem can be solved, that is, the delayed cluster consensus can be reached. Further, ct⟶c∗ and qt⟶q∗, and the proof is completed.Remark 5.
Since the connected undirected graph does not exist vertex with zero in-degree, the control protocol (8) for the network with undirected graph can be simplified as follows:(21)uit=−γi^thi^t−h1t−ςi^+ct∑j=1Nlijhj^t−ctdiξit,vi∈V˜i^,−γi^thi^t−h1t−ςi^,vi∈Vi^∖V˜i^.
Under (21), a corollary similar to the previous theorem can be derived, and it is omitted here.Remark 6.
MAS-related problems have been widely analyzed in many fields, such as robotic systems [39], sensor networks [40], and neural networks [41]. Since different networks may contain the same or similar graph structures, the study of one network may shed light on other related networks. One may consider some classical graph structures of the networks, such as neural network, to study the similarity and do some enlightening works in the future research.


## 4. Numerical Examples

In this part, simulation examples are given to check the correctness of the theorem. Consider a MAS composed of three clusters with 9 nodes, and the corresponding graph is shown in Figure1.The nodes are labelled with an ascending sequence from 1 to 9. The purple arrows describe the pinning control that thekth leader put on the nodes with specific topological properties (k=1,2,3). The blue edges denote the interactions among clusters, and the black edges represent the communication links inside each cluster. Through (8) and Figure 1, we know that the red vertexes (labelled by 3,5,7,9) belong to V1i, the yellow vertexes (2,4,6,8) belong to V3i, and the green vertex (labelled by 1) belongs to V2i. It is assumed that the dimensional dynamical system of each agent is one-dimensional.Example 1.
The nonlinear functionf is described by fxit,xit−ςt=0.5sinxi−0.5cosxit−ςt, where xit,xit−ςt∈R, and ςt=0.25tanht. The elements lij of graph Laplacian in (5) are chosen as follows: l21=−1,l22=2,l23=−1,l31=−1,l33=2,l34=−1,l44=1,l45=−1,l55=2,l56=−1,l59=−1,l64=−1,l66=1,l71=−1,l76=1,l86=−1,l87=2,l88=−1,l93=−1,l99=1, and if the subscripts of lij are not involved above, then set lij=0.
The parameters in the adaptive controller (8) with updating law (9) are selected as follows: γ1=0,γ2=2,γ3=1;ς1=0,ς2=0.5,ς3=1;d1=2,d3=1.5,d5=2,d7=1,d9=0.5,andα=0.05.
In view of Assumption1, let β=0.2, then IN−D−L<0 holds, set θ=0.3,η=0.7,andϵ=0.3, one has θ−c∗+η/1−ε<0, choose a1=0.2,a2=0.1, and thus, the conditions of Theorem 1 can be satisfied. The states of all nodes are shown in Figures 2–4, and their initial conditions are as follows: x10=−1.5,x20=1.75,x30=3.5,x40=−0.5,x50=−2.5,x60=0.8,x70=1.5,x80=−0.8,x90=1.2;h10=0.8,h20=−1.2,andh30=0.6. The state changes of leaders are shown in Figure 5. The initial value of ct is chosen as c0=0.1, the orbit of coupling strength ct is shown in Figure 6, one can see that ct adaptively adjusts with ct⟶2, the two error states of the two subnetworks, i.e., ξit and ξwt both convergence to zero, and the delayed cluster consensus is indeed achieved.Figure 2
State trajectories of cluster 1.Figure 3
State trajectories of cluster 2.Figure 4
State trajectories in cluster 3.Figure 5
State trajectories of leaders.Figure 6
The orbit of adaptive coupling strengthc(t).Example 2.
This example shares the same nonlinear functionf with Example 1. Choose the same Laplacian matrix L=lij9×9 with the previous example, and set the initial values as h10=0.8,h20=−1.2,h30=0.6;γ1=0,γ20=0.02,γ30=0.05;x10=−1.5,x20=1.75,x30=3.5,x40=−0.5,x50=−2.5,x60=0.8,x70=1.5,x80=−0.8,x90=1.2;ς0=0;c0=0.1,q0=0.01. Let the parameters in the control protocol be ς1=0,ς2=0.5,ς3=1,d1=2,d3=1.5,d5=2,d7=1,d9=0.5;α=0.05,δ=0.1;p1=1,p2=0.5,p3=0.2; in view of Assumptions 1 and 2, set a1=0.2,a2=0.15,ρ=0.2,andε=0.25, and thus, the conditions of Theorem 2 can be satisfied. The description of the state changes of all nodes is shown in Figures 7–10. The change curves of the adaptive couplings ct and γkt are shown in Figure 11, and one can see that ct,γkt adapt themselves to certain values.
It can be seen that the simulation verifies the effectiveness well, and the newly defined multitracking problem is indeed solved.Figure 7
The trajectories of state changes in cluster 1.Figure 8
The trajectories of state changes in cluster 2.Figure 9
The state trajectories in cluster 3.Figure 10
The state trajectories of leaders.Figure 11
The orbits ofct and γkt,k=2,3.


## 5. Conclusion

In this work, a type of multitracking problem with adaptive coupling strength named delayed cluster consensus has been investigated. By considering the graph properties of the clustered network, a new pinning-like control scheme with adaptive approach has been designed. Some sufficient criteria for solving the multitracking problem have been obtained. There still exist lots of works on the similar multitracking problem with MAS deserving further study, for example, multitracking issue with higher order system, multitracking with impulsive control, multitracking with switching topology, etc., and some of the problems might be done in one’s future research.



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