Research Article On Multitracking of First-Order MASs with Adaptive Coupling Strength

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.


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.
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. e 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.
rough 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) is 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 paper In 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. e rest part is organized as follows. Model description and some preliminaries are given in Section 2. 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 R n denotes the n-dimensional Euclidean space and R M×N denotes the M × N real matrices. O N×N denotes the zero matrix, and I n is the n-dimensional identity matrix. For a real matrix A ∈ R N×N , let A T be its transpose and denote its symmetric matrix as A s � (A + A T )/2, and λ 2 (A) denotes the maximum eigenvalue of A. e norm of a vector is denoted by ‖x‖ � (x T x) 1/2 , for x ∈ R n . For a real symmetric matrix F, denote F < 0(F > 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.

Model Description
In this research, the graph structure of a MAS is denoted by a digraph , v N denotes the agents, E⊆V × V is the edge set, and A � [a ij ] N is the weighted adjacency matrix which denotes the linking structure of the MAS. A directed edge of B denoted by (v i , v j ) means that there is a directed information link from v i to v j , which can be understood as v j can receive information from v i . e elements of A are defined as follows: e digraph is supposed to be simple [37] and weakly connected. e in-degree of node v i is defined as Let us consider a first-order multiagent network consisting of N followers and m leader nodes. e dynamics of the follower with time-varying delay is modeled by where x i (t) ∈ R n is the state of the ith node, u i (t) ∈ R n is the control input, and f(x i (t), x i (t − ς(t))) ∈ R n is the intrinsic nonlinear dynamics of the ith agent. c(t) > 0 is the timevarying coupling strength. ς(t) > 0 is the inherent timevarying delay. Suppose the networked system has m clusters with 2 ≤ m < N, and each cluster has one leader. Let V r be the 2 Discrete Dynamics in Nature and Society node set of the rth cluster, thus Let i be the subscript of the cluster that the ith vertex belongs, i.e., v i ∈ V i . Let the leader set be e followers of the rth leader are the nodes in V r , r ∈ 1, 2, . . . , m { }. Let V r ⊆V r be subset of nodes which can receive information from other clusters, i.e., for any vertex v i ∈V r , there exists at least one node v j ∈ V∖V p such that a ij ≠ 0. e leaders of clusters for system (1) are described by the following equation: where h w (t) is the state of v * k , c w > 0 is the coupling coefficient between v * w and v * 1 , and ς w is the time delay between the v * w and v * 1 , ς 1 � 0 and ς w > 0, w � 2, . . . , m.
Remark 1. We can deduce from (2) that the linking structure of the term −c w (h w (t) − h 1 (t − ς w )) implies a star coupled digraph, which has v * 1 as its center vertex and v * w as its leaf vertex. e 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) e main goal of the research is to impose suitable control effects u i (t) 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.

Definition 1.
e MAS with delayed inherent dynamics (1) and (2) is said to reach the delayed cluster consensus, if the solutions of (1) and (2) Remark 2. One can see from the dynamics that the movement of v * 1 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. e entire network is designed to achieve the delayed cluster consensus via applying adaptive control strategies.

Assumption 1.
ere exist two positive constants θ and η, such that for any φ(t), ϕ(t) ∈ R n , where ς(t) is the time-varying delay of intrinsic dynamic.

Main Results
Since l ij � −a ij , i ≠ j, and l ii � N j�1,j ≠ i a ij , (1) can be rewritten by e consensus error in each cluster is as follows: en the error system with (1) and (2) can be obtained: By the network structure and Lemma 1, the control input u i (t) can be constructed as follows: where c(t) > 0 is the adaptive coupling strength, d i > 0 is the feedback control gain, and the adaptive updating law is as follows: where α > 0 is the adaptive gain. For simplicity, the nodes in V i can be classified as 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 , the first term is used for counteracting the interaction between the leaders v * k and v * 1 , while the third term −c(t)d i ξ i (t) is a feedback term exert on the ith node, and it is applied to make the nodes reach the consensus intra one cluster. e second term applied to the nodes in v i ∈ 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: us one can acquire the following theorem.

Theorem 1. Based on Assumptions 1 and 2, if the following conditions hold:
en 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: where β and c * are positive constants.
On the other aspect, for achieving the delayed consensus among leader nodes, the error for system (2) is denoted by en the error system is denoted by Discrete Dynamics in Nature and Society e Lyapunov function candidate is established as follows: Denote ϱ * (t) � (ϱ T 2 (t), . . . , ϱ T m (t)) T ; then system (2) can reach the LDC by the following derivation: where λ � λ max (a 1 I N − Θ) and Θ � diag(α 2 , . . . , α m ). erefore, by Lemma 3 and the condition λ < − a 2 , we have where ϵ satisfies ϵ + 2λ + 2a 2 exp ϵς { } � 0 and ϵ > 0. (17) en 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 c(t) adapt itself to some certain values, i.e., c(t) ⟶ c * ; thus, the multitracking problem under the adaptive approach is solved, and this completes the proof. □ Remark 4. In view of subsystem (2), if c w is time varying and it is described by the form c w (t) � p w q(t), in which q(t) is the coupling strength and p w is the corresponding coupling weights. Set the adaptive updating law as the form: where δ > 0 is the adaptive gain, and thus the second theorem can be obtained.

Theorem 2. Suppose Assumptions 1 and 2 hold, based on Remark 4, if q(t) satisfies the following updating law:
and if the following conditions hold: (1) and (2) under controller (8) with adaptive updating law (9) can solve the multitracking problem. Proof.
e proof of the first part is similar to that of eorem 1, and to the leading system, we condiser the following Lyapunov functional candidate: where the constant q * is positive. Denote ϱ * (t) � (ϱ T 2 (t), . . . , ϱ T m (t)) T , Ξ � diag(q 2 , q 3 , . . . , q m ), and choose ρ ≥ a 2 , and then we have 6 Discrete Dynamics in Nature and Society Some sufficiently large q * can be chosen so that ( (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, c(t) ⟶ c * and q(t) ⟶ 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: 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.

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 Figure 1. e nodes are labelled with an ascending sequence from 1 to 9. e purple arrows describe the pinning control that the kth leader put on the nodes with specific topological properties (k � 1, 2, 3). e blue edges denote the interactions among clusters, and the black edges represent the communication links inside each cluster. rough (8) and Figure 1, we know that the red vertexes (labelled by 3,5,7,9) belong to V i 1 , the yellow vertexes (2,4,6,8) belong to V i 3 , and the green vertex (labelled by 1) belongs to V i 2 . It is assumed that the dimensional dynamical system of each agent is onedimensional.

Example 1.
e nonlinear function f is described by e elements l ij of graph Laplacian in (5)  e parameters in the adaptive controller (8) with updating law (9) are selected as follows: e state changes of leaders are shown in Figure 5. e initial value of c(t) is chosen as c(0) � 0.1, the orbit of coupling strength c(t) is shown in Figure 6, one can see that c(t) adaptively adjusts with c(t) ⟶ 2, the two error states of the two subnetworks, i.e., ξ i (t) and ξ w (t) both convergence to zero, and the delayed cluster consensus is indeed achieved.

Example 2.
is example shares the same nonlinear function f with Example 1. Choose the same Laplacian matrix L � (l ij ) 9×9 with the previous example, and set the initial values as x 5 (t) x 6 (t)     Discrete Dynamics in Nature and Society a 1 � 0.2, a 2 � 0.15, ρ � 0.2, and ε � 0.25, and thus, the conditions of eorem 2 can be satisfied. e description of the state changes of all nodes is shown in Figures 7-10. e change curves of the adaptive couplings c(t) and c k (t) are shown in Figure 11, and one can see that c(t), c k (t) 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.

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.
ere 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.

Data Availability
Some or all data, models, or code generated or used during the study are available from the corresponding author by request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.