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Observer-Based Coordinated Tracking Control for Nonlinear Multi-Agent Systems with Intermittent Communication under Heterogeneous Coupling Framework

In this article, the observer-based coordinated tracking control problem for a class of nonlinear multi-agent systems(MASs) with intermittent communication and information constraints is studied under dynamic switching topology. First, a state observer is designed to estimate the unmeasurable actual state information in the system. Second, adjustable heterogeneous coupling weighting parameters are introduced in the dynamic switching topology, and the distributed coordinated tracking control protocol under heterogeneous coupling framework is proposed. Then, a new Lemma is constructed to realize the cooperative design of observer gain, state feedback gain and heterogeneous coupling gain matrices. Furthermore, the stability of the system is further proved, and the range of communication rate is obtained. On this basis, the intermittent communication mode is extended to three time interval cases, namely normal communication, leader-follower communication interruption and all agents communication interruption, and then the distributed coordinated tracking control method is improved to solve this problem. Finally, simulation experiments are conducted with nonlinear MASs to verify the correctness of methods.


I. INTRODUCTION
T HE problem of cooperative control of multi-agent systems (MASs) has attracted increasing attention in various scientific communities recently including synchronization of complex networks, cooperative control of multiple vehicles, coordination of multiple robotic fish, teaming of robots, and so on [1][2][3][4][5][6]. According to the control objectives of distributed control of MASs, the existing coordination control problems are divided into regulation cooperative control [7] and coordinated tracking control [8]. However, it is important to design a suitable control strategy for whichever control goal you want to achieve. In the meantime, an appropriate framework should be chosen to address the phenomenon of information interaction within and outside the system. state information to achieve a specific number of interests into agreement, which can categorized into leaderless control protocol [9,10] and leader-follower control protocol [11,12] for MASs. In the latter case, which is the concern of this paper, all agents are expected to follow the trajectory of expectations generated by the leader.
In recent years, many meaningful results have been generated on studies related to behavioral consensus. State feedback based on linear combinations of relative state measurements between neighboring agents was used to implement the design of the proposed protocol in [13][14][15]. However, in practical or engineering applications, many state quantities in the system are not directly accessible because of economic costs, environmental constraints, etc., and thus the measurable output information is used to design a state observer, thus providing an elegant alternative that can be used for observation, diagnosis, and estimation purposes.
In order to estimate the leader's velocity, a new neighborbased protocol was proposed for first-order followers in [16], and it was improved to second-order followers [17,18] and high-order systems with input saturation [19], time delays [20], and external disturbance [21]. Li et al. proposed a distributed observer-based consensus protocol [22] for general linear MASs and further investigated corresponding reducedorder case [23]. A fixed-time distributed observer was developed in [24] with high-order integrator dynamics and matched external disturbances. In addition, the cooperative output regulation problem of linear discrete-time time-delay MASs was developed in [25] utilizing adaptive distributed observers. Nevertheless, most of the current research on observers was focused on low-order integral systems or general linear systems. The study of observer problems in nonlinear MASs is more complicated and requires dealing with nonlinear dynamics through Lipschitz conditions. It is noteworthy that the above distributed control problems are solved based on the fact that the communication between neighboring agents are continuous all the time. However, communication bandwidth, physical device failures and economic costs lead to communication constraints in real systems, which means that communication between neighboring agents may be intermittent or interrupted [26][27][28][29]. How to address observer-based coordinated tracking control for nonlinear MASs with intermittent communication and dynamic switching topology via heterogeneous coupling framework is worthwhile to research, and this further motivates our work.
This article mainly studies observer-based consensus control problems for nonlinear MASs with intermittent communications under heterogeneous coupling network. The main contributions are summarized as follows: (1) A class of nonlinear observers is investigated based on each agent only temporally sharing its relative output with neighbors to estimate the actual and unavailable states for nonlinear MASs. (2) The distributed tracking controllers based heterogeneous coupling framework with nonlinear observers are designed for considering intermittent constrained and intermittent coordinated constrained, in which some additional adjustable weighting parameters are constructed via dynamic interaction topology and heterogeneous coupling network. (3) Utilizing the switching system theory, Lyapunov function and LMI technology, the stability of the system is proved and communication conditions are given, meanwhile, the co-design of heterogeneous coupling weighting, feedback gain and observer gain matrices are realized. (4) The result presented in this work can resolve the coordinated tracking consensus for higher order nonlinear MASs with unavailable states under intermittent communication and dynamic switching topology.

II. GRAPH THEORY AND PROBLEM FORMULATION A. Graph Theory And Problem Formulation
The communication topology between agents is described by the directed graph G(ν, ζ,Ã), in whichν = {ν 1 ,ν 2 , ...,ν N } denotes the set of nodes, ζ ⊆ν ×ν denotes the set of edges, andÃ = [a ij ] N ×N is the weighted adjacency matrix with elements a ij . If (ν i ,ν j ) ∈ ζ, the information in the jth node can be passed to the ith one. then a ij = 1. If (ν i ,ν j ) / ∈ ζ, a ij = 0. The directed dynamic switching topologyG consists of N + 1 nodes, i.e., generated by the directed graph G of the follower and the leader. Let L ϑ(t) be the Laplace matrix of the directed graphG and D ϑ(t) = diag d 1 denote the diagonal matrix of the different communication paths between the followers and the leader, where ϑ (t) : [0, +∞) → {1, 2, . . . , p} denotes the switching signal. If the followers can obtain the leader information at time t ≥ 0, then d i ϑ(t) = 1, otherwise d i ϑ(t) = 0. In addition,G ϑ(t) ∈Ḡ, whereḠ = G 1 , . . . ,G p denotes the set of topologies composed of different interaction methods under the switching signal.

B. Problem Formulation
The nonlinear leader-following MASs considered in this research composes of N agents named followers and one more agent as a leader, the dynamic model iṡ x 0 (t) = Ax 0 (t) + f (x 0 (t)) where i = 1, 2, ..., N , x i (t) ∈ R n , y i (t) ∈ R z and u i (t) ∈ R m represent the ith follower's state, output and control input, respectively. In addition, x 0 (t) ∈ R n and y 0 (t) ∈ R z denote the leader's state and output. A, B and C are matrices of appropriate dimensions. f (x i (t)) and f (x 0 (t)) are continuously differentiable vector-valued functions, represent the intrinsic nonlinear dynamics of the ith agent and the leader.
Assumption 1 : The digraphG contains a directed spanning tree, and the leader as one agent is the root.
Assumption 2 : For a nonlinear function f (x i (t)), there exists a nonnegative constant such that: then it means that f (x i (t)) satisfies the Lipschitz condition. Some lemmas will be provided here to derive the proposed criterion.
Lemma 1 [30]: Define L ϑ(t) be a structure matrix of derivative directed graphG. Then,G has a spanning tree if and only if L ϑ(t) has a nonzero eigenvalues with multiplicity 1 and all the other nonzero eigenvalues have positive real parts.
Definition 1: Let Z n ∈ R n×n be the set of all square matrices of dimension n with nonpositive off-diagonal entries. A matrix A ∈ R n×n is said to be a nonsingular M-matrix if A ∈ R n×n and all eigenvalues of A have positive real parts.
Lemma 2 [31]: Suppose that a matrix A = [a ij (t)] ∈ R n×n satisfies a ij (t) < 0 for any i = j. Then, the following statements are equivalent: 1) A is a nonsingular M-matrix; 2) there exists a positive definite n × n diagonal matrix Θ such that A T Θ + ΘA > 0; and 3) all eigenvalues of A have positive real parts.
Lemma 3 [32]: Suppose that M ∈ R n×n is the symmetric positive-define matrix andR ∈ R n×n is the symmetric matrix, then, for any vector x ∈ R n , the following formula holds where ∀a, b ∈ R n and l > 0.
is satisfied for any initial states, the nonlinear MASs (1) achieve consensus tracking.

III. STABILITY ANALYSIS
This session introduces the problem about observer-based tracking consensus control with intermittent communication and dynamic switching topology for nonlinear MASs, which can ensure the system (1) stable and the heterogeneous coupling gain and feedback gain matrices derived.
In order to estimate the unmeasurable states of nonlinear MASs, by using the measurable output information, a type of nonlinear observers can be designed as follows: Here, the communication mode is considered as periodic intermittent communication. Suppose that t ∈ [kw, kw + δ) = T m represents normal communication time period, while t ∈ [kw + δ, (k + 1) w) = T n represents communication interruption time period, where w > δ > 0, k ∈ Z. t ∈ [kw, (k + 1) w) indicates that within each periodic intermittent communication time period w, the normal communication time period δ and the communication interruption time period w − δ are included, where k refers to the first few periodic intermittent communication periods. In order to achieve consensus tracking for nonlinear MASs under the dynamic switching topologies with intermittent communications, the following protocol under heterogeneous coupling framework is presented: the feedback gain matrix K ∈ R m×n is designed later. Before proposing the stable criterion, the following errors are defined: . . .
In order to make the system achieve the tracking target (2), we give the assumption for constructing dynamic interactive network topology.
Assumption 3 : Assume that there exists an infinite sequence of consistent bounded non-overlapping time intervals t ∈ [kw, (k + 1)w) , k ∈ N , where (k − 1) w = t k , kw = t k+1 , t 1 , t 2 , . . . , t N denotes the switching sequence of communication network topology in MASs. When Suppose that the switching signal is ϑ (t) : [0, +∞) → {1, . . . , p}. LetG ϑ(t) ∈Ḡ be the digraph for nonlinear MASs (1), and the set of all directed graphs can be represented as G = G 1 , . . . ,G p , where p ≥ 1. After the above analysis, Further, the following Lemma is proposed to realize the design of heterogeneous coupling gain matrix, observer gain matrix and state feedback gain matrix.
Lemma 5 : DefineG ϑ(t) ,L ϑ(t) and Γ ϑ(t) as the switched directed graph, its corresponding switching structure matrix and heterogeneous coupling gain matrix, respectively. Since the leader trajectory is not affected by the followers, we can obtain:L and there exists a bounded scalar 0 < β ≤ λ min Λ ϑ(t) can make the following LMIs holds is the appropriate positive number to be selected. Matrices M =ḠCP 2 and M = KP 2 , and feedback gain matrix K = −B T P −1 1 . Proof: It follows from Lemmas 1 and 2 that there exists a matrix Ξ i ϑ(t) > 0 such that: and It thus follows (16) that Φ 1 ϑ(t) > 0 for any positive number π 1 ϑ(t) . Choosing a suitable π i+1 ϑ(t) guarantees Φ i+1 ϑ(t) > 0, in addition by Schur complement lemma and (16), we can get , the following equation holds: where Λ ϑ(t) > 0 andΞ ϑ(t) = √ Π ϑ(t) Ξ ϑ(t) . Assuming that (A, B) is stable, there exists 0 < β ≤ λ min Λ ϑ(t) , matrix P 1 > 0, P 2 > 0, P 3 > 0 and P 4 > 0 such that the following inequality is constructed: This completes the proof. Stability analysis and proofs are performed for the designed distributed coordinated tracking controllers (4) based state observers (3), and the main results are given by the following theorem.
Theorem 1: Suppose Assumptions 1-3 hold, the nonlinear MASs (1) under switching topology with intermittent communication can achieve cooperative consensus tracking by the proposed Lemma 5, if the communication condition (20) holds: whereγ min = βηminλmin(P −1 , λ min(max) (·) indicates the minimum (maximum) value among eigenvalues of the matrix.
Proof: Set Lyapunov function candidate as: where Π ϑ(t) and Ξ ϑ(t) are the adjustable parameters which can be defined by Lemma 5, P 1 and P 2 are the solutions of (12) and (13). Next, the stability is analyzed in two time intervals: normal communication t ∈ [kw, kw + δ) and communication interruption t ∈ [kw + δ, (k + 1)w).
For t ∈ [kw, kw + δ) Based on Lemma 4, we can obtain Substituting (23) and (24) into (22) proceedṡ According to Lemma 3, we have (27) can be written aṡ According to Lemmas 2, 3 and 5, we can obtain 0 < β ≤ λ min Λ ϑ(t) , and the time derivative of V (t) for the communication network topologyG ϑ(t) under any switching signal ϑ (t) is taken as: Combining (12) with (13) in Lemma 5,(29) can be written asV To obtain the maximum value of δ max , define According to (19) and Lemma 3, substituting , P 1 and P 2 are the positive-definition matrix, so we can getγ > 0. Similar to , and according For t ∈ [kw + δ, (k + 1)w), Similarly, based on Lemmas 3 and 4, and some transformations of the necessary form, equation (32) is rewritten aṡ can be written aṡ wherel = λ max Q 3 , Q 4 λ max P −1 1 , P −1 2 , P 1 and P 2 are the positive-definition matrix, so we can getl > 0. According to the minimum valueγ min and the maximum valueγ max , we can obtain whereΥ 1 = −γδ0+l(w−δ0) . The range ofΥ 1 can be obtained by (20), that isΥ 1 > 0. For any positive integer k, Set sw ≤ t ≤ (s + 1) w for any s ∈ R. Define [kw, (k + 1)w), k ∈ N + is a uniformly bounded time sequence, ω max = max k∈+ {(k + 1) w, kw} andΥ = min j∈N+ Υ j . Finally, according to the recursion principle, we can obtain where Ω 0 = e ω maxl V (0) and Ω 1 =Υ ω max . Through the above process, it is demonstrated that the nonlinear MASa (1) under a dynamically switched communication network topology subject to the combined constraints of intermittent communication and information unpredictability is able to achieve the consensus tracking goal (2) under the combined action of a state observer (3) and a distributed coordinated tracking controller (4). This completes the proof. Theorem 1 and Lemma 5 mainly address the distributed coordinated tracking control problem for nonlinear MASs when intermittent communication is divided into normal communication t ∈ [kw, kw + δ) and communication interruption t ∈ [kw + δ, (k + 1) w). However, in some specific leader-follower MASs, communication interruption is extended to leader-follower communication interruption t ∈ [kw + δ, kw + h) and all agents communication interruption t ∈ [kw + h, (k + 1) w). When the intermittent communication method is extended as described above, the distributed coordinated tracking control protocol can be further improved and extended: In order to design the heterogeneous coupling gain matrix Γ i ϑ(t) and state feedback gain matrix K ∈ R m×n in the distributed coordinated tracking control protocol (38) and achieve the tracking target (2), we construct the following Lemma and Corollary.
Lemma 6 : Similar to Lemma 5, there exists a bounded positive scalar β, and the switching consensus protocol (38) under heterogeneous coupling framework with intermittent communication and dynamic switching topology for nonlinear MASs (1) can be constructed by following matrices: Corollary 1: Suppose Assumptions 1-3 hold, the nonlinear HMASs (1) under switching topology with intermittent communications can achieve cooperative consensus tracking by the proposed Lemma 6, if the following condition (45) holds: , λ min(max) (·) indicates the minimum (maximum) value among eigenvalues of the matrix.
Proof: Set Lyapunov function candidate as: where Π ϑ(t) and Ξ ϑ(t) are the adjustable parameters which can be defined by Lemma 6, P 1 and P 2 are the solutions of (39) and (40).
IV. NUMERICAL EXAMPLES In this section, two simulation examples are provided to verify the effectiveness of the obtained results. Both nonlinear MASs consist of four followers and a leader, with the directed dynamic interaction topology as shown in Fig. 1, where 0 is the leader and 1-4 are the followers. The dynamic of the ith agent for nonlinear MASs (1) can be written with  (1) compose of four followers and a leader, whose dynamic switching communication network is shown in Fig. 1, where node 0 represents the leader and the remaining nodes 1-4 are the followers. For the convenience of analysis, suppose that the underlying derivative directed network topologies can be constructed as shown in Fig. 1(a), (b) and (c).
Example 1: First, we consider the intermittent constrained consensus for cooperative tracking control. The intermittent communication mode is shown in Fig.2, in which, if y=1, it indicates that the communication between adjacent agents in the system is working, and y=0 shows the neighboring agents can not communicate with each other.
According to Theorem 1, there exists an infinite time sequence of uniformly bounded and nonoverlapping time intervals t ∈ [kw, (k + 1)w), where w = 5. To further analyze the stability, the switching sequence within the time interval t ∈ [kw, kw + δ) is defined asG 1 →G 2 →G 3 →    (3), the positive definite matrices P 1 , P 2 , the controller feedback gain matrix and observer gain matrix can be obtained as: Example 2: Second, we consider the observer-based consensus control under second intermittent coordinated constrained with three types of intervals. The intermittent communication mode is shown in Fig. 11, in which, if y=2, it indicates that the communication between adjacent agents and the leader in the system is working, if y=1, it indicates that each follower evolves only based on the information of itself and   its neighbors, and y=0 shows the neighboring agents can not communicate with each other. Three types of time intervals t = [5k, 5k + 4) = T m , t = [5k, 5k + 4.5) = T q and t = [5k + 4.5, 5k + 5) = T n respectively are considered for simulation. of consensus control based on improved distributed coordinated tracking control protocol (38) with second intermittent coordinated constrained. We can summarize that the proposed distributed control protocol based on heterogeneous coupling framework and observers are effective and asymptotically stable for nonlinear MASs with intermittent communication and dynamic switching topology.

V. CONCLUSIONS
In our work, the coordinated tracking control problem for nonlinear MASs under heterogeneous coupling network with intermittent communication has been studied. Especially, considering the unavailable states, nonlinear observers have been constructed for each agents. Then, a lemma has been correspondingly constructed to calculate the heterogeneous coupling gain, feedback gain and observer gain matrices. The system stability has been analyzed utilizing Lyapunov stability theory, switching system theory and LMI technology, and the allowed maximum communication threshold have been obtained. The results have been extended to the case of the intermittent communication with three types of intervals. Finally, two simulation examples have been provided to prove the effectiveness and correctness of the proposed methods.