Bipartite-tracking quasi-consensus of nonlinear uncertain multi-agent systems: neural network-based adaptive state-constraint impulsive control approach

For the application of multi-agent systems (MASs), the uncertain dynamics and the actuator saturation are inevitable. Considering these factors, the bipartite-tracking quasi-consensus of nonlinear uncertain MASs over signed graph is discussed in this paper. Two kinds of neural network-based adaptive state-constraint impulsive control protocols are proposed, where the communication among the leader and followers only takes place at some discrete instants. Based on the superior approximation ability of radial basis function neural networks (RBFNNs), we design an adaptive weight updating strategy to compensate the uncertain nonlinear dynamics of the system. By means of linear matrix inequality (LMI), convex analysis, impulsive system theory and Lyapunov stability theory, we derive some simple sufficient conditions for the bipartite-tracking quasi-consensus under the assumption that the communication topology is structurally balanced. Furthermore, we estimate the region of attraction of the error system. Three simulation examples are also provided to show the effectiveness of the proposed protocols.


Introduction
With the rapid development of computer science and communication engineering in the past 20 years, multiagent systems (MASs) as a kind of distributed system have attracted wide attention and been deeply studied because of their important applications in network control system [1,2] and other engineering fields. As the foundation of the distributed cooperative control technology, the consensus problem has been deeply and widely investigated on the coordinate tracking [3,4], the formation control [5,6] and etc. It aims at designing an appropriate control protocol by the local and relative state information to make all agents achieve an agreement [7][8][9][10]. Among the tremendous researches on the consensus, the issues can be generally divided into three categories according to the number of the leader, which are leaderless consensus (without leader) [11][12][13], tracking consensus (only one leader) [14,15], and containment control (two and more leaders) [16,17]. In [12], Wen designed a kind of control protocol via data-sampling and zero-order hold technology to deal with the average consensus of nonlinear MASs under different communication topologies. In [15], Xu et al. utilized the discontinuous communication control to achieve the quantized tracking-consensus with a leader at a fixed time. In [17], Li et al. established three kinds of fractional-order models for MASs and obtained equivalent conditions to realize containment control.
It is noteworthy that most researches above are focused on the cooperation of MASs, which indicates the communication topology between agents is described by unsigned graph. However, in some practical circumstances, there is not only cooperation but also competition within the system, such as multi-robot systems [18]. Based on that, Altafini developed the concept of the signed graph in [19] where the positive and negative weights on network topology are used to represent the cooperation and the competition among agents, respectively. In recent years, more and more researchers have begun to pay attention to the collective behavior among agents on cooperative and competitive network [20][21][22][23][24][25][26]. The bipartite average tracking of linear MASs with bounded disturbances was addressed in [23], in which the conditions for the finite-time and fixed-time average tracking were obtained. In [24], the bipartite containment problem was investigated for linear singular MASs with multiple leaders. As far as we know, there are few studies about the bipartite consensus for nonlinear MASs on the signed network. Besides, conservative assumption on the nonlinear function is proposed in the previous results. So these two factors greatly stimulate us to efficiently address the bipartite consensus problem with mild conditions.
In the literature mentioned above, the continuous control method is used mostly. However, considering the energy problem for most cyber-physical systems in real circumstances, how to reduce the communication cost is an important and meaningful research topic. At present, the discontinuous control is one way to achieve the goal. As a kind of discontinuous control, impulsive control plays an important role in complex systems because of its low cost and high robustness [27][28][29][30][31][32]. In [27], Han et al. designed two kinds of impulsive consensus protocol with encoding-decoding in order to save communication energy. By combining impulsive control and event-triggered control technology, a class of impulsive event-triggering controller was designed to deal with consensus problem of nonlinear MASs in [28]. In [29], the authors designed a discontinuous controller with time-varying impulsive gain to realize impulsive synchronization of reactiondiffusion system. Until now, there are few important researches on the realization of the bipartite-tracking consensus of nonlinear MASs by impulsive control, which also makes this study meaningful.
Furthermore, many practical systems have limited information processing capability due to the limited input of physical devices. If the control signals are out of the capability of actuators, unexpected feedback will be taken by the system. This could lower down the performance directly, or even make the system crush. In recent years, this problem has also attracted the attention of researchers [33][34][35]. In [33], the authors combined sampled data control and impulsive control to obtain synchronization conditions for coupled systems with input saturation. The leader-based formation for linear MASs with or without time delays was addressed in [34] with asymmetric saturated feedback. In fact, considering the bipartite consensus of MASs with actuator saturation under signed topology also has strong significance.
Because neural networks have excellent approximation and learning ability, they have become very effective tool for nonlinear system control. There are also a lot of literature about neural network-based uncertain nonlinear control strategies [36][37][38][39]. In [38], neural networks are used to approximate the unknown nonlinear functions for second-order nonlinear systems. In [39], the authors studied robust quasi-containment and asymptotic containment control issues for multi-leader MASs with unknown dynamics. However, there is little research on the bipartite consensus of MASs with unknown model and disturbance, especially in the case that information transmission is discontinuous, which spurs us to explore it.
Based on the above discussion, it is not difficult to find that there are few researches on the bipartite consensus of nonlinear MASs, especially those with uncertain dynamics. Furthermore, how to reduce the communication energy cost in the process of achieving bipartite consensus is also a important problem to be considered in the design of control protocol. Therefore, this paper pays attention to the bipartitetracking quasi-consensus of nonlinear MASs via neural network-based adaptive state-constraint impulsive control protocols. Compared with the existing works, this paper has the following contributions.
(1) This paper considers the bipartite-tracking quasiconsensus of MASs with uncertain nonlinear dynamics, which is more complicated than the related paper [22,23]. More importantly, the constraints on nonlinear MASs are less conservative and more general than the existing results [40]. Furthermore, the bipartite-tracking quasi-consensus for both homogeneous and heterogeneous systems can be achieved. (2) Compared with the continuous control method used in [20,21], we propose two kinds of stateconstrained impulsive control protocols for actuator saturation. The communication among the leader and followers only takes place at some discrete time instants in these methods, which improves the system robustness and reduces the communication energy consumption. (3) It is more convenient to analyze error system with saturation constraint by auxiliary matrix. Furthermore, the difference from the analysis method used in [5,6,13] is that we further consider the existence and estimation of attractive region of the error system.
The outline of this paper is as follows: The knowledge related to graph theory, saturation and RBFNNs approximation is given in Sect. 2. The problems to be addressed in this paper are also introduced in Sect. 2. In Sect. 3, two kinds of neural network-based adaptive state-constraint impulsive control strategies are designed, and the region of attraction of error systems is also estimated. The theoretical results will be verified by two simulation examples in Sect. 4. Finally, Sect. 5 will present the conclusion. Notations: throughout this paper, denote ndimensional Euclidean space and a set of n × m matrix by R n and R n×m , respectively. R and N + represent the set of real numbers and the set of positive integers, respectively. For A ∈ R n×n , A i is the ith row of it and tr(A) stands for the trace of A. A −1 and A T represent inverse and transpose of A, respectively. Meanwhile, A > 0 (A ≥ 0) means the A is a positive-definite (positive semi-definite) matrix. λ i (A) are the ith eigenvalue of A and satisfies λ 1 (A) ≤ λ 2 (A) ≤ · · · ≤ λ n (A).
· is the Euclidean norm of a vector. I n is the ndimensional identity matrix. Define a diagonal matrix by diag(h 1 , h 2 , . . . , h n ). The notation co{·} and ⊗ stands for the convex hull of a group of vectors and Kronecker product, respectively. The sgn(·) represents the symbolic function.

Preliminaries and problem statement
In this section, we will introduce fundamental knowledge including graph theory, saturation nonlinearity theory and function approximation by RBFNNs.

Graph theory
In this paper, the signed undirected graph defined by G = {V , E , A } represents communication topology among agents. V = {ν 1 , ν 2 , . . . , ν N } is a set of nodes, E ⊆ V × V represents the set of edges and A = [a i j ] N ×N is defined as the weighted adjacency matrix. The notation N i represents all neighbors of ith agent, i.e., N i = j|(ν i , ν j ) ∈ E . The adjacency matrix A has the property that a i j = 0 if there exists an edge (ν i , ν j ) between node i and j, otherwise, a i j = 0. Specially, let a ii = 0, which means the graph G does not exist the self-loop. Further, a i j > 0 and a i j < 0 represent the cooperation and competition between node i and j, respectively. If there is a sequence of adjacent edge (ν i , ν m ), (ν m , ν p ), . . . , (ν q , ν j ) between nodes i and j, then we say a path exists between these two nodes. Besides, if there exists a path for any two nodes, then we say the undirected graph is connected. The Laplacian matrix L = [l i j ] N ×N of the graph G is defined as Divide the set V into two subsets V 1 and V 2 satisfying V 1 ∩ V 2 = 0 and V 1 ∪ V 2 = V . If the elements of the adjacency matrix satisfy a i j > 0 for ν i , ν j ∈ V p , p ∈ {1, 2}, and a i j < 0 for ν i ∈ V p and ν j ∈ V 3− p , we can say the signed undirected graph is structurally balanced, otherwise it is structurally unbalanced. D = {diag{d 1 , d 2 , . . . , d N }} with d i = ±1 represents the set consisting of diagonal matrices. Generally speaking, for ith nodes, For structurally balanced graph G , there exists a matrix D ∈ D such that L D = DL D We can add a new node to the graph G to represent the leader. This added node indexed by 0 only sent information to followers and cannot accept the information from followers, which is called root node. The topology after adding the leader can be denoted by graphḠ . Define the Laplacian matrix ofḠ bȳ L = L + A 0 , in which A 0 = diag{a 10 , a 20 , . . . , a N 0 } describes the communication between the leader and followers. a i0 = 0 indicates there is a communication link between the leader and the ith follower, otherwise a i0 = 0.

Saturation
The saturation commonly exists in practical system, and its mechanism diagram can be presented in Fig. 1. We can describe this by the following mapping: where r is the input of the system and r max (−r max ) represents the maximum (minimum) saturation output. For r = [r 1 , r 2 , . . . , r n ] T , define sat r = [sat (r 1 ),sat (r 2 ), . . . , sat (r n )] T . Ψ = {Ψ i : i ∈ ℵ} represents the set composed of n ×n-dimensional diagonal matrices, where ℵ is the number of elements in this set. For each matrix Ψ i , its diagonal elements can only be 1 or 0. Thus, the number of elements of the set Ψ is 2 n . Besides, for any Ψ i ∈ Ψ , define the complemen- There are the following two lemmas related to saturation based on the above definition.

Function approximation based on RBFNNs
As is well known, RBFNNs are widely used in the field of control science and engineering because they have strong learning ability and can approximate any nonlinear smooth function. Any continuous smooth function χ(z 1 , z 2 , . . . , z n ) : R n → R m can be approximated by RBFNNs on a compact set Ω z by the following form: where [z 1 , z 2 , . . . , z n ] T ∈ Ω z ⊆ R n is the input vector, W * ∈ R q×m represents the ideal weight matrix with p being the number of neurons of RBFNNs. ε err is the approximation error satisfying ε err ≤ max and max > 0, which can be considered as the minimum possible deviation between the ideal approximation (W * ) T φ(z 1 , z 2 , . . . , z n ) and continuous smooth function The ideal weight matrix W * cannot be known accurately, and it is just an artificial quantity for analytical purposes, which often needs to be estimated in practical applications. Actually, W * can be defined as: . . , q is the basis function of ith neuron with the following form: where represents the center of the receptive field; i > 0 is defined the width of the basis function.

Problem statement
The MASs consist of N followers and one leader in this paper, whose interaction topology is represented by the graphḠ . The dynamics of the ith follower and the leader x 0 are described, respectively, as: where x i (t) ∈ R n and u i (t) ∈ R n represent the state vector and control input of ith agent, respectively. Similarly, x 0 ∈ R n is the state vector of the leader. The initial value of the leader and the ith follower is represented by x 0 (0) and x i (0), respectively. f i (·), f l (·) : R n → R n are the uncertain nonlinear function. The research goal of this paper is designing a reasonable control protocol u i (t), i = 1, 2, . . . , N so that followers can realize the bipartite-tracking quasiconsensus to track the leader, whose exact definition is following Definition 1 For an arbitrary given small positive constant ω, if the following formulate is satisfied, then it is called that the MASs (4) achieve bipartite-tracking quasi-consensus where the definition of d i has been shown in 2.1. Further, if ω = 0, the bipartite-tracking quasi consensus will reduce to bipartite-tracking consensus; in other words, lim Before proceeding, important lemmas and assumptions are provided, which will be used in subsequent proof.
Lemma 3 [42] For any two vectors α ∈ R n , β ∈ R n , a constant k > 0 and matrix where 1 and 2 are two positive constants.

Assumption 1
The subgraph G among followers is connected and structurally balanced. Meanwhile, there exists at least one follower who can accept information from the leader in graphḠ .

Main results
According to the property of matrix D, define p i (t) = d i x i (t) as a new variable. Then, the MASs (4) can be converted to other new system as shown below: In this way, the bipartite-tracking quasi-consensus of the MASs (4) can be transformed into the tracking quasi-consensus of the MASs (7). Denote the disagreement between the ith follower and the leader in terms of e i (t) = p i (t)−x 0 (t), which is called the tracking error. Further, denote the relative measurement state between the ith agent and its neighbor by the following variable.
Based on e i (t) = p i (t) − x 0 (t), formula (8) can be rewritten as: According to algebraic graph theory, it is easy to know L D is a semi-positive-definite matrix, whose N eigenvalues are denoted by 0, λ 2 , λ 3 , . . . , λ N . Denote the eigenvectors of the matrix L D ⊗ I n by m 11 , . . . , m 1n , m 21 , . . . , m 2n , · · · , m N 1 , . . . , m N n , which correspond to the eigenvalues 0, λ 2 , λ 3 , · · · , λ N , respectively. Actually, these eigenvectors of the matrix L D ⊗ I n can be chosen in orthogonal based on R nN . Further, we can get Select a class of smooth scalar function as follows: According to the previous analysis, we can get the following results . . , λ N I n },Φ = diag{λ 2 I n , λ 2 I n , . . . , λ N I n }, and Λ = M TΦ−1 M. Therefore, the following results can be obtained: In MASs (7), f i (x) ∈ R n is difficult to be known by other agents. According to RBFNNs approximation, it can be approximated to arbitrary accuracy in the below form: where W * i ∈ R m×n is the ideal RBFNNs weight matrix, m represents the neurons number, φ(x i ) ∈ R m is the basis function vector and In order to achieve the goal of this paper, we will propose two kinds of state-constraint impulsive protocols and obtain sufficient conditions achieving tracking quasi-consensus for the MASs (7).

Full state-constraint impulsive control protocol
The following full state-constraint impulsive control protocol is designed: where Δ(t) is the Dirac function and impulsive time t k is given by a time sequence where τ a > τ b > 0. b i is impulsive coupling gain of ith agent, which are designed later. The control parameter ρ i is designed as where γ i is the convergence gain to be designed.Ŵ i (t) is the estimation matrix variables of the ideal weight W * i of RBFNNs, which obeys the updating laws as follows.
where P i = P T i ∈ R n×n is a positive-definite matrix; σ i is the positive constant to be designed.

Remark 1
The control protocol (14) obtains three parts. The term ρ i j∈N i |a i j |(x i (t) − sgn(a i j )x j (t)) plays a role in facilitating the communication between followers, while the termŴ T i (t)φ(x i ) is used to eliminate uncertain nonlinearity by RBFNN. Only the third term sat (b i a i0 (d i x i (t) − x 0 (t))) has contribution to communicate between the leader and the follower.
Using the protocol u i (t), we can obtain the error system of the ith agent as follows: Where Based on Lemma 2, there exists a auxiliary matrix H i ∈ R n×n , i = 1, 2, . . . , N such that −r max ≤ H i e i (t) ≤ r max . Then, the whole error system is as follows: where

Remark 2
The asymptotic stability of the error system (17) at the origin indicates the bipartite-tracking quasi-consensus of system (7) and then the designed controller can achieve our goal. In addition, according to Lemma 2, we introduce an auxiliary matrix H to analyze the state constraints. In other words, at time t = t k , this requires that the states of the error system (18) should be located in a special area called the error attraction region. We will discuss the problem of the error attraction region in Sect. 3.3. (4) satisfy Assumptions 1 and 2. The full state-constraint impulsive control protocol and the updating law ofŴ i (t) are presented by (14) and (16), respectively. For given constants b i , γ i > 0, σ i > 0, η > 0, τ a > τ b > 0, a constant matrix H and positive-definite matrix P i such that the following conditions hold:

Theorem 1 Suppose the MASs
(1) The initial error conditions of the MASs (4) are in the error attraction region Ω; Then, the bipartite-tracking quasiconsensus of the MASs (4) can be realized in exponential form with rate K − ln η τ a .
Proof The following functions are selected as Lyapunov candidate functions where V 1 (t) is shown in Eq. (10),W i (t) =Ŵ i (t) − W * i is the estimated error betweenŴ i (t) and W * i , and When t ∈ [t k−1 , t k ) for given k ∈ N, the total derivative of V (t) along the trajectories of (17) is follows: According to Assumption 2 and Cauchy inequality, one can obtain that Substituting (13) and (21) into (20), we can obtain the following resulṫ Further, according to Lemma 3 and Young's inequality, the following inequalities can be derived According to the properties of trace, substituting (14), (16), (23) and (24) into (22) yieldṡ Further, according to the facts thatW T Substituting (26) into (25), we can obtaiṅ When t = t k , we can obtain according to LMI (3) in Theorem 1 Combining (27) and (28), it leads to Then, according to Lemma 4, it is not hard to obtain the following result.
Based on (30), we can obtain the following results for t ∈ [t k , t k+1 ).
Further, for t ∈ [t k , t k+1 ), we have and Then, according to (31), (32) and (33), when 0 < η < 1, we can further get when η = 1, we can derive when η > 1, we can have Based on inequality (12), we can get that . (38) Further, according to the definition of ξ(t), we can obtain 2 . Then, there exists T > 0 such that where ω ≥ 2θ K λ min (Λ)λ min (L 2 D ) . According to the above analysis, the constant ω finally depends on the controller parameters γ i , P i and σ i and approximation error ε i . Increasing the gain γ i , gain σ i and the number of nodes of RBFNNs can make ω be a very small constant. Then, (39) indicates that d i x i (t)−x 0 (t) ≤ ω as t → +∞. According to definition 1, the MASs (4) can achieve bipartite-tracking quasi-consensus. The proof is completed.

Remark 3
Although the bipartite consensus of MASs has also taken into considered in [22,23], there are too many constraints on the MASs; for example, the system is required to be linear, or the nonlinear function in the system needs to meet the like-Lipschitz condition. However, the MASs in this paper can be the homogeneous or heterogeneous nonlinear system, and there exist less constraints for nonlinear function. That is to say that the research object of this paper is more general.

Remark 4
The consensus of MASs can be realized by state-constraint impulsive protocol in [6,13], but the dynamics of system is certain. Further, bipartite consensus of nonlinear MASs is involved in [25,26], but the designed protocol adopts continuous control. Compared with the above works, the nonlinearity and uncertainty of the dynamics and the actuator constraint have brought difficulties to bipartite consensus for MASs. However, this paper provides a kind of effective approach and further discusses error attraction region.

Partial state-constraint impulsive control protocol
In actual system, not all agents are required to be limited to a specific area. It seems more reasonable to consider partial state saturation. Therefore, considering the MASs (4), we design the following partial stateconstraint impulsive control protocol for the ith agent.
The control parameter ρ i and estimation matrix vari-ableŴ T i (t) are the same as that in controller (16). b i and b i are the other two control parameters to be designed.
We can get the following error system under the controller (40).
Based on Lemma 2, choosing a auxiliary matrixH ∈ R n×n with appropriate dimensions such that −r max ≤ He i (t) ≤ r max , we can obtain the error system of the ith agent as follows: Theorem 2 Suppose the MASs (4) satisfy Assumption 1 and 2. The partial state-constraint impulsive control protocol and the updating law ofŴ i (t) are presented by (40) and (16), respectively. For the given constants a constant matrixH ∈ R n×n and positive-definite matrix P i such that the following conditions hold: (1) The initial error conditions of the MASs (4) are in the error attraction region Ω; For i = 1, 2, . . . , m, the inequality (I n + Ψ i d i b i a i0 I n +Ψ − iH ) T (I n +Ψ i d i b i a i0 I n +Ψ − iH ) ≤ η 1 I n holds; (4) For i = m + 1, m + 2, . . . , N , the inequality (I n + d ibi a i0 I n ) T (I n + d ibi a i0 I n ) ≤ η 2 I n holds; } andη = min{η 1 , η 2 }. Then, the bipartite-tracking quasi-consensus for MASs (4) can be realized in exponential form with consensus rate K − lnη τ a .
Proof We still choose (19) as the candidate function. When t ∈ [t k , t k+1 ), V (t) still satisfies relation (27) and the proof is omitted. When t = t k , according to the error system (42), we can obtain Combining conditions (3) and (4) in Theorem 2, the equality (43) can be converted into Then, combining (27) and (44), it is easy to get the following result: The remaining proof is similar to Theorem 1. When t ∈ [t k , t k+1 ), one has Similar to the analysis in Theorem 1, based on the inequality (12), we can obtain 2 , then, there also existsT > 0 such that where According to definition 1, it indicates that e i (t) ≤ ω as t → +∞. Then, the bipartite-tracking quasiconsensus of MASs (4) can be realized via the partial state-constraint impulsive control protocol (40). The proof is also completed.

Estimation of error attraction region
In this subsection, the error attraction region mentioned in the previous section will be further discussed. First, taking the error system (18) as an example, we give the definition of error attraction region, which can be expressed in mathematical teRema2rms as follows.

Definition 2
The attraction region of the error system (18) with saturation feedback is given by: where e 0 = [e T 1 (t 0 ), . . . , e T N (t 0 )] T is the compact vector of the initial states of the error system (18).
From the mathematical point, the error attraction regions are a subset of a compressed invariant set. Usually, the shape of the attraction region is irregular and difficult to express accurately, so it needs to be analyzed and estimated. For simplicity, we take the theorem 1 as an example to illustrate the estimation of error attraction regions.
Define a set of ellipsoids by E(G, ) = {e(t) ∈ R n : e T (t)Ge(t) ≤ }, in which matrix G = G T and is the positive-definite matrices and positive constant, respectively. The ellipsoid E(G, ) can be called a contractive invariant set for the error system (18), if the following constraints are satisfied: (i) conditions (2) and (3) in Theorem 1, Further, in order to estimate the error attraction region in the invariant set E(G, ), we define an ellipsoid H 0 by the following definition where R ∈ R N n×N n is the positive-definite matrices. Therefore, the problem of estimation of the error attraction region can be converted the other problem that to find appropriate positive constant δ make the following constraints satisfied: (2) and (3) in Theorem 1, (iii) −r max ≤ He(t) i ≤ r max .
Then δH 0 is the estimated attraction region for the error system (18). Unfortunately, it is still difficult to estimate it only depending on the above three constraints. However, the above constraints can be simplified as an optimization problem. Firstly, constraint (i) δH 0 ⊂ E(G, ) for the fixed shaped ellipsoid H 0 is equivalent to By the Schur complements, it can be expressed by the following LMI Secondly, the constraint (iii) indicates that all hyperplanes Γ i e(t) = ±r max , i = 1, 2, . . . , N n lie completely outside of the ellipsoid E(G, ), where Γ i is the ith row of H . Obviously, all points on the hyperplane Γ i e = ±r max will satisfy e T (t)Ge(t) ≥ . Then, the constraint (iii) becomes an optimization problem, which is to separately seek out the minimum of e T (t)Ge(t) when Γ i e(t) = ±r max . Using the Lagrange multiplier method, we can get Consequently, constraint (iii) is equivalent to Similarly, we have the following LMI by the Schur complements.
To sum up, we can transform the problem of the error attraction regions into an optimization problem satisfying the following constraints.
Conditions (2) and (3) in Theorem (1), The above discussion shows the existence of the error attraction region Ω and the needed constraints to estimate it.

Remark 5
The conditions in Theorems 1 and 2 are independent on matrix G and constant . Formula (55) is added to ensure that the errors between the leader and follower are in the specific regions. Obviously, the initial state of each agent is not independently constrained. On the contrary, only the relative state is limited by admissible region. At the same time, once the initial condition is given, the condition for existence of regions only depends on the matrix G. Meanwhile, according to −r max ≤ He(t) i ≤ r max , if the matrix H is selected small enough, the error will have a very large variation range.

Simulation examples
In this section, three simulation examples are carried out to illustrate the effectiveness of the proposed control protocol and its application to complex network.
696. The adaptive weight matrixŴ i (t) is updated by (16), and the initial matrixesŴ i (0) are randomly distributed in the appropriate space.

Example 1
The bipartite-tracking quasi-consensus of Chua's chaotic circuit is verified in this example. We consider the MASs composed of 4 nodes and one leader, whose communication topology is shown in Fig. 2. Each agent represents a Chua's chaotic circuit, whose schematic is shown in Fig. 3. Based on it, the dynamics of the Chua's chaotic system can be described as: where x 1 , x 2 , x 3 are the voltage of capacitors C 1 , C 2 and the current of the inductor L c , respectively. The value of negative resistance G r depends on g(  Fig. 4, which indicates the whole circuit system is chaotic. In Fig. 2, the solid line and the dashed line represent the cooperative and competitive relation among the agents, respectively. It also shows that the nodes can be divided into two subsets V 1 = {0, 1, 3} and V 2 = {2, 4}, which indicates D = diag(1, −1, 1, −1). Obviously, Assumption 1 is satisfied, which indicates A 0 = diag(1, 0, 1, 0). The Laplacian matrix L is The maximum (minimum) saturation output is r max = 4 (−r max = −4). The controller gains ρ i and σ i are designed as ρ i = 6, i = 1, 2, 3, 4 and σ i = 6, i = 1, 2, 3, 4, respectively. The impulsive gain b i is chosen as b i = −0.8, and the impulsive interval is set as τ a = t k+1 − t k = 0.15s. The constant η is selected as η = 0.85. Select a smaller auxiliary matrix H to satisfy the condition 1 and 3 in Theorem 1. Further, the condition 2 in Theorem 1 is fulfilled after calculating.
The error trajectories of the system are presented in Fig. 5, which indicates the tracking errors converge to 0 with t → +∞. The state trajectories of the system are shown in Fig. 6, which indicates the MASs (56) can achieve the bipartite-tracking quasi-consensus by the control protocol (14). Example 2 In this example, we will verify the effectiveness of the control protocol (40). We consider the two-dimensional Hopfield-type MASs, whose dynamics are described as: (57) The initial values of the followers are randomly distributed in range [−3, 3] × [−3, 3]. The communication topology is shown in Fig. 7. The dynamics of the leader are described as: ẋ 01 (t) = 4cos(4t) + 2cos(6x 01 (t)), The initial value of the leader is x 0 = [1.5, 1.6] T . Obviously, the leader and the followers are heterogeneous system. According to Fig. 7, we know that the nodes can be divided into two subsets V 1 = {0, 1, 2, 3} and V 2 = {4, 5, 6}. Further, it is easy to know the gauge transfor- The maximum (minimum) saturation output is r max = 2 (−r max = −2). The controller gains ρ i and σ i are designed as ρ i = 8, σ i = 5, i = 1, 2, 3, 4, 5, 6. The impulsive gains b i andb i are chosen as b i = −1.2,b i = −1.6, and the impulsive interval is set as τ a = t k+1 − t k = 0.1s, The constants η 1 and η 2 are selected as η 1 = 0.92 and η 2 = 0.9. Similarly, select a smaller auxiliary matrixH to satisfy the condition 1 and 3 in Theorem 2. Further, the condition 2 in Theorem 2 is fulfilled after calculating.
The state trajectories of the system are presented in Figs. 8 and 9, respectively, and the error trajectories of the system are shown in Fig. 10. They can prove that the partial state-constraint impulsive control protocol (40) can achieve the bipartite-tracking quasi-consensus of MASs (57).  In this example, we will compare the communication cost under different communication methods. We choose the MASs with the following dynamics, whose communication topology is presented in Fig. 2, x i2 (t) = 0.56x i1 (t)sin(x i2 (t)), i = 0, 1, 2, 3, 4. (59) The initial value of MASs (59) is also randomly distributed in the range [−3, 3] × [ −3, 3]. Similar to the Han and Zeng [38], the calculation method of communication cost is N i=1 +∞ t 0 u 2 i (t)dt for the continuous communication and N i=1 +∞ k=1 u 2 i (t k ) for the discontinuous communication. We choose continuous com-munication in [40], intermittent communication in [43] and the control protocol (14) in this paper to compare the communication cost. The parameter settings of the three communication methods are as follows. The feedback control gain k i , i = 1, 2, 3, 4 is set as k i = 6 in continuous communication and intermittent communication. Further, the proportion of interaction time is set as 0.6 in intermittent communication. As for the control protocol (14), the maximum (minimum) saturation output is r max = 3 (−r max = −3) and the feedback control gain ρ i and σ i are designed as ρ i = 6, i = 1, 2, 3, 4 and σ i = 6, i = 1, 2, 3, 4, respectively. The impulsive gain b i is chosen as b i = −0.8, and the impulsive interval is set as τ a = t k+1 − t k = 0.25s. It can be verified that the above settings satisfy the conditions of Theorem 1. The error trajectory of the MASs (59) under these three communication modes is shown in Fig. 11, which indicates that the MASs (59) can achieve bipartite-tracking consensus under three communication methods. Further, the total communication cost under three communication modes is shown in Fig. 12. Through comparison, it can be found that the designed control strategy can reduce the communication cost of the system. Figures 11 and 12 also further show that different communication modes make the system have a trade-off problem between convergence speed and communication cost.

Conclusion
This paper has studied the bipartite-tracking quasiconsensus of nonlinear uncertain MASs on cooperativecompetitive network via neural network-based adaptive state-constraint impulsive control. The RBFNNs have been used to approximate the uncertain nonlinear dynamic with small allowable error. Two kinds of adaptive impulsive control protocols with full saturationconstraint or partial saturation-constraint have been designed, which is of certain significance to the real system. The less conservative estimation of the error attraction regions has been formulated by convex optimization method. Finally, three simulation examples have verified the effectiveness of the designed protocols. In the future work, we will consider the consensus problem and the containment control problems of the MASs with fuzzy model via the fully asynchronous impulsive control with saturation constraint.