Distributed Adaptive Bipartite Synchronization of Networks with Competitive Relationships and Disturbances

Abstract

This paper studies the problem of under what conditions the general nonlinear networks with directional competition and perturbations to achieve fully distributed BS (bipartite synchronization). For the norm bounded disturbances with unknown upper bound, under the assumption that the signed digraph is structurally balanced and contains a directed spanning tree, a fully distributed observer-based adaptive protocols is designed for BS problem of general nonlinear networks with no leader and neural networks approximation. For the disturbances generated by exosystems, based on a disturbance observer, a fully distributed disturbance observerbased adaptive protocol is proposed to make network achieve bounded BS. In order to reduce the chattering phenomenon, some adaptive gains and adaptive parameter vector of neural networks approximation are proposed. Finally, the theoretical results are verified by two numerical simulation examples.

Headnote

Abstract-This paper studies the problem of under what conditions the general nonlinear networks with directional competition and perturbations to achieve fully distributed BS (bipartite synchronization). For the norm bounded disturbances with unknown upper bound, under the assumption that the signed digraph is structurally balanced and contains a directed spanning tree, a fully distributed observer-based adaptive protocols is designed for BS problem of general nonlinear networks with no leader and neural networks approximation. For the disturbances generated by exosystems, based on a disturbance observer, a fully distributed disturbance observerbased adaptive protocol is proposed to make network achieve bounded BS. In order to reduce the chattering phenomenon, some adaptive gains and adaptive parameter vector of neural networks approximation are proposed. Finally, the theoretical results are verified by two numerical simulation examples.

Index Terms-Bipartite synchronization, competitive relationships, disturbances, nonlinear networks.

I. Introduction

SINCE competition and cooperation are ubiquitous in ocial, biological, physical and other fields, in the last decade, the research on the impact of competition and cooperation on network synchronization (consensus) behavior has attracted more and more attention from scholars in various related fields [1], [2], [3], [4], [5], [6], [7], [8]. A network with competition and cooperation can be represented by a signed digraph, where the positive edge represents cooperation and the negative edge represents competition. Thus, the consensus problem in a network with competitive and cooperative relationships becomes the problem of studying how to design consensus agreements based on signed graphs [9]. Since Altafinis pioneering work, there has been many literature studies on the problem of consensus on signed graphs, such as BS [1], [10], [11], [12], [13], [14], [15], [16], modulus synchronization [17], interval BS [3] and so on.

Depending on whether there is a leader, existing work on BS can be divided into two categories, that is, leaderless BS [18], [19], [20], [21], [22] and leader-following BS [23], [11], [24], [12], [25], [26]. In the absence of a leader, if the system has its own dynamics, it is generally necessary to obtain the coupling strength condition for all nodes to achieve BS. Since these conditions generally depend on the Laplacian matrix of the signed graph, the resulting conditions are not fully distributed. In order to overcome this problem, few fully distributed BS algorithms observerbased were proposed in [19], [21], [22]. Under the undirected signed graph, a adaptive BS strategy has been proposed to guarantee BS for leaderless networks [19]. The finitetime BS problem of first-order networks has been studied in [20]. Some adaptive BS strategies have been designed for general networks with input saturation [21], and general nonlinear networks[22]. Leader-follower BS of linear uncertain networks has been studied in [11]. The BS tracking for networks with leaders unknown input was addressed in [24], and for higher-order heterogeneous nonlinear uncertain networks, the same problem was studied in [12], where agent nonlinear dynamics (including leader dynamics) are general and unknown. When the topology is switching, [25] addressed the leader-following BS for the nonlinear networks subject to exogenous disturbances.

Due to the fact that actual systems can be affected by various disturbances, it is very necessary to study the impact of disturbances on the system. There are generally two types of disturbance, one is the bounded disturbance, and the other is the disturbance generated by external systems. For the BS problem of networks, [20], [23], [11] addressed the condition that the system is affected by bounded disturbance. Norm bounded disturbances may come from inputs [20], [23], [11] or from the system itself, such as in the case of nonlinear uncertainties [19], [24], [22]. Under the Lipschitz-like nonlinear condition, [25] addressed the BS problem of networks with exogenous disturbances, and a leader-following BS method based on disturbance observer is proposed. Based on the above literature review, in the absence of leadership, there is little literature on the impact of various disturbances on the BS, especially the fully distributed protocol design.

Motivated by the above literature review, we will research the leaderless BS problem of general nonlinear networks with directed competitive relationships and disturbances. In this paper, the situation that network nodes are subject to the norm bounded disturbance and the external system disturbance is considered respectively. The main contributions of this paper are the following three areas:

* First, unlike the works in [19], [24], [22], we consider the bounded disturbances and disturbances generated by exosystems, and obtain some fully distributed disturbance observer-based adaptive protocols for the leaderless BS problem.

* Second, compared with the Lipschitz-like nonlinear condition [25], the general nonlinear systems with neural networks approximation are considered in this paper and a modified adaptive gain vector is proposed to estimate the unknown constant vector of approximation.

* Third, some adaptive gains and adaptive parameter vectors of neural networks approximation are designed for reducing the chattering phenomenon. For the disturbances generated by exosystems, the bounded BS is achieved by a fully distributed disturbance observerbased adaptive protocol.

The paper is organized as follows. Section II presents notations, some properties for digraphs, and the problem formulation. Section III provides a fully distributed observerbased adaptive protocol for the BS of general nonlinear networks with bounded disturbances. Section IV gives a fully distributed disturbance observer-based adaptive protocol for the bounded BS of general nonlinear networks with disturbances produced from the external system. Section V gives two numerical examples to verify the theorems deduced in Sections III and IV. Section VI summarizes the conclusions of this paper and describes the direction of future work.

II. Preliminaries

A. Notations

The symbol ® represents the Kronecker product. diag(r1, * * * , rN) represents a diagonal matrix consisting of A, * * * , Гм. II * 111, II * II and HxIL represent the 1-norm, 2-norm and oo-norm respectively. Let In denote an ndimensional column vector with each element equal to 1. 0 represents a matrix with elements all 0. X > (<)o represents X is positive (negative) definite matrix, AminCX) and Лтах(Х) represent the minimum and maximum eigenvalues for the symmetric matrix X respectively. In represents the identity matrix of dimension n. The sgn(-) function is defined as: when X > 0, sgn(x) = 1; X < 0, sgn(x) = -1; X = 0, sgn(x) = 0.

B. Directed interaction graphs

In this subsection, we introduce some concepts of signed digraph and some lemmas that will be used in the sequel.

LetV = {Vi, * * * , i/n} be the node set. The interactions among nodes can be represented by the signed digraph G = (V, E , A), where E c V x V represents the edge set and A = (a,/) e RwxTv represents the weighted adjacency matrix. The edge Ey e E represents a node set (ł//, Vy) means V/ can receive information from v,. The coupling configuration information for each edge in G is recorded in A. 1/, and 1/, is cooperative if the weight a,y > o, Vi and Vy is competitive if the weight ay < o, and a¡y = о if (w Vy) Z E. For any pair of nodes Vi, Vj e V that are not identical, there exists a directed path from Vy to v¡ in V which consists of a set of directed edges, that is, (Vi, Vu), * * *, (y¡,k-i, v¡k), (Vik, Vy) e E. The digraph is strongly connected, if there always exists directed path from v, to 1/, for any two different agents in V. For a node Vy e V, there always exists directed path for any not identical node Vt e V,the topology is called containing a directed spanning tree. Let Lc = (Ay) e rwxw define the Laplacian matrix of the signed digraph G, where hi = -Эу, i/= j and In = L

Definition 1. [1] The signed digraph G is structurally balanced if the node set can be separated into Va and Vb, satisfying Va и Vb = V and Va ci Vb = 0 such that V Vi, v j E Vh(h E (a, b}), a,y > o and V Vi e V^, v j e V^^kt, = (a, b}), a.y < o.

The gauge transformation matrices set is expressed as

Lemma 1. [1] The signed digraph G is structurally balanced f and only if 3P e P such that PAP has all nonnegative elements.

Assumption 1. The signed digraph G is structurally balanced and contains a directed spanning tree.

Lemma 2. [27] With Assumption 1, the node set V can be separated into V Q = (1/ ъ ***,vand V b = {1/ к+ъ'" ,v n}, sudy that

1 ) The subdigraphV a is strongly connected.

2) The node in V a has no neighbors inV b.

Without losing the generality, this paper assumes that V a = (Vi, * * *, V/,} and V b= (Vk+i, * * * , i/n}. Based on Lemma 2, the Laplacian matrix Lc can be written as

where Lei e R^x· is the Laplacian matrix of V a, LC3 e k)x(N-k)

Lemma 3. ]21], [24] The signed digraph Gi with the Laplacian matrix Lei is strongly connected, there exists a matrix R = diag(A, * * *, a) > o, such that

where L cl = R Lei + Lļ/? with r = [a, * * * , Гк]т as the left zero eigenvector for P1Lc1P1, A2(LC1) z's the second minimum eigenvalue cf Lcl, PiZ represents vector with all positive elements.

Lemma 4. [28] For the Laplacian matrix LC3/ there exists a_diagonal matrix G = diag(g,, * * * , ди-k) > O such that GLc3 + L^G > o.

C. Problem formulation

Consider the following general nonlinear network with N agents defined on a signed digraph G

where i = 1, * * * , N, Xie Rn is the state, Ui e Rm is the control input, and di e Rm is the disturbance. ffXi, t) is the smooth function, A E RnXn an(J В e RnXrn are consţanţ matrices.

Assumption 2. [29], ]19] The function ffXi, ť) satiifies

where v\h e R? is an unknown constant parameter vector, and ф,(х,, t) e RmX4 z's a bounded continuous function matrices, e R- is a bounded approximation error satisfying IIçAOIL < и with и being a positive constant.

Assumption 3. The pair (A, B) is controllable.

Remark 1. Note that Assumption 2 holds, many well known practical physical systems satisfy this condition, such as robot manipulator dynamics [30], and yaw dynamics cf ship [31]. In fact, any smoothed nonlinear function can be expressed as an approximate form cf a neural network. With Assumption 3,for the following Riccati inequality equation, there exists a solution with Q > o.

Definition 2. НхД) - ppx.(t)\\ = o, k/ij e V, then the general nonlinear networks (1) can achieve BS.

Definition 3. The general nonlinear networks (1) can achieve bounded BS if there exists a control input such that llev(OII < 1ß(llev(o)ll, f)+E, V/J e V, where E is a positive constant, eij(t) = xftj-ppjXft), and ß(-, *) is a class KL function.

III. Bounded disturbances with unknown upper

BOUND

In this section, we will focus on designing a fully distributed observer-based adaptive protocol for the BS problem of a general nonlinear networks (1) with no leader and neural network approximation, while consider the bounded disturbances dft) with an unknown upper bound.

Assumption 4. Each agent is subject to a bounded disturbance, such that

where W > О is an unknown bounded constant.

To solve the BS problem, we first design the state observer as follows

the observer state, zft) denotes the adaptive gain with z,(o) > o, 6ft) = qj UfQqfť), К = -Вт Q with Q > о being the solution of (3) and Г = QBBTQ.

The following fully distributed observer-based adaptive protocol is proposed to solve the BS problem

where vG(t) is the estimation of W/, К = -BTQ with Q > 0 is the solution of (3), i#df) is the solution of following equation

where m, and Si are positive constants, y(f) is the adaptive gain defined by

where s / is a positive constant

Let x t = Xi - K t be the observer error, w i = Wi - и/ = Wi - Wi, the closed-loop dynamics can be expressed as

where d. = ę + d ¡ and Ild II < w + и.

Let = \Я т'***,£ т ŕ, £ = ]ЯТ .***,ЯТУ with 1 _ 1 _ k · J<+1 N K = , q^T, q", = [^,..., q? y with q = [f] f, q2 r]r, we have

Let x = VxI, , хт ]r, and у = diag(yi, * * *, yA), then the closed-loop dynamics can be expressed as

[wt,-,wTf,W = [w т,*** ,w T K d = [d r,*** ,d r]r, IN IN IN Ф = diag(0i, * * *, фи). Zt = diag(Zi, * * *, Zk), Z2. = diag(Zk+i, * * *, Zw)_yvith Z = diag(Zi, Z2), and 6 ļ = diag(ö1, * * v, 6k), 6 2 = diagØ+i, * , 6jv) with б = diag(6 !,6 2)-Theorem

1. Suppose that Assumptions 1-4 hold. Under the fully distributed observer-based adaptive protocol (5), the general nonlinear networks (1) on the antagonistic digraph can achieve BS.

Procf: Considering the following Lyapunov function

with

where p, ßi, ß2 are positive constants, n and g, are given by Lemma 3 and Lemma 4.

Take the time-derivative of Vn along closed-loop system where L C1 = R L + L^R. Define ? = [(Z, + 6 ߮ln\q ? we have - _

(14) where we use the fact that rTP,L clPt = o, with Pi = diag(pi, * * *, pk). Because each element of P,r is positive, we can get each element of P/Zi + б 1)~1Р1Г ® In is also positive. By Lemma 3, we have

Substituting fl 5) into Ç13), we have

Taking the derivative of along closed-loop system (10) N N

(17) where L c3 = GU3 + L^G with Ao as the smallest eigenvalue.

By Lemma 4, we have , , ,

By Young's Inequality, we have

Choosing ц = and combining (16)-(19), we have

Choosing ßt > and ß2 > By Young's Inequality, we have

Considering the following Lyapunov function

where ß3 being a positive constant that need to determine.

The derivative of V2 along closed-loop system (10) as follows

Note that

On the one hand,

On the other hand,

Substituting (24)-(26) to (23), we have

By choosing ß3 > 6, we can get

where 9 = w + u, x T QBsgn(/<X/) = -IIKx/llz. Thus, we can derive that , . , .

Therefore, we can get Vi is þounded, q i, q 2jmd z, are also bounded. We know that V , e о represents q z = 0 and q 2 = o, so q e o. Through LaSalle Invariance principle, we have that q asymptotically converges to zero. In addition, we can get V2is bounded and ^x T xdt exists and is bounded, x, И/, y are also bounded, that x r x is finite. By Barbalat's lemma [32], we can get that x asymptotically converges to zero. Therefore, the problem is solved. *

Remark 2. Note that the observer-based adaptive protocol

(5) is fully distributed. Compared to literature [19], we consider directed graphs and bounded disturbances. In this paper, we provide the adaptation laws (6) and (7) which can regard as adaptive form for the O-modf cation [33]. The adaptation laws (6) and (7) can be seen as extensions cf laws in [24], [22], which avoid the overlarge gains and reduce the chattering phenomenon.

IV. Disturbances from external systems

In this section, we consider the bounded BS problem for a general nonlinear networks described by equation (1), with no leader and neural network approximation. We assume that the disturbances di(t) experienced by each agent are produced by an external system described by the following dynamics

where ξi e R is the state of the external system (30), S e RnXn anc) q E RmXn are constant matrices.

Assumption 5. Suppose that (BD, 5) is observable.

Remark 3. Note that in Assumption 2 there exist nonlinear small residual errors. However, in this section we do not consider the nonlinear small residual error, that is

To solve the external disturbances, we design a disturbance observer as shown below

Hurwitz. K¡, öi, Zi and q¡ are the same as they were defined before. Assumption 5 ensures that there exists a matrix F.

To solve the bounded BS problem, the following fully distributed disturbance observer-based adaptive protocol is proposed

where d (ř) is the estimation of d (t), К = -BTQ with Q > 0 being the solution of (3), i/¡h(t) is the estimation of Wz(r) as follows

(33) where m, and Ki are positive constants.

Let x í = Xi - Ki, et = ξi- ξ t and И7 / = Wi - fh, then one can get the closed-loop system as follows

get the following form of closed-loop system

Theorem 2. Suppose that Assumptions 1-3 and Assumption 5 are satisfied. Under the fully distributed disturbance observer-based adaptive protocol (32), bounded BS cf the general nonlinear networks (1) on the antagonistic digraph can be achieved.

Proof: Considering the Lyapunov function

where Ц is a positive constant to be determined, Q > о is the solution of (5 + FBD)T Q + Q (5 + FBD) < o.

Taking the derivative of V3 along closed-loop system (35), we have

Note that

where T1 = Ämax(DT BT Q^BDY

where 1/1/ = \wT, · " / и/т ]T and Ki is to be determined later.

where T2 = Лmin( - 2Q(S + FBD)).

where т3 = Q F В\\2фтах, and фтах is the upper bound of lløzll2.

By choosing [1 = Ki = Ц T3 + a^, then we have

where аъ a2 are positive numbers, and a3 = rnaxMmgţ^V^^^Âmax ( Q)}. Let «4 = K i У=1 Then the following inequality holds

which implies

Hence, one has

and bounded BS of the general nonlinear networks (1) is achieved. Therefore, the problem is solved. *

Remark 4. When the disturbances are generated by exosystems (30), BS is difficult to achieve, and bounded BS can be achieved by the fully distributed disturbance observer-based adaptive protocol (32). The references [24], [22] did not consider this situation and only addressed the BS problem.

V. TWO NUMERICAL EXAMPLES

This section will provide two numerical simulation examples to demonstrate the effectiveness of the theoretical results obtained. We consider a third-order nonlinear network consisting of six agents. The system matrix for the network is expressed as follows

and nonlinear functions f¿(x¿, t) are assumed to be

Then, we choose Q as the solution to the Riccati equation (3)

It follows that К = [-2.1169, -4.3302, -3.0545].

Fig. 1. The signed digraph with six agents, the digraph contains a directed spanning tree and a strongly connected subdigraph. Edges without markers all have a weight of 1.

Fig. 2. Simulation results for the states хц of the network (1) under the fully distributed observer-based adaptive protocol (5).

Example 1. This example considers the case where each node is effected by bounded disturbances, which is expressed as

Through the distributed state observer (4) and the adaptive controller (5), the adaptive gain Zi is mentioned in (4) with Zi(o) > 0, the adaptive control gain 16 is mentioned in (7) with S / = 1. In addition, the initial values cf Zi and 16 are random. The basic function is

and и// = [-1, 1, 1? with Si = 1. The interaction digraph is shown in Fig. 1. According to Theorem 1, the network (1) with the fully distributed observer-based adaptive protocol (5) achieves the BS.

The state trajectories cfXi are shown in Fig. 2, Fig. 3 and Fig. 4 that achieve BS under the fully distributed observerbased adaptive protocol (5). The states cf agents diverge to two sides. The tracking errors X ¡ are shown in Fig. 5, Fig. 6 and Fig. 7. V7e can see that under the bounded disturbance condition, the tracking error cfx ¡ asymptotically approaches zero, which is consistent with our theoretical analysis. The adaptive gain // is shown in Fig. 8.

Example 2. In this example, we consider a scenario where each agent in the third-order nonlinear network is effected by external disturbances described by equation (30), where

The basic function is

0/ = [sİn(Xz1Xz3), sin(Xzi - Xz3), COS(Xz3) + 1], and и// = [0.5/, / - 0.5, 2]T. Choosing

to_ solve the linear matrix inequality (S + FBD)T Q + Q (5 + FBD) < o. According to Theorem 2, the state trajectories of Xi are shown in Fig. 9, Fig. 10 and Fig. 11 that achieve bounded BS under the fully distributed observer-based adaptive protocol (32). The states of agents diverge to two sides. The tracking errors x/ are shown in Fig. 12, Fig. 13 and Fig. 14. We can see that under the condition of external perturbation, the absolute value of the tracking errors x i is less than a bounded constant close to zero, the bounded BS is achieved, which is consistent with our theoretical analysis.

VI. Concluding remarks

This paper addressed the BS problem for general nonlinear networks with neural networks approximation and external disturbances in signed digraphs. For the agents subject to bounded disturbances with an unknown upper bound, when the signed digraph contains a directed spanning tree, designed the fully distributed observer-based adaptive protocol to make network achieve leadless BS. When the agents are affected by disturbances generated by a external system, the fully distributed disturbance observer-based adaptive protocol is designed to the general nonlinear nonlinear networks with no leader and neural networks approximation, and the closed-loop network achieves bounded BS. Finally, the theoretical results are verified by two numerical simulation examples. Future research can be carried out on switching topologies or cluster BS.

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