Practical bipartite synchronization via pinning control on a network of nonlinear agents with antagonistic interactions

Abstract

This paper studies the synchronization phenomena in a network of heterogeneous nonlinear systems over signed graph, which can be considered as the perturbation of a network of homogeneous nonlinear systems. Assume that the signed graph is structurally balanced, the nonlinear system satisfies one-sided Lipschitz condition, and a leader pins a subset of agents. Under some proper initial conditions of the leader, we derive some conditions under which bipartite synchronization error can be kept arbitrary small by choosing a proper pining scheme. This property is called practical pinning bipartite synchronization as an alternative synchronization notion for network of heterogeneous nonlinear systems over signed graph. Finally, we present a numerical example to illustrate the effectiveness of the obtained results.

Appendices

Appendix 1: Signed graphs

Let $\mathcal {G}=(\mathcal {V},\mathcal {E},A)$ denotes a weighted directed signed graph, where $\mathcal {V}=\{1,2,\ldots ,N\}$ is the set of all nodes, $\mathcal {E}\in \mathcal {V}\times \mathcal {V}$ is the set of edges and A is an adjacency matrix which assigns real numbers to the edges. The notation $\mathcal {G}(A)$ denotes a graph with adjacency matrix A. An edge $(i,j)\in \mathcal {E}$ is directed from node i to node j, where nodes i and j are called parent node and child node, respectively. The entry $a_{ij}>0(or<0)$ is the weight corresponding to the edge (i, j). We define $\mathcal {E}^{+}=\{(i,j)|a_{ij}>0\}$ and $\mathcal {E}^{-}=\{(i,j)|a_{ij}<0\}$. Then, $\mathcal {E}=\mathcal {E}^{+}\cup \mathcal {E}^{+}$. We assume that $a_{ij}a_{ji}\ge 0$. For directed signed graph $\mathcal {G}(A)$, let $L=C_r-A$ denotes the Laplacian matrix of $\mathcal {G}(A)$, where $C_r=\mathrm{{diag}}\{\sum _{j=1}^N |a_{1j}|,\ldots ,\sum _{j=1}^N |a_{Nj}|\}$. There exists an undirected graph $\mathcal {G}(A_u)$ with adjacency matrix $A_u=(A+A^\mathrm{T})/2$, and the corresponding Laplacian is $L_u=(L+L^\mathrm{T})/2=C_r-A_u$. $C_c=\mathrm{{diag}}\{\sum _{i=1}^N |a_{i1}|,\ldots ,\sum _{i=1}^N |a_{iN}|\}$. It is obvious that $C_r=C_c$ when $A^\mathrm{T}=A$. A directed signed graph $\mathcal {G}(A)$ is said weight balanced if $C_r=C_c$. A directed path (length $l-1$) is a sequence of directed edges of the form $(i_1,i_2),(i_2,i_3),\ldots ,(i_l1,i_l)$ with distinct nodes. A directed signed graph has a spanning tree if there is a root node, which has directed paths to all other nodes. A directed signed graph is said to be strongly connected if there is a directed path between any two distinct nodes.

Definition 3

[7] A signed graph $\mathcal {G}(A)$ is structurally balanced if it has a bipartition of the nodes $\mathcal {V}_1,\mathcal {V}_2,\mathcal {V}_1\cup \mathcal {V}_2=\mathcal {V},\mathcal {V}_1\cap \mathcal {V}_2=\varnothing$ such that $a_{ij}\ge 0,\forall i,j\in \mathcal {V}_p (p\in \{1,2\})$, $a_{ij}\le 0,\forall i\in \mathcal {V}_p,j\in \mathcal {V}_q,p\ne q (p,q\in \{1,2\})$. It is said structurally unbalanced otherwise.

Denote the signature matrices set as

$$\begin{aligned} \ominus =\{\Psi =\mathrm{{diag}}(\psi _1,\ldots ,\psi _N)| \psi _i\in \{1,-1\}\}. \end{aligned}$$

We have following results about undirected (directed) signed graph.

Lemma 1

[11] Suppose that the signed digraph $\mathcal {G}(A)$ has a spanning tree. Then, the following statements are equivalent.

• 1. $\mathcal {G}(A)$ is structurally balanced;

• 2. $a_{ij}a_{ji}\ge 0$, and the associated undirected graph $\mathcal {G}(A_u)$ is structurally balanced, where $A_u=0.5(A^\mathrm{T}+A)$;

• 3. $\exists \Psi \in \ominus$, such that ${\bar{A}}=\Psi A\Psi$ is a nonnegative matrix;

• 4. All eigenvalues of its signed Laplacian matrix have nonnegative real parts and 0 is a simple eigenvalue.

Lemma 2

[28] Suppose that directed signed graph $\mathcal {G}(A)$ has a spanning tree and structurally balanced. If all nodes of $\mathcal {G}(A)$ can be partitioned into two subset $V_1=\{1,2,\ldots ,l\},V_2=\{l+1,l+2,\ldots ,N\}$ such that $a_{ij}\ge 0,\forall i,j\in V_p,p=1,2$ and $a_{ij}\le 0,\{(i,j)|i\in V_p,j\in V_q,p,q\in \{1,2\},p\ne q\}$, then the matrix $\Psi$ in Lemma 1 can be chosen as $\displaystyle \mathrm{{diag}}\{\underbrace{1,1,\ldots ,1}_l,\underbrace{-1,\ldots ,-1}_{N-l}\}$.

Appendix 2: Some definitions and notations about dynamic systems

Consider an autonomous system ${\dot{x}}=g(x)$, where g is a smooth nonlinear function. Let $\varphi (t)$ be the trajectory of ${\dot{x}}=g(x)$ with initial condition $\varphi (0)=x_0$. If $x^{*}$ is an equilibrium of ${\dot{x}}=g(x)$, then the eigenvalues of the Jacobian matrix $Dg_{(x^{*})}$ are called the eigenvalues of $x^{*}$. If the real parts of all the eigenvalues of $Dg_{(x^{*})}=\frac{\partial g(x^{*})}{\partial x}$ are nonzero, then the $x^{*}$ is called hyperbolic. The stable manifold of the equilibrium $W_g^s(x^{*})$ is the set of all the points that tend to the equilibrium as t goes to plus infinity, that is, $W_g^s(x^{*})=\{p_0|\varphi (t,p_0)\;\text {tends to}\;x^{*}\;\text {as}\;t\rightarrow \infty \}$. The unstable manifold of the equilibrium $W_g^u(x^{*})$ is the set of all the points that tend to the equilibrium as t goes to minus infinity, that is, $W_g^u(x^{*})=\{p_0|\varphi (t,p_0)\;\text {tends to}\;x^{*}\; \text {as}\;t\rightarrow -\infty \}$.