Pinning control for synchronization of nonlinear complex dynamical network with suboptimal SDRE controllers

Abstract

Based on the state-dependent Riccati equation (SDRE) technique, in this paper, a suboptimal pinning control scheme is proposed to synchronize linearly coupled complex networks. The Lyapunov direct method is used to analyze the stability of the closed-loop control system, where it leads to a LMI criterion for pinning synchronization. It is shown that the time interval for synchronization of the proposed SDRE controllers is faster comparing with the results in the latest literatures. It is also shown that the minimum required coupling weights for the network synchronization in a finite desired time is decreased when some specified nodes in the network are pinned with the SDRE controllers. Based on the proposed criterion for pinned nodes selection, the network performances for different topological structures are investigated and the results are compared. The results indicate that the coupling weights for network synchronization in finite desired time in random Erdos Reiny networks are minimum when the pinned nodes are selected based on the minimum matching theorem. In small-world and scale-free networks, the minimum required coupling weight for network synchronization in finite desired time decreases when the highest degree nodes are pinned.

Appendix

Complete proof of Theorem 1

Consider the Lyapunov functional candidate:

$$\begin{aligned} V(t)=\frac{1}{2}\mathop {\sum }\limits _{i=1}^N e_i ^{T}(t)e_i (t) \end{aligned}$$

(35)

The derivative of V(t) along the trajectories of (6) is;

$$\begin{aligned} {\dot{V}}(t) = \mathop \sum \limits _{i = 1}^N e_{i}^{T}(t){\dot{e}_i}(t) \end{aligned}$$

(36)

Substitution of Eqs. (9) and (21) into (36) in expanded form forK, gives:

$$\begin{aligned} \dot{V}(t)= & {} \mathop \sum \limits _{i = 1}^N e_{i}^{T}(t)\left[ {\left( {f\left( {{e_i}(t),t} \right) + c\mathop \sum \limits _{j = 1}^N {l_{ij}}\varGamma {e_j}(t)} \right) }\right] \nonumber \\&-\,c\mathop \sum \limits _{i = 1}^N e_{i}^{T}(t){K_i}\left( e \right) \varGamma {e_i}(t) \end{aligned}$$

(37)

where, $K_i \in R^{n\times n}$ is a matrix whose elements are calculated by Eq. (22) and link to $e_j (t)$ with linking coupled variable $\varGamma$. $K_i$ equal to unpinned nodes is zero matrix.

Based on the Lipchitz condition in Sect. 2 and Kronecker product algebra, Eq. (37) may be rewritten as following form:

$$\begin{aligned} {\dot{V}}(t)\le & {} {e^T}(t)\big [ ( {{I_N} \otimes \Lambda \varGamma }) \nonumber \\&+\, {c}( {\mathrm{L} \otimes \varGamma } ) - {c}( {K \otimes \varGamma } ) \big ]{e_j}(t) \end{aligned}$$

(38)

We must have;

$$\begin{aligned} \left( {I_N \otimes \Lambda \varGamma }\right) +\hbox {c}\left( {\hbox {L}\otimes \varGamma } \right) -\hbox {c}\left( {K\otimes \varGamma } \right) <0 \end{aligned}$$

(39)

By substituting Eq. (21) into (30), the following criterion has been established:

$$\begin{aligned}&\left( {I_N \otimes \Lambda \varGamma } \right) +\hbox {c}\left( {\hbox {L}\otimes \varGamma } \right) \nonumber \\&\quad -\,\hbox {c}\left( {-R^{-1}(e)\bar{B} ^{T}(e)P(e)\otimes \varGamma } \right) <0 \end{aligned}$$

(40)