Adaptive Control of Dynamically Complex Networks with Saturation Couplings

Guoliang Wang,Tingting Yan

doi:10.1155/2016/4614856

Abstract

This paper considers the control problem of dynamically complex networks with saturation couplings. Two novel control schemes in terms of adaptive control are presented to deal with such saturation couplings. Based on the robust idea, the underlying complex network is firstly transformed into a strongly connected network having time-varying uncertainty. However, the upper bound of uncertainty is unknown. Because of such an unavailable bound, a kind of adaptive controller added to each node is proposed such that the closed-loop auxiliary network is uniformly bounded. In particular, the original system states are asymptotically stable. Moreover, in order to avoid adding the desired controller to every node, another different kind of adaptive controller based on the pinning control idea is proposed. Compared with the former method, it is only applied to a part of nodes and has a good operability. Finally, a numerical example is provided to show the effectiveness of our results.

Highlights

This paper considers the control problem of dynamically complex networks with saturation couplings
Complex network is composed of a large number of interconnected dynamical units, which is ubiquitous in nature and human society, such as ecosystem network, biological network, food network, social network, and transportation network
A kind of adaptive controller guaranteeing the states of the closed-loop complex network asymptotically stable is proposed as u (t) = −c (K ⊗ In) x (t) c2δ2 (t) x (t) xT cδ (t) xT (t)

Summary

Introduction

Complex network is composed of a large number of interconnected dynamical units, which is ubiquitous in nature and human society, such as ecosystem network, biological network, food network, social network, and transportation network. In many practical applications especially in the real networks with a limited communication capacity, it is difficult to allow them varying abruptly in terms of being randomly large. In this case, saturation phenomenon is a common problem existing in most network systems, which could affect the stability in a large part. To the best of authors’ knowledge, no related results are available and could be used to study the related problems of complex networks with saturation couplings. All these observations motivate the current study. IN represents an identity matrix being of N dimensions. λmax(⋅) is the maximum eigenvalue of matrix. ‖ ⋅ ‖ denotes the Frobenius norm of a matrix. ∗ denotes an ellipsis for the term induced by symmetry

Problem Formulation

Main Results

Numerical Example

Conclusions