**1. Introduction**

With the rapid development of science and technology, human beings have marched into the network era, and complex network has become a hot topic. Complex network is an important

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

method to describe and study complex systems, and all complex systems can be abstracted from practical background by different perspectives and become a complex network of interacting individuals, such as ecological network, food network, gene regulation network, social network, and distributed sensor network. Research on complex network has become a frontier subject with many subjects and challenges. Over the past few years, studies on complex network have received more and more attention from various fields of scientific research See [1–5]. The popularization of complex network has also caused a series of important problems about the network structures and studies of the network dynamic behaviors. Particularly, special attention has been paid to the studies of synchronization control problems of complex dynamical networks. As one of the significant dynamic behaviors of complex dynamical network, synchronization is widely used in neural network, public transit scheduling, laser system, secure communiza‐ tion system, information science, etc. [6–11]. So it is concerned by more and more scholars. In real networks, because of the complex dynamical network having a great many nodes, and every node has its dynamical behavior, it is hard for the complex dynamical network itself to make the states of the network to desired trajectory. Thus, the studies on the control strategy of complex dynamical network will be meaningful. So far, many control methods for complex dynamical network have been reported in refs. [12–17]. Pinning control such as in refs. [18–20] is widely welcomed for its advantages. It is easy to be realized and can save the cost effectively. The main idea of pinning control is to control a part of nodes in the complex networks to realize the whole network to the expected states and to reduce the number of the controllers effectively. When there exist some unknown parameters, the adaptive control method could be exploited, some of which was mentioned in refs. [21–23].

On the other hand, there are many factors that affect the stability of complex network, where time delay and network topology are two important factors. First, time delay is an objective phenomenon in nature and human society. In the process of transmission and response of complex network, it is inevitable to produce time delay, which is because of the physical limitations of the speed of transmission and the existence of network congestion, such as the existence of time delay in communication network and virus transmission. There are some typical time delay network systems such as circuit system [24], satellite communication system [25], and laser array system [26]. It is noticed that the majority of the studies on complex network have been performed on some absolute assumptions. For example, the stabilization referred to state feedback control is realized only by a non-delay or delay controller, which is relied on some absolute assumptions [18, 19, 27]. However, in many practical applications, these assumptions do not accord with the peculiarities of the real networks. Based on these facts, we may design a kind of controller that contains non-delay and delay states simultane‐ ously. Second, the topology of the network plays an important role in determining the network characteristics and the synchronization control. The research of coupling delay also plays a significant role in complex networks. In most of the above papers, it is seen that the topologies are fixed. But in practical applications, the topological structure of the complex network is not constant and may be changed randomly. That is because of the influence of various stochastic factors. In this case, how to ensure the stabilization of networks by the proposed controller when the topologies related to couplings change is worth discussing.

Motivated by the above discussions, in this chapter, the stabilization problem of complex networks with non-delayed and delayed couplings over random exchanges is studied by exploiting the robust method to describe the topologies exchanging randomly. A kind of stochastic pinning controller being partially delay-dependent is developed, which contains non-delay and delay terms simultaneously but occur asynchronously. Here, the probability distributions are taken into account in the proposed controller design. The rest of this chapter is organized as follows: In Section 2, the model of complex dynamical networks with nondelayed and delayed couplings over random exchanges is established. In Section 3, the stabilization of the underlying complex networks is considered, which is realized by partially delay-dependent controller and adaptive controller respectively. A numerical example is demonstrated in Section 4; the conclusion of this chapter is given in Section 5.

method to describe and study complex systems, and all complex systems can be abstracted from practical background by different perspectives and become a complex network of interacting individuals, such as ecological network, food network, gene regulation network, social network, and distributed sensor network. Research on complex network has become a frontier subject with many subjects and challenges. Over the past few years, studies on complex network have received more and more attention from various fields of scientific research See [1–5]. The popularization of complex network has also caused a series of important problems about the network structures and studies of the network dynamic behaviors. Particularly, special attention has been paid to the studies of synchronization control problems of complex dynamical networks. As one of the significant dynamic behaviors of complex dynamical network, synchronization is widely used in neural network, public transit scheduling, laser system, secure communiza‐ tion system, information science, etc. [6–11]. So it is concerned by more and more scholars. In real networks, because of the complex dynamical network having a great many nodes, and every node has its dynamical behavior, it is hard for the complex dynamical network itself to make the states of the network to desired trajectory. Thus, the studies on the control strategy of complex dynamical network will be meaningful. So far, many control methods for complex dynamical network have been reported in refs. [12–17]. Pinning control such as in refs. [18–20] is widely welcomed for its advantages. It is easy to be realized and can save the cost effectively. The main idea of pinning control is to control a part of nodes in the complex networks to realize the whole network to the expected states and to reduce the number of the controllers effectively. When there exist some unknown parameters, the adaptive control method could be exploited, some

On the other hand, there are many factors that affect the stability of complex network, where time delay and network topology are two important factors. First, time delay is an objective phenomenon in nature and human society. In the process of transmission and response of complex network, it is inevitable to produce time delay, which is because of the physical limitations of the speed of transmission and the existence of network congestion, such as the existence of time delay in communication network and virus transmission. There are some typical time delay network systems such as circuit system [24], satellite communication system [25], and laser array system [26]. It is noticed that the majority of the studies on complex network have been performed on some absolute assumptions. For example, the stabilization referred to state feedback control is realized only by a non-delay or delay controller, which is relied on some absolute assumptions [18, 19, 27]. However, in many practical applications, these assumptions do not accord with the peculiarities of the real networks. Based on these facts, we may design a kind of controller that contains non-delay and delay states simultane‐ ously. Second, the topology of the network plays an important role in determining the network characteristics and the synchronization control. The research of coupling delay also plays a significant role in complex networks. In most of the above papers, it is seen that the topologies are fixed. But in practical applications, the topological structure of the complex network is not constant and may be changed randomly. That is because of the influence of various stochastic factors. In this case, how to ensure the stabilization of networks by the proposed controller

when the topologies related to couplings change is worth discussing.

of which was mentioned in refs. [21–23].

26 Robust Control - Theoretical Models and Case Studies

**Notation**: ▯*<sup>n</sup>* denotes the *n* dimensional Euclidean space, ▯*<sup>m</sup>* × *<sup>n</sup>* is the set of all *m* × *n* real matrices. E{⋅} is the expectation operator with respect to some probability measure. diag{⋅⋅⋅} stands for a block-diagonal matrix. *IN* is an identity matrix being of *N* dimensions. S=S<sup>ℓ</sup> <sup>∪</sup>S¯ <sup>ℓ</sup>, where <sup>S</sup>*<sup>l</sup>* ={1, 2, …, *<sup>l</sup>*}, S¯ <sup>ℓ</sup> ={*l* + 1, *l* + 2, …, *N* }. *λ*max(*M*) is the maximum eigenvalue of *M*, while *σ*max(*M*) is the maximum singular value of *M*. ‖*G*‖ denotes the 2-norm of matrix *G*. \* stands for an ellipsis for the term induced by symmetry.
