Decentralized guaranteed cost dynamic control for synchronization of a complex dynamical network with randomly switching topology
Introduction
During last half a century, complex dynamical networks have been received much attention from researchers working in different fields and their important research results have been noted [1], [2], [3], [4]. Complex dynamical networks are a set of interconnected nodes, in which a node is a basic unit with specific contents or dynamics. As well known, the properties of such a complex dynamical network depend on the structure of coupling matrix (or network topology), so network topology is most important information in the field of complex dynamical networks. There are some useful network topologies such as small-world network, scale-free network and so on which have been defined from random-graph model built by Erdös and Rényi [5], [6]. After above works, these network topologies have been extensively investigated and become basic assumption in complex dynamical networks [7], [8]. In particular, the synchronization of complex dynamical networks with these network topologies is one of the key issues that has been extensively addressed in several books and reviews [9], [10], [11]. Therefore, many researchers have focused on this topic and have developed several efficient synchronization techniques for complex dynamical networks.
The classical constant connection topology is of course very restrictive and only reflects a few ideal situations. Time-varying connection topology is more realistic and covers various situations in practice. Further, the study about the synchronization of complex dynamical networks with switching topology have been attracting increasing research attention recently. In [12], a sensor network has been shown to have jumping behavior due to the network’s working environment and the mobility of sensor node. In [13], the synchronization of complex dynamical networks with arbitrary switching topology has been studied by using multiple Lyapunov functions. In [14], the global and local synchronization criteria in the form of the second smallest eigenvalue of a switching coupling matrix have been proposed for complex dynamical networks with switching topology. More recently, the concept of Markovian jumping topology is adopted to synchronization problems of complex dynamical networks. The Markovian jump systems have the advantage of modeling the complex dynamical networks subject to abrupt variation in their communication topologies, such as component failures or repairs, sudden environmental disturbance, changing subsystem interconnections, and operating in different points of a nonlinear plant. In [15], the exponential synchronization of complex dynamical networks with Markovian jump and mixed delay has been studied. In [16], the asymptotic synchronization problem has been investigated for a class of discrete-time stochastic Markovian complex dynamical networks with discrete and distributed time delays. In [17], the synchronization of Markovian jumping stochastic complex dynamical networks with distributed and probabilistic interval time delays was treated.
Furthermore, in many real situations, networks can not synchronize by themselves. Therefore, some control schemes are adopted to design controllers, such as linear sate feedback control [18], adaptive control [19], state observer-based control [20], impulsive control [21], control [22] and pinning control [23]. However, most of works in the literature on the controller design problem has focused on static controllers. Unlikely the static controllers, the dynamic control method means that the controller has their own dynamics. In some real control situations, there are some strong needs to construct dynamic feedback controllers in order to obtain a better performance and dynamical behavior of the state response. The dynamic controller will provide more flexibility compared to the static controller and the apparent advantage of this type of controller is that it provides more free parameters for selection [24]. In addition, designing a stabilizing controller for a complex network with N nodes is often difficult if not impossible, because of their complexity and strong interconnection. Therefore, it is more efficient to construct decentralized controllers to achieve control objectives for the network. So it is very worth to consider the design problem of dynamic and decentralized controller for synchronization in a complex dynamical network.
On the other hand, when controlling a real plant, it is also desirable to design a control system which is not only stable but also guarantees an adequate level of performance. There is a solution so called guaranteed cost control approach [25], [26], [27]. This approach has the advantage of providing an upper bound on a given linear quadratic cost function. Up to date, some researchers in diverse fields have applied the approach to achieve both stability and performance of dynamic systems. Unfortunately, there are a few paper about the topic of guaranteed cost control for complex dynamical network [28].
In this paper, we consider the synchronization of a complex network with randomly switching topology. From above mentioned, most of researches about synchronization of complex dynamical networks with switching topology have assumed that switching trajectory is known or satisfies Markovian process. The randomly switching topology, however, means that the topology is switched at the random time. Unlikely other switching concept, randomly switching concept have not any switching rules, so it is said that this concept has more generality in switching system. To the best of authors’ knowledge, the concept of randomly switching topology for complex dynamical network is still an open area and worth considering. Furthermore, in order to consider both stability and performance of the network, the guaranteed cost control scheme for the synchronization problem is also applied. Additionally, we apply the problem of decentralized control via dynamic feedback controllers. The existence condition of such controller is derived in terms of linear matrix inequalities which can be easily solved by standard convex optimization algorithms [29].
This paper is organized as follows. A problem statement is described in Section 2. Section 3 provides the design method of a stabilizing controller for synchronization of a complex network with randomly switching topology. Two numerical examples are given in Section 4 to show the effectiveness of the derived results. Conclusions are drawn in Section 5.
Notation: is the n-dimensional Euclidean space, denotes the set of real matrix. (respectively, 0) means that the matrix X is a real symmetric positive definite matrix (respectively, positive semi-definite). denotes the n-dimensional identity matrix. refers to the Euclidean vector norm and induced matrix norm. in a matrix represents the elements below the main diagonal of a symmetric matrix. and , respectively, mean the expectation of the stochastic variable x and the expectation of the stochastic variable x conditional on the stochastic variable y. denotes the block diagonal matrix. means the occurrence probability of the event .
Section snippets
Problem formulation
Consider following a complex dynamical network with randomly switching topology:where is the state vector of the ith node, A is a known constant matrix, is a smooth nonlinear vector field, is the control input of ith node, is a switching signal, and is the coupling matrix function of the network, where the coupling configuration parameter, , is defined as
Main results
In this section, the existence criterion for a decentralized guaranteed cost dynamic controller (6) for error system (5) will be derived by use of Lyapunov theory and linear matrix inequality framework with convex optimization.
The following is a main result of this paper. Theorem 1 For given a positive constant and a known Lipschitz constant l, the dynamic controller (6) is the guaranteed cost synchronization controller for the complex network (3) if there exist positive-definite matrices
Numerical examples
In this section, two simulation results are presented to show the effectiveness of the proposed controller for synchronizing all nodes of a complex network to a target node. In two examples, the parameters associated with cost function are chosen as and . Example 1 Chua’s circuit system The first example is about synchronization of a complex dynamical network with five linearly coupled identical nodes which are Chua’s circuit [34] which is typical benchmark three dimensional chaotic system. Fig. 1 depicts its
Conclusions
In this paper, the decentralized guaranteed cost dynamic controller for synchronization of a complex dynamical network with randomly switching topology has been designed based on the Lyapunov method and linear matrix inequality framework. Unlike other works, randomly switching topology was considered for the synchronization problem instead of known switching topology trajectory or Markovian jumping topology, so it is said that this concept has more generality in switching system. Then, two
Acknowledgement
This work was supported by Yeungnam University Research Grant.
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