Structural Balance Control of Complex Dynamical Networks Based on State Observer for Dynamic Connection Relationships

Abstract

This paper investigates the observer-based structural balance control for a class of complex dynamical networks. Generally speaking, a complete complex dynamical network is composed of two coupled subsystems, which are called node subsystem (NS) and connection relationship subsystem (CS), respectively. Similar to synchronization and stabilization of networks, the structural balance is another phenomenon of networks and determined by the state of connection relationships. However, it is not feasible to design the controller for the CS directly because the states of the connection relationships are difficult to be measured accurately in practical applications. In order to solve this problem, a state observer for the CS has been designed. Thus, the structural balance controller in the CS can be directly designed by using the estimation information of the state observer. Then, with the help of the Lyapunov stability theory, it is proved that the CS can asymptotically track a given structural balance matrix under the influence of the observer-based controller. Finally, the results derived from this paper are demonstrated by performing a numerical example.

Double Tracking Control for the Directed Complex Dynamic Network via the Observer of Outgoing Links

“…(iii) Fig.5 shows that the given tracking signal L * i does not converge to zero, which means that the eventual network structure (t → +∞) is shown as all the nodes are not isolated when synchronization happens, in other words, the connections are still connected in eventual complex network. This is different from the results in [19][20][21] where the connections are disconnected in eventual complex network.…”

“…(b).The curves of tracking error norm ∥Ei∥ of OLS with the controller in [19][20][21] and this paper (i) From Fig.2 and Fig.4, it can be seen that the tracking controllers in [19][20][21] are designed by utilizing the observer's state of LS to control NS to track the reference state, but tracking error curves show not only the larger chattering problem but also the slower convergence speeds than ones in this paper. Similarly, it can be observed from Fig.3 and Fig.4 that the tracking controllers are synthesize by employing the observer's sate of LS to control LS to track the differentiable bounded reference goal, but the tracking response speeds of OLS in [19][20][21] are also slower than that in this paper.…”

“…T is the state vector of the ith Chua's circuit, (ii) Solving for the positive definite matrices M i and K i such that linear matrix inequality where θ is an adjustable parameter, and its value is randomly generated real numbers in the interval Fig.4 (a).The curves of tracking error norm ∥ei∥ of NS with the controller in [19][20][21] and this paper;…”

“…The states of LS are more complex and difficult to be measured than the states of NS in the complex dynamic network, so that it is necessary to employ the measurable outputs of LS to achieve the estimation of uncertain state variables for LS. At present, some research results have been achieved on the state observer [9,[19][20][21] design of LS. In [19], the state observer of LS is synthesized for the first time for the complex dynamic network.…”

“…In [19], the state observer of LS is synthesized for the first time for the complex dynamic network. In [20,21], the state observer employs the output of LS as its input to obtain the state observation information for LS, and then the LS tracks on a reference relation matrix by utilizing the state observation information to synthesize the controller of LS, without considering the dynamic behavior of NS. In view of this, the results in [9] extend the continuous state observer to the discrete state observer for LS.…”