Complex Dynamical Network Control for Trajectory Tracking Using Delayed Recurrent Neural Networks

In this paper, the problem of trajectory tracking is studied. Based on the V-stability and Lyapunov theory, a control law that achieves the global asymptotic stability of the tracking error between a delayed recurrent neural network and a complex dynamical network is obtained. To illustrate the analytic results, we present a tracking simulation of a dynamical network with each node being just one Lorenz’s dynamical system and three identical Chen’s dynamical systems.


Introduction
The analysis and control of complex behavior in complex networks, which consist of dynamical nodes, have become a point of great interest in the recent studies, [1][2][3].The complexity in networks comes not only from their structure and dynamics but also from their topology, which often affects their function.
Recurrent neural networks have been widely used in the fields of optimization, pattern recognition, signal processing and control systems, among others.They have to be designed in such a way that there is one equilibrium point that is globally asymptotically stable.In biological and artificial neural networks, time delays arise in the processing of information storage and transmission.Also, it is known that these delays can create oscillatory or even unstable trajectories, [4].Trajectory tracking is a very interesting problem in the field of theory of systems control; it allows the implementation of important tasks for automatic control such as: high speed target recognition and tracking, real-time visual inspection, and recognition of context sensitive and moving scenes, among others.We present the results of the design of a control law that guarantee the tracking of general complex dynamical networks.

Mathematical Models
2.1.General Complex Dynamical Networks.Consider a network consisting of  linearly and diffusively coupled nodes, with each node being an -dimensional dynamical system, described by where   = ( 1 ,  2 , . . .,   )  ∈ R  are the state vectors of the node ,   : R   → R  represents the self-dynamics of the node , and the constants   > 0 are the coupling strengths between node  and node , with ,  = 1, 2, . . ., .Γ = (  ) ∈ R × is a constant internal matrix that describes the way of linking the components in each pair of connected node vectors (  −   ): this means that for some pairs (, ) with 1 ≤ ,  ≤  and   ̸ = 0, the two coupled nodes are linked through their th and th substate variables, respectively, while the coupling matrix  = (  ) ∈ R × denotes the coupling configuration of the entire network: this means that if there is a connection between node  and node  ( ̸ = ), then   =   = 1; otherwise,   =   = 0.

Tracking Error Stabilization and Control Design
In order to establish the convergence of (10) to   = 0,  = 1, 2, . . ., , which ensures the desired tracking, first, we propose the following Lyapunov function: The time derivative of (11), along the trajectories of (10), is Reformulating ( 12), we get Next, let us consider the following inequality, proved in [9,10]: We define ũ  = ũ(1) ,  = 1, 2, . . ., , and then (20) becomes Now, we propose the use of the following control law: Then, V  () < 0 for all  ̸ = 0.This means that the proposed control law (22) can globally and asymptotically stabilize the th error system (10), therefore ensuring the tracking of ( 1) by (2).Finally, the control action driving the recurrent neural networks is given by
The experiment is performed as follows.Both systems, the delayed neural network (2) and the dynamical networks (24) and (25), evolve independently until  = 10 seconds; at that time, the proposed control law (23) is incepted.Simulation results are presented in Figures 4, 5, and 6 for sub-sates of node 1.As can be seen, tracking is successfully achieved and error is asymptotically stable, as it is shown in Figures 7, 8, and 9 for sub-states of node 4.

Conclusions
We have presented the controller design for trajectory tracking determined by a general complex dynamical network.This framework is based on dynamic delayed neural networks and the methodology is based on V-stability and Lyapunov theory.The proposed control is applied to a dynamical network with each node being a Lorenz and Chen's dynamical systems, respectively, being able to also stabilize in asymptotic

Figure 3 :
Figure 3: Structure of the network with each node being a Lorentz and Chen's system.