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

Authors: Jose P. Perez; Joel Perez Padron; Angel Flores Hemandez; Santiago Arroyo
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

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.


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## Body

## 1. 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, [1–3]. 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.


## 2. Mathematical Models

### 2.1. General Complex Dynamical Networks

Consider a network consisting ofN linearly and diffusively coupled nodes, with each node being an n-dimensional dynamical system, described by(1)x˙i=fi(xi)+∑j=1j≠iN‍cijaijΓ(xj-xi),i=1,2,…,N,
where xi=(xi1,xi2,…,xin)T∈ℝn are the state vectors of the node i, fi:ℝn↦ℝn represents the self-dynamics of the node i, and the constants cij>0 are the coupling strengths between node i and node j, with i, j=1,2,…,N. Γ=(τij)∈ℝn×n is a constant internal matrix that describes the way of linking the components in each pair of connected node vectors (xj-xi): this means that for some pairs (i,j) with 1≤i, j≤n and τij≠0, the two coupled nodes are linked through their ith and jth substate variables, respectively, while the coupling matrix A=(aij)∈ℝN×N denotes the coupling configuration of the entire network: this means that if there is a connection between node i and node j(i≠j), then aij=aji=1; otherwise, aij=aji=0.


### 2.2.  Delayed Recurrent Neural Networks

Consider a delayed recurrent neural network in the following form:(2)x˙ni=Anixni+Wniσ(xni(t-τ))+uni+∑j=1j≠iN‍cninjaninjΓ(xnj-xni),i=1,2,…,N,
where τ is the fixed known time delay [5, 6], xni=(xni1,xni2,…,xnin)T∈ℝn is the state vector of the neural network i, uni∈ℝn is the input of the neural network i, Ani=-λniIn×n, i=1,2,…,N, is the state feedback matrix, with λni being a positive constant, Wni∈ℝn×n is the connection weight matrix with i=1,2,…,N, and σ(·)∈ℝn is a Lipschitz sigmoid vector function [7, 8], such that σ(xni)=0 only at xni=0, with Lipschitz constant Lσi, i=1,2,…,N, and neuron activation functions σi(·)=tanh(·), i=1,2,…,n.


## 2.1. General Complex Dynamical Networks

Consider a network consisting ofN linearly and diffusively coupled nodes, with each node being an n-dimensional dynamical system, described by(1)x˙i=fi(xi)+∑j=1j≠iN‍cijaijΓ(xj-xi),i=1,2,…,N,
where xi=(xi1,xi2,…,xin)T∈ℝn are the state vectors of the node i, fi:ℝn↦ℝn represents the self-dynamics of the node i, and the constants cij>0 are the coupling strengths between node i and node j, with i, j=1,2,…,N. Γ=(τij)∈ℝn×n is a constant internal matrix that describes the way of linking the components in each pair of connected node vectors (xj-xi): this means that for some pairs (i,j) with 1≤i, j≤n and τij≠0, the two coupled nodes are linked through their ith and jth substate variables, respectively, while the coupling matrix A=(aij)∈ℝN×N denotes the coupling configuration of the entire network: this means that if there is a connection between node i and node j(i≠j), then aij=aji=1; otherwise, aij=aji=0.


## 2.2.  Delayed Recurrent Neural Networks

Consider a delayed recurrent neural network in the following form:(2)x˙ni=Anixni+Wniσ(xni(t-τ))+uni+∑j=1j≠iN‍cninjaninjΓ(xnj-xni),i=1,2,…,N,
where τ is the fixed known time delay [5, 6], xni=(xni1,xni2,…,xnin)T∈ℝn is the state vector of the neural network i, uni∈ℝn is the input of the neural network i, Ani=-λniIn×n, i=1,2,…,N, is the state feedback matrix, with λni being a positive constant, Wni∈ℝn×n is the connection weight matrix with i=1,2,…,N, and σ(·)∈ℝn is a Lipschitz sigmoid vector function [7, 8], such that σ(xni)=0 only at xni=0, with Lipschitz constant Lσi, i=1,2,…,N, and neuron activation functions σi(·)=tanh(·), i=1,2,…,n.


## 3. Trajectory Tracking

The objective is to develop a control law such that theith neural network (2) tracks the trajectory of the ith dynamical system (1). We define the tracking error as ei=xni-xi, i=1,2,…,N, whose derivative with respect to time is
(3)e˙i=x˙ni-x˙i,i=1,2,…,N.
Substituting (1) and (2) in (3), we obtain
(4)e˙i=Anixni+Wniσ(xni(t-τ))+uni-fi(xi)+∑j=1j≠iN‍cninjaninjΓ(xnj-xni)-∑j=1j≠iN‍cijaijΓ(xj-xi),i=1,2,…,N.
Adding and subtracting Wniσ(xi(t-τ)), αi(t), i=1,2,…,N, to (4), where αi will be determined below, and considering that xni=ei+xi, i=1,2,…,N, then
(5)e˙i=Wni(σ(xni(t-τ))-σ(xi(t-τ)))+(uni-αi(t))+Aniei+(Anixi+Wniσ(xi(t-τ))+αi(t))-fi(xi)+∑j=1j≠iN‍cninjaninjΓ(xnj-xni)-∑j=1j≠iN‍cijaijΓ(xj-xi),i=1,2,…,N.In order to guarantee that theith neural network (2) tracks the ith reference trajectory (1), the following assumption has to be satisfied:Assumption 1.
There exist functionsρi(t) and αi(t), i=1,2,…,N, such that
(6)dρi(t)dt=Aniρi(t)+Wniσ(ρi(t))+αi(t),ρi(t)=xi(t),i=1,2,…,N.
Let us define
(7)u~ni=(uni-αi(t)),ϕσ(t-τ)=σ(xni(t-τ))-σ(xi(t-τ)),i=1,2,…,N.
Considering (6) and (7), equation (5) is reduced to
(8)e˙i=Aniei+Wniϕσ(t-τ)+u~ni+∑j=1j≠iN‍cninjaninjΓ(xnj-xni)-∑j=1j≠iN‍cijaijΓ(xj-xi),i=1,2,…,N.
Writing the summations as
(9)∑j=1j≠iN‍cninjaninjΓ(xnj-xni)=Γ(∑j=1j≠iN‍cninjaninjxnj-xni∑j=1j≠iN‍cninjaninj)×∑j=1j≠iN‍cijaijΓ(xj-xi)=Γ(∑j=1j≠iN‍cijaijxj-xi∑j=1j≠iN‍cijaij),i=1,2,…,N
and using that cninj=cij and aninj=aij, then, using the equations above, (8) becomes
(10)e˙i=Aniei+Wniϕσ(t-τ)+u~ni+Γ(∑j=1j≠iNcijaijej‍-ei∑j=1j≠iN‍cijaij)=Aniei+Wniϕσ(t-τ)+u~ni+∑j=1j≠iN‍cijaijΓ(ej-ei),i=1,2,…,N.
It is clear that ei=0, i=1,2,…,N, is an equilibrium point of (10), when u~ni=0, i=1,2,…,N. In this way, the tracking problem can be restated as a global asymptotic stabilization problem for the system (10).


## 4. Tracking Error Stabilization and Control Design

In order to establish the convergence of (10) to ei=0, i=1,2,…,N, which ensures the desired tracking, first, we propose the following Lyapunov function:
(11)VN(e)=∑i=1N‍V(ei)=∑i=1N‍(12∥ei∥2+∫t-τt‍(ϕσT(s)WniTWniϕσ(s))ds),e=(e1T,…,eNT)T.
The time derivative of (11), along the trajectories of (10), is
(12)V˙N(e)=∂VN(e)∂ee˙=∑i=1N‍∂VN(e)∂eie˙i=∑i=1N‍((∑j≠ij=1N‍)ϕσT(t)WniTWniϕσ(t)-ϕσT(t-τ)WniTWniϕσ(t-τ)+eiT(∑j≠ij=1NcijaijΓ(ej-ei)Aniei+Wniϕσ(t-τ)+u~ni+∑j=1j≠iNcijaijΓ(ej-ei)‍)).Reformulating (12), we get
(13)V˙N(e)=∑i=1N‍(+eiTu~ni+∑j=1j≠iN‍cijaijeiTΓ(ej-ei)-λni∥ei∥2+eiTWniϕσ(t-τ)+eiTu~ni+∑j=1j≠iN‍cijaijeiTΓ(ej-ei)+ϕσT(t)WniTWniϕσ(t)-ϕσT(t-τ)WniTWniϕσ(t-τ)+eiTu~ni+∑j=1j≠iN‍cijaijeiTΓ(ej-ei)∥ei∥2).
Next, let us consider the following inequality, proved in [9, 10]:
(14)XTY+YTX≤XTΛX+YTΛ-1Y,
which holds for all matrices X, Y∈ℝn×k and Λ∈ℝn×n with Λ=ΛT>0. Applying (14) with Λ=In×n to the term eiTWniϕσ(t-τ), i=1,2,…,N, we get
(15)eiTWniϕσ(t-τ)≤12eiTei+12ϕσT(t-τ)WniTWniϕσ(t-τ)×12∥ei∥2+12ϕσT(t-τ)WniTWniϕσ(t-τ),i=1,2,…,N.
Then, we have that
(16)V˙N(e)≤∑i=1N(-λni∥ei∥2+12∥ei∥2+12ϕσT(t-τ)WniTWniϕσ(t-τ)+eiTu~ni+∑j=1j≠iN‍cijaijeiTΓ(ej-ei)+ϕσT(t)WniTWniϕσ(t)-ϕσT(t-τ)WniTWniϕσ(t-τ)12).
By simplifying (16), we obtain
(17)V˙N(e)≤∑i=1N‍(+∑j=1j≠iN‍cijaijeiTΓ(ej-ei)-λni∥ei∥2+12∥ei∥2-12ϕσT(t-τ)WniTWniϕσ(t-τ)+eiTu~ni+∑j=1j≠iN‍cijaijeiTΓ(ej-ei)+ϕσT(t)WniTWniϕσ(t)+∑j=1j≠iN‍cijaijeiTΓ(ej-ei)12)≤∑i=1N‍(∑j≠ij=1N‍-λni∥ei∥2+12∥ei∥2+ϕσT(t)WniTWniϕσ(t)+eiTu~ni+∑j=1j≠iN‍cijaijeiTΓ(ej-ei)).
Since ϕσ is Lipschitz with Lipschitz constant Lϕσi [7], then
(18)∥ϕσ(t)∥=∥σ(xni(t))-σ(xi(t))∥≤Lϕσi∥xni(t)-xi(t)∥=Lϕσi∥ei(t)∥,i=1,2,…,N.
Applying (18) to ϕσT(t)WniTWniϕσ(t), we obtain
(19)ϕσT(t)WniTWniϕσ(t)≤∥ϕσT(t)WniTWniϕσ(t)∥≤(Lϕσi)2∥Wni∥2∥ei∥2,i=1,2,…,N.
Now, (17) is reduced to
(20)V˙N(e)≤∑i=1N‍(∑j≠ij=1N‍-λni∥ei∥2+12∥ei∥2+(Lϕσi)2∥Wni∥2∥ei∥2+eiTu~ni+∑j=1j≠iN‍cijaijeiTΓ(ej-ei))=∑i=1N‍eiT(-λniei-∑j=1j≠iN‍cijaijΓei+(12+Lϕσi2∥Wni∥2)ei+∑j=1j≠iN‍cijaijΓej+u~ni).
We define u~ni=u~ni(1)+u~ni(2), i=1,2,…,N, and then (20) becomes
(21)V˙N(e)≤∑i=1N‍eiT(-λniei-∑j=1j≠iN‍cijaijΓei+(12+Lϕσi2∥Wni∥2)ei+u~ni(1)+∑j=1j≠iN‍cijaijΓej+u~ni(2)).
Now, we propose the use of the following control law:(22)u~ni=-(12+Lϕσi2∥Wni∥2)ei-∑j=1j≠iN‍cijaijΓej,i=1,2,…,N.
Then, V˙N(e)<0 for all e≠0. This means that the proposed control law (22) can globally and asymptotically stabilize the ith error system (10), therefore ensuring the tracking of (1) by (2). Finally, the control action driving the recurrent neural networks is given by(23)uni=fi(xi)+λnixi-Wniσ(xi(t-τ))-(12+Lϕσi2∥Wni∥2)ei-∑j=1j≠iN‍cijaijΓej,i=1,2,…,N.


## 5. Simulations

In order to illustrate the applicability of the discussed results, we consider a dynamical network with just one Lorenz’s node and three identical Chen’s nodes. The single Lorenz system is described by(24)(x˙1x˙2x˙3)=(10x2-10x1-x2-x1x2+28x1x1x2-(83)x3)xi(0)=(10,0,10)T,i=1.And the Chen’s oscillator is described by(25)(x˙i1x˙i2x˙i3)=(p1(xi2-xi1)+∑j=1j≠iN‍cijaij(xj1-xi1)(p3-p2)xi1-xi1xi3+p3xi2+∑j=1j≠iN‍cijaij(xj2-xi2)xi1xi2-p2xi3+∑j=1j≠iNcijaij(xj3-xi3)‍)xi(0)=(-10,0,37)T,i=1,2,3,4.If the system parameters are selected asp1=35, p2=3, and p3=28, then the Lorenz’s system and Chen’s system are shown in Figures 1 and 2, respectively. In this set of system parameters, one unstable equilibrium point of the oscillator (25) is x=(7.9373,7.9373,21)T [11].Figure 1
Substate of Lorentz’s attractor with initial conditionX1(0)=(10;0;10)T.Figure 2
Sub-States of Chen’s attractor with initial conditionX2,3,4(0)=(-10;0;37)T.Suppose that each pair of two connected Lorenz and Chen’s oscillators are linked together through their identical substate variables; that is,Γ=diag(1,1,1), and the coupling strengths are c12=c21=π, c13=c31=π, c23=c32=π, c14=c41=2π, c24=c42=2π, and c34=c43=2π. Figure 3 visualizes this entire dynamical network.Figure 3
Structure of the network with each node being a Lorentz and Chen’s system.The neural network is selected as(26)Ani=(-1000-1000-1),Wni=(120-340023),σ(xni(t-τ))=(tanh(xni1(t-τ))tanh(xni2(t-τ))tanh(xni3(t-τ))),τ=10seconds,Lϕσi≜n=3,xni(0)=(20,20,-10)T,i=1,2,3,4.Theorem 2.
For the unknown nonlinear system modeled by (1), the on-line learning law tr{WTW}=-eTWσ(x) and the control law (23) ensure the tracking of nonlinear reference model (4), [12].Remark 3.
From (21), we have V˙N(e)≤∑i=1N‍eiT(-λniei-∑j=1j≠iNcijaijΓei+((1/2)+Lϕσi2∥Wni∥2)ei + u~ni(1)+∑j=1j≠iNcijaijΓej+u~ni(2))<0, ∀e≠0, ∀W, and therefore V is decreasing and bounded from below by V(0). Since VN(e)=∑i=1N‍((1/2)∥ei∥2+∫t-τt‍(ϕσT(s)WniTWniϕσ(s))ds), then we conclude that e, W∈L1; this means that the weights remain bounded.The experiment is performed as follows. Both systems, the delayed neural network (2) and the dynamical networks (24) and (25), evolve independently until t=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.Figure 4
Time evolution for sub-state 1 with initial stateXn1(0)=(10;0;10)T.Figure 5
Time evolution for sub-state 1 with initial stateXn1(0)=(10;0;10)T.Figure 6
Time evolution for sub-state 2 with initial stateXn1(0)=(10;0;10)T.Figure 7
Time evolution for sub-state 4 with initial stateXn4(0)=(20,20,-10)T.Figure 8
Time evolution for sub-state 4 with initial stateXn4(0)=(20,20,-10)T.Figure 9
Time evolution for sub-state 4 with initial stateXn4(0)=(20,20,-10)T.


## 6. 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 form the tracking error between the two systems. The results of the simulation clearly show clearly the desired tracking. In future work, we will consider the stochastic case for the complex dynamical network.



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*Source: 162610-2014-01-02.xml*