State Reconstruction for Complex Dynamical Networks with Noises *

The state reconstruction problem is addressed for complex dynamical networks coupled with states and outputs respectively, in a noisy transmission channel. By using Lyapunov stability theory and H∞ performance, two schemes of state reconstruction are proposed for the complex dynamical networks with the nodes coupled by states and outputs respectively, and the estimation errors are convergent to zeros with H∞ performance index. A numerical simulation demonstrates the effectiveness of the proposed observers.


Introduction
Complex dynamical networks have recently been a hot topic in science and engineering fields because it can describe many phenomena in nature and engineering [1][2][3][4].For instance, power grid is a complex network if the electrical equipments are treated as nodes and the interactions between the equipments as edges; the individuals are treated as nodes and the interactions between the individuals as edges in a community then the community can be described as a network.
Recently, topology identification, fault diagnosis and parameter identification [13][14][15][16][17][18][19][20] of complex networks have become hot topic in complex network applications, and network synchronization has found applications in these fields.For topology identification, it is assumed that all of the states are available for a complex network in [15,[17][18][19][20].For monitoring topology change of network, it is assumed that the partial state variables are required for the network in [14,16].
In the above study and other fields of complex net-work, all or partial state variables are needed for design.However, for a large scale network, measuring all state variables is not easy or even impossible in practice, and locating many sensors costs much.Therefore, it is very important to estimate or reconstruct all state variables based on some limited available network information.
For discrete complex networks, state reconstruction has been reported in [21].For continuous time complex dynamical networks with transmission noise, there has been little theoretical work on state estimation in the literature.Motivated by the above observations, in this paper we study the state reconstruction or state estimation problem for a complex network with transmission channel noise.By using Lyapunov stability theory and H ∞ control theory, some state reconstruction schemes are derived for complex dynamical networks including state coupling and outputs coupling.For suppressing noise in the channel, the integral observers [22,23] are applied and the estimation errors are bounded with H ∞ performance.Some numerical examples are given to shown the effecttiveness of the proposed schemes.
The rest of this paper is organized as follows.In Section 2, the state reconstruction of state coupling networks is studied and some estimation criteria are derived in the form of linear matrix inequality.In Section 3, the state reconstruction of output coupling networks is studied and some estimation criteria are given.Some examples are given in Section 4 and conclusions are drawn in Section 5.

State Reconstruction of Networks Coupled with States
ing of N identical nodes with states couplings, which is described as follows where , , , is the output vector of the i th node, A R   n n is the system matrix of node i, is the coupling matrix of node i, node dynamics is is the coupling configuration matrix.
If there is a link from node i to node , then ; otherwise .Assume that C is a diffusive . It is noted that the con-  figuration matrix C does not need to be irreducible and symmetric.
Hypothesis 1: (H1) Suppose that   f x is Lipschitz continuous.That is, there exists a positive number constant α satisfying represents the Euclidean normal.For most networks, all of the states are generally not available.To reconstruct the states of network (1), outputs i are transmitted from (1) to the observer through the transmission channel.In the practical engineering, there exists noise in the transmission channel.Therefore, the measurements received by the observer are characterized by where m i y R  is the actual measurement outputs and are the noises in the transmission channel.To reconstruct all the states of (1), the following observer is presented   and are the observer gain matrices to be determined.
Remark 1: The observer (3) is different from the tradiional proportional observer because its controller is the t integral of the measurements [22,23].The states reconstructed by (3) can better converge to the states of (1) since the disturbance is not amplified if a large proportional gain i K is used [8].The aim is to determine appropriate observer gain matrices and such that the reconstructed states x approach the network states i x .Define the state errors ˆ1, 2, , Then it follows from ( 1) and ( 3) that and .Then ( 5) can be rewritten as Then we will derive i K to guarantee that i are convergent to zeros when E 0 i w  , and i are convergent to zeros with E H  performance  , characterized by the following inequality, when and 0 Theorem 1: Suppose that H1 and H2 hold.If there exist matrices 0  N and a constant 0   such that the following inequality holds where , , , then the error dynamical system (6) will converge to zeros with H ∞ performance γ when  .Consequently, network (3) can estimate the state of network (1) with H ∞ performance γ when 0 i w  and Proof: Define a Lyapunov function .
1 Differentiating V along the error dynamical system (6) nd using Hypothesis 1, one obtains

V E PB B P DK P PK D E E PF x x E P c GE c GE PE E PDw E PB B P DK P PK D I P I DD P E w w E P c GE c GE PE
Copyright © 2012 SciRes.IJMNTA Define .Using (8), one obtains From ( 10) and Lyapunov stability theorem, are exponentially convergent to zeros when i .Under , integrating (10) The proof is completed.
To easily solve the matrix inequality, Schur complements lemma [9] is used here.Then ( 8) is transformed into the following linear matrix inequality where , and .

State Reconstruction of Networks Coupled with Outputs
Next, we consider a complex dynamical network consisting N identical nodes coupled with the outputs characterized by where .
n m L R   The outputs i are disturbed by noise when they are transmitted from network (12) to the observer.Therefore information received by the observer is characterized by (2).The observer is designed as the following form where are the observer gain matrices to be determined.Then one can obtain the error dynamics He w le Let , then ( 14) can be rewritten as where 0 0 0 . Consequently, atrices determine m i K such that error dynamics ( 15) is H  stable with pe ance, that is, observer (13) re-rform f construct the states o network ( 12) with H  performance.The fo ing theorem gives the criteria of determining matrices i llow K .Theorem 2: Suppose that H1 and H2 hold.If there exist matrices P P 0  and a constant 0   such that the following inequality holds where , ,

T T T M D PB B P I P I DD P I
1 then the error dynamical system (15 the zeros with H  performance γ when 0 i w  .Consequently, observer (13) can estimate the state of network ( 12) with H  performance γ when 0 i w  and . The proof of Theorem 2 is sim r to that of Theorem 1, omitted ila so it is here.Using Schur complement lemma, one obtains the following linear matrix inequality , , diag P P    .Remark 2: In this section, the complex dynamical netw ed ork coupl with the outputs is considered because this kind of networks is practical in engineering for saving communications and sensors.When the transmission channel is ideal, the observer (13) can reconstruct the states of network (12) with exponential convergence.When the transmission channel is noisy, the observer can reconstruct the states of network (12) with H  performance.

Numerical Simulations
ples are given to demsed state reconstruct-In this section, some numerical exam onstrate the effectiveness of the propo tion scheme for complex dynamical networks.In the network, chaotic Lorenz system is taken as the node dynamics.
Lorenz chaotic system is a well-known typical benchmark chaos, which can be described by the following [10] where a, b and c are parameters.When a = 10, b = 28 and .Then one gets the following Then the Lorenz system satisfies the Hypothesis H1.
The complex dynamical network is assumed to contain by G (H2) Suppose that the disturbances in the transmission channel are bounded, i.e., there exists a positive constant d such that i w d  .

1
The simulation results are shown in Figure1.From Figure1, one can see that the error dynamics converge to zeros although there is transmission noise.In this paper, we study the problem of state reconstructtion for a complex dynamical network under noisy circumstances.The state coupling network a coupling network a in the transmission and estimation errors with H ∞ performance index are obtained.Some examples are given to demonstrate the effectiveness of the proposed scheme.