FINITE-TIME H∞CONTROL FOR A CLASS OF MARKOVIAN JUMPING NEURAL NETWORKS WITH DISTRIBUTED TIME VARYING DELAYS-LMI APPROACH∗

P.BASKAR

New Horizon College of Engineering,Marathhalli,Bangalore 560103,India

S.PADMANABHAN

RNS Institute of Technology,Channasandra,Bangalore 560098,India

M.SYED ALI

Department of Mathematics,Thiruvalluvar University,Vellore,Tamilnadu 632115,India

E-mail:syedgru@gmail.com

1 Introduction

In recent years,neural networks(especially switched neural networks,recurrent neural networks,Hop field neural networks,and cellular neural networks)have been successfully applied in many areas such as pattern recognition,associative memory,image processing,fault diagnosis,and combinatorial optimization.Many researchers focused on studying the existence,uniqueness,and global robust asymptotic stability of the equilibrium point in the presence of time delays and parameter uncertainties for various classes of nonlinear neural networks(see[1–6]).When a neural network incorporates abrupt changes in its structure,a Markovian jump system is very appropriate to describe its dynamics.Thus,the problem of stochastic robust stability for uncertain delayed neural networks with Markovian jumping parameters is investigated via LMI technique in[7–13].

The linear matrix inequality approach is one of the most extensively used in recent publications.For instance,in[14–18],a class of switched Hop field neural networks with time-varying delay by integrating the theory of switched systems with neural networks with time-varying delay are reported,global exponential stability conditions for switched Hop field neural networks with time-varying delay are addressed on the basis of the Lyapunov-Krasovskii functional approach,and a delay-dependent robust stability criteria are presented by employing LMIs and free weighting matrices methods.

In practical implementations,uncertainties are inevitable in neural networks because of the existence of modeling errors and external disturbance.It is important to ensure the neural networks system is stable under these uncertainties(see[19–21]).Both time delays and uncertainties can destroy the stability of neural networks in an electronic implementation.Therefore,it is of great theoretical and practical importance to investigate the robust stability for delayed neural networks with uncertainties(see[22–26]).

In the recent years,H∞concept was proposed to reduce the effect of the disturbance input on the regulated output to within a prescribed level.Hence,H∞finite time boundness for switched neural networks with time varying delays takes considerable attention[27–33].Usually,some performance indexes,that is,H∞and L∞,are employed to deal with external disturbances.Many results are developed with these performance indexes(see[34–39]references therein).

To the best of authors knowledge,there is no results available on the H∞control for Markovian jumping neutral-type neural networks(MJNNs)time varying delays.Motivated by this,in this article,we analyze the adaptive finite time stability for the MJNNs.LMI-matrixbased criteria for determining finite time stability for the MJNNNs are developed.

This article is organized as follows.In Section 2,we presents problem formulation,notations,definitions,and technical lemmas.In Section 3,stability conditions are proposed for delayed neural network systems;a reliable H∞controller is derived to guarantee the exponential stability with uncertainties of the resulting closed-loop neural network systems.Section 4 illustrative examples and comparison results are given to show the conservativeness and effectiveness of the proposed results.

2 Notations

Throughout this article,we will use the notation A＞0 to denote that the matrix A is a symmetric and positive definite matrix.Let(⊗,F,{Ft}t≥0,P)be a complete probability space with a filtration{Ft}t≥0satisfying the usual conditions(it is right continuous and F0contains all P-null sets);be the family of all bounded,F0-measurable,C([−τ,0];Rn)-valued random variables ξ={ξ(θ): −τ≤ θ ≤ 0}such thatThe mathematical expectation operator with respect to the given probability measure P is denoted by E{·}.The shorthand diag{···}denotes the block diagonal matrix. ‖ ·‖ stands for the Euclidean norm.Moreover,the notation∗always denotes the symmetric block in one symmetric matrix.Let{ηt,t≥ 0}be a homogeneous, finite-state Markovian process with right continuous trajectories and taking values in finite set S={1,2,···,s}with a given probability space(⊗,F,P)and the initial model η0. Π =[πij]s×s,i,j ∈ S,which denotes the transition rate matrix with transition probability

2.1 Problem formulation and preliminaries

Consider the following Markovian jumping neural networks of neutral type with distributed time varying delays

where x(t)=[x1(t),x2(t),···,xn(t)]T∈ Rnis the state,u(t) ∈ Rlis the control input,w(t)∈ Rnis the disturbance input which belongs to L2[0,∞),and z(t)∈ Rqis the controlled output.f(x(t))is the neuron activation function,and φ(θ)is a continuous vectorvalued initial function.For eachin which Diare a positive diagonal matrices,Ai,Bi,Ci,Di,Ei,F1i,F2i,Gi,J1i,J2i∈ Rn×nare the weight connection matrices with appropriate dimensions,and ΔAi(t),ΔBi(t),ΔCi(t),ΔDi(t),ΔJ1i(t)are uncertain real-valued matrices.The variables τ(t),σ(t),and ρ(t)represent respectively the time varying delay,distributive,and neutral delays satisfyingandand d2are positive constants,and

Assumption 1Assume that the uncertainties are norm-bounded and of the form

where M1i,M2i,M3i,M4i,M5iare known real-valued constant matrices with appropriate dimensions,Ωiare unknown and possibly time-varying matrix with Lebesgue measurable elements satisfying

Assumption 2For a given time constant Tf,the external disturbance w(t)satisfies

Assumption 3The activation functions satisfy the following condition,for any i=1,2,···,n,there exist constantssuch that

For presentation convenience,we denote

Definition 2.1([8]Finite-time stability) For a given time constant τ＞0,neural networks(2.1)is said to be stochastically finite-time stable with respect toif

where c2＞ c1＞ 0,R is a positive definite matrix,and η(t)is a switching signal.

Remark 2.2Consider neural network(2.1)with u(t)≡0 and w(t)≡0,the neural networks is said to be uniformly finite-time stable with respect toif(2.4)holds.

Definition 2.3([37]Finite-time boundedness) For a given time constant Tf,neural networks(2.1)with u(t)≡ 0 is said to be finite-time bounded with respect toif condition(2.4)holds,where c2＞ c1＞ 0,R is a positive definite matrix,η(t)is a switching signal,and w(t)satisfy(2.3).

Definition 2.4([37]Finite-time H∞performance) For a given time constant Tf,neural networks(2.1)is said to be robust finite-time H∞performance with respect toR,η(t))if the networks is finite-time bounded and the following inequality holds

Definition 2.5([33]Robust finite-time H∞control) For a given time constant Tf,neural networks(2.1)is said to be robust finite-time stabilizable with H∞disturbance attenuation level γ,if there exists a controller u(t)=Kη(t)x(t),t∈ (0,Tf],such that(i)The corresponding closed-loop neural network is finite-time bounded;(ii)Under zero initial condition,inequality(2.5)holds for any w(t)satisfying Assumption 2.

Lemma 2.6([25]) For any constant matrix M ∈ Rn×n,M=MT＞ 0,scalar η2＞ η1≥ 0,and vector function w:[η1,η2] → Rnsuch that the integrations concerned are well defined,then we have

Lemma 2.7([40]) Let U,V,W,and X be real matrices of appropriate dimensions with X satisfying X=XT,then for all VTV≤I,X+UV W+UTVTWT＜0,if and only if there exists a scalar δ＞ 0 such that X+ δUUT+ δ−1WTW ＜ 0.

3 Main Results

3.1 Finite-time boundedness analysis

In this section,we consider the problem of finite time boundedness for the following neural networks

Theorem3.1Consider the neural networks(3.1),and letIf there exist positive scalars α,λk,(k=1,2,···,7),positive definite matrices Pi,Q1,Q2,Q3,S1,S2,T,and positive diagonal matrices U1,U2with appropriate dimensions,such that the following LMI holds,

with

ProofChoose the following Lyapunov-Krasovskii functional

where

Taking the derivative of the V(t)along the trajectory of(3.1),we have

Using Lemma 2.6,we obtain

By Assumption 3,it is obtained that

can be compactly written as

Then,for any positive diagonal matrices U1and U2,the following inequalities hold true

Combining and adding(3.4)–(3.11),we obtain

here

with

Using Schur complement lemma,we have

pre-multiply and post multiply by the term of(3.14)by diagand then by using fact that,and replacingwe get Λ＜0.From(3.2),we have

Then,

On the other hand,

We obtain

then it holds that

here

By Definition 2.1,we have

According to Definition 2.3,we know that neural networks(3.1)is finite-time bounded with respect toThus,the proof is completed.

Corollary 3.2Consider neural networks(3.1)withThen,the networks is finite-time bounded,if there exist positive scalars α,λk,k=(1,2,···,7),positive definite matrices Pi,Q1,Q2,Q3,S1,S2,and positive diagonal matrices U1,U2with appropriate dimensions,such that the following LMI holds.

ProofThe proof is similar to that of Theorem 3.1,so it is omitted here.

3.2 Finite time H∞control

Consider the neural networks(2.1),under the controller u(t)=Kη(t)x(t),t∈ (0,Tf],the corresponding neural network is given by

Theorem 3.3Consider the neural networks(3.23)–(3.25),and letIf there exist positive scalars α,τ,σ,ρ,δ1,δ2,positive definite symmetric matrices Pi,Q1,Q2,Q3,S1,S2with appropriate dimensions and positive diagonal matrices U1,U2,such that the following inequalities hold,

where

ProofReplacing Di,Ai,Bi,Ci,Giin the left side of(3.26)withJ2iKi,and using Schur complement lemma,we can obtain

By Assumption 1,we have

with

By Lemma 2.7,we can get

Using Schur complement lemma,from(3.26)we obtain Ψi＜ 0.Thus,the proof is completed.

4 Numerical Example

In this section,we present numerical examples to illustrate the effectiveness and advantage of the obtained theoretical results.

Example 1Consider neural networks(3.1)with parameters as follows

X1=diag{0,0},X2=diag{−2,2}.For the given values of Tf=0.7,τ=0.3,=0.5,d1=0.6,d2=0.4,=0.8,α=1.5,c1=0.01,d=0.2,by Theorem 3.1 we know that the optimal value ofdepends on parameter α.By solving the matrix inequalities(3.2)–(3.3),we can get the optimal bound ofwith different value of α in each subsystems.The smallest bound can be obtained as=0.4315 when α=1.5.

Example 2Consider the neural networks(2.1)with parameters as follows:

M1i=M2i=M3i=M4i=M5i=0.3I,X1=diag{0,0},X2=diag{−2,2}.For the given values of Tf=5,¯τ=0.5,¯ρ=0.2,d1=0.9,d2=0.3,¯σ=1,α=4,c2=4,d=2,µ=2,R=I and taking γ2as optimized variable with a fixed α,by solving the optimal value problem for each subsystem according to Theorem 3.4,we get the following conclusion.For subsystem 1,when α ∈ [1.618,11.512],the LMI in Theorem 3.3 has feasible solution,and γmin=0.6000.

By solving the matrix equalities in Theorem 3.3,we obtain

Example 3Consider the class of Markovian jumping neural networks[41]

with parameters as follows:

X1=diag{0,0},X2=diag{1,1}.For the given values of Tf=5,=0.26,d1=0.2,=1,c1=0.5,d=2,R=I,by solving the optimal value problem for each subsystem according to Corollary 3.2,we get the minimal value of c2=1.8101.It should be mentioned that the minimal c2in[41]is 5.4296.Thus delayed Markovian jumping neural networks is stochastically finite-time bounded with respect to(c1,c2,Tf)with minimal c2smaller than that in[41].Therefore,the stability conditions proposed in this article are less conservative than[41].

Example 4Consider the class of Markovian jumping neural networks[42],

with parameters as follows:

the activation function X1=diag{0.2,0.2,0.2},X2=diag{0.4,0.4,0.4}.For the given values of Tf=5,d1=0.2,c1=0.5,c2=0.3,d=2,R=I,by solving the problem for each subsystem according to Corollary 3.2,we get the upper bound of¯τ=4.450,whereas for the upper bound of¯τ=2.5 in[42],it shows that delayed Markovian jumping neural networks is stochastically finite-time bounded with respect to(c1,c2,T).Hence the results in this article are less conservative than that in[42].

Conclusion

In this article,we investigate the problem of robust finite-time H∞control for a class of uncertain neutral-type neural networks with distributed time varying delays.The finite-time boundedness analysis and finite-time H∞controller design for neural networks system with H∞disturbance attenuation level γ are studied.Numerical examples are provided to show the effectiveness of the proposed method.The results are compared with the existing results to show the conservativeness.

Figure 1 State trajectories of the system in Example 1

Figure 2 State trajectories of the system in Example 2

Figure 3 State trajectories of the system in Example 3

Figure 4 State trajectories of the system in Example 4

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