Passivity Analysis of Impulsive Complex Networks

International Journal of Automation and Computing8(4), November 2011, 484-489DOI: 10.1007/811633-011-0607-zPassivity Analysis of Impulsive Complex NetworksJin-Liang Wang'Huai-Ning Wu1Zhi-Chun Yang2Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University,Beijing 100191, PRC2College of Mathematice Science, Chongqing Normal Universit, Chongqing 400047, PRCAbstract: In this paper, we first investigate input passivity and output passivity for a class of impulsive complex networks withtime varying delays. By constructing suitable Lyapunov functionals, some input passivity and output passivity conditions are derivedfor the impulsive complex networks. Finally, an example is given to show the effectiveness of the proposed criteria.Keywords: Impulsive complex networks, time varying delays, input p88sive, output passive, Lyapunov functional.1 Introductionsystems and neural networks, and many interesting resultshave been derived!5-12. Song et al.l8l investigated the pas-In the past ten years, there have been much activity tosivity for a cla8s of discrete-time stochastic neural networkswards studying the topology and dynamical behavior ofwith time varying delays. A generalized activation functioncomplex networks across manyfields of science and enwas considered, where the traditional assumptions on thegineering, such as power grids, communication networks,boundedness, monotony and differentiability of the activa-and Internet, the World Wide Web, metabolic systems,tion functions were removed. A delay-dependent pa88ivityfood webs, etc. One of the most remarkable phenomenacondition was derived in terms of linear matrix inequali-in the dynamics of complex networks is their spontaneousties (LMIs) by constructing a proper Lyapunov-Krasovskiisynchronizationl, which has been extensively investigated.functional and stochastic analysis technique. To our knowl-It is well known that impulsive effects are common pheedge, there are few studies on the passivity of complex dy-nomena in many systems such as computer networks, au-namical networks(13,14, in which Yao et al. obtained sometomatic control systems, signal processing systems andsuficient conditions on passivity properties for linear (ortelecommunications. Therefore, the study of complex netlinearized) complex dynamical networks with and withoutworks with impulsive effects is important for understandingcoupling delays (constant delay). However, in the practi~the dynamical behaviors of real networks. Recently, syn-cal evolutionaryprocesses of networks, absolute constantchronization of impulsive complex dynamical networks hasdelay may be scarce and delays frequently vary with time.been investigated by researchers. For instance, Cai et al.lTherefore, such simplifcation does not match the peculiari-investigated robust impulsive synchronization of complexties of real networks in many circumstances, and it is neces-delayed dynamical networks with nonsymmetrical couplingsary to further investigate the passivity properties of com-from the view point of dynamics and control. Some criplex delayed dynamical networks. In [11], Chen and Leeteria for robust impulsive synchronization were establishedfirst developed input passivity to study both input-outputand internal stability problems for feedback connections ofsystems. In [3], Li and Lai studied the adaptive impulsivediscrete-time linear descriptor systems. By applying thesynchronization of uncertain complex dynamical networks.input passivity and Lyapunov theory, Chang et al.L2] de-On the basis of the stability analysis of impulsive system,rived the relaxed stability conditions to guarantee the sta-several network synchronization criteria for local and globalbility and passivity property of closed-loop systems. Toadaptive-impulsive synchronization were established. Zhouthe best of our knowledge, the input passivity of impul-et al.(4) investigated the synchronization of complex delayeddynamical networks with impulsive effects. Some criteriabeen established. Therefore, it is important and interestingwere derived for global synchronization based on the im-to study the input passivity of impulsive complex networkspulsive stability theory.with time-varying delays.On the other hand, passivity is also an important conceptMotivated by the above discussions, we investigate theof system theory and plays an important role in both elec-input passivity of complex delayed dynamical networks withtrical networks and nonlinear control systems, and providesimpulsive effects. As a natural extension of input passivity,a nice tool for analyzing the stability of systems. Therefore,we also introduce the output passivity in order to studythe passivity theory has received a lot of attention. In recentthe dynamical behavior of impulsive complex networks in ayears, some researchers have studied the passivity of fuzzymuch better way.The rest of thThis work was supported by National Natural Science FaSection 2, ou中国煤化工opulsive complexdation. of. China_ (No. 10971240，No. 6100404，No 61074057),networks withCN M H Gsented and some(No. CSTC2008BB2364)，Foundation of Science and Technologypreliminaries ac g:YH∈ai ipui auu output passivityProject of Chongqing Education Commission(No. KJ080806), Fun-criteria are established in Section 3. In Section 4, a numer-nd& for the Central Universities, China(No. YWF- 10-01-A19).ical example and its simulation are given to illustrate theJ. L. Wang et al. / Passivity Analysis of Impulsive Complex Networks485effectiveness of the theoretical results. Finally, conclusionsfor all tp≥0 and for all solutions x(t,0) of (1).are presented in Section 5.Definition 2Impulsive complex network (1) iscalled output passive if there exists a scalar ry > 0 such that2 Networks model and preliminariesLet R+ = [0, +∞), R" be the n-dimensional Euclideanspace and rnxm be the space of nx m real matrices.P∈Rnxn≥0(P∈R"xn≤0)meansthatmatrixPIt is well known that the matrix inequality is an impor-is symmetrical and semi-positive (semi- negative) definite.tant tool for studying the behavior of dynamical systems.P∈R"xn>0(P∈Rnxn<0)meansthatmatrixPisWe also give the following lemma on matrix inequality.symmetrical and positive (negative) definite. In denotesLemma 1. For any vectors r,y∈R" and n x n squarethe n x n real identity matrix. BT denotes the transposematrix W > 0, the following matrix inequality holds:of a square matrix B. C([- -r,0)],R") is a Banach space ofcontinuous functions mapping the interval [-r,0] into R"xTy+y"x≤xTWx+y"W~'ywith the norm |l| = sup_τ≤e≤o |(0)|, wherel|.ll is theEuclidean norm.3 Main resultsIn this paper,we consider a dynamical network consist-In this section, we shall investigate the input pasivitying of N identical nodes with difusive and delay coupling,in which each node is an n-dimensional dynamical system.and output passivity of impulsive complex network.The mathematical model of the impulsive complex networkIn (15, 17], the authors make the following assumption onthe function f(-), which is called "Function class QUAD".can be described 88 follows:Assumption 1. There exists a positive definite diago-nal matrix P = diag{p1, p2, ... , Pn} and a diagonal matrixi(t)= f(x())+ E 2 LqjTxj(t- +()+E = diag{E1,E2,... ,En} such that f :R"→R”satisfiesBru(t),tttk,(1)the following inequality:yi(t) = Citi(t) + Di:ue(t),(x-y)T P[f(x)- f(y)-e(x-y川≤Oxi= Ie(xi)，t=tk-n(x-y)"(x-y)where i = 1,2,... ,N, k = 1,2,.. ru(t) is the time-for someη> 0 and allx,y∈R".varying delay with 0≤Tu(t)≤T,l= 1,2,.. ,n.It can be verified that Assumption 1 holds for many of theThe function f()， describing the local dynamics ofbenchmark chaotic systems including the Lorenz system,the nodes, is continuously differentiable and capable 。the Chen system, the Lv system and the unified chaoticproducing various rich dynamical behaviors, xi(t) =system. In this paper, we assumef(0)=0, and let y= 0,(zi1(t), xi2((t) ...., cin(t)T∈R”is the state variableof node i, and x(t-T:() = (xi1(t - Trl()), xi2(t -Assumption 2. There exists a positive definite diago-nal matrix P = diag{p1, pz, ... , pn} and a diagonal matrixnode i, and u:(t)∈R”is the input vector of nodei, B, Ciε = diag{ε1,E2, ... , En} such that f satisfes the followingand Di are known matrices with appropriate dimensions,inequality:T∈Rnxn describes the coupling relations between twonodes, a is a positive real number which represents the over-xT P(f(x) -ex)≤-nxTxall coupling strength, k; is the degree of node i and w isfor someη> 0 and all x∈R”a tunable weight parameter, the real matrix L = (Li)NxN .For convenient analysis, we letis a symmetric matrix with diagonal entries Lui= -ks andoff-diagonal entries Lij = 1 if node i and node j are con-Gij=，E= diag{e,E...,e}nected by a link, and Lij = 0 otherwise. The fixed mo-9mentstksatisfy0=to 0, Q= diag{Q1, Qz,.. ,QN},Qi =sponding trajectory, wizero initial condition.diag{9i1, 9i2, ... . Qin},Qu .> 0. and a positive constant 7Defnition 113, 14.161. Impulsive complex network (1) issuch that中国煤化工called input passive if there exists a scalar γ > 0 such that-2nInN +2YHCNMHGB-CT)T+a2P(G8T)Q-'(G8T)TF≤0(21-.oσ486International Journal of Automation and Computing 8(4), Novermber 2011D+DT -γInN-z≥0(3)By (2), (3), (6)-(8), we obtainwhere i=1,2,... ,N,l= 1,2,.. ,n, k=1, 2,...V(t) - 2y"()u(t) + ruT (t)u(t)≤Proof. First, we rewrite network (1) in a compact formx"(t)[- 2nInN +Q + 2Pe+(PB-cT)z-'(PB-as follows:CTT+ P(G&r)Q -'(G⑧DFPr(t)≤。<(t) = F()) + a(G8 T)x(t- r()) + Bu(t)1-σt≠tk，t≠te.(9)(4)y(t) = Cx(t) + Du(t)By integrating (9) with respect to t over the time periodOx = Ir(x),t= tk0totp(t≤tpxing，construct Lyapunov functional for system (4) asfollows:u()ds+ (.' "V(a)- 2()u()+V(t) =x(t)Px(t) +年1 EJt- Tri(t)quxu(a)da. (5ru"(s)u(8)]ds≤0.Furthermore,The derivative of V(t) along the solution x(t) of system(4) is given as follows:f。"IV() - zy()() + x()()ds=V(t)≤2x"(t)Pr(1) + x"(t)Qx(t) -(1 -o):(t-+()) Qx(t-+(t))=if* Vld+f," v()ds+2x"(t)PF(x(t)) + 2xT ()PBu(t) +2ax~(t)P(G 8 T)z(t- +()) +xT(t)Qx() -。”[-2w()u(x)+ r ()()]ds=(1-o)(t-()) Qx(t-())，t≠tk.EV(5)- V(1-1)]+ (V(z)- V(u)]-Then, we haveV() -2y"(t)u(t) + ru"(t)u(t) <2[" v()u(o)ds+r [" u()u()ds=2xT (t)PF(x() + 2xT ()(PB -CT)u(t) +2ax' ()P(G 8 r)(t - (1) + xT()Qx(t)-EIV(t)- V(t2)+ [V(t)- V(O) -(1-o)(-()) Qx(t-r()) -u"()(D+ DT - γIn)u(),t≠th.2]。”r()()ds+心]。u(ode.According to Assumption 2 and Lemma 1, we getTherefore, it follows that(x()F(x()==12f。” r()()d> Ev[V(tT)- V()]+ .[V(tp)- V(0)]+2[-nxT ()x() + x? (t)ex:(t)=r/("(oul()de.xT (t)(-ηInN + Pe)x(t)6)On the other hand,2axT(t)P(G@ r)x(t-T(M≤V(tx)=2撬T(幻)Px(t)+(1 - o)x(t- ()) Qx(t-r()) +a2rT(t)P(G r)Q-(G r)TPx(t)M pe7)au;xH(a)da.中国煤化工2x"(t)(PB - CT)u(t)≤r"()(PB-CT)z -\(PB-CT)"<(t)+Because r;YHCNMHGuT (t)Zu(t).8)V(tk)≤V(k).J. L. Wang et al. / Passivity Analysis of Impulsive Complex Networks487Let x(t) = x(t, 0) with the zero initial condition for anyx()(PB-CT +rCT D)z~'(PB-CT+γCTxgiven u(t).From the construction of V(), we haveV(0) = 0D)F+ aP(GQr)q-(G&D)TP- 2nInN +2Pe +and V(tp)≥0. Thus,1-σQ +rCTC]x(t)≤0.(13)2f["()()ds>By integrating (13) with respect to t over the time period0totp (t≤tp()-2u"()u()+ r()()ds++r，u"(e)u()ds[. ' (V(a)- 2("a)u()+ r"()(o)]da≤0.for alltp≥0.Theorem 2. Let Assumption 2 hold, ru(t)≤σ 0,Q =["IV()- 2o()u() + r(ny()ds=diag{Q1, Q2,.. , QN}, Qi = diag{qi1, q62, ... , 9in},qi >0, and a positive constant ry such that二广voeda+ /(" v()da+ .(PB-CT +rC"D)z-(PB-CT +xCTD)T+a?P(G@r)Q-'(Ggr)TP- 2nInN + 2PE+[ 1-2)y()(6) + ry(y(a)ds=Q+γCTC≤0(10)EIV(t7)- V(-1-1)]+ IV(t)- V(t)] -D+DT-γDTD-Z≥0,(11)2[" v()u(o)ds+r f”y"()y()d=where i= 1,2,... ,N, l=1,2,... ,n, k= 1,2,.. .Proof. Constructing the same Lyapunov functional asE1V(5)- V(&)] + [V(tp)- V(0)]-(5) for system (4), we getV(t) < 2xT()PF(x()) + 2xT()PBu(t) +2[" y"()u()do+r ["F()()d.2ax'(t)P(G 8 r)x(t - r()) + x"(t)Qx() -(1 -o)x(t -()) Qx(t-()).Therefore, it fllows thatThen, we have2[" y()u()ds≥ZV(C)- V()]+V(t) - 2y"(t)u(t) + ry"(t)y(t)≤2x"(t)PF()) + 2xT(t)PBu(t) +[V(p)- V(0)]+2axT (t)P(G⑧r)x(t - r()) +xT (t)Qx(t) -rf° v(yv()de(1 -)x(t- r() Qx(t- r(I) - 2x"(t)CTu(t) -uT()(D + D")u(t) + r([Cx() + Du(t)]TOn the other hand,[Cx(t) + Du()] =V(tx)='场(tF)Px;(tF)+2x"(t)PF(x()) + 2axT(t)P(G 8 r)x(t- r(切+2xT(t)(PB-CT + rC' D)u(t)+Qux对(a)da.xT(t)(Q + rCTC)x(t)- (1-σ)x(t-r()T xru(t )Qx(t-t()) - uT(t)(D + DT- γDT D)u(t).Because r场≤1 for all i and k, we haveAccording to Lemma 1, we obtainV(tk)≤V(t%).2xT(t)(PB- cT + rCT D)u(t)≤Let x(t) = x(t,0) with the zero initial condition for anyxT(t)(PB-CT + rCTD)z-1(PB-cT+rCTD)"() +u[()Zu(). (12)given u(t). Fro中国煤化工haveV(0)=0and V(x)≥0.By (6),(7),(10)- (12) we getTYHCNMH G2[ y"()u(a)ds≥ 2 C[V(t7)- V(2)]+V(t) - 2y"(t)u(t) + rv"()y(t) <188International Journal of Automation and Computing 8(4), November 2011[V(z)- V(0)]+ .0.4r/%。”r()()d>0.3s 0.27f。”r(ny(odoofor alltp≥0.Crif -0.14 Example-0.2In this section, we give an example and its simulation to-0.3show the effectiveness of our obtained theoretical results.Consider a complex dynamical network, in which each2.02.5 3.0node is a three-dimensional nonlinear system described by ;Time (s)0.0510(x2 - x1)0.04i2| =| 2x1 - x1x3 - 10x20.03$1Z2 -号工3 .0.020.01Takea=0.1, Bu=1, D:= Is and0.3 0.2 0.1-0.02r=| 00.1 0.2-0.030.1 0.2 0.20.1 0.2 0.3-0.05B:=Ci=| 0.2 0.4 0.20.:1.2.53.00.3 0.2. 0.60.Clearly, we can takeη=g，P = diag{1,1,1}, e = diag{0,0,0}.The matrix L is chosen as-0.1受-0.2-0.42.Fig.1 Time response of the input variables, state variables andoutput variables in the impulsive complex network, where thenetwork sizeis N = 10.Setrik=号, tk=0.1k, Tu(t)= 1-击e-;, then we have0≤rik≤1,0≤Tu()≤τ=1, and iu(t)= e≤{<5 Conclusions1,fort≥0, i=1,2...,10, l= 1,2,3.We can fnd two matrices Q = 0.5I30，= L30, stistsingWe have studied the input passivity and, output passiv-), (3), (10) and (11) with 2hγ:; A30According to Theoremsity of impulsiv中国煤化rime varying de1 and 2, we know that network (1) with the above givenlays.. Some inassivity criteriaparameters is input passive and output passive. Then, sethave been eYHC N M H Gpropriate Lyau(t) = [in(t)/(i+2), sin(πt)/(i+ 3), sin(πt)/(i+ 4)]T andpunov functionals. An illustrative example was presentedthe simulation results are shown in Fig. 1.to show the efficiency of the derived results.