1、Proceedings of the 6th International Conference on Differential Equations and Dynamical Systems, (2009) 200206DCDIS A Supplement, Copyright c 2009 Watam PressSTABILIZING NON-DIAGONALIZABLE NETWORKSLinying Xiang1, Zengqiang Chen1, Zhongxin Liu1, Kai Chang2 and Zhuzhi Yuan11Department of Automation, N
2、ankai University, Tianjin, 300071, China, Email: 2Department of Automatic Control, Beijing Institute of Technology, Beijing, 100083, ChinaAbstract. The V-stability problem is investigated for non-diagonalizable networks with nonidentical node dynamics. Pin-ning control is suggested to stabilize such
3、 dynamical networks. Thecomplicated stabilization problem is reduced to measure the semi-positive property of a characteristic matrix which embodies not onlythe network topology, but also the node self-dynamics and the feed-back control gains. The network controllability is defined in termsof the sm
4、allest eigenvalue of the characteristic matrix. Numericalsimulations of two representative non-diagonalizable networks com-posing of non-chaotic systems and chaotic systems, respectively, areshown for illustration and verification.Keywords. Complex network, Non-diagonalizable network, Pin-ning contr
5、ol, Stabilize, V-stability, Controllability.AMS (MOS) subject classification: 93C15, 93D15.1 IntroductionThe last decades have witnessed the birth of a new move-ment of interest and research in the study of complexnetworks 1-15. This is partially due to the fact that anylarge-scale and complicated s
6、ystem in nature and soci-eties can be modeled by a complex network, where ver-tices are the elements of the system and edges representthe interactions between them. Examples of complex net-works in real-world life include the WWW, the Internet,metabolic networks, neural networks, food webs, electri-
7、cal power grids, social networks, and many others. Thenew discoveries of small-world phenomena 1 and scale-free features 2 in such natural and artificial complexsystems have attracted increasing attention on the com-plexity of graph topology by scientists from various fieldsof science and engineerin
8、g, yielding fruitful studies whichenrich and deepen the understanding of real networkingsystems.Very recently, more and more attempts have been paid tocontrolling the aggregate dynamics of a complex networkand guiding it to a desired state, where pinning controlwas proposed as a viable and attractiv
9、e control scheme9-15. The essential ideal behind pinning control is a self-feedback action with a limited number of controllers plac-ing at the nodes. It expends less cost on the network thanthe way to control all the nodes. In most realistic net-works, the dynamics placed at the nodes may be differ
10、ent.For instance, in social networks, different nodes denotepeople with different nationalities, areas, age and earn-ing. The subjects of synchronization and control for suchnetworks have received much less attention because ofsome technical difficulties. More recently, the V-stabilityapproach, intr
11、oduced in 14, has been successfully ap-plied to synchronizing and controlling symmetrical net-works with nonidentical node dynamics. In Ref.15, theV-stability of asymmetricalnetworks with different nodeswas addressed by introducing an extra virtual node (orreference node) into the original network.
12、The networksconsidered therein 14-15 are all assumed diagonalizable.However, most optimal networks are non-diagonalizable(A network is diagonalizable if the corresponding Lapla-cian matrix is diagonalizable. Otherwise, the network iscalled non-diagonalizable 7), in particularly when thenetworks are
13、directed. Hitherto, research on stabilizingnon-diagonalizable networks is quite rare and is thereforeof great interest and importance.In this paper, an extension of the original V-stabilityframework to the case of non-diagonalizable networkswith different node dynamics is developed. Pinning con-trol
14、 as a popular and desirable scheme is still appliedto stabilize the network. A quantity to assess the con-trollability of the pinned network, i.e., its propensity tobeing controlled onto a given reference state, is derivedin terms of the smallest eigenvalue of the characteristicmatrix. It is founded
15、 that the larger the smallest eigen-value is, the more the network is pinning controllable.Note that differing from the Laplacian or adjacent ma-trix characterizing the network topology in most previ-ous eigenvalue-based studies 4-6, the characteristic ma-trix herein encompasses not only the network
16、 topologyin terms of the connections and the weights over them,but also the node self-dynamics (described by passivitydegrees) and the control gains over them.The rest of the paper is outlined as follows. A non-diagonalizable network composing of nonidentical nodesis presented and some preliminaries
17、 are introduced inSection 2. In Section 3, the V-stability conditions areobtained and an index characterizing network controlla-bility is given. Some numerical simulations for demon-200strating the effectiveness and feasibility of the proposedV-stability criterion and control scheme are given in Sec
18、-tion 4. Finally, Section 5 concludes the investigation.2 Model Description and Prelim-inariesConsider a network consisting of N different nodes inter-acting through a diffusive-type coupling, with each nodebeing an m-dimensional dynamical system, described byxi = fi(xi)Nsummationdisplayj=1Lijxj, i
19、= 1,2, ,N, (1)where xi Rm represents the state vector of the ithnode, and the function fi(), governing the self-dynamicsof node i, is capable of producing various rich dynamicalbehaviors, including periodic orbits and chaotic states.The parameter is positive ruling the overall couplingstrength. Also
20、, Rmm is a constant matrix linkingcoupledvariables, whilethe realmatrixL = (Lij) is calledthe Laplacian matrix of the directed weighted network,satisfying zero row-sum. The topological information onthe network in terms of the connections and the weightsis contained in the Laplacian matrix L, whose
21、entries arezero if node i is not connected to node j (j negationslash= i), but arenegative if there is a directed influence from node j tonode i. In addition, L is not necessarily diagonalizableand symmetric because the network is not constrained tobe undirected and unweighted.Note that (1) encompas
22、ses a general and large class ofnetworks, allowing for nonlinearity in the function f andleaving complete freedom for the choice of the weightsover the network connections. The constraint is repre-sented by the condition Lii = summationtextjnegationslash=i Lij, which is nec-essary in order to ensure
23、 that the Laplacian matrix L iszero row-sum.Assume that all the network nodes have a commonequilibrium x, satisfying fi(x) = 0 for i = 1,2, ,N.The task here is to stabilize network (1) onto the refer-ence state x1 = x2 = = xN = x.Before proceeding with the analytic treatment, the fol-lowing assumpti
24、on and Lemma introduced in 14 aremade throughout of the paper.Assumption 1 There exists a continuously differen-tiable Lyapunov function V(x) satisfying V(x) = 0 withx D, such that for each node function fi(xi), there is ascalar i called passivity degree holdingV(xi)xi (fi(xi)i(xxi) 0, D =Nuniontext
25、i=1Di.Lemma 1 Define D = D1 D2 DN RmN.Consider the following Lyapunov function for the network(1):VN(X) =Nsummationdisplayi=1V(xi), X = (xT1 ,xT2 , ,xTN)T. (3)If there exists a function G(X) 0 such that VN(X) G(X) for all X D 0, then the network (1) is locallyasymptotically stable about its equilibr
26、ium point. More-over, the region of attraction is given by = X : VN(X) 0actingon nodeiis need tobe designed.Thus, the self-dynamics of the pinned nodes becomes xi =fi(xi)ki(xi x), i = 1,2, ,l. Worth noting herethat the networks are divided into two different layers ofdynamical nodes: the un-pinned n
27、odes and the pinnedones. In particular, the latter play the role of networkleaders leading the entire network toward a given desiredstationary state. The direct control action is propagatedto the rest of the network through the coupling among thenodes. Naturally, the controlled networkcan be describ
28、edby the following set of equations:xi = fi(xi)ki(xi x)Nsummationtextj=1Lijxj,i = 1,2, ,N, (5)with ki 0 for i = 1,2, ,l and ki = 0 otherwise.Theorem 1 Assume that there exists functions V(xi) =12xTi Qxi,i = 1,2, ,N, with Q being a symmetric andpositive definite matrix, satisfying Assumption 1 with p
29、as-sivity degree value i, such that the following inequalityholds:Q+TQ 0. (6)Then, the controlled network (5) is V-stable if the follow-ing inequality is satisfied:+K + 12(L +LT) 0, (7)where = diag(1,2, ,N) RNN and K =diag(k1,k2, ,kN) RNN. Moreover, if D = RmN,the above stability is global.201Proof.
30、 Without loss of generality, assume X = 0 and con-sider the following Lyapunov function for the controllednetwork (5):VN(X) =Nsummationdisplayi=1V(xi), X = (xT1 ,xT2 , ,xTN)T. (8)Its time derivative along trajectory X is given byVN(X) =Nsummationdisplayi=1V(xi)xidxidt=Nsummationdisplayi=1V(xi)xi (fi
31、(xi)ki(xi x) Nsummationdisplayi=1V(xi)xiNsummationdisplayj=1Lijxj. (9)It is easy to say that VN( X) = 0 and VN( X) = 0. Also,Assumption 1 implies that, for X negationslash= X, one hasVN(X) 0 (i = 1,2, ,N), the networkitself is always stable without external control. It can alsobe seen that the large
32、r the passivity degree value i, themore likely the condition (7) is held.Remark 3 When L is symmetric, the stability condition(7) is consistent with Theorem 2 given by 14.Remark 4 Let j is the set of eigenvalues of the char-acteristic matrix C, ordered in such a way that 1 2 N. Similar to the concep
33、t of network syn-chronizability defined in terms of the range of the globalcoupling strength for which the network synchronizes4, here, a scale ”1” denoting the smallest eigenvalueof the characteristic matrix is presented to evaluate thenetwork pinning controllability, for which the referencestate x
34、1 = = xN = x is stable. Note that this defi-nition is dependent of the choice of the function f, quitediffering from the definition given in Ref.12, where allthe dynamical systems at the network nodes are identical.Specifically, the larger the 1 is, the more the network ispinning controllable. The n
35、etwork controllability can beenhanced by an appropriate choice of the reference nodesand of the feedback control gains over them.4 Numerical Simulations4.1 Non-chaotic NetworksTo validate the theoretical results, consider a simplenon-diagonalizable network with four different nodes, asshown in Fig.1
36、. The node local dynamics are given by14x1 = x12, x2 = 3x2, x3 = sin(x3), x4 = |x4|.(12)Obviously, x = 0 is the common equilibrium point. Notethat the function in the 4-th node is not differential at theequilibrium, so any linearizationmethod is not preferable.However, the limitation of linearizatio
37、n is avoided by in-troducing an important scale, , as a measure of the effectof the node self-dynamics on the network stability.Take the common Lyapunov function V(X) = XTX,then the correspondingpassivity degrees of the four nodesare 1 = ,2 = 2.999,3 = 1.001 and 4 = 1.001with 0. The correspondingfea
38、sible domains areD1 =x1 : |x1| and D2 = D3 = D4 = R. The region ofattraction is then given by = (x1,x2,x3,x4) : x21 +x22 +x23 +x24 2. (13)Thus, the characteristic matrix of the controlled networkcan be rewritten asC = +K + 12(L +LT), (14)where = diag(,2.999,1.001,1.001) andL =0 0 0 01 1 0 01 0 1 00
39、1 0 1.202Note that all the eigenvalues of C are real and sorted as1 2 3 4. In the sequel, the effects of the cou-pling strength and the control gain on its controllabilityare discussed.The relation between 1 and in the network withoutcontrol is plotted in Fig.2. It is clearly observed that thenetwor
40、k is unstable without external control regardlessof and . In order to stabilize the network, pinningcontrol is the best choice.In view of = 10.0001, the node 1 is selected to bepinned and the control law is designed asu1 = kx1. (15)As a comparison, the smallest eigenvalue 1 has beenplotted versus k
41、and corresponding to different andk respectively in Figs.3 and 4. It is concluded that in-creasing or k can enhance the network controllability.Specifically, an interesting phenomenon is found that theincrease of 1 is shown to saturate when k is increasedto above a certain value, i.e., further incre
42、ase in the con-trol gain does not lead to further improvements in thenetwork controllability. Notice again that this network isunstable without external control (see the changes of 1when k = 0 in Fig.4), which is in accordance with theobservation in Fig.2.Figure 5 shows the evolution process of the
43、networkstates. The networkdivergesfromthe initialstateX(0) =(0.5,2,1,3)T without control but converges quickly tothe origin after control (k = 20).4.2 Chaotic NetworksIn this subsection, a chaotic network is simulated to fur-ther illustrate the proposed V-stability scheme.Consider a 22-node non-diag
44、onalizable network (asshown in Fig.6) consisting of two kinds of node self-dynamics: the Lorenz system L(x)16 and the Chen sys-tem C(x)17, whereL(x) :x1x2x3 =L(x2 x1)Lx1 x2 x1x3x1x2 Lx3, (16)C(x) :x1x2x3 =c(x2 x1)(c c)x1 +cx2 x1x3x1x2 cx3.(17)When L = 10,L = 8/3 and L = 28, the Lorenz systemis in a
45、chaotic state, so is the Chen system with c =35,c = 3 and c = 28. The nodes labeled by 1 to 5are Lorenz-type and the remaining ones are Chen-type.Consider = diag(0,1,1) and V(x) = xTQx with Q =diag(0.1,1,1). From Assumption 1, one has L = 29and c = 2 regarding to the common equilibrium x =(0,0,0)T.
46、The initial values of the network are in theuniform distribution on the interval (0, 1).Choose the first l nodes (indexed by 1 to l in Fig.6)to be pinned. Now, in the case often considered in theliterature of all control gains being the same, i.e., k1 = = kl = k. The 1 has been plotted against in Fi
47、gs.7and 8 respectively, as varying both the control gain kand the number of controlled nodes l. As Figs.7 and 8show, using a larger number of controllers is effective inenhancing the network controllability, while only a smallimprovement is achieved by increasing the control gain k.Figure 9 shows th
48、e stabilization process with the first5 nodes (indexed by 1 to 5 in Fig.6) pinned. It can beobservedthat increasing the overall coupling strength willaccelerate the convergence of the network stabilization.The colors of the node states in Fig.9 are consistent withthoseofnodes in Fig.6. It is clearlyshownthat the controlpower is spread layer by layer from the controlled nodesto the whole network.Note that once the number of pinned nodes l is given,there are ClN different possibilities of choosing the nodesto control. It is natur
