噪聲擾動下時滯復(fù)雜網(wǎng)絡(luò)動力學(xué)參數(shù)及拓撲結(jié)構(gòu)辨識

衛(wèi)亭1，楊曉麗1，孫中奎2

(1.陜西師范大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院,西安710062；2.西北工業(yè)大學(xué)應(yīng)用數(shù)學(xué)系，西安710072)

摘要：針對隨機噪聲及時間滯后普遍存在于耦合網(wǎng)絡(luò)，且其結(jié)構(gòu)往往未知或部分未知問題，基于網(wǎng)絡(luò)間隨機廣義投影滯后同步原理，通過合理設(shè)計控制器與自適應(yīng)更新規(guī)則，構(gòu)建辨識網(wǎng)絡(luò)模型未知動力學(xué)參數(shù)及拓撲結(jié)構(gòu)的識別方案；結(jié)合隨機時滯微分方程LaSalle型不變性原理，從數(shù)學(xué)上嚴格證明識別方案的準(zhǔn)確性。通過具體網(wǎng)絡(luò)模型，借助計算仿真驗證識別方案的有效性。數(shù)值模擬結(jié)果表明，網(wǎng)絡(luò)未知動力學(xué)參數(shù)及拓撲結(jié)構(gòu)不但能準(zhǔn)確辨識，且識別方案不依賴耦合時滯、更新增益及網(wǎng)絡(luò)拓撲結(jié)構(gòu)等選取。

關(guān)鍵詞：網(wǎng)絡(luò)結(jié)構(gòu)識別；網(wǎng)絡(luò)同步；耦合時滯；隨機噪聲

中圖分類號:O322文獻標(biāo)志碼：A

基金項目：國家自然科學(xué)基金資助項目(11372081)

收稿日期：2014-09-22修改稿收到日期：2014-11-06

基金項目：國家自然科學(xué)基金重大研究計劃重點支持項目“古建木構(gòu)的狀態(tài)評估、安全極限與性能保持”(51338001)；北京交通大學(xué)人才基金項目(2014RC011)

收稿日期：2014-07-29修改稿收到日期：2014-10-17

Identification of system parameters and network topology of delay-coupled complex networks under circumstance noise

WEITing1,YANGXiao-Li1,SUNZhong-kui2(1. Shanxi Normal University, Xi’an 710062, China; 2. Northwestern Polytechnical University, Xi’an 710072, China)

Abstract:Noting that the random noise and time delay are prevalent in complex networks and the topology of a network is often unknown or partially unknown, based on the principle of random generalized projective lag synchronization, an approach was proposed to estimate the system parameters and topological structure of delay-coupled complex networks under circumstance noise. By constructing an appropriate controller and adaptive updating rules, the unknown network parameters and topological structure of the concerned networks were identified simultaneously. The accuracy of the method was rigorously proved by the LaSalle-type theorem for stochastic differential delay equations．An example of network with chaotic oscillator was provided to illustrate the method. The numerical results indicate that the unknown network parameters and topological structure can be accurately identified, and yet the proposed method is robust against the time delay, the update gain and the network topology.

Key words:topology identification; network synchronization; coupled delay; random noise

在網(wǎng)絡(luò)結(jié)構(gòu)已知條件下，有關(guān)其統(tǒng)計特征(如平均路徑長度、度分布、聚類系數(shù))的動力學(xué)行為及控制、網(wǎng)絡(luò)結(jié)構(gòu)對動力學(xué)影響等獲得廣泛研究[1-7]。然而，由于各種因素的不確定性，諸多(如蛋白質(zhì)相互作用、生物神經(jīng)、電力等)網(wǎng)絡(luò)動力學(xué)參數(shù)或拓撲結(jié)構(gòu)往往未知或部分未知，因此，辨識網(wǎng)絡(luò)模型的動力學(xué)參數(shù)及拓撲結(jié)構(gòu)在復(fù)雜動力網(wǎng)絡(luò)研究中具有重要的理論意義與應(yīng)用價值。

Yu等[8]首次提出利用網(wǎng)絡(luò)動力學(xué)演化信息，通過構(gòu)造含控制器的新網(wǎng)絡(luò)，并基于兩網(wǎng)絡(luò)間完全同步追蹤原網(wǎng)絡(luò)的拓撲結(jié)構(gòu)。該思想在具有相同[9-11]、不同節(jié)點動力學(xué)[12-13]網(wǎng)絡(luò)拓撲結(jié)構(gòu)及動力學(xué)參數(shù)識別中廣泛研究。文獻[14]采用攝動法反演網(wǎng)絡(luò)模型的連接矩陣，由于信息傳遞速度的有限性、交通堵塞及腦神經(jīng)系統(tǒng)與其它網(wǎng)絡(luò)系統(tǒng)信息傳播路徑長短不同，網(wǎng)絡(luò)節(jié)點之間進行信息傳輸時存在時間滯后。因此，對網(wǎng)絡(luò)結(jié)構(gòu)識別研究需拓展到含耦合時滯的復(fù)雜網(wǎng)絡(luò)情形。基于網(wǎng)絡(luò)間完全同步，文獻[15-21]通過設(shè)計自適應(yīng)反饋控制器，分別研究含常數(shù)時滯或時變時滯、節(jié)點動力學(xué)相同或不同、加權(quán)網(wǎng)絡(luò)或等權(quán)復(fù)雜網(wǎng)絡(luò)的動力學(xué)參數(shù)或拓撲結(jié)構(gòu)識別。網(wǎng)絡(luò)間其它同步類型如投影同步[22]、超前同步[23]及滯后同步[24]在時滯耦合網(wǎng)絡(luò)結(jié)構(gòu)識別中也發(fā)揮重要作用。而基于最優(yōu)化法[25]、穩(wěn)態(tài)控制法[26]也用于時滯復(fù)雜網(wǎng)絡(luò)的拓撲結(jié)構(gòu)識別研究。因現(xiàn)實網(wǎng)絡(luò)系統(tǒng)會受隨機噪聲影響，如何識別復(fù)雜網(wǎng)絡(luò)的拓撲結(jié)構(gòu)及動力學(xué)參數(shù)仍為具有挑戰(zhàn)性的前沿課題。對含噪聲的網(wǎng)絡(luò)模型, Ren等[27]通過定義網(wǎng)絡(luò)節(jié)點動力學(xué)信息間相關(guān)性,推導(dǎo)信息相關(guān)矩陣與決定網(wǎng)絡(luò)拓撲結(jié)構(gòu)的Laplacian矩陣關(guān)系。吳曉群等[28]通過設(shè)計自適應(yīng)反饋控制器，基于網(wǎng)絡(luò)間隨機同步研究節(jié)點動力學(xué)含隨機噪聲的復(fù)雜動力網(wǎng)絡(luò)模型拓撲結(jié)構(gòu)識別。文獻[29-33]采用基于最優(yōu)化法、格林因果檢驗法、ROC曲線分析法及反復(fù)性理論法研究噪聲擾動下復(fù)雜網(wǎng)絡(luò)的拓撲結(jié)構(gòu)識別。

考慮隨機噪聲及耦合時滯普遍存在于復(fù)雜網(wǎng)絡(luò)，尤其利用網(wǎng)絡(luò)間同步原理反演網(wǎng)絡(luò)結(jié)構(gòu)時，耦合網(wǎng)絡(luò)亦會受時間滯后、隨機噪聲影響。針對噪聲擾動下含耦合時滯的復(fù)雜網(wǎng)絡(luò)模型，采用反復(fù)性理論法[34]、時滯反饋控制法[35]及信息論法[36]研究網(wǎng)絡(luò)模型的拓撲結(jié)構(gòu)識別。而基于同步法辨識噪聲擾動下時滯復(fù)雜網(wǎng)絡(luò)模型的動力學(xué)參數(shù)及網(wǎng)絡(luò)拓撲結(jié)構(gòu)研究較少。本文構(gòu)建含耦合時滯及噪聲擾動的驅(qū)動-響應(yīng)網(wǎng)絡(luò)模型，通過合理設(shè)計控制器與自適應(yīng)更新規(guī)則，基于網(wǎng)絡(luò)間隨機廣義投影滯后同步辨識網(wǎng)絡(luò)模型的動力學(xué)參數(shù)與拓撲結(jié)構(gòu)，并數(shù)值仿真驗證理論推理的有效性。

1網(wǎng)絡(luò)模型及預(yù)備知識

1.1網(wǎng)絡(luò)模型

考慮含N個節(jié)點的一般復(fù)雜動力網(wǎng)絡(luò)，其動力學(xué)方程為

(1)

式中：xi(t)=(xi1(t),xi2(t),…,xin(t))T∈Rn為第i個節(jié)點狀態(tài)變量；f∈Rn×1，F(xiàn)∈Rn×m1為光滑向量函數(shù)及矩陣函數(shù)；ξ∈Rm1為未知或不確定的動力學(xué)參數(shù)； B=(bij)N×N為耦合矩陣，表示未知或不確定的網(wǎng)絡(luò)拓撲結(jié)構(gòu)，bij定義為：若從節(jié)點i到節(jié)點j有一個連接，則bij≠0，否則，bij=0；?！蔙n×n為決定變量間相互關(guān)系的內(nèi)部耦合矩陣；τ(t)為網(wǎng)絡(luò)內(nèi)部節(jié)點間時變耦合時滯。

將方程(1)作為驅(qū)動網(wǎng)絡(luò)，構(gòu)建響應(yīng)網(wǎng)絡(luò)為

dyi(t)=[g(yi(t))+G(yi(t))θ+

σi(yi(t)-γxi(t-δ),yi(t-τ(t))-

γxi(t-τ(t)-δ),t)dW(t)，(i=1,2,…,N)

(2)

式中：yi(t)=(yi1(t),yi2(t),…,yin(t))T∈Rn為響應(yīng)網(wǎng)絡(luò)第i個節(jié)點狀態(tài)變量；g∈Rn×1，G∈Rn×m2為光滑向量函數(shù)及矩陣函數(shù)，其中驅(qū)動網(wǎng)絡(luò)與響應(yīng)網(wǎng)絡(luò)可具有不同節(jié)點動力學(xué)；θ∈Rm2為未知或不確定的動力學(xué)參數(shù)； D=(dij)N×N為耦合矩陣，表示對耦合矩陣B的估計；Ui(t)∈Rn為控制器；δ為網(wǎng)絡(luò)間耦合時滯；σi:Rn×Rn×R+→Rn×m為噪聲強度函數(shù)；W(t)=(w1(t),w2(t),…,wm(t))T∈Rm為定義在完備概率空間(Ω,H,P)的m維布朗運動，σidW(t)用于刻畫響應(yīng)網(wǎng)絡(luò)與驅(qū)動網(wǎng)絡(luò)的耦合過程中，會受外界環(huán)境浮動、耦合強度設(shè)計不精確性等不確定因素影響。本文設(shè)m=1。

1.2預(yù)備知識

引理對任意向量x，y∈Rn及正定矩陣Q∈Rn×n，有2xTy≤xTQx+yTQ-1y成立。

2網(wǎng)絡(luò)動力學(xué)參數(shù)及拓撲結(jié)構(gòu)識別方案

通過設(shè)計適當(dāng)?shù)目刂破髋c自適應(yīng)更新規(guī)則,基于網(wǎng)絡(luò)間隨機廣義投影滯后同步,研究網(wǎng)絡(luò)模型的動力學(xué)參數(shù)與拓撲結(jié)構(gòu)識別。

定理對驅(qū)動-響應(yīng)網(wǎng)絡(luò)模型式(1)、(2),在假設(shè)1-3下，所用控制器及自適應(yīng)更新規(guī)則為

(3)

(4)

(5)

(6)

(7)

dei(t)=dyi(t)-γdxi(t-δ)=

ki(t)ei(t)]dt+σi(ei(t),ei(t-τ(t)),t)dW(t)

(8)

建立V函數(shù)為

式中：V∈C1,2(R+;G)，G=Rm1+m2+N2+N+nN ；k*為可確定的足夠大正常數(shù)。

(9)

由假設(shè)1知

peTi(t)ei(t)

式(9)進一步變?yōu)?/p>

令e(t)=(eT1(t),eT2(t),…,eTN(t))T∈RnN，A=B?Γ，將lV寫成緊積形式為

peT(t)e(t)+qeT(t-τ(t))e(t-τ(t))

(10)

由引理知

將式(10)改寫為

qeT(t-τ(t))e(t-τ(t))?-ω1(x)+ω2(y)

據(jù)隨機時滯微分方程LaSalle型不變性原理[37-38]，可得

由假設(shè)3、更新規(guī)則(7)、狀態(tài)誤差系統(tǒng)(8)，得

0,ki(t)=const,ei(t)=0}

至此，在控制器、更新規(guī)則作用及幾乎必然漸近穩(wěn)定性意義下，網(wǎng)絡(luò)模型未知動力學(xué)參數(shù)與拓撲結(jié)構(gòu)能得到正確識別、反饋強度能自適應(yīng)調(diào)整到常數(shù)、驅(qū)動網(wǎng)絡(luò)與響應(yīng)網(wǎng)絡(luò)間亦能實現(xiàn)隨機廣義投影滯后同步。

3數(shù)值仿真

對具體網(wǎng)絡(luò)系統(tǒng)進行數(shù)值仿真，驗證推論推理的有效性。設(shè)驅(qū)動網(wǎng)絡(luò)局部動力學(xué)為四維超混沌系統(tǒng)[39],節(jié)點數(shù)N=6,構(gòu)造含未知動力學(xué)參數(shù)及未知拓撲結(jié)構(gòu)的驅(qū)動網(wǎng)絡(luò)為

dxi(t)=[f(xi(t))+F(xi(t))ξ+

式中：

構(gòu)造含未知動力學(xué)參數(shù)的響應(yīng)網(wǎng)絡(luò)[40]為

dyi(t)=[g(yi(t))+G(yi(t))θ+

σi(yi(t)-γxi(t-δ),yi(t-τ(t))-

γxi(t-τ(t)-δ),t)dW(t)，(i=1,2,…,6)

式中：

選噪聲強度函數(shù)為

σi(ei(t),ei(t-τ(t)),t)=(ei1(t)+ei1(t-τ(t)),

ei2(t)+ei2(t-τ(t)),ei3(t)+ei3(t-τ(t)),

ei4(t)+ei4(t-τ(t)))T

易驗證該函數(shù)滿足假設(shè)1,即

eTi(t-τ(t))ei(t-τ(t))

數(shù)值仿真時，為使網(wǎng)絡(luò)節(jié)點動力學(xué)呈混沌行為，驗證所提識別方案的有效性，選動力學(xué)參數(shù)為

ξ=(a1,b1,c1,k1,g1)T=(35,3,35,-8,-10)T

θ=(a2,b2,c2,k2)T=(-1.0,0.25,0.5,0.05)T

選決定驅(qū)動網(wǎng)絡(luò)拓撲結(jié)構(gòu)的耦合矩陣為

為進一步刻畫驅(qū)動網(wǎng)絡(luò)與響應(yīng)網(wǎng)絡(luò)間的同步動力學(xué)，引入兩網(wǎng)絡(luò)間同步總誤差為

式中：wk∈Ω；h為樣本軌道。

取內(nèi)部耦合矩陣Γ=diag(1,0,0,0)、網(wǎng)絡(luò)內(nèi)部與網(wǎng)絡(luò)之間的耦合時滯分別為τ(t)=0.02與δ=0.03、比例因子γ=2.0、更新增益λi=30.0，樣本軌道h=10。易驗證構(gòu)造的驅(qū)動-響應(yīng)網(wǎng)絡(luò)模型滿足假設(shè)3。

圖1　響應(yīng)網(wǎng)絡(luò)耦合矩陣d ij(t)的演化曲線 Fig.1 The evolution of the topological structure d ij(t) of response network

圖2　網(wǎng)絡(luò)未知動力學(xué)參數(shù)的估計值 ξ(t)與θ(t)的演化曲線 Fig.2 The evolution of the unknown parameter’s estimation ξ(t) and θ(t)

圖3　反饋強度k i(1≤i≤6)的演化曲線 Fig.3 The evolution of the feedback strength k i(1≤i≤6)

圖4　不同更新增益λ i下網(wǎng)絡(luò)同步總誤差Δ(t)演化曲線 Fig.4 The evolution of the total synchronization error Δ(t) for different λ i

圖5　不同耦合時滯δ下網(wǎng)絡(luò)同步總誤差Δ(t)演化曲線 Fig.5 The evolution of the total synchronization errorΔ(t) for different δ

圖6　響應(yīng)網(wǎng)絡(luò)耦合矩陣d ij(t)的演化曲線 Fig.6 The evolution of the topological structure d ij(t) of response network

由6圖計算結(jié)果看出，隨時間演化，dij(t)仍能分別收斂到預(yù)設(shè)的bij，即驅(qū)動網(wǎng)絡(luò)耦合矩陣B通過響應(yīng)網(wǎng)絡(luò)耦合矩陣D得到正確識別。

4結(jié)論

(1)針對噪聲擾動下含耦合時滯的驅(qū)動-響應(yīng)網(wǎng)絡(luò)模型，基于網(wǎng)絡(luò)間隨機廣義投影滯后同步原理，提出辨識網(wǎng)絡(luò)未知動力學(xué)參數(shù)及拓撲結(jié)構(gòu)的研究方案。

(2)通過設(shè)計合理的控制器及自適應(yīng)更新規(guī)則，利用隨機時滯微分方程的LaSalle型不變性原理，嚴格證明識別方案不僅能使網(wǎng)絡(luò)未知動力學(xué)參數(shù)及拓撲結(jié)構(gòu)得到正確識別，亦能使驅(qū)動網(wǎng)絡(luò)、響應(yīng)網(wǎng)絡(luò)在幾乎必然漸近穩(wěn)定性意義下實現(xiàn)隨機廣義投影滯后同步。

(3)通過具體網(wǎng)絡(luò)模型，利用計算機仿真驗證理論推理的有效性，且數(shù)值模擬結(jié)果也表明識別方案對耦合時滯、更新增益、拓撲結(jié)構(gòu)等的選取具有魯棒性。

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第一作者鄒廣平男，博士，教授，博士生導(dǎo)師，1963年生

第一作者高延安男，博士生，1986年生

通信作者楊慶山男，博士，教授，1968年生