噪聲擾動下時滯復雜網絡動力學參數及拓撲結構辨識

噪聲擾動下時滯復雜網絡動力學參數及拓撲結構辨識

衛亭1，楊曉麗1，孫中奎2

(1.陜西師范大學數學與信息科學學院,西安710062；2.西北工業大學應用數學系，西安710072)

摘要：針對隨機噪聲及時間滯后普遍存在于耦合網絡，且其結構往往未知或部分未知問題，基于網絡間隨機廣義投影滯后同步原理，通過合理設計控制器與自適應更新規則，構建辨識網絡模型未知動力學參數及拓撲結構的識別方案；結合隨機時滯微分方程LaSalle型不變性原理，從數學上嚴格證明識別方案的準確性。通過具體網絡模型，借助計算仿真驗證識別方案的有效性。數值模擬結果表明，網絡未知動力學參數及拓撲結構不但能準確辨識，且識別方案不依賴耦合時滯、更新增益及網絡拓撲結構等選取。

關鍵詞：網絡結構識別；網絡同步；耦合時滯；隨機噪聲

中圖分類號:O322文獻標志碼：A

基金項目：國家自然科學基金資助項目(11372081)

收稿日期：2014-09-22修改稿收到日期：2014-11-06

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

在網絡結構已知條件下，有關其統計特征(如平均路徑長度、度分布、聚類系數)的動力學行為及控制、網絡結構對動力學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(如蛋白質相互作用、生物神經、電力等)網絡動力學參數或拓撲結構往往未知或部分未知，因此，辨識網絡模型的動力學參數及拓撲結構在復雜動力網絡研究中具有重要的理論意義與應用價值。

Yu等[8]首次提出利用網絡動力學演化信息，通過構造含控制器的新網絡，并基于兩網絡間完全同步追蹤原網絡的拓撲結構。該思想在具有相同[9-11]、不同節點動力學[12-13]網絡拓撲結構及動力學參數識別中廣泛研究。文獻[14]采用攝動法反演網絡模型的連接矩陣，由于信息傳遞速度的有限性、交通堵塞及腦神經系統與其它網絡系統信息傳播路徑長短不同，網絡節點之間進行信息傳輸時存在時間滯后。因此，對網絡結構識別研究需拓展到含耦合時滯的復雜網絡情形?；诰W絡間完全同步，文獻[15-21]通過設計自適應反饋控制器，分別研究含常數時滯或時變時滯、節點動力學相同或不同、加權網絡或等權復雜網絡的動力學參數或拓撲結構識別。網絡間其它同步類型如投影同步[22]、超前同步[23]及滯后同步[24]在時滯耦合網絡結構識別中也發揮重要作用。而基于最優化法[25]、穩態控制法[26]也用于時滯復雜網絡的拓撲結構識別研究。因現實網絡系統會受隨機噪聲影響，如何識別復雜網絡的拓撲結構及動力學參數仍為具有挑戰性的前沿課題。對含噪聲的網絡模型, Ren等[27]通過定義網絡節點動力學信息間相關性,推導信息相關矩陣與決定網絡拓撲結構的Laplacian矩陣關系。吳曉群等[28]通過設計自適應反饋控制器，基于網絡間隨機同步研究節點動力學含隨機噪聲的復雜動力網絡模型拓撲結構識別。文獻[29-33]采用基于最優化法、格林因果檢驗法、ROC曲線分析法及反復性理論法研究噪聲擾動下復雜網絡的拓撲結構識別。

考慮隨機噪聲及耦合時滯普遍存在于復雜網絡，尤其利用網絡間同步原理反演網絡結構時，耦合網絡亦會受時間滯后、隨機噪聲影響。針對噪聲擾動下含耦合時滯的復雜網絡模型，采用反復性理論法[34]、時滯反饋控制法[35]及信息論法[36]研究網絡模型的拓撲結構識別。而基于同步法辨識噪聲擾動下時滯復雜網絡模型的動力學參數及網絡拓撲結構研究較少。本文構建含耦合時滯及噪聲擾動的驅動-響應網絡模型，通過合理設計控制器與自適應更新規則，基于網絡間隨機廣義投影滯后同步辨識網絡模型的動力學參數與拓撲結構，并數值仿真驗證理論推理的有效性。

1網絡模型及預備知識

1.1網絡模型

考慮含N個節點的一般復雜動力網絡，其動力學方程為

(1)

式中：xi(t)=(xi1(t),xi2(t),…,xin(t))T∈Rn為第i個節點狀態變量；f∈Rn×1，F∈Rn×m1為光滑向量函數及矩陣函數；ξ∈Rm1為未知或不確定的動力學參數； B=(bij)N×N為耦合矩陣，表示未知或不確定的網絡拓撲結構，bij定義為：若從節點i到節點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系的內部耦合矩陣；τ(t)為網絡內部節點間時變耦合時滯。

將方程(1)作為驅動網絡，構建響應網絡為

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為響應網絡第i個節點狀態變量；g∈Rn×1，G∈Rn×m2為光滑向量函數及矩陣函數，其中驅動網絡與響應網絡可具有不同節點動力學；θ∈Rm2為未知或不確定的動力學參數； 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為網絡間耦合時滯；σi:Rn×Rn×R+→Rn×m為噪聲強度函數；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)用于刻畫響應網絡與驅動網絡的耦合過程中，會受外界環境浮動、耦合強度設計不精確性等不確定因素影響。本文設m=1。

1.2預備知識

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網絡動力學參數及拓撲結構識別方案

通過設計適當的控制器與自適應更新規則,基于網絡間隨機廣義投影滯后同步,研究網絡模型的動力學參數與拓撲結構識別。

定理對驅動-響應網絡模型式(1)、(2),在假設1-3下，所用控制器及自適應更新規則為

(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。

(9)

由假設1知

peTi(t)ei(t)

式(9)進一步變為

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)

據隨機時滯微分方程LaSalle型不變性原理[37-38]，可得

由假設3、更新規則(7)、狀態誤差系統(8)，得

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

至此，在控制器、更新規則作用及幾乎必然漸近穩定性意義下，網絡模型未知動力學參數與拓撲結構能得到正確識別、反饋強度能自適應調整到常數、驅動網絡與響應網絡間亦能實現隨機廣義投影滯后同步。

3數值仿真

對具體網絡系統進行數值仿真，驗證推論推理的有效性。設驅動網絡局部動力學為四維超混沌系統[39],節點數N=6,構造含未知動力學參數及未知拓撲結構的驅動網絡為

dxi(t)=[f(xi(t))+F(xi(t))ξ+

式中：

構造含未知動力學參數的響應網絡[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

易驗證該函數滿足假設1,即

eTi(t-τ(t))ei(t-τ(t))

數值仿真時，為使網絡節點動力學呈混沌行為，驗證所提識別方案的有效性，選動力學參數為

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

選決定驅動網絡拓撲結構的耦合矩陣為

為進一步刻畫驅動網絡與響應網絡間的同步動力學，引入兩網絡間同步總誤差為

式中：wk∈Ω；h為樣本軌道。

取內部耦合矩陣Γ=diag(1,0,0,0)、網絡內部與網絡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。易驗證構造的驅動-響應網絡模型滿足假設3。

圖1　響應網絡耦合矩陣d ij(t)的演化曲線 Fig.1 The evolution of the topological structure d ij(t) of response network

圖2　網絡未知動力學參數的估計值 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下網絡同步總誤差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下網絡同步總誤差Δ(t)演化曲線 Fig.5 The evolution of the total synchronization errorΔ(t) for different δ

圖6　響應網絡耦合矩陣d ij(t)的演化曲線 Fig.6 The evolution of the topological structure d ij(t) of response network

由6圖計算結果看出，隨時間演化，dij(t)仍能分別收斂到預設的bij，即驅動網絡耦合矩陣B通過響應網絡耦合矩陣D得到正確識別。

4結論

(1)針對噪聲擾動下含耦合時滯的驅動-響應網絡模型，基于網絡間隨機廣義投影滯后同步原理，提出辨識網絡未知動力學參數及拓撲結構的研究方案。

(2)通過設計合理的控制器及自適應更新規則，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絡未知動力學參數及拓撲結構得到正確識別，亦能使驅動網絡、響應網絡在幾乎必然漸近穩定性意義下實現隨機廣義投影滯后同步。

(3)通過具體網絡模型，利用計算機仿真驗證理論推理的有效性，且數值模擬結果也表明識別方案對耦合時滯、更新增益、拓撲結構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