Adaptive algorithms for synchronization, consensus of multi-agents and anti-synchronization of direct complex networks
aa r X i v : . [ n li n . AO ] J u l Adaptive algorithms for synchronization, consensus ofmulti-agents and anti-synchronization of direct complexnetworks ✩ Wenlian Lu a,b,c,d , Xiwei Liu e , Tianping Chen ∗ ,f,a a School of Mathematical Sciences, Fudan University, Shanghai 200433, China b Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University,Shanghai 200433, China c Shanghai Key Laboratory for Contemporary Applied Mathematics and Laboratory ofMathematics for Nonlinear Science, Fudan University, Shanghai 200433, China d Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China e Department of Computer Science and Technology, Tongji University, Shanghai 201804,China f School of Computer Science, Fudan University, Shanghai 200433, China
Abstract
In this paper, we discuss distributed adaptive algorithms for synchronizationof complex networks, consensus of multi-agents with or without pinning con-troller. The dynamics of individual node is governed by generalized QUADcondition. We design new algorithms, which can keep the left eigenvectorof the adaptive coupling matrix corresponding to the zero eigenvalue invari-ant. Based on this invariance, various distributive adaptive synchronization,consensus, anti-synchronization models are given. ✩ This work is jointly supported by the National Key R & D Program of China (No.2018AAA010030), National Natural Sciences Foundation of China under Grant (No.61673119 and 61673298), STCSM (No. 19JC1420101), Shanghai Municipal Science andTechnology Major Project under Grant 2018SHZDZX01 and ZJLab, the Key Project ofShanghai Science and Techonology under Grant 16JC1420402. ∗ Corresponding author. E-mail address: [email protected]
Email addresses: [email protected] (Wenlian Lu), [email protected] (Xiwei Liu), [email protected] (Tianping Chen)
Preprint submitted to Elsevier July 30, 2020 ey words:
Adaptive, Distributed algorithm, Consensus, Synchronization,Anti-synchronization.
1. Introduction
In more than twenty years, synchronization of complex networks, as a spe-cial case, consensus of multi-agents attracted many researchers. The generalmodel is ˙ z i ( t ) = F ( z i ( t )) + c m X j =1 l ij Γ z j ( t ) , (1)where z i ( t ) ∈ R n is the state variable, i = 1 , · · · , m , f : R n → R n , l ij ≥ i = j , and l ii = − P mj =1 l ij , and Γ ∈ R n × n .In [1], authors proved that network (1) can synchronize if c is large enough.However, large c is impractical. [2] also provided an example, showing that toreach synchronization, the theoretical value c min = 0 .
7. However, numericalcomputation shows that if c > .
06, the synchronization has reached. Thus,in [2] the authors hoped to find a sharp bound. It was also pointed that it istoo difficult to prove it theoretically.An effective approach is adaptive control. Adaptive control stability hasbeen studied for a long time. It can be traced to the book [3]. The core is todesign updating laws. Early works on the adaptive control for synchroniza-tion of complex networks, readers can see [4]-[9] and others. For example, [6]proposed distributed adaptive schemes for synchronization.In this paper, we propose novel general adaptive algorithms, which aredistributed and can be applied to synchronization, consensus of multi-agentsand anti-synchronization of direct complex networks [10]-[18].2 . Some basic concepts and background
Before giving main theoretical results, we need following result given in[1].
Proposition 1.
Assume L is connected, then for eigenvalue , [1 , . . . , T isthe right eigenvector, θ = [ θ , · · · , θ m ] T ∈ R m is the left eigenvector, θ i > .In the following, we let m P i =1 θ i = 1 . Notation 1.
Define
Θ = diag { θ } , then the eigenvalues of (Θ L ) s = (Θ L + L T Θ) are sorted as: λ > λ ≥ · · · ≥ λ m . Assumption 1.
QUAD-Function class: Function F ( · )( · ) satisfies QUAD-condition, denoted as F ( · ) ∈ (Φ , P, η ) , if there exist positive definite matrix P ∈ R n × n , matrix Φ ∈ R n × n , and scalar η > , for any z , z ∈ R n , [ z − z ] T P (cid:8) [ F ( z ) − F ( z )] − Φ[ z − z ] (cid:9) ≤ − η [ z − z ] T [ z − z ] . (2) Remark 1.
The concept of QUAD was first introduced in [1] for the casethat P and Φ are positive definite diagonal matrices. The QUAD conditionintroduced here is a natural generalization, which means that after some ro-tations, reflections or other transformations, not only enlargement, we canestimate the nonlinearity of F ( · ) by matrices P, Φ and constant η . Readerscan also refer to [8]. Here, we compare two definitions of QUAD-condition.Let P = Q T J Q be its eigenvalue decomposition. Denote ˜ z i = Qz i ,˜ F (˜ z i ) = Q F ( z i ) = Q F ( Q T ˜ z i ), ˜Φ = Q Φ Q T . Then, (2) can be writtenas [˜ z i − ˜ z j ] T J (cid:8) [ ˜ F (˜ z i ) − ˜ F (˜ z j )] − ˜Φ[˜ z i − ˜ z j ] (cid:9) ≤ − η [˜ z i − ˜ z j ] T [˜ z i − ˜ z j ] .
3f ˜Φ is also a positive diagonal matrix, which is equivalent to that P andΦ have same eigenvectors. In this case, ˜ F ( · ) satisfies the QUAD-conditionintroduced in [1], where J and ˜Φ are positive diagonal matrices. Notation 2.
The kernel of a matrix A is defined as ker ( A ) = { u | Au =0 , u ∈ R n } , while the orthogonal complement of ker ( A ) is denoted by ker ( A ) ⊥ . Assumption 2.
If Assumption 1 holds, P Γ is semi-positive definite on R n ,where Γ is defined in (1), and P Γ is positive definite on subspace ker ( P Φ) ⊥ .
3. Global synchronization analysis for distributed adaptive algo-rithm of complex networks
In this section, we discuss following adaptive algorithm model ˙ z i ( t ) = F ( z i ( t )) + P mj =1 w ij ( t )Γ z j ( t ) , ˙ w ij ( t ) = θ − i [ z i ( t ) − z j ( t )] T P Γ[ z i ( t ) − z j ( t )] , j ∈ N ( i ) ,w ij (0) = l ij , (3)where N ( i ) means the neighborhood of i , w ij ( t ) is the directed adaptive cou-pling weight at time t > w ii ( t ) = − P j = i w ij ( t ), and ˙ w ii ( t ) = − P j ∈ N ( i ) ˙ w ij ( t ) . Theorem 1.
Under Assumptions 1 and 2, algorithm (3) can reach synchro-nization.
Firstly, we give two lemmas.
Lemma 1.
Under the assumption of Theorem 1, any u ∈ R n can be decom-posed to u = u + u , where u ∈ ker ( P Φ) ⊥ , u ∈ ker ( P Φ) , and u ⊥ u .Therefore, there is a constant c , such that u T ( P Φ) u = u T ( P Φ) u ≤ c u T ( P Γ) u ≤ c u T ( P Γ) u. (4)4 emma 2. (see [19] equation (19)) If A = ( a ij ) is a symmetric matrixsatisfying a ij ≥ , if i = j and P mi =1 a ij = 0 . Then, we have m X i,j =1 a ij u i T v j = − m X i,j =1 a ij [ u i − u j ] T [ v i − v j ] . (5)Lemma 2 was given in [19] to discuss synchronization of nonlinearly cou-pled networks. It plays a key role in discussing synchronization of the dis-tributive adaptive algorithm. Lemma 3.
Under algorithm (3), for all t > , m X i =1 θ i w ij ( t ) = 0 , m X i =1 θ i ˙ w ij ( t ) = 0 , ∀ j ; m X j =1 w ij ( t ) = 0 , m X j =1 ˙ w ij ( t ) = 0 , ∀ i. Proof of Theorem 1:
Define V ( t ) = 12 m X i =1 θ i ( z i ( t ) − ¯ z ( t )) T P ( z i ( t ) − ¯ z ( t )) + 14 X i = j ( θ i w ij ( t ) − cθ i l ij ) , where ¯ z ( t ) = m P i =1 θ i z i ( t ).Differentiating it, one can get˙ V ( t ) = m X i =1 θ i [ z i ( t ) − ¯ z ( t )] T P [ F ( z i ( t )) − F (¯ z ( t ))]+ m X i =1 [ z i ( t ) − ¯ z ( t )] T m X j =1 θ i w ij ( t ) P Γ[ z j ( t ) − ¯ z ( t )] − X i = j [ cθ i l ij − θ i w ij ( t )][ z i ( t ) − z j ( t )] T P Γ[ z i ( t ) − z j ( t )] . By QUAD-condition and Lemma 1, we have m X i =1 θ i [ z i ( t ) − ¯ z ( t )] T P [ F ( z i ( t )) − F (¯ z ( t ))]5 m X i =1 θ i [ z i ( t ) − ¯ z ( t )] T P Φ[ z i ( t ) − ¯ z ( t )] − η m X i =1 [ z i ( t ) − ¯ z ( t )] T [ z i ( t ) − ¯ z ( t )] ≤ c m X i =1 θ i [ z i ( t ) − ¯ z ( t )] T P Γ[ z i ( t ) − ¯ z ( t )] − η m X i =1 [ z i ( t ) − ¯ z ( t )] T [ z i ( t ) − ¯ z ( t )] . Based on Lemma 2 and Lemma 3, we have2 m X i =1 [ z i ( t ) − ¯ z ( t )] T m X j =1 θ i l ij P Γ[ z j ( t ) − ¯ z ( t )]= m X i,j =1 [ z i ( t ) − ¯ z ( t )] T ( θ i l ij + θ j l ji ) P Γ[ z j ( t ) − ¯ z ( t )]= − m X i,j =1 [ z i ( t ) − z j ( t )] T ( θ i l ij + θ j l ji ) P Γ[ z i ( t ) − z j ( t )]= − m X i,j =1 [ z i ( t ) − z j ( t )] T θ i l ij P Γ[ z i ( t ) − z j ( t )] , and m X i =1 [ z i ( t ) − ¯ z ( t )] T m X j =1 θ i w ij ( t ) P Γ[ z j ( t ) − ¯ z ( t )]= − m X i,j =1 [ z i ( t ) − z j ( t )] T θ i w ij ( t ) P Γ[ z i ( t ) − z j ( t )] . Therefore, we have˙ V ( t ) ≤ c m X i =1 θ i [ z i ( t ) − ¯ z ( t )] T P Γ[ z i ( t ) − ¯ z ( t )] − η m X i =1 [ z i ( t ) − ¯ z ( t )] T [ z i ( t ) − ¯ z ( t )]+ c m X i,l =1 [ z i ( t ) − ¯ z ( t )] T ( θ i l ij + θ j l ji ) P Γ[ z i ( t ) − ¯ z ( t )] . Since c m X i =1 ( z i ( t ) − ¯ z ( t )) T m X j =1 ( θ i l ij + θ j l ji ) P Γ( z j ( t ) − ¯ z ( t ))6 cλ max i θ i m X i,j =1 θ i ( z i ( t ) − ¯ z ( t )) T P Γ( z j ( t ) − ¯ z ( t )) . In case that c > c max i θ i | λ | − , we have˙ V ( t ) ≤ − η m X i =1 θ i [ z i ( t ) − ¯ z ( t )] T [ z i ( t ) − ¯ z ( t )] , (6)and Z t m X i =1 θ i [ z i ( κ ) − ¯ z ( κ )] T [ z i ( κ ) − ¯ z ( κ )] dκ < η [ V (0) − V ( t )] . Therefore, lim t →∞ P mi =1 θ i [ z i ( t ) − ¯ z ( t )] T [ z i ( t ) − ¯ z ( t )] = 0. Theorem is provedcompletely. Remark 2.
It has been explored that the left and right eigenvectors corre-sponding to the eigenvalue of the coupling matrix plays key roles in dis-cussing synchronization. Therefore, to keep the left and right eigenvectorscorresponding to the eigenvalue of all coupling matrices W ( t ) = [ w ij ( t )] , t > , invariant is the most important. Our algorithm ensures that all cou-pling matrices W ( t ) = [ w ij ( t )] have same left and right eigenvectors corre-sponding to the eigenvalue . Remark 3.
It is clear that QUAD condition and the equality (5) in Lemma2 (see [19] equation (19)) play key role in proof of Theorem.
Corollary 1.
Assume L is symmetric or node balanced. Thus all θ i = m , i = 1 , · · · , m . Under the assumptions made in Theorem 1, the followingalgorithm ˙ x i ( t ) = F ( z i ( t )) + P mj =1 w ij ( t )Γ z j ( t ) , ˙ w ij ( t ) = [ z i ( t ) − z j ( t )] T P Γ[ z i ( t ) − z j ( t )] , j ∈ N ( i ) ,w ij (0) = l ij , i, j = 1 , · · · , m an reach synchronization.
4. Adaptive pinning synchronization of complex networks with asingle controller
We discuss adaptive synchronization with a single pinning controller,which is identical to synchronization of leader-follower systems, see [19].Consider the following pinning control model [19], ˙ z ( t ) = F ( z ( t )) + c m P j =1 l j Γ z j ( t ) − cε Γ[ z ( t ) − s ( t )] , ˙ z i ( t ) = F ( z i ( t )) + c m P j =1 l ij Γ z j ( t ) , i = 2 , · · · , m (7)where s ( t ) be a solution of ˙ s ( t ) = F ( s ( t )). Its adaptive algorithm ˙ z ( t ) = F ( z ( t )) + m P j =1 w j ( t )Γ z j ( t ) − w ( t )Γ[ z ( t ) − s ( t )] , ˙ z i ( t ) = F ( z i ( t )) + m P j =1 w ij ( t )Γ z j ( t ) , i = 2 , · · · , m ˙ w ij ( t ) = θ − i [ z i ( t ) − z j ( t )] T P Γ[ z i ( t ) − z j ( t )] , j ∈ N ( i ) , ˙ w ( t ) = θ − [ z ( t ) − s ( t ))] T P Γ[ z ( t ) − s ( t )] ,w ij (0) = l ij , i, j = 1 , · · · , m, w (0) = ε (8)Similarly, we have Theorem 2.
Under Assumptions 1 and 2, distributive adaptive algorithm(8) can synchronize to s ( t ) .Proof. Define the following candidate Lyapunov function V ( t ) = 12 m X i =1 θ i [ z i ( t ) − s ( t )] T P [ z i ( t ) − s ( t )]8 14 m X i = j [ θ i w ij ( t ) − cθ i l ij ] + 12 [ θ w ( t ) − cθ ε ] , where c will be given later.Differentiating V ( t ), one can get˙ V ( t ) = m X i =1 θ i [ z i ( t ) − s ( t )] T P [ F ( z i ( t )) − F ( s ( t ))]+ m X i =1 [ z i ( t ) − s ( t )] T m X j =1 θ i w ij ( t ) P Γ[ z j ( t ) − s ( t )] − θ w ( t )[ z ( t ) − s ( t )] T P Γ[ z ( t ) − s ( t )] − m X i,j =1 [ cθ i l ij − θ i w ij ( t )][ z i ( t ) − z j ( t )] T P Γ[ z i ( t ) − z j ( t )]+ θ [ w ( t ) − cε ][ z ( t ) − s ( t )] T P Γ[ z ( t ) − s ( t )]= m X i =1 θ i [ z i ( t ) − s ( t )] T P [ F ( z i ( t )) − F ( s ( t ))] − cθ ε [ z ( t ) − s ( t )] T P Γ[ z ( t ) − s ( t )] − c m X i,j =1 θ i l ij [ z i ( t ) − z j ( t )] T P Γ[ z i ( t ) − z j ( t )] . By Lemma 2, we have − c m X i,j =1 θ i l ij [ z i ( t ) − z j ( t )] T P Γ[ z i ( t ) − z j ( t )]= c m X i,j =1 θ i l ij [ z i ( t ) − s ( t )] T P Γ[ z i ( t ) − s ( t )] . By similar derivations given in Theorem 1, we have˙ V ( t ) ≤ − η m X i =1 θ i [ z i ( t ) − s ( t )] T [ z i ( t ) − s ( t )]9 c m X i =1 θ i [ z i ( t ) − s ( t )] T P Γ[ z i ( t ) − s ( t )]+ c m X i =1 [ z i ( t ) − s ( t )] T m X j =1 θ i ˜ l ij P Γ[ z j ( t ) − s ( t )] , where c is the constant given in (4) in Lemma 1, ˜ l ij = l ij , except ¯ l = l − ǫ . Define a matrix ˜ L = (˜ l ij ) mi,j =1 . [19] pointed out that all eigenvalues µ i , i = 1 , · · · , m of (Θ ˜ L ) s = [Θ ˜ L + ˜ L T Θ] satisfy 0 > µ ≥ µ ≥ · · · ≥ µ m .If c > c max i θ i | µ | − , we have c m X i =1 θ i [ z i ( t ) − s ( t )] T P Γ[ z i ( t ) − s ( t )]+ cd m X i =1 [ z i ( t ) − s ( t )] T m X j =1 θ i ˜ l ij P Γ[ z j ( t ) − s ( t )] < , which implies ˙ V ( t ) ≤ − η m X i =1 θ i [ z i ( t ) − s ( t )] T [ z i ( t ) − s ( t )] , and Z t m X i =1 θ i [ z i ( κ ) − s ( κ )] T [ z i ( κ ) − s ( κ )] dκ < η [ V (0) − V ( t )] . Therefore, lim t →∞ P mi =1 θ i [ z i ( t ) − s ( t )] T [ z i ( t ) − s ( t )] = 0. Theorem is provedcompletely.
5. Applications to Consensus of Multi-agents Systems
As applications, we discuss consensus of multi-agents, dz i ( t ) dt = Az i ( t ) + c m X j =1 l ij Γ z j ( t ) , i = 1 , · · · , m (9)10onsider the model ˙ z ( t ) = Az ( t ) + Bu ( t ) (10)If the system (10) is controllable, there are positive definite matrix P , suchthat P A + A T P − BB T < , (11)which is equivalent to the following QUAD condition[ z − z ] T P (cid:8) [ Az − Az ] − P − BB T [ z − z ] (cid:9) ≤ − η [ z − z ] T [ z − z ] . By the results obtained in previous section, we can give
Theorem 3.
If the system (10) is controllable, the distributive adaptive sys-tem ˙ z i ( t ) = Az i ( t ) + m P j =1 w j ( t ) P − BB T z j ( t )˙ w ij ( t ) = θ − i [ z i ( t ) − z j ( t )] T BB T [ z i ( t ) − z j ( t )] , j ∈ N ( i ) ,w ij (0) = l ij , can reach consensus.If ˙ s ( t ) = As ( t ) , then following adaptive algorithm ˙ z ( t ) = Az ( t ) + m P j =1 w j ( t ) P − BB T z j ( t ) − w ( t ) P − BB T [ z ( t ) − s ( t )] , ˙ z i ( t ) = Az i ( t ) + m P j =1 w ij ( t ) P − BB T z j ( t ) , i = 1˙ w ij ( t ) = θ − i [ z i ( t ) − s ( t )] T BB T [ z i ( t ) − s ( t )] , j ∈ N ( i ) , ˙ w ( t ) = θ − [ z ( t ) − s ( t ))] T BB T [ z ( t ) − s ( t )] ,w ij (0) = l ij , w (0) = ε can reach z i ( t ) − s ( t ) → , for all i = 1 , · · · , m . P − BB T . Theorem is a direct consequence of Theorem 1 andTheorem 2.
6. Anti-synchronization
Next, we discuss anti-synchronization [20, 21, 22, 23],˙ z i ( t ) = F ( z i ( t )) + c m X j = i | l ij | Γ[sign( l ij ) z j ( t ) − z i ( t )] , (12)where all nodes are connected and can be split into two subgroups V and V , such that l ij ≥ i, j ∈ V p ( p = 1 , l ij ≤ i ∈ V p , j ∈ V q , p = q .Let ˆ z i ( t ) = z i ( t ), if i ∈ V , and ˆ z i ( t ) = − z i ( t ), if i ∈ V , then (12) can berewritten as ˙ˆ z i ( t ) = F (ˆ z i ( t )) + c P mj =1 l ∗ ij Γˆ z j ( t ) , i ∈ V , j ∈ N ( i ) , ˙ˆ z i ( t ) = − F ( − ˆ z i ( t )) + c P mj =1 l ∗ ij Γˆ z j ( t ) , i ∈ V , j ∈ N ( i ) , (13)where l ∗ ij = | l ij | , if i = j , and l ∗ ii = − P mj = i | l ij | . Therefore, anti-synchronizationof (12) is equivalent to synchronization of ˆ z i ( t ) for system (13).For matrix L ∗ = ( l ∗ ij ), θ = [ θ , · · · , θ m ] T is the left eigenvector in Propo-sition 1. Assumption 3.
For F ( · ) , suppose ( z − z ) T Q (cid:8) [ F ( z ) − F ( z )] − Φ( z − z ) (cid:9) ≤ − η ( z − z ) T ( z − z ) , ( z + z ) T Q (cid:8) [ F ( z ) + F ( z )] − Φ( z + z ) (cid:9) ≤ − η ( z + z ) T ( z + z ) , (14) where Q is a positive definite matrix.
12n [23], it was proved that under Assumption 3, (13) can reach synchro-nization if c is sufficiently large.Based on previous discussion, the adaptive algorithm takes following form ˙ˆ z i ( t ) = F (ˆ z i ( t )) + P mj =1 w ij ( t )Γ[ˆ z j ( t ) − ˆ z i ( t )] , ˙ w ij ( t ) = θ − i [ˆ z i ( t ) − ˆ z j ( t )] T P Γ[ˆ z i ( t ) − ˆ z j ( t )] , if i ∈ V , j ∈ N ( i ) , ˙ˆ z i ( t ) = − F ( − ˆ z i ( t )) + P mj =1 w ij ( t )Γ[ˆ z j ( t ) − ˆ z i ( t )] , ˙ w ij ( t ) = θ − i [ˆ z i ( t ) − ˆ z j ( t )] T P Γ[ˆ z i ( t ) − ˆ z j ( t )] , if i ∈ V , j ∈ N ( i ) ,w ij (0) = l ∗ ij , i, j = 1 , · · · , m (15) Theorem 4.
Under Assumptions 2 and 3, anti-synchronization can be reachedfor adaptive system (15).Proof.
Define anti-synchronization state as¯ˆ z ( t ) = m X i =1 θ i ˆ z i ( t ) = X j ∈V θ j z j ( t ) − X j ∈V θ i z j ( t ) , (16)and Lyapunov function takes form V ( t ) = 12 m X i =1 θ i [ˆ z i ( t ) − ¯ˆ z ( t )] T P [ˆ z i ( t ) − ¯ˆ z ( t )] + 14 m X i,j =1 [ θ i w ij ( t ) − cθ i l ∗ ij ] . For i ∈ V , from (14),[ˆ z i ( t ) − ¯ˆ z ( t )] T P (cid:2) F (ˆ z i ( t )) − F (¯ˆ z ( t )) (cid:3) ≤ [ˆ z i ( t ) − ¯ˆ z ( t )] T P Φ[ˆ z i ( t ) − ¯ˆ z ( t )] − η [ˆ z i ( t ) − ¯ˆ z ( t )] T [ˆ z i ( t ) − ¯ˆ z ( t )] . For i ∈ V , from (14),[ˆ z i ( t ) − ¯ˆ z ( t )] T P (cid:2) − F ( − ˆ z i ( t )) − F (¯ˆ z ( t )) (cid:3) [ˆ z i ( t ) − ¯ˆ z ( t )] T P Φ[ˆ z i ( t ) − ¯ˆ z ( t )] − η [ˆ z i ( t ) − ¯ˆ z ( t )] T [ˆ z i ( t ) − ¯ˆ z ( t )] . By the same derivations as before, we can prove ˙ V ( t ) ≤ − η P mi =1 θ i [ˆ z i ( t ) − ¯ˆ z ( t )] T [ˆ z i ( t ) − ¯ˆ z ( t )], and Z t m X i =1 θ i [ˆ z i ( κ ) − ¯ˆ z ( κ )] T [ˆ z i ( κ ) − ¯ˆ z ( κ )] dκ < η [ V (0) − V ( t )] . Therefore, lim t →∞ P mi =1 θ i [ˆ z i ( t ) − ¯ˆ z ( t )] T [ z i ( t ) − ¯ˆ z ( t )] = 0, anti-synchronizationis realized. Remark 4.
In case F ( − x ) = − F ( x ) , such as F ( x ) = Ax , (2) and (14)will be identical. The adaptive algorithm (15) becomes ˙ˆ z i ( t ) = F (ˆ z i ( t )) + P mj =1 w ij ( t )Γ[ˆ z j ( t ) − ˆ z i ( t )] , ˙ w ij ( t ) = θ − i [ˆ z i ( t ) − ˆ z j ( t )] T P Γ[ˆ z i ( t ) − ˆ z j ( t )] , j ∈ N ( i ) ,w ij (0) = l ∗ ij , i, j = 1 , · · · , m
7. Global synchronization analysis for distributed adaptive algo-rithm of nonlinear coupled complex networks
In practice, we can not observe the state z i ( t ) directly. Instead, we onlycan obtain data G ( z i ( t )), i = 1 , · · · , m . We need to use G ( z i ( t )) to synchro-nize the uncoupled system, which means that the synchronization scheme isnonlinear, dz i ( t ) dt = F ( z i ( t ) , t ) + m X j =1 ,j = i a ij [ G ( z j ( t )) − G ( z i ( t ))] , where G ( z ) = [ g ( z ) , · · · , g n ( z n )] T , for z = [ z , · · · , z n ] T , and satisfying[ g i ( u ) − g i ( v )] > β [ u − v ], i = 1 , · · · , m , β >
0, for any u = v .14 heorem 5. Suppose matrix L is symmetric, the QUAD-condition ( I n , I n , η ) holds, the following algorithm ˙ x i ( t ) = F ( z i ( t )) + P mj =1 w ij ( t )[ G ( z j ( t )) − G ( z i ( t ))];˙ w ij ( t ) = ρ [ G ( z i ( t )) − G ( z j ( t )] T [ G ( z i ( t )) − G ( z j ( t ))] , j ∈ N ( i ); w ij (0) = l ij (17) can realize the synchronization, where ρβ < . Clearly, ˙ w ij ( t ) = ˙ w ji ( t ), which implies that ( w ij ( t )) is symmetric for all t > Proof.
Define V ( t ) = 12 m X i =1 [ z i ( t ) − ¯ z ( t )] T [ z i ( t ) − ¯ z ( t )] + 14 m X i =1 [ w ij ( t ) − cl ij ] . Based on the equality (5), we have m X i =1 [ z i ( t ) − ¯ z ( t )] T m X j =1 w ij ( t )[ G ( z j ( t )) − G (¯ z ( t ))]= − m X i,j =1 [ z i ( t ) − z j ( t )] T w ij ( t )[ G ( z i ( t )) − G ( z j ( t ))] , and differentiate it,˙ V ( t ) = m X i =1 [ z i ( t ) − ¯ z ( t )] T [ F ( z i ( t )) − F (¯ z ( t ))]+ m X i,j =1 [ z i ( t ) − ¯ z ( t )] T w ij ( t )[ G ( z j ( t )) − G (¯ z ( t ))]+ ρ m X i,j =1 w ij ( t )[ G ( z i ( t )) − G ( z j ( t ))] T [ G ( z i ( t )) − G ( z j ( t ))] − cρ X i = j l ij [ G ( z i ( t )) − G ( z j ( t )] T [ G ( z i ( t )) − G ( z j ( t ))]15 I ( t ) + I ( t ) + I ( t ) + I ( t ) . By (2), we have I ( t ) ≤ m X i =1 [ z i ( t ) − ¯ z ( t )] T [ z i ( t ) − ¯ z ( t )] − η m X i =1 [ z i ( t ) − ¯ z ( t )] T [ z i ( t ) − ¯ z ( t )] ,I ( t ) = − m X i,j =1 [ z i ( t ) − z j ( t )] T w ij ( t )[ G ( z i ( t )) − G ( z j ( t ))] ≤ − β m X i,j =1 [ z i ( t ) − z j ( t )] T w ij ( t )[ z i ( t ) − z j ( t )]= β m X i,j =1 [ z i ( t ) − ¯ z ( t )] T w ij ( t )[ z i ( t ) − ¯ z ( t )] ,I ( t ) ≤ − ρβ m X i,j =1 [ z i ( t ) − ¯ z ( t )] T w ij ( t )[ z i ( t ) − ¯ z ( t )] ,I ( t ) ≤ cρβ m X i,j =1 [ z i ( t ) − ¯ z ( t )] T l ij [ z i ( t ) − ¯ z ( t )] . In case ρβ <
1, we have I ( t ) + I ( t ) <
0. Pick a sufficiently large c , we canprove ˙ V ( t ) < I ( t ) + I ( t ) < − η m X i =1 [ z i ( t ) − ¯ z ( t )] T [ z i ( t ) − ¯ z ( t )] . The rest follows the proof of Theorem 1.
8. Conclusions
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