Adaptive Synchronization via State Predictor on General Complex Dynamic Networks

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 415734, 6 pages http://dx.doi.org/10.1155/2015/415734 Research Article Adaptive Synchronization via State Predictor on General Complex Dynamic Networks Lijun Wang,1 Pingnan Ruan,1 Shuang Li,2 and Xiao Han2 1 2 School of Economics and Management, Beijing University of Technology, Beijing 100022, China College of Science, North China University of Technology, Beijing 100144, China Correspondence should be addressed to Shuang Li; Received 25 June 2014; Accepted 14 August 2014 Academic Editor: Michael Chen Copyright © 2015 Lijun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper considers the adaptive synchronization of general complex dynamic networks via state predictor based on the fixed topology for nonlinear dynamical systems. Using Lyapunov stability properties, it is proved that the complex dynamical networks with state predictor are asymptotically stable. Moreover, it is also shown that the rate of convergence of complex dynamical networks with state predictor is faster than the complex dynamical networks without state predictor. 1. Introduction In recent years, the synchronization of complex dynamical networks has received more and more attention. The synchronous research can be applied in many fields, such as biology, smart city, computer, and the traffic [1–23]. It is well known that the complex network has a lot of nodes; however, in order to save an increasing number of energies, the pinning control is introduced to study the synchronization of complex dynamical networks. So far, the pinning control is a main tool by controlling a small number of nodes to steer the whole network. In [2], the pinning control of a continuous-time complex dynamical network with general coupling topologies was researched. The speed of synchronization is a significant issue, so, in [6], a state predictor was introduced. In [7], the adaptive synchronization of complex dynamical networks with state predictor was studied, therefore, this paper studies the problem using the pinning control. This paper considers the adaptive synchronization of general complex dynamic networks via state predictor based on the fixed topology for nonlinear dynamical systems. With the limited information, state predictor can predict the future state of the nodes and its neighbors; therefore, general complex dynamic networks via state predictor can be faster to achieve synchronization. This paper is organized as follows. Section 2 gives a model of the complex dynamical network. In addition, some preliminaries are introduced to prove the adaptive synchronization. Section 3 gives the main results and the theoretical analysis. The simulations of the theoretical results are given in Section 4. Finally, the conclusion is drawn in Section 5. 2. Preliminaries and Problem Statement Consider a complex dynamical network described by 𝑁 𝑥𝑖̇ (𝑡) = 𝑓 (𝑥𝑖 (𝑡)) + ∑𝑎𝑖𝑗 𝑐𝑖𝑗 (𝑡) [𝑥𝑗 (𝑡) − 𝑥𝑖 (𝑡)] 𝑗=1 (1) 𝑝 𝑝 + 𝛾 ∑ 𝑎𝑖𝑗 𝑐𝑖𝑗 (𝑡) (𝑥𝑖̇ (𝑡) − 𝑥𝑗̇ (𝑡)) + 𝑢𝑖 (𝑡) , 𝑗∈N𝑖 where 𝑥𝑖 (𝑡) = (𝑥𝑖1 (𝑡), 𝑥𝑖2 (𝑡), . . . , 𝑥𝑖𝑛 (𝑡))𝑇 ∈ 𝑅𝑛 (𝑖 = 1, 2, . . . , 𝑁) is the state vector of the 𝑖th node at time 𝑡, where 𝑡 is the continuous time; 𝑓𝑖 : 𝑅𝑛 → 𝑅𝑛 is a continuous function; 𝑁𝑖 represents the neighbor node of 𝑖; 𝑎𝑖𝑗 typify the coupling weight between any two nodes, where 𝑎𝑖𝑗 ≥ 0 and 𝑎𝑖𝑖 = 0; 𝑐𝑖𝑗 (𝑡) stands for the coupling strengths between node 𝑖 and node 𝑗; 2 Mathematical Problems in Engineering define the matrix of the weighted coupling configuration of the system as 𝑎11 𝑐11 [ 𝑎21 𝑐21 [ 𝑈 = [ .. [ . 𝑎12 𝑐12 ⋅ ⋅ ⋅ 𝑎1𝑁𝑐1𝑁 𝑎22 𝑐22 ⋅ ⋅ ⋅ 𝑎2𝑁𝑐2𝑁 ] ] 𝑁×𝑁 , .. .. ] ∈ 𝑅 ] . d . [𝑎𝑁1 𝑐𝑁1 𝑎𝑁2 𝑐𝑁2 ⋅ ⋅ ⋅ 𝑎𝑁𝑁𝑐𝑁𝑁] Δ = diag {𝛿1 , . . . , 𝛿𝑛 } , 𝑝 𝑝 𝑃 = diag {𝑝1 , . . . , 𝑝𝑛 } (2) Lemma 3 (see [8]). For any vectors 𝑥, 𝑦 ∈ 𝑅𝑛 and positivedefinite matrix 𝐺 ∈ 𝑅𝑛×𝑛 , the following matrix inequality holds: 2𝑥𝑇 𝑦 ≤ 𝑥𝑇 𝐺𝑥 + 𝑦𝑇 𝐺−1 𝑦. (3) 𝑝 ̇ )𝑇 , 𝛾 represents the impact factor where 𝑋̇ 𝑝 = (𝑥1̇ , 𝑥2̇ , . . . , 𝑥𝑁 of the state predictor. Under the state predictor (3), network (1) can be written as −2𝑎𝑇 𝑏 ≤ inf {𝑎𝑇 𝐸𝑎 + 𝑏𝑇 𝐸−1 𝑏} . 𝐸>0 𝑥𝑖̇ (𝑡) = 𝑓 (𝑥𝑖 (𝑡)) + ∑ 𝑎𝑖𝑗 𝑐𝑖𝑗 (𝑡) [𝑥𝑗 (𝑡) − 𝑥𝑖 (𝑡)] 𝑁 𝑖=1 − ∑ ∑ 𝑎𝑖𝑗 𝑎𝑗𝑝 𝑐𝑖𝑗 (𝑡) 𝑐𝑗𝑝 (𝑡)(𝑥𝑗 (𝑡) − 𝑥𝑝 (𝑡))] 𝑗∈N𝑖 𝑝∈N𝑗 ] + 𝑢𝑖 (𝑡) . (4) The control input is designed as (5) where ℎ𝑖 is a binary number; if the 𝑖th agent is controlled, ℎ𝑖 = 1; otherwise ℎ𝑖 = 0. 𝑐𝑖 is the feedback gain of position. Definition 1. Network (4) is said to achieve synchronization if 󵄩 󵄩 lim 󵄩󵄩𝑥 (𝑡) − 𝑥 (𝑡)󵄩󵄩󵄩 = 0, 𝑖 = 1 ⋅ ⋅ ⋅ 𝑁, (6) 𝑡→∞ 󵄩 𝑖 where the homogeneous state satisfies (8) where 𝑐𝑖𝑗 (0) ≥ 0. In the following, some necessary assumptions and lemmas are stated. Assumption 2 (see [10]). The continuous function 𝑓𝑖 : 𝑅𝑛 × [0, +∞] → 𝑅𝑛 satisfies 𝑇 (𝑥 − 𝑦) 𝑃 {[𝑓 (𝑥, 𝑡) − 𝑓 (𝑦, 𝑡)] − Δ (𝑥 − 𝑦)} 𝑇 ≤ −𝜔(𝑥 − 𝑦) (𝑥 − 𝑦) , 𝑁 𝑗=1,𝑗=𝑖̸ 𝑇 1𝑁 𝑁 = ∑ ∑ 𝑎𝑖𝑗 𝑐𝑖𝑗 (𝑥𝑖 − 𝑥𝑗 ) 𝑃 (𝑥𝑖 − 𝑥𝑗 ) . 2 𝑖=1 𝑗=1,𝑗=𝑖̸ (13) Lemma 6 (see [18]). For a connected graph which is undirected, the Laplace matrix is positive semidefinite matrix, and the minimum nonzero eigenvalue is the algebraic connectivity of 𝐿, as follows: 𝜆 2 (𝐿) = 𝑥𝑇 𝐿𝑥 . 2 𝑥=0,1 ̸ 𝑥=0 ‖𝑥‖ min 𝑇 (14) Lemma 7 (see [18]). For a system which is similar to 𝑥𝑖̇ = 𝑢𝑖 (𝑖 = 1, 2, . . . , 𝑛), the evolution rate associated with the minimum nonzero eigenvalue 𝜆 2 . 𝜆 2 describes the lower bound of convergence rate. Generally, the bigger the 𝜆 2 is, the faster the system converges. 3. Main Results (7) The adaptive control at node 𝑖 is designed as 𝑇 𝑇 ∑(𝑥𝑖 − 𝑥) 𝑃 ∑ 𝑎𝑖𝑗 𝑐𝑖𝑗 (𝑥𝑖 − 𝑥𝑗 ) − 𝛾 [ ∑ ∑ 𝑎𝑖𝑗 𝑎𝑖𝑘 𝑐𝑖𝑗 (𝑡) 𝑐𝑖𝑘 (𝑡) (𝑥𝑖 (𝑡) − 𝑥𝑘 (𝑡)) [𝑗∈N𝑖 𝑘∈N𝑖 𝑐𝑖𝑗̇ (𝑡) = 𝑎𝑖𝑗 𝑘𝑖𝑗 [𝑥𝑖 (𝑡) − 𝑥𝑗 (𝑡)] 𝑃 [𝑥𝑖 (𝑡) − 𝑥𝑗 (𝑡)] , (12) Lemma 5 (see [10]). The following equation holds: 𝑗=1 𝑥̇ (𝑡) = 𝑓 (𝑥 (𝑡) , 𝑡) = 0. (11) Lemma 4 (see [9]). Suppose that 𝑎 and 𝑏 are vectors; then for any positive-definite matrix 𝐸, the following inequality holds: 𝑁 𝑢𝑖 = −ℎ𝑖 𝑐𝑖 (𝑥𝑖 (𝑡) − 𝑥 (𝑡)) , (10) are positive constant matrices, for the constant 𝜔 > 0. with 𝑎𝑖𝑖 𝑐𝑖𝑖 = − ∑𝑁 𝑗=1,𝑗=𝑖̸ 𝑎𝑖𝑗 𝑐𝑖𝑗 . Introduce the state predictor as 𝑋̇ 𝑝 = −𝐿𝑋, for ∀𝑥, 𝑦 ∈ 𝑅𝑛 . And (9) In the following, we will give the main result. Theorem 8. Consider network (4) with the state predictor (3) and 𝑁 nodes steered by adaptive control (8), under Assumption 2, and at least one node is selected to be controlled. Then, all nodes asymptotically synchronize with the given homogeneous stationary state: 󵄩󵄩 󵄩󵄩 lim 󵄩𝑥 (𝑡) − 𝑥 (𝑡)󵄩󵄩 = 0. 𝑡→∞ 󵄩 𝑖 (15) Proof. Let 𝑥̃𝑖 (𝑡) ≜ 𝑥𝑖 (𝑡) − 𝑥(𝑡). Construct the following Lyapunov function: 𝑉 (𝑡) = 𝑉1 (𝑡) + 𝑉2 (𝑡) , (16) Mathematical Problems in Engineering 3 where 𝑁 𝑁 𝑖=1 𝑖=1 ≤ −𝜔∑𝑥̃𝑖𝑇 𝑥̃𝑖 + ∑𝑥̃𝑖𝑇 𝑃Δ𝑥̃𝑖 1𝑁 𝑉1 (𝑡) = ∑𝑥̃𝑖𝑇 (𝑡) 𝑃𝑥̃𝑖 (𝑡) , 2 𝑖=1 𝑁 𝑁 𝑖=1 𝑗=1 + ∑𝑥̃𝑖𝑇 𝑃 ∑𝑎𝑖𝑗 𝑐𝑖𝑗 (𝑡) (𝑥𝑗 − 𝑥𝑖 ) (17) 2 [𝑐𝑖𝑗 (𝑡) − 𝑚] 1𝑁 𝑉2 (𝑡) = ∑ ∑ . 2 (...truncated)