Foreword It gives us great pleasure to introduce this collection of papers to be presented at the 2012 International Conference on Computer Design (CDES’12), July 16 through 19, 2012, at Monte Carlo Resort, Las Vegas, USA. The Academic Co-Sponsors, Corporate Co-Sponsors, Co-Sponsors At-Large and Organizers of this year's conference include (separated by semicolons): Bioinformatics & Computational Biology Program, George Mason University, Virginia, USA; Biomedical Cybernetics Laboratory, HST of Harvard University and MIT, USA; Minnesota Supercomputing Institute, University of Minnesota, USA; Center for Cyber Defense, NCAT, USA; Argonne's Leadership Computing Facility of Argonne National Laboratory, Illinois, USA; The Center for Advanced Studies in Identity Sciences (CASIS: NC A&T, Carnegie Mellon, Clemson, UNC Wilmington), USA; Knowledge Management & Intelligent System Center (KMIS) of University of Siegen, Germany; Intelligent Cyberspace Engineering Lab., ICEL, Texas A&M University, Commerce, Texas, USA;UMIT, Institute of Bioinformatics and Translational Research, Austria; Hawkeye Radiology Informatics, Department of Radiology, College of Medicine, University of Iowa, Iowa, USA;The International Council on Medical and Care Compunetics, Europe; US Chapter of World Academy of Science (http://www.world-academy-of-science.org/); Supercomputer Software Department (SSD), Institute of Computational Mathematics & Mathematical Geophysics, Russian Academy of Sciences, Russia; International Society of Intelligent Biological Medicine, USA; NDSU-CIIT Green Computing and Communications Laboratory, USA; Medical Image HPC & Informatics Lab (MiHi Lab), University of Iowa, Iowa, USA; High Performance Computing for Nanotechnology, USA; Manx Telecom, Europe; Computer Science Research, Education, and Applications Press; World Academy of Biomedical Sciences and Technologies; HoIP Telecom, Europe; Super Micro Computer, Inc., San Jose, California, USA; Intel Corporation; Hodges Health, UK; and OMG ™ . In addition, a number of university faculty members and their staff (names appear below and also on the cover of the proceedings), several publishers of computer science and computer engineering books and journals, chapters and/or task forces of computer science associations/organizations from 6 countries, and developers of high-performance machines and systems provided significant help in organizing the conference as well as providing some resources. An important mission of WORLDCOMP (a federated congress to which this conference is affiliated with) includes "Providing a unique platform for a diverse community of constituents composed of scholars, researchers, developers, educators, and practitioners. The Congress makes concerted effort to reach out to participants affiliated with diverse entities (such as: universities, institutions, corporations, government agencies, and research centers/labs) from all over the world. The congress also attempts to connect participants from institutions that have teaching as their main mission with those who are affiliated with institutions that have research as their main mission. The congress uses a quota system to achieve its institution and geography diversity objectives." The program committee would like to thank all those who submitted papers for consideration. About 65% of the submissions were from outside the United States. Each paper was peer-reviewed by two experts in the field for originality, significance, clarity, impact, and soundness. In cases of contradictory recommendations, a member of the conference program committee was charged to make the final decision; often, this involved seeking help from additional referees by using a double-blinded review process. In addition, papers whose authors included a member of the conference program committee were evaluated using the double-blinded review process. The only exception to the above evaluation process was for papers that were submitted directly to chairs/organizers of approved sessions/workshops; in these cases, the chairs/organizers were responsible for the evaluation of such submissions. The overall paper acceptance rate for regular papers was 28%; 17% of the remaining papers were accepted as poster papers (at the time of this writing, we had not yet received the acceptance rate for one individual track.) We are very grateful to the many colleagues who helped in organizing the conference. In particular, we would like to thank the members of the CDES’12 Program Committee who we hope will offer their help

again in organizing the next year's conference (CDES’13). The CDES’12 Program Committee members were:

• Prof. Babak Akhgar (WC Steering Committee), PhD, FBCS, CITP, Professor of Informatics, Sheffield Hallam University, Sheffield, UK

• Prof. Naji Masned Irshyd AlQbailat, Assistant Dean for Planning, Developing and Quality, Princess Alia University College, Al-Balqa' Applied University, Shmeisani, Amman, Jordan

• Prof. Hamid R. Arabnia (WC General Chair & Coordinator), Elected Fellow, ISIBM; Editor-in-Chief, The Journal of Supercomputing (Springer); Member, Advisory Board, IEEE TC on Scalable Computing; University of Georgia, Georgia, USA

• Prof. Baharuddin Aris, Professor and Director, Universiti Teknologi Malaysia, Johor Bahru, Malaysia • Dr. Ezendu Ariwa (WC Publicity Co-Chair), Chartered Fellow of The British Computer Society; Fellow of

Institute of Information Technology Training: Fellow of Higher Education Academy; Chair, IEEE Consumer Electronics Chapter and IEEE Broadcast Technology Chapter (UK&RI); Associate Prof./Senior Lecturer, Strategic Information Systems, London Metropolitan University, London, UK

• Dr. Waqas Haider Khan Bangyal, Iqra University Islamabad, Pakistan • Prof. H-P. Bischof, Rochester Institute of Technology, Rochester, New York, USA • Prof. Juan-Vicente Capella-Hernandez, Universitat Politecnica de Valencia, Valencia, Spain; Executive

Manager, Wireless Sensor Networks Valencia, Spain • Prof. Victor Clincy, Computer Science Department, College of Science and Mathematics, Kennesaw State

University, Kennesaw, Georgia, USA • Prof. Kevin Daimi (WC Steering Committee), Director, Computer Science and Software Engineering

Programs, Department of Mathematics, Computer Science and Software Engineering, University of Detroit Mercy, Detroit, Michigan, USA

• Prof. Gerry Vernon Dozier (WC Steering Committee), Chair, Department of Computer Science; Director, Center for Advanced Studies in Identity Sciences; Center for Cyber Defense; North Carolina A&T State University, North Carolina, USA

• Prof. Madjid Fathi (WC Steering Committee), Director, Knowledge Management and Intelligent Systems Center, University of Siegen, Germany

• Dr. Bilal Gonen, University of Alaska, Anchorage, Alaska, USA • Prof. Michael R. Grimaila (WC Steering Committee), Air Force Institute of Technology, Systems

Engineering; Fellow of ISSA; CISM, CISSP, IAM/IEM; Editorial Board of ISSA Journal; Air Force Center of Cyberspace Research; Advisor to the Prince of Wales Fellows & Prince Edward Fellows at MIT and Harvard Universities; PC member, NATO Cooperative Cyber Defence Centre of Excellence (CCD COE) & Int'l Conf. on Information Warfare and Security

• Dr. Shaikh Abdul Hannan, Department of Computer Science, Vivekanand College, Aurangabad, India • Prof. Houcine Hassan, Universitat Politecnica de Valencia, Spain • Dr. Shahram Javadi, Electrical Engineering Department, Azad University, Central Tehran Branch, Tehran,

Iran; Director in Chief, International Journal of Smart Electrical Engineering • Prof. Guillermo Botella Juan, Computer Architecture and Automation Department, Faculty of Computer

Science (Facultad de Informatica), Universidad Complutense de Madrid, Madrid, Spain • Prof. D. V. Kodavade, Head, Computer Science & Engineering Department, D.K.T.E Society's Textile &

Engineering Institute, Maharashtra State, India • Dr. Praveen Koduru, Electrical & Computer Engineering, Kansas State University, USA • Dr. B. V. Durga Kumar, Taylors University, Malaysia • Dr. A. V. Senthil Kumar, Director, Department of MCA, Hindusthan College of Arts and Science, Hindusthan

Gardens, India • Prof. Kun Chang Lee (WC Steering Committee), Professor of MIS and WCU Professor of Creativity Science,

Business School and Department of Interaction Science, Sungkyunkwan University, Seoul, South Korea • Prof., Dr., Dr.h. Victor Malyshkin (WC Steering Committee), Head, Supercomputer Software Department

(SSD), Institute of Computational Mathematics and Mathematical Geophysics, Russian Academy of Sciences, Russia

• Prof. George Markowsky (WC Steering Committee), Associate Director, School of Computing and Information Science; Chair International Advisory Board of IEEE IDAACS; Director 2013 Northeast Collegiate Cyber Defense Competition; Chair Bangor Foreign Policy Forum; Cooperating Professor Mathematics and Statistics Department UMaine; Cooperating Professor School of Policy & International Affairs UMaine; University of Maine, Orono, Maine, USA

• Prof. Andy Marsh (WC Steering Committee), Director HoIP; Director HoIP Telecom, UK; Secretary-General WABT; Vice-president ICET; Visiting Professor University of Westminster, UK

• Farhad Mehran, Saman Sanat Jahan Gostar Co., Tehran, Iran

• Dr. Sara Moein, Editorial board, International Journal of Science and Technology, Faculty of Engineering, MultiMedia University, Malaysia

• Dr. Ali Mostafaeipour, Industrial Engineering Department, Yazd University, Yazd, Iran • Dr. Mohammad Hossein Nadimi-Shahraki, Head, Research Department, Artificial Intelligence, Faculty of

Computer Engineering, Najafabad branch, Islamic Azad University, Iran • Prof. Max M. North, Professor of Management Information Systems; Pioneer of Virtual Reality Therapy;

Director of Visualization & Simulation Research Center; School of Engineering Technology & Management; Southern Polytechnic State University; Marietta, Georgia, USA

• Dr. Sarah M. North, Distance Learning Coordinator, Kennesaw State University, Kennesaw, Georgia, USA • Prof. James J. (Jong Hyuk) Park (WC Steering Committee), Department of Computer Science and

Engineering, Seoul National University of Science and Technology (SeoulTech), Korea; President, KITCS; President, FTRA; Editor-in-Chiefs: HCIS, JoC and IJITCC Journals

• Prof. Yongyuth Permpoontanalarp, Logic and Security Lab, Department of Computer Engineering, King Mongkut's University of Technology Thonburi, Bangkok, Thailand

• Dr. Kadiyala Ramana, Annamacharya Institute of Technology and Sciences, Andhra Pradesh, India • Dr. Hassan Reza (WC Steering Committee), UND Aerospace, University of North Dakota, Department of

Computer Science, Grand Forks, North Dakota, USA • Dr. Won Woo Ro, Yonsei University, Seoul, South Korea • Dr. Yong Shi, Kennesaw State University, Georgia, USA • Dr. Akash Kumar Singh, IT Architect, IBM, Sacramento, California, USA • Ashu M. G. Solo (WC Publicity Chair), Fellow of British Computer Society, Principal/R&D Engineer,

Maverick Technologies America Inc. • Prof. Sang C. Suh (WC Steering Committee), Head and Professor, Department of Computer Science; Vice

President, Society for Design and Process Science (SDPS); Director, Intelligent Cyberspace Engineering Lab (ICEL); Texas A&M University, Commerce, Texas, USA

• Dr. Sim Kok Swee, Faculty of Engineering and Technology, Jalan Ayer Keroh Lama, Melaka, Malaysia • Prof. Ousmane Thiare, Department of Computer Science, Gaston Berger University, Senegal • Prof. Keshav D. Verma, Chairman, Department at S.V. (P.G.) College, Aligarh, India; Founder and

Director, MS Research Laboratory (MSRL), India; Editor-in-Chief: IJNMC Journal + IJBRE Journal + MSRJ Journal

• Prof. Layne T. Watson (WC Steering Committee), IEEE Fellow; NIA Fellow; ISIBM Fellow; Fellow of The National Institute of Aerospace; Virginia Polytechnic Institute & State University, Virginia, USA

• Dr. Wei Wei, Xi'an University of Technology, Xi'an, P. R. China • Prof. Jeff Zadeh, Collegiate Professor and Program Chair, University of Maryland, University College

Europe, Germany; University of Maryland, USA We express our gratitude to keynote, invited, and individual conference/tracks and tutorial speakers - the list of speakers appears on the conference web site. We would also like to thank the followings: UCMSS (Universal Conference Management Systems & Support, California, USA) for managing all aspects of the conference; Dr. Tim Field of APC for managing the printing of the proceedings; and the staff of Monte Carlo Resort in Las Vegas for the professional service they provided. Last but not least, we would like to thank the Co-Editors of CDES’12: Prof. Hamid R. Arabnia, Prof. Leonidas Deligiannidis, Prof. Andy Marsh, and Ashu M. G. Solo. We present the proceedings of CDES’12. Steering Committee, CDES 2012 http://www.world-academy-of-science.org/worldcomp12/ws

CDES'12

The 2012 International Conference on Computer Design

Foreword

SESSION: PERFORMANCE ISSUES AND ENHANCEMENT METHODS + LOW POWERCOMPUTING AND ANALYSIS

Reducing Energy Usage of NULL Convention Logic Circuits using NULL Cycle ReductionCombined with Supply Voltage ScalingBrett Sparkman, Scott SmithNew Single-Phase Adiabatic Logic FamilyMihail Cutitaru, Lee A. Belfore, IIOptimising Energy Management of Mobile Computing DevicesMartin J. Johnson, Ken A. HawickDesign and Low Power Implementation of a Reorder BufferJon David Fisher, Claudia Romo, Eugene John, Wei-Ming Lin

SESSION: ALGORITHMS, LOGIC, CIRCUIT/HARDWARE DESIGN, AND TOOLS

Area-Time Efficient Digit-Serial-Serial Two's Complement MultiplierEssam Elsayed, Hatem El-BoghdadiRB_DSOP: A Rule Based Disjoint Sum of Products Synthesis MethodP. Balasubramanian, R. Arisaka, H. R. ArabniaAnalysis of Notebook Computer Chassis Design for Hard Disk Drive and Speaker MountingJ. Q. Mou, Fukun Lai, I. B. L. See, W. Z. LinAccurate Throughput Derivation of Pipelined NULL Convention Logic Asynchronous CircuitsLiang Zhou, Scott SmithEffect of Channel Lengthening and Threshold Voltage Variation on a Nanometric Gate�sDelay and PowerAzam Beg, Amr Elchouemi, Raahim Beg

SESSION: OPERATING SYSTEMS TOOLS AND DESIGN + COMMUNICATION SYSTEMSAND DESIGN

A Survey on Computer System Memory ManagementQi ZhuSynchronization of a Complex Dynamical Network with Nonidentical Nodes via DynamicFeedback ControlT.H. Lee, J.H. Park, H.Y. Jung, S.M. Lee

SESSION: HPC AND MULTI-PROCESSOR MULTI-CORE SYSTEMS + DESIGN ISSUES + FPGA+ GPU + NOC + EMBEDDED SYSTEMS

A Novel Branch Predictor Using Local History for Miss-Prediction BiasLin Meng, Katsuhiro Yamazaki, Shigeru OyanagiA Modular Processor Architecture for High-Performance Computing Applications on FPGAFritz Mayer-LindenbergField Programmable Gate Arrays for Computational Acceleration of Lattice-OrientedSimulation ModelsAndrew Gilman, Ken A. HawickParallel Counter Formulation using a Generator Polynomial ExpansionLee A. Belfore IIPerformance Bound Energy Efficient L2 Cache Organization for Emerging Workload forMulti-Core Processor: A Comparison of Private and Shared CacheRamya Arun, Eugene JohnSimplified FPGA Design with RobeiGuosheng WuSD-MARC: A New Multi-Processor ArchitectureSomdip DeyA New Technique To Use A Parallel Compiler for Multi- core MicrocontrollersSomdip DeyData Center Design and Implementation at the UniversityAskar Boranbayev, Sergey Belov

Synchronization of a complex dynamical network with nonidenticalnodes via dynamic feedback control

T.H. Lee1, J.H. Park1, H.Y. Jung1, S.M. Lee21Nonlinear Dynamics Group/Dept. EE/ICE, Yeungnam University, Kyongsan, Republic of Korea.

2Department of Electronic Engineering, Daegu University, Gyungsan, Republic of Korea.

Abstract— This paper considers synchronization problemof a complex dynamical network with nonidentical nodes.For this problem, a dynamic feedback controller is de-signed to achieve the synchronization of the network. Basedon Lyapunov stability theory and linear matrix inequalityframework, the existence condition for feasible controllers isderived in terms of linear matrix inequalities. The conditioncan be solved easily by the application of convex optimiza-tion algorithms. Finally, the proposed method is applied toa numerical example in order to show the effectiveness ofour result.

Keywords: Complex dynamical network, synchronization, non-identical node, dynamic feedback control.

1. IntroductionDuring the last decade, complex dynamical networks,

which are a set of interconnected nodes with specific dy-namics, have been attracted increasing attention in variousfields such as physics, biology, chemistry and computerscience [1]. As science and society develop, our everydaylives have been closed to complex networks, for instance,transportation networks, World Wide Web, coupled biolog-ical and chemical engineering systems, neural networks,social networks, electrical power grids and global economicmarkets. Many of these networks exhibit complexity in theoverall topological and dynamical properties of the networknodes and the coupled units. Recently, one of the significantand interesting phenomena in complex dynamical network isthe synchronization. Synchronization of complex dynamicalnetworks can be divided into two points of view. One isthe synchronization of a complex network that is called’inner synchronization’ [2]-[5]. It means that all the nodesin a complex network eventually approach to trajectory of atarget node. Another is called ’outer synchronization’ [6]-[8]which considers the synchronization between two or morecomplex networks. In this paper, a new control problem forinner synchronization will be investigated.

The random-graph model had become a basics of modernnetwork theory since proposed by Erdös and Renyi [9]-[10].In a random network, each pair of nodes is connected witha certain probability. Watts and Strogatz [11] introduceduseful network model to translate from a regular network toa random network, it is called small-world network. Then,

Newman and Watts [12] modified it to generate anothervariant of the small-world model. And then, Barabasi andAlbert [13] proposed a scale-free network model, in whichthe degree distribution of the nodes follows a power-lawform . Thereafter, small-world and scale-free networks havebeen extensively investigated.

Synchronization of a complex dynamical network havebeen well noticed that many researchers adopt the assump-tion that all nodes dynamics are identical [2]-[3]. However,this assumption about identical nodes is unlikely environ-ment in most of complex dynamical networks. For example,in a swarm robot system, every individual robots have differ-ent dynamics of them, and even if the swarm robot system isconsisted of same robots, it has possibility to be nonidenticalnetwork system due to uncertainties, saturation and so on.When the nodes of a complex network is nonidentical, theywill show different dynamics. It should be noted that thesynchronization schemes for networks with identical nodes isgood for nothing. Therefore, the further investigation of newsynchronization schemes for a complex dynamical networkwith nonidentical nodes is necessary. In this regard, only afew papers have been reported until now [4]-[5].

In this paper, we will investigate the synchronization of acomplex network with nonidentical node via dynamic feed-back control unlike previous works which usually treated acomplex network with identical nodes. Until now, in orderto treat this kind of problem for a complex network, severalcontrol schemes such as adaptive control ([2], [7]-[8]) andpinning control ([2]-[3], [6]) are applied. However, to thebest of the authors’ knowledge, the synchronization problemvia dynamic feedback controller for complex networks hasnot been investigated up to now. In some real controlsituations, there is a strong need to construct a dynamicfeedback controller instead of a static feedback controllerin order to obtain a better performance and dynamicalbehavior of the state response. The dynamic controller willprovide more flexibility compared to the static controllerand the apparent advantage of this type of controller is thatit provides more free parameters for selection [14]. So itis very worth to consider the design problem of dynamiccontroller for synchronization in a complex network. Theexistence condition of such controller is derived in terms oflinear matrix inequalities (LMIs) which can be easily solvedby standard convex optimization algorithms [15].

Notation: X > 0 (respectively, X ≥0) means that the matrixX is a real symmetric positive definite matrix (respectively,positive semi-definite). In denotes the n-dimensional iden-tity matrix. ⊗ denotes the notation of Kronecker product.

2. Problem statement and preliminariesConsider a delayed complex dynamical network con-

sisting of N linearly coupled nonidentical nodes describedby

xi(t) = fi(xi(t)) +N∑j=1

cijxj(t) + ui(t), i = 1, . . . , N (1)

where xi = (xi1, xi2, . . . , xin)T ∈ Rn is the state vector of

the ith node, fi : Rn → Rn is a smooth nonlinear vectorfield, ui(t) is the control input of ith node, and cij is thecoupling configuration parameter representing the couplingstrength and the topological structure of the network, inwhich cij is nonzero if there is a connection from nodei to node j(i = j), and is zero otherwise. For simplicity,let us define C = (cij)N×N . Also, the diagonal elements ofmatrix C are assumed that

cii = −N∑

j=1,j =i

cij , i = 1, . . . , N. (2)

Definition 1. A complex network is said to achieve asymp-totical inner synchronization, if

x1(t) = x2(t) = · · · = xN (t) = s(t) as t → ∞,

where s(t) ∈ Rn is a solution of a target node, satisfyings(t) = fs(s(t)).

Here, define error vectors as follows :

ei(t) = s(t)− xi(t). (3)

From Eq. (3), the error dynamics is given to

ei(t) = fs(s(t))− fi(xi(t))−N∑j=1

cijej(t)− ui(t)

= fi(ei(t))−N∑j=i

cijej(t)− ui(t), (4)

where fi(ei(t)) = fs(s(t))− fi(xi(t)).Also, a vector-matrix form of Eq. (4) is described by

e(t) = F (t)− C ⊗ Ine(t)− U(t) (5)

where F =[fT1 (e1(t)), f

T2 (e2(t)), . . . , f

TN (eN (t))

]T , e =[eT1 (t), . . . , e

TN (t)

]T, and U =

[uT1 (t), u

T2 , . . . , u

TN (t)

].

3. Controller designIn this section, a dynamic feedback controller will be

designed to achieve the synchronization goal.In order to stabilize the error system given in Eq. (5), let’sconsider the following dynamic feedback controllers:

ζ(t) = Ac ⊗ Inζ(t) +Bc ⊗ Ine(t),U(t) = Cc ⊗ Inζ(t) + F (t), ζ(0) = 0,

(6)

where ζ(t) ∈ RNn is the controller state, and Ac, Bc andCc are constant gain matrices of N ×N dimensions.

Applying this controller (6) to error system (5) results inthe following closed-loop system

z(t) = H ⊗ Inz(t), (7)

where

z(t) =

[e(t)ζ(t)

]∈ R2Nn, H =

[−C −Cc

Bc Ac

]∈ RN×N .

Then we have following main result.Theorem 1. There exists a dynamic feedback controllergiven in Eq. (6) for synchronization of the complex networkEq. (1) if there exist positive-definite matrices S, Y ∈ RN×N

and matrices X1, X2, X3 ∈ RN×N satisfying the followingLMIs :[

−Y CT − CY −X1 −XT1 −C +X3

⋆ (2, 2)

]< 0 (8)

and [Y ININ S

]> 0. (9)

where (2, 2) = −CTS − STC +X2 +XT2 .

Proof. Consider the following Lyapunov function: V (t) =zT (t)P ⊗ Inz(t) where P ∈ R2N×2N > 0. Then, the timederivative of the Lyapunov function is

V (t) = zT (t)(HTP + PH)⊗ Inz(t). (10)

Here, let us define

Σ ≡ HTP + PH. (11)

Thus, if the inequality Σ < 0 holds, then it can be saidthat synchronization of a complex network with nonidenticalnode is achieved by our proposed dynamic controller. How-ever, in the matrix Σ, the matrix P > 0 and the controllerparameters Ac, Bc and Cc, which included in the matrixH , are unknown and occur in nonlinear fashion. Hence, theinequality Σ < 0 cannot be considered as an linear matrixinequality problem. In the following, we will use a methodof changing variables such that the inequality can be solvedas convex optimization algorithm [16].

First, partition the matrix P and its inverse as

P =

[S JJT T

], P−1 =

[Y MMT W

], (12)

where S, Y are positive-definite matrices, and M,N ∈RN×N are invertible matrices. It should be pointed out thatthe equality P−1P = I2N gives that

MJT = IN − Y S. (13)

Define two matrices as

F1 =

[Y INJT 0

], F2 =

[IN S0 JT

]. (14)

Then, it follows that

PF1 = F2, FT1 PF1 = FT

1 F2 =

[Y ININ S

]> 0. (15)

Now, postmultiplying and premultiplying the matrix in-equality, Σ < 0, by the matrix FT

1 and by its transpose,respectively, gives

FT2 HF1 + FT

1 HTF2 < 0. (16)

By utilizing the relation Eqs. (12)-(15), it can be easilyobtained that the inequality Eq. (16) is equivalent to[

Γ1 Γ2

⋆ Γ3

]< 0 (17)

where Γ1 = −Y CT − CY − MCTc − CcM

T ,Γ2 = −C − Y CTS + Y XT

2 − X1S + MATc J

T ,Γ3 = −CTS − STC + JBc + BT

c JT . By defining a new

set of variables as follows: X1 = MCTc , X2 = JBc,

X3 = −Y CTS + Y XT2 −X1S +MAT

c JT , then, Eq. (17)

is simplified to LMI (8). And the LMI (9) is equivalent tothe positiveness of P . This completes the proof. �

Remark 1. Given any solution of the LMIs given in Eqs.(8) and (9) in Theorem 1, a corresponding controller of theform Eq. (6) will be constructed as follows:

• Compute the invertible matrices M and J satisfyingEq. (13) using matrix algebra.

• Utilizing the matrices M and J obtained above, solvethe equations Xi for Bc, Cc and Ac (in this order).

4. Numerical exampleIn order to show the effectiveness of the proposed method,

we present a numerical example which is inner synchro-nization of a complex network with five nonidentical nodes.Each nodes are different chaotic systems such as well-known Lorenz, Chen, Lü, Chen-Lee and Genesio-Tesi sys-tems. They are typical benchmark three dimensional chaoticsystems and their chaotic behavior are displayed in Fig. 1.Thus, the complex network system consisting of five nodesis described by:

xi(t) = fi(xi(t)) +

N∑j=1

cijxj(t) + ui(t), (18)

−100

10

−20

0

20

10

20

30

40

50

x11

Lorenz system

x12

x 13

−200

20

−50

0

500

20

40

x21

Chen system

x22

x 23

−200

20

−20

0

200

20

40

60

x31

Lu system

x32

x 33

−500

50

−50

0

50−20

−10

0

x41

Chen−Lee system

x42

x 43

−100

10

−10

0

10−5

0

5

10

x51

Genesio−Tesi system

x53

x 52

Fig. 1: Original chaotic behavior of each nodes

where i = 1, . . . , 5 and

f1(x1(t)) =

p1(x12 − x11)p3x11 − x12 − x11x13

x11x12 − p2x13

f2(x2(t)) =

p4(x22 − x21)(p6 − p4)x21 + p6x21 − x21x23

x21x21 − p5x23

f3(x3(t)) =

p7(x32 − x31)p9x32 − x31x33

x31x32 − p8x33

f4(x4(t)) =

q1x41 − x42x43

−q2x42 + x41x43

−q3x43 + (1/3)x41x42

f5(x5(t)) =

x52

x53

−q4x51 − q5x52 − q6x53 + x251,

with the parameters p1 = 10, p2 = 8/3, p3 = 28, p4 = 35,p5 = 3, p6 = 28, p7 = 36, p8 = 3, p9 = 20, q1 = 5,q2 = 10, q3 = 3.8, q4 = 6, q5 = 2.92, and q6 = 1.2.Each nodes represent chaotic behavior as follows :f1(x1(t))− Lorenz System, f2(x2(t))− Chen System,f3(x3(t))− Lü System, f4(x4(t))− Chen-Lee System,f5(x5(t))− Genesio-Tesi System. And a target node is alsoselected same one as first node, Lorenz system. In thisexample, random function, |d(t)| < 10, is used to everyinitial conditions of xi(0) = (d(t), d(t), d(t)), s(0) =(d(t), d(t), d(t)), (i = 1, 2, . . . , 5) and coupling matrix,C, is given by

C = 0.2×

−3 1 1 0 11 −4 1 1 10 1 −2 1 00 1 1 −3 11 0 0 1 −2

. (19)

In order to show original behavior of the complex network(18) with nonidentical node, the trajectories of the complexnetwork (18) without controller is depicted in Fig 2.

0 2 4 6 8 10 12 14 16 18 20−60

−40

−20

0

20

40

60

time

e

Fig. 2: Error signals of Example without control input

Now, we design a suitable dynamic feedback controller ofthe form Eq. (6) for system Eq. (18), which guarantees theasymptotic stability of the closed-loop system. By applica-tion of Theorem 1 to the system Eq. (18) and checking thefeasibility of the LMIs given in Eqs (8) and (9), we can findthat the LMIs are feasible by use of LMI control toolbox andobtain a possible set of solution of the LMIs. But, due tolimitation of space, the solutions are omitted here. Then, byfurther calculation in light of Remark 1, we have a possiblestabilizing dynamic feedback controller for the system Eq.(18):

Ac =

−1.6000 0.2667 0.0376 0.0580 −0.06380.3141 −1.6010 0.0853 0.3113 0.15370.0212 0.2735 −1.6185 0.0896 −0.19720.0639 0.2963 0.0465 −1.4779 0.17950.1297 0.1701 −0.1499 0.1795 −1.5026

Bc =

−1.1000 0.2000 0.1000 0.0000 0.20000.2904 −0.8863 −0.1097 0.5507 −0.51200.0294 0.8606 −0.0314 −0.0612 −0.75670.0609 −0.0716 0.2112 −1.0067 0.14880.0330 0.4988 −0.9123 0.0974 0.0563

Cc =

−1.1000 0.2904 0.0294 0.0609 0.03300.2000 −0.8863 0.8606 −0.0716 0.49880.1000 −0.1097 −0.0314 0.2112 −0.91230.0000 0.5507 −0.0612 −1.0067 0.09740.2000 −0.5120 −0.7567 0.1488 0.0563

The simulation result with control input is presented in

Fig. 3. As seen in Fig. 3, the trajectories of error systemsapproach to zero as expected. This achieves asymptoticsynchronization of the complex network (18).

0 2 4 6 8 10 12 14 16 18 20−15

−10

−5

0

5

10

15

time

e

Fig. 3: Error signals of Example

5. ConclusionsIn this paper, the asymptotic inner synchronization of a

complex dynamical network has been studied. The noniden-tical node was considered in the sense of reality. Unlikeother works, a dynamic feedback controller was designedfor the our synchronization scheme based on the Lyapunovmethod. Then, a criterion expressed by LMIs for stability oferror dynamics was derived. Finally, one numerical examplewas illustrated to show the effectiveness of the designedcontroller.

AcknowledgementsThis work was supported in part by MEST & DGIST (12-BD-0101,

Renewable energy and intelligent robot convergence technology develop-ment.) This research was also supported by Basic Science Research Programthrough the National Research Foundation of Korea (NRF) funded by theMinistry of Education, Science and Technology (2010-0009373).

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