Life System Modeling and Intelligent Computing: International Conference on Life System Modeling and Simulation, LSMS 2010, and International ... II (Lecture Notes in Computer Science, 6329) 9783642155963, 3642155960

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Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max Planck Institute for Informatics, Saarbruecken, Germany

6329

Kang Li Minrui Fei Li Jia George W. Irwin (Eds.)

Life System Modeling and Intelligent Computing International Conference on Life System Modeling and Simulation, LSMS 2010 and International Conference on Intelligent Computing for Sustainable Energy and Environment, ICSEE 2010 Wuxi, China, September 17-20, 2010 Proceedings, Part II

13

Volume Editors Kang Li The Queen’s University of Belfast, Intelligent Systems and Control School of Electronics, Electrical Engineering and Computer Science Ashby Building, Stranmillis Road, Belfast BT9 5AH, UK E-mail: [email protected] Minrui Fei Shanghai University, School of Mechatronical Engineering and Automation P.O. Box 183, Shanghai 200072, China E-mail: [email protected] Li Jia Shanghai University, School of Mechatronical Engineering and Automation P.O. Box 183, Shanghai 200072, China E-mail: [email protected] George W. Irwin The Queen’s University of Belfast, Intelligent Systems and Control School of Electronics, Electrical Engineering and Computer Science Ashby Building, Stranmillis Road, Belfast BT9 5AH, UK E-mail: [email protected]

Library of Congress Control Number: 2010933354 CR Subject Classification (1998): J.3, F.1, I.4, F.2, I.6, C.3 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN ISBN-10 ISBN-13

0302-9743 3-642-15596-0 Springer Berlin Heidelberg New York 978-3-642-15596-3 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2010 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper 06/3180

Preface

The 2010 International Conference on Life System Modeling and Simulation (LSMS 2010) and the 2010 International Conference on Intelligent Computing for Sustainable Energy and Environment (ICSEE 2010) were formed to bring together researchers and practitioners in the fields of life system modeling/simulation and intelligent computing applied to worldwide sustainable energy and environmental applications. A life system is a broad concept, covering both micro and macro components ranging from cells, tissues and organs across to organisms and ecological niches. To comprehend and predict the complex behavior of even a simple life system can be extremely difficult using conventional approaches. To meet this challenge, a variety of new theories and methodologies have emerged in recent years on life system modeling and simulation. Along with improved understanding of the behavior of biological systems, novel intelligent computing paradigms and techniques have emerged to handle complicated real-world problems and applications. In particular, intelligent computing approaches have been valuable in the design and development of systems and facilities for achieving sustainable energy and a sustainable environment, the two most challenging issues currently facing humanity. The two LSMS 2010 and ICSEE 2010 conferences served as an important platform for synergizing these two research streams. The LSMS 2010 and ICSEE 2010 conferences, held in Wuxi, China, during September 17–20, 2010, built upon the success of two previous LSMS conferences held in Shanghai in 2004 and 2007 and were based on the Research Councils UK (RCUK)-funded Sustainable Energy and Built Environment Science Bridge project. The conferences were jointly organized by Shanghai University, Queen's University Belfast, Jiangnan University and the System Modeling and Simulation Technical Committee of CASS, together with the Embedded Instrument and System Technical Committee of China Instrument and Control Society. The conference program covered keynote addresses, special sessions, themed workshops and poster presentations, in addition to a series of social functions to enable networking and foster future research collaboration. LSMS 2010 and ICSEE 2010 received over 880 paper submissions from 22 countries. These papers went through a rigorous peer-review procedure, including both pre-review and formal refereeing. Based on the review reports, the Program Committee finally selected 260 papers for presentation at the conference, from amongst which 194 were subsequently selected and recommended for publication by Springer in two volumes of Lecture Notes in Computer Science (LNCS) and one volume of Lecture Notes in Bioinformatics (LNBI). This particular volume of Lecture Notes in Computer Science (LNCS) includes 55 papers covering 7 relevant topics.

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Preface

The organizers of LSMS 2010 and ICSEE 2010 would like to acknowledge the enormous contributions from the following: the Advisory and Steering Committees for their guidance and advice, the Program Committee and the numerous referees worldwide for their significant efforts in both reviewing and soliciting the papers, and the Publication Committee for their editorial work. We would also like to thank Alfred Hofmann, of Springer, for his continual support and guidance to ensure the highquality publication of the conference proceedings. Particular thanks are of course due to all the authors, as without their excellent submissions and presentations, the two conferences would not have occurred. Finally, we would like to express our gratitude to the following organizations: Chinese Association for System Simulation (CASS), IEEE SMCS Systems Biology Technical Committee, National Natural Science Foundation of China, Research Councils UK, IEEE CC Ireland chapter, IEEE SMC Ireland chapter, Shanghai Association for System Simulation, Shanghai Instrument and Control Society and Shanghai Association of Automation. The support of the Intelligent Systems and Control research cluster at Queen’s University Belfast, Tsinghua University, Peking University, Zhejiang University, Shanghai Jiaotong University, Fudan University, Delft University of Technology, University of Electronic Science Technology of China, Donghua University is also acknowledged.

July 2010

Bohu Li Mitsuo Umezu George W. Irwin Minrui Fei Kang Li Luonan Chen Li Jia

LSMS-ICSEE 2010 Organization

Advisory Committee Kazuyuki Aihara, Japan Zongji Chen, China Guo-sen He, China Frank L. Lewis, USA Marios M. Polycarpou, Cyprus Olaf Wolkenhauer, Germany Minlian Zhang, China

Shun-ichi Amari, Japan Peter Fleming, UK Huosheng Hu,UK Stephen K.L. Lo, UK Zhaohan Sheng, China

Erwei Bai, USA Sam Shuzhi Ge, Singapore Tong Heng Lee, Singapore Okyay Kaynak, Turkey Peter Wieringa, The Netherlands

Cheng Wu, China Guoping Zhao, China

Yugeng Xi, China

Kwang-Hyun Cho, Korea

Xiaoguang Gao, China

Shaoyuan Li, China Sean McLoone, Ireland Xiaoyi Jiang, Germany Kok Kiong Tan, Singapore Tianyuan Xiao, China Donghua Zhou, China

Liang Liang, China Robert Harrison, UK Da Ruan, Belgium Stephen Thompson, UK Jianxin Xu, Singapore Quanmin Zhu, UK

Steering Committee Sheng Chen, UK Tom Heskes, The Netherlands Zengrong Liu, China MuDer Jeng, Taiwan, China Kay Chen Tan, Singapore Haifeng Wang, UK Guangzhou Zhao, China

Honorary Chairs Bohu Li, China Mitsuo Umezu, Japan

General Chairs George W. Irwin, UK Minrui Fei, China

International Program Committee IPC Chairs Kang Li, UK Luonan Chen, Japan

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Organization

IPC Regional Chairs Haibo He, USA Wen Yu, Mexico Shiji Song, China Xingsheng Gu, China Ming Chen, China

Amir Hussain, UK John Morrow, UK Taicheng Yang, UK Yongsheng Ding, China Feng Ding, China

Guangbin Huang, Singapore Qiguo Rong, China Jun Zhang, USA Zhijian Song, China Weidong Chen, China

Maysam F. Abbod, UK Vitoantonio Bevilacqua, Italy Yuehui Chen, China

Peter Andras, UK Uday K. Chakraborty, USA Xinglin Chen, China

Costin Badica, Romania

Minsen Chiu, Singapore Kevin Curran, UK Jianbo Fan, China

Michal Choras, Poland Mingcong Deng, Japan Haiping Fang, China Wai-Keung Fung, Canada Xiao-Zhi Gao, Finland Aili Han, China Pheng-Ann Heng, China Xia Hong, UK Jiankun Hu, Australia

Tianlu Chen, China Weidong Cheng, China Tommy Chow, Hong Kong, China Frank Emmert-Streib, UK Jiali Feng, China Houlei Gao, China Lingzhong Guo, UK Minghu Ha, China Laurent Heutte, France Wei-Chiang Hong, China Xiangpei Hu, China

Peter Hung, Ireland

Amir Hussain, UK

Xiaoyi Jiang, Germany Tetsuya J. Kobayashi, Japan Xiaoou Li, Mexico Paolo Lino, Italy Hua Liu, China Sean McLoone, Ireland Kezhi Mao, Singapore Wasif Naeem, UK Feng Qiao, China Jiafu Tang, China Hongwei Wang, China Ruisheng Wang, USA Yong Wang, Japan Lisheng Wei, China Rongguo Yan, China Zhang Yuwen, USA Guofu Zhai, China Qing Zhao, Canada Liangpei Zhang, China Shangming Zhou, UK

Pingping Jiang, China

IPC Members

Huijun Gao, China Xudong Guo, China Haibo He, USA Fan Hong, Singapore Yuexian Hou, China Guangbin Huang, Singapore MuDer Jeng, Taiwan, China Yasuki Kansha, Japan Gang Li, UK Yingjie Li, China Hongbo Liu, China Zhi Liu, China Fenglou Mao, USA John Morrow, UK Donglian Qi, China Chenxi Shao, China Haiying Wang, UK Kundong Wang, China Wenxing Wang, China Zhengxin Weng, China WeiQi Yan, UK Wen Yu, Mexico Peng Zan, China Degan Zhang, China Huiru Zheng, UK Huiyu Zhou, UK

Aim`e Lay-Ekuakillel, Italy Xuelong Li, UK Tim Littler, UK Wanquan Liu, Australia Marion McAfee, UK Guido Maione, Italy Mark Price, UK Alexander Rotshtein, Ukraine David Wang, Singapore Hui Wang, UK Shujuan Wang, China Zhuping Wang, China Ting Wu, China Lianzhi Yu, China Hong Yue, UK An Zhang, China Lindu Zhao, China Qingchang Zhong, UK

Organization

IX

Secretary-General Xin Sun, China Ping Zhang, China Huizhong Yang, China

Publication Chairs Xin Li, China Wasif Naeem, UK

Special Session Chairs Xia Hong, UK Li Jia, China

Organizing Committee OC Chairs Shiwei Ma, China Yunjie Wu, China Fei Liu, China OC Members Min Zheng, China Yijuan Di, China Qun Niu, UK

Banghua Yang, China Weihua Deng, China Xianxia Zhang, China

Yang Song, China Tim Littler, UK

Reviewers Renbo Xia, Vittorio Cristini, Aim'e Lay-Ekuakille, AlRashidi M.R., Aolei Yang, B. Yang, Bailing Zhang, Bao Nguyen, Ben Niu, Branko Samarzija, C. Elliott, Chamil Abeykoon, Changjun Xie, Chaohui Wang, Chuisheng Zeng, Chunhe Song, Da Lu, Dan Lv, Daniel Lai, David Greiner, David Wang, Deng Li, Dengyun Chen, Devedzic Goran, Dong Chen, Dongqing Feng, Du K.-L., Erno Lindfors, Fan Hong, Fang Peng, Fenglou Mao, Frank Emmert-Streib, Fuqiang Lu, Gang Li, Gopalacharyulu Peddinti, Gopura R. C., Guidi Yang, Guidong Liu, Haibo He, Haiping Fang, Hesheng Wang, Hideyuki Koshigoe, Hongbo Liu, Hongbo Ren, Hongde Liu, Hongtao Wang, Hongwei Wang, Hongxin Cao, Hua Han, Huan Shen, Hueder Paulo de Oliveira, Hui Wang, Huiyu Zhou, H.Y. Wang, Issarachai Ngamroo, Jason Kennedy, Jiafu Tang, Jianghua Zheng, Jianhon Dou, Jianwu Dang, Jichun Liu, Jie Xing, Jike Ge, Jing Deng, Jingchuan Wang, Jingtao Lei, Jiuying Deng, Jizhong Liu, Jones K.O., Jun Cao, Junfeng Chen, K. Revett, Kaliviotis Efstathios, C.H. Ko, Kundong Wang, Lei Kang,

X

Organization

Leilei Zhang, Liang Chen, Lianzhi Yu, Lijie Zhao, Lin Gao, Lisheng Wei, Liu Liu, Lizhong Xu, Louguang Liu, Lun Cheng, Marion McAfee, Martin Fredriksson, Meng Jun, Mingcong Deng, Mingzhi Huang, Minsen Chiu, Mohammad Tahir, Mousumi Basu, Mutao Huang, Nian Liu, O. Ciftcioglu, Omidvar Hedayat, Peng Li, Peng Zan, Peng Zhu, Pengfei Liu, Qi Bu, Qiguo Rong, Qingzheng Xu, Qun Niu, R. Chau, R. Kala, Ramazan Coban, Rongguo Yan, Ruisheng Wang, Ruixi Yuan, Ruiyou Zhang, Ruochen Liu, Shaohui Yang, Shian Zhao, Shihu Shu, Yang Song, Tianlu Chen, Ting Wu, Tong Liang, V. Zanotto, Vincent Lee, Wang Suyu, Wanquan Liu, Wasif Naeem, Wei Gu, Wei Jiao, Wei Xu, Wei Zhou, Wei-Chiang Hong, Weidong Chen, WeiQi Yan, Wenjian Luo, Wenjuan Yang, Wenlu Yang, X.H. Zeng, Xia Ling, Xiangpei Hu, Xiao-Lei Xia, Xiaoyang Tong, Xiao-Zhi Gao, Xin Miao, Xingsheng Gu, Xisong Chen, Xudong Guo, Xueqin Liu, Yanfei Zhong, Yang Sun, Yasuki Kansha, Yi Yuan, Yin Tang, Yiping Dai, Yi-Wei Chen, Yongzhong Li, Yudong Zhang, Yuhong Wang, Yuni Jia, Zaitang Huang, Zhang Li, Zhenmin Liu, Zhi Liu, Zhigang Liu, Zhiqiang Ge, Zhongkai Li, Zilong Zhao, Ziwu Ren.

Table of Contents – Part II

The First Section: Advanced Evolutionary Computing Theory and Algorithms A Novel Ant Colony Optimization Algorithm in Application of Pheromone Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peng Zhu, Ming-sheng Zhao, and Tian-chi He Modelling the Effects of Operating Conditions on Motor Power Consumption in Single Screw Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chamil Abeykoon, Marion McAfee, Kang Li, Peter J. Martin, Jing Deng, and Adrian L. Kelly Quantum Genetic Algorithm for Hybrid Flow Shop Scheduling Problems to Minimize Total Completion Time . . . . . . . . . . . . . . . . . . . . . . . Qun Niu, Fang Zhou, and Taijin Zhou Re-diversified Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . Jie Qi and Shunan Pang Fast Forward RBF Network Construction Based on Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jing Deng, Kang Li, George W. Irwin, and Minrui Fei A Modified Binary Differential Evolution Algorithm . . . . . . . . . . . . . . . . . . Ling Wang, Xiping Fu, Muhammad Ilyas Menhas, and Minrui Fei

1

9

21

30

40

49

Research on Situation Assessment of UCAV Based on Dynamic Bayesian Networks in Complex Environment . . . . . . . . . . . . . . . . . . . . . . . . Lu Cao, An Zhang, and Qiang Wang

58

Optimal Tracking Performance for Unstable Processes with NMP Zeroes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianguo Wang, Shiwei Ma, Xiaowei Gou, Ling Wang, and Li Jia

69

Typhoon Cloud Image Enhancement by Differential Evolution Algorithm and Arc-Tangent Transformation . . . . . . . . . . . . . . . . . . . . . . . . . Bo Yang and Changjiang Zhang

75

Data Fusion-Based Extraction Method of Energy Consumption Index for the Ethylene Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhiqiang Geng, Yongming Han, Yuanyuan Zhang, and Xiaoyun Shi

84

XII

Table of Contents – Part II

Research on Improved QPSO Algorithm Based on Cooperative Evolution with Two Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Longhan Cao, Shentao Wang, Xiaoli Liu, Rui Dai, and Mingliang Wu

93

Optimum Distribution of Resources Based on Particle Swarm Optimization and Complex Network Theory . . . . . . . . . . . . . . . . . . . . . . . . . Li-lan Liu, Zhi-song Shu, Xue-hua Sun, and Tao Yu

101

The Model of Rainfall Forecasting by Support Vector Regression Based on Particle Swarm Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . Shian Zhao and Lingzhi Wang

110

Constraint Multi-objective Automated Synthesis for CMOS Operational Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jili Tao, Xiaoming Chen, and Yong Zhu

120

Research on APIT and Monte Carlo Method of Localization Algorithm for Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jia Wang and Fu Jingqi

128

Quantum Immune Algorithm and Its Application in Collision Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jue Wu, LingXi Peng, LiXue Chen, and Lei Yang

138

An Artificial Bee Colony with Random Key for Resource-Constrained Project Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yan-jun Shi, Fu-Zhen Qu, Wang Chen, and Bo Li

148

The Second Section: Advanced Neural Network and Fuzzy System Theory and Algorithms Combined Electromagnetism-Like Mechanism Optimization Algorithm and ROLS with D-Optimality Learning for RBF Networks . . . . . . . . . . . . Fang Jia and Jun Wu

158

Stochastic Stability and Bifurcation Analysis on Hopfield Neural Networks with Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xuewen Qin, Zaitang Huang, and Weiming Tan

166

EMD-TEO Based Speech Emotion Recognition . . . . . . . . . . . . . . . . . . . . . . Xiang Li, Xin Li, Xiaoming Zheng, and Dexing Zhang A Novel Fast Algorithm Technique for Evaluating Reliability Indices of Radial Distribution Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mohammoud M. Hadow, Ahmed N. Abd Alla, and Sazali P. Abdul Karim

180

190

Table of Contents – Part II

An Improved Adaptive Sliding Mode Observer for Sensorless Control of PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ran Li and Guangzhou Zhao Clustering-Based Geometric Support Vector Machines . . . . . . . . . . . . . . . . Jindong Chen and Feng Pan A Fuzzy-PID Depth Control Method with Overshoot Suppression for Underwater Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhijie Tang, Luojun, and Qingbo He Local Class Boundaries for Support Vector Machine . . . . . . . . . . . . . . . . . . Guihua Wen, Caihui Zhou, Jia Wei, and Lijun Jiang

XIII

199

207

218

225

Research on Detection and Material Identification of Particles in the Aerospace Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shujuan Wang, Rui Chen, Long Zhang, and Shicheng Wang

234

The Key Theorem of Learning Theory Based on Sugeno Measure and Fuzzy Random Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minghu Ha, Chao Wang, and Witold Pedrycz

241

Recognition of Fire Detection Based on Neural Network . . . . . . . . . . . . . . Yang Banghua, Dong Zheng, Zhang Yonghuai, and Zheng Xiaoming

250

The Design of Predictive Fuzzy-PID Controller in Temperature Control System of Electrical Heating Furnace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ying-hong Duan

259

Stability Analysis of an Impulsive Cohen-Grossberg-Type BAM Neural Networks with Time-Varying Delays and Diffusion Terms . . . . . . . . . . . . . Qiming Liu, Rui Xu, and Yanke Du

266

The Third Section: Modeling and Simulation of Societies and Collective Behaviour Characterizing Multiplex Social Dynamics with Autonomy Oriented Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lailei Huang and Jiming Liu

277

A Computational Method for Groundwater Flow through Industrial Waste by Use of Digital Color Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Takako Yoshii and Hideyuki Koshigoe

288

A Genetic Algorithm for Solving Patient- Priority- Based Elective Surgery Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yu Wang, Jiafu Tang, and Gang Qu

297

XIV

Table of Contents – Part II

A Neighborhood Correlated Empirical Weighted Algorithm for Fictitious Play . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hongshu Wang, Chunyan Yu, and Liqiao Wu

305

Application of BP Neural Network in Exhaust Emission Estimatation of CAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linqing Wang and Jiafu Tang

312

Dynamic Behavior in a Delayed Bioeconomic Model with Stochastic Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yue Zhang, Qingling Zhang, and Tiezhi Zhang

321

The Fourth Section: Biomedical Signal Processing, Imaging, and Visualization A Feature Points Matching Method Based on Window Unique Property of Pseudo-Random Coded Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hui Chen, Shiwei Ma, Hao Zhang, Zhonghua Hao, and Junfeng Qian

333

A Reconstruction Method for Electrical Impedance Tomography Using Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Min-you Chen, Gang Hu, Wei He, Yan-li Yang, and Jin-qian Zhai

342

VLSI Implementation of Sub-pixel Interpolator for AVS Encoder . . . . . . . Chen Guanghua, Wang Anqi, Hu Dengji, Ma Shiwei, and Zeng Weimin

351

The Fifth Section: Intelligent Computing and Control in Distributed Power Generation Systems Optimization of Refinery Hydrogen Network . . . . . . . . . . . . . . . . . . . . . . . . . Yunqiang Jiao and Hongye Su Overview: A Simulation Based Metaheuristic Optimization Approach to Optimal Power Dispatch Related to a Smart Electric Grid . . . . . . . . . . Stephan Hutterer, Franz Auinger, Michael Affenzeller, and Gerald Steinmaurer

360

368

Speed Control for a Permanent Magnet Synchronous Motor with an Adaptive Self-Tuning Uncertainties Observer . . . . . . . . . . . . . . . . . . . . . . . . Da Lu, Kang Li, and Guangzhou Zhao

379

Research on Short-Term Gas Load Forecasting Based on Support Vector Machine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chao Zhang, Yi Liu, Hui Zhang, Hong Huang, and Wei Zhu

390

Network Reconfiguration at the Distribution System with Distributed Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gao Xiaozhi, Li Linchuan, and Xue Hailong

400

Table of Contents – Part II

XV

The Sixth Section: Intelligent Methods in Power and Energy Infrastructure Development An Autonomy-Oriented Computing Mechanism for Modeling the Formation of Energy Distribution Networks: Crude Oil Distribution in U.S. and Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benyun Shi and Jiming Liu

410

A Wavelet-Prony Method for Modeling of Fixed-Speed Wind Farm Low-Frequency Power Pulsations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel McSwiggan and Tim Littler

421

Direct Torque Control for Permanent Magnet Synchronous Motors Based on Novel Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sizhou Sun, Xingzhong Guo, Huacai Lu, and Ying Meng

433

The Seventh Section: Intelligent Modeling, Monitoring, and Control of Complex Nonlinear Systems A Monitoring Method Based on Modified Dynamic Factor Analysis and Its Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xueyan Yin and Fei Liu

442

A Novel Approach to System Stabilization over Constrained Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weihua Deng, Minrui Fei, and Huosheng Hu

450

An Efficient Algorithm for Grid-Based Robotic Path Planning Based on Priority Sorting of Direction Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aolei Yang, Qun Niu, Wanqing Zhao, Kang Li, and George W. Irwin

456

A Novel Method for Modeling and Analysis of Meander-Line-Coil Surface Wave EMATs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shujuan Wang, Lei Kang, Zhichao Li, Guofu Zhai, and Long Zhang

467

The Design of Neuron-PID Controller for a Class of Networked Control System under Data Rate Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lixiong Li, Rui Ming, and Minrui Fei

475

Stochastic Optimization of Two-Stage Multi-item Inventory System with Hybrid Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuli Zhang, Shiji Song, Cheng Wu, and Wenjun Yin

484

Iterative Learning Control Based on Integrated Dynamic Quadratic Criterion for Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Li Jia, Jiping Shi, Dashuai Cheng, Luming Cao, and Min-Sen Chiu

493

XVI

Table of Contents – Part II

Impedance Measurement Method Based on DFT . . . . . . . . . . . . . . . . . . . . . Xin Wang A 3D-Shape Reconstruction Method Based on Coded Structured Light and Projected Ray Intersecting Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hui Chen, Shiwei Ma, Bo Sun, Zhonghua Hao, and Liusun Fu

499

505

Erratum Modelling the Effects of Operating Conditions on Motor Power Consumption in Single Screw Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chamil Abeykoon, Marion McAfee, Kang Li, Peter J. Martin, Jing Deng, and Adrian L. Kelly

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E1

515

Table of Contents – Part I

The First Section: Intelligent Modeling, Monitoring, and Control of Complex Nonlinear Systems Stabilization of a Class of Networked Control Systems with Random packet Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minrui Fei, Weihua Deng, Kang Li, and Yang Song

1

Improved Nonlinear PCA Based on RBF Networks and Principal Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xueqin Liu, Kang Li, Marion McAfee, and Jing Deng

7

Application of Partical Swarm Optimization Algorithm in Field Holo-Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guangrui Wen, Xining Zhang, and Ming Zhao

16

Analyzing Deformation of Supply Chain Resilient System Based on Cell Resilience Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yonghong Li and Lindu Zhao

26

Multi-objective Particle Swarm Optimization Control Technology and Its Application in Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Li Jia, Dashuai Cheng, Luming Cao, Zongjun Cai, and Min-Sen Chiu

36

Online Monitoring of Catalyst Activity for Synthesis of Bisphenol A . . . . Liangcheng Cheng, Yaqin Li, Huizhong Yang, and Nam Sun Wang

45

An Improved Pyramid Matching Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jun Zhang, Guangzhou Zhao, and Hong Gu

52

Stability Analysis of Multi-channel MIMO Networked Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dajun Du, Minrui Fei, and Kang Li

62

Synthesis of PI–type Congestion Controller for AQM Router in TCP/AQM Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Junsong Wang and Ruixi Yuan

69

A State Identification Method of Networked Control Systems . . . . . . . . . . Xiao-ming Yu and Jing-ping Jiang Stabilization Criterion Based on New Lyapunov Functional Candidate for Networked Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qigong Chen, Lisheng Wei, Ming Jiang, and Minrui Fei

77

87

XVIII

Table of Contents – Part I

Development of Constant Current Source for SMA Wires Driver Based on OPA549 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yong Shao, Enyu Jiang, Quanzhen Huang, and Xiangqiang Zeng High Impedance Fault Location in Transmission Line Using Nonlinear Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Min-you Chen, Jin-qian Zhai, Zi-qiang Lang, Ju-cheng Liao, and Zhao-yong Fan Batch-to-Batch Iterative Optimal Control of Batch Processes Based on Dynamic Quadratic Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Li Jia, Jiping Shi, Dashuai Cheng, Luming Cao, and Min-Sen Chiu

95

104

112

Management Information System (MIS) for Planning and Implementation Assessment (PIA) in Lake Dianchi . . . . . . . . . . . . . . . . . . . Longhao Ye, Yajuan Yu, Huaicheng Guo, and Shuxia Yu

120

Integration Infrastructure in Wireless/Wired Heterogeneous Industrial Network System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Haikuan Wang, Weiyan Hou, Zhaohui Qin, and Yang Song

129

Multi-innovation Generalized Extended Stochastic Gradient Algorithm for Multi-Input Multi-Output Nonlinear Box-Jenkins Systems Based on the Auxiliary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jing Chen and Xiuping Wang

136

Research of Parallel-Type Double Inverted Pendulum Model Based on Lagrange Equation and LQR Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jian Fan, Xihong Wang, and Minrui Fei

147

A Consensus Protocol for Multi-agent Systems with Double Integrator Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fang Wang, Lixin Gao, and Yanping Luo

157

A Production-Collaboration Model for Manufacturing Grid . . . . . . . . . . . . Li-lan Liu, Xue-hua Sun, Zhi-song Shu, Shuai Tian, and Tao Yu

166

The Second Section: Autonomy-Oriented Computing and Intelligent Agents Parallel Computation for Stereovision Obstacle Detection of Autonomous Vehicles Using GPU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhi-yu Xu and Jie Zhang

176

Framework Designing of BOA for the Development of Enterprise Management Information System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shiwei Ma, Zhaowen Kong, Xuelin Jiang, and Chaozu Liang

185

Table of Contents – Part I

XIX

Training Support Vector Data Descriptors Using Converging Linear Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hongbo Wang, Guangzhou Zhao, and Nan Li

196

Research on Modeling and Simulation of an Adaptive Combat Agent Infrastructure for Network Centric Warfare . . . . . . . . . . . . . . . . . . . . . . . . . . Yaozhong Zhang, An Zhang, Qingjun Xia, and Fengjuan Guo

205

Genetic Algorithm-Based Support Vector Classification Method for Multi-spectral Remote Sensing Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yi-nan Guo, Da-wei Xiao, and Mei Yang

213

Grids-Based Data Parallel Computing for Learning Optimization in a Networked Learning Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lijun Xu, Minrui Fei, T.C. Yang, and Wei Yu

221

A New Distributed Intrusion Detection Method Based on Immune Mobile Agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yongzhong Li, Chunwei Jing, and Jing Xu

233

Single-Machine Scheduling Problems with Two Agents Competing for Makespan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guosheng Ding and Shijie Sun

244

Multi-Agent Asynchronous Negotiation Based on Time-Delay . . . . . . . . . . LiangGui Tang and Bo An

256

The Third Section: Advanced Theory and Methodology in Fuzzy Systems and Soft Computing Fuzzy Chance Constrained Support Vector Machine . . . . . . . . . . . . . . . . . . Hao Zhang, Kang Li, and Cheng Wu

270

An Automatic Thresholding for Crack Segmentation Based on Convex Residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chunhua Guo and Tongqing Wang

282

A Combined Iteration Method for Probabilistic Load Flow Calculation Applied to Grid-Connected Induction Wind Power System . . . . . . . . . . . . Xue Li, Jianxia Pei, and Dajun Du

290

Associated-Conflict Analysis Using Covering Based on Granular Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shuang Liu, Jiyi Wang, and Huang Lin

297

Inspection of Surface Defects in Copper Strip Based on Machine Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xue-Wu Zhang, Li-Zhong Xu, Yan-Qiong Ding, Xin-Nan Fan, Li-Ping Gu, and Hao Sun

304

XX

Table of Contents – Part I

BIBO Stability of Spatial-temporal Fuzzy Control System . . . . . . . . . . . . . Xianxia Zhang, Meng Sun, and Guitao Cao

313

An Incremental Manifold Learning Algorithm Based on the Small World Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lukui Shi, Qingxin Yang, Enhai Liu, Jianwei Li, and Yongfeng Dong

324

Crack Image Enhancement of Track Beam Surface Based on Nonsubsampled Contourlet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chunhua Guo and Tongqing Wang

333

The Class-2 Linguistic Dynamic Trajectories of the Interval Type-2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liang Zhao

342

The Fourth Section: Computational Intelligence in Utilization of Clean and Renewable Energy Resources Strategic Evaluation of Research and Development into Embedded Energy Storage in Wind Power Generation . . . . . . . . . . . . . . . . . . . . . . . . . . T.C. Yang and Lixiong Li

350

A Mixed-Integer Linear Optimization Model for Local Energy System Planning Based on Simplex and Branch-and-Bound Algorithms . . . . . . . . Hongbo Ren, Weisheng Zhou, Weijun Gao, and Qiong Wu

361

IEC 61400-25 Protocol Based Monitoring and Control Protocol for Tidal Current Power Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jung Woo Kim and Hong Hee Lee

372

Adaptive Maximum Power Point Tracking Algorithm for Variable Speed Wind Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moo-Kyoung Hong and Hong-Hee Lee

380

Modeling and Simulation of Two-Leaf Semi-rotary VAWT . . . . . . . . . . . . . Qian Zhang, Haifeng Chen, and Binbin Wang

389

The Fifth Section: Intelligent Modeling, Control and Supervision for Energy Saving and Pollution Reduction Identification of Chiller Model in HVAC System Using Fuzzy Inference Rules with Zadeh’s Implication Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yukui Zhang, Shiji Song, Cheng Wu, and Kang Li An Improved Control Strategy for Ball Mill Grinding Circuits . . . . . . . . . Xisong Chen, Jun Yang, Shihua Li, and Qi Li

399 409

Table of Contents – Part I

Sliding Mode Controller for Switching Mode Power Supply . . . . . . . . . . . . Yue Niu, Yanxia Gao, Shuibao Guo, Xuefang Lin-Shi, and Bruno Allard

XXI

416

The Sixth Section: Intelligent Methods in Developing Vehicles, Engines and Equipments Expression of Design Problem by Design Space Model to Support Collaborative Design in Basic Plan of Architectural Design . . . . . . . . . . . . Yoshiaki Tegoshi, Zhihua Zhang, and Zhou Su Drive Cycle Analysis of the Performance of Hybrid Electric Vehicles . . . . Behnam Ganji, Abbas Z. Kouzani, and H.M. Trinh

425 434

The Seventh Section: Computational Methods and Intelligence in Modeling Genetic and Biochemical Networks and Regulation Supply Chain Network Equilibrium with Profit Sharing Contract Responding to Emergencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ating Yang and Lindu Zhao

445

Modeling of the Human Bronchial Tree and Simulation of Internal Airflow: A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yijuan Di, Minrui Fei, Xin Sun, and T.C. Yang

456

Robust Semi-supervised Learning for Biometrics . . . . . . . . . . . . . . . . . . . . . Nanhai Yang, Mingming Huang, Ran He, and Xiukun Wang

466

Research on Virtual Assembly of Supercritical Boiler . . . . . . . . . . . . . . . . . Pi-guang Wei, Wen-hua Zhu, and Hao Zhou

477

Validation of Veracity on Simulating the Indoor Temperature in PCM Light Weight Building by EnergyPlus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chun-long Zhuang, An-zhong Deng, Yong Chen, Sheng-bo Li, Hong-yu Zhang, and Guo-zhi Fan Positive Periodic Solutions of Nonautonomous Lotka-Volterra Dispersal System with Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ting Zhang, Minghui Jiang, and Bin Huang An Algorithm of Sphere-Structure Support Vector Machine Multi-classification Recognition on the Basis of Weighted Relative Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shiwei Yun, Yunxing Shu, and Bo Ge Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

486

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A Novel Ant Colony Optimization Algorithm in Application of Pheromone Diffusion Peng Zhu1, Ming-sheng Zhao2, and Tian-chi He3 1

Department of Information Management, Nanjing University, 210093 Nanjing, China [email protected] 2 Nanjing Forest Police College, 210046 Nanjing, China [email protected] 3 Nanjing University of Financial and Economics, 210046 Nanjing, China [email protected]

Abstract. Ant Colony Optimization (ACO) Algorithm is a novel stochastic search technology, which simulates the social behavior of ant colony. This paper firstly analyzes the shortcomings of basic ACO, then presents an enhanced ACO algorithm which is more faithful to real ants’ behavior in application of pheromone diffusion. By setting up the pheromone diffusion model, the algorithm improves the collaboration among the nearby ants. The simulation results show that the proposed algorithm can not only get much more optimal solutions but also greatly enhance convergence speed. Keywords: ACO algorithm, ant colony system, pheromone diffusion.

1 Introduction According to bionomists’ study, ant individual transmit information by the substance called pheromone, which ants leave in their paths. What’s more, they can feel the concentration of it and it guides their direction of movement. Ants are inclined to move towards the high concentration of the substance. Therefore, ant colony formed by the large number of collective action suggests information on the phenomenon of positive feedback: the more ants walk on a certain path, the greater possibility that ant behind choose the same one. Inspired by the true ant colony’s action in the nature, ACO algorithm was first put forward by Italian scholar Dorigo M. et al. [1, 2]. They apply it to complex combinatorial optimization problems and achieve good results. However, this algorithm has some disadvantages, such as slow evolution, falling into local optimum easily and so on. In the light of genetic algorithm thoughts in the aspect of improving the solution convergence speed and global optimal area, a number of scholars have made some improvements in many sides of basic ACO algorithm, for instance, Holden el al. [3] proposed an Ant colony system(ACS), Stutzle et al. [4] proposed ACO algorithm with mutation features, these improved methods have done a good job about answering some particular questions, but from the view of ant colony’s information system, no good biological basis lays in these improvements. On the bases of analyzing ACO K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 1–8, 2010. © Springer-Verlag Berlin Heidelberg 2010

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P. Zhu, M.-s. Zhao, and T.-c. He

algorithm model defects, this article put forward an ACO algorithm of applying the spread of pheromone which matches better with the true ant colony system.

2 Analysis of Basic ACO Algorithm 2.1 Brief Introduction of Basic ACO Algorithm ACO algorithm is usually helpful in complex combinatorial optimization problems. The definition of ACO algorithm model varies when it is applied to problems with different features [5]. This thesis will explain basic ACO algorithm model by the following example which is about Traveling Salesman Problems (TSP) in m cities. The question is to find out the shortest path on the condition of traveling m cities only for each one time. Presume n is the number of ants among the ant colony, d ij (i, j = 1, 2, , m) stands for the distance between city i and city j , τ ij (t ) is the concentration of pheromone between city i and city j . At the beginning, each path has the same concentration of pheromone. Set τ ij (t ) ( C is a constant). Ant k (k = 1, 2, , n) keeps moving, deciding the transmitting direction by the concentration of pheromone on each path, Pijk (t ) is the probability of ant k ’s moving from city i to city j at time t , the equation is as (1).

⎧τ ijα (t )ηijβ (t ) / ∑ τ isα (t )ηisβ (t ), j ∉ tabuk ⎪ s∈allowed k . Pijk = ⎨ ⎪⎩0, otherwise Among them, d ij (i, j = 1, 2,

(1)

, m) is the congregation that ant k has passed

through. At the beginning, tabuk has only one element that is the departure city. With the progress of evolution, the elements of tabuk was in increasing. The elements leaved on every way before disappeared gradually over time. Regard parameter 1 − ρ as the degree of volatile pheromone level, the ant finished its circulation through the time numbered m . The concentration of pheromone on every path should adjust to equation (2).

τ ij (t + m) = ρ ⋅τ ij (t ) + Δτ ij , n

Δτ ij = ∑ Δτ ijk , ρ ∈ (0,1)

.

(2)

k =1

Δτ means the concentration of pheromone that ant k leaves on the path ij . Δτ ij k ij

is the sum of all the concentration of pheromone that every ants leaves on path ij . Dorigo M has ever given three different models. They were called: ant cycle system, ant quantity system and ant density system [6], their difference is that they have different calculation expression for Δτ ijk .

A Novel Ant Colony Optimization Algorithm in Application of Pheromone Diffusion

3

In ant cycle system Δτ ijk is:

⎧Q / Lk , if the kth ant uses edge(i, j ) Δτ ijk = ⎨ . otherwise ⎩0,

(3)

In ant quantity system model and ant density system, Δτ ijk are: ⎧⎪Q / dij , if the kth ant uses edge(i, j ) Δτ ijk = ⎨ . otherwise ⎪⎩0,

(4)

⎧Q, Δτ ijk = ⎨ ⎩Q,

(5)

if the kth ant uses edge(i, j ) . otherwise

The difference is that local information was used in the latter two, but public information was used in the former one. When solve the TSP, basic ACO algorithm has advantage in performance of ant cycle system. When it comes to the truth, ant quantity system model is better. So in this paper we choose ant quantity system as foundaβ Q and ρ could be tion, and then improve algorithm proposed. Parameter α test or through evolutionary learning to decide the best combination. Fixed evolution algebra could be used as a signature to stop or you can stop when the evolutionary trends are not apparent. Due to the theory of complexity of the algorithm, we can notice that the complexity is O(nc ⋅ n3 ) , and nc is the number of circulation. These introductions are concentrating on the solution of TSP. It can be use on other aspects with a little amend.

、 、

2.2 Defects of the Basic ACO Algorithm

Essentially, ACO Algorithm is still a random search algorithm. It searches for the best solution through candidate solution colonies. Algorithm is finished by many ants, every ant search dependently in the candidate solution area, and leave stated information element. The better the solution’s performance is the larger amount of information element they leave. And then the solution will be chosen more possible than others. At the beginning, information elements on every solution are the same. With the algorithm’s progress, information elements on the better solutions become more, the algorithm tend to convergence gradually. For TSP, its solution is consists of a series of sub solution, which is the paths between cities. The cooperation among ants is work through the information on sub solution. But there are some problems exist. Suppose that one sub solution has a quite high element concentration, it will regulate the ants in two sub solution cities, and let the ant choose it. When it starts, the concentrations are the same. Due to different path search, ant changes the concentration. Because the insufficient collaboration and not in time among the ants in the basic ACO algorithm, the convergence rate slow down, and be limited to local optimum easily.

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P. Zhu, M.-s. Zhao, and T.-c. He

The ant colonies’ action in nature is that they exchange their information through the spread of pheromone. When former ant has an influence on the latter ants, latter ants didn’t move on the locus of former ants. And it has something to do with the distance. The short the distance is, the big the impact is. Otherwise, the impact is small. But in present ACO algorithm, there is no consideration of the spread of pheromone. It should be point out that present ACO algorithm is still in simulation phase, its validity were not have been analyzed. Due to this situation, this paper is faithful to the natural ant colonies’ actions. Inspired by this thought, this article supposes a novel search algorithm which simulates the social behavior of ant colony depending on pheromone's communication and adjusts it to the TSP solution.

3 Improved ACO Algorithm Based on Pheromone Diffusion 3.1 The Theory of the ACO Algorithm Based on Pheromone Diffusion

This article still uses the TSP solution as the example to introduce. When an ant searches on a path, if it finds a sub solution, it releases a corresponding concentration of pheromone. On the one hand the pheromone has an impact on the ant in two sub cities, on the other hand, it use the path as a center to expand and affect other ants, so that the ant may more likely to choose this path at next part. Through this collaborative approach use the pheromone diffusion, the interference of the ants to choose next city will be descended, and the speed of the convergence will be proved. Figure 1 illustrates the theory of ACO algorithm used pheromone diffusion. In figure 1, assume that ant 2 wants go to E from H , FG is a short path. The concentrations of pheromone on GH and IH are the same, but GH is shorter than IH , so, ant 2 may more likely to choose G as the next destination. Before this, if ant 1 just get to F from G , it leaves the same concentrations of pheromone and expand nearby. Due to the different distance, concentrations of pheromone on GH will be much larger than those on IH . When it is large to a certain extent, ant 2 may choose G as the next destination, so that there is little disturbances of ant 2 to choose the shortest path H − G − F − E .

Fig. 1. Schematic diagram of ACO algorithm in application of pheromone diffusion

3.2 ACO Algorithm Based on Pheromone Diffusion

This article uses ant quantity system as the basic model, and pulls in the pheromone diffusion model.

A Novel Ant Colony Optimization Algorithm in Application of Pheromone Diffusion

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The pheromone diffusion model is that concentrations of pheromone become less when the distance goes further in atmosphere, and lay as a normal distribution. In order to facilitate the calculation, the diffusion model has been simplified. In the simplified pheromone diffusion model, concentrations of pheromone at 0 is Dmax , parameter θ is a acute angle and keep state, h is the height of cone, r is the radius of expand, its size is h ⋅ tgθ . Use C point as example, the concentrations of pheromone that the ant get on this point is Dc , which is described by equation (6), σ is the distance between C and 0 . It is easy to draw a conclusion from equation (6) that the far it is to the source, the less pheromone the ant gets. Dc = Dmax ⋅ ((h ⋅ tgθ − σ ) /(h ⋅ tgθ )) .

(6)

The pheromone diffusion model used to solve TSP. Assume that ant k has just pass city i and j , the distance is d ij . Ant uses i and j as the center to expand the pheromone. Concentrations of pheromone at i and j are both Dmax . The result is cone like in figure 3, which regard i and j as the center of bottom. To any other one city, which is in the range of ant k ’ pheromone, the concentrations of pheromone ant k expand to this city are Dilk and D kjl , they can be calculate. This article has done some simplify, use Dilk and D kjl to replace the concentrations of pheromone on path il and jl . Then it could get the concentrations of pheromone that ant k has leaved on every related paths in one circulation. The equations are (7) (8) (9).

Δτ ijk = Q / dij .

(7)

Δτ ik1 = Dilk .

(8)

Δτ kj1 = D kjl .

(9)

Among then, equation (7) is the path that ant k has passed in time t to time t + 1 , equation (8) shows if the ant k has passed city i in time t to time t + 1 , but do not pass city l , equation (9) shows if ant k has passed city j in time t to time t + 1 , but do not pass city l . that is to say in former ant quantity system model, every ants’ single step can only change the very path’s concentrations of pheromone, but in this model, ant can change concentrations of pheromone on more than one path. Writer thinks it will improve the cooperation between ant colonies and strengthen the effectiveness of algorithm at the same time, the man-made ant colony will much like to nature ant colony. Next is to derived the equation of Dilk and D kjl . Assume h = d

ω +1

/(dij ) w , w is the

adjustable constant than 1, d is the average distance between every cities. This article assume Dmax = γ ⋅ Δτ ijk , γ is the adjustable number smaller than 1. when use i as the

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P. Zhu, M.-s. Zhao, and T.-c. He

center to expend the pheromone, σ = d il in equation (6), and when use j as the center to expend the pheromone, σ = d jl in equation (6), pull Δτ ijk k il

h

Dmax and σ

k jl

into equation (6) so that to get the equations of D and D , that are equation (10) and (11). w +1 ⎧ dil ⋅ (dil )ω ctgθ ,if dil < d /(dij )ω ⋅ tgθ ⎪γ ⋅ Q / dij (1 − w +1 D =⎨ . d ⎪0, otherwise ⎩

(10)

⎧ d jl ⋅ (d jl )ω w +1 ctgθ ,if d jl < d /(d ij )ω ⋅ tgθ ⎪γ ⋅ Q / d ij (1 − w +1 . D =⎨ d ⎪ otherwise ⎩0,

(11)

k il

k il

The processes of the ACO algorithm in application of pheromone diffusion are as follows: Step 1: initialization, set Q = 100 , C = 3 , θ = π / 8 , the maximum evolution generation is set to be 400 . Step 2: randomly choose initial position of each ant. Step 3: use equation (1) to calculate the position which each ant k will transfer to, and assume the position as j , the former position as i . Step 4: calculate a new generated pheromone concentrations on the path ij by equation (7). Step 5: calculate the concentration of pheromone spread from the path i to other path il by equation (8). Step 6: calculate the concentration of pheromone spread from the path j to other path il by equation (9). Step 7: if this cycle of ants are implemented Step 3 ~ 6, then turn to Step 8, otherwise turn to Step 3. Step 8: update the concentration of pheromone on each path, then in the equation (2), m = 1 . Step 9: if each ant has completed a full path, then turned to Step 10, otherwise turn to Step 3. Step 10: whether the appointed evolution algebra reaches, or the result obtained no significant improvement in the last several generations, if so, then turned to Step 11, otherwise turn to Step 3 for steering a new round of evolution. Step 11: output evolution results.

4 Experimental Result and Analyzes This article chooses Oliver 30 as an example to do the experiment. The reason of choosing it is that TSP problem is a classical one of NP-hard, which is often used to

A Novel Ant Colony Optimization Algorithm in Application of Pheromone Diffusion

7

verify the validity of an algorithm. Writers figure out spread application of pheromone ACO algorithm by Visual C + + language programming, Table 1 shows the results. In Table 1, the shortest path length is the optimal path length, and evolutionary generation is obtained by taking an average of 10 experiments. Table 1. Experimental results under different parameters of this algorithm

α

β

ρ

ω

γ

The length of the shortest path

2 2 4 4

4 4 2 2

0.6 0.9 0.5 0.9

1 1 1 1

0.4 0.4 0.6 0.6

432.77053 432.87607 437.56275 437.42469

Needed evolutional generation before reach convergence 73 62 67 71

To illustrate the validity of this algorithm, this article has done experiments based on basic ACO algorithm. Owing to the base of this algorithm is ant quantity system model, on which writers’ base to get solution of Oliver 30, Table 2 shows the result. From comparison between Table 1 and Table 2, whether the speed or the results, the algorithm of this article is obviously better than the basic ACO algorithm. Table 2. Experimental results under different parameters of basic ACO algorithm α

β

ρ

2 2 4 4

4 4 2 2

0.6 0.9 0.5 0.9

The length of the shortest path 451.94774 450.75633 459.45779 442.53267

Needed evolutional generation before reach convergence 386 357 394 375

5 Conclusions This article firstly analyzes basic ACO algorithm model and points out the defects among ants’ cooperation. On this base, inspired by ant colony’s information systems in nature, the thesis puts forward an ACO algorithm which is more faithful to true ants’ behavior in application of pheromone diffusion. From the results of TSP problem solving, the overall solution and convergence speed of the algorithm in this thesis have been greatly improved.

References 1. Dorigo, M., Maniezzo, V., Colorni, A.: The ant system: Optimization by a colony of cooperating agents. IEEE Transactions on Systems, Man, and Cybernetics, Part B 26(1), 29–41 (1996)

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2. Dorigo, M., Gambardella, L.M.: Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Transactions on Evolutionary Computation 1(1), 53–66 (1997) 3. Holden, N., Freitas, A.A.: Web Page Classification with an Ant Colony Algorithm. In: Yao, X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervós, J.J., Bullinaria, J.A., Rowe, J.E., Tiňo, P., Kabán, A., Schwefel, H.-P. (eds.) PPSN 2004. LNCS, vol. 3242, pp. 1092–1102. Springer, Heidelberg (2004) 4. Stutzle, T., Dorigo, M.: A short convergence proof for a class of ACO algorithms. IEEE Transactions on Evolutionary Computation, 358–365 (2002) 5. Blum, C., Dorigo, M.: Search bias in ant colony optimization: On the role of competitionbalanced systems. IEEE Trans. on Evolutionary Computation, 159–174 (2005) 6. Dorigo, M., Thomas, S.: Ant Colony Optimization, pp. 223–244. Prentice-Hall of India Publication, New Delhi (2005)

Modelling the Effects of Operating Conditions on Motor Power Consumption in Single Screw Extrusion Chamil Abeykoon1, , Marion McAfee2 , Kang Li3 , Peter J. Martin1 , Jing Deng3 , and Adrian L. Kelly4 1

School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast, BT9 5AH, UK Tel.: +44289097236, +442890974164 [email protected] 2 Department of Mechanical and Electronic Engineering, Institute of Technology Sligo, Sligo, Ireland 3 School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, BT9 5AH, UK 4 IRC in Polymer Science and Technology, School of Engineering, Design and Technology, University of Bradford, Bradford, BD7 1DP, UK

Abstract. Extrusion is one of the most important production methods in the plastics industry and is involved in the production of a large number of plastics commodities. Being an energy intensive production method, process energy efficiency is of major concern and selection of the most energy efficient processing conditions is a key aim to reduce operating costs. Extruders consume energy through motor operation (i.e. drive to screw), the barrel heaters and also for cooling fans, cooling water pumps, gear pumps, screen pack changing devices etc. Typically the drive motor consumes more than one third of the total machine energy consumption. This study investigates the motor power consumption based on motor electrical variables (only for direct current (DC) motors) and new models are developed to predict the motor power consumption from easily measurable process settings for a particular machine geometry. Developed models are in good agreement with training and unseen data by representing the actual conditions with more than 95% accuracy. These models will help to determine the effects of individual process settings on the drive motor energy consumption and optimal motor energy efficient settings for single screw extruders. Keywords: Single Screw Extrusion, Energy Efficiency, Modelling.

1

Introduction

Polymer materials are becoming more popular as a raw material for the production of a large number of components in various industrial sectors such as: packaging, household, automotive, aerospace, marine, construction, electrical and 

Corresponding author.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 9–20, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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electronic, and medical applications. For example, use of polymer materials in the UK industry has a 2.5% annual average growth rate [1]. Usually, extruders are energy intensive production machines. Therefore, energy efficient operation of extruders is critical for producers to survive in a highly competitive market. However, achieving both an energy efficient operation and a quality output with desirable output rates are still challenging despite significant developments in the extrusion field over last few decades. Under these circumstances, it is very important to study the effects of processing conditions on the extruder energy consumption. By consuming about 1/3 of the total extruder plant energy consumption [2], machine rated efficiency and efficient operation of drive motors are crucial for overall extrusion process efficiency. Most existing extrusion plants are still using DC motors although some new plants are fitted with alternating current (AC) drives. Barlow argues that DC motors are operated at only about 90% efficiency, even with full load and full speed. Motor efficiency is further reduced when the motor is running outside its rated speed and also when the plant becomes older. Thus, minimising unnecessary energy usage by selecting optimum processing conditions is important to achieve a better overall process efficiency as machine attributed inefficiencies may not be controlled or eliminated. [3] argues that motors are often neglected from energy usage considerations within extrusion plants and although motors in the main processing equipment, such as extruders and injection moulding machines are obvious, the majority of motors are hidden in other equipment such as compressors, pumps and fans. [4] reveals that over 65% of the average UK industrial electricity bill in year 1994 accounted for motor operations which cost about £3 billion. However, more than 10% of motor energy consumption is wasted, costing about £460 million per annum in the UK. Although this is the overall motor energy usage, the contribution of the plastic industry may be considerable as the major power consumer in plastic materials processing machines are the electric motors. The UK plastics industry is a one of the major industries within the UK and has a considerable contribution to the UK economy which accounts approximately £17.5 billion of annual sales (approximately 2.1% of UK GDP)[1]. The same trend applies to most of the developed countries in the world. A small improvement in process energy efficiency will therefore considerably reduce global energy costs. 1.1

Effects of Process Settings on Extruder Power Consumption

Extruder power consumption increases as the screw speed increases while the extruder specific energy consumption (i.e. the energy consumed by the motor to produce 1g of extrudate) decreases [5]. Furthermore, the same group found that the extruder power consumption is dependent on the screw geometry and the material being processed within the same operating conditions [6]. Work presented by [7] found that the feed zone barrel temperature has the greatest influence on the power consumption of the extruder. They investigated the effects of the individual barrel zone temperatures on the power consumption in a single screw extruder.

Modelling the Effects of Operating Conditions on Motor Power Consumption

11

Previous work by the present authors [8] discussed the effects of process settings on the motor power consumption and motor specific energy consumption in a single screw extruder. It was found that the motor power consumption increases as the screw speed increases while the motor specific energy consumption decreases. Moreover, it was found that the barrel set temperatures had a slight effect on the motor power consumption and the motor specific energy consumption. The motor specific energy consumption reduces as the barrel zone temperatures increase. However, running an extruder at a higher screw speed with higher energy efficient conditions may not be realistic as the required quality of melt output may not be achieved due to the reduction of material residence time. The selection of an optimum operating point in terms of energy efficiency and thermal quality may be the most important requirement for the current industry. Thus, it is important to understand how to select the most suitable process settings for an energy efficient operation while achieving the required melt quality and output rates. 1.2

Modelling the Extruder Power Consumption

From the review of literature it is clear that there is little reported work that has attempted to develop a model to predict the extruder or extruder motor power consumption. [9] proposed a mathematical model to calculate the power consumption per channel in single screw extruders based on screw speed, material viscosity and a few other machine geometrical parameters. However, no details were provided regarding the model performance or predictions. [10] presented a computer model for single-screw extrusion and states that the model takes into account five zones of the extruder (i.e. hopper, solids conveying, delay zone, melting zone, and melt conveying) and the die. The model predicts the mass flow rate, pressure and temperature profiles along the extruder screw channel and in the die, the solid bed profile, and the power consumption based on the given material and rheological properties of the polymer, the screw, the hopper and die geometry and dimensions, and the extruder operating conditions (i.e. screw speed and barrel temperature profile). However, no details were given of the predicted motor power consumption. The development of models to predict the power consumption based on the processing conditions may help operators to select the most desirable operating conditions by eliminating excessive power consumption (i.e. situations in which the power is more than that required for the process). Particularly, models based on the motor power consumption may be very useful for selecting the most desirable and highest screw speed (higher energy efficiency at higher screw speeds) with suitable barrel set temperatures to run the process while achieving the required melt quality which is still a challenging task within the industry. In this work, a method is proposed to calculate the motor power consumption for DC motors by measuring the motor electrical variables. An attempt is made to model the effects of process settings on motor power consumption. The study is focused to a single screw extruder and two processing materials.

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Equipment and Procedure

All measurements were carried out on a 63.5mm diameter (D) single screw extruder (Davis Standard BC-60). A tapered compression screw with 3:1 compression ratio (Feed-4D, Compression (or Melting)-10D, Metering-10D) was used to process the polymer materials. The extruder was fitted with an adaptor prior to a short cylindrical die with a 12mm bore. The barrel has four separate temperature zones equipped with Davis Standard Dual Therm controllers. The extruder drive is a horizontal type separately excited direct current (SEDC) motor which has ratings: 460Vdc, 50.0 hp (30.5kW) at 1600rpm. The motor and screw are connected through a fixed gearbox with a ratio of 13.6:1, hence the gearbox efficiency is relatively constant at all speeds (∼96%). Motor speed was controlled by a speed controller (MENTOR II) based on speed feedback obtained through a DC tachometer generator. The extruder was instrumented with two high voltage probes to collect armature and field voltage data (Testoon GE8115) and two current probes were used to measure armature and field currents (Fluke PR430 and PR1001). A LabVIEW software programme was developed to communicate between the experimental instruments and a PC. All signals were acquired at 10kHz using a 16-bit DAQ card (National Instruments PCMCIA 6036E) through a SC-2345 connector box. Amplification was applied to the armature and field current signals. A high sampling speed was necessary as the electrical signals contain high frequencies associated with rectification of the a.c. supply. Experimental trials were carried out on a virgin high density polyethylene (HDPE), HM5411, from BP Chemicals Ltd (MFI - 0.12g/10min and density 0.952g/cm3) and a recycled extrusion grade black HDPE (MFI -0.16g/10min, density - 0.967g/cm3, and ∼2.5% carbon black) provided by Cherry Pipes Ltd. The melt flow index (MFI) values are presented according to the ISO 1133 standard (190◦ C, 2.16kg). From here onwards, recycled black HDPE, (RH), and virgin HDPE, (VH), are referred as recycled material and virgin material respectively. The extruder temperature settings were fixed as described in Table 1 and three experimental trials were carried out with each material and denoted as A (low temperature), B (medium temperature), and C (high temperature). The screw speed was adjusted from 10rpm to 90rpm in steps of 40rpm in tests A and C and in steps of 20rpm in test B, with the extruder running for about nine minutes at each speed. Table 1. Extruder barrel temperature settings Temperature settings/◦ C Barrel Zones Clamp Ring Adapter 1 2 3 4 A 130 155 170 180 180 180 B 140 170 185 200 200 200 C 150 185 200 220 220 220

Test

Die 180 200 220

Modelling the Effects of Operating Conditions on Motor Power Consumption

3

13

Calculation of Motor Power

A three phase a.c. supply was connected to the motor which is converted to a d.c. current via full wave rectification. Root mean square (rms) values of the armature current, field current, armature voltage, and field voltage signals were calculated from the measured instantaneous signals. The original power supply frequency was 50Hz and r.m.s. values were calculated over each period where one period is equal to 0.02s from the data measured at the 10kHz sampling rate. Afterwards, the calculated power signals at 50Hz were down sampled to 10Hz by calculating the average values of each of the five data points. Both armature and field power consumptions were calculated. Finally, the total motor power consumption was given by the sum of the field and armature power consumptions. Figure 1.a shows the average motor power consumption over last five minutes (4-9 minutes) at different screw speeds and barrel set temperatures. In general, motor power consumption increases as screw speed increases during processing of both materials. The rate of increase of motor power reduces at higher screw speeds. This was expected due to a reduction in the polymer viscosity with shearthinning, resulting in lower back-pressure than would otherwise occur. The motor specific energy consumption (SECmotor ) was also calculated from the average motor power data over the same five minute period and the measured melt output rate (m) ˙ according to equation (1). ˙ SECmotor = M otor Power /m

(1)

Figure 1.b shows the variations in the motor specific energy demand over different processing conditions. The virgin material consumed relatively high power per gram of extrudate at 10 and 30rpm; this may be due to severe conveying problems during those tests as evident by the very low mass throughput rates. However, in general the SECmotor reduces as the screw speed increases.

         

3.5 A-RH

3.0

8 7 A-RH A-VH

5

B-RH B-VH

3

C-RH

SEC of the motor (kJ/g)

Motor power consumption (kW)

11

A-VH

2.5

B-RH B-VH

2.0

C-RH C-VH

1.5 1.0 0.5

C-VH

1

0

20

40

60

Screw speed (rpm) a

80

100

0

0

20

40 60 Screw speed (rpm) b

80

100

Fig. 1. Motor power consumption for RH and VH materials processing from 10-90rpm

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The lowest energy consumption was at 90rpm for both materials. Moreover, the SECmotor decreases as barrel temperature increases, this may be due to the reduction of material viscosity particularly close to the barrel wall which results in lower frictional forces on the screw. Also, back pressure generation is lower with lower viscosity conditions which increases the throughput rate.

4 4.1

Modelling System Model Identification

The main aim of this work was to model the effects of process settings on the motor power consumption. Firstly the model inputs and outputs were identified. Five input parameters (u1-u5 ) and one output parameter (y1 ) were considered for modelling as illustrated in Figure 2.

Die

Adapter

Zone-4

T4

Clamp Ring

Zone-3

Zone-2

T1 T2 T3

Zone-1

N

 Mp   Motor 

Fig. 2. Extruder model with selected inputs and output

Inputs: screw speed (N ), barrel set temperatures (Tb ) at each zone (T1 , T2 , T3 , T4 ). The set temperatures of the clamp ring, the adapter, and the die were always equal to T4 in this study. If these values are different from T4 , it is possible to add them as three different model input parameters. Output: Motor power consumption (Mp ). 4.2

Model Development

Each experimental trial was run over nine minutes for each set of process settings. Process signals were unsteady and contain transients within the first few minutes. Therefore, the data collected over the 7th and 8th minutes by the 10Hz sampling rate were used for model validation and development respectively. According to Figure 1.a, the extruder motor power consumption can be assumed as a function of N and Tb ; Mp = f (N, Tb )

(2)

This study was focused on developing a static model based on the above relationship. Firstly, an attempt was made to identify a linear model to correlate

Modelling the Effects of Operating Conditions on Motor Power Consumption

15

the output with the inputs by approximating the function f. However, the linear model did not predict the motor power consumption values accurately due to the significant nonlinearities in the process. Secondly, nonlinear polynomial models were adopted to approximate the function f. The predicted nonlinear relationships were shown to give reasonable agreements with the experimental data. Due to the strong nonlinearity of the polymer process, the maximum power was selected as 4 for the models of both materials. As a result, a large number of terms were included in the models, which may limit their practical application. However, only a few terms in these models were found to provide significant contribution to the outputs. Sub-model selection algorithms, such as orthogonal least squares (OLS) [11] and fast recursive algorithm (FRA) [12] can be applied to construct a parsimonious model with satisfactory generalisation capability. Due to the lower computational complexity and improved stability, an FRA was used as a sub-model selection algorithm here [13].

5

Discussion

Linear and nonlinear models were identified for both recycled and virgin materials to investigate the effects of process settings on motor power consumption. Moreover, the modelling errors (ME) and the normalised prediction percentage errors (NPPE) of the models were determined by equations (3) and (4) respectively. M E = y(t) − yˆ(t) (3) N N   NPPE  [ (ˆ y (t) − y(t))2 / y(t)2 ]1/2 × 100% t=1

(4)

t=1

Where y(t) is the measured motor power consumption and the yˆ(t) is the model estimated motor power consumption. 5.1

Linear Models

Firstly, an attempt was made to identify more general linear models to predict the motor power consumption. However, the identified linear models for recycled and virgin materials had only about an 80% fit with training data and were not accurate enough to represent the actual motor power consumption over different processing conditions. The major problem attributed to the linear models were predicting the changes of the power consumption over different barrel set temperatures as shown in Figures 3 and 4. Therefore, nonlinear models were developed and the prediction capability was sufficient to represent the actual situations more closely, and hence the detailed information is presented only for the nonlinear models.

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Table 2. The NPPE of the studied nonlinear models with different orders and number of terms for recycled material Selected model order 3 4 5

5.2

1 TD 12.71 VD 12.84 TD 12.71 VD 12.84 TD 12.71 VD 12.84

2 2.11 2.04 2.11 2.04 2.11 2.04

3 2.08 2.02 2.07 2.00 2.06 2.00

NPPE with different number of terms 4 5 6 7 8 2.05 2.04 1.90 1.66 1.62 2.00 1.90 1.64 1.60 1.57 1.96 1.77 1.72 1.63 1.58 1.92 1.71 1.69 1.58 1.55 2.02 1.88 1.74 1.60 1.41 1.90 1.72 1.54 1.52 1.45

9 1.61 1.57 1.35 1.22 1.40 1.42

10 1.61 1.57 1.14 1.19 1.40 1.40

11 1.62 1.57 1.09 1.19 1.08 1.18

12 1.60 1.57 1.09 1.18 1.08 1.18

Nonlinear Models

Recycled Material For the nonlinear model selection, a number of different model combinations (i.e. with different orders and number of terms) were studied and the details of all of the models studied for the recycled material are shown in Table 2 along with their normalised prediction percentage errors with the training data (TD) and the validation data (VD). Finally, a 4th order model with 11 terms was selected as the final model since further increase in the order or number of terms did not improve the model performance considerably. Although, the training and test errors were very low in some cases, the model was unable to predict the power consumption at some different processing conditions properly (e.g. prediction of the 4th order 10 terms model was poor for the conditions, test B-50rpm). The selected nonlinear static model showed a 97.63% fit with the training data and a 97.40% fit with the unseen data. The model normalised prediction percentage error (NPPE) with training data was 1.09% as shown in Table 2. The linear and nonlinear models fit with the training data along with the nonlinear modelling error corresponding to the each data point are shown in Figure 3. The model equation for the recycled material in terms of screw speed and barrel set temperatures is shown in (5). MRH = −(1227.616 ∗ N ) +(42.290 ∗ N ∗ T2 ) +(0.216 ∗ N 2 ∗ T4 ) 2

+ (0.346 ∗ N 3 )

− (2.271e − 03 ∗ N 3 ∗ T2 )

− (0.297 ∗ N ∗ T12 ) − (0.03734 ∗ N 2 ∗ T12 ) 4

+ (3.294e − 05 ∗ T44 )

+(0.02666 ∗ N ∗ T1 ∗ T3 ) + (1.890e − 04 ∗ N )

− (4.720e − 05 ∗ T1 ∗ T43 ) (5)

Furthermore, each model term was closely examined to explore the effects of the individual processing parameters on the motor power consumption. The screw speed (N ) was identified as the most influential processing parameter. The temperatures of the feed zone (T1 ) was recognised as the critical temperature which influence the motor power consumption. The temperatures of barrel zones two, three, and four (T2 , T3 and T4 ) also show slight effects.

Modelling the Effects of Operating Conditions on Motor Power Consumption

  

11 Motor power consumption (kW)

  

B A

9



50rpm

A

B

C

B C 90rpm

 7 30rpm

B

5 3 A 1

NL Modelling Error

70rpm

17

0.35 0.25 0.15

Measured Linear Model Nonlinear Model

10rpm B C

0

1,200

2,400

3,600

4,800

6,600

0

1,200

2,400

3,600 Data points

4,800

6,600

0

-0.15 -0.25 -0.35

Fig. 3. Estimated and measured motor energy consumptions with nonlinear (NL) modelling error for the recycled material Table 3. The NPPE of the studied nonlinear models with different orders and number of terms for virgin material Selected model order 3 4 5

1 TD 10.80 VD 10.35 TD 10.80 VD 10.35 TD 10.80 VD 10.35

2 9.72 9.47 9.64 9.39 9.59 9.34

3 8.76 8.51 8.48 8.15 8.43 8.09

NPPE with different number of terms 4 5 6 7 8 8.22 6.62 6.27 5.75 5.41 7.97 6.23 5.83 5.28 5.05 7.86 7.11 5.79 5.26 4.04 7.52 6.64 5.56 5.01 3.77 7.93 7.26 5.80 5.13 3.85 7.59 6.80 5.58 4.91 4.66

9 4.98 4.56 3.40 3.72 3.31 3.62

10 4.98 4.56 2.95 3.70 2.45 2.64

11 4.98 4.56 2.31 2.39 2.20 2.35

12 4.98 4.56 2.12 2.12 2.11 2.10

Virgin Material A nonlinear model for the virgin material was also selected by following the same procedure with the recycled material. Details of all of the models studied prior to selection of the virgin material model are shown in Table 3 along with their normalised prediction percentage errors with the training data (TD) and the validation data (VD). A 4th order model with 12 terms was selected as the final model. The model shows a 95.82% fit with training data and a 95.80% fit with the unseen data. The normalised prediction percentage error (NPPE) was identified as 2.12% with training data as shown in Table 3.

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Motor power consumption (kW)

11 9 50rpm A

B

C

70rpm B C

7 30rpm

5

A

B

C

Measured Linear Model Nonlinear Model

10rpm A B C

3 1

NL Modelling Error

90rpm B

0

1,200

2,400

3,600

4,800

6,000

7,200

0

1,200

2,400

3,600 Data points

4,800

6,000

7,200

0.7 0.4 0 -0.4 -0.7

Fig. 4. Estimated and measured motor energy consumptions with nonlinear (NL) modelling error for the virgin material

The linear and nonlinear models fit with the training data along with the nonlinear modelling error corresponding to the each data point are shown in Figure 4. The model equation for the virgin material in terms of the screw speed and barrel set temperatures is given in (6). MV H = −(3256.96679 ∗ N ) +(1958.282428 ∗ T1 )

− (4.49708e − 04 ∗ N 4 )

− (0.02547192 ∗ N 3 ∗ T4 )

− (2113.603062 ∗ T2 )

− (0.103493 ∗ N 2 ∗ T12 )

+(66.297976 ∗ N ∗ T4 ) − (0.38109 ∗ N ∗ T1 ∗ T3 ) 3

2

+ (172.16861 ∗ N 2 )

+(0.0303845 ∗ N ∗ T2 ) + (0.077861 ∗ N ∗ T1 ∗ T2 ) + (3.3009 ∗ T1 ∗ T3 )

(6)

The screw speed was identified as the most critical processing parameter for the virgin material as it was for the recycled material. Similarly the feed zone temperature (T1 ) showed considerable effects with the virgin material. Evaluation of Nonlinear Energy Models The screw speed was the most influential processing parameter for the extruder motor power consumption with both materials. Changes to the barrel set temperatures had only slight effects on the motor power consumption with both materials as shown in Figures 3 and 4. Of the barrel zone temperatures, the feed zone (T1 ) temperature showed more significant effects than the other three zones with both materials. The solid friction is highly temperature dependent [14, 15], therefore changes in T1 can cause significant changes in the solid frictional forces. Changes in solid friction change the screw load and hence the motor power. Solid

Modelling the Effects of Operating Conditions on Motor Power Consumption

19

friction has a greater resistance to the screw rotation than the viscous frictional forces [16]. Therefore, minor changes in the coefficient of solid friction can have a significant impact on the motor power consumption. Moreover, the temperature dependency of the material viscosity may be another factor which affects the motor power consumption. According to Figure 1.a, recycled material consumed higher power than the virgin material in almost all the processing conditions. Usually, viscosity decreases as temperature increases and this is a material dependent property. Therefore, it might expect to be that the recycled material’s viscosity may remain higher than the virgin material (as shown by MFI values) at similar operating conditions and hence resistance to the screw rotation is higher with recycled material. Usually, this is being checked by off-line material testing methods. Likewise, recycled polymer contains around 2.5% of component of carbon black which may affect the rheological properties (e.g. thermal, frictional etc. Reduction in motor power consumption at higher barrel temperatures may be due to the reduction of material viscosity particularly close to the barrel wall. However, this is not always true as evidenced by Figures 3 and 4 which show higher power consumption with higher barrel temperatures (e.g. RH-B-50rpm, VH-B-50rpm, VH-C-90rpm). One of the possible reasons for such a high power demand even with higher barrel zone temperatures may be that the material viscosity remains high at higher screw speeds with poor melting conditions because of the reduction of material residence time. These poor melting conditions may cause the load on the screw to increase as poorly melted high viscous material pushes through the converging metering section of the extruder. Therefore, increases in both screw speeds and barrel temperatures with the aim of achieving good melting and energy efficiency may not provide good thermal stability or better energy efficiency. Therefore, it is better to have an idea of the combined effect of process settings on process thermal stability and energy efficiency.

6

Conclusions

New static nonlinear polynomial models have been presented to predict the motor power consumption in extrusion with different processing conditions and materials. The screw speed was identified as the most critical parameter affecting the extruder motor power consumption while the barrel set temperatures also show a slight effect. Of the barrel zone temperatures, the effects of the feed zone temperature was more significant than other three zones. Moreover, the models developed can be used to find out the significance of individual processing conditions and optimum process settings to achieve better energy efficiency. However, selection of energy efficient process settings should coincide with good thermal stability as well. Thus, studies to identify the combined effect of process settings on both energy efficiency and thermal stability would be more desirable to select a more attractive operating point with better overall process efficiency.

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Acknowledgments. This work was supported by the UK’s Engineering and Physical Sciences Research Council (EPSRC) under grant number EP/F021070/1. The authors would like to thank PPRC staff of Queen’s University Belfast and Dr. E. C. Brown with all IRC staff from the University of Bradford, for their assistance. Also, the support provided by Cherry Pipes Ltd in providing materials is greatly acknowledged.

References [1] BEF.: About the industry, http://www.bpf.co.uk, (viewed April 2010) [2] Barlow, S.: Reducing electrical energy costs for extrusion processes. SPE-ANTEC Tech. papers, 1157–1162 (2009) [3] Kent, R.: Energy Management in Plastics Processing - Part 10: Motors and drives, http://www.tangram.co.uk (viewed February 22, 2010) [4] Falkner, H.: Energy Cost Savings in Motive Power. Water Environ. J. 11, 225–227 (1997) [5] Kelly, A.L., Brown, E.C., Coates, P.D.: The effect of screw geometry on melt temperature profile in single screw extrusion. Polym. Eng. Sci. 12, 1706–1714 (2006) [6] Brown, E.C., Kelly, A.L., Coates, P.D.: Melt temperature homogeneity in single screw extrusion: effect of material type and screw geometry. SPE-ANTEC Tech. papers, 183–187 (2004) [7] Rasid, R., Wood, K.: Effect of process variables on melt temperature profiles in extrusion using single screw plastics extruder. Plast. Rubb. Comp. 32, 193–198 (2003) [8] Abeykoon, C., McAfee, M., Thompson, S., Li, K., Kelly, A.L., Brown, E.C.: Investigation of torque fluctuations in extrusion through monitoring of motor variables. In: PPS 26th Europe/Africa Regional Meeting, Larnaca, Cyprus, vol. 22-O, pp. 315–333 (2009) [9] Lai, E., Yu, D.W.: Modeling of the plasticating process in a single screw extruder: A fast track approach. Polym. Eng. Sci. 40, 1074–1084 (2000) [10] Wilczynski, K.: Single screw extrusion model for plasticating extruders. Polym. Plast. Technol. Eng. 38, 581–608 (1999) [11] Chen, S., Billings, S.A., Luo, W.: Orthogonal least squares methods and their application to non-linear system identification. Int. J. Control 50, 1873–1896 (1989) [12] Li, K., Peng, J., Irwin, G.W.: A fast nonlinear model identification method. IEEE Trans. Autom. Control 50, 1211–1216 (2005) [13] Abeykoon, C., Li, K., McAfee, M., Martin, P.J., Deng, J., Kelly, A.L.: Modelling the effects of operating conditions on die melt temperature homogeneity in single screw extrusion. In: UKACC int. Conf. Control, Coventry, UK (2010) [14] Chung, C.I., Hennessey, W.J., Tusim, M.H.: Frictional behaviour of solid polymers on a metal surface at processing conditions. Polym. Eng. Sci. 17, 9–20 (1977) [15] Spalding, M.A., Hyun, K.S.: Coefficients of dynamic friction as a function of temperature, pressure, and velosity for several polyethylene resins. Polym. Eng. Sci. 35, 557–562 (1995) [16] Menning, G.: The torque in the melting zone of single screw extruders. Polym. Eng. Sci. 27, 181–185 (1987)

Quantum Genetic Algorithm for Hybrid Flow Shop Scheduling Problems to Minimize Total Completion Time Qun Niu, Fang Zhou, and Taijin Zhou Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering and Automation, Shanghai University, 200072 Shanghai, China [email protected]

Abstract. This paper investigates the application of the quantum genetic algorithm (QGA) for Hybrid flow shop problems (HFSP) with the objective to minimize the total completion time. Since HFSP has shown to be NP-hard in a strong sense when the objective is to minimize the makespan in case of two stages, an efficient QGA is proposed to solve the problem. A real number representation is used to convert the Q-bit representation to job permutation for evaluating the solutions and quantum rotation gate is employed to update the population. Two different types of crossover and mutation operators are investigated to enhance the performance of QGA. The experimental results indicate that QGA is capable of producing better solutions in comparison with conventional genetic algorithm (GA) and quantum algorithm (QA). Keywords: Hybrid flow shop scheduling, Genetic algorithm, Quantum algorithm, total completion time.

1 Introduction The hybrid flow shop problem (HFSP) has been attracting many researchers over the last few decades since it is quite common in production systems. For example, manufacturing, oil food, paper, chemical and cosmetic industry can be modeled as hybrid flow shop scheduling problems. This paper describes HFSP which consists of g production workshops or stages and M jobs. Each stage has several parallel machines. These machines have the same effect. Some stages have only one machine, but at least one stage must have more than one machine. At the same time, a job is processed on any one machine at each stage. At any time, each machine also only processes one job. One machine must process another job after the former completed. The processed time pt (i, t ) ( i = 1, 2,3,..., M ; t = 1, 2, 3,..., g ) is given for each job at each stage. In this research, the scheduling objective is to find a job schedule that minimizes the total completion time. This criterion is more realistic than the common used makespan minimization (Cmax) as it is known to increase productivity while at the same time reduce the work-in progress(WIP). K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 21–29, 2010. © Springer-Verlag Berlin Heidelberg 2010

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Q. Niu, F. Zhou, and T. Zhou

HFSP has been proved to be NP-Hard [1] when the objective is to minimize makespan in case of two stages. Moreover, Hoogeveen et al. proved that the HFSP remains NP-hard even if the preemption is allowed [2]. Many approaches have been used to solve HFSP, such as branch-and-bound (B&B), Artificial immune systems (AIS), Genetic Algorithm(GA), Simulated Annealing Algorithm (SA), Tabu Search (TS), and etc. Among these intelligent optimization algorithms, GA is a well known stochastic search technique [3] [4] and has been frequently applied to many problems in scheduling. Der-Fang Shiau [5] presented a hybrid constructive genetic algorithm to solve the proportionate flow shop problem. Recently, a genetic algorithm (GA) has been developed for HFSP, with such studies including those of Şerifoğlu and Ulusoy [6] and Oğuz and Ercan [7]. Chang, Chen, and Lin [8] introduced a two-phase sub population GA to solve the parallel machine-scheduling problem with multiple objectives (makespan and total tardiness). Although GA has been successfully applied to various optimization problems, there are several drawbacks for GA as well, such as, the global convergence is not enough and sometimes is also easy to get into local minimum [9]. In this paper, a Quantum Genetic Algorithm (QGA) is proposed to solve the HFSP. Quantum algorithm (QA) is a recently developed evolutionary algorithm based on probability [10] to search the best solution. Q-bit can overcome premature and maintain solution diversity and quantum rotation gate is used to update the individuals. Although Han and Kim [11] [12] have proposed a QEA formulation for a class of combinatorial optimization, it is not suitable to solve HFSP. In this research, a new quantum representation has been adopted to generate solution string according to the encoding of HFSP [13]. That is, a real number representation is used to convert the Q-bit representation to a job permutation. In addition, different crossover and mutation operators are investigated to improve the performance of QGA. To evaluate the performance of the QGA for HFSP, a series of experiments have been conducted in this paper. In order to show the superiority of the proposed QGA, we also compare it with GA and QA.

2 Quantum Algorithm (QA) QA maintains a population of individuals in qubits and uses a qubit to represent all the states in the search space. A qubit may be in the “1” state, or in the “0” state, or in the any superposition between “0” and “1”. A Q-bit is described by a superposition of basic states (“0” state and “1” state): |ψ >= α | 0 > + β |1 > , |α |2 +|β |2 =1

Where |ψ > , | 0 > , | 1 > is the quantum state, α and β are complex number that denote the probability amplitudes of the corresponding states. Note that α and β also satisfy the normalization condition |α |2 +|β |2 =1 , |α |2 and |β |2 give the probability that the Q-bit will be found in the‘0’ state and ‘1’ state. A Q-bit chromosome as a string of m-qubit is defined as: ⎡ α 1 α 2 ... α m ⎤ ⎢ ⎥ ⎣ β 1 β 2 ... β m ⎦

QGA for Hybrid Flow Shop Scheduling Problems to Minimize Total Completion Time

23

n

Hence, a qubit string with n bits represents a superposition of 2 binary states and it can search the entire space. Quantum rotation gate is used to update the individuals and makes a current solution approach to the best solution gradually. The quantum rotation gate can be represented as follows: ⎡α i' ⎤ ⎡ cos(θ i ) − sin(θ i ) ⎤ ⎡α i ⎤ ⎡α i ⎤ ⎢ β ' ⎥ = ⎢sin(θ ) cos(θ ) ⎥ ⎢ β ⎥ = U (θ i ) ⎢ β ⎥ i i ⎣ i⎦ ⎣ ⎦⎣ i⎦ ⎣ i⎦

Where: U (θi ) is the quantum rotation gate and θi is the rotation angle of the ith Q-bit, α and ⎡⎢ i ⎤⎥ is the ith Q-bit. β ⎣ i⎦

After a qubit string is generated, we need to observe the qubit string. A string including the same number of random numbers between 0 and 1 (R) is generated. If R > α 2 , P will be set to 0 and 1 otherwise. Table 1 presents the observation of qubit string on the above instance. In this paper, the individual is represented by a real number encoding method. In order to reduce the calculation time and complexity of QGA, we proposed another observational manner. We will describe it in details in the following section.

3 QGA for Hybrid Flow Shop Problem In the following, we will describe QGA for HFSP in detail including initial population, convert mechanism, rotation operation, selection, crossover and mutation operations. 3.1 Presentation

A job permutation represents the processing sequence. For example, a three-job flow shop problem, permutation [3 1 2] denotes the processing sequence is job-3, job-1, job-2. But the Hybrid Flow Shop uses a real number coding whose fractional part is used to sort the jobs assigned to each machine and whose integer part is the machine number to which the job is assigned. For example, considering a problem with four jobs ( n = 4 ), three processes ( g = 3 ), two machines at stage one ( m1 = 2 ), three machines at stage two ( m 2 = 3 ) and one machine at stage three ( m3 = 1 ). About this problem we must generate four random numbers from uniform distribution [ 1,1 + mt ] ( t = 1, 2,3 ) in each stage of process. As shown in Table 1, at stage one, job1 and job2 are assigned to machine 1; job 3 and job 4 are assigned to machine 2. The order of jobs to be scheduled on machine 1 is job 2 followed by job 1. Because the fractional part of job 2 is greater than the fractional part of job 1. At stage i ( i > 1 ), if two jobs are assigned to the same machine, we can arrange the jobs according to another rule. The rule is described as follows: Every job is based on the completion time of the (i-1) stage to ensure its processing sequence, that is, the job which first completes the former stage will be first processed. If the completion time of the former stage is also the same, we can compare the values of genes. The job whose value of gene is smaller will be first processed. If the values are also equal, we can choose the job processing sequence randomly.

24

Q. Niu, F. Zhou, and T. Zhou Table 1. Representation of candidate solutions

1 .24

Stage One 1 2 .75 .35

2 .98

Stage Two 2 3 .38 .28

1 .80

1 .25

Stage Three 1 1 .20 .57

1 .96

1 .01

3.2 Initial Population

Randomly generate an initial population PQ (t ) , PQ (t ) = { p1 (t ), p2 (t ), p3 (t ),..., p N (t )}

Where pi (t ) has m×n Q-bits and is defined as follows: ⎡ cos( t i1 ) cos(t i 2 ) ... cos( tin ) ⎤ p i (t ) = ⎢ ⎥ ⎣ sin(t i1 ) sin(t i 2 ) ... sin( tin ) ⎦

Where tij = 2 * π * rand and rand is a random between 0 and 1; i = 1, 2,..., m ; j = 1, 2,..., n ; m is a number of work stages and n is a number of jobs. 3.3 Convert PQ (t ) to Job Permutation Pp (t ) ⎡ cos( t i1 ) cos(t i 2 ) ... cos(t in ) ⎤ pi (t ) = ⎢ ⎥ is generated randomly. Because the coding of ⎣ sin( t i1 ) sin(t i 2 ) ... sin( tin ) ⎦

hybrid flow shop is a real number which is between 1 and mt + 1 , Pp (t ) is defined as follows: 2

2

Ppi (t ) = [ cos(tk1) * mt (k ) + 1 ... cos(tkj ) * mt (k ) + 1 ...] Where Ppi (t ) is a job permutation of the ith individual at the t time; k = 1, 2,..., m ; j = 1, 2,..., n . m is the number of work stages and n is the number of jobs. This converting method can guarantee each solution is feasible and is used to evaluate the objective value easily. 3.4 Rotation Operation

Due to the real number coding, rotation operation used in this paper is different from that used in other papers such as [12]. Rotation operation is defined as follows: ⎡α ⎤ ⎡ cos(θ ) − sin(θ ) ⎤ ⎡α ⎤ ⎡α ⎤ ⎢ β ⎥ = ⎢ sin(θ ) cos(θ ) ⎥ ⎢ β ⎥ = U (θ ) ⎢ β ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ '

i

i

i

i

i

i

'

i

i

i

i

i

⎡α i ⎤ ⎡αi' ⎤ Where ⎢ ⎥ denotes the present solution. U ( Δθ i ) is a Q-gate. The ⎢ ⎥ denotes the ⎣ βi ⎦ ⎣ βi' ⎦ present solution which is performed by rotation operation, Δθ i is a rotation angle

QGA for Hybrid Flow Shop Scheduling Problems to Minimize Total Completion Time

25

Table 2. Lookup table of rotation angle

xi > besti

x i = best i

x i < best i

f(x) ≥ f(best)

Δθ i

false false false false true true

false false true true false false

true true false false false false

false true false true false true

-0.01π -0.001π 0.001π -0.001π 0.01π 0.001π

whose lookup table is shown in Table 2. In this Table, f ( r ) < f (b) means that the solution b is preceded the solution r; the best solution respectively.

xi and besti are the ith bit of the x solution and

3.5 Selection Individuals from the current population are selected based on their fitness value to form the next generation. The roulette wheel selection is used widely. This selection strategy is widely used in standard GA. Chromosomes for reproduction are selected based on their fitness. Chromosomes with higher fitness will have a higher probability to be selected to enter into the next generation.

3.6 Crossover Operators This paper uses two crossovers, which are single point crossover and double-point crossover respectively.

(1) Single Point Crossover Each of processes has a different crossover point and changes genes after the crossover point between two parents. Randomly generate one point and regard this point as crossover point. If it satisfies the conditions of crossover, the two parents will change genes to generate the offspring. (2) Double-Point Crossover In a double-point crossover, two cutting points are selected at random and then the genes between two cutting points are exchanged between the two parents. As an example, Fig. 1 shows a representation of the double-point crossover operation. 3.7 Mutation Operators This paper uses two mutation operators, which are inversion mutation and Left&Right Swap mutation respectively.

(1) Inversion Mutation Mutation selects two points at each process. A mutation operator is used to change the machine assignment. Mutation one is described as follows: Randomly generate two points (point A and point B) and reverse genes between point A and point B.

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Q. Niu, F. Zhou, and T. Zhou

Fig. 1. An example of double-point Crossover

(2) Left & Right Swap Mutation Randomly generate two points and do not change the genes between the two points (including gene of the left point but not including gene of the right point). Interchange the genes in front of the left point with the genes after the right point (including the right point).

4 Computational Results In this section, QGA, GA and QA are implemented on a Pentium Dual 1.60Ghz computer and in MATLAB software. The total completion time is used to evaluate the performance of the methods proposed.

4.1 Comparison of Different Crossover and Mutation Operators In section 3, we have introduced two different crossover and mutation operators respectively. In order to examine the performance of these operators, one instance is randomly generated where the number of jobs is 10 and the number of process stages is 5. The job processing times and the amount of the machine number were assumed to be uniformly distributed integers from U{1,2,…,40}and U{1,2,…,5}. QGA and GA are tested on the instance. For QGA and GA, the crossover probability is set to 0.85, and the mutation probability is 0.1. The population size is chosen to be 40 and both algorithms are run ranging from 200 to 500 generations. All the algorithms will be run 10 times. Table 3 presents the detailed comparison results for QGA and GA on the above mentioned instance. The information shown in Table 3 is as follows: Size means the generation and the population size; Type represents the type of Crossover and mutation operators; Min and Average denote the minimum and the average values of 10 runs. From Table 3, it can be seen that double-point crossover operator outperforms single point crossover in terms of both minimum and average values. With regard to mutation operators, Left&Right Swap mutation can obtain better results than

QGA for Hybrid Flow Shop Scheduling Problems to Minimize Total Completion Time

27

Table 3. Comparison of QGA and GA in terms of diffrent Crossover and mutation operators Size

Type

200×40 200×40 500×40 500×40

Single Point Crossover Double-point Crossover Inversion Mutation Left&Right Swap Mutation

GA Min/Average 1644/1685.9 1641/1673.3 1643/1672.1 1638/1662.5

QGA Min/Average 1633/1662.8 1620/1656.5 1630/1667.8 1628/1657.2

Table 4. Comparison results of 30 instances Problem size

10×3

10×5

20×3

20×5

30×3

30×5

40×3

40×5

50×3

50×5

Example A1 A2 A3 B1 B2 B3 C1 C2 C3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3 G1 G2 G3 H1 H2 H3 I1 I2

GA Min/Average 805/823.2 776/811.7 969/987.2 1635/1671.7 1382/1426.0 1532/1557.3 2436/2485.2 2608/2681.0 2124/2189.0 4422/4550.9 4472/4625.3 5136/5200.6 5504/5778.4 5985/6219.2 6404/6555.6 7421/7581.3 6730/6967.5 6734/7001.6 9267/9475.4 8561/8823.2 8523/8705.5 10741/11030.4 11285/11637.4 11550/11746.0 8964/9261.1 8016/8257.9 8860/9179.2 16128/16355.5 17347/17677.1

QA Min/Average 834/859.8 801/836.7 994/1018.2 1672/1698.5 1401/1459.8 1580/1620.1 2500/2560.1 2774/2828.0 2247/2311.1 4605/4704.4 4684/4787.4 5202/5307.4 5829/5971.7 6216/6475.2 6524/6812.4 7565/7757.1 6829/7134.9 7090/7374.5 9322/9777.4 8671/8959.4 8574/8984.9 11138/11386.4 11614/11851.1 11868/12152.8 9312/9653 8347/8804.6 9333/9763.6 16536/16866.2 17532/17952.0

QGA Min/Average 788/819.1 756/786.3 945/978.8 1616/1653.3 1364/1414.0 1526/1542.0 2302/2460.2 2464/2627.3 2043/2161.3 4382/4539.3 4417/4527.4 4954/5091.6 5310/5603.7 5861/6170.4 6255/6480.2 7000/7479.5 6257/6889.3 6697/6942.1 8677/9304.1 8265/8659.2 8007/8594.7 10512/10957.9 10744/11418.9 11075/11613.9 8443/9273.3 7508/8210.2 8395/9064.7 15271/15996.5 16380/17556.0

I3

17557/18067.3

18362/18707.3

16668/17611.5

Best Value 788 756 945 1616 1364 1526 2302 2464 2043 4382 4417 4954 5310 5861 6255 7000 6257 6697 8677 8265 8007 10512 10744 11075 8443 7508 8395 15271 16380 16668

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inversion mutation. These indicate that double-point crossover operator and Left & Right Swap Mutation are more suitable for HFSP than other operators used in this paper. In addition, it is obvious, as shown in Table 3 that the proposed QGA performs better than GA in terms of solution quality.

4.2 Comparison Results of 30 Instances In order to validate the performance of the proposed method, 30 instances are randomly generated. The number of jobs n is 10, 20, 30, 40, and 50. For each number of jobs, the process stage number is set to 3 and 5. For each problem size, 3 instances are generated. The processing time and the amount of the machine number are normally generated from a discrete uniform distribution, U{1,2,…,40}and U{1,2,…,5}. The solution obtained by QGA is compared with GA and QA. For QGA and GA, we use double-point crossover with a crossover probability of 0.8, and Left & Right Swap Mutation with mutation probability 0.1. The population size is chosen to be 40 and the generation number is set to 400 for all the methods. For each instance, all the algorithms will be run 10 times. Table 4 indicates the results of the 30 instances in different problem size. It can be seen that the range of processing time distribution does not have a significant impact on problem hardness. QGA is the best algorithm in comparison with GA and QA, while GA performs better than QA, which means QA, is easy to trap in a local optimal without the help of other evolutionary algorithms. QGA can avoid premature convergence and escape the local optima. Therefore, QGA results in rapid convergence, and more robustness.

5 Conclusions In this paper, the proposed QGA has been used for solving hybrid flow shop scheduling problems with the objective of total completion time minimization. The proposed QGA inspired by concept of quantum computer and genetic algorithm has been applied to solve the HFSP. Double-point crossover and Left&Right Swap Mutation operators are selected to improve the performance of GA through the simulations. The experimental results have showed the superiority of the QGA compared with GA and QA by allowing GA and QA to have the same generations. The proposed QGA provides the advantage of giving a greater diversity by using qubit coding, which can avoid premature convergence and obtain better solutions. Therefore, the proposed QGA possesses the merits of global exploration, fast convergence, and robustness. Future work will focus on enhancing the efficiency of the proposed approach and applying the proposed method to different scheduling problems. We will also want to verify the potential of QGA for other real-world optimization problems.

Acknowledgments. This work is supported by the National Natural Science Foundation of China (Grant No. 60804052), Shanghai University "11th Five-Year Plan" 211 Construction Project and the Graduate Innovation Fund of Shanghai University.

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29

References 1. Gupta, J.N.D.: Two-Stage Hybrid Flowshop Scheduling Problem. Oper. Res. Soc. 39, 359–364 (1988) 2. Hoogeveen, J.A., Lenstra, J.K., Veltman, B.: Preemptive Scheduling in a Two-Stage Multiprocessor Flow Shop is NP-Hard. European Journal of Operational Research 89, 172–175 (1996) 3. Goldberg, D.E.: Genetic Algorithms in Search Optimization and Machine Learning. Addison-Wesley, Reading (1989) 4. Swarup, K.S., Yamashiro, S.: Unit Commitment Solution Methodology Using Genetic Algorithm. IEEE Transactions on Power Systems 17(1), 87–91 (2002) 5. Der-Fang Shiau, D.F., Cheng, S.C., Huang, Y.M.: Proportionate Flexible Flow Shop Scheduling Via a Hybrid constructive genetic algorithm. Expert Systems with Applications 34, 1133–1143 (2008) 6. Serifoglu, F.S., Ulusoy, G.: Multiprocessor Task Scheduling in Multistage Hybrid FlowShops: a Genetic Algorithm Approach. Journal of the Operational Research Society 55(5), 504–512 (2004) 7. Oguz, C., Ercan, M.: A Genetic Algorithm for Hybrid Flow-Shop Scheduling with Multiprocessor Tasks. Journal of Scheduling 8(4), 323–351 (2005) 8. Chang, P.C., Chen, S.H., Lin, K.L.: Two-Phase Population Genetic Algorithm for Parallel Machine-Scheduling Problem. Expert Systems with Applications 29, 705–712 (1999) 9. Rudolph, G.: Convergence Analysis of Canonical Genetic Algorithms. IEEE Transactions on Neural Networks 5, 96–101 (1994) 10. Narayanan, A.: Quantum Computing for Beginners. In: Proceedings of the 1999 Congress on Evolutionary Computation, pp. 2231–2238 (1999) 11. Han, K.H., Kim, J.H.: Quantum-Inspired Evolutionary Algorithms With a New Termination Criterion, Hε Gate, and Two-Phase Scheme. IEEE Trans. Evol. Comput. 8(2), 156–169 (2002) 12. Han, K.H., Kim, J.H.: Quantum-Inspired Evolutionary Algorithm for a Class of Combinatorial Optimization. IEEE Trans. Evol. Comput. 6, 580–593 (2002) 13. Niu, Q., Zhou, T.J., Ma, S.W.: A Quantum-Inspired Immune Algorithm for Hybrid Flow Shop with Makespan Criterion. Journal of Universal Computer Science 15(4), 765–785 (2009)

：

Re-diversified Particle Swarm Optimization Jie Qi and Shunan Pang College of Information Science and Techenology, Donghua University, Shanghai, China [email protected]

Abstract. The tendency to converge prematurely is a main limitation which affects the performacne of evolutionary computation algorithm, including particle swarm optimization (PSO). To overcome the limitation, we propose an extended PSO algorithm, called re-diversified particle swarm optimization (RDPSO). When population diversity is small, i.e., particles’s velocity approches zero and the algorithm stagnates, a restart approach called diversification mechanism begins to work, which disperses particles and lets them leave bad positions. Based on the diversity calculated by the particles’ current positions, the algorithm decides when to start the diversification mechanism and when to return the usual PSO. We testify the performance of the proposed algorithm on a 10 benchmark functions and provide comparisons with 4 classical PSO variants. The numerical experiment results show that the RDPSO has superior performace in global optimization, especially for those complex multimodal functions whose solution is difficult to be found by the other tested algorithm. Keywords: Re-diversified particle swarm optimization, population diversity, benchmark function, swarm intelligence, local optima.

1 Introduction Since particle swarm optimization (PSO) was introduced by Kennedy and Eberhart [1], due to its effectiveness and simplicity in implementation, many successful applications have been seen in solving real world optimization problems [2-6]. As the original PSO may easily get trapped in the local optima when solving complex multimodal problems, many modified versions have been proposed to improve the performance [7-11]. Some of the major variants of PSO are summarized as follows: (1) Control strategies for the PSO parameters. Time-varying inertia coefficient and the acceleration coefficients are usually used in the evolution of the algorithm. Also, the PSO parameters can be dynamically adjusted in response to feedback information from current swarm and environment [7, 10]. (2) Swarm topology. Swarm topology can change the algorithm’s convergence properties by influencing the information transfer mode among particles [7]. (3) Restart mechanism. Through re-initializing particles’ velocities and positions or perturbation of the current best particle, restart mechanism is introduced into the algorithm to prevent the algorithm from stagnation. [6, 8]. (4) Hybrid particle swarm. Techniques from other evolutionary computations have also been borrowed by PSO researchers [7, 11, 12]. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 30–39, 2010. © Springer-Verlag Berlin Heidelberg 2010

Re-diversified Particle Swarm Optimization

31

In this paper, we propose a new extended version called re-diversified PSO(RDPSO). Based on the constricted PSO, a new restart mechanism is introduced. The RDPSO uses a dispersion strategy to diversify the stagnated swarm. Contrary to the idea of basic PSO that particles fly towards the best position, the dispersion strategy lets particles leave the worst current position. Because of its properties, we call this strategy diversification mechanism, while call the basic PSO’s update rule optimization mechanism. For the purpose of measuring swarm diversity, we define a parameter Div for RDPSO called diversity factor. When Div is small, particles will be gathered together, and the algorithm will trend to stagnate. In this case, the algorithm starts diversification mechanism, which helps the PSO obtain new vitality and make it less vulnerable to local optima. This disperse effect increases the swarm diversity Div again. When Div increases to a certain amount, the algorithm will return to basic PSO optimization mechanism. The alternative use of diversification and optimization mechanisms can balance the exploration and exploitation behavior of the algorithm. Therefore, the RDPSO has obtained good performance of global optimization to avoid the local optima. The paper is organized as follows. In section 2, the PSO and some variants used to compare with our algorithm are briefly introduced. In section 3, we present the RDPSO and describe it in detail. Section 4 gives experimental results of the RDPSO and other 4 PSO variants to solve 10 benchmark functions. Finally, conclusions are drawn in section 5.

2 Particle Swarm Optimization and Some Variants PSO is a population based optimization technique, where a swarm of particles I = {1,..., i ,..., n} is randomly initialized and the algorithm searches for optima by updating generations. For an objective function to be optimized f : R → R , four vectors are defined for a particle at each iteration t: position vector xi = [ xi1 ,..., xid ,..., xiD ] expressing a potential solution of the objective function; D

vi = [vi1 ,..., vid ,..., viD ] describing direction and distance of one step move; history best position vector pbi = [ pbi1 ,..., pbid ,..., pbiD ] recording the best

velocity vector

position particle i ever

visited; and the

neighbor best position vector

→

pgi = [ pgi1 ,..., pgid ,..., pgiD ] representing the best position in the neighborhood. During the iteration process, the velocity and position of particle i on dimension d are updated as [1]

vid = vid + c1rand1 ( pbid − xid ) + c2 rand 2 ( pgid − xid )

(1)

xid = xid + vid .

(2)

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J. Qi and S. Pang

Where c1 and c2 are the acceleration coefficients, and rand1 and rand2 are two uniformly distributed random number which are independently generated within [0,1]. This update rule illustrates that (I) a particle has inertia which can keep its original velocity; (II)a particle has memory which can record the best position found so far; (III) particles can communicate each other to obtain their neighborhood’s information. In addition, a maximum velocity parameter Vmax is defined to prevent the algorithm from divergence. In the following paragraphs, we describe some classical variants that will be used in RDPSO or compared with the RDPSO. A. Constricted PSO A constriction factor γ is introduced into the equation (1) to avoid the unlimited growth of the particles’ velocity. The velocity update equation is modified to

vid = γ (vid + c1rand1 ( pbid − xid ) + c2 rand 2 ( pgid − xid )) with

γ =

2 2 − ϕ − ϕ 2 − 4ϕ

the algorithm[14]. Usually,

,

ϕ = c1 + c2 ,

and

ϕ > 4 to

(3)

ensure convergence of

c1 and c2 are set to 2.05, thus γ = 0.72984 [13]. In the

RDPSO, we adopt the constricted PSO as the optimization mechanism. B. The Gbest and Lbest Swarm Topology In the gbest topology, the swarm forms a network of complete graph. Thus, a particle can communicate with any particle of the population. On the contrary, the lbest swarm forms a network of a ring lattice where each individual is only connected to 2 adjacent members in the population array. As subpopulations could converge in diverse regions, the ring topology has the advantage of allowing to parallel search by narrowing the scope of communication. So lbest PSO is less vulnerable to the attraction of local optima than gbest [7]. C. Full Informed Particle Swarm Optimization Mendes et al.[14] proposed the fully informed particle swam (FIPS), in which a particle uses best position information from all its topological neighbors. The velocity update rule becomes

vid = γ (vid + ∑ k∈N ck rand k ( pbkd − xid ))

(4)

i

N i is the neighborhood of particle i. And all acceleration coefficients ck = ϕ / | N i | .

Where

Re-diversified Particle Swarm Optimization

33

D. Self- organized Hierarchical PSO with Time-Varying Acceleration Coefficients Ratnaweera et al[8] proposed the self-organized hierarchical PSO with time-varying acceleration coefficients (HPSOTVAC). HPSOTVAC has three main extensions based on the standard PSO, (1) The inertia velocity term in equation (1) is eliminated, so the update formula becomes

vid = c1rand1 ( pgid − xid ) + c2 rand 2 ( pg d − xid )

(5)

(2) If any component of a particle’s velocity is very close to zero, it is reinitialized to a value proportional to current value of Vmax. Additionally, the maximum velocity is linearly decreased from Vmax to 0.1 Vmax during a run. (3) A linearly adapting strategy is used to change the value of the acceleration coefficients c1 and c2 . During a run of the algorithm, c1 is decreased from 2.5 to 0.5, and

c2 is increased from 0.5 to 2.5.

3 The Re-diversified Particle Swarm Optimization The tendency to converge prematurely on local optima is a main disadvantage of PSO, especially for complex multimodal functions. So in this paper, we proposed a novel restart approach called diversification mechanism to help the prematurely stagnated swarm to jump out of the local optima. The restart approach can increase both the swarm diversity and the chance of a particle to move to a better position. When the swarm diversity is large enough, the constricted PSO called optimization mechanism begins to work. When the swarm diversity becomes small, the diversification mechanism starts to reinvigorate the stagnated algorithm. Adopting optimization or diversification mechanism depends on the value of the diversity factor Div which measures the degree of the population diversity. (1) Diversification Mechanism The diversification mechanism can disperse the concentrated swarm and let particles leave the current worst position by taking the following velocity update rule

vid = χ (vid + c3rand 3 ( xid − xmd ) + c4 rand 4 ( xid − lwid )) where

xmd =

tion, and

(6)

1 N x is the dth dimension coordinate of the swarm center posi∑ i =1 id D

lwid is the dth dimension coordinate of the particle with the worst fitness

among particle i's neighbors. The second term of Equation (6) plays a role to disperse the concentrated swarm, so parameter c3 is called dispersal coefficient. The third term gives particles a repulsion velocity to let them leave the worst position, so we call c4 the repulsion factor.

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J. Qi and S. Pang

(2) Diversity Factor The diversity factor Div is defined to measure the degree of the swarm diversity

Div =

1 N

∑

N i =1

∑

D d =1

( xid − xmd )2

(7)

Div is the average distance between each particle and the swarm center position. (3) The Transformation Process between the Diversification and the Optimization Mechanism A small Div means poor swarm diversity. In this case, particles gather around the swarm center and thus the velocities become slower and slower. If the current center is very close to a local optimum, all particles might stagnate in the center and the algorithm might be trapped in this local optimum. As a restart approach, the diversification mechanism will be applied in the algorithm when Div is reduced to a certain amount. Due to the effect of the diversification mechanism, the swarm diversity increases again and the particle swarm scatters in a wider region. So it is the time for the algorithm to start the optimization mechanism again. The transformation process of these two (optimization and diversification) mechanism is described as follows: The transformation process flow Initiation of particles and calculation of current diversity factor Div; set Div1=Div and Div2=Div. While an end criterion condition is unsatisfied If Div1>=T1 Constricted PSO (optimization mechanism); Calculation of current diversity factor Div; Div1=Div; Div2=Div; Else if Div2=T2 Div1=Div; End if End if End if End While In the flow, we assume two diversity factors Div1 and Div2 for the purpose of implementing the alternation between the optimization and diversification mechanism. T1 is a lower threshold value defining the work range of the optimization mechanism. And T2 is an upper threshold value defining the work range of the diversification mechanism. In the algorithm, we set T2>>T1.

Re-diversified Particle Swarm Optimization

35

4 Numerical Experiments and Results In order to demonstrate the performance of the proposed RDPSO, we use 10 benchmark functions shown in table 1 for the experimental tests. Table 1. Benchmark Functions Equation

D

Feasible Bounds

f1 = ∑ i =1 xi2

30

[−100,100]D

f 2 = ∑ i =1 (∑ j =1 x j )2

30

[−100,100]D

f3 = ∑ i =1 xi + ∏ i =1 xi

30

[−10,10]D

f 4 = ∑ i =1 (100( xi +1 − xi2 )2 + ( xi − 1)2 )

30

[−30,30]D

D x D 1 ∑ xi2 − ∏ i=1 cos( ii ) + 1 4000 i =1

30

[ −600, 600]D

30

[−5.12,5.12]D

30

[−5.12,5.12]D

30

[ −32, 32]D

30

[ −50,50]D

30

[ −50,50]D

Unimodal

D

D

i

D

D

D

f5 =

f 6 = ∑ i =1 ( xi2 − 10 cos(2π xi ) + 10) D

f 7 = ∑ i =1 ( yi2 − 10 cos(2π yi ) + 10) D

Multimodal

| xi |< 0.5 ⎧ xi yi = ⎨ round (2 x ) / 2 | xi |≥ 0.5 i ⎩ f 8 = −20exp( −0.2

1 D

∑

D i =1

xi2 )

− exp( D1 ∑ i =1 cos(2π xi )) + 20 + e D

f9 =

π D

{10sin 2 (π y1 ) + ∑ i =1 ( yi − 1) 2 [1 + 10sin 2 (π yi +1 )] D −1

+ ( y D − 1) 2 } + ∑ i =1 μ ( xi ,10,100, 4) D

yi = 1 + 14 ( xi + 1) ⎧ k ( xi − a ) ⎪

m

μ ( xi , a, k , m ) = ⎨0

xi > a −a ≤ xi ≤ a

⎪ k ( − x − a )m x < − a i i ⎩

f10 = 0.1{sin 2 (3π x1 ) + ∑ i =1 ( xi − 1)2 [1 + sin 2 (3π xi +1 )] D −1

+ ( x D − 1) 2 (1 + sin 2 (2π xD ))} + ∑ i =1 μ ( xi ,5,100, 4) D

According to the literature [15]’s standard for performance testing, the population should be initialized within a subspace of the entire feasible search space which does not contain the global optimum. The setting of asymmetric initialization range can test the

36

J. Qi and S. Pang

algorithm global optimization ability to expand its search beyond its initial limits. This impact is great especially for function f8 whose landscape is very different between the region near the center of the feasible bounds (which is the global optimum) and the region far away from the center. Therefore, we do experiments respectively for two f8 initialization settings; they are expressed by f8 and f8’, which are listed in table 2. Table 2. Optima and initialization ranges for ten functions

function f1

Initialization

f2

[50,100]D [5,10]D

f3

[50,100]D

D

optimum 0D

Optimal fitness 0

0D

0

0D

0

D

0

f4

[15, 30]

1

f5

[300, 600]D

0D

0

0

D

0

0

D

0

0D

0

f6 f7 f8

D

[2.15,5.12]

D

[2.15,5.12] [ −32, 32]D D

D

0

f8’

[16,32]

f9

[25,50]D

-1D

0

f10

[50,100]D

1D

0

0

The experimental results, in terms of the mean and standard deviation of the fitness value obtained from 20 independent trials, are displayed in table 3. We compare other 4 PSO variants (gbest, lbest, FIPS and HPSOTVAC) with the proposed RDPSO. Note that all the PSO algorithms use the same population size of 30, the same number of 300000 fitness evaluations (FEs) and the same maximum velocity Vmax = Xmax (half of the feasible range) in experiments. Gbest, lbest and RDPSO’s optimization mechanism use the constricted PSO’s update rules. In FIPS, the URing topology structure is applied and the parameter ϕ is set to 4.1. HPSOTVAC uses the parameters setting defined in section 2. In addition to the above settings, the other parameters of RDPSO are set as follows: c3=1, c4=1, T1=0.001Xmax, T2=Xmax, and χ linearly decreased from 0.7 to 0.0001. The best results obtained by the 5 PSO algorithms are shown in bold. The results indicate that the RDPSO has already obtained good results although it does not show the obvious advantage on finding the solution of unimodal function over other 4 algorithm. However, for multimodal functions, the RDPSO demonstrates very well performance in term of jump out of the local optima, i.e., it can obtain better solution than the other algorithms for functions f6, f7, f8’, f9, f10, especially for function f8’ whose initialization range does not contain the global optimum, the solution found by the RDPSO is much better than that found by the other 4 PSO algorithms because the other 4 PSOs are trapped in a local optimum far away the global optimum. The improvement of the global optimization ability shows that the diversification mechanism does work in the RDPSO algorithm.

Re-diversified Particle Swarm Optimization

37

Table 3. Search results comparisons among different algorithms function

RDPSO

1.27E-3 2

2.34E-51

1.26E-10 6

lbest

FIPS

7.07E-9 0

f1

Mean

1.53E-18 0 0

0

3.2E-32

4.46E-52

0

f2

Std.De v Mean Std.De v

1.27E-2

0.204

0.176

1.18E-4

3.34E-5

1.98E-2

0.14

0.128

3.58E-5

6.46E-5

1.37E-61

9.39E-5 3

2.21E-20

5.29E-39

6.06E-61

2.5E-52

8.95E-21

6.37E-40

16.58

30.61

1.32E-6 2 6.23E-6 3 24.53

0.247937

2.68

29.53

43.24

12.33

0.577217

1.95

0.0223

0.0041

8.91E-4

0.005913

0.008287

0.0255

0.0076

3.27E-3

0.00936

0.008622

109.47

104.4

30.21

0.00372

3.02E-7

26.41

20.24

8.83

0.05433

7.25E-5

7.30

5.55

7.96

0.6

9.52E-10

3.95

3.14

3.82

0.680557

9.54E-10

Mean

1.69

0.0031

3.45E-7

3.7E-7

Std.De v

1.10

0.1701

9.73E-8

1.73E-7

Mean

18.11

15.60

15.92

17.91297

1.82E-5

Std.De v

5.32

8.01

7.45

5.085797

8.25E-5

Mean

0.364

0.052

2.43E-5

1.76E-15

2.115E-1 6

Std.De v

0.453

0.128

3.24E-6

7.44E-16

1.82E-16

4E-14

5.13E-15

1.89E-14

4.72E-15

f3

f4 f5 f6 f7

f8

f8 ¶

f9

f1 0

HPSOTVA C

gbest

Mean Std.De v Mean Std.De v Mean Std.De v Mean Std.De v Mean Std.De v

Mean

0.049

0.0016

Std.De v

0.141

0.0040

7.87E-1 5 8.31E-1 5

5.33E-1 2 7.64E-1 3

The convergence characteristics of the evolution processes are shown in Fig 1 (f1, f6, f7 and f8’ selected for illustration). From Fig.1, we can find an interesting result that the RDPSO converges in a ladder form. This is because the diversification mechanism restarts optimization when the convergence stops for a while. Therefore, the convergence curve of the RDPSO is not smooth like other algorithms. Fig. 1 reveals that the RDPSO generally offers a high speed of convergence, especially for the functions f6, f7 and f8’ whose global optimum is difficult to be found.

38

J. Qi and S. Pang

f1

20

10

10

0

10

10

-20

10

10

-40

10

10

-60

10 Mean fitness

Mean fitness

10

-80

10

-100

10

10

-120

10

10

-140

RDPSO HPSO FIPS lbest gbest

10

-160

10

-180

10

10

0

10

10

0.5

1

1.5 FEs

2

2.5

10

3

f6

3

2

1

0

-1

-2

-3

-4

-5

RDPSO HPSO FIPS lbest gbest

-6

-7

0

0.5

1

5

x 10

(a) f1

10

Mean fitness

10

10

f7

4

2.5

3 x 10

5

f8'

2

10

1

2

10

0

0

10

-2

RDPSO HPSO

10

2

(b) f6

Mean fitness

10

1.5 FEs

FIPS

-4

lbst

-1

10

-2

10

gbest 10

10

10

-3

-6

10

RDPSO HPSO FIPS lbest gbest

-4

-8

10

-5

-10

0

0.5

1

1.5 FEs

(c) f7

2

2.5

3 x 10

5

10

0

0.5

1

1.5 FEs

2

2.5

3 x 10

5

(d) f8’

Fig. 1. Algorithm convergence for selected functions f1, f6, f7 and f8’

5 Conclusion In this paper, we propose an extended PSO algorithm by introducing a restart mechanism when the algorithm stagnates. The progress has been made by the diversification mechanism, which utilizes the information of stagnate swarm and generates a force to drive particles disperse. In the disperse process, each particle finds a way to leave bad position based on the information obtained from its neighbors. This process helps the swarm jump out of the local optima and thus the RDPSO can overcome the premature convergence of the algorithm. With the increase of the fitness evaluations, the algorithm can keep the capability of optimization so as to find better solution. In addition, the RDPSO is easy to be implemented because only a little change is required based on the standard PSO. Compared with gbest, lbest, FIPS and HPSOTVAC on benchmark functions, our approach has demonstrated good performance in terms of global search ability to jump out the local optima. In general, the RDPSO offers the best accuracy on functions f2, f6, f7, f8’, f9 and f10, especially for function f8’ whose initialization range

Re-diversified Particle Swarm Optimization

39

does not contain the global optimum, the solution found by the RDPSO is much better than that found by the other 4 PSO algorithms. Acknowledgments. This work is supported by the Natural Science Foundation of Shanghai (No. 09ZR1401800) and the Shanghai education development foundation (No. 2008CG38).

References 1. Kennedy, J., Eberhart, R.: Particle Swarm Optimization. In: 4th IEEE International Conference on Neural Networks, pp. 1942–1948. IEEE Press, Piscataway (1995) 2. Tasgetiren, M.F., Liang, Y.C., Sevkli, M., Gencyilmaz, G.: A Particle Swarm Optimization Algorithm for Makespan and Total Flowtime Minimization in the Permutation Flowshop Sequencing Problem. European Journal of Operational Research 177, 1930–1947 (2007) 3. Franken, N., Engelbrecht, A.P.: Particle Swarm Optimization Approaches to Coevolve Strategies for the Iterated Prisoner’s Dilemma. IEEE Trans. Evol. Comput. 9, 562–579 (2005) 4. Ho, S.Y., Lin, H.S., Liauh, W.H., Ho, S.J.: OPSO: Orthogonal Particle Swarm Optimization and Its Application to Task Assignment Problems. IEEE Trans. Syst., Man, Cybern. A, Syst., Humans. 38, 288–298 (2008) 5. Tchomte, S.K., Gourgand, M.: Particle Swarm Optimization: A Study of Particle Displacement for Solving Continuous and Combinatorial Optimization Problems. Int. J. Production Economics. 121, 57–67 (2009) 6. Dong, J., Yang, S., Ni, G., Ni, P.: An Improved Particle Swarm Optimization Algorithm for Global. International Journal of Applied Electromagnetics and Mechanics 25, 723–728 (2007) 7. Poli, R., Kennedy, J., Blackwell, T.: Particle Swarm Optimization: An Overview. Swarm Intelligence 1, 33–57 (2007) 8. Ratnaweera, A., Halgamuge, S., Watson, H.: Self-Organizing Hierarchical Particle Swarm Optimizer with Time-Varying Acceleration Coefficients. IEEE Trans. Evol. Comput. 8, 240–255 (2004) 9. Oca, M.A.M., Stützle, T., Birattari, M., Dorigo, M.: Frankenstein’s PSO: A Composite Particle Swarm Optimization Algorithm. IEEE Trans. Evol. Comput. 13, 1120–1132 (2009) 10. Zhan, Z.H., Zhang, J., Li, Y., Chung, H.S.H.: Adaptive Particle Swarm Optimization. IEEE Trans. Syst., Man, Cybern. B, Cybern. 39, 1362–1380 (2009) 11. Arumugam, M.S., Rao, M.V.C., Tan, A.W.C.: A Novel and Effective Particle Swarm Optimization like Algorithm with Extrapolation Technique. Applied Soft Computing 9, 308–320 (2009) 12. Poli, R., Langdon, W.B., Holland, O.: Extending Particle Swarm Optimization via Genetic Programming. In: Keijzer, M., Tettamanzi, A.G.B., Collet, P., van Hemert, J., Tomassini, M. (eds.) EuroGP 2005. LNCS, vol. 3447, pp. 291–300. Springer, Heidelberg (2005) 13. Clerc, M., Kennedy, J.: The Particle Swarm-Explosion, Stability and Convergence in a Multidimensional Complex Space. IEEE Trans. Evol. Comput. 6, 5–73 (2002) 14. Mendes, R., Kennedy, J., Neves, J.: The Fully Informed Particle Swarm: Simpler, Maybe Better. IEEE Trans. Evol. Comput. 8, 204–210 (2004) 15. Bratton, D., Kennedy, J.: Defining a Standard for Particle Swarm Optimization. In: Proceedings of IEEE Swarm Intelligence Symposium, Honolulu, pp. 120–127 (2007)

Fast Forward RBF Network Construction Based on Particle Swarm Optimization Jing Deng1 , Kang Li1 , George W. Irwin1 , and Minrui Fei2 1 2

School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, BT9 5AH, UK Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronical Engineering and Automation, Shanghai University, Shanghai 200072, China

Abstract. The conventional forward RBF network construction methods, such as Orthogonal Least Squares (OLS) and the Fast Recursive Algorithm (FRA), can produce a sparse network with satisfactory generalization capability. However, the RBF width, as a nonlinear parameter in the network, is not easy to determine. In the aforementioned methods, the width is always pre-determined, either by trail-and-error, or generated randomly. This will inevitably reduce the network performance, and more RBF centres may then be needed to meet a desired modelling specification. This paper investigates a new forward construction algorithm for RBF networks. It utilizes the Particle Swarm Optimization (PSO) method to search for the optimal RBF centres and their associated widths. The efficiency of this network construction procedure is retained within the forward construction scheme. Numerical analysis shows that the FRA with PSO included only needs about two thirds of the computation involved in a PSO assisted OLS algorithm. The effectiveness of the proposed technique is confirmed by a numerical simulation example. Keywords: Forward selection, Radial basis function, Nonlinear modelling, Particle swarm optimization.

1

Introduction

Due to the simple topological structure and universal approximation ability, the radial basis function network has been widely used in data mining, pattern recognition, signal processing, system modelling and control [10]. The main issues involved in constructing RBF network are the optimization of basis functions and the estimation of output layer weights. The conventional strategy is to handle these issues separately by selecting the RBF centres using unsupervised clustering [15], optimizing the basis function parameters by gradient-based searches, and estimating the output weights by a least-squares method. Clearly,the network parameters are interdependent, and a better strategy is to optimize them simultaneously [11]. For a single hidden layer RBF neural network, it is possible to formulate its weight estimation as linear-in-the-parameters. A sub-set selection technique, K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 40–48, 2010. c Springer-Verlag Berlin Heidelberg 2010 

Fast Forward RBF Network Construction Based on PSO

41

such as Orthogonal Lease Squares (OLS) [2], the Fast Recursive Algorithm (FRA) [9] and two-stage stepwise selection [8], can then be applied to produce a sparse network with satisfactory accuracy. To further improve the generalization capability and prevent over-fitting with noisy data, Leave-one-out cross validation and Bayesian learning framework were also introduced to these algorithms [3]. The main outstanding issues with these methods is that the width of the radial basis function needs to be pre-determined, and the placement of the RBF centres is limited to the data samples. Further, all hidden nodes share the same RBF width. As a result, the network model obtained is not optimal, and more RBF nodes are often needed to achieve a satisfactory accuracy. Recently swarm intelligence has been learned as a robust and efficient technique for solving nonlinear optimization problems [1]. Unlike conventional calculusbased methods, swarm intelligence introduces a large number of unsophisticated entities that cooperate to exhibit a global behaviour. Though a single member of these societies may be an unsophisticated individual, collectively they are able to achieve complex tasks by working in cooperation. Particle Swarm Optimization is a popular version of swarm intelligence that was originally proposed in 1995 [7]. It has been widely applied to optimization problems ranging from classical problems such as scheduling, neural network training and task assignment, to highly specialized applications [1]. The popular OLS technique has also been revised to utilize PSO to optimize the nonlinear parameters [4]. However, the computation complexity involved remains higher than the FRA based alternatives. In this paper, the Particle Swarm Optimization is effectively integrated with our Fast Recursive Algorithm, leading to a new forward construction scheme for RBF networks. The method involves continuous optimization of both RBF centres and widths, and a discrete optimization of network structure. Unlike the original FRA technique which selects the centres from a candidate pool, the new algorithm randomly generate some initial points (particles in the swarm) from the training data as the starting point. PSO is then adopted to optimize these parameters according to their contribution to the cost function. The best global solution found by the entire swarm becomes the new RBF centre and is added to the network. This procedure continues until a satisfactory network has been constructed. The result from a simulation experiment are included to confirm the efficacy of the approach.

2 2.1

Problem Formulation Radial Basis Function Networks

Consider a general RBF neural network with m inputs, n − 1 hidden nodes and a scalar output that can be expressed by the linear-in-the-parameters model: y(t) =

n 

θk ϕk (x(t); wk ) + ε(t)

(1)

k=1

there y(t) is the actual output at sample time t, x(t) ∈ m is the input vector, ϕk (x(t); wk ) denotes the radial basis function (e.g. a Gaussian basis function),

42

J. Deng et al.

and wk = [σk , cTk ]T ∈ m+1 is the hidden layer parameter vector which includes the width σk ∈ 1 and centres ck ∈ m . Finally, θk represents the output layer weight for each RBF node, and ε(t) is the network error at sample time t. If a set of N data samples {(x)(t), y(t)}N t=1 is used for network training, (1) can be written in matrix form as y = Φθ + e

(2)

where Φ = [φ1 , . . . , φn ] ∈ N ×n is known as the regression matrix with column vectors φi = [ϕi (x(1)), . . . , ϕi (x(N ))]T , i = 1, . . . , n, y = [y(1), . . . , y(N )]T ∈ N is the actual output vector, θ = [θ1 , . . . , θn ]T ∈ n and e = [ε(1), . . . , ε(N )]T ∈ N denotes the network residual vector. The network training aims to build a parsimonious representation based on optimized RBF centres, width parameter σ and output layer weights θ with respect to some appropriate cost function, e.g. Sum Squared Error (SSE). In this paper the RBF network is built using forward construction such that one RBF centre is optimized and added to the network at each step. Suppose that k (k xmax (j), then xi (j) = xmax (j)

(8)

if xi (j) < xmin (j), then xi (j) = xmin (j) f or i = 1, · · · , s; and j = 1, · · · , m.

(9)

where i is the particle index and j is the index of an element in the input vector xi . For velocity, the maximum value is normally obtained from the solution search space and is given by vmax =

1 (xmax − xmin ) 2

(10)

and the search space is defined as [−vmax , vmax ]. The rule for velocity can be set similarly.

3

Fast Forward RBF Network Construction

In a forward network construction scheme one RBF centre is added to the network at each step. The RBF node is optimized based on its contribution to the cost function which is given by: J(c, σ, θk ) = (y − Pk θk )T (y − Pk θk )

(11)

Here it is proposed that the nonlinear parameters c and σ are optimized by PSO, while the output layer weights θk are estimated using Least Squares θˆk = (PTk Pk )−1 PTk y

(12)

where θˆk is the estimated output layer weights. Equation (12) is not normally used in practice because the noise on the data usually causes the matrix P to be ill-conditioned, and the estimated θˆk from (12) can be inaccurate. The Fast Recursive Algorithm (FRA) [9] has proved to be an effective and efficient method to overcome this problem. 3.1

PSO Assisted Forward Selection

The Fast Recursive algorithm (FRA) is introduced by defining a recursive matrix Mk and a residual matrix Rk . Thus Mk  ΦTk Φk Rk  I −

k = 1, · · · , n

T Φk M−1 k Φk

R0  I

(13) (14)

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J. Deng et al.

where ΦTk ∈ N ×k contains the first k columns of the regression matrix Φ in (2). According to [9] and [8], the matrix terms Rk , k = 0, · · · , n can be updated recursively: Rk+1 = Rk −

Rk pk+1 pTk+1 RTk , k = 0, 1, · · · , n − 1 pTk+1 Rk pk+1

(15)

The cost function in (11) can now be rewritten as: J(Pk ) = yT Rk y

(16)

In forward stepwise construction, the RBF centres are optimized one at a time. Thus, suppose at the k-th step, one more centre pk+1 is to be added. The net contribution of pk+1 to the cost function can then be calculated as: (k)

ΔJk+1 (Pk , pk+1 ) = yT (Rk − Rk+1 )y =

(yT pk+1 )2 (k)

pTk+1 pk+1

(17)

(k)

where pk+1  Rk pk+1 . According to (15), this net contribution can be further simplified by defining an auxiliary matrix A ∈ n×n and a vector b ∈ n×1 with elements given by: (i−1) T

ai,j  (pi bj 

) pj , 1 ≤ i ≤ j

(j−1) T (pj ) y,

1≤j≤n

(18) (19)

(0)

where (pj = pj ). The efficiency of the FRA then follows from updating these terms recursively as: ai,j = pTi pj −

i−1 

al,i al,j /al,l

(20)

l=1

bj =

pTj y

−

j−1 

(al,j bl )/al,l

(21)

l=1

Now, substituting (18) and (19) into (17), the net contribution of a new RBF centre pk+1 to the cost function can then be expressed as: ΔJk+1 (pk+1 ) =

a2k+1,y ak+1,k+1

(22)

This provides a formula for selecting the best particle in the swarm at each iteration. When a pre-set number of updating cycles is reached, the best solution from the entire swarm will be added to the network. This continues until some termination criterion is met (e.g., Akaike’s information criterion (AIC) [13]) or until a maximum number of centres have been added. The initial particle values

Fast Forward RBF Network Construction Based on PSO

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at each stage can be chosen from the data points rather than randomly generated in an attempt to improve convergence. Finally, after a satisfactory network has been constructed, the output layer weights are computed recursively as: ⎞ ⎛ k  θˆj = ⎝aj,y − (23) θˆi aj,i ⎠ /aj,j , j = k, k − 1, · · · , 1. i=j+1

3.2

Construction Algorithm

The proposed algorithm for RBF network construction can be summarized as follows: step 1. Initialization: Set the network size k = 0, and assign initial values for the following terms: – s: The size of swarm; – lmax : The maximum iteration; – [xmin , xmax ]: The search space of the particles; – [vmin , vmax ]: The speed range of the particles; – w: The inertia weight in velocity updating; – x0 :s data samples randomly selected as the starting points; – v0 :Randomly generated inside the velocity range; step 2. At the kth step, (a) Compute the RBF output for each particle using all the data samples; (b) Calculate ai,k (1 ≤ i ≤ k) and ak,y for each particle using (20) and (21); (c) Compute the contribution of each particle to the cost function using (22), update the best position that each particle has visited so far and the best position for the entire swarm; (d) Update the velocity and position for each particle using (4) and (5); (e) Check the value of velocity and position for each particle using (8)(9); (f) If l < lmax , let l = l + 1 and go to 2(a); otherwise, go to the next step (g) Update the matrix elements ai,k for 1 ≤ i ≤ k, and ak,y with the best centres found in the swarm. If the chosen pre-defined criterion is met, terminate the procedure; otherwise, let k = k + 1 and go to step 2(a); step 3. Use equation (23) to calculate the output layer weights.

4

Simulation Example

In this section, the proposed algorithm is compared to several other popular alternatives for RBF network construction, including OLS, OLS with the leaveone-out cross validation [6] and the original FRA.

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Consider a nonlinear dynamic system defined by [12] z(t) =

z(t − 1)z(t − 2)z(t − 3)u(t − 2)[z(t − 3) − 1] + u(t − 1) 1 + z 2 (t − 2) + z 2 (t − 3)

(24)

where u(t) is the random system input which is uniformly distributed in the range [−1, 1]. The output at sample time t was given by y(t) = z(t) + ε(t), where ε(t) was from a Gaussian white noise sequence with zero mean and variance 0.12 . A total of 400 data samples were generated with the first 200 data samples used for network training and the remaining 200 points for testing. The RBF network employed the Gaussian kernel function φ(x, ci ) = exp(− 21  x − ci 2 /σ 2 ) as the basis function. For the conventional forward selection methods, the width of the Gaussian basis function was fixed and chosen as σ = 4. The input vector was predetermined as x(t) = [y(t − 1), y(t − 2), y(t − 3), u(t − 1), u(t − 2)]T . The OLS method was first applied to construct the RBF network with the assist of the AIC. The algorithm was terminated when 36 RBF centres had been selected as the AIC value started to increase resulting in an over-fitted model. The OLS with LOO improved the construction procedure, it had chosen 14 RBF centres when the LOO error started to increase. Finally for the algorithm described in this paper, the swarm size was set at 15, and the maximum number of iterations was 20. The inertia weight in the velocity updating was 0.8. As the network size continuously increased, this meant that only 2 RBF centres were sufficient to approximate the nonlinear dynamic system in (24). The Root Mean-Squared Error (RMSE) over the training and test data sets are given in Table 1. Table 1. The comparison of network size and performance (RMSE) Algorithm OLS OLS with LOO The proposed

5

Network size 36 14 2

Training Error 0.0185 0.0729 0.0954

Testing error 0.0360 0.0837 0.0840

Discussion and Future Work

Though it has been shown that the proposed algorithm is able to produce a good network compared to conventional alternatives, there still exist some problems. Firstly, the forward construction procedure is not optimal. It involves a constrained minimization, since the calculation of a new particle’s contribution depends on the previously optimized RBF centres. Secondly, the cost function is the Sum Squared Error (SSE) which easily causes the network to be over-fitted. Thus, the proposed algorithm (or the PSO assisted OLS method) only produce an undesirable network. To overcome the first problem, a network refinement

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procedure can be adopted, leading to our two-stage stepwise construction algorithm [8]. The original two-stage method removes the optimization constraint involved in the forward selection by reviewing the significance of each selected RBF centre with those remaining in the candidate pool, insignificant centres being replaced. For the PSO assisted method, each RBF centre can thus be re-optimized based on other centres in the network refinement procedure. For the second problem, cross-validation can be introduced to the selection process where the RBF centres are optimized based on their testing error instead of training error. A special case of this technique is known as the Leave-One-Out (LOO) cross validation [6]. However, if the testing data has the same type of noise as the training data, the resultant network can still be over-fitted. The Bayesian learning framwork [16] has proven useful preventing the over-fitting problem. Specifically, the output layer weights are assigned priors known as hyperparameters, and the most probable values of these hyperparameters are iteratively estimated from the data. In practice, the posterior distribution of irrelevant weights are sharply peaked around zero [16]. Therefore, those centres that are mainly determined by the noise will have large value of hyperparameters, and their corresponding layer weights are forced to be near to zero. Sparsity is then achieved by removing such irrelevant centres from the trained network. Future work is to efficiently integrate such techniques into the construction to produce even better RBF network model.

6

Conclusions

This paper has proposed a novel hybrid forward construction approach for RBF networks. This integrates our Fast Recursive Algorithm [9], with Particle Swarm Optimization, to produce in a sparse network with only a few significant RBF centres. The computation complexity was greatly reduced compared to PSO assisted OLS by introducing a residual matrix which could be updated recursively. Simulation results confirmed the effectiveness of the proposed algorithm compared to conventional OLS and its extended version. Though there still exist some problems in the proposed method, they can be tackled by adopting other techniques, such as a second stage of network refinement procedure, the Leave-one-out cross validation and the Bayesian regularisation. Further, this new algorithm can also be easily generalized to a wide range of nonlinear models that have a linear-in-the-parameters structure, such as Nonlinear Autoregressive with Exogenous input (NARX) model. Acknowledgments. Jing Deng wishes to thank Queens University Belfast for the award of an ORS scholarship to support his doctoral studies. This work is also partially supported by EPSRC under UK-China Science Bridge grant EP/G042594/1 and EP/F021070/1 and the Key Project of Science and Technology Commission of Shanghai Municipality under grant 08160512100 and 08160705900.

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Reference [1] Blum, C., Merkle, D.: Swarm intelligence: introduction and applications. Springer, New York (2008) [2] Chen, S., Cowan, C.F.N., Grant, P.M.: Orthogonal least squares learning algorithm for radial basis function networks. IEEE Transactions on Neural Networks 2(2), 302–309 (1991) [3] Chen, S., Hong, X., Harris, C.J.: Sparse kernel density construction using orthogonal forward regression with leave-one-out test score and local regularization. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 34(4), 1708–1717 (2004) [4] Chen, S., Hong, X., Luk, B.L., Harris, C.J.: Non-linear system identification using particle swarm optimisation tuned radial basis function models. International Journal of Bio-Inspired Computation 1(4), 246–258 (2009) [5] Eberhart, R.C., Shi, Y.: Comparing inertia weights and constriction factors in particle swarm optimization. In: Proc. Congress on Evolutionary Computation, vol. 1, pp. 84–88 (2000) [6] Hong, X., Sharkey, P.M., Warwick, K.: Automatic nonlinear predictive modelconstruction algorithm using forward regression and the press statistic. IEE Proceedings: Control Theory and Applications 150(3), 245–254 (2003) [7] Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proc. IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948. IEEE, Australia (1995) [8] Li, K., Peng, J.X., Bai, E.W.: A two-stage algorithm for identification of nonlinear dynamic systems. Automatica 42(7), 1189–1197 (2006) [9] Li, K., Peng, J.X., Irwin, G.W.: A fast nonlinear model identification method. IEEE Transactions on Automatic Control 50(8), 1211–1216 (2005) [10] Li, Y.H., Qiang, S., Zhuang, X.Y., Kaynak, O.: Robust and adaptive backstepping control for nonlinear systems using RBF neural networks. IEEE Transactions on Neural Networks 15(3), 693–701 (2004) [11] McLoone, S., Brown, M.D., Irwin, G.W., Lightbody, G.: A hybrid linear/nonlinear training algorithm for feedforward neural networks. IEEE Transactions on Neural Networks 9(4), 669–684 (1998) [12] Narendra, K.S., Parthasarathy, K.: Identification and control of dynamical systems using neural networks. IEEE Trans. Neural Networks 1(1), 4–27 (1990) [13] Nelles, O.: Nonlinear System Identification. Springer, Heidelberg (2001) [14] Rajakarunakaran, S., Devaraj, D., Suryaprakasa Rao, K.: Fault detection in centrifugal pumping systems using neural networks. International Journal of Modelling, Identification and Control 3(2), 131–139 (2008) [15] Sutanto, E.L., Mason, J.D., Warwick, K.: Mean-tracking clustering algorithm for radial basis function centre selection. International Journal of Control 67(6), 961–977 (1997) [16] Tipping, M.E.: Sparse baesian learning and the relevance vector machine. Journal of Machine Learning Research 1(3), 211–244 (2001)

A Modified Binary Differential Evolution Algorithm Ling Wang, Xiping Fu, Muhammad Ilyas Menhas, and Minrui Fei Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronics and Automation, Shanghai University, 200072, Shanghai, China [email protected]

Abstract. Differential evolution (DE) is a simple, yet efficient global optimization algorithm. As the standard DE and most of its variants operate in the continuous space, this paper presents a modified binary differential evolution algorithm (MBDE) to tackle the binary-coded optimization problems. A novel probability estimation operator inspired by the concept of distribution of estimation algorithm is developed, which enables MBDE to manipulate binaryvalued solutions directly and provides better tradeoff between exploration and exploitation cooperated with the other operators of DE. The effectiveness and efficiency of MBDE is verified in application to numerical optimization problems. The experimental results demonstrate that MBDE outperforms the discrete binary DE, the discrete binary particle swarm optimization and the binary ant system in terms of both accuracy and convergence speed on the suite of benchmark functions. Keywords: Differential Evolution, Binary Encoding, Probability Estimation Operator, Multidimensional Knapsack Problem.

1 Introduction Differential Evolution (DE), an emerging population-based stochastic optimization technique first proposed by Storn and Price in 1995 [1], has become a new hotspot in evolutionary computation. The standard DE algorithm, which is simple yet efficient in global optimization, has been successfully applied in scientific and engineering fields. As a versatile evolutionary algorithm, DE does not need any gradient information so that it is capable of solving non-convex, nonlinear, non-differentiable and multimodal problems. Moreover, there are only two control parameters in the update formulas of DE, thus it is easy to implement and tune parameters. Literatures have reported that DE is superior to particle swarm optimization (PSO) and genetic algorithm (GA) in some real-world applications [2-4]. Due to its simplicity and effectiveness, DE has attracted much attention in recent years, and a number of improved variants have been proposed [5-7]. However, the standard DE and many of its improved variants operate in the continuous space, which are not suitable for solving discrete combinational optimization problems. Therefore, several binary DE algorithms are proposed to tackle this drawback. Inspired by angle modulated PSO algorithm [8], Pampará [9] proposed a new binary DE K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 49–57, 2010. © Springer-Verlag Berlin Heidelberg 2010

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called angle modulated DE (AMDE). In AMDE, standard DE was adopted to update the four real-coded parameters of angle modulated function and the angle modulated functions with different parameter values were used to generate the binary-coded solution by sampling till the global best solution was found. So it actually operated in the continuous space. He[10] presented a binary DE based on artificial immune system(AIS-DE), in which the scaling factor were treated as a random bit-string and trial individuals were generated by Boolean operators. Further, an extra control parameter was introduced, which weakened the flexibility of algorithm as it was significantly dependent on the problem. Gong et al. [11] proposed a binary-adapted DE (BADE) where the scaling factor was regarded as the probability of the scaled difference bit to take on “1”. However, both AIS-DE and BADE discarded the update formulas of the standard DE and generated the new individuals based on different Boolean operating. Chen [12] developed a discrete binary differential evolution (DBDE) where the sigmoid function used in DBPSO [13] was borrowed to convert the real individuals to bit strings. DBDE directly search in the binary space so that it is easy to implement, but it is very sensitive to the setting of control parameters. Moreover, the value transformed by the sigmoid function is not symmetrical resulting in deterioration of global searching ability. In this work, an improved binary DE (MBDE) is proposed which develops a novel probability estimation operator to generate the offspring individuals. MBDE reserves the updating strategy of the standard DE so that the characteristics of DE, such as ease of implementing and tuning parameters are well maintained as well. This paper is organized as follows. Section 2 gives a brief review of the standard DE. The proposed MBDE is introduced in detail in section 3. Section 4 presents the application of MBDE into numerical optimization, where the comparison of MBDE with DBDE, DBPSO and the binary ant system (BAS) [14] on a suite of 16 well-know benchmark functions is conducted. Finally, conclusions are remarked in section 5.

2 The Standard DE DE is a population-based evolutionary algorithm for global optimization, whose population is consisted of a group of floating-point encoded individuals randomly initialized in the continuous space. Three commonly used evolutionary operators, i.e., mutation operator, crossover operator and selection operator are used for DE to update the population. In the evolutionary process, mutation operator and crossover operator are used to generate the new trail individual, while selection operator chooses the better one for the next generation by comparing it with the target individual. Mutation: There are several mutant schemes in DE, where “DE/rand/1”, as Eq. (1), is the most popular one. The mutated individual ui is produced according to Eq. (1),

uij t +1 = xrt 1, j + F *( xrt 2, j − xrt 3, j )

(1)

where F, a positive real constant, is the scaling factor ; t is the index of generation; xr1, j , xr 2, j and xr 3, j are three random chosen individuals with indexes r1≠r2≠r3≠i.

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Crossover: The trial individual vi is generated by crossing the target individual xi with its mutant counterpart ui. The commonly used binomial crossover is defined as Eq. (2),

⎧⎪u t +1 , if (randj ≤ CR) or (j =rand (i )) vij t +1 = ⎨ ij t ⎪⎩ xij , otherwise

(2)

where CR is the crossover probability ranged in (0, 1). The randj is a uniform stochastic number distributed within [0,1); rand(i) is a random integer within {1, 2, , N } where N is the length of individual; j is the index of the dimensionality with j=1, 2, …, N. Selection: The selection operator is defined as Eq. (3), t +1 t +1 t ⎪⎧v , if f (vi ) < f (xi ) xi t +1 = ⎨ i t ⎪⎩ xi , otherwise

(3)

As shown above, if the fitness value of the trial individual is better than the target one, it replaces the target individual. Otherwise, the target individual is reserved in the next generation. Till now, the whole population has been updated.

3 The Modified Binary DE Inspired by the estimation of distribution algorithms (EDAs) [15], this paper presents a modified binary differential evolution algorithm (MBDE) based on probability model. In the proposed modified binary DE (MBDE), each individual is represented as a bit-string and denoted as pxi = { pxij , pxij ∈ {0,1}; i = 1, 2, , NP, j = 1, , N } , where NP is the population size and N is dimensionality of solutions. So MBDE can be directly used to solve discrete binary problems. MBDE reserves the updating strategy of the standard DE, including the mutation operator, crossover operator and selection operator. Since the standard mutant operator generates real-coded vectors not binary-coded strings, an effective probability estimation operator is proposed to tackle this problem in MBDE, into which the mutant operator has been integrated. 3.1 Probability Estimation Operator

Similar to the probability model used to create new offspring in the population based incremental learning (PBIL) algorithm [16], a novel probability estimation operator P( px) is proposed and utilized in MBDE to generate the mutated individuals according to the information of parent population. The probability estimation operator P( px) is shown as follows,

1

P( pxijt ) = 1+ e

−

2 b ∗[ pxrt 1, j + F ∗( pxrt 2, j − pxrt 3, j ) − 0.5] 1+ 2 F

(4)

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where b is a positive real constant, usually is set as 6; F is the scaling factor just the same as that of the standard DE; pxrt 1, j , pxrt 2, j and pxrt 3, j are the j-th bits of three randomly chosen individuals at generation t. As seen, the mutant operator of the standard DE has been embedded into the probability estimation operator of MBDE. According to the probability estimation vector P( pxit ) = [ pxit,1 , pxit,2 , , pxit, N ] created by Eq. (4), the corresponding offspring puit of the current target individual pxit is generated as Eq. (5).

⎧ 1, if rand () ≤ P (pxijt ) puijt = ⎨ ⎩0, otherwise

(5)

where rand () is a random number; P(pxijt ) is the j-th component of the probability vector of the i-th individual. 3.2 Crossover Operator

The crossover operator is used to mix the target individual and its mutated individual. The trial individual pvi t = ( pv t i ,1 , pv t i ,2 ,... pv t i , N ) can be obtained by the crossover operator as follows,

⎧⎪ pu t , if (randj ≤ CR ) or (j =randi ) pvij t = ⎨ ij t ⎪⎩ pxij , otherwise

(6)

where randj, CR and randi ∈ {1, 2, ..., N } are as same as those in standard DE. When a random number is less than the predefined CR or the index j is equal to the random index randi, pvi t takes the value of puij t , otherwise, it is set equal to pxij t . From Eq. (6), it is obvious that at least one bit of the trial individual is inherited from the mutant individual so that MBDE is able to avoid duplicating individuals and therefore effectively search within the neighborhood. 3.3 Selection Operator

The selection operator is adopted to determine whether a trial individual can survive in the next generation in MBDE. If the fitness value of the trial individual pvi t +1 is better than the target individual pxi t , then pvi t +1 replaces pxi t and passes to the next population, otherwise, the original individual pxi t is reserved as shown in Eq. (7). So this selection operator is also called one to one elitism selection. t t +1 t ⎪⎧ pv , if f (pvi ) < f (pxi ) pxi t +1 = ⎨ it ⎪⎩ pxi , otherwise

(7)

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In summary, the procedure of MBDE can be stated as follows: Step 1: Initialize the population randomly in the binary space; Step 2: Generate the binary mutant individuals according to the probability estimation operator as Eq. (4)( 5); Step 3: Generate the trial individuals by the binary crossover operator as Eq. (6); Step 4: Evaluate the target individual and its trial individual, choose the better one to survive into the next generation according to formula Eq. (7); Step 5: If the stop condition is met, then terminate the iteration; otherwise, go to step 2.

The evolution process terminates if the global solution is found or some exit condition is satisfied, such as the maximum iteration number is run out, the maximum fitness evaluation times or minimum fitness error is satisfied.

4 Numerical Optimization 4.1 Parameter Settings

To demonstrate its optimization ability, MBDE are adopted to solve in the numerical optimization problems. Meanwhile, DBDE, DBPSO and BAS are also made on standard functions for a comparison. The control parameters of DBDE were set the same as [12], i.e., CR=0.1, F=0.9. The inertia weight w and the learning factors of DBPSO were respectively taken as w=0.8, c1=c2=2.0 in [17], the maximum velocity Vmax=6.0. The parameters of BAS were taken from [14], i.e., the number of ants m=N (N is the length of individuals), the initial pheromone level τ 0 = 0.5 , the amount of pheromone intensified Δτ = 1 , the evaporation parameter ρ = 0.1 , the probability of pheromone re-initialization cf = 0.9 . The parameters of MBDE is set as CR=0.2, F=0.8. The maximum generation and the population size of MBDE, DBDE and DBPSO were set as 3000, 40 respectively. 4.2 Results and Discussion

Firstly, MBDE, DBDE, DBPSO and BAS are adopted to optimize the benchmark functions. Each dimension of benchmark functions was encoded by 20 bits. The experiment was conducted for 100 times independently. Numerical results of MBDE, DBDE, DBPSO and BAS on the 16 benchmark functions are shown in Table 1. From Table 1, we observe that MBDE outperforms DBDE, DBPSO and BAS on 15, 16 and 13 functions respectively in terms of success rate and convergence speed. Specially, for DBDE, MBDE only performs poorer than DBDE on f2. Though MBDE

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Algorithm f1 MBDE DBDE DBPSO BAS f2 MBDE DBDE DBPSO BAS f3 MBDE DBDE DBPSO BAS f4 MBDE DBDE DBPSO BAS f5 MBDE DBDE DBPSO BAS f6 MBDE DBDE DBPSO BAS f7 MBDE DBDE DBPSO BAS f8 MBDE DBDE DBPSO BAS

SR 86% 1% 18% 52% 95% 98% 28% 33% 100% 81 % 15% 10% 95% 81% 9% 6% 100% 94% 85% 100% 100% 27% 90% 98% 100% 100% 89% 98% 100% 100% 100% 99%

MF 5.36e-11 1.63e-9 7.06e-9 2.61e-10 3+1.26e-9 3+5.03e-10 3+1.81e-8 3+1.68e-8 0 3.54e-10 9.21e-7 8.69e-7 5.92e-10 4.03e-9 2.39e-5 2.58e-8 0 2.97e-4 5.95e-4 0 -1 -0.993543 -0.999411 -0.999922 -186.730904 -186.730904 -186.730897 -186.730894 0 0 0 7.81e-8

MG 1855 1547 2213 307 556 1085 1122 83 65 954 1468 61 516 1250 1410 45 341 1028 1158 269 865 1683 1300 635 1035 1131 1430 270 134 451 840 84

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Algorithm MBDE DBDE DBPSO BAS MBDE DBDE DBPSO BAS MBDE DBDE DBPSO BAS MBDE DBDE DBPSO BAS MBDE DBDE DBPSO BAS MBDE DBDE DBPSO BAS MBDE DBDE DBPSO BAS MBDE DBDE DBPSO BAS

SR 100% 100% 99% 99% 100% 100% 100% 99% 100% 83% 100% 100% 100% 98% 25% 71% 100% 99% 49% 99% 100% 100% 99% 100% 100% 38% 32% 46% 100% 74% 67% 96%

MF 0 0 1.51e-4 5.90e-4 0 0 0 0.006195 -176.137539 -176.135701 -176.137531 -176.137540 -1.03162845 -1.03162845 -1.03162645 -1.03162834 -7.885600 -7.885600 -7.870738 -7.885599 -176.54177 -176.54176 -176.54173 -176.54177 0 2.53e-10 6.36e-10 6.53e-8 -3.8627809 -3.862771 -3.8627711 -3.8627792

MG 175 1031 1237 123 101 437 908 58 431 950 674 221 273 856 1665 587 603 854 674 460 639 777 757 215 784 2225 1869 188 433 1386 1603 384

Note: SR, MF, MG are performance indicators that represent the success rate, the main fitness value, the main generation number respectively; MG only accounts the mean generation number of successful runs.

converges a little slower than BAS on f5, f11 and f14, they are very close from the view of iteration number. The curves of the mean fitness value are depicted in Fig. 1, where the vertical axis is logarithmic scaled. Note that where the curve halts when the fitness value is zero as its logarithmic scaled value is -∞ in subfigures (a-f) in Fig.1, which demonstrates that MBDE converges to the global optima faster than DBDE, DBPSO and BAS. As for f1 and f4, MBDE can reach better accuracy of solutions than the other algorithms. In addition, BAS is obviously apt to trap into the local optima compared with MBDE, as the probability estimation operator can provide better global searching ability for MBDE and prevent MBDE from trapping in the local optima, so that MBDE is superior to DBDE, DBPSO and BAS on the suite of benchmark numerical functions.

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5 Conclusion This paper proposed a modified binary DE (MBDE) to solve binary-valued optimization problems, in which a novel probability estimation operator is developed and enables DE to manipulate directly with binary individuals. The updating strategy of the standard DE is reserved in MBDE so that the advantages of DE were well maintained such as easy implementation and parameter tuning. The developed MBDE is then adopted to tackle the numerical optimization problems and the comparisons with discrete binary PSO, binary Ant System and DBDE are also made. The experiment results demonstrate that MBDE is superior to DBDE, DBPSO and BAS on the wellknown standard numerical functions in terms of accuracy and convergence speed. Acknowledgement. This work is supported by ChenGuang Plan (2008CG48), the Projects of Shanghai Science and Technology Community (10ZR1411800, 08160705900 & 08160512100), Shanghai University “11th Five-Year Plan” 211 Construction Project, Mechatronics Engineering Innovation Group project from Shanghai Education Commission, and the Graduate Innovation Fund of Shanghai University.

Reference 1. Storn, R., Price, K.V.: Differential Evolution – A simple and efficient adaptive scheme for global optimization over continuous spaces. Technology Report. Berkeley, CA, TR-95-012 (1995) 2. Vesterstrom, J., Thomsen, R.: A Comparative Study of Differential Evolution, Particle Swarm Optimization, and Evolutionary Algorithms on Numerical Benchmark Problems. In: IEEE Congress on Evolutionary Computation, vol. 2, pp. 1980–1987. IEEE Press, Los Alamitos (2004) 3. Rekanos, I.T.: Shape Reconstruction of a Perfectly Conducting Scatterer Using Differential Evolution and Particle Swarm Optimization. IEEE Transaction on Geoscience and Remote Sensing 46, 1967–1974 (2008) 4. Ponsich, A., Coello, C.A.: Differential Evolution performances for the solution of mixed integer constrained Process Engineering problems. Applied Soft Computing (2009), doi: 10.1016/j.asoc.2009.11.030 5. Liu, J., Lampinen, J.: A Fuzzy Adaptive Differential Evolution Algorithm. Soft Comput. 9, 448–462 (2005) 6. Qin, A.K., Huang, V.L., Suganthan, P.N.: Differential Evolution Algorithm with Strategy Adaptation for Global Numerical Optimization. IEEE Transaction on Evolutionary Computation 13, 398–417 (2009) 7. Das, S., Abraham, A., Chakraborty, U.K., Konar, A.: Differential Evolution Using a Neighborhood-Based Mutation Operator. IEEE Transaction on Evolutionary Computation 13, 526–553 (2009) 8. Pampara, G., Franken, N., Engelbrecht, A.P.: Combining Particle Swarm Optimisation with Angle Modulation to Solve Binary Problems. In: The 2005 IEEE Congress on Evolutionary Computation, pp. 89–96. IEEE Press, Los Alamitos (2005) 9. Pampará, G., Engelbrecht, A.P., Franken, N.: Binary Differential Evolution. In: Proceedings of IEEE Transaction on Evolutionary Computation, pp. 1873–1879 (2006)

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10. He, S.X., Han, L.: A novel binary differential evolution algorithm based on artificial immune system. In: IEEE Congress on Evolutionary Computation, pp. 2267–2272. IEEE Press, Los Alamitos (2007) 11. Gong, T., Tuson, A.L.: Differential Evolution for Binary Encoding. Soft Computing in Industrial Applications, ASC 39, 251–262 (2007) 12. Chen, P., Li, J., Liu, Z.M.: Solving 0-1 Knapsack Problems by a Discrete Binary Version of Differential Evolution. In: Second International Symposium on Intelligent Information Technology Application, IITA 2008, pp. 513–516. IEEE Press, Los Alamitos (2008) 13. Kennedy, J., Eberhart, R.: A Discrete Binary Version of the Particle Swarm Optimization. In: The 1997 Conference on System, man and Cybernetics, pp. 4104–4108. IEEE Press, Los Alamitos (1997) 14. Kong, M., Tian, P.: A Binary Ant Colony Optimization for the Unconstrained Function Optimization Problem. In: Hao, Y., Liu, J., Wang, Y.-P., Cheung, Y.-m., Yin, H., Jiao, L., Ma, J., Jiao, Y.-C. (eds.) CIS 2005. LNCS (LNAI), vol. 3801, pp. 682–687. Springer, Heidelberg (2005) 15. Pelikan, M., Goldberg, D.E., Lobo, F.G.: A Survey of Optimization by Building and Using Probabilistic Models. Computational Optimization and Applications 21, 5–20 (2002) 16. Baluja, S.: Population-Based Incremental Learning: A Method for Integrating Genetic Search Based Function Optimization and Competitive Learning. Technical Report CMUCS-94-163, Pittsburgh, PA: Carnegie Mellon University (1994) 17. Wang, L., Wang, X.T., Fu, J.Q., Zhen, L.L.: A Novel Probability Binary Particle Swarm Optimization Algorithm and Its Application. Journal of Software 3, 28–35 (2008)

Research on Situation Assessment of UCAV Based on Dynamic Bayesian Networks in Complex Environment Lu Cao, An Zhang, and Qiang Wang Department of Electronic and Information, Northwestern Ploytechnical University, Xi’an 710129, China [email protected], [email protected], [email protected]

Abstract. UCAV is an inevitable trend of the future intelligent and uninhabited flight platform. Situation assessment (SA) is an effective method to solve the problem of the autonomous decision-making in UCAV investigation. The concepts, contents and process of SA are put forward and the methods about the implementation of SA are analyzed. Then the concept and inference of dynamic Bayesian networks (DBN) are introduced, and SA configuration of UCAV autonomous decision system is given. Finally, the SA is applied to the UCAV autonomous decision system, especially SA based on DBN is used and the model is propounded. The simulation result indicates that the inference results are consistent with the theoretical analysis. The subjectivity of the assessment is reduced and the accuracy is greatly improved. Keywords: situation assessment; UCAV; dynamic Bayesian networks; complex environment.

1 Introduction The modern battlefield has increasingly progressed towards the use of automated systems and remotely controlled devices to perform a variety of missions. UCAV (Unmanned Combat Aerial Vehicle), as an integrated air combat system, can be used to accomplish all mission of manned fighter plane [1]. Because of no pilot, the casualties are reduced and the flexibility of UCAV is enhanced without considering the fact of person. But, the signal between the operator and UCAV may be disturbed by the other electromagnetic wave and it may be miss the attack chance in the complex battlefield environment at present, so UCAV strengthens the capacity of the autonomous attack and develops the autonomous decision system [2]. The part of SA plays a very important role in the autonomous decision system and it is the principal part of the autonomous decision system. The function of SA is estimating the threat and providing the proof for the decision-making according to the external information. So in the autonomous decision system, SA is an effective method to solve the problem of the autonomous decision-making. And it becomes the key point of the victory or defeat in the future combat. SA is a complex problem and discussed by many scholars at present. As to the technology of estimating, the methods of SA are divided into three different kinds [3]. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 58–68, 2010. © Springer-Verlag Berlin Heidelberg 2010

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First, mathematic models are used to assess the situation. Second, expert systems are adopted to assess the situation. Finally, SA is based on the fuzzy technology and the neural network. The three methods have different characters and are applied in a certain extent. DBN method is an effective approach to model uncertainties with network topology and to inference based on Bayesian Formula in a low visibility and uncertainty context. DBN can be utilized to model the uncertainties in the inference process, to depict the relations between all the relative factors according to the experiences obtained from the experts and then compute in a probabilistic format. It is extremely suitable for the description and computation of SA problem because of the continuity, the cumulativeness and the mathematical stability of the inference process [4]. In the paper, SA based on DBN is applied in UCAV autonomous decision system. DBN can enhance the autonomous decision system to conduct probabilistic reasoning to deal with the uncertainty associated with the rapid changes taking place.

2 Summary of SA 2.1 Concept of SA The concept of SA is generally accepted as referring to “the perception of the elements in the environment within a volume of time and space, the comprehension of their meaning, and the projection of their status in the near future” [5]. From the concept of the technology and system, SA has some characters such as steadiness, investigation summary, coherence consequence, depending on the early experiences, information fusion, expansibility and the power of learning. The famous JDL (Joint Directors of Laboratories) consider that SA is a view based on the maneuver, event, time, position and troops [6]. In JDL data fusion model, level 1 processing estimates and predicts the situations and attributes of the target on the basis of observation-to-track association; level 2 processing estimates the situation of battlefield according to the target information from the level 1 processing; level 3 processing estimates the ability, purpose and threat degree of opposite according to the level 1 processing and the result of the SA; level 4 processing estimates capability and efficiency of the whole system to optimize the processing and controlling. Combining the distributing of strength, the environment of battlefield and opposite flexibility, we can analyze and confirm the reason of the happened events, obtain the forecast about the configuration of opposite troops and using characters, then gain the integrated situation of battlefield. SA is considered describing and explaining the connection with targets and events in the current combat environment [3]. The result of SA is refining and estimating the situation of battlefield and campaign. The main content of SA includes three aspects. First, it is the target clustering. Second, it is the event or action clustering. Third, relationship is explained and fused. So the process of SA is a process of thinking to the different levels of situation. It’s divided into three stages including perception, comprehension and projection [7]. 2.2 Method of SA SA is one of the most important processes in the military decision. As a component of battlefield data fusion and decision support, SA is not easy to be realized ideally by

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using one particular technology in practice; like any other complex military process, it requires the cooperation of lots of information processing technology. Many methods are considered as follows. As the farther development of fuzzy set, the fuzzy logic becomes a powerful tool to research the complex system such as aeronautic and astronautic system, socioeconomic system etc [8]. Fuzzy logic applied to SA is mapping the illation of some events through the collected state vector of course of action (COA). We can gain the different value estimation of opposite situation according to the fuzzy illation. The SA based on neural networks including four parts is given [9]. They are the model of simulation generation, the analysis model of troop deployment, the analysis model of time fusion and the analysis model of tactical situation. We can use the four parts to analyze movement changes of troop with time by the fusion of the battlefield situation. The optimization based on genetic algorithm has been widely applied in combat mission planning and data fusion areas [10]. Charles River’s ACTOGA system based on the genetic algorithm optimize the decision-making aerial battles both offline and online. It also establishes the SA decision-making system based on the genetic algorithm, and it can be used in air, marine and missile defense.

3 Dynamic Bayesian Networks 3.1 Concept of DBN Besides three methods we introduced above, DBN method can be used in SA in the paper. Bayesian networks (BN) are a computational framework for the representation and the inference of uncertain knowledge. A BN can be represented graphically as a directed acyclic graph. DBN are a way to extend Bayesian networks to model probability distributions over semi-infinite collections of random variables [11]. With a DBN, it is possible to model dynamic processes: Each time the DBN receives new evidence a new time slice is added to the existing DBN. In principle, DBN can be valuated with the same inference procedures as normal BN; but their dynamic nature places heavy demands on computation time and memory. Therefore, it is necessary to apply roll-up procedures that cut off old time slices without eliminating their influence on the newer time slices. DBN is a model of simulating the stochastic evolution of any random variables aggregates in the axis of time. It’s assume that Z = {Z1 , Z 2 , , Z n } is an attribute aggregate changed with time, where Z i [t ] defines the value of Z i on the time t and

Z [t ] is the aggregate of random variables Z i [t ] . To express the degree of the networks configuration change, we need to work out the probability distribution of random variables. Although the distribution is complex, we assume that the whole change process satisfy the Markov models in the paper. And we assume that the whole change process is static, namely, the transition probability P( Z [t + 1] Z [t ]) is not correlation with the time t . With the two assumptions, A DBN is defined to be a pair, ( B0 , B→ ) , where B0 is a BN which defines the prior probability P( Z (0)) , and

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B→ is a two-slice temporal Bayesian net which defines transition probability P( Z [t + 1] Z [t ]) . The set Z (t ) is commonly divided into two sets: the unobserved state variables

X (t ) and the observed variables Y (t ) . The observed variables Y (t ) are assumed to depend only on the current state variables X (t ) . The joint distribution represented by a DBN can then be obtain by unrolling the two-time-slice BN:

P( X (0),…, X (T ), Y (0),…, Y (T )) = T

P( X (0)) P(Y (0) | X (0))∏ P( X (t ) | X (t − 1)) P(Y (t ) | X (t ))

(1)

t =1

3.2 Inference of DBN Since in the DBN, only a subset of states can be observed at each time slice, we have to calculate all the unknown states in the network. This is done by procedure called inference [12]. Different types of DBN request different types of estimations and calculations based on their specific structure. Therefore, we could choose significant states of a state, and estimate only their values for different time slices. In the paper, junction trees algorithm will be adopted in DBN inference. First we change a DBN to static BN with the time slice because the algorithm is a static networks inference algorithm. To make sure of the coherence between Bayesian network and corresponding junction trees algorithm, the state of Bayesian network is same to the junction trees. After the junction trees constructed successfully, it is initializing for some time and then they will achieve the same state. If we suppose some nodes becoming evidence variables, the coherence state will be break. Then through adjusting the states of every nodes and transmitting information in each another nodes, the junction trees achieves new coherence state. It is the principle of the junction trees algorithm. The process of junction trees algorithm is as follows. First we can change a BN with known configuration and parameter to junction trees. Second the junction trees should be initialized. And then the information will be transmitted in junction trees. Finally the result will be obtained through the inference.

4 SA Based on DBN of UCAV Autonomous Decision System 4.1 SA Configuration of UCAV Autonomous Decision System SA of UCAV autonomous decision system is a evaluate process of adaptability for the situation of support decision. The opposite situation can’t observe directly, and we can get the information directly including opposite action. To assess the opposite situation, decision-maker usually acquires the more information from the sensors and AWACS. But we will be faced with the more difficulties for situation assessment. First, modern military platforms can get the huge information from sensors and big capability of communication link, but it’s important for decision-maker to select useful information to reflect the opposite situation. So the information over loading is a

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hard work. Second, in the modern combat, the opposite adopts more measures including the jamming and fraudulence action, so the collected information isn’t inadequacy. So the integrated information will be a problem and we need a better reasoning method. Finally, the collected information comes from different platforms and it’s hard for decision-maker to deal with so much different information in limited time. In the process of situation assessment, uncertain information of expressing and reasoning are two problems needed to solve. Because of the consistency inference ability and uncertain express ability, the difficulties in SA of UCAV autonomous decision system can be solved better with DBN. SA of UCAV autonomous decision system is proposed by the definition, nature and research of manned combat aircraft. It is closely related with manned combat aircraft and has its distinctive characteristics. UCAV can take the autonomous decision-making to plan and re-plan trails/task online, establish goals and deliver weapons according to the development of the battlefield situation. The SA for UCAV involves many subsystems, in addition to the sensor system, the opposite situation and friend’s situation involved in the SA for manned aircraft, the and friendship, but also involves fire-control systems, flight control systems, and so on. Figure 1 shows the SA configuration of UCAV autonomous decision system.

Fig. 1. SA configuration of UCAV autonomous decision system

In the system configuration, the UCAV platform using its own sensor (such as radar, IFF, electronic support measures, etc.) to collect the information from the battlefield, and the information is treated by all the corresponding sensor to create a different expression of battlefield information, such as the type, characteristic and attribute of battlefield targets. After these battlefield information processed by sensor information and knowledge expressed, data vectors which the UCAV can use DBN to fuse are gained. The UCAV platform use DBN to fuse the sensor information and the information in the database (such as the course of action, complex environment, etc.) and finally gain the opposite situation. 4.2 SA Model Configuration Based on DBN In the decision-making process, the opposite intention and activity provide the possible information of opposite situation. The opposite main situations are attack, defense and retreat. The information of nodes is from the opposite intention and activity. The opposite intention relating to the opposite current national defense system can be

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divided into attack and defense, and the activity in complex environment can be divided into active reconnaissance, conducting electronic warfare, mount weapon and securing target. The opposite intentions relate to the military strength, command intention and political pressure. Military strength is both sides of the comparison of strength, and it is divided into advantage, similar and disadvantage. Political pressure is the both sides of political pressure, which can be divided into strength, common and small. To judge the information of action nodes, evidence nodes are added including the opposite weapons type, communications and mobility. Evidence nodes of the opposite situation can detect from the following aspects, and they’re the opposite weapons type, communications and mobility. The weapon type can be divided into tanker, gun, missile and vehicle. The communication can be divided into radio silence, communication base and jamming. The activity can be divided into immobile, slow and rapid. The weapon type and activity are used sensors and reconnaissance to apperceive. Because these two methods of access to information may be redundant, DBN can fuse the redundant information to reduce the uncertainty of collected information. If there is a conflict about the information through these two methods, DBN use conditional probability to fuse conflict information, and gain the probability distribution of situation information of nodes. Means of communication use sensor to apperceive. DBN can also fuse the complementary information, for example, the opposite actions can be estimated through the data fusion of opposite types of weapons, communications and mobility characteristics. Based on the above analysis, the SA model configuration based on DBN of UCAV autonomous decision system is shown in Figure 2. 4.3 SA Model Based on Continuous Time Slice DBN In DBN, the configuration of network and relation of random variables are same in every time slice. A DBN includes a prior network and a transition network. Figure 3 shows SA model based on continuous time slice DBN which evolved from figure 2.

Fig. 2. SA model configuration based on DBN of UCAV autonomous decision system

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Fig. 3. SA model of continuous time slice DBN

With the lapse of time, more and more information can be obtained. On the current time T, the Bayesian network will be constructed according to the causality. We can conclude the situation on the time T by the prior information and history information. With the lapse of time to T+1, the situation information on the time T can be used for evidence on the time T+1. According to information, the prior information can be complemented and thus we can obtain the model of SA based on DBN.

5 Result of Simulation Once the structure of a DBN completed, the next step is to construct the CPT of nodes in the model of SA of UCAV autonomous decision system. In this step, experts’ opinions, experiences and more information are considered. The probability of node in the

Table 1. The CPT of node “Situation Assessment” Intention

Attack

Defense

Retreat

Activity ActiveRecons ConductEW MountWeapon SecureTarget ActiveRecons ConductEW MountWeapon SecureTarget ActiveRecons ConductEW MountWeapon SecureTarget

Situation Assessment Attack Defense Retreat 0.75 0.25 0 0.85 0.15 0 0.95 0.05 0 0.7 0.2 0.1 0.45 0.45 0.1 0.4 0.5 0.1 0.25 0.7 0.05 0.2 0.75 0.05 0.45 0.3 0.25 0.25 0.3 0.45 0.45 0.3 0.25 0.15 0.2 0.65

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Table 2. The transition probability from time T to time T+1 T+1/T

Attack

Defense

Retreat

Attack ActiveRecons ConductEW MountWeapon SecureTarget ActiveRecons ConductEW MountWeapon SecureTarget ActiveRecons ConductEW MountWeapon SecureTarget

Defense

Attack

Defense

Retreat

Attack

Defense

Retreat

0.75 0.85 0.95 0.7 0.45 0.4 0.25 0.2 0.45 0.25 0.45 0.15

0.75 0.85 0.95 0.7 0.45 0.4 0.25 0.2 0.45 0.25 0.45 0.15

0.75 0.85 0.95 0.7 0.45 0.4 0.25 0.2 0.45 0.25 0.45 0.15

0.25 0.15 0.05 0.2 0.45 0.5 0.7 0.75 0.3 0.3 0.3 0.2

0.25 0.15 0.05 0.2 0.45 0.5 0.7 0.75 0.3 0.3 0.3 0.2

0.25 0.15 0.05 0.2 0.45 0.5 0.7 0.75 0.3 0.3 0.3 0.2

T+1/T

Attack

Defense

Retreat

Retreat ActiveRecons ConductEW MountWeapon SecureTarget ActiveRecons ConductEW MountWeapon SecureTarget ActiveRecons ConductEW MountWeapon SecureTarget

Attack

Defense

Retreat

0 0 0 0.1 0.1 0.1 0.05 0.05 0.25 0.45 0.25 0.65

0 0 0 0.1 0.1 0.1 0.05 0.05 0.25 0.45 0.25 0.65

0 0 0 0.1 0.1 0.1 0.05 0.05 0.25 0.45 0.25 0.65

model can be acquired from the experiences of experts and the data learning through the computer. Table 1 shows the conditional probability table of node “Situation Assessment”. And the transition probability from time T to time T+1 is shown in table 2. At first, we have no information about the situation. So it’s assumption that the prior probabilities of nodes in DBN obey uniform distribution. After initialization, the SA system is waiting for the refresh data. Once new information obtained, the reasoning of network will be start-up. The probability distribution of node state in network will be refreshed, and we can obtain the probability distribution of root node. It is assumed that the weapon type “missile” is detected by the sensor of UCAV on the time T. According to the evidence, and through the reasoning of network, the probability of “attack, defense, retreat” about situation of UCAV on the time T is [0.625, 0.285, 0.09] . It is shown in figure 4.

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Fig. 4. State of SA based on DBN on the time T

Fig. 5. State of SA based on DBN on the time T+1

On the time T+1, the sensor detect that the missile is moving rapid. So according to the evidence, every information of network node will be refreshed and the probability of “attack, defense, retreat” on the time T+1 is [0.655, 0.276, 0.069] . It is shown in figure 5. Above results are reasoning from Bayesian network based on single time slice. If we use the reasoning method of DBN, and consider two evidences on the time T and T+1, the probability of “attack, defense, retreat” on the time T+1 is [0.694, 0.246, 0.06] . It is shown in figure 6. We conclude that the probability of “attack” on the time T through reasoning from DBN raise than the probability through reasoning from Bayesian network based on single time slice. And another probability is falling. Obviously, history assess result

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Fig. 6. State of SA based on continuous time slice DBN

affects the current result. The current result is not only involved latest information, but also considered history information. And the DBN is capacity of cumulating information. The method of SA based on DBN is that the causality of combat information element on different time or same time can be obtained according to the target information on different time slice. With the change of time, the DBN can adopt reasoning according to the history and latest evidence information, and it is proved to have the ability to dynamically modify and improve the knowledge base and model of the situation assessment. The simulation results show that the results of SA based on DBN is same to the experts’ results. It concludes that the method is a scientific and logical algorithm. The subjectivity of the assessment is reduced and the accuracy is greatly improved.

6 Conclusions In common sense, SA is the perception, comprehension and prediction to the elements in the environments within a volume of time and space. It is very essential to the control, decision-making and management issues of UCAV. In the paper, the methods of SA and their application of UCAV autonomous decision system are considered. And the DBN model of UCAV autonomous decision system is established. From the above analysis, we can see that DBN having the consistency inference ability and uncertain express ability can solve the problem of uncertain information of expressing and reasoning. But it’s important and hard to obtain the prior information, so the military knowledge and experts’ experiences are the key point for DBN. According to military experts’ ways of thinking and experiences, multi-source data are analyzed and judged to make reasonable explanation on the current battlefield scenarios and provide the quantificational situation information for autonomous decision-making system. Correct and rapid SA has important research value and actual combat significance on tactical flight path, mission planning and distribution of tasks.

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References 1. Chao, X.L., Chen, Q.: UCAV’s development and key technologies. Flight Dynamic 19, 1–6 (2001) 2. Tang, Q., Zhu, Z.Q., Wang, J.Y.: Survey of foreign researches on an autonomous flight control for unmanned aerial vehicles. Systems Engineering and Electronics 26, 418–422 (2004) 3. Ba, H.X., Zhao, Z.G., Yang, F., et al.: Concept, Contents and Methods of Situation Assessment. Journal of PLA University of Science and Technology 5, 10–16 (2004) 4. Yu, Z.Y., Chen, Z.J., Zhou, R.: Study on Algorithm of Threat Level Assessment Based on Bayesian Network. Journal of System Simulation 17, 555–558 (2005) 5. Endsley, M.R.: SAGAT: A methodology for the measurement of situation awareness (NOR DOC 87-83). Hawthorne, CA, Northrop Corporation (1987) 6. White, F.E.: Joint directors of laboratories-technical panel for C3I. In: Data Fusion Subpanel. Naval Ocean Systems Center, San Diego (1987) 7. Howard, C., Stumptner, M.: Probabilistic reasoning techniques for situation assessment. In: Third International Conference on Information Technology and Applications, ICITA 2005, vol. 1, pp. 383–386 (2005) 8. Hinman, M.L.: Some computational approaches for situation assessment and impact assessment. In: Proceedings of Fifth International Conference on Information Fusion, vol. 1, pp. 687–693 (2002) 9. Jan, T.: Neural network based threat assessment for automated visual surveillance. In: Proceedings IEEE International Joint Conference on Neural Networks, vol. 2, pp. 1309–1312 (2004) 10. Gonsalves, P.G., Burge, J.E., Harper, K.A.: Architecture for genetic algorithm-based threat assessment. In: Proceedings of the Sixth International Conference of Information Fusion, pp. 965–971 (2003) 11. Murphy, K.: Dynamic Bayesian Networks: Representation, Inference and Learning. Ph.D. thesis. U.C. Berkeley (2002) 12. Sumit, S., Pedro, D., Daniel, W.: Relational Dynamic Bayesian Networks. Journal of Artificial Intelligence Research 24, 759–797 (2005)

Optimal Tracking Performance for Unstable Processes with NMP Zeroes Jianguo Wang1,2,∗, Shiwei Ma1,2, Xiaowei Gou1,2, Ling Wang1,2, and Li Jia1,2 1

Shanghai Key Lab of Power Station Automation Technology, Shanghai 200072, China 2 School of Mechatronical Engineering and Automation, Shanghai University, Shanghai 200072, China {jgwang,swma,xwgou,wangling,jiali}@shu.edu.cn

Abstract. This paper has investigated optimal tracking performance for unstable processes with non-minimum phase (NMP) under control energy constraint. Firstly, based on prime factorization of unstable process, a performance index containing tracking error and plant input energy is defined, which is represented by sensitivity function and control sensitivity function. Then, utilizing spectral factorization to minimize the performance criterion we derive an optimal controller design method for unstable processes and furthermore study the optimal tracking performance under control energy constraint. The validity of the obtained result can be illustrated by the simulation research.

，

Keywords: optimal tracking performance, control energy constraint, unstable process, spectral factorization.

1 Introduction Since the seminal work of Bode during the 1940s related to feedback amplifier design [1], fundamental performance limitations of feedback control loops have been a topic of interest [2]. In recent years, there has been growing attention devoted to the studies of performance limitations in practical application [3]. With a view to the fact that the input to the plant must be finite energy in a more realistic setting, the paper [4] has studied optimal tracking and regulation control problems under control energy constraint and found that the performance limits depend not only on the plant non-minimum phase (NMP) zeros, time delays, and unstable poles, but also on the plant gain in the entire frequency range. However, the paper has not presented the corresponding optimal controller design method and investigated the optimal performance for the plant in the simultaneous presence of NMP zeros and the unstable poles under control energy constraint. ∗

Sponsored by Shanghai Municipal Science & Technology Commission Key Project (08160512100,08160705900,09JC1406300,8DZ2272400) and Shanghai University "11th Five-Year Plan" 211 Construction Project.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 69–74, 2010. © Springer-Verlag Berlin Heidelberg 2010

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The unstable system is difficult to be controlled in industry process control domain [2, 5]. The stabilization become worse for the presence of unstable poles and the simultaneous presence of NMP zeros greatly constrain the available optimal control performance [6]. Therefore, the control for unstable processes with NMP zeros has been given great attention. Optimal performance for unstable processes with NMP zeros has been investigated under control energy constraint. Utilizing spectral factorization to minimize the performance criterion containing tracking error and plant input energy, we derive an optimal controller design method for unstable processes and furthermore study the optimal tracking performance under control energy constraint.

2 Performance Index and Optimal Design Unity feedback control system is shown in Fig. 1. We consider linear SISO system with step reference input and assume that the plant is unstable, the system is initially at rest and there is no output disturbance.

Fig. 1. Unit feedback control system

In step reference tracking problems, a standard assumption is that the plant has no zero at s = 0. In order for performance index to be finite, another assumption is that the plant must contain an integral item [4]. Denote a class of stable, proper and rational function as Φ and the plant G ( s ) can be prime factorized as G (s) =

Find X ( s ), Y ( s ) ∈ Φ

N (s) M (s)

， N ( s), M ( s) ∈ Φ

(1)

which satisfy

N ( s ) X ( s ) + M ( s )Y ( s ) = 1

(2)

Then, all the controllers C ( s ) which can stabilize the feedback system can be expressed as a set ⎧ ⎫ X ( s ) + M ( s )Q ( s ) , Q (s) ∈ Φ ⎬ ⎨C ( s ) = Y ( s ) − N ( s ) Q ( s ) ⎩ ⎭

(3)

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Based the parameterization, the sensitivity function and control sensitivity function can be given by

S ( s ) = M ( s )(Y ( s ) − N ( s ) Q ( s ))

(4)

S u ( s ) = M ( s )( X ( s ) + M ( s )Q ( s ))

(5)

For simplification transfer function expression is given as M (s) = M (s) / s

(6)

Since the control energy is to be considered, we adopt an integral square criterion to measure the tracking error and the plant input energy. Therefore, we consider performance index as follows [4] ∞

∞

0

0

J = (1− ε )∫ e(t)2dt + ε ∫ u(t )2 dt

(7)

where, 0 ≤ ε ≤ 1 , may be used to weight the relative importance of tracking objective and the input control energy. For the system reference input being unit step signal, making use of Parseval’s theorem, the performance can be expressed as J = (1 − ε )

1 2π

∞

S ( jω )

−∞

ω2

∫

2

dω + ε

1 2π

∞

Su ( jω)

−∞

ω2

∫

2

dω

(8)

Next, the task is to determine the optimal Q( s) to minimize the performance index (8). Substituting (4) and (5) into (8) and making use of expression (6), the performance index can be written as 2 1 ∞ J = (1 − ε ) ∫ M ( jω)(Y ( jω) − N ( jω)Q( jω)) dω 2π −∞ (9) 2 1 ∞ +ε ω ω + ω ω ω M ( j )( X ( j ) M ( j ) Q ( j )) d 2π ∫−∞ When completing square operation, we find that the coefficient item of Q( jω ) 2 can be written as COEF ( s ) = M ( s ) M (− s )[(1 − ε ) N ( s ) N (− s ) + ε M ( s ) M (− s )]

(10)

It can be proven that COEF ( s ) has a spectral factor, which we label as H , then H ( s ) H ( − s ) = C OEF ( s )

(11)

Utilizing the procedure outlined in [7], J can be written as J=

1 2π

∫

∞

−∞

2

Q( jω ) H ( jω ) +

2

F ( jω ) F ( jω ) dω + D( jω ) − H (− jω ) H (− jω )

(12)

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Where F (s) = M (s)M (−s)[−(1 − ε )Y (s) N (−s) + ε X (s)M (−s)]

(13)

D(s) = M (s)M (−s)[(1 − ε )Y (s)Y (−s) + ε X (s) X (−s)]

(14)

Thus, the minimum of (12) depends only on the first term and can be achieved by making the choice of Q( s) as follows Qopt ( s ) = −

1 ⎧ F (s) ⎫ ⎨ ⎬ H ( s ) ⎩ H ( − s ) ⎭ stable

(15)

part

Substituting Qopt ( s) into parameterization (3), the optimal controller can be obtained.

3 Some Discussion With the increase of ε , the constraint on the input control energy become stricter and the tracking performance become poorer. In the two limit cases, when ε = 0 , there is no any constraint on the control energy has no and the best tracking performance can be obtained; when ε = 1.0 , the control system has little ability to track the objective signal and the control energy is only used to stabilize the unstable plant, which is the minimal energy to make the system stable. As is well known, the sensitivity function describes the responses of both system output to disturbance and tracking error to reference input. And the responses of plant input to both output disturbance and reference input can be described by the control sensitivity function. Therefore, in the same way, the system will possess the optimal performance of disturbance rejection in the presence of step output disturbance. The controller expressed in (3) is stable and so this design method only applies to the plant that can be strongly stabilized. If the number of real poles of plant between every pair of NMP zeros is odd, then the presented design become invalid [8]. For example, the controller can’t be used to stabilize the SISO Inverted Pendulum system.

4 Simulation Research To illustrate the preceding results and study the optimal tracking performance, we consider unstable plant with a NMP zero and the model is given as G (s) =

s−3 s ( s − 1)( s + 4)

(16)

According to (1) and (6), N ( s) , M ( s) and M ( s ) can be obtained. According to (2), X ( s) and Y ( s) can be obtained. For a given weigh coefficient ε , the optimal controller can be obtained. And for ε = 0.0 , 0.33 , 0.67 and 1.0 , we can obtain four corresponding optimal controllers.

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73

4.1 Time Response For each optimal controller and the unstable plant, the dynamic responses of u and y of closed loop system are shown in Fig. 2 and Fig. 3 respectively. From the response curves, we can find that with the increase of the value of ε (relative importance of control energy constraint), the magnitude of plant input u decreases, the response speed slows down and the tracking error increases. It is obvious that good tracking performance is at the cost of large control energy and the increasing relative importance of control energy constraint will result in poor tracking performance. For different values of ε , the obtained closed loop system always have a great overshoot which is determined by the plant itself, and there is an minimal value of overshoot for the simultaneous presence of unstable poles and NMP zeros.

Fig. 1. Time response curves of system output

Fig. 2. Time response curves of control input

4.2 Optimal Performance By simulation and numerical integral square measure of the tracking error e(t) and plant input u from t=0 to 20s, we can obtain the actual tracking performance and control energy, denoted as Pe and Pu respectively, for each optimal controller.

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Performance

ε = 0.0

ε = 0.33

ε = 0.67

ε = 1.0

Pe

2.480

7.305

12.85

--

Pu P

85.84

21.62

16.36

10.98

2.480

12.06

15.16

10.98

The optimal performance can be obtained as follows

P = (1 − ε ) Pe + ε Pu

(17)

According to the results of simulation, we can obtain optimal performances for different values for ε = 0.0 , 0.33 , 0.67 and 1.0 , which are listed in Tab.1. By simulation, we can find that when the optimal controller is substituted by other controller selected randomly, the obtained performance will be greater than the values listed in Tab.1, so the presented design method can make the system possess optimal performance under control energy constraint.

5 Conclusion and Future Research This paper has investigated optimal tracking performance for unstable processes with NMP zeros under control energy constraint. Based on prime factorization of unstable process, a performance index containing tracking error and plant input energy is defined. Utilizing spectral factorization to minimize the performance criterion we derive an optimal controller design method for unstable processes, which can be used to obtain optimal tracking performance under control energy constraint. It is desirable to obtain analytical expressions and give additional insight into the impact of the various factors on the optimal performance. A natural extension is to investigate similar issues for plants that can’t be strongly stabilized. These should be future research directions.

，

References 1. Bode, H.W.: Network Analysis and Feedback Amplifier Design. Van Nostrand, New York (1945) 2. Seron, M.M., Braslavsky, J.H., Goodwin, G.C.: Fundamental Limitations in Filtering and Control. Springer, London (1997) 3. Chen, J.: Control Performance Limitation: to Achieve? or not to Achieve? In: SICE Annual Conference, Fukui, pp. 4–6 (2003) 4. Chen, J., Hara, S., Chen, G.: Best Tracking and Regulation Performance under Control Energy Constraint. IEEE Trans. Automat. Contr., 1320–1336 (2003) 5. Morari, M., Zafiriou, E.: Robust Process Control. Prentice-Hall, New York (1989) 6. Freudenberg, J., Middleton, R., Stefanopoulou, A.: A Survey of Inherent Design Limitations. In: Proceedings of the American Control Conference, Chicago, pp. 2987–3001 (2000) 7. Goodwin, G.C., Graebe, S.F., Salgado, M.E.: Control System Design. Prentice-Hall, Upper Saddle River (2002) 8. Doyle, J.C., Francis, B.A., Tannenbaum, A.R.: Feedback Control Theory. Macmillan, New York (1992)

Typhoon Cloud Image Enhancement by Differential Evolution Algorithm and Arc-Tangent Transformation Bo Yang1 and Changjiang Zhang1,2 1

College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, 321004, Jinhua, China 2 State Key Laboratory of Remote Sensing Science, Jointly Sponsored by the Institute of Remoe Sensing Applications of Chinese Academy of Sciences and Beijing Normal University, Beijing, China [email protected]

Abstract. This paper proposed an image enhancement method based on the differential evolution algorithm (DEA) and arc tangent transformation for typhoon cloud images. Because of the effect of sensors or other factors, the contrast of the satellite cloud images received directly by satellite was not acceptable. In view of the features of typhoon cloud images, especially the feature of gray level distribution of typhoon eye’s surrounding area, this algorithm can choose the most suitable parameter for arc tangent transformation to enhance the overall contrast of eyed-typhoon cloud image. To examine the validity of the proposed method, we used the partial differential equation (PDE) based on geodesic activity contour GAC to extract the typhoon eye. The experimental results indicated that the proposed method could improve the overall contrast of typhoon cloud images directly, and make the typhoon eye differ distinctively from the surrounding area.

（

）

Keywords: typhoon cloud image; typhoon eye; contrast enhancement; differential evolution; arc tangent transformation.

1 Introduction At present, using the satellite cloud image analysis has already become one of the important technological means for typhoon forecast, analysis quality does influence to typhoon forecast precision. And using stationary satellite cloud materials has large superiority in the aspect of typhoon service forecast, because compare the cloud materials obtained by stationary satellite with the polar orbiting satellite, they are comparatively continuous, thus it is much more advantageous to analyze time-variant weather event such as typhoon. But as a result of the components’ resolution of the satellite image formation system, and the sensitivity as well as the sensor itself, and the effect that made by the atmosphere in the long-distance and the weather condition in the process of satellite image transmission and the transformation, the quality of satellite cloud image perhaps reduces, or certain essential regions such as the eye of a typhoon become fuzzy, so it will be difficult to carry on the next step of segmentation as well as the further analysis. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 75–83, 2010. © Springer-Verlag Berlin Heidelberg 2010

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There are many image enhancement methods for the satellite images, in recent years some researchers proposed some views of the enhancement method for satellite cloud images, for instance Ref. [1] developed an approach which merged the data from several images of the same area and the same time in order to improve the image resolution; In Ref. [2], satellite image enhancement and smoothing towards automatic feature extraction is accomplished through an effective serial application of anisotropic diffusion processing and alternating sequential filtering; Ref. [3] proposed a method to enhance the detail and sharp the edge of colored satellite cloud images, this method is similar to the wavelet decomposition, it divides the image into the two layers, each layer processed alone, and finally reconstructed; Ref. [4] and [5] proposed the improved gamma correction curve method, although this method has improved the high light and shadow region illumination condition, the effect of the enhancement of the typhoon eye is not obvious, so the typhoon eye cannot be effectively segmented. .Ref. [6]and Ref. [7]proposed a method based on incompleteness Beta transformation, this method has improved the image contrast obviously, but it has two parameters to be optimized, so it is complex to use the optimization algorithm. The proposed method based on only single parameter and normalized arc tangent grey level transformation to enhance the typhoon cloud image, and we use DEA to optimize the parameter. After processed, the image has high overall contrast, and the typhoon eye has been extruded obviously from surrounding. It is advantageous to segment the typhoon eye in the enhanced satellite image.

2 Parameter Optimization Based on DEA and Enhancement on Typhoon Cloud Image Based on Arc-Tangent Transformation Differential evolution algorithm was proposed by R.Storn and K.Price in 1995. At first, the main purpose is to solve the continual global optimization question with the differential evolution algorithm. Its basic philosophy is to obtain the middle population from recombining the difference of current population individual. It uses fathers and sons mixture individual adaptive value to compete directly to obtain the new population. The most attractive part of the differential evolution algorithm is its variant operation. When an individual is selected, the algorithm adds the weighted difference of other two individuals to the selected individual to complete the variation. In the initial period of the algorithm, because the differences between individuals are so large that such variant operation can make the algorithm itself have the strong ability of global search. In the later period of the algorithm, when the algorithm tends to restrain, the individual differences in the population tend to be small, this enables the algorithm to have the strong local search ability. Compared to the other similar methods, this kind of variant operation enables the algorithm to have the unexampled merit in function optimization. The algorithm has three advantages as follows: (1) few undetermined parameters; (2) it is not easy to fall into local optimum; (3) the convergence rate is quick. After the differential evolution being proposed soon, there are many kinds of improvements to the differential evolution algorithm, but they are essentially the same, the main difference between them is the producing method of

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77

the middle test vector, this paper has only one parameter to be optimized and the expression is so simple, so the most primitive differential evolution algorithm is enough to satisfy the requirement. As the pixels in the typhoon eye are generally dark and the surrounding is bright, we should make the eye of the typhoon much darker and the surrounding much brighter. The transformation curve must have the property of compressing high gray level and low gray level and stretching middle gray level. The arc tangent transformation curve proposed by this paper has the property and can satisfy the requirement above. Its expression is shown as follows:

y=

arctan(kx) arctan(k )

(1)

where arctan (k) is the normalization factor, k shows transformation parameter whose value scope is [0, 80]. Different values of the parameter k can obtain different transformation curve, but if the value k is regulated by artificial, it is difficult to obtain the optimal value. Figure 1 shows that different values of k will produce different curves of gray transformation. In order to solve the problem, DEA is used to optimize the parameter k to get a good enhancement result to the typhoon cloud image. 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fig. 1. Different k results in different curve

Generally speaking, large information entropy means that much information is contained in the image. Moreover, the image standard deviation is also one of key indicators of the image contrast evaluation. Finally, we choose the objective function as follows:

f =

1 g ×h 5

(2)

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2500 2000 1500 1000 500 0

0

10

20

30

40

Fig. 2. Curve of

Here

g = −∑ pi log 2 pi

is

50

60

70

80

g5 × h

image

information

entropy,

and

i

h=

1 ( xi − μ ) 2 is image standard deviation, μ is the mean value of image N −1

pixels, N is the number of the pixels in an image, pi is probability of pixels of gray level i. Our goal is to make the g × h have the largest value. The objective function determined in equation (2) is a multi-peak function, figure 2 shows the curve of 5

g 5 × h , and the DEA is used to solve the global optimization problem in the initial, therefore we can avoid falling into the local optimal solution. We discover from figure 1 that the normalized arc-tangent transformation gray level curve is symmetrical about the zero point. We will define the mapping interval in [- 1, 1]. Therefore, before the image enhancement, we need to make a preprocessing to the original cloud image, that is to map the original image grey levels to the interval [- 1,1], and after the enhancement, we must carry on the opposite mapping to the image to make the gray level of the image restore in interval [0, 255]. The positive map is as follows:

g=

f × 2 −1 255

(3)

Where f is the original image, g is the image whose gray level is in interval [-1, 1] after mapping.

3 Principle of Image Segmentation Based on GAC The geodesic activity contour (GAC) model without any free parameter was proposed by V.Caselles, R.Kimmel and G. Sapiro in 1997, and the GAC model is an important breakthrough in the image segmentation based on PDE method [12].

Typhoon Cloud Image Enhancement by Differential Evolution Algorithm

79

In optics, Fermat principle tells us that when light propagates in the nonhomogeneous media, “the light” will not be a straight line, but the way determined depends on the shortest optical path. If the media’s index of refraction is n (x, y, z), then the actual path which the light propagates from A to B should make the optical path B

L R = ∫ n( s )ds

(4)

A

achieve the minimum value, ds in the expression indicates the Euclidean length. That is, light is always propagating through the partial minimum of n. The Fermat principle may explain the geometrical optics problems such as how refraction and reflection produces. But the train of thought this principle is similar to that the activity contour curve must go through the partial minimum of g (| ∇I |) . So we can use the energy functional below to determine the activity contour:

LR (C ) = ∫

L (C )

0

g (| ∇I [C ( s )] | ds

(5)

Where L(C) in the equation (5) indicates the length of closed curve C, and the LR(C) is the weighted length. The gradient downflow which it corresponds is as follows:

∂C = g (C )κN − (∇g • N ) N ∂t

(6)

When the curve evolutes according to equation (6), it will be controlled by two kinds of “force”: one is from the geometrical deformation, the curvature movement, it is called the internal force. The internal force becomes very little or even stops in the neighboring of image edge. Therefore we often call the edge function g (| ∇I |) as

（，） because g ( x, y ) = g (| ∇I ( x, y ) |) , and ∇y is generated from I（x，y）, so we “edge stop function”. The other force comes from the gradient

∇g of g x y ,

call the second force external force. The external force will make C turn toward the edge of the objects in the image and finally stabilizes in the edge. 3.1 Detail Steps of Proposed Enhancement Algorithm Detail steps of proposed enhancement algorithm based on DEA and arc-tangent transformation curve are as follows: Step1: Map the range of the gray level of the input image to the interval [-1, 1] using the method introduced in section 2; Step2: Initialize every parameter of the differential evolution algorithm and the range of the parameter k of the arc-tangent transformation curve; Step3: Differential evolution algorithm is used to optimize the parameter of the arc-tangent transformation curve; Step4: The optimal k which is obtained by Step 3 is used to enhance contrast of the satellite cloud image;

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Step5: Use the segmentation method of PDE based on GAC model to examine the effect to the typhoon eye segmentation.

4 Experimental Results and Discussion There are four methods to be used to enhance typhoon cloud image. These four methods are called as histogram equalization, improved gamma correction curve, incompleteness Beta transformation, and normalized arc-tangent grey level transformation respectively. Figure 3(a) is the original image, figure 3 (b) is the arc-tangent gray level transformation curve based on the parameter k obtained by the differential evolution algorithm, the final value of k is 8.1474. 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1

(a) Original image

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

（k=8.1474）

(b) Gray level transformation curve

Fig. 3. Original image, gray level transformation curve

Figure 4(a), (b), (c), (d) show the enhanced images by histogram equalization, improved gamma correction curve, incompleteness Beta transformation, and normalized arc-tangent grey level transformation respectively and their effect of segmentation by PDE method. We have discovered from the experiments above that except the gamma correction curve method, other three methods all have obvious effect to the typhoon eye segmentation. But after having observed the enhancement results carefully, we also discover that only the enhanced image, which has been enhanced by the arc-tangent transformation curve method, the typhoon eye has similar size as the original cloud image. Histogram equalization and incompleteness Beta transformation method cause the typhoon eye bigger or smaller than actual size in certain extent. Even if the typhoon eye has been extracted accurately, because the size has been changed, the size of typhoon eye will be inaccurate. Finally, we need to compare each running rate of the above four algorithms. All experiments are run on the computer based on Duo dicaryon processor 1.73G, memory 2G, windows XP platform, matlab2009a. The running time of all the experiments are shown in the table 1:

Typhoon Cloud Image Enhancement by Differential Evolution Algorithm

(a) Histogram equalization

(b) Improved gamma correction

(c) Incompleteness Beta transformation

(d) Normalized arc-tangent

81

Fig. 4. Enhanced images with different methods and their affection to typhoon eye segmentation Table 1. Compare of running time with different methods Enhancement methods

Time (s)

Histogram equalization

0.5930

Improved gamma correction

1.5470

Incompleteness Beta transformation

20.5630

Normalized arc-tangent

25.391

We may see by table 1 that it takes much time in using the differential evolution algorithm, but it obtains the best enhancement effect. Histogram equalization and improved gamma correction curve do not have to use optimization algorithm, so they cost little time, but it is obvious that they take the bad enhancement effect as the cost. We have done many other experiments to improve a fact that when the size of the image becomes smaller, the time that Normalized arc-tangent costs also becomes less, but Incompleteness Beta transformation also costs much time. Taking all factors into consideration, if we choose the enhancement method based on differential evolution algorithm and normalized arc-tangent transformation curve, we can get the best original data from the segmentation of the typhoon eye.

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5 Conclusion This paper proposed an efficient enhancement method based on differential evolution algorithm and normalized arc-tangent transformation curve to enhance the typhoon cloud image. The experiments indicate that different image may have different gray level transformation curve. The proposed algorithm can stretch the global contrast in the enhancement of typhoon cloud image. It can make the typhoon eye more obvious without distortion. Our final goal is to measure the size of the typhoon eye, we need to segment the eye accurately from the typhoon cloud image before we measure the size of the eye. Our work is to do some preprocessing for the segmentation of the typhoon eye, which will make it easier and more accurate to segment the typhoon eye. From this paper, we can see that this efficient enhancement method could not only be used in the area of typhoon cloud images but could also be used in other many areas such as general satellite image enhancement or medical image enhancement, it will make good visual effect in these fields. Acknowledgments. Part of the research is supported by the Grants for China Natural Science Foundation (40805048), Typhoon Research Foundation of Shanghai Typhoon Institute/China Meteorological Administration (2008ST01), Research Foundation of State Key Laboratory of Remote Sensing Science, Jointly sponsored by the Institute of Remote Sensing Applications of the Chinese Academy of Sciences and Beijing Normal University (2009KFJJ013), Research foundation of State Key Laboratory of Severe Weather/Chinese Academy of Meteorological Sciences (2008LASW-B03). China Meteorological Administration, China National Satellite Meteorological Center is acknowledged for providing all the typhoon cloud images in this manuscript.

References 1. Albertz, J., Zelianeos, K.: Enhancement of satellite image data by data cumulation. ISPRS Journal of Photogrammetry and Remote Sensing 45, 161–174 (1990) 2. Karantzalos, K.G.: Combining Anisotropic Diffusion and Alternating Sequential Filtering for Satellite Image Enhancement and Smoothing. In: Proceedings of SPIE - Image and Signal Processing for Remote Sensing IX, Barcelona, Spain, September 9-12, 2003, vol. 5238, pp. 461–468 (2004) 3. Attachoo, B., Pattanasethanon, P.: A new approach for colored satellite image enhancement. In: 2008 IEEE International Conference on Robotics and Biomimetics, ROBIO 2008, Bangkok, Thailand, February 21-26, 2009, pp. 1365–1370 (2008) 4. Xu, G., Su, J., Pan, H., Zhang, Z., Gong, H.: An Image Enhancement Method Based on Gamma Correction. In: 2009 Second International Symposium on Computational Intelligence and Design, pp. 60–63 (2009) 5. Yang, J., Shi, Y., Xiong, X.: Improved Gamma Correction Method in Weakening Illumination. Journal of Civil Aviation 24, 39–42 (2006) 6. Zhang, C., Wang, X., Zhang, H.: Non-Linear Gain Algorithm to Enhance Global and Local Contrast for Infrared Image. Journal of Computer &Aided Design & Computer Graphics 18, 844–848 (2006) 7. Zhou, J.L., Hang, L.V.: Image Enhancement Based on a New Genetic Algorithm. Chinese Computers 24, 959–964 (2001) (in Chinese)

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8. Hu, Z.B.: The Study of Differential Evolution Algorithm for the Function Optimization. Wuhan University of Technology, Wuhan (2006) 9. Wang, D., Hou, Y., Peng, J.Y.: Image processing based on PDE, pp. 80–109. Science Press, Beijing (2008) 10. Liang, Y.M., Li, Y., Fan, H.L.: Image enhancement for liver CT images. In: 2009 International Conference on Optical Instruments and Technology, vol. 7513, pp. 75130K-1– 75130K-8 (2008) 11. Yang, C., Huang, L.: Contrast enhancement of medical image based on sine grey level transformation. Optical Technique 28, 407–408 (2002) 12. Wang, D., Hou, Y., Peng, J.: Image processing based on PDE, pp. 80–109. Science Press, Beijing (2008) 13. Hu, Z.B.: The Study of Differential Evolution Algorithm for the Function Optimization. Wuhan University of Technology, Wuhan (2006)

Data Fusion-Based Extraction Method of Energy Consumption Index for the Ethylene Industry* Zhiqiang Geng , Yongming Han, Yuanyuan Zhang, and Xiaoyun Shi

，

School of Information Science and Technology Beijing University of Chemical Technology, Beijing 100029, China {gengzhiqiang,hanym,zhangyy,shixy}@mail.buct.edu.cn

Abstract. The assessment target of energy consumption for ethylene equipment is based on Special Energy consumption (SEC) index currently without considering the differences among the raw materials, process technology and equipment scales. Because the standards of the traditional energy consumption statistical methods are not uniform, it affects the comparability of energy consumption. Aiming at the lack of energy consumption evaluation methods for existing ethylene industrial equipments, the data fusion method is researched to obtain the energy consumption indexes of the ethylene industrial devices. The data variance fusion method of multivariate series is proposed based on cluster analysis and variance analysis, and then consumption indexes about water, steam, electricity, fuel and virtual benchmark of SEC are extracted respectively in ethylene industrial process. It can objectively evaluate the energy consumption status of the ethylene equipments, analyze the actions and opportunities of energy saving, and then suggest the direction of the energy saving and the consumption reduction of ethylene equipments. Keywords: Data fusion; Ethylene process; Cluster analysis; Energy saving.

1 Introduction Ethylene process equipment is one of high energy consumption in petrochemical units. With the developing of continued large-scale device, its energy consumption attracts people’s more attention, and it has become an extremely important integrated performance index to measure the technical performance of the devices [1]. Currently the widespread use of ethylene Special Energy consumption (SEC) per ton is regarded as the evaluation index of the devices. However, the SEC is caused by many factors, such as process technology, production scale, geographical environment, raw materials and so on, including raw materials which affect the greatest among all the factors, yet SEC evaluation method takes no account of the impact of the raw materials. For the energy consumption index of enterprise products, due to the different situations of various regions and sectors, formulating uniform standards for energy consumption *

This work is partly supported by “the fundamental research funds for the central universities” (JD0906). K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 84–92, 2010. © Springer-Verlag Berlin Heidelberg 2010

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indexes are not objective and unfair to evaluate the energy consumption of some enterprises, affect the comparability of the results, but also unscientific. As time goes by, the energy consumption-related data of the ethylene devices form a time series stereoscopic data sheet table according to the chronological sequence of graphic data. The paper collects ten years data of energy consumption for the whole ethylene industry, including relevant production data, economic data, design data, expertise, operating procedures and other massive data and information. The research uses data fusion technology [2-6] to obtain the energy consumption indexes of the ethylene industry devices. According to the characteristics of energy consumption, meanwhile taking into account the ethylene device technology, size differences, different materials, different length of historical data of different devices, gradual extension of the devices, the data variance fusion of multivariate time series is proposed to extract energy consumption indexes based on cluster analysis[7] of the ethylene industry devices. And then energy consumption indexes are respectively extracted from ethylene industrial water, steam, electricity, fuel and virtual benchmarks and quantitative indexes of SEC. The method overcomes the disadvantages of the past SEC index calculated uniformly and unreasonably, and makes the results more comparable to find out the direction of energy saving and consumption reducing, and guides energy saving and consumption reducing, for the decomposition and the implementation of quantitative energy saving targets of great strategic importance.

2 The Comprehensive Processing Method of Energy Consumption Data Ethylene process is a typical complex industrial process, including various types of data, such as the data of production and operation, process simulation data, process design data, expertise data and so on. The data characteristics demonstrate a complex nonlinear time series relationship, including noise, abnormal data and etc. Therefore it will take advantage of comprehensive data processing method to obtain the real energy consumption data of the ethylene device, ensuring the accuracy and consistency of the data. 2.1 Abnormal Data Detection Abnormal data samples are either very different or inconsistent from other data samples, expressing the abnormity of the spatial position and data items relationship, those are isolated points or outliers in the spatial position deviating from the groups. As known by the probability theory, when the variables X obey a normal distribution or an approximate normal distribution, the random variables with high probability appear in the vicinity of E ( X ) and small probability appear in the distance of the

E ( X ) ; so when X far deviates from E ( X ) , X can be regarded as outliers and be removed. Specific formulas are written as follows in formula (1)

| xij (t k ) − x ij |> 2 Si j ， then remove xij (t k ) .

(1)

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Where,

xij =

1 N

N

∑ x (t ) k =1

ij

k

i = 1, 2,

1 N ∑ ( xij (tk ) − xij )2 N − 1 k =1

Sij =

, m; j = 1, 2,

,n

， i = 1, 2, , m; j = 1, 2, , n

In the formula (1), when the data which need to be detected are less, there will be a greater miscarriage of justice. So the consistency test of the same time data uses Grubbs criteria in different devices. Specific formula (2) is showed as follows: If T j (tk ) ≥ T (n, α ) , n is the number of data, α is the significant level, then remove

xij (tk ) . Where,

x j (t k ) =

1 p ∑ xij (tk ) p < m; j = 1, 2, p i =1

, n; k = 1, 2,

,N

(2)

1 p ∑ ( xij (tk ) − x j (t k ))2 p − 1 i =1

(2) S j (tk ) =

p < m; j = 1, 2, T j (tk ) =

| V j (tk ) | S j (t k )

, n; k = 1, 2, =

,N

| xij (tk ) − x j (t k ) | S j (tk )

2.2 Data Normalization Data normalization refers to the proportion of data at a certain scale, so that the data can fall into a small specific region. For this theme, different variables may work in different types, some variables have the positive effect on the theme by using the following transformation formula (3):

xij' =

xij − x min j

(3)

x max − x min j j

Some variables have the negative effect on the theme, using the following transformation formula (4):

xij' = 1 −

xij − x min j x max − x min j j

=

x max − xij j

(4)

x max − x min j j

In Formula (3) and (4):

xj xj

min

max

= max{x1 j , x2 j ,

= min{x1 j , x2 j ,

, xtj } ,

, xtj } , i = 1, 2, , t ； j = 1, 2,

,m .

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3 Energy Consumption Index Extraction of Ethylene Process Device 3.1 Clustering-Based of Multivariate Data Variance Fusion Because time-series data of the dimension of different variable are generally not the same, the values are not comparable among the variables. The time-series data sets need to be normalized before using the cluster analysis. Normalized formulas are formula (3) and (4). Time-series data set X is including m variables, as the formula (5) illustrates, t is the length of time series.

X1 X = X2 Xt

x1

x2

xm

⎡ x11 ⎢x ⎢ 21 ⎢ ⎢ ⎣ xt1

x12

x1m ⎤ x2 m ⎥⎥ ⎥ ⎥ xtm ⎦

x22 xt 2

(5)

Through the transformation above, the k-means cluster analysis is used to cluster the time series data, ensure the information matrix to be a positive definite matrix. K centre cluster algorithm is used to classify, compared to other clustering algorithm, it does not produce a noise point. Firstly, k objects are selected in all objects as the initial cluster centers. According to the mean of class objects, each object is assigned to the most similar category, and update the class means (calculating the mean of each class of objects). And then iteration is repeated again and again until no change of the class centre happens, then stops to get the clustering results. The core of K center cluster is showed in formula (6). p

E =∑

∑

r =1 xij ( tk )∈C r

In the formula

xij (tk ) − mr

2

where, m = 1 r nr

∑

(6)

xij (tk ) ；

xij ( tk )∈C r

E is the squared error sum of all the objects in data set, p is the num-

ber of all the class,

xij (tk ) is the point in the space, n r is the number of C r class,

mr is the mean of C r class. '

By using the algorithm above, normalized multivariate time series data X uses kmeans cluster algorithm to obtain cluster centers results of n classes, the results are in equation (7) below. K1 K = K2 Kn

x1

x2

xm

⎡ k11 ⎢k ⎢ 21 ⎢ ⎢ ⎣ k n1

k12 k 22

k1m ⎤ k2 m ⎥ ⎥ ⎥ ⎥ knm ⎦

kn 2

(7)

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According to the calculation formula of mean square deviation, equation (7) including n classes of cluster centers which including all kinds of time-series data. We can get the mean square deviation calculation formula based on the cluster centre as the formula (8) illustrates to calculate the mean deviation of all the variables on every cluster centre. 1 Ni

σ ij =

Where, i

= 1, 2,

Ni

∑ (x l =1

, n; j = 1, 2,

' lj

− x' )2 = lj

1 Ni

Ni

∑ (x l =1

− kij ) 2

' lj

(8)

, m , N i is the length of time series data contained

th

in the i center of a cluster. The n dimension of the vector covariance matrix COR: ⎡σ 11 σ 12 ⎢σ σ 22 21 COR = ⎢ ⎢ ⎢ ⎣⎢σ i1 σ i 2

σ1j ⎤ σ 2 j ⎥⎥ ⎥ ⎥

σ ij ⎦⎥

For the n dimension symmetric matrix, according to the product of the square root method (geometric mean method), you can obtain the feature vectors, that is,

W = ( w1 , w2 ,

wi =

σi σ

, wn )T , the specific process is as follows: （

i = 1, 2,

1

n

, n ） Where, σ i = (∏ σ ij ) n , i = 1, 2,

,n

j =1

n

σ = ∑σ i i =1

（ i = 1, 2, , n ）

By using W to fuse the multi-variables and get the final data fusion results Y, which is showed in formula (9): T

⎡ y1 ⎤ ⎡ x11 ⎢y ⎥ ⎢x Y = ⎢ 2 ⎥ = X T W = ⎢ 21 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎣ ym ⎦ ⎣ xn1

x12 x22 xn 2

x1m ⎤ x2 m ⎥⎥ ⎥ ⎥ xnm ⎦

T

⎡ w1 ⎤ ⎢w ⎥ ⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎣ wn ⎦

(9)

Multivariate time series data fusion based on K-means clustering is actually a description of the same theme, meanwhile the various variables fusion of different time, so each cluster center can be regarded as a "sensor", and then selecting a comprehensive weighted vector W T = [ w1 w2 wn ] to every cluster centre with weighted fusion,

Y = W T K , Y = [ y1 y 2

y m ] ,which the fusion result is multi-variable vector.

3.2 Energy Consumption Index Extraction of Ethylene Device Currently the common method to show the energy consumption [8] level of ethylene device is converting the energy consumption relative parameters of the fuel energy,

Data Fusion-Based Extraction Method of Energy Consumption Index

89

steam, water, electricity units of measurement unification to GJ, and then sum (SEC = fuel consumption + steam consumption + water consumption + electricity consumption) , although the method can reflect the energy consumption level of the devices in some level, yet only considering the fuel and dynamical variables, without considering other relative parameters of ethylene device energy consumption, such as raw materials(light diesel oil, naphtha and etc), ethylene production, ethylene yield and device loading rates and so on. So it can’t be used to reveal the same device using different raw materials or different procedures of the impact on energy consumption of ethylene device. Because there are certain restrictions in the process of actual using, it is necessary to use time series data of all relative variables of the ethylene device energy consumption to synthesized a virtual benchmarking as the energy efficiency Benchmarking, so that through the “Benchmark” we can get more information about affecting device energy consumption level, it can reveal the real reason for the impact of device energy consumption. The ethylene device [9] in China, no matter in size or the use of the process technology route, there are great differences. In order to make the fusion results more pointed, and improve the availability of the fusion results, it is essential to classify the same scale of different device data with different technology. Firstly, using k-means clustering algorithm to cluster the actual production of ethylene, (C1 , C 2 , , C p . p = 6) , various types of cluster centers are obtained in Table 1. Table 1. Various cluster center of device scale Category

I class

II class

III class

IVclass

V class

VI class

(10 thousand ton /year )

554153

992238

403411

147575

796931

219512

Approximate

55

100

40

15

80

22

35 30 e n e 25 l y 20 h t e t 15 / J G 10

Fuel Steam Water electricity SEC

5 0

9 9 9 1

0 0 0 2

1 0 0 2

2 0 0 2

3 4 0 0 0 0 2 2 Year

5 0 0 2

6 0 0 2

7 0 0 2

8 0 0 2

I W

Fig. 1. The comparisons of indexes every year

Then fusing the data of all kinds of different devices, the results of an industry can be obtained in a certain time. At last the results are synthesized for each category to get an annual energy consumption parameters index of various categories and whole industry (WI). The results of all the categories and whole industry in the nearly ten years are synthesized, the energy consumption parameters of ethylene industry can be obtained in Table 2 and Table 3.

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Time 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 WI

Fuel 23.58411 23.31697 23.32371 23.41733 22.42492 22.33542 21.86399 22.14065 21.92217 21.25513 22.60862

Steam 3.291627 3.680596 2.925814 2.112513 2.188256 2.809711 2.501022 1.6106 3.94892 3.59662 2.737578

Water 2.361319 2.160454 2.296988 2.332972 2.358964 2.195339 2.261988 2.104972 1.950608 2.24307 2.243674

electricity 1.495922 1.490465 1.277374 1.124713 1.207989 1.336623 1.367917 1.394879 1.379317 1.241479 1.310054

SEC 31.72016 32.23439 31.22217 29.35906 28.70127 29.16086 28.50626 28.18823 29.27635 28.93197 29.64936

Table 3. The results of various scales and WI (unit: GJ/t ethylene) Scale

total fuel

total steam

total water

electricity

SEC

100

ethylene yield 0.30355

24.96236

1.17122

2.342821

0.835042

26.05036

80

0.308427

22.36089

1.709225

2.931578

0.9795

28.14677

55

0.313079

23.70051

3.075524

2.212257

1.075339

30.77399

40

0.321684

22.35227

1.269662

2.707031

0.981048

27.28043

22

0.326123

21.4851

2.818949

1.700215

1.878129

28.63107

15

0.311631

22.16183

5.232646

1.865978

1.895455

33.10422

WI

0.317792

22.60862

2.737578

2.243674

1.310054

29.64936

Note: The scale is the ethylene annual production scale, the unit is 10thousand ton per year.

35 30 e 25 n e l y 20 h t e 15 t / J 10 G

ethylene yield total fuel total steam total water electricity SEC

5 0 100

80 55 40 22 15 Scale(t theylene/Year)

WI

Fig. 2. The comparisons between various categories and total SEC

Data Fusion-Based Extraction Method of Energy Consumption Index

91

From Table 2 and Figure 1, we can see that the overall SEC industry presents a downward trend, from 1999 to 2008, the range of SEC is 28.18 ~ 32.25GJ / t ethylene, SEC ranges from 28.18 to 29.36GJ / t ethylene since 2002, which was corresponding to the result proposed in the literature [1] that average comprehensive energy consumption is about 26-31GJ / t ethylene around the world until 2006 in average industrial naphtha cracking. This proves the effectiveness of cluster-based multivariate time series data variance fusion method. In addition, from 2002 to 2008, the annual SEC is lower than the industry SEC in the recent ten years, and the results of the energy-related parameters (such as steam, fuel, electricity, etc.) are obtained at the same time. This provides more information for the operations management and decision-making on energy saving and consumption reducing in ethylene industry. From Figure 2 and Table 3, we can see the comparison between ethylene device scale and the ethylene industry of SEC, it is easy to find that the yield of Class 1 and Class IV devices are higher than industry SEC, while the yield of class II, III, V, VI are lower than the industry SEC. It can be seen from the overall that the output of ethylene is inversely proportional to the SEC device, with the constant transformation of the devices and the expansion of the scale, the device of SEC has also been declining. With Table 3, we can further identify the various situations of the energy consumption parameters, and can further identify the real cause of high energy consumption. However, from Figure 2 and Table 3, it can be seen that in section I (the production scale of 550,000 ton) the SEC is to be higher than Class III (the production scale of 400,000 ton) and Class VI (the production scale of 220,000 ton). We find that the main reason for this result is that the scale of production of 550,000 tons equipment device is normally provided by the production scale of 400,000 tons and production scale of 150,000 tons for simple synthetic devices, rather than to expand the production scale of a single device transformation. The SEC of Class III (the production scale of 400,000 tons) is lower than Class V (the scale of production of 80 million tons), the main reason we know that the class III is used Linde technology and ethane SS & W technology equipment, it is of high efficiency, low energy consumption in these two types of ethylene process technology. These results are corresponding to the operation results of actual device in ethylene industry of China.

4 Conclusion This paper is based on data variance fusion of multivariate time series based on clustering analysis, to extract the energy consumption indexes of ethylene industry in China. From the analysis results, it proves the validity of the proposed data fusion method. At the same time, by extracting energy consumption parameter information related to energy consumption of similar devices and the comprehensive analysis in the whole industry, we can obtain the quantitative characteristics and reasonable energy efficiency virtual benchmark. It can provide reliable basis for management and decision-making to improve energy efficiency and energy saving, and it can be further identified the direction of energy saving and consumption reducing to guide the operation and management.

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References 1. Tao, R., Martin, P., Kornelis, B.: Olefins from conventional and heavy feedstocks: Energy use in steam cracking and alternative processes. Energy 31, 425–451 (2006) 2. Wang, J.Q., Zhou, H.Y., Yi, W.: Data fusion based on optimal estimation theory. Applied Mathematics 20, 392–399 (2007) 3. Zhao, Z.G., Wei, Z.: Dynamic measurement of uncertainty based on the best linear theory of fusion. Instrumentation Technology 5, 928–932 (2007) 4. De Vin, L.J., Ng, A.H.C., Oscarsson, J., Andler, S.F.: Information fusion for simulation based decision support in manufacturing. Robotics and Computer-Integrated Manufacturing 22, 429–436 (2006) 5. Eric, G., Sebastien, K.: Logic-based approaches to information fusion. Information Fusion 7, 4–18 (2006) 6. Anthony, H., Weiru, L.: Fusion rules for merging uncertain information. Information Fusion 7, 97–134 (2006) 7. Sun, J.G., Liu, J., Zhao, L.: Clustering Algorithms Research. Journal of Software 19(1), 48–61 (2008) 8. Zong, Y.G.: The Status of China’s Ethylene Industry and its Development. China Petroleum and Chemical Economic Analysis 13, 47–54 (2006) 9. Gielen, D., Taylor, P.: Indicators for industrial energy efficiency in India. Energy 27, 1–8 (2008)

Research on Improved QPSO Algorithm Based on Cooperative Evolution with Two Populations∗ Longhan Cao1,2, Shentao Wang1, Xiaoli Liu1, Rui Dai1, and Mingliang Wu1 1

Key Laboratory of Control Engineering, Chongqing Institute of Communication, Chongqing, 400035 2 Key Laboratory of Manufacture and Test Techniques for Automobile Parts (Chongqing University of Technology), Ministry of Education, Chongqing, 400050 [email protected]

Abstract. This paper presents a Cooperative Evolutionary Quantum-behaved Particle Swarm Optimization (CEQPSO) algorithm with two populations to tackle the shortcomings of the original QPSO algorithm on premature convergence and easily trapping into local extremum. In the proposed CEQPSO algorithm, the QPSO algorithm is used to update individual and global extremum in each population; the operations of absorbing and cooperation are used to exchange and share information between the two populations. The absorbing strategy makes the worse population attracted by the other population with a certain probability, and the cooperation strategy makes the two populations mutually exchange their respective best information. Moreover, when the two populations are trapped into the same optimum value, Cauchy mutation operator is adopted in one population. Four benchmark functions are used to test the performance of the CEQPSO algorithm at a fixed iteration, and the simulation results showed that the proposed algorithm in this paper had a better optimization performance and faster convergence rate than PSO and QPSO algorithms. Keywords: Cooperative Evolution; Quantum-behaved Particle Swarm Optimization; Cauchy Mutation.

1 Introduction PSO (Particle Swarm Optimization) algorithm is a global optimization algorithm based on swarm intelligence[1], which has characteristics of simple realization, strong currency and good optimization performance. At present, it is applied to many fields, such as neural network training, fault diagnosis, chemical system and robot area and so on[2-5]. But the basic PSO algorithm has disadvantages of limited search space, prematurity and easily trapping into local extremum. For these shortcomings of basic PSO algorithm, some improved methods were presented such as self-adapting weight, combining with SA thought or GA thought or chaos thought et al[6-9]. In a word, the ∗

This work is supported by Project of International Science Cooperation of MOST (No.2007DFR10420) and Open fund of Key Laboratory of the Ministry of Education of Manufacture and Test Techniques for Automobile Parts of Chongqing University of Technology (No.2009-10).

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 93–100, 2010. © Springer-Verlag Berlin Heidelberg 2010

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purposes of all these improved algorithms are to keep population’s diversity and to improve local search ability. The QPSO algorithm proposed by Sun et al in 2004 is a new algorithm based on basic PSO, in which all particles have the characteristic of quantum behavior[10]. QPSO algorithm can overcome disadvantages of limited search space and easily trapping into local extremum, so it has higher optimization performance. However, it still has problems of premature convergence and trapping into local extremum. So a cooperative evolutionary QPSO algorithm with two populations is proposed in this paper in order to improve QPSO algorithm’s local search ability.

2 Basic PSO Algorithm PSO algorithm was proposed by American psychologist James Kennedy and electrical engineer Russell Eberhart in 1995. In basic PSO algorithm, suppose that search space is D -dimension, size of population is m . The position and velocity of i -th particle are represented as X i = [ xi1 , xi 2 , ⋅⋅ ⋅ xiD ] and Vi = [vi1 , vi 2 , ⋅⋅⋅viD ] respectively, and X and V are both m ∗ D matrices. Particle’s position X i is substituted in object function to calculate fitness value f ( i ) . Then by evaluating f ( i ) , the individual extremum Pi = [ pi1 , pi 2 , ⋅⋅⋅ piD ] and global extremum Pg = [ pg1 , pg 2 , ⋅⋅⋅ pgD ] are obtained. All particles’ velocities and positions can be updated according to equations (1) and (2):

Vi (t + 1) = wVi (t ) + c1r1[ Pi (t ) − X i (t )] + c2 r2 [ Pg (t ) − X i (t )]

(1)

X i (t + 1) = X i (t ) + Vi (t )

(2)

Where w is inertia weight, c1 and c2 are positive acceleration constants, r1 and r2 are random numbers valued between 0 and 1. From equations (1) and (2), it can be seen that all particles are tending to the optimal value under the attraction of individual and global extremum in every evolutionary process, and this means that in the latter process of PSO algorithm, each particle's velocity will tend to 0, and the positions of all particles will tend to the same point. If the optimal value at this time is a local extremum, particles’ certain trajectory will make the algorithm difficult to escape from the local extremum. Thereby the basic PSO algorithm’s optimization performance will degrade and its latter search efficiency will slow down.

3 Co-evoluionary QPSO Algorithm With Two Popultions 3.1 QPSO Model

In QPS algorithm, the state of particles can be described only by position vector. Suppose that search space is D -dimension, population size is m . The position of i -th particle is represented as X i = [ xi1 , xi 2 , ⋅⋅ ⋅ xiD ] , particle’s individual extremum is Pi = [ pi1 , pi 2 , ⋅⋅⋅ pid , ⋅⋅⋅ piD ] , and the population’s global extremum is Pg = [ pg1 , pg 2 , ⋅⋅⋅ pgd , ⋅⋅⋅ pgD ] . The updating formulas of particles’ positions are shown as follows:

Research on Improved QPSO Algorithm Based on Cooperative Evolution

xid ( k + 1) =

⎧ ⎪ paveid − a⋅ mbestd − xid ⎪ ⎨ ⎪ ⎪ paveid + a⋅ mbestd − xid ⎩

paveid =

⎛

⎞

( k ) ⋅ln ⎜⎜ u1 ⎟⎟

u >0.5

(k )

u ≤0.5

⎝ ⎠ ⎛ ⎞ ⋅ln ⎜⎜ 1 ⎟⎟ ⎝u ⎠

r1 pid + r2 pgd

95

(3)

(4)

r1 + r2 m

mbest = ∑ i =1

Pi

m

(5)

Where u, r1 , r2 are random numbers valued between 0 and 1; k is current iterations; a is a contraction-expansion coefficient. The selection of a is very important as it is in relation to the algorithm’s convergence rate; mbest is an average of individual best positions; pavei is a random position between individual best position Pi and global best position Pg . From the QPSO optimization model, we can find that there is only one control parameter a , so the realization of QPSO algorithm is simpler than basic PSO algorithm. Furthermore, the QPSO algorithm breaks limitation of search space in basic PSO algorithm, and ensures particles can search in the whole feasible solution space, even in a position far away from the point pave , and then find the optimal value better than the current value. 3.2

Co-Evolutionary QPSO Algorithm with Two Populations

Although QPSO algorithm’s global optimization ability is higher than basic PSO algorithm’s, but from formula (3) it can be seen that QPSO algorithm still faces the problems of prematurity and easily trapping into local extremum. When mbest − Pg tends to be 0, that is, when all particles have the tendency to gathering, like basic PSO algorithm, QPSO algorithm will lose the population’s diversity, and if the current global optimal value is local extremum, the algorithm will eventually trap into the local extremum. Therefore, in order to overcome these shortcomings, this paper proposed an improved QPSO algorithm, that is co-evolutionary QPSO algorithm with two populations. This algorithm's primary idea can be separated into two parts: first is to adopt QPSO algorithm to search optimal value in each population and to find out individual 1 2 extrema fi1 , fi 2 , individual best positions Pi1 , Pi 2 , global optimal value fbest , fbest 1 2 and global best positions Pg , Pg respectively; Second is to enhance mutual optimization performance through competition and cooperation between populations. In order to achieve competition and cooperation between populations, the annexing operator and cooperative operator are introduced. The realization of annexing operator is to annex poor performance of population by the other population. Concretely, 1 2 population 2 will be annexed by population 1 if fbest is better than fbest , conversely, 1 2 population 1 will be annexed by population 2 if fbest is worse than fbest . Take for example, population 2 is annexed by population 1, and the annexing strategy is shown in formula (6):

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rik =

( )( ( )

⎧ 1 1 2 ⎪ pgk + rand 0,1 ⋅ pgk − Pik ⎪ ⎪ ⎨ p1 + rand 0,1 ⋅θ ⎪ gk 2 ⎛ ⎞ ⎪ π ⋅⎜⎜ p1gk +θ 2 ⎟⎟ ⎪ ⎝ ⎠ ⎩

( )

)

if

1 f Pi 2

and

(7)

(

( ))

u < exp f ( ri ) − f Pi 2 else

Where u is a random number valued between 0 and 1. From formula (7) we can find that when fitness of ri is better than Pi 2 , Pi 2 is replaced by ri ; when fitness of ri is worse than Pi 2 , Pi 2 is still replaced by ri with a certain probability. Moreover, the closer the two fitnesses, the larger the replaced probability. This replacement strategy includes two operations: ‘exploitation’ and ‘exploration’. Exploitation is to accept better position with the probability 1 and make full use of winner’s information. Exploration is to retain worse (loser’s) information with a certain probability and to do a favour to develop new space. So using the replacement strategy to update population 2 can help to improve the algorithm’s speed of convergence and keep population’s diversity. The realization of cooperation operator: if rand ( 0,1) < pc , in which pc is a pre-set number valued between 0 and 1, two new individuals are generated by cooperation strategy 1, as shown in formulas (8) and (9); Or else two new individuals are generated by strategy 2, as shown in formulas (10) and (11). 2 qk = β ⋅ p1gk + (1 − β ) ⋅ pgk , k = 1, 2,

,D

(8)

2 rk = (1 − β ) ⋅ p1gk + β ⋅ pgk , k = 1, 2,

,D

(9)

Where β is a random number valued between 0 and 1. q = p1g1 ,

(

p1g ( i1 −1) , pgi2 1 , pg2 ( i1 +1) ,

pgi2 2 , p1g ( i2 +1) , p1g ( i2 + 2 ) ,

p1gD

)

(10)

(

pg2 ( i1 −1) , pgi2 1 , p1g ( i1 +1) ,

p1gi2 , pg2 ( i2 +1) , pg2 ( i2 + 2) ,

2 pgD

)

(11)

r = pg21 ,

Where i1 and i2 are pre-set, and 1 < i1 < D , 1 < i2 < D , i1 < i2 .

Research on Improved QPSO Algorithm Based on Cooperative Evolution

97

From the two cooperation strategies it can be seen that the mutual optimization performances are improved through the exchange of information of optimal position between populations. The cooperation strategy 1 is spoken to the whole global optimal position vectors, and the cooperation strategy 2 is spoken to partial dimensions of global optimal position vectors. These cooperation strategies not only realize the exchange and sharing of information, but also keep the populations' diversity. The updating formulas of two populations are shown in (12) and (13).

{

⎧⎪ p1 , p1 , nP1 = ⎨ 1 2 1 ⎪⎩ P

{

pi1−1 , q, pi1+1 , pi1+ 2 ,

⎧⎪ p 2 , p 2 , p 2 , r , p 2j +1 , p 2j + 2 , nP 2 = ⎨ 1 2 j −1 2 ⎪⎩ P

, p1m

}

( )

(12)

( )

(13)

f ( q ) < f pi1 else

, pm2

}

f ( r ) < f p 2j else

In a word, the operations annexing of and cooperation play different roles respectively. The annexing operation makes full use of information of optimal position, and the annexed population run hill-climbing search near by the position with best fitness value, so its role is local searching. The role of the cooperative operation is to improve mutual fitnesses by exchange information, that is to say, a population not only use its own optimal information to search, but also use the other population’s. The implementation of co-evolutionary QPSO algorithm with two populations is described as follows: (1) Initialization of populations, suppose that search space are both D -dimension, size of populations are both m , set contraction-expansion coefficient a , max iterations K , set θ and pc , initializing particles’ positions X 1 and X 2 ( X 1 and X 2 are both m * d matrices) randomly; (2) Making current positions X 1 and X 2 are individual best positions P1 and P 2 , current individual fitnesses are individual extrema F 1 and F 2 . Through evaluation, obtaining two populations’ global best positions p1g , pg2 and global extrema 1 2 fbest , fbest separately; k = 0 ; (3) Updating the two populations’ positions according to formulas (3), (4) and (5); (4) Calculating every particle’s fitness; Obtaining personal best positions p1 , p 2 and individual extrema f 1 , f 2 by evaluating every particle; Obtaining global best 1 2 positions p1g , pg2 and global extrema fbest , fbest by evaluating all particles; (5) Choosing annexing operator or cooperative operator randomly. If annexing operator is chosen, the annexed population is updated based on formulas (6) and (7); If cooperative operator is chosen, two new individuals which contain mutual best information are generated based on formulas (8), (9) or (10), (11), and then determine whether to replace one particle’s best position vector with the new individuals based on formulas (12) (13); (6) Judging the algorithm whether to meet termination conditions, if it does, the algorithm is over, and output the global extremum as a final result; conversely goes back to step (3).

、

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4 Experiments and Simulation This paper adopted 4 optimization functions to evaluate the performance of CEQPSO algorithm in comparision with PSO, PSO combined with simulated annealing thought (SAPSO) and QPSO algorithm. The test functions are shown as follows:

： f = ∑x D

(1) Sphere function

1

i =1

：f

(2) Griewank function

2

i D

2

=1+ ∑ i =1

xi 2

x − ∏ cos ⎛⎜ i ⎞⎟ i⎠ ⎝ 4000 i =1 D

： f = −20 exp(−0.2 ∑ x

(3) Ackley function

n

3

：f

(4) Rosenbrock function

i =1

N −1

4

(

i

2

/ n ) − exp( 1

= ∑ ⎡100 xi +1 − xi 2 ⎢ i =1 ⎣

n

n∑ i =1

cos(2πxi )) + 20 + e

) + ( x − 1) ⎤⎦⎥ 2

2

i

These functions are variable dimension functions and they have the same global optimal value 0. These optimization algorithms run in different dimensions which are 10, 20, 50, 100 respectively, and each algorithm run 10 times independently. The average optimization results are shown in Table 1. The main parameters used in these algorithms are shown as follows: m = 50 ; k = 2000 ; w : 0.9 → 0.4 ; c1 = 2.8 ; c2 = 1.3 ; χ = 0.98 ; a : 1 → 0.2 ; θ = 0.01 ; pc = 0.8 . Table 1. The optimization results of different algorithms for different functions

Dimensions PSO SAPSO QPSO CEQPSO PSO SAPSO QPSO CEQPSO PSO SAPSO QPSO CEQPSO PSO SAPSO QPSO CEQPSO

10

20 50 Sphere function 8.2601e-004 4.3732e-003 2.3317e-001 2.0539e-005 3.6254e-004 1.5965e-002 9.5730e-050 2.4275e-017 4.6106e-009 1.4942e-156 7.5046e-026 6.5795e-011 Griewank function 2.4662e-003 2.9255e-003 9.5480e-002 1.5353e-003 1.0353e-003 1.1498e-002 1.2306e-003 1.9702e-004 1.2501e-002 2.4662e-004 2.4676e-005 2.4662e-003 Ackley function 1.9388e-005 3.7743e-002 4.8346e-001 1.4625e-006 5.5061e-004 2.9911e-001 1.4251e-014 7.4341e-013 7.0700e-002 6.5963e-015 6.4837e-014 3.0336e-004 Rosenbrock function 7.0380e+000 1.8329e+001 7.3082e+001 4.0534e+000 1.7441e+001 5.8643e+001 5.0909e+000 1.7784e+001 6.0619e+001 2.3698e+000 1.4307e+001 4.7354e+001

100 1.4933e+000 9.4486e-001 4.5002e-001 9.0879e-002 7.8114e-002 3.7127e-002 6.5555e-002 2.9809e-003 1.1005e+000 8.2814e-001 2.9061e-001 1.7268e-002 2.9990e+002 2.1867e+002 2.6161e+003 2.0263e+002

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From Table 1 it can be seen that the CEQPSO algorithm had the best optimization performance for different functions in different dimensions, especially in high dimensions. Take Ackley function for example, the convergence curves lines of these algorithms are shown as follows:

Fig. 1. When n = 10 , The convergence curve lines of different PSO algorithms

Fig. 2. When n = 20 , The convergence curve lines of different PSO algorithms

Fig. 3. When n = 50 , The convergence curve lines of different PSO algorithms

Fig. 4. When n = 100 , The convergence curve lines of different PSO algorithms

From fig.1 to 4, it can be seen that the PSO algorithm easily trapped into local extremum in the later stage of evolution process and it hardly escaped from the local extremum; In the same iteration number, the SAPSO algorithm had a better performance than PSO algorithm due to its mutation occurring at a certain probability during the process of temperature drop.And the CEQPSO algorithm proposed in this paper both had best optimization performance. In figure 4, the value of the CEQPSO is considerably lower at step 0, and the reason of this phenomenon was because that the random initial values of particles positions were better than other algorithms .

’

’

5 Conclusions The key to solve problems of prematurity and trapping into local extremum in QPSO is to keep the population's diversity and to avoid all particles trapping into the same local extremum. The CEQPSO algorithm proposed in this paper uses two populations to

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co-evolve, and the advantages of this algorithm are that the two populations not only make full use of its own information, but also utilize optimal information of the other population. The simulation results showed that the convergence rate of CEQPSO algorithm is quicker, and the optimization performance is better in solving different dimension of optimization problems.

Reference 1. Kennedy, J., Eberhart, R.: Particle Swarm Optimization. In: Proceedings of IEEE International Conference on Neural Networks, pp. 1942–1948 (1995) 2. Wei, G., Hai, Z., Jiuqiang, X., Song., C.: A Dynamic Mutation PSO algorithm and its application in the Neural Networks. In: First International Conference on Intelligent Networks and Intelligent Systems, ICINIS 2008, vol. 11, pp. 103–106 (2008) 3. Pan, H.X., Ma, Q.F., Wei, X.Y.: Research on Fault Diagnosis of Gearbox Based on Particle Swarm Optimization Algorithm. J. Journal of Donghua University (English Edition) 23(6), 32–37 (2006) 4. Ourique, C.O., Biscaia, J.E.C., Pinto, J.C.: The use of particle swarm optimization for dynamical analysis in chemical processes. J. Computers & Chemical Engineering 26(12), 1783–1793 (2002) 5. Pugh, J., Martinoli, A.: Multi-robot learning with particle swarm optimization. In: AAMAS 2006: Proceedings of the Fifth International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 441–448. ACM Press, New York (2006) 6. Chen, D., Wang, G.F., Chen, Z.Y., et al.: A method of self-adaptive inertia weight for PSO. In: 2008 International Conference on Computer Science and Software Engineering, pp. 1195–1197 (2008) 7. Fu, A., Lei, X.J., Lei, X.X.: The Aircraft Departure Scheduling Based on Particle Swarm Optimization Combined with Simulated Annealing Algorithm. In: IEEE Congress on Evolutionary Computation, CEC 2008, vol. 6, pp. 1393–1398 (2008) 8. Du, S.Q., Li, W., Cao, K.: A Learning Algorithm of Artificial Neural Network Based on GA-PSO. In: Proceedings of the 6th World Congress on Intelligent Control and Automation, vol. 6, pp. 3633–3636 (2006) 9. Zhang, H., Shen, J., Zhang, T.N.: An Improved Chaotic Particle Swarm Optimization and Its Application in Investment. In: 2008 International Symposium on Computational Intelligence and Design, vol. 12, pp. 124–128 (2008) 10. Jun, S., Bin, F., Xu, W.: Particle swarm optimization with particles having quantum behavior. In: USA: Proceedings of 2004 Congress on Evolutionary Computation, pp. 321–325 (2004)

Optimum Distribution of Resources Based on Particle Swarm Optimization and Complex Network Theory∗ Li-lan Liu, Zhi-song Shu, Xue-hua Sun, and Tao Yu Shanghai Enhanced Laboratory of Manufacturing Automation And Robotics, Shanghai University, Shanghai, 200072, China [email protected]

Abstract. The multi-project allocation with constrained resources problems is quite common in manufacturing industry. While relationship and data in enterprise has become complex and bulky along with the leaping development, this makes it far beyond the human experience to optimize the management. Particle Swarm Optimization (PSO) algorithm is then introduced to optimize resources allocation to products. Due to the deficiency of PSO dealing with large scale network, Complex Network theory, good at statistics but not optimization, is firstly introduced to simulate and help analyze the Collaborative Manufacturing Resource network (CMRN) as a complementation. Finally, an optimization is successfully applied to the network with the results presented. Further, these methods could be used for similar researches which integrate PSO with complex network theory. Keywords: Resource constrained projects, Collaborative Manufacturing Resource network, Complex network theory, Particle Swarm Optimization, Optimum distribution.

1 Introduction Manufacturing industry is among the most important industry in China. A manufacturing project consists of many activities is always requiring many kinds of resources such as labors, energy and equipments. However, since the manufacturing engineering are managed through experience alone. Their management of allocation is usually inefficient and ineffective [1, 2]. The waste and shortage of resource both happen quite frequently since the resource are allocated to each project improperly, thus, how to improve the resource management by reasonable resource-project allocation is an important economic issue for manufacturing industry. The most common problem in the allocation of resources is the resource attribution of several sets of identical equipments. Any equipment of the same resource set is capable for the same manufacturing mission. Obviously, such distribution problem, with the constraints relaxed and the optimization space expanded than usual ones, is actually the most realistic and common resource allocation problem [3]. And in such a ∗

Supported by NSFC (No. 50805089) and Shanghai Science and Technology Committee(NO. 09DZ1122502).

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 101–109, 2010. © Springer-Verlag Berlin Heidelberg 2010

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static multi-product resource allocation problem, we assume all resources are allocated to several products at the same time in form of a collaborative network without taking the schedule into consideration. In fact, since the manufacturing industry has leapt forward in recent years, the quantity of resources in enterprise has increased sharply to turn their relationship into a complex and bulky network showing full of uncertainty, which has make it far beyond the human experience to make an excellent management. Complex network theory can be used to complete large-scale network data statistics and analysis, and research on common characteristics and the universal processing methods of a variety of different complex systems with the help of graph theory and physical theory, including the analysis of the omnibus feature to simulate the generation of networks. However, complex network theory could do help with dealing with large scale networks, but it’s not good at making an optimization after a statistic analysis, while there was no research to bring in any optimization algorithm to achieve an optimization based on a large scale network base on complex network theory. The Particle Swarm Optimization (PSO), a popular algorithm by simulating flock flying bionics[4-5], have achieved many good results [6] in every aspects of optimization problems on continuous multi-dimensional space for its simply calculation, robustness and so on. That’s why it’s hot to apply PSO more than Ant Colony or Genetic algorithm to solve the resource allocation optimization currently [7], with many researches such as shop scheduling on limited resources [7-9] has been done. But the deadly deficiency of these above is the limited efficiency while dealing with large scale data, not to mention the Genetic algorithm[10] on regional water resource allocation, Ant Colony algorithm[11] on resource optimization allocation or any else. No research has succeeded in making an optimization on large scale, realistic allocation problem with solely algorithm. So in this article PSO and Complex network theory may be both brought in for complementary advantages. Firstly an allocation model of constrained resources in multi-project is built and analyzed with the complex network theory, and then an optimization on large scale network generalized by the allocation model is done with the PSO algorithm to ultimately achieve the follows: Firstly the Optimum allocation of resources ensures that every resource is under a normal load, that is, by making full play to its performance, maintaining high utilization, and precluding the phenomenon of producing cheap products with expensive equipments, finally try to achieve the maximum benefit while using the limited resources. Secondly, it is also important to ensure the entire production network keeping normal in the whole production process. So it does be important to enhance the robustness of production networks to stop the whole chain collapsing in a chain reaction just due to some certain devices broke down in the beginning. Fortunately the robustness of production network could also be analyzed by using complex network theory to study the degree distribution.

2 Complex Network Theory and CMRN Based on BA Model It’s hot to do research of the complex network since Watts and Strogatz proposed the small-world network model [12] in 1998 while Albert and Barabasi published their

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research findings of the scale-free network in the next year [13], henceforth, in order to reveal the topology, formation and evolution mechanism of complex network, there has been a great deal of empirical researches identifying the common statistical properties of all sorts of complex systems. Then based on the plenitude of analysis of the evolution of the network, modeling is done to validate its rationality by simulating the dynamic nature in a larger amount of data, comparing with the empirical results and analyzing the dynamic mechanism leading to such common elaborate. All above is done to finally make a deeper understanding of complex systems. There are sectors of the business enterprises, and all manner of resources belong or detached to the business in Collaborative network. With the aim to directly use the resources belong to some other enterprises or to share their own idle resources, more and more companies have joined in the network. Obviously the idle resources can be quickly allocated to the manufacturing enterprises which is lack of to fulfill their project through the establishment of this collaboration subnet, and then numerous collaboration sub-networks make interconnections to form a huge network namely Collaborative Manufacturing Resource network (CMRN). So as can be seen there are three main parts of the CMRN: the collaboration activities constitute the upper node, sets of resources constitute the lower node and the relationship between resources and collaborative activities. The collaborative activities refer to projects in the enterprises joined in the network. As has been illustrated above, enterprises join in the network due to their specific resource requirements, or some kind of idle resources taking intangible losses. Resource nodes is another important component of the CMRN, however, the resource concerned in CMRN is much more complex than in other networks. Such as manufacturing equipment, geographical location, knowledge and skills of employees, management, corporate structure, culture, control systems, brand and so on. The relationship between the participants is the way how they get linked with each other, ignoring the owner of every resource node, the collaboration activities and the resources needed will be connected, and here the same resource nodes would be sorted into the same set, then we may get the structure of CMRN as figure 1 below:

Fig. 1. Bi-particle graph of CMRN

As shown in figure 1, node b is used more than once, that is to say there are at least two resources (used by the 1st and 2nd project and maybe some more existed ones were idle) in set b in the network, and the model based on BA model we build to simulate the actual case is as below:

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1. There would be continuous increasing new projects in network with 3-5 resources adding to the network to complete. The value of each project will be set randomly between 50 and 70; 2. When a new project is setup, the needed resource will either be in a new set for possibility p or else a randomly copy of anyone in some existed set; 3. When the resource is a copy of exited one, it will be a preferential chosen proportional to the size of each set. Also the resource belong to the same set will be set at the same cost randomly between 20 and 30; 4. The real resource will be randomly placed in a two-dimensional space size of 100 by 100; 5. Randomly add some detached resources as a preferential attach to the existed sets. So after several times of the simulating iteration, there would be 200 projects, 1000 resources belong to 237 sets, 912 of the 1000 resources are actually used. Figure 2 is the degree distribution of the network:

Fig. 2. The degree distribution of the network

This is in truly an summarize of the characteristic of network structure: this is typically a scale-free network, with characteristics of scale-free network such as being robust to randomly break down but fragile to the break down on the core equipment and so on, this may provide some advice on how to enforce the robustness of CMRN. Now return to the point, there are two facts may lead to unnecessary cost in total: 1. The resources belong to the same project are far located from each other, which may lead to an extra cost of transportation, time and labor, so the first goal is to minimize

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range sum of all projects, by interchanging their access of resources so as to make their subordinate resources more close to each other, namely the adaptive value F1: (1)

2. It does be unreasonable to allocate costly resources to manufacturing valueless product, so the second goal is to ensure there is no such deployment, we weigh the rationality by comparing the value of product to the total cost of subordinate resources, namely the adaptive value F2: (2) In order to simplify this Multi-objective Optimization into a Single-objective Optimization, we add F1 and F2 together proportionally, that is to set the final adaptive value F as below: (3) Obviously, there are plenty of improper distribution (allocation) of resources existed in this network, as listed above, the resources belong to the project are in a great distance, or some high costly resource are assigned to manufactory valueless product. However, complex network was only used to make a statistic analysis such as robustness but no more further studied such as to make any optimization. The next step in this article is trying to use PSO algorithm to make an optimization on the network.

3 PSO Algorithm and Application in Resource Distribution Optimization Particle Swarm Optimization, a new swarm intelligence algorithm, originates from the investigation of the bird swarm preying behavior [14]. It is an optimization technology based on iteration. System would be initialized into a group of random solutions and the optimization value is searched by iteration. PSO algorithm keeps searching by continuous update its particles and swarm properties at each step. And each particle in the M-dimensional space is actually a solution to the optimization problem keeping being dynamically affected by its own and group experience [15], means to modify its speed and direction according to the best record of its own (pbest) and the swarm (gbest). The space location of particle i could be described as Xi(t) ={xi1(t),xi2(t),…,xiM(t)} , sameness the space velocity of particle i could be described as Vi(t) = {vi1(t),vi2(t),…,viM(t)} . So the position and velocity could be updated as below: Vi(t) =w(t)Vi(t-1) +c1r1(XLi-Xi(t-1)) +c2r2(XG-Xi(t-1)) . Xi(t) =Vi(t) +Xi(t-1) .

(4) (5)

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The parameter i in the formula varies from 1 to p, p is the size of the swarm. The parameter t in the formula varies from 1 to T, T is the number of the iteration to update the state. X ={ xLi1,xLi2,…,xLiM } is the best recorded location of particle i , XG= { xG1,xG2,…,xGM } is the best recorded location of the swarm. If w 0 in formula(4) then the velocity is only depends on the best record of its own(pbest) and group(gbest), when a particle gets to the best recorded location in overall, it may stay still while the other moves towards it, this may end the algorithm as an local optimum. If c1 0, this means the particle is a social-only one without a self-cognitive, this may also end the algorithm as an local optimum. While if c2 =0, this is even more horrible, all particle are cognition-only and searching the solution separately without any social information sharing, might hardly obtain a solution. So c1 and c2 are used to scale the effect of its own experience and group experience, low value allow particles lingering around the target area before being pulled back, and the high value may lead particles to a sudden rush towards or over the target area. In this article we’ll set c1=c2 =2. While r1 and r2 keeps varying randomly between 0 and 1. While w keeps decreasing smoothly from 0.9 at the beginning to 0.4 at the end. Generally, the solution procedure of optimization in this article is down below as figure 3:

Fig. 3. The solution procedures of optimization

However, the particle location and velocity are expressed as continuous parameter in the traditional particle swarm algorithm; so the application of the particle swarm algorithm in discrete combinatorial optimization problems is greatly limited by using the common location-speed calculation model in continuous real number field. To solve this problem, the encoding method firstly proposed by [9] will be improved, which may successfully applied PSO for continuous optimization problems to the discrete particle swarm optimization, make it feasible to be implemented easily with only a simple operation,

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For example, now here is a resource constrained projects including 3 products and 10 resources, as persumed each product requires 3 resources to complete, Table 1 is the expression form of a resolution. As can be seen in this table, the quantity of dimension is the quantity of resources, that is, 10. The 3 elements in minimum position are allocated to the 1st product, and the less three minimum ones are allocated to the 2nd product and so on. Then by get a position serials X=(7,6,3,9,1,2,8,4,5,10), from which a product-resource array (3213113220) might be generalized. obviously the second 3 in the array means the 4th resource is allocated to the 3rd product, and the third 2 means the 9th resource is allocated to the 2nd product, also, the 0 in the table means no product requires the 10th resource. Table 1. Expression form of coding for particles

id of dimensions

1

2

3

4

5

6

7

8

9

10

Position of particles

2.40

1.71

-0.89

3.10

-2.34

-1.20

2.45

0.26

1.20

3.89

serials of resources

7

6

3

9

1

2

8

4

5

10

product of allocation

3

2

1

3

1

1

3

2

2

0

This expression form successfully correspond each dimensional element to an allocation of resource to the product. That is to mean, a suggestion of allocation may be obtained from the position of each particle.

4 Simulated Optimization of CMRN Based on PSO Then according to the simulated network in the second chapter, we received an initial solution of allocation, that is, an initial product-resource allocation as below in Table 2. A particle maybe positioned by make a conversion using the method in chapter 3; Table 2. A realistic allocation

Product identifier 1 2 3 198 199 200

Resources identifier(number in bracket is the set id of the resource) (158)390 (32)320 (77)241

(6)560 (76)16 (1)220 (75)446 (23)529 (1)35 (1)287 (1)373 …… (1)191 (65)64 (1)666 (1)76 (34)17 (1)888 (37)763 (1)173 (74)565 (3)298 200 products with 912 of 1000 resources in 212 sets

(85)305

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Then a particle cluster including other 9 particles will be randomly initialized on the basis of this solution for both position and velocity. And each adaptive value F of 10 particles before iteration is calculated with formula 3 as follow: F1=2281.2253, F2=2274.6416, F3=2222.2522, F4=2241.592, F5=2236.763, F6=2234.2651, F7=2268.665, F8=2229.128, F9=2261.8296, F10=2281.375, so the best location is the 3rd particle, so gbest = F3=2222.2522, and after iteration for 1000 times, we may get follows in Table 3: Table 3. The final optimized allocation

Product identifier 1 2 3 198 199 200

Resources identifier(number in bracket is the set id of the resource) (31)4 (39)20 (4)72

(1)8 (21)10 (1)22 (1)26 (6)52 (82)81 (1)92 (1)89 …… (19)131 (1)133 (23)135 (137)159 (1)147 (27)148 (188)149 (36)161 (1)184 (22)189 200 products with 912 of 1000 resources in 212 sets

(21)162

And each adaptive value F of 10 particles after iteration is calculated with formula 3 as follow: F1=F2=F3=F4=F5=F6=F7=F8=F9=F10=2045.7863, as can be seen from the comparison, all particles have been convergent into the same location, obviously gbest = 2045.7863. We’ve made an optimization to the allocation, though not so marked.

5 Conclusions and Prospect The multi-project allocation with resource constrained problem is popular in manufacturing engineering. Its optimization can improve the productivity of manufacturing industry greatly It is in the first instance that Complex Network theory is tried to be brought in to combine with PSO to fix an optimization on large scale network. Particle Swarm Optimization, a new swarm intelligence algorithm, is computationally suitable for treating optimizing problems. However, the efficiency of PSO may decreases sharply as a geometric progression when quantity of nodes increases. As in this article, there do be more than 1000 resources to allocate in the network; while most realistic collaboration networks contain far more than 10000 nodes. So it does be a problem while dealing with a common, realistic and large scale network. Complex network theory is mostly used to complete large-scale network data statistics and analysis, which is to mean we could also improve our PSO algorithm by taking into Complex network theory. That’s why the model is firstly built to generalize a network of real multiproject with constrained resources, and then an optimization would successfully be done to the allocation with PSO algorithm, though the effect is not so remarkable with

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only 10% saved after optimization. Also we obtain a suggestion that the key to enforce the robustness of network is to only pay more attention to the core resources. We’ve succeeded in fixing an optimization on large scale network by combine PSO and Complex network theory in this paper. However, this is only the beginning of our research, and our finding has laid down a very good basis for a further research to make PSO far more efficient for large scale network optimization by bringing in more algorithm of Complex network theory in future.

References 1. Li, H., Love, P.E.D., Gunasekaran, A.: A conceptual approach to modeling the procurement process of construction using petri-nets. Journal of Intelligent Manufacturing 10, 347–353 (1999) 2. Gupta, S.: The effect of bid rigging on prices: a study of the highway construction industry. Review of Industrial Organization 19, 453–467 (2001) 3. Wang, F.R., Xu, W.W., Xu, H.F.: Solving Nonstandard Job-Shop Scheduling Problem by Efficiency Scheduling Algorithm. J. Computer Integrated Manufacturing Systems 7(7), 12–15 (2001) 4. Kennedy, J., Eberhart, R.: Particle Swarm Optimization. In: Proc. IEEE Int. Conf. on Neural Networks, pp. 1942–1948 (1942) 5. Gavish, B., Pirkul, H.: Algorithms formulti-resource generalized assignment problem. J. Management Science 37(6), 695–713 (1991) 6. Eberhart, R.C., Shi, Y.: Tracking and optimizing dynamic systemswith particle swarms. In: Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2001), pp. 94–97. IEEE, Seoul (2001) 7. Cheng, X.M.: Research on Multi-mode Resource Constrained Project Scheduling Problem Base on Particle Swarm Optimization. D. Hefei University of Technology (2007) 8. Yang, Z.: Solving Robust Flow-Shop Scheduling Problems with Uncertain Processing Times Based on Hybrid Particle Swarm Optimization Algorithm. D. Shangdong University (2008) 9. Chang, H.J.: Research and application of PSO algorithm on shop scheduling. Qingdao University (2008) 10. Wu, A.H.: The Multi-Object Ant-Genetic Algorithm and its Application in Regional Water Resource Allocation. D. Hunan University, Hunan (2008) 11. Li, J., Hu, W.B.: Research on the System of Resource Optimization Allocation Based on Ant Colony Algorithm. J. Journal of zhongyuan university of technology, 06-0008-05 (2008) 12. Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘Small world’ networks. J. Nature 393(6684), 440–442 (1998) 13. Barabási, A.L., Albert, R.: Emergence of scaling in random networks. J. Science 286(5439), 509–512 (1999) 14. Kennedy, J., Eberhart, R.: Particle Swarm Optimization. In: Proc. IEEE Int. Conf. on Neural Networks, pp. 1942–1948 (1995) 15. Bell, C.E., Han, J.: A new heuristic solution method in resource- constrained project scheduling. J. Naval Research Logistics 38, 315–331 (1991) 16. Carlos, A.C.: Handling Multiple Objectives With Particle Swarm Optimization. IEEE Transactions on Evolutionary Computation 8(3), 264–280 (2004)

The Model of Rainfall Forecasting by Support Vector Regression Based on Particle Swarm Optimization Algorithms Shian Zhao1 and Lingzhi Wang2 1

2

Department of Mathematics and Computer Science, Baise University Baise, 533000, Guangxi [email protected] Department of Mathematics and Computer Science, Liuzhou Teachers College Liuzhou, 545004, Guangxi, China [email protected]

Abstract. Accurate forecasting of rainfall has been one of the most important issues in hydrological research. In this paper, a novel neural network technique, support vector regression (SVR), to monthly rainfall forecasting. The aim of this study is to examine the feasibility of SVR in monthly rainfall forecasting by comparing it with back–propagation neural networks (BPNN) and the autoregressive integrated moving average (ARIMA) model. This study proposes a novel approach, known as particle swarm optimization (PSO) algorithms, which searches for SVR’s optimal parameters, and then adopts the optimal parameters to construct the SVR models. The monthly rainfall in Guangxi of China during 1985–2001 were employed as the data set. The experimental results demonstrate that SVR outperforms the BPNN and ARIMA models based on the normalized mean square error (NMSE) and mean absolute percentage error (MAPE). Keywords: Particle Swarm Optimization, Neural Network, Support Vector Regression.

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Introduction

Accurate rainfall forecasting is regarded as a challenging task of weather prediction, special in severe weather forecasting, which is important for many catchment management applications, in particular for flood warning systems [1,2]. The variability of rainfall in space and time, however, renders quantitative forecasting of rainfall extremely difficult by traditional statistical methods, such as multiple regression, exponential smoothing, autoregressive integrated moving average, etc [3,4]. The main reason is that the system of rainfall and its distribution in the temporal and spatial dimensions depends on many variables, such as pressure, temperature, and wind speed and direction. Due to the complexity of the atmospheric process by which rainfall is generated and the lack of available K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 110–119, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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data on the necessary temporal and spatial scales, it is not feasible generally to forecast rainfall using a physically based process model [5,6]. There have been many studies using artificial neural networks (ANN) in this area. A large number of successful application have shown that ANN can be a very tool for rainfall forecasting modeling [1,3,7,8]. Some of these studies, however, showed that ANN had some limitations in forecasting rainfall because rainfall data has tremendous noise and complex dimensionality [9,10]. ANN often exhibits inconsistent and unpredictable performance on noisy data. However, backpropagation (BP) neural network, the most popular neural network model, suffers from difficulty in selecting a large number of controlling parameters which include relevant input variables, hidden layer size, learning rate and momentum term. Recently, support vector machine (SVM), a novel neural network algorithm, was developed by Vapnik and his colleagues [11]. Many traditional neural network models had implemented the empirical risk minimization principle, SVM implements the structural risk minimization principle [12,13]. Additionally, the solution of SVM may be global optimum based on quadratic programming problems, while other neural network models may tend to fall into a local optimal solution. Originally, SVM has been presented to solve pattern recognition problems. However, with the introduction of Vapniks ε-insensitive loss function, SVM has been developed to solve nonlinear regression estimation problems, such as new techniques known as support vector regression (SVR) [14,15], which have been shown to exhibit excellent performance. At present, SVR has been emerging as an alternative and powerful technique to solve the nonlinear regression problem. It has achieved great success in both academic and industrial platforms due to its many attractive features and promising generalization performance. When using SVR, the main problem is confronted: how to set the best kernel paraments. In application SVR, proper parameters setting can improve the SVR regression accuracy. However, inappropriate parameters in SVR lead to over–fitting or under–fitting. Different parameter settings can cause significant differences in performance. Therefore, selecting optimal hyper-parameter is an important step in SVR design [16,17,18]. In this paper, a novel method of rainfall forecasting is presented by Support Vector Regression Based on Particle Swarm Optimization Algorithms. This approach is the use of the Particle Swarm Optimization (PSO) algorithms to determine free parameters of SVR, known as PSO–SVR, which optimizes all SVR’s parameters simultaneously from the training data. Then, the monthly rainfall data in Guangxi of China, was is used as a case study for the development of rainfall forecasting model. The proposed approach was compared with the back– propagation neural networks (BPNN) and traditional time series models, such as ARIMA, so as to show that the SVR model is substantially featured with an excellent forecasting capacity. The rest of the paper is organized as follows. Section 2 describes the model building process. For further illustration, this work employs the method to set up a prediction model for monthly rainfall forecasting in Section 3. Finally, some concluding remarks are drawn in Section 4.

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The Building Process of the PSO-SVR Model

SVR have three significant features. Firstly, it can model nonlinear relationships. Secondly, the SVR training process is equivalent to solving linearly constrained quadratic programming problems and the SVR embedded solution meaning is unique, optimal and unlikely to generate local minima. Finally, it chooses only the necessary data points to solve the regression function, which results in the sparseness of solution. 2.1

Support Vector Regression N

Suppose we are given training data (xi , di )i=1 , where xi ∈ Rn is the input vector; di is the output value and n is the total number of data dimension. The modelling aim is to identify a regression function, y = f (x), that accurately predicts the outputs di corresponding to a new set of input–output examples, (xi , di ). The linear regression function (in the feature space) is described as follows: f (x) = ωφ(x) + b φ : Rn → F, ω ∈ F

(1)

where ω and b are coefficients; φ(x) denotes the high dimensional feature space, which is nonlinearly mapped from the input space x. Therefore, the linear regression in the high-dimensional feature space responds to nonlinear regression in the low-dimension input space, disregarding the inner product computation between ω and φ(x) in the high-dimensional feature space. The coefficients ω and b can thus be estimated by minimizing the primal problem of SVR as follows: ⎧ N  ⎪ ⎪ minR(ω, ξ, ξ ∗ ) = 12 ω T ω + C (ξ + ξ ∗ ) ⎪ ⎪ ⎨ i=1 (2) s.t. di − f (xi ) ≤ ε + ξ ∗ ⎪ ⎪ f (x ) − d ≤ ε + ξ ⎪ i i ⎪ ⎩ ξ, ξ ∗ ≥ 0, i = 1, 2, · · · , N, ε ≥ 0 where C is the regulator, ξ and ξ ∗ are slack variables that measure the error of the up and down sides, respectively. The ε-insensitive loss function means that if f (xi ) is in the range of di ± ε, no loss is considered. This primal optimization problem is a linearly constrained quadratic programming problem [19], which can be solved by introducing Lagrangian multipliers and applying Karush-KuhnTucker (KKT) conditions to solve its dual problem: ⎧ N N   ⎪ ⎪ minR(α, α∗ ) = di (αi − α∗i ) − ε di (αi + α∗i ) ⎪ ⎪ ⎪ i=1 i=1 ⎪ ⎪ N  N ⎪  ⎨ − 12 (αi − α∗i )(αj − α∗j )K(xi , xj ) (3) i=1 j=1 ⎪ ⎪ N ⎪  ⎪ ⎪ (αi − α∗i ) = 0 s.t. ⎪ ⎪ ⎪ i=1 ⎩ 0 ≤ α, α∗ ≥ C, i = 1, 2, · · · , N

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where αi and α∗i are the Lagrangian multipliers associated with the constraints, the term K(xi , xj ) is defined as kernel function, where the value of kernel function equals the inner product of two vectors xi and xj in the feature space φ(xi ) and φ(xj ), meaning that K(xi , xj ) = φ(xi ) × φ(xj ). The typical examples of kernel function are the polynominal kernel and the Gaussian kernel. In this paper, the Gaussian kernel function is selected for SVR model as follows: K(xi , xj ) = exp{− 2.2

x − xi 2 } σ2

(4)

SVR Parameters

Despite its superior features, SVR is limited in academic research and industrial applications, because the user must define various parameters appropriately. An excessively large value for parameters in SVR lead to over-fitting or while a disproportionately small value leads to under-fitting [16,17]. Different parameter settings can cause significant differences in performance. Therefore, selecting the optimal hyper-parameter is an important step in SVR design [18,20]. The parameters have three as follows: (1) Regularization parameter C: C determines the trade-off cost between minimizing the training error and minimizing the models complexity. (2) Bandwidth of the kernel function (σ 2 ): σ 2 represents the variance of the Gaussian kernel function. (3) The tube size of e-insensitive loss function (ε): It is equivalent to the approximation accuracy placed on the training data points. SVR generalization performance (estimation accuracy) and efficiency depends on the hyper-parameters (C, ε and kernel parameters σ 2), being set correctly. Therefore, the main issue is to locate the optimal hyper–parameters for a given data set. 2.3

Apply PSO to SVR Parameters

PSO is an emerging population-based meta-heuristic that simulates social behavior such as birds flocking to a promising position to achieve precise objectives in a multidimensional space. Each individual in PSO is assigned with a randomized velocity according to its own and its companions’ flying experience. The individuals, called particles, are then flown through hyperspace. Successful applications of PSO in some optimization problems such as function minimization and NNs design, demonstrate its ability of global search [21,22]. The fitness of the training data set is easy to calculate, but is prone to overfitting. In this paper, this problem can be handled by using a cross validation technique, which the training data set is randomly divided into k mutually exclusive subsets of approximately equal size. The parameters of SVR could be given by using k − 1 subsets as the training set. The performance of the parameter set is measured by the RMSE (root mean square error) on the last subset. The above procedure is repeated k times, so that each subset is used once for testing. Averaging the RMSE over the k trials can be computed as follows:

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n

1  RM SE =  (f (xi ) − di )2 n i=1

(5)

where n is the number of training data samples; di is the actual value and f (xi ) is the predicted value. The fitness is defined as follows: 1 (6) RM SE Details of this paper proposed PSO–SVR are described as follows: First, the population of particles is initialized, each particles is represented as (τ1 , τ2 , τ3 ), where τ1 , τ2 and τ3 denote the regularization parameters C, σ 2 and ε, respectively. The initial group is randomly generated, each group having a random position and a random velocity for each dimension. Second, each particles fitness for the SVM is evaluated. The each particles fitness in this study is the regression accuracy. If the fitness is better than the particle’s best fitness, then the position vector is saved for the particle. If the particle’s fitness is better than the global best fitness, then the position vector is saved for the global best. Finally the particle’s velocity and position are updated until the termination condition is satisfied. The basic flow diagram can be shown in Fig.1. f itness2 =

Random ly initialize population position and ve locitie s

Initial value of SVR param e nte rs

Calculate fitne ssvalure by the original datd se ts

Data se ts

Ye s S atis fy s toping criteria?

Ge ne rate ne w SVR param e nte rs

No Update particle be st Update global be st Update particle ve locity Update particle position

Train SVR m ode l

PSO-SVR fore cast

Fig. 1. A Flow Diagram of The Proposed Nonlinear Ensemble Forecasting Model

3

Experiments Analysis

The platform adopted to develop the PP–PSO approach and the SVR–PSO approach is a PC with the following features: Intel Celeron M 1.86 GHz CPU, 1.5 GB RAM, a Windows XP operating system and the Visual C++ 6.0 development environment. Table 1 gives overview of PSO parameter settings.

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Table 1. PSO parameter settings Iteration times Population size The minimum inertia weight The maximum inertia weight The minimum velocity The maximum velocity Learning rate

3.1

100 60 0.1 0.9 0.1 0.9 2.0

Empirical Data

Real–time ground rainfall data have been obtained in monthly rainfall from 1961 to 2008 in Guangxi. Thus the data set contained 576 data points in time series. Method of modeling is one-step ahead prediction, that is, the forecast is only one sample each time and the training samples is an additional one each time on the base of the previous training.average precipitation on. When artificial intelligence technology is applied to the forecasting of time series, the number of input nodes (order of autoregressive terms) critically affects the forecasting performance [23], since the number of input nodes precisely describes the correlation between lagged-time observations and future values. According to the literature [24,25], this paper experimented with a relatively larger number (6) for the order of autoregressive terms. Thus, we can make form couples for 576 observation values to become 570 input patterns. The prior 540 in 570 input patterns were employed for the training sample to model SVR; the other 30 input patterns were employed for testing samples to estimate the forecasting capacity of SVR models. Each data point was scaled by Eq. (10) within the range of (0,1). This scaling for original data points helps to improve the forecasting accuracy: xi =

3.2

xi − min(xi ) · 0.7 + 0.15 max(xi ) − min(xi )

(7)

Performance Evaluation of Model

In order to measure effectiveness of the proposed method, three types of errors, such as, normalized mean squared error (NMSE), the mean absolute percentage error (MAPE) and the Pearson Relative Coefficient (PRC), which have been found in many papers, are also used here. Interested readers can be referred to [12] for more details. 3.3

Parameters Determination of Model

When all the training results satisfy the request of error, all the new factor matrix and the paraments of SVR have been gentled well by learning data. Fig.2

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Fitnaess value

0.7

0.6 average fitness best fitness worst fitness

0.5

0.4

0.3

0.2

0.1

0

1

10

20

30

40

50

60

70

80

90

100

Training

Fig. 2. Fitness Alternation During Training Process

shows the curve of fitness of training samples in the learning stage for SVR–PSO approach. One can see that the maximum, average and the minimum fitness and convergent speed are tending towards stability with increase of iteration number. Thus, the individual at Generation 46 produced optimal parameters, which were C = 19, σ 2 = 0.7821, and ε = 0.0921. These optimal parameter sets were applied to construct the SVR models with the input node number set to 6. According to the literature, BP–NN parameters are obtained by the experimental verification. The results show that the {6 − 8 − 1} topology with learning rate of 0.07 yields the best forecasting result (minimum testing NRSE). Here, {ni − nh − no } represents the number of nodes in the input layer, number of nodes in the hidden layer and number of nodes in the output layer, respectively. The authors used Eviews statistical packages to formulate the ARIMA model. Akaike information criterion (AIC) was used to determine the best model. The model generated from the data set is AR(4). The equation used is presented in Eq. (8): xt = 1 − 0.7558xt−1 − 0.1953xt−2 + 0.05685xt−3 + 0.05175xt−4 3.4

(8)

Analysis of the Results

Fig.3. shows the forecasting results of different models for 60 testing samples, we can see that the forecasting results of PSO–SVR model are best in all models. Table 2 shows that the forecasting performance from three various forecasting models are all appropriate. From Table 2, for the rainfall testing case, the NMSE and MAPE of the PSO– SVR model models is superior to those from the other two models. The minimum values of NMSE and MAPE indicate that the deviations between original values and forecast values are very small. At the same time, from Table 2, the PRC of the PSO–SVR model is the highest in all model. The accurate efficiency of the proposed model is measured as PRC, The higher values of PRC (maximum

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500 Actual PSO−SVR AR BR−NN

Rainfall(mm)

450 400 350 300 250 200 150 100 50 0

5

10

15

20

25

30

35

40

45

50

55

60

Month

Fig. 3. Forecasting of Different Models for Monthly Rainfall Table 2. Comparison of The Forecasting Results from Each Model. Moel

NMSE

MAPE

PRC

PSO-SVR AR BR–NN

0.0697 0.4075 0.4897

0.3426 1.0887 1.5410

0.9820 0.8231 0.7579

value is 1) indicate that the forecasting performance of the PSO–SVR model is effective, which can capture the average change tendency of the cumulative rainfall data.

4

Conclusions

Accurate rainfall forecasting is crucial for a frequent unanticipated flash flood region to avoid life losing and economic loses. This paper applied SVR to the forecasting fields of monthly rainfall time series. To build stable and reliable forecasting models, the parameters of SVR must be specified carefully. This study applied PSO to obtain the optimal parameters of SVR. Thereafter, these optimal parameters were employed to build the actual PSO–SVR forecasting models. In terms of the different forecasting models, empirical results show that the developed model performs the best for monthly rainfall series time on the basis of different criteria. Moreover, the experimental results also suggest that within the forecasting fields of rainfall, the PSO–SVR is typically a reliable forecasting tool, with the forecasting capacity more precise than that of BP–NN.

Acknowledgment The authors would like to express their sincere thanks to the editor and anonymous reviewer’s comments and suggestions for the improvement of this paper.

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This work was supported in part by the Natural Science Foundation of Guangxi under Grant No. 0832092, and in part by the Department of Guangxi Education under Grant No. 200807MS098 and 200707MS061.

References 1. Jiansheng, W., Long, J., Mingzhe, L.: Modeling Meteorological Prediction Using Particle Swarm Optimization and Neural Network Ensemble. In: Wang, J., Yi, Z., ˙ Zurada, J.M., Lu, B.-L., Yin, H. (eds.) ISNN 2006. LNCS, vol. 3973, pp. 1202–1209. Springer, Heidelberg (2006) 2. Nasseri, M., Asghari, K., Abedini, M.J.: Optimized Scenario for Rainfall Forecasting Using Genetic Algorithm Coupled with Artificial Neural Network. Expert Systems with Application 35, 1414–1421 (2008) 3. French, M.N., Krajewski, W.F., Cuykendal, R.R.: Rainfall forecasting in space and time using a neural network. Journal of Hydrology 137, 1–37 (1992) 4. Burlando, P., Rosso, R., Cadavid, L.G., Salas, J.D.: Forecasting of short-term rainfall using ARMA models. Journal of Hydrology 144, 193–221 (1993) 5. Valverde, M.C., Campos Velho, H.F., Ferreira, N.J.: Artificial neural network technique for rainfall forecasting applied to The S¨ o Paulo Region. Journal of Hydrology 301(1-4), 146–162 (2005) 6. Luk, K.G., Ball, J.E., Sharma, A.: An application of artificial neural network for rainfall forecasting. Mathematical and Computer Modeling 33, 683–693 (2001) 7. Lin, G.F., Chen, L.H.: Application of an artificial neural network to typhoon rainfall forecasting. Hydrological Processes 19, 1825–1837 (2005) 8. Luk, K.G., Ball, J.E., Sharma, A.: Study of optimal lag and statistical inputs to artificial neural network for rainfall forecasting. Journal of Hydrology 227, 56–65 (2000) 9. Jiansheng, W., Enhong, C.: A Novel Nonparametric Regression Ensemble for Rainfall Forecasting Using Particle Swarm Optimization Technique Coupled with Artificial Neural Network. In: Yu, W., He, H., Zhang, N. (eds.) ISNN 2009. LNCS, vol. 5553, pp. 49–58. Springer, Heidelberg (2009) 10. Yingni, J.: Prediction of Monthly Mean Daily Diffuse Solar Radiation Using Artificial Neural Networks and Comparison with other Empirical Models. Energy Policy 36, 3833–3837 (2008) 11. Vapnik, V.N.: Statistical learning theory. Wiley, New Yourk (1998) 12. Tay, F.E.H., Cao, L.: Modified support vector machines in financial time series forecasting. Neurocomputing 48(1-4), 847–861 (2002) 13. Vapnik, V., Golowich, S., Smola, A.: Support vector method for function approximation, regression estimation and signal processing. In: Mozer, M., Jordan, M., Petsche, T. (eds.) Advance in Neural Information Processing System, pp. 281–287. MIT, Cambridge (1997) 14. Sch¨ okopf, B., Smola, A.: Learning with Kernels: Support Vector Machines, Regularization, Optimization and Beyond, pp. 465–479. MIT Press, Cambridge (2002) 15. Hastie, T., Tibshirani, R., Friedman, J.H.: The elements of statistical learning: data mining, in- ference, and prediction, pp. 314–318. Springer, Heidelberg (2001) 16. Keerthi, S.S.: Efficient tuning of SVM hyper–parameters using radius/margin bound and iterative algorithms. IEEE Tranaction of the Neural Network 13(5), 1225–1229 (2000)

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17. Duan, K., Keerthi, S., Poo, A.: Evaluation of simple performance measures for tuning SVM hyperparameters. Technical report, National University of Singapore, Singapore (2001) 18. Lin, P.T.: Support vector regression: systematic design and performance analysis. Doctoral Dissertation, Department of Electronic Engineering, National Taiwan University of Science and Technology, Taipei (2001) 19. Sch¨ olkopf, B., Smola, A., Williamson, R.C., Bartlett, P.L.: New support vector algorithms. Neural Computation 5, 1207–1245 (2000) 20. Cristianini, N., Shawe-Taylor, J.: An introduction to support vector machines and other kernel-based learning methods. Cambridge University Press, Cambridge (2002) 21. Lin, S.W., Ying, K.C., Chen, S.C., Lee, Z.J.: Particle swarm optimization for parameter determination and feature selection of support vector machines. Expert Systems with Applications 35, 1817–1824 (2008) 22. Kennedy, J., Eberhart, R.C.: Swarm Intelligence. Morgan Kaufmann Press, San Francisco (2001) 23. Smeral, E., Witt, S.F., Witt, C.A.: Econometric forecasts: Tourism trends to 2000. Annals of Tourism Research 19(3), 450–466 (1992) 24. Box, G.E.P., Jenkins, G.M.: Time series analysis: Forecasting and control. HoldenDay, San Francisco (1976) 25. Zhang, G., Hu, Y.: Neural network forecasting of the British Pound/US Dollar exchange rate. Omega 26(4), 495–506 (1998)

Constraint Multi-objective Automated Synthesis for CMOS Operational Amplifier Jili Tao, Xiaoming Chen, and Yong Zhu Ningbo Institute of Technology, Zhejiang University, Ningbo Zhejiang, China [email protected]

Abstract. The synthesis of CMOS operational amplifier (Op-Amp) can be translated into a constrained multi-objective optimization problem, in which a large number of specifications have to be taken into account, i.e., gain, unity gain-bandwidth (GBW), slew-rate (SR), common-mode rejection ratio (CMRR) and bias conditions. A constraint handling strategy without penalty parameters for multi-objective optimization algorithm is proposed. A standard operational amplifier is then designed, the results show the proposed methodology is very effective and can obtain better specifications than other methods. Keywords: multi-objective evolution algorithm; CMOS Op-Amp; constraint handling.

1 Introduction Op-Amp is of great importance in lots of analog circuit design, such as high-speed A/D, RF communication integrated circuits (IC), LED driver IC, etc. [1]. The acquisition of low-power, high-speed, large open loop gain and high unity gain bandwidth Op-Amp is not an easy task, because some performances are contradictory, e.g., the ascent of high DC gain is at the price of the ascent of dissipation, the decent of speed and slew rate [2]. The use of efficient multiple-objectives optimization algorithms is, therefore, of great importance to the automatic design of Op-Amp. Accuracy, ease of use, generality, robustness, and reasonable run-time are necessary for a circuit synthesis solution to gain acceptance by using optimization methods [3]. Many evolution computing based parameter-level design methods and tools have been published in recent years [4]-[8], where circuit synthesis is translated into single-objective minimization problem and a performance evaluator based on equations or simulators is introduced to solve the optimization problem. The optimization problems of Op-Amp design are always highly constrained, and the constraint handling method is very important. Most reported synthesis methods use penalty functions to handle constraints[9][10], where the results of the methods based on penalty functions are very sensitive to the penalty coefficients, and may not meet the designer’s specifications in many cases. Although several penalty strategies have been developed, there are not general rules for penalty coefficient K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 120–127, 2010. © Springer-Verlag Berlin Heidelberg 2010

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tuning till now. Herein, a constraint handling method without penalty coefficients is proposed in the paper. After constraints handling, the constrained multi-objective optimization problem is transferred into unconstrained one. Many MOEAs have been developed to handle the problem [11]. NSGA-II is a typical MOEA, which is superior to several other representative algorithms [12] and achieved a number of successful applications [13][14]. However, population diversity of NSGA-II remains to be further improved. Hence, the author proposed an elitists maintaining scheme to guarantee the diversity preservation [15], which is introduced to perform cell sizing, i.e., search for transistors sizes, biasing current and compensating capacitance values in order to meet a set of specifications. This methodology is tested in the synthesis of a standard two-stage Op-Amp and our results are compared with other evolution algorithm based method.

2 Multi-objective Optimization for Op-Amp Design In the particular case of Op-Amp optimization design, a large number of specifications must be achieved by the design process. Ideally, all specifications should be included, but, usually, only the most important ones are taken into account. Human design relies on the solution of a set of equations. The main problem comes from the fact that there is not an exact solution when many objectives are included. In this case, Op-Amp design is expressed as the minimization of two objectives (e.g., power consumption and chip area), subject to some constraints (e.g., open loop gain larger than a certain value). They can be formulated as follows:

min subject to

f1 ( x ), f 2 ( x ) g m ( x ) ≥ lm

(1)

h( x ) = 0 xL ≤ x ≤ x H In (1), the objective functions f1(x) is the power consumption and f2(x) is the chip area to be minimized. Vector x corresponds to design variables, xL and xH are their lower and upper bounds, respectively. The vector g(x) corresponds to user-defined specifications, m is the number of specifications. h(x) are the equality constraints. In analog circuit design, the equality constraints mainly refer to Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL) equations. 2.1 Miller Compensated Two- Stage Amplifier

A typical Miller compensated two- stage amplifier, shown in Fig. 1, is chosen to illuminate the design processes. Specifications in most Op-Amp synthesis problems include gain, GBW, SR and CMRR etc.

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Fig. 1. The miller-compensated two-stage amplifier

The design parameters are: widths and lengths of the given transistors, compensation capacitor and bias current. The load capacitance CL is 3pF. The performances for two-stage amplifier can be estimated by the following equations [16]:

f1 ( x ) = (Vdd − Vss )( I bias + I 5 + I 7 )

(2)

8

f 2 ( x ) = a0 + a1cc + a2 ∑ Wi Li i =1

W5 L8 WL I bias , I 7 = 5 7 I bias , α0 gives the fixed area, α0=100, α1 is the ratio of L5W8 L5W7 capacitor area to capacitance, α1=1000, and the constant α2 can take into account wiring in the drain and source area, α2=8. More accurate polynomial formulas for the amplifier die area can be developed, if needed. The specifications are listed as follows [16]: Where I 5 =

The constraints for symmetry and matching are: W1 = W2 , L1 = L2 ,W3 = W4 , L3 = L4 , L5 = L7 = L8

(3)

The open-loop voltage gain is Av =

2Cox (λn + λ p )2

μ p μn

W2W6 L2 L6 I1 I 7

I1 =

I5 2

(4)

The unity gain bandwidth (GBW) is Av ω3 dB =

g m1 Cc

(5)

Constraint Multi-objective Automated Synthesis for CMOS Operational Amplifier

The standard value of the compensation resistor is used, i.e., Rc = 1 g m6

where g m 6 = 2μ p Cox

123

(6)

W6 L6

The slew rate (SR) can be expressed as SR =

I5 Cc

(7)

The common-mode rejection ratio (CMRR) is 2Cox WW μn μ p 1 32 CMRR = (λn + λ p )λ p L1 L3 I 5

(8)

Bias conditions to guarantee transistors in saturation are listed as follows: I1 L3

μn Cox 2W3

(9)

< Vcm,min + Vss − VTP − VTN

I1 L3 I 5 L5 + < Vdd − Vcm,max + VTP μn Cox 2W3 μ P Cox 2W5

(10)

I 7 L6 ≤ Vout ,min − Vss μn Cox 2W6

(11)

I 7 L7 ≤ Vdd − Vout ,max μ p Cox 2W7

(12)

Where Vss=0, Vdd=5V, Vcm,min is the lowest value of common-mode input voltage, Vcm,max is the highest value of common-mode input voltage, Vcm is fixed at 2.5V. Vout,min is the lowest value for the drain voltage of transistor M6, Vout,min=0.5V. Vout,max is the highest value for the drain voltage of transistor M7, Vout,max=4.5V. According to level 1 model, VTP is set to 0.7V, VTN is set to 0.9V, un=600, up=200, Cox=1.73e-5. The requirements are shown in Table 1. Table 1. Specifications and results of proposed method and GA+SPF Specifications DC gain (dB) GBW(MHz) Slew rate(V/us) CMRR(dB) Power(mw)

constraints

≧80 ≧40 ≧30 ≧60

minimize

GA+SPF 80 72 80 74 1.73

Proposed 82 127 80 70 1.59

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2.2 Proposed Constraint Handling Approach

High performance circuits are always highly constrained. Most reported synthesis methods use penalty functions to handle constraints. Since the results are very sensitive to the penalty coefficients, and may not meet the designer’s specifications. Other constraint handling methods according to the following criteria is imposed [17]: (1) Any feasible solution is preferred to any infeasible. (2) Among two feasible solutions, the one having better objective function value is preferred. (3) Among two infeasible solutions, the one having smaller constraint violation is preferred. Although there exist a number of implementations where criteria similar to the above, some of these implementations still need a penalty parameter for each constraint [18], and some need calculate separately with infeasible and feasible solutions [19]. To solve the above problems, the constrained optimization problem is transformed into an unconstrained one by the following formula: f ' ( x ) = f ( x ) + f max ( g ) ⋅ violation

(13)

where violation = ∑ m =1 min(0, g m ( x ) − lm ) / ξ , ξ is a coefficient to guarantee feasible solution superior to infeasible solution when 00 , object A and B are not collided. This problem can be converted to the optimization problem with constraint conditions. We use Collision (A, B) to present the collision between two objects. 0 means that A and B are collided. 1 means that A and B are separated. We use an equation (5), (6) and (7) to describe this situation. ⎧ ⎪0 iff Mind A, B = Min ⎪ Coldect ( A, B ) = ⎨ ⎪ ⎪1 iff Mind A, B = Min ⎩ n

∑α i =1

i =1

n

i =1

i i

j =1

m

n

i =1

j =1

j

yj = 0

(5)

∑ αi xi − ∑ β j y j > 0

i

= 1 and α i ≥ 0

(6)

i

= 1 and β i ≥ 0

(7)

n

∑β

m

∑α x − ∑ β

2.2 Algorithm Description

In this paper, the problem of collision detection is converted to the non-linear programming problem with restricted conditions. The CDAQIG is adopted to solve the problem [6-7].

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2.2.1 CDAQIG Encoding Q-bit as the information storage unit is a two-state quantum system. It is a unit vector which is defined in a two-dimensional complex vector space. The space is composed of standard orthogonal basis { 0 , 1 } . Therefore, it can be in two quantum superposition at the same time. The state can be represented as below.

ϕ =α 0 +β 1

(8)

Where α and β are complex numbers that specify the probability amplitudes of the corresponding sates. The α

2

gives the probability that the Q-bit will be found in the

“0” state, and β gives the probability that the Q-bit will be found in the “1” state. 2

Normalization of the state to unity guarantees.

α + β =1 2

2

(9)

If there is a system of m Q-bits, the system can represent 2m states at the same time. However, in the act of observing a quantum state, it collapses to a single state. A number of different representations can be used to encode the solutions for individual in genetic algorithm. The representations can be classified broadly as: binary, numeric, and symbolic. The quantum genetic algorithm uses a new representation, called a Q-bit, for the probabilistic representation that is based on the concept of Q-bits. This encoding has a better characteristic of population diversity than other representation, hence it can represent linear superposition of states probabilistically. In the CDAQIG algorithm we can directly use quantum bit of probability amplitude to encode. Taking into account the population initialization is encoded randomness and quantum state of probability amplitude should meet the criteria described by formula (9), we use dual chain encoding scheme. A minimum distance equation of quantum to qi represents as follows. ⎡ cos(ti1 ) cos(ti 2 ) ... cos(tin ) ⎤ qi = ⎢ ⎥ ⎣ sin(ti1 ) sin(ti 2 ) ... sin(tin ) ⎦

(10)

Where tij=0.5*pi*rand(); pi is the circumference ratio. Rand()is a random number between 0 and 1. i=1,2,…,n; j=1,2,…m, m is the size of population. n is the number of the Q-bit individual. Each Q-bit’s probability amplitude can be seen as an upper and lower side-by-side gene, each gene chain represents an optimal solution. Therefore, each chromosome represents two optimization solutions of the search space. Where i=1,2,...,m. This encoding method can avoid the randomness of the measurement, and prevents frequent decoding process from binary to decimal. The two solutions can be updated during each iteration. This can enhance the ergodicity of search space in population with fixed size. At the same time this can expand the number of global optimization, and increase probability of global optimization [8].

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2.2.2 Clone Operator The Cloning operator is a random map, and it is induced by the affinity of antibodies. Assumed that the antibody population size is n, and the antibody population A={a1, a2,…, an}, the immune clone operator can be defined as below.

Tc(A)=[Tc(a1), Tc(a2),…, Tc(an)]T

(11)

Where Tc(ai)=Ii*ai, i=1,2,…,n, Ii is a qi dimension row vector, and its element value is 1.qi is the clone size of antibody ai, which means that the antibody realizes the biological double under the antigenic stimulation. qi can be calculated by the following formula. ⎡ ⎤ ⎢ ⎥ f ( a ( k )) i ⎥ , i = 1, 2,..., n qi = Int ⎢ nc × n ⎢ ⎥ f ( a ( k )) ∑ j ⎢ ⎥ = 1 j ⎣ ⎦

(12)

Int is expressed as the integral function, Int(x) represent the smallest integer greater than or equal x. nc is the scale of the cloning of the settings and meet nc>n. The clone size qi can adjust itself adaptively. And this function is based on the relative size of antibody-antigen affinity f(ai(k)) throughout the antibody, which means when the antigen simulates the antibody, the clone size is decided by the influence of the simulation. The clone operator generates multiple images of a single antibody to achieve adaptive expansion of individual space. 2.2.3 Crossover and Mutation Strategy In the quantum-inspired immune algorithm, the immune mutation operator is realized by the rotation gate. The quantum rotation gate updates the Q-bit. This operation makes the population develop to the best individual. The Q-gate is defined by formula (13). Where Δθ is a rotation angle of each Q-bit.

⎡ cos(Δθ ) − sin(Δθ ) ⎤ U (Δθ ) = ⎢ ⎥ ⎣ sin(Δθ ) cos(Δθ ) ⎦

(13)

Q-bit is updated as follows. ⎡cos(Δθ ) − sin(Δθ ) ⎤ ⎡cos(t ) ⎤ ⎡cos(t + Δθ ) ⎤ ⎢ sin(Δθ ) cos(Δθ ) ⎥ ⎢ sin(t ) ⎥ = ⎢ sin(t + Δθ ) ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

(14)

We can see from the formula (14) that this update operation can only change phase of the Q-bit, but can not change the length of the Q-bit. The magnitude of Δθ has an effect on the speed of convergence, but if it is too big, the solution may diverge or converge prematurely to a local optimum. The sign of Δθ determines the direction of convergence. The Δθ can be determined by the following method. Assumed that the α0 β0 is the probability amplitude of the global optimal solution in the current search, α1β1 is the probability amplitude of Q-bit in current solution. Defined A as formula (15).

Quantum Immune Algorithm and Its Application in Collision Detection

A=

α 0 α1 β 0 β1

143

(15)

The direction of Δθ can be determined as follows. If A≠0, the direction is –sign(A), if A=0 the direction can be selected randomly[7]. In order to avoid the premature convergence, the size of Δθ can be determined as formula (16) described. This method is a dynamic adjustment strategy and has nothing to do with the problem. pi is the circumferential ratio. Δθ = 0.5* pi * exp( − gen / max Gen)

(16)

2.2.4 Clone Selection Operator The immune clonal selection is different from the selection in the evolution algorithm. The immune selection is selecting the best antibody from the offspring and its corresponding parents to generate a new population. That is A(k+1)=Ts(A(k)+A’’(k)), where the A’’(k) represents the antibody population which has been through the immune clone and immune mutation genetic operator.

3 CDAQIG Convergence Analysis Theorem 1. The CDAQIG algorithm population sequence is the finite homogeneous Mankov chains.

Proof: The Q-bits of antibody. A is used in the algorithm. In the evolution algorithm, the value of antibodies is discrete. Assumed that the length of the antibody is m, the population size is n, so the state space size is 2mn. Because of the continuity of the value of A, the size of the state space size is infinite in theory, but the accuracy is finite during the calculating. Assumed that the dimension is V, then the state size of the population is Vmn, and so the population is finite. In the algorithm, the immune operation has nothing to do with the generation number, so the transition between the states is only affiliated to the antibody constituting the state. Therefore the population sequence is a finite homogeneous markov chain. The state transition matrix of sequence {CN(t)} can be represented by state matrix. If the state is sorted from big to small based on the affinity, the 1-step transition matrix of the finite homogenous markov chain can be represented as formula (17). P = { Pij

}

ΩΩ

⎡P ⎢ 11 = ⎢ P21 P22 ⎢ P P ⎣⎢ Ω 1 Ω 2

⎤ ⎥ ⎥ ⎥ PΩ1 Ω1 ⎥ ⎦

(17)

P is lower triangular stochastic matrix. Pij>0; P11=1. The convergence of the algorithm is defined as following before the proof of the convergence.

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Definition 4. fk=max{f(xi), i=1,2,..., N} is a random variant sequence. The variant represents the best affinity in the k generation. If the condition meets the formula (18), then we can declare that the method is convergent with probability 1 to the global optimal solution.

{

lim probability f k = f * } = 1

k →∞

(18)

(where f * = max{ f (b) | b ∈ Ω} ). According to reference 8, supposed that the P is a matrix can be of statute, Cm×n is a fundamental random matrix, R and T are not equal to 0, and the P∞ is a stable random matrix. The P∞ is described by formula (19). ⎛ Ck 0 ⎞ ⎜ k −1 ⎟ P = lim P = lim ⎜ i k −1 k ⎟ k →∞ k →∞ T ⎟ ⎜ ∑ T RC ⎝ i =0 ⎠ ∞

k

(19)

Besides the formula (20) and (21) are correct. p ∞ = [1,1,...,1]

T

n

pij( k ) ≥ 0, { p1 , p2 ,..., pn } , ∑ pij = 1, p j = klim →∞ i =1

p j > 0(1 ≤ j ≤ m), p j = 0( m + 1 ≤ j ≤ n)

(20) (21)

Theorem 2. CDAQIG algorithm is convergent with probability 1 to the global optimal solution.

Proof: Supposed that the X is a point in the research space. The population can be treated as a point in the state space, |S| is the size of S, si S is a population, Vki represents a random variant V, and V is on state si in the k generation. Noted that S* is the optimal state set. Assumed that Pij(k) is the probability of the transition from state Xk to the state Xk+1.

∈

S * = {x ∈ X ; f ( x) = max f ( xi )} xi ∈ X

(22)

The optimal antibody has been kept for the clone selection. This operation ensured that each generation antibody would not degenerate. That is, for any k≥0, f(Xk+1) ≥ f(Xk). If f(Xj)>f(Xi), Pij is determined by the normal distribution probability density function. C is the status space which composed by the states. If the state is satisfied with the inequality f(Xj)>f(Xi), the Pij is calculated by formula (x, y ≠i,j) (23). If f(Xj) 0.5

(5)

That is, for a given individual, we judge a random rSGS to decide a schedule of SGS, where the fitness of the individual is defined as the makespan of the schedule. For any random key representation, both the serial and parallel SGS can be directly used to derive a schedule from ρ . On each stage g of SGS, we must select an activity j from the decision set Dg following some rules. Since the random keys play the role of priority values, what we need to do is just to select an efficient priority rule. The latest finish time (LFT)[9]heuristic is used in this study because it is reported to be an outstanding heuristic for evolutionary algorithm. Therefore, on the stage g , we will select activity j with the minimum random key rj = min {ri | i ∈ Dg } from the decision set Dg .

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3.3 The ABC-RK Algorithm for RCPSP

Using the mentioned-above solution representation, the ABC-RK algorithm for RCPSP can be seen in Fig. 2. Firstly, the ABC-RK generates a randomly distributed initial population of SN solutions, i.e., food source positions. Each solution xi ( i ∈ [1, SN ] ) is a D-dimensional vector. Then, we evaluate the initial population after random key based solution encoding is done. Moreover, an iterative process with max generation N is searched through the employed bees, the onlooker bees and scout bees: (1) An employed bee produces a modification on the position (solution) in her memory depending on the local information (visual information) and tests the nectar amount (fitness value) of the new source (new solution). (2) An onlooker bee evaluates the nectar information taken from all employed bees and chooses a food source with a probability related to its nectar amount. (3) A scout bee investigates the food source if the nectar is abandoned. Then, we decide that the bee is replaced with a new food source by the scouts or not. Furthermore, when the Best solution is the ideal solution or we have run out of generation N, the iterative process is terminated.

Fig. 2. The proposed ABC-RK algorithm

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In the above interactive process, the ABC-RK algorithm produce new solutions )

vi,j for the employed bees using (see Fig. 2 symbol

①

vi j = xij + ψ ij ( xi j − xkj ) ,

(6)

where k ∈ [1, SN ] , j ∈ [1, D ] are randomly chosen indexes, and ψ ij ∈ [ −1,1] is a random number that controls the production of neighbor food sources around xij and represents the comparison of two food positions visually by a bee. Note if a parameter value produced by this operation exceeds its predetermined limit, the parameter can be set to an acceptable value. The probability for an onlooker bee choosing a food source can be calculated by SN

p i = fitn ess i / ∑ fitn ess n ,

②

(7)

n =1

(See Fig. 2 symbol ) where fitnessi is the function fitness value of the solution i. Additionally, the value of predetermined number of generations (cycles) is an important control parameter of the ABC algorithm, which is called “limit” for abandonment.

4 Experimental Study We applied ABC-RK to the standard instance sets J 30 (480 instances each with 30 activities), J 60 (480 instances each with 60 activities), and J 120 (600 instances each with 120 activities) generated by the problem generator ProGen devised by Kolisch and Sprecher[13]. These sets are available in the well known PSPLIB (http://129.187.106.231/psplib/) along with the optimum, or best known values that have been obtained by various authors over the years. Only in J 30 are the optimal makespans for all instances known. In J 60 and J 120 the optimal makespans for some instances are not known and only upper bounds (current best solutions) and lower bounds are provided. The lower bounds are determined by the length of a critical path in the resource relaxation of the problem using a critical path method. For a fair machine-independent comparison, we limited the number of generated and evaluated schedules in ABC-RK to 1000, 5000, and 50000, respectively. This restriction allowed us to compare our results with those reported in the evaluation study of Kolisch and Hartmann[12] for several RCPSP heuristics proposed in the literature. ABC-RK was implemented in JAVA 1.6 and all the experiments were performed on a PC with 1.86GHz CPU and 1G RAM running the Windows XP operating system. Tables 1-3 give the performances of ABC-RK and several RCPSP heuristics from the literature with 1000, 5000, and 50000 evaluated schedules. This study compared ABC-RK with priority-rule based sampling methods[14], genetic algorithms[4][15][19], scatter search[10], ant colony algorithms[20 ,tabu search[21] , etc. Table 1 gives the average percentage deviation from the optimal makespan for J 30 . ABC-RK ranks 9th for 1000 schedules and 7th and 7th for 5000 and 50000

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schedules, respectively. Table 2 showed the lower bound results for instances with 60 activities. ABC-RK ranks 10th and 7th for 1000 and 5000 schedules, respectively, and 7th for 50000 schedules. Table 3 gives the results for J 120 , where ABC-RK ranks 8th for 1000 schedules, and 7th and 8th for 5000 and 50000 schedules, respectively. To highlight the actual performance, Table 4 lists the results of the heuristics together with the average and maximum computation times as well as the clock-cycle of the computer used. The performances of the reported methods were obtained from surveys[12] and the original papers. The performance of ABC-RK as shown in the table was limited to 5000 evaluated schedules. From Table 4, we can see that ABCRK requires a shorter average computation time for J 30 , J 60 , and J 120 . Table 1. Average deviation (%) from optimal makespan-ProGen set J = 30

1000

Schedules 5000

50000

0.10

0.04

0.00

0.06 0.22

0.02 0.09

0.01 -

Debels et al.[10]

0.27

0.11

0.01

Debels et al.[15] Valls et al.[19] Valls et al.[4] This study

0.15 0.27 0.34 0.35

0.04 0.06 0.20 0.12

0.02 0.02 0.02 0.04

Tormos and Lova[14]

0.25

0.13

0.05

Nonobe and Ibaraki[21]

0.46

0.16

0.05

Algorithm GA, TS-path relinking GAPS ANGEL Scatter SearchFBI GA-DBH GA-hybrid, FBI GA-FBI ABC-RK Sampling-LFT, FBI TS-activity-list

References Kochetov Stolyar[18] Mendes et al.[22] Tseng et al.[20]

and

Table 2. Average deviation (%) from critical path lower bound-ProGen Set J = 60

Algorithm

References

GAPS GA-DBH Scatter Search-FBI GA-hybrid, FBI GA, TS-path relinking ANGEL GA-FBI ABC-RK GA-self-adapting GA-activity list

Mendes et al. [22] Debels et al.[15] Debels et al.[10] Valls et al. [19] Kochetov and Stolyar[18] Tseng et al.[20] Valls et al.[4] This study Hartmann[17] Hartmann[16]

1000 11.72 11.45 11.73 11.56 11.71 11.94 12.21 12.75 12.21 12.68

Schedules 5000 50000 11.04 10.67 10.95 10.68 11.10 10.71 11.10 10.73 11.17 10.74 11.27 11.27 10.74 11.48 11.18 11.70 11.21 11.89 11.23

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Table 3. Average deviation (%) from critical path lower bound-ProGen Set J = 120

Algorithm

References

GA-DBH GA-hybrid, FBI GAPS Scatter Search-FBI GA-FBI GA, TS-path relinking GA-self-adapting ABC-RK Sampling-LFT, FBI ANGEL

Debels et al.[15] Valls et al.[19] Mendes et al. [22] Debels et al.[10] Valls et al.[4] Kochetov and Stolyar[18] Hartmann[17] This study Tormos and Lova[14] Tseng et al. [20]

1000 34.19 34.07 35.87 35.22 35.39 34.74 37.19 36.29 35.01 36.39

Schedules 5000 50000 32.34 32.54 33.03 33.10 33.24 33.36 35.39 34.18 34.41 34.49

30.82 31.24 31.44 31.57 31.58 32.06 33.21 33.69 33.71 -

Table 4. Average and maximum computation times for J 30 , J 60 and J 120

Algorithm (a) J = 30 Decompos. & local opt. Local search-critical ABC-RK ANGEL Population-based (b) J = 60 Decompos. & local opt. Population-based ABC-RK ANGEL Local search-critical Tabu search (c) J = 120 Population-based Decompos. & local opt. ABC-RK ANGEL Local search-critical Tabu search

CPU-time(seconds) Average Max.

Reference

Result

Computer

[23] [24] This study [20] [5]

0.00 0.06 0.12 0.09 0.10

10.26 1.61 1.15 0.11 1.16

123.0 6.2 4.17 5.5

2.3 GHz 400 MHz 1.86 GHz 1 GHz 400 MHz

[23] [5] This study [20] [24] [25]

10.81 10.89 11.18 11.27 11.45 12.05

38.8 3.7 2.12 0.76 2.8 3.2

223.0 22.6 13.5 14.6 -

2.3 GHz 400 MHz 1.86 GHz 1 GHz 400 MHz 450 MHz

[5] [23] This study [20] [24] [25]

31.58 32.41 33.18 34.49 34.53 36.16

59.4 207.9 30.8 4.79 17.0 67.0

264.0 501.0 239.8 43.9 -

400 MHz 2.3 GHz 1.86 GHz 1 GHz 400 MHz 450 MHz

These results show that ABC-RK is capable of providing near-optimal solutions for the instance set J 30 , and can produce good solutions for the large scale problems J 60 and J 120 . The reason for the showed performance is that ABC-RK algorithm

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combines local search methods, carried out by employed and onlooker bees, with global search methods, managed by onlookers and scouts, and then balance exploration and exploitation process to provide a good solution.

5 Conclusion In this study, we proposed an artificial bee colony (ABC for short) algorithm with random key for resource-constrained project scheduling. The computational results of ABC-RK were compared with state-of-the-art heuristics on the standard instance sets J 30 , J 60 and J 120 from the PSPLIB. The preliminary experimental results indicated that, with a limited number of generated schedules, ABC-RK is capable of providing near-optimal solutions for a small scale RCPSP and large scale problems. Further studies will focus on enhancing the applicability and efficiency of the proposed ABC-RK algorithm for large-scale instances of RCPSP.

Acknowledgements This work was supported by National Natural Science Foundation of P R China (Grant No. 50975039) and National Defense Basic Scientific Research Project of PR China (Grant No. B0920060901).

References 1. Karaboga, D.: An idea based on honey bee swarm for numerical optimization. Technical Report TR06, Computer Engineering Department, Erciyes University, Turkey (2005) 2. Karaboga, D., Akay, B.: A comparative study of Artificial Bee Colony algorithm. Applied Mathematics and Computation 214, 108–132 (2009) 3. Karaboga, D., Akay, B.: Artificial Bee Colony (ABC) Algorithm on training artificial neural networks. In: IEEE 15th Signal Processing and Communications Applications, Inst. of Elec. and Elec. Eng. Computer Society, Eskisehir, Turkey (2007) 4. Valls, V., Ballestín, F., Quintanilla, S.: Justification and RCPSP: A technique that pays. European Journal of Operational Research 165, 375–386 (2005) 5. Valls, V., Ballestín, F., Quintanilla, S.: A Population-Based Approach to the ResourceConstrained Project Scheduling Problem. Annals of Operations Research 131, 305–324 (2004) 6. Chen, W., Shi, Y.-j., Teng, H.-f., Lan, X.-p., Hu, L.-c.: An efficient hybrid algorithm for resource-constrained project scheduling. Information Sciences 180, 1031–1039 (2010) 7. Christofides, N., Alvarez-Valdes, R., Tamarit, J.M.: Project scheduling with resource constraints: A branch and bound approach. European Journal of Operational Research 29, 262–273 (1987) 8. Kolisch, R., Hartmann, S.: Heuristic Algorithms for Solving the Resource-Constrained Project Scheduling Problem: Classification and Computational Analysis. In: Weglarz, J. (ed.) Project Scheduling: Recent Models, Algorithms and Applications, pp. 147–178. Kluwer Academic Publishers, Berlin (1999)

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9. Hartmann, S., Kolisch, R.: Experimental evaluation of state-of-the-art heuristics for the resource-constrained project scheduling problem. European Journal of Operational Research 127, 394–407 (2000) 10. Debels, D., De Reyck, B., Leus, R., Vanhoucke, M.: A hybrid scatter search/electromagnetism meta-heuristic for project scheduling. European Journal of Operational Research 169, 638–653 (2006) 11. Sprecher, A., Kolisch, R., Drexl, A.: Semi-active, active, and non-delay schedules for the resource-constrained project scheduling problem. European Journal of Operational Research 80, 94–102 (1995) 12. Kolisch, R., Hartmann, S.: Experimental investigation of heuristics for resourceconstrained project scheduling: An update. European Journal of Operational Research 174, 23–37 (2006) 13. Kolisch, R., Sprecher, A.: PSPLIB - A project scheduling problem library: OR Software ORSEP Operations Research Software Exchange Program. European Journal of Operational Research 96, 205–216 (1997) 14. Tormos, P., Lova, A.: Integrating heuristics for resource constrained project scheduling: One step forward. Technicalreport, Department of Statistics and Operations Research, Universidad Politecnica de Valencia (2003) 15. Debels, D., Vanhoucke, M.: A Decomposition-Based Genetic Algorithm for the ResourceConstrained Project-Scheduling Problem. Operations Research 55, 457–469 (2007) 16. Hartmann, S.: A competitive genetic algorithm for resource-constrained project scheduling. Naval Research Logistics 45, 733–750 (1998) 17. Hartmann, S.: A self-adapting genetic algorithm for project scheduling under resource constraints. Naval Research Logistics 49, 433–448 (2002) 18. Kochetov, Y., Stolyar, A.: Evolutionary local search with variable neighborhood for the resource constrained project scheduling problem. In: Proceedings of the 3rd International Workshop of Computer Science and Information Technologies, Russia (2003) 19. Valls, V., Ballestin, F., Quintanilla, M.S.: A hybrid genetic algorithm for the RCPSP. Department of Statistics and Operations Research, University of Valencia (2003) 20. Tseng, L.-Y., Chen, S.-C.: A hybrid metaheuristic for the resource-constrained project scheduling problem. European Journal of Operational Research 175, 707–721 (2006) 21. Nonobe, K., Ibaraki, T.: Formulation and tabu search algorithm for the resource constrained project scheduling problem. In: Hansen, P. (ed.) Essays and Surveys in Metaheuristics, pp. 557–588. Kluwer Academic Publishers, Dordrecht (2001) 22. Mendes, J.J.M., Goncalves, J.F., Resende, M.G.C.: A random key based genetic algorithm for the resource constrained project scheduling problem. Computers & Operations Research 36, 92–109 (2009) 23. Palpant, M., Artigues, C., Michelon, P.: LSSPER: Solving the Resource-Constrained Project Scheduling Problem with Large Neighbourhood Search. Annals of Operations Research 131, 237–257 (2004) 24. Valls, V., Quintanilla, S., Ballestín, F.: Resource-constrained project scheduling: A critical activity reordering heuristic. European Journal of Operational Research 149, 282–301 (2003) 25. Artigues, C., Michelon, P., Reusser, S.: Insertion techniques for static and dynamic resource-constrained project scheduling. European Journal of Operational Research 149, 249–267 (2003)

Combined Electromagnetism-Like Mechanism Optimization Algorithm and ROLS with D-Optimality Learning for RBF Networks Fang Jia1 and Jun Wu2 1

Department of Control Science and Engineering, Zhejiang University, 310027, Hangzhou, Zhe Jiang, China 2 National Key Laboratory of Industrial Control Technology, Institute of Cyber Systems and Control, Zhejiang University, Hangzhou, 310027, China {fjia,jwu}@iipc.zju.edu.cn

Abstract. The paper proposed a new self-constructed radial basis function network designing method via a two-level learning hierarchy. Aiming at getting stronger generalization ability and robustness, an integrated algorithm which combines the regularized orthogonal least square with learning with Doptimality experimental design method was introduced at the lower level, while electromagnetism-like mechanism algorithm for global optimization was employed at the upper level to search the optimal combination of three important learning parameters, i.e., the radial basis function width, regularized parameter and D-optimality weight parameter. Through simulation results, the effectiveness of the proposed algorithm was verified. Keywords: radial basis function network, two-level learning hierarchy, electromagnetism-like mechanism algorithm, generalization ability.

1 Introduction Radial basis function (RBF) neural network has advantages of simple structure, strong nonlinear approximation ability and fast learning speed, and it has widely used in areas of function approximation [1], systems modeling [2], and density estimate [3]. In RBF neural network modeling, the generalization performance of RBF network is crucially dependent on the hidden node number. Among RBF network designing methods, Chen, Cowan, Grant (1991) proposed an orthogonal least squares (OLS) forward selection algorithm [4], which can avoid large network size and some numerical ill-posed problems [5]. In order to construct an RBF network with parsimonious structure and good generalization, regularized orthogonal least squares (ROLS) algorithm [6] was proposed. ROLS can effectively improve the generalization ability of RBF network and avoid over-fitting problem caused by noise data of the samples. After that, D-Optimality experimental design (D-opt) method was added to ROLS [8][9], which can enhance the model efficiency and robustness through maximizing the determinant of design matrix. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 158–165, 2010. © Springer-Verlag Berlin Heidelberg 2010

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In ROLS algorithm, the regularization parameter by iterations using Bayesian formula, however, may not be the best one since the iterative procedure is based on a gradient algorithm essentially. To further improve the generalization of RBF network, an effective population-based intelligent searching method was introduced to search the proper combination of the RBF width, regularization parameter and D-optimality weight. Electromagnetism-like mechanism optimization algorithm (EM) [10] is a population-based heuristic for global optimization, originally proposed by BİRBİL and FANG in 2002. EM algorithm has no demand for the task model information, and possesses excellent optimization performance. A new two-level learning hierarchy for designing an RBF network based on ROLS+D-optimality method and EM algorithm is proposed in this paper. Electromagnetism-like mechanism optimization algorithm was employed at the upper level, which can optimize the RBF network parameters. Meanwhile, an integrated algorithm (ROLS+D-opt) is introduced at the lower level to construct a parsimonious RBF model automatically. The proposed algorithm can improve greatly the network performance through searching the optimal combination of learning parameters of the ROLS+D-opt integrated algorithm.

2 Network Construction The basic RBF network considered in this paper has a simple MISO structure, with three layers, m inputs, n nodes and one output. The network output is defined by the following equation: n ) y (k ) = Fr ( x ) = ∑ θi exp(- x - ci

2

/ ρ)

(1)

i =1

where x = [ x1...xm ]T is the network input vector, θ i is RBF network weights, ci = [c1,i ...cm,i ]T is the center vectors, ⋅ as the Euclidean norm. ρ is the uniform width of Gaussian nonlinearity function. 2.1 ROLS with D-Optimality Network Construction Assume a training set of N samples { y(k ), x(k )}k =1 is available, where y ( k ) is the desired network output, x( k ) is the network input vector. ci = x (i) for 1 ≤ i ≤ N , which is considered as candidate centers. By introducing: N

φi (k ) = exp(− x (k ) − ci / ρ ) 2

(2)

The network output can be written as N ) y (k ) = y (k ) + e (k ) = ∑ θiφi (k ) + e (k ) 1 ≤ i ≤ N

(3)

i =1

)

where e(k ) is the error between y ( k ) and the network output y ( k ) .

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：

By defining

y = [ y (1),L , y ( N )]T Φ = [Φ 1 ,... ,Φ N ]

Φi = [φi (1),L ,φi ( N )]T Θ = [θ1,L,θ N ]T e = [e(1),L , e( N )]T

We can get y =ΦΘ+e

(4)

To construct a smaller subset model, the ROLS algorithm introduced an orthogonal decomposition to the regression matrix as Φ = WA , where W is an orthogonal matrix. and A is an upper triangular matrix with unit diagonal elements. The original system can be rewritten as y = (ΦA -1 )( AΘ ) + e = Wg + e

(5)

where g = [ g1 ,L, g N ]T = AΘ . To improve the RBF network’s generalization, efficiency and robustness, the Doptimality design criterion [1] was incorporated into the regularized orthogonal least squares algorithm. D-Optimality criterion is straightforward to verify the following JD: Ns ⎧ ⎫ max ⎨ J D = det ( ΦT Φ ) = ∏ λk ⎬ = 1 k ⎩ ⎭

(6)

or equivalently, minimize − log det (W N TW N ) . s

s

Combine with ROLS method, the cost function became: J CR ( g , λ , β ) = eT e + λ g T g + β ∑ − log ( wi T wi ) N

(7)

i =1

where β is the D-optimality weight. Then the error reduction ratio became:

(

)

[crerr ]i = ( wiT wi + λ ) gi2 + β log ( wiT wi ) / yT y

(8)

When [crerr ]l ≤ 0, n s + 1 ≤ l ≤ N once met, the selection process stops automatically. ROLS+D-optimality algorithm requires choosing a proper D-optimality weight parameter. In fact, the RBF width, regularization parameter and D-optimality weight are significant parameters related to the performance of RBF network, so we should use a global optimization method to search the combination of these three parameters.

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161

2.2 Electromagnetism-Like Mechanism Algorithm

（ ）

Electromagnetism-like Mechanism EM algorithm [10] is a population-based heuristic for global optimization. Consider the optimization problems with bounded variables in the form of: min f ( x )

x ∈ [l , u ]

s.t .

(9)

where [l,u] := { x ∈ ℜ | lk ≤ xk ≤ uk , k = 1,..., n} , uk and l k are the upper and lower bound in the kth dimension, f ( x) is fittness function that is minimized. The EM algorithm consists the following four phases: n

─ ─ ─ ─

Initialization of the algorithm; Application of neighborhood search to exploit the local minima; Calculation of the total force exerted on each particle; Movement along the direction of the force.

The force Calculation on each particle is abided by the following function:

(

q i = exp − n ( f ( x i ) − f ( x best ) )

∑( f (x

k

) − f ( x best ) )

)

i = 1, 2,...P

(10)

where n is the dimension of the problem, P is the population size. F i = ∑ j ≠ i sign ( f ( x j ) − f ( x i ) )( x i − x j ) qi q j m

x j − xi

2

(11)

Each particle movement is calculated by the following function: x i = x i + λEM

Fi Fi

( RNG )

i = 1, 2,..., m

(12)

where λ EM is the uniformly distributed between 0 and 1, RNG is a vector whose components denote the allowed feasible movement toward the upper bound, uk , or the lower bound, l k , for the corresponding dimension. 2.3 The Combined EM and ROLS+D-Optimality Learning Aiming at searching the optimal combination of the RBF width ρ , regularized parameter λ and D-optimality weight parameter β , which crucially influent the network generalization performance, a two-level learning hierarchy for designing an RBF network based on ROLS+D-optimality algorithm and EM optimization method is proposed. The schematic of two-level learning hierarchy is illustrated in Fig. 1. In this two-level learning hierarchy, EM algorithm is introduced at the upper level, with a population size of P , based on fitness function values feedback from the lower level. At the lower level, ROLS+D-optimality was presented in parallel scheme. Each provides the specific fitness function value f i for the given ρi , λi and βi .

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Fig. 1. Schematic of two-level learning hierarchy based on EM for RBF networks

Generalization performance is usually indicated by the MSE over the testing validation data set. Divide the candidate model bases into training subset and testing subset. For each Ki = [λi , βi , ρi ] , the RBF network can be self-constructed through ROLS+D-optimality algorithm using training subset. And the mean square error (MSE) of testing subset can be considered as the fitness function as following [11]: min f ( K ) =

1 n ) ∑ c ( y( x (k )) − y ( x (k )))2 nc k =1

(13)

)

where y( x(k )) is the network output over the testing data, y( x(k )) is the desired output of testing data, nc is the number of training data, K indicates [ λ, β , ρ ] , the smaller the fitness function value is ,the better the generalization of the RBF network.

3 Modeling Example This example used a radial basis function (RBF) network to model the scalar function [11]: y ( x) = 1.1(1 − x + 2 x 2 ) exp(− x 2 / 2)

(14)

The training set is generate as following: the training data number N = 100 ; and y is the model output generated from y( x) + e , where the input x was uniformly distributed in [-4,4] and the noise e was Gaussian with zero mean and variance 0.5. And 50 noise-free data y ( x ) with equally spaced x with distance Δ = 0.16 were generated as the testing data set. The curve of function (14) and the training data is shown in Fig. 2. Assuming the the parameters of EM algorithm are: the problem dimension n = 3 ; population size P=10; local search iterations LSITER=3; and local search parameter δ = 0.001 After 100 generations, ⎡⎣λopt , βopt , ρopt ⎤⎦ = [0.2178, 0.4807, 0.9974] , the performance of the constructed RBF network was shown in Fig 3, where the black solid line is the network output, and little circle are the chosen RBF centers. It verifies the construction method adopted here is efficient in function approximation.

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Fig. 2. Simple scalar function modeling problem with noisy training data (dots) and underlying function y(x) (curve)

Fig. 3. Model mapping (curve) produced by the ROLS + D-optimality algorithm with EMbased two level learning method.

Fig. 4. EM algorithm convergence trend

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Fig. 4 shows the convergence of the EM algorithm optimization process, where the horizontal axis represents the number of generation, the vertical axis represents function fitnness defined by function (13). To compare the generalization performance and model sparsity, the simulation result is shown in Table. 1, In the comparison methods, assume ρ = 0.866 ( 2 ρ 2 = 1.5 ), D-optimality weight parameter β = 5 × 10 −4 .50 points was taken as testing data. As was shown in Table 1, by proposing EM-based two level learning method, the MSE over testing dating are both smaller than the result of OLS+D-optimality method and the ROLS+D-optimality method, which indicates better generalization capability. It also shows that the new method simplifies the RBF network’s construction. To summarize, simulation result verified that the combination of EM optimization and ROLS learning with D-Optimality can improve the RBF network’s generalization capability and model sparsity. Table 1. Performance comparison of RBF network designed with three methods MSE over MSE over number of terms training data testing data 0.28737 0.15019 19 OLS+D-opt 0.32303 0.03495 12 ROLS+D-opt 0.34315 0.02671 7 EM based ROLS+D-opt

4 Conclusions The paper proposed a two-level learning hierarchy for RBF networks by combining Electromagnetism-like Mechanism algorithm and the ROLS+D-optimality algorithm. And the new method has the following characteristics: First of all, the generalization performance and model sparsity of the original RBF network are improved by introducing regularized orthogonal least squares (ROLS) algorithm, which was proposed aiming the problem of network overfitting. Secondly, since poor selection of RBF network centers leads to ill-conditioned model, DOptimality (D-opt) experimental design theory was added to optimize the design matrix. Thus, the performance of RBF network became more efficient and robust. Thirdly, a new two-level learning hierarchy has been proposed to construct the RBF network, the upper level was based on electromagnetism-like mechanism (EM) algorithm, which was added to search the global optimal value of the RBF width, regularized parameter and D-optimality weight parameter. At the lower level, the regularized orthogonal least squares learning with D-optimality method was presented in parallel scheme. The simulation verified superior generalization properties and sparsity level of the combined EM and ROLS+Dopt learning approach. In this method, the RBF width is a constant value learned by EM algorithm. In the future, tunable-node RBF model should be worth considering.

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References 1. Chen, S., Hong, X., Harris, C.J.: Sparse Kernel Regression Modeling Using Combined Locally Regularized Orthogonal Least Squares and D-Optimality Experimental Design. IEEE Transactions on Automatic Control 48(6), 1029–1036 (2003) 2. Wei, H.K., Ding, W.M., Song, W.Z., Xu, S.X.: Dynamic method for designing RBF neural networks. Control Theory and Applications 19(5), 673–680 (2002) 3. Chen, S., Hong, X., Harris, C.J.: Probability Density Function Estimation Using Orthogonal Forward Regression. In: Proceedings of International Joint Conference on Neural Networks, Orlando, Florida (2007) 4. Chen, S., Cowan, C.F.N., Grant, P.M.: Orthogonal Least Squares Learning Algorithm for Radial Basis Function Neural Networks. IEEE Trans. Neural Network 2(2), 302–309 (1991) 5. Zhao, W.B., Yang, L.Y., Wang, L.M.: The Hybrid Structure Optimization Algorithms of Radial Basis Probabilistic Neural Networks. Acta Simulata Systematica Sinica 16(10), 2175–2184 (2004) 6. Chen, S., Chng, E.S., Alkadhimi, K.: Regularized Orthogonal Least Squares Algorithm for Constructing Radial Basis Function Networks. Int. J. Contr. 64(5), 829–837 (1996) 7. MacKay, D.J.C.: Bayesian interpolation. Neural Compute 4(3), 415–447 (1992) 8. Hong, X., Harris, C.J.: Nonlinear Model Structure Design and Construction Using Orthogonal Least Squares and D-Optimality Design. IEEE Trans. Neural Networks 13, 1245–1250 (2002) 9. Hong, X., Harris, C.J.: Experimental Design And Model Construction Algorithms For Radial Basis Function Networks. International Journal of Systems Science 34(14-15), 733–745 (2003) 10. Bi̇rbi̇l, Ş.İ., Fang, S.C.: A Multi-point Stochastic Search Method for Global Optimization. Journal of Global Optimization 25, 263–282 (2003) 11. Chen, J.F., Ren, Z.W.: A New Two-level Learning Design Approach for Radial Basis Function Neural Network. Computer Simulation 26(6), 151–155 (2009)

Stochastic Stability and Bifurcation Analysis on Hopfield Neural Networks with Noise Xuewen Qin, Zaitang Huang , and Weiming Tan School of Mathematics and Physics, Wuzhou University, Wuzhou 543002, P. R. China [email protected]

Abstract. A stochastic differential equation modelling a Hopfield neural network with two neurons is investigated. Its dynamics are studied in terms of local stability analysis and Hopf bifurcation analysis. By analyzing the Lyapunov exponent, invariant measure and singular boundary theory , its nonlinear stability is investigated and Hopf bifurcations are demonstrated. The stability and direction of the Hopf bifurcation are determined from the dynamical and phenomenological points of view.

1

Introduction

In some artificial neural network applications, such as content-addressable memories, information is stored as stable equilibrium points of the system. In 1984, Hopfield [1] considered a simplified neural network model in which each neuron is represented by a linear circuit consisting of a resistor and a capacitor and is connected to the other neurons via nonlinear sigmoidal activation functions. Since then, dynamical characteristics of neural networks has become a subject of intensive research activity[2-13,17-23]. We also mention that noise always arise in neural networks due to the processing of information. Hence, most of the models of neural networks are described by systems of stochastic differential equations (see, for example, [7-13]). For the general theory of delay differential equations (SDEs) we refer to [14-16]. Due to the complexity of the analysis, most work has focused on the situation where all connection terms in the network have the time-delay. For neural networks with noise, however, the analysis is usually simplified by considering networks with small number of neurons or with simple architectures. In this paper, we consider a two-unit ring network modeled by the following system of SDEs in a parameter space consisting of the noise, the internal decay rate and the connection strength. More precisely, we shall give conditions on the linear or nonlinear stability of the trivial solution of the system x˙ i = −axi +

2 

cij f (xj (t)) + αi xi (t)ξ(t) + Ii ηi (t)

j=1 

Corresponding author.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 166–179, 2010. c Springer-Verlag Berlin Heidelberg 2010 

(1.1)

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167

where xi denote the potential(or voltage) of the cell i at time t; a is positive constants, they denote the rate with which the cell i will reset their potential to the resting state in isolation when isolated form the other cells and inputs; cji is the first-order connection weights of the neural network; we assume that ξ(t) is the multiplicative random excitation and η(t) is the external random excitation directly(namely additive random). ξ(t) and η(t) are independent, in possession of zero mean value and standard variance Gauss white noises. i.e. E[ξ(t)] = E[η(t)] = 0, E[ξ(t)ξ(t + τ )] = δ(τ ), E[η(t)η(t + τ )] = δ(τ ), E[ξ(t)η(t + τ )] = 0. And (βi , αi ) is the intensities of the white noise. Our purpose in this paper is to investigate the stochastic bifurcation and stability for (1.1) by applying the singular boundary theory, Lyapunov exponent and the invariant measure theory, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are also determined.

2

Preliminary Results

In the section, we present some preliminary results to be used in a subsequent section to establish the stochastic averaging system. Let the Taylor expansion of fj at 0. Then we can rewrite (1.1) as the following equivalent system dx1 = −ax1 (t) + a21 x2 (t) + a22 x22 (t) + a23 x32 (t) + a24 x21 (t) + α1 x1 ξ(t) + I1 η(t) dt dx2 = −ax2 (t) + a11 x1 (t) + a12 x21 (t) + a13 x32 (t) + α2 x2 ξ(t) + I2 η(t) (2.1) dt Set x = r cos θ, y = r sin θ, and by substituting the variable in (2.1), we obtain ⎧ r(t) ˙ = r(b1 cos2 θ + b2 sin2 θ) + r 2 (b11 cos3 θ + (b12 + a21 ) cos2 θ sin θ ⎪ ⎪ ⎪ ⎪ +(b13 + a22 ) cos θ sin2 θ + b23 sin3 θ) + r 3 (b14 cos4 θ + (b15 + b24 ) cos3 θ sin θ ⎪ ⎪ ⎪ ⎪ +(b16 + b25 ) cos2 θ sin2 θ + (b17 + b26 ) cos3 θ sin θ + b27 sin4 θ) ⎪ ⎪ ⎪ ⎪ +r(k11 cos2 θ + (k12 + k21 ) cos θ sin θ + k22 sin2 θ)ξ(t) ⎪ ⎪ ⎨ +(r1 cos θ + r2 sin θ)η(t) ˙ ⎪ θ(t) = (b2 − b1 ) cos θ sin θ + r[b21 cos θ + (b22 − b11 ) cos2 θ sin θ ⎪ ⎪ ⎪ ⎪ +(b23 − b12 ) cos θ sin2 θ − b13 sin3 θ] + r 2 [(b24 cos4 θ + (b25 − b14 ) cos3 θ sin θ ⎪ ⎪ ⎪ ⎪ +(b26 − b15 ) cos2 θ sin2 θ + (b27 − b16 ) cos3 θ sin θ − b17 sin4 θ)] ⎪ ⎪ ⎪ ⎪ +(k21 cos2 θ + (k22 − k11 ) cos θ sin θ − k12 sin2 θ)ξ(t) ⎪ ⎩ + 1r (r2 cos θ − r1 sin θ)η(t). (2.2)

where the coefficient are omitted. Through the stochastic averaging method, we obtained the Itˆ o stochastic differential equation (2.2) the process satisfied    1  1 dr = (μ1 + μ82 )r + μr3 + μ87 r3 dt + μ3 + μ84 r2 2 dWr + (rμ5 ) 2 dWθ , (2.3) 1   1 dθ = μ88 r2 dt + (rμ5 ) 2 dWr + μr23 + μ86 2 dWθ . where the coefficient are omitted. From the diffusion matrix[24-27], we can find that the averaging amplitude r(t) is a one-dimensional Markov diffusing process as follows:   μ3 μ7 3  μ2 μ4 2  12 dr = (μ1 + )r + + r dt + μ3 + r dWr . (2.4) 8 r 8 8

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This is an efficient method to obtain the critical point of stochastic bifurcation through analyzing the change of stability of the averaging amplitude r(t) in the meaning of probability.

3 3.1

Stochastic Stability Local Stochastic Stability

Firstly, we consider the stability of the linear Itˆ o stochastic differential equation, i.e. the stability at μ3 = 0, μ7 = 0. System (2.4) becomes   μ  12 μ2  4 2 dr = (μ1 + )r dt + r dWr . 8 8

(3.1)

Thus, the approximation of Lyapunov exponent of the linear Itˆo stochastic differential equation is: λ = lim

t→+∞

μ4 1 μ2 ln r(t) = μ1 + − . t 8 16

Thus we have: when μ1 + μ82 − μ164 < 0, that is λ < 0, thus the trivial solution of the linear Itˆ o stochastic differential equation r = 0 is stable in the meaning of probability. when μ1 + μ82 − μ164 > 0, that is λ > 0. Thus the trivial solution of the linear Itˆo stochastic differential equation r = 0 is unstable in the meaning of probability. When μ1 + μ82 − μ164 = 0, that is λ = 0. Whether μ1 + μ82 − μ164 = 0, or not can be regarded as the critical condition of bifurcation at the equilibrium point. And whether the Hopf bifurcation could occur or not are what we will discuss in the next section. 3.2

Global Stochastic Stability

The global stochastic stability at μ3 = 0, μ7 = 0. When μ3 = 0, μ7 = 0, the system (2.4) can be rewritten as follows:  μ  12  μ7 3  μ2 4 2 dr = (μ1 + )r + r dt + r dWr . 8 8 8

(3.2)

Thus r = 0 is the first kind of singular boundary of system (3.2). When r = +∞, we can find mr = +∞; thus r = +∞ is the second kind of singular boundary of system (3.2). According to the singular boundary theory, we can calculate the diffusion exponent, drifting exponent and characteristic value of boundary r = 0 and the results are as follows: αr = 2, βr = 1, cr = lim

r→0+

2[(μ1 + 2mr (r − 0)(αr −βr ) = lim 2 + σ11 r→0

μ2 )r + μ87 r 3 ]r 8 μ4 2 r 8

=

2(8μ1 + μ2 ) . (3.3) μ4

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2 So if cr > 1, i.e. 8μ1μ+μ > 12 , the boundary r = 0 is exclusively natural. 4 8μ1 +μ2 1 If cr < 1, i.e. μ4 < 2 , the boundary r = 0 is attractively natural. 2 If cr = 1, i.e. 8μ1μ+μ = 12 , the boundary r = 0 is strictly natural. 4 We can also calculate the diffusion exponent, drifting exponent and characteristic value of boundary r = +∞, and the results are as follows:

αr = 2, βr = 3, 2((μ1 + 2mr (r − 0)(αr −βr ) = − lim 2 r→+∞ r→+∞ σ11

cr = − lim

μ2 )r + μ87 r 3 )r −1 8 μ4 2 r 8

2μ7 =− .(3.4) μ4

So if cr > −1, i.e. μμ74 < 12 , the boundary r = +∞ is exclusively natural. If cr < −1, i.e. μμ74 > 12 , the boundary r = +∞ is attractively natural. If cr = 1, i.e. μμ74 = 12 , the boundary r = +∞ is strictly natural. As we know, if the singular boundary r = 0 is attractively natural boundary and r = +∞ is entrance boundary, this situation is all the solve curves enter the inner system from the right boundary and is attracted by the left boundary, the equilibrium point is global stable. From the analysis above, we can draw a conclusion that the equilibrium point is global stable when the singular boundary r = 0 is attractively natural boundary and r = +∞ is entrance boundary. Combine the condition of local stability, 2 the equilibrium point r = 0 is stable when 8μ1μ+μ < 12 , and μμ74 < 12 . 4 The global stochastic stability at μ3 = 0, μ7 = 0. When μ3 = 0, μ7 = 0, the system (2.4) can be rewritten as follows:   μ3 μ7 3  μ2 μ4 2  12 )r + + r dt + μ3 + r dr = (μ1 + dWr . 8 r 8 8

(3.5)

One can find σ11 = 0 at r = 0, so r = 0 is a nonsingular boundary of system (3.5). Through some calculations we can find that r = 0 is a regular boundary(reachable). The other result is mr = ∞ when r = ∞, so r = ∞ is second singular boundary of (3.5). The details are presented as follows: αr = 2, βr = 3, 2((μ1 + μ82 )r + μr3 + μ87 r 3 )r −1 2μ7 2mr (r)(αr −βr ) =− = − lim .(3.6) 2 r→+∞ r→+∞ σ11 (μ3 + μ84 r 2 ) μ4

cr = − lim

So if cr > −1, i.e. μμ74 < 12 , the boundary r = +∞ is exclusively natural. If cr < −1, i.e. μμ74 > 12 , the boundary r = +∞ is attractively natural. If cr = 1, i.e. μμ74 = 12 , the boundary r = +∞ is strictly natural. Thus we can draw the conclusion that the trivial solution r = 0 is unstable, i.e. the stochastic system is unstable at the equilibrium point Q no matter whether the deterministic system is stable at equilibrium point Q or not.

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Stochastic Bifurcation D-Bifurcation

Case I: When μ3 = 0, μ7 = 0. Then system (2.4) becomes  μ  12  μ2  4 2 )r dt + r dr = (μ1 + dWr . 8 8

(4.1)

When μ4 = 0, equation (4.1) is a determinate system, and there is no bifurcation phenomenon. Here we discuss the situation μ4 = 0, let  μ4  μ2 − r, m(r) = μ1 + 8 16

σ(r) =

 μ  12 4

8

r.

The continuous random dynamic system generate by (4.1) is  t  t ϕ(t)x = x + m(ϕ(s)x)ds + σ(ϕ(s)x) ◦ dWr . 0

0

where ◦dWr is the differential at the meaning of Statonovich, it is the unique strong solution of (4.1) with initial value x. And m(0) = 0, σ(0) = 0, so 0 is a fixed point of ϕ. Since m(r) is bounded and for any r = 0, it satisfy the ellipticity condition: σ(r) = 0; it assure that there is at most one stationary probability density. According to the Itˆ o equation of amplitude r(t), we obtain its FPK equation corresponding to (4.1) as follows: ∂  ∂ 2  μ4 2   μ2    ∂p =− μ1 + r p + 2 r p . ∂t ∂r 8 ∂r 8 Let

∂p ∂t

= 0, then we obtain the solution of system (4.2)   t 2m(u) −1 du . p(t) = c|σ (r)| exp 2 0 σ (u)

(4.2)

(4.3)

The above dynamical system(4.2) has two kinds of equilibrium state: fixed point and non-stationary motion. The invariant measure of the former is δ0 and it’s probability density is δx . The invariant measure of the latter is ν and it’s probability density is (4.3). In the following, we calculate the lyapunov exponent of the two invariant measures. Using the solution of linear Itˆ o stochastic differential equation, we obtain the solution of system (4.1)     t  t   σ(r)σ (r) ds + σ (r)dWr . m (r) + (4.4) r(t) = r(0) exp 2 0 0 The lyapunov exponent with regard to μ of dynamic system ϕ is defined as: λϕ (μ) = lim

t→+∞

1 ln ||r(t)||, t

(4.5)

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substituting(4.4) into (4.5), note that σ(0) = 0, σ (0) = 0, we obtain the lyapunov exponent of the fixed point:    t  t   1 λϕ (δ0 ) = lim ds + σ (0) dWr (s) ln ||r(0)|| + m (0) t→+∞ t 0 0 μ4 μ2 = μ1 + − . (4.6) 8 16 For the invariant measure which regard (4.4) as its density, we obtain the lyapunov exponent:    1 t  λϕ (ν) = lim m (r) + σ(r)σ (r) ds t→+∞ t 0 " ! √ 3 16 μ4 2 μ4 μ2 μ2 2 − − ) . (4.7) = −32 2μ4 μ1 + exp (μ1 + 8 16 μ4 8 16 Let α = μ1 + μ82 − μ164 . We can obtain that the invariant measure of the fixed point is stable when α < 0, but the invariant measure of the non-stationary motion is stable when α > 0, so α = αD = 0 is a point of D-bifurcation. Simplify Eq.(4.3), we can obtain pst (r) = cr

2(8μ1 +μ2 −μ4 ) μ4

,

(4.8)

r −→ 0,

(4.9)

where c is a normalization constant, thus we have pst (r) = o(rv )

2 −μ4 ) where v = 2(8μ1 +μ . Obviously when v < −1, that is μ1 + μ82 − μ164 < 0, μ4 pst (r) is a δ function. when −1 < v < 0, that is μ1 + μ82 − μ164 > 0, r = 0 is a maximum point of pst (r) in the state space, thus the system occur D-bifurcation when v = −1, that is μ1 + μ82 − μ164 = 0, is the critical condition of D-bifurcation at the equilibrium point. When v > 0, there is no point that make pst (r) have maximum value, thus thus the system does not occur P-bifurcation.

Case II: When μ3 = 0, μ7 = 0. then Eq(2.4) can rewrite as following

# Let φ =

  μ  12 μ7 3  μ2 4 2 )r + r dt + r dr = (μ1 + dWr . 8 8 8 −μ7 8 r, μ7

(4.10)

< 0, then we consider the system (4.10) becomes

 1   −μ4 2 μ2 dφ = (μ1 + )φ − φ3 dt + φ ◦ dWt 8 μ7

(4.11)

which is solved by  φ exp φ → ψμ1 (t, ω)φ =  1 + 2φ2

$

t 0

exp

(μ1 +

μ2 8

  2 (μ1 +

 )t + μ2 8

−μ4 μ7

 )t +



1 2

−μ4 μ7

Wt (ω) 

1 2

Ws (ω)

 1/2 . ds

(4.12)

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We now determine the domain Dμ1 (t, ω), where Dμ1 (t, ω) := {φ ∈  : (t, ω, φ) ∈ D}X(D =  × Ω × X) is the (in general possibly empty) set of initial values φ ∈  for which the trajectories still exist at time t and the range Rμ1 (t, ω) of ψμ1 (t, ω) : Dμ1 (t, ω) → Rμ1 (t, ω). We have % , t ≥ 0, (4.13) Dμ1 (t, ω) = (−dμ1 (t, ω), dμ1 (t, ω)), t < 0, We can now determine Eμ1 (ω) := ∩t∈ Dμ1 (t, ω) and obtain % Eμ1 (ω) =

− (−d− μ1 (t, ω), dμ1 (t, ω)), {0},

μ1 + μ82 > 0, μ1 + μ82 ≤ 0,

(4.14)

The ergodic invariant measures of system (4.10) are (i) For μ1 + μ82 ≤ 0, the only invariant measures is μμω1 = δ0 . (ii) For μ1 + μ82 > 0, we have the three invariant forward Markov measures μ1 μ1 μω = δ0 and υ±,ω = δ±kμ1 (ω) , where   kμ1 (ω) := 2



0

μ2 )t + 2 exp 2(μ1 + 8 −∞



−μ4 μ7

 12

− 12

Wt (ω) ds

.

We have Ekμ2 1 (ω) = α. Solving the forward Fokkwer-planck equation L∗ pμ1 = −

    μ4 μ2 μ4  2 )φ − φ Pν1 (φ) = 0 (μ1 + φ − φ3 Pμ1 (φ) − 8 2μ7 2μ7

yield (i) pμ1 = δ0 for all (μ1 + (ii) for pμ1 > 0  qμ+1 (φ)

=

μ2 8 ),

Nμ1 φ− 0,

μ 2μ7 (μ1 + 2 ) 8 −1 μ4

2

exp( 2μμ74φ ),

μ (μ +

μ2

)

φ > 0, φ ≤ 0, μ μ7 (μ1 + 2 ) 8

μ4 and qμ+1 (φ) = qμ+1 (−φ), where Nμ−1 = Γ (− 7 μ1 4 8 )(− μμ47 ) . μ1 Naturally the invariant measures υ±,ω = δ±kμ1 (ω) are those corresponding to the stationary measures qμ+1 . Hence all invariant measures are Markov measures. We determine all invariant measures(necessarily Dirac measure) of local RDS χ generated by the SDE

 1   −μ4 2 μ2 3 )φ − φ dt + φ ◦ dW. dφ = (μ1 + 8 μ7

(4.15)

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 12  4 on the state space , μ1 + μ82 ∈  and −μ ≥ 0. We now calculate the μ7 Lyapunov exponent for each of these measure. The linearized RDS χt = DΥ (t, ω, φ)χ satisfies the linearized SDE  1   −μ4 2 μ2 dχt = (μ1 + ) − 3(Υ (t, ω, φ))2 χt dt + χt ◦ dW. 8 μ7 hence 

μ2 DΥ (t, ω, φ)χ = χ exp (μ1 + )t + 8



−μ4 μ7

 12



t

Wt (ω) − 3

(Υ (s, ω, φ))2 ds . 0

Thus, if νω = δφ0 (ω) is a Υ - invariant measure, its Lyapunov exponent is 1 log DΥ (t, ω, φ)χ t→∞ t μ2 = (μ1 + ) − 3Eφ20 , 8

λ(μ) = lim

provided the IC φ20 ∈ L1 (P) is satisfied. μ1 (i) For μ1 + μ82 ∈ , the IC for ν,ω = δ0 is trivially satisfied and we obtain λ(ν1μ1 ) = (μ1 +

μ2 ). 8

So ν1μ1 is stable for μ1 + μ82 < 0 and unstable for μ1 + μ82 > 0. μ1 0 (ii) For μ1 + μ82 > 0, ν2,ω = δdμω1 is F−∞ measurable, hence the density pμ1 of μ1 μ1 ρ = Eν2 satisfies the Fokker-Planck equation L∗ν1

    μ4 μ2 μ4  2 3 )φ − φ pμ1 (φ) = 0 =− (μ1 + φ − φ pμ1 (φ) − 8 2μ7 2μ7

which has the unique probability density solution  2  μ 2μ (μ + 2 ) φ μ7 − 7 μ1 8 −1 μ1 4 P (φ) = Nμ1 φ exp , φ>0 μ4 

Since Eν2μ1 φ2 = E(dμ−1 )2 =

∞

φ2 pμ1 (φ)dφ < ∞,

0

the IC is satisfied. The calculation of the Lyapunov exponent is accomplished by observing that    12  4 exp 2(μ1 + μ82 )t + 2 −μ W (ω) t μ7 Ψ (t)   dμ−1 (ϑt ω)2 = , = 1   $t 2Ψ μ2 −μ4 2 2 −∞ exp 2(μ1 + 8 )s + 2 μ7 Ws (ω) ds

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Hence by the ergodic thoerem E(dμ−1 )2 =

1 1 μ2 lim log Ψ (t) = μ1 + , 2 t→∞ t 8

finally

μ2 ) < 0. 8 measurable. Since L(dμ+1 ) = L(dμ−1 )

λ(ν2μ1 ) = −2(μ1 + (iii) For (μ1 +

μ2 8 )

μ1 0 > 0, ν2,ω = δdμω1 is F−∞

E(−dμ−1 )2 = E(dμ−1 )2 = μ1 +

μ2 8

thus

μ2 ) < 0. 8 The two families of densities (qμ+1 )μ1 >0 clearly undergo a P-bifurcation at the μ4 - which is the same value as the transcritical case, parameter value μ1P = − 2μ 7 since the SDE linearized at φ = 0 is the same in the next section. in both case. Hence, we have a D-bifurcation of the trivial reference measure δ0 at μD = 0 μ (μ + 2 )2 and a P-bifurcation of μP = 1 2 8 . λ(ν2μ1 ) = −2(μ1 +

4.2

P-Bifurcation

o equation of amplitude r(t), Case I: When μ3 = 0, μ7 = 0. According to the Itˆ we obtain its FPK equation as follows: ∂p ∂  μ3   ∂ 2  μ2  μ4 2   μ3 + p =− μ1 + r+ p + 2 r ∂t ∂r 8 r ∂r 8

(4.16)

with the initial value condition μ7 = 0, p(r, t|r0 , t0 ) → δ(r − r0 ), t → t0 , where p(r, t|r0 , t0 ) is the transition probability density of diffusion process r(t). The invariant measure of r(t) is the steady-state probability density pst (r) which is the solution of the degenerate system as follows: 0=−

μ3   ∂ 2  μ2  μ4 2   ∂  μ3 + p . μ1 + r+ p + 2 r ∂r 8 r ∂r 8

(4.17)

Through calculation, we can obtain & pst (r) = 4

2 −3v 2−v 2 μ3 π



μ4 μ3

3 2

−1    v−2 1 −v Γ (2 − v) Γ r 2 μ4 r 2 + 8μ3 ,(4.18) 2

$ ∞ x−1 −t where v = (8μ1 + μ2 )μ−1 e dt. 4 , Γ (x) = 0 t According to Namachivaya’s theory, we now calculate the most possible amplitude r∗ of system (2.4)., i.e. pst (r) has a maximum value at r∗ . So we have ' ' d2 pst (r) '' dpst (r) '' = 0, 0

r=0

2

= r=˜ r

2+(8μ1 +μ2 −μ4 )μ−1

= 26+3(8μ1 +μ2 −μ4 )μ μ3  (8μ1 + μ2 − μ4 )3 8μ3 − −16 (8μ1 +

8μ3 μ4 8μ1 +μ2 −μ4 μ2 − μ4 )3

 8μ1μ+μ2 4

< 0.

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X. Qin, Z. Huang, and W. Tan

Thus what we need is r∗ = r˜. In the meantime, pst (r) is 0 (minimum) at r = 0. This means that the system subjected to random excitations is almost unsteady at the equilibrium point (r = 0) in the meaning of probability. The conclusion is to go all the way with what has been obtained by the singular boundary theory. The original nonlinear stochastic system has a stochastic Hopf bifurcation at r = r˜. −8μ3 , (i.e. r = r˜). x21 + x22 = 8μ1 + μ2 − μ4 We now choose some values of the parameters in the equations, draw the graphics of pst (r) . The curves in the graph belonging to the cond1,2,3,4 in turn are shown in Fig.1a. It is worth putting forward that calculating the Hopf bifurcation with the parameters in the original system is necessary. If we now have values of the original parameters in system (2.1), that a = 0.6, a11 = 0.6, a12 = −1, a13 = 0, a21 = 0.8, a23 = 1, a24 = −1, α1 = 0.4, α2 = 0.5, I1 = 0.3, I2 = 0.6. After further calculations we obtain μ1 = −0.6, μ2 = 1.8125, μ3 = 0.4388, μ4 = 1.8075, ν=

8μ1 + μ2 1 = −2.41287 < , μ4 2

p(r) =

150.435r2 . (3.5104 + 1.8075r2 )3.65284

What is more is that r˜ = 0.855626 where pst (r) has the maximum value(see Fig.1b). Case II: When μ3 = 0, μ7 = 0. then Eq(2.4) can rewrite as following  μ  12  μ7 3  μ2 4 2 )r + r dt + r dWr . dr = (μ1 + 8 8 8

(4.19)

According to the Itˆ o equation of amplitude r(t), we obtain its FPK equation form (4.19) as follows: " ( %!  ∂p ∂ μ4 μ2 μ4 ∂ 2  2 3 =− )r − r p(r) (4.20) r − r p(r) − (μ1 + ∂t ∂r 8 2μ7 2μ7 ∂r2 with the initial value condition μ7 = 0, p(r, t|r0 , t0 ) → δ(r − r0 ), t → t0 , where p(r, t|r0 , t0 ) is the transition probability density of diffusion process r(t). The invariant measure of r(t) is the steady- state probability density pst (r) which is the solution of the degenerate system as follows: " ( %!  μ4 ∂ μ2 μ4 ∂ 2  2 3 )r − r p(r) . (4.21) 0=− r − r p(r) − (μ1 + ∂r 8 2μ7 2μ7 ∂r2 Through calculation, we can obtain  exp pst (r) =

r 2 μ7 μ4



r−1−

μ 2(μ1 + 2 )μ7 8 μ4 μ

  (μ1 +μ 82 )μ7  μ (μ1 + 82 )μ7 4 μ7 − μ4 Γ − μ4

.

(4.22)

Stochastic Stability and Bifurcation Analysis on Hopfield Neural Networks

177

According to Namachivaya’s theory, we now calculate the most possible amplitude r∗ of system (4.19)., i.e. pst (r) has a maximum value at r∗ . So we have ' ' dpst (r) '' d2 pst (r) '' = 0, 0) . If the points in error are penalized quadratically with a penalty factor C ' , then, it has been shown that the problem reduces to that of a separable case with C = ∞ . The kernel function is modified as K ' ( xi , x j ) = K ( xi , x j ) +

1 δ i, j C'

(5)

where δ i , j = 1 if i = j and δ i , j = 0 otherwise. The advantage of this formulation is that the SVM problem reduces to that of a linearly separable case. It can be seen that training the SVM involves solving a quadratic optimization problem which requires the use of optimization routines from numerical libraries. This step is computationally intensive, can be subject to stability problems and is nontrivial to implement

3 Geometric SVMs The Geometric SVM [11]improves the scaling behavior of the Direct SVM [6]. The Direct SVM is an intuitively appealing algorithm, which builds the support vector set incrementally. Recently it has been proved that the closest pair of points of the opposite class is always support vectors[12]. Direct SVM starts off with this pair of points in the candidate Support Vector set. It has been conjectured that the maximum violator during each by iteration is a support vector. The algorithm finds the maximum violator during each iteration and rotates the candidate support plane to make the maximum violator a Support Vector. In case the dimension of the space is exceeded or all the data points are used up, without convergence, the algorithm reinitializes with the next closest pair of points from opposite classes. The advantage of the Direct SVM algorithm is that it is geometrically motivated and simple to understand. The Geometric SVM using an optimization-based approach to add points to the candidate Support Vector set. This algorithm uses a greedy algorithm, which picks the next immediately available violating point for inclusion in the candidate support vector set. Neither of the algorithms has a provision to backtrack, i.e. once they decide to include a point in the candidate support vector set they cannot discard it. During each step, both the algorithms spend their maximum effort in finding the maximum violator. If S is the current candidate support vector set, and N is the size of the dataset, the

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algorithms spend

( N − s ) S kernel evaluations to locate the maximum violator.

Caching schemes have been proposed to increase the efficiency of this step. The reasons lead us to a greedy algorithm that picks the next immediately available violating point for inclusion in the candidate support vector set. Some stray nonsupport vectors may be picked up for inclusion in the candidate set because of this greedy approach. This algorithm backtracks later by pruning away such stray points from the candidate support vector set. In order to satisfy the KKT conditions for all points it has to make repeated passes though the dataset. Recently some work has also been done on incremental SVM algorithms that can converge to exact solutions and also efficiently calculate leave one out errors [7]. The algorithm works by adding a point at a time to the current dataset. It is shown that the addition of a point converges in a finite number of steps. The algorithm calculates the change in the KKT conditions of each one of the current data points in order to do book keeping. In case the size of the current data set is N and the number of Support Vectors is

S , then, this algorithm has to perform N S kernel operations for addi-

tion of the next point. The authors suggest a practical online variant where they introduce a δ margin and concentrate only those points that are within the δ margin of the boundary. But, it is clear that the results may vary by varying the value of δ . Instead of concentrating on the entire current dataset, which may be costly, the algorithm focuses its attention only on the current support vector set. It then picks data points greedily for inclusion in the current support vector set. While augmenting the current support vector set, some already existing support vectors may become well classified because of the addition of a new support vector. We prune away those points using the decremental technique described in[7]. It may also happen that other points in the dataset may become violators or Support Vectors. The Incremental algorithm handles such points by doing costly book keeping in the current iteration itself and hence does not need to make repeated passes over the dataset. On the other hand our algorithm does not handle such points in the current iteration, it makes repeated passes over the dataset to identify and satisfy such points.

4 CBGSVM Model and Algorithm Design In this part, first, introduce clustering feature and k-means clustering, and then we develop the algorithm of clustering-based Geometric SVM. 4.1 Clustering Feature

A subclass of training sample is S = { xi , i = 1, , N } , xi ∈ R d , then define the subclass’s clustering feature as: →

CF = ( N , S L , Ss ) →

N

N

i =1

i =1

where S L = ∑ xi , Ss = ∑ xi2 .

(6)

Clustering-Based Geometric Support Vector Machines

211

A clustering feature is a triple summarizing the information that we maintain about a cluster, embody compression of the class’s information. It doesn’t include all samples, instead the information of 0 order moment, 1-order moment, and 2-order moment. Meanwhile, define the center and radius of a subclass: (7)

N

C= N

R=

∑ i =1

i =1

i

N xi − C

(8)

2

N

Definition 1 (CF Additively Theorem) and S2 = { xi , i = 1,

∑x

[8]

: Given subclass S1 = { xi , i = 1, , N1} , xi ∈ R d ,

, N 2 } , xi ∈ R d , S1 ∩ S2 = ∅ , their clustering feature are

→

→

CF1 = ( N1 , S L1 , Ss1 ) , CF2 = ( N 2 , S L 2 , Ss2 ) , S = S1 + S 2 , then the CF vector of S that is formed by merging the two clusters, is: →

→

(9)

CF1 + CF2 = ( N1 + N 2 , S L1 + S L 2 , Ss1 + Ss2 )

From the CF definition and additively theorem, we know that the CF vectors of clusters can be stored and calculated incrementally and accurately as clusters are merged. 4.2 k-Means Clustering

Suppose sample set

S = { x1 ,

, xN } , separate the set S into k clusters, samples in

one cluster have high similarity, samples in different cluster are dissimilarity. The algorithm of K-means clustering is as following: Input: Training sample set S and the number of cluster k (k 0 ∀p ∈ S . But, this condition may be violated if any point in S blocks c. When we say that a point p ∈ S is blocking the addition of c to S what we mean is that α p of that point may become negative due to the addition of c to S. What it physically implies is that p is making a transition from S to the well-classified set R. Because of the presence of such centers we may not be able to update α c by the amount specified by Equation 8. In such a case we can prune away p from S by using [7] Rij = Rij − R−ss1 Ris Rsj ∀Rij ∈ R

(21)

and then remove the alpha entry corresponding to p from S so that all the other points in S continue to remain support vectors. We now try to add c to this reduced S. We keep pruning centers from S till c can actually become a support vector. Algorithm steps Using the ideas we discussed above an iterative algorithm can be designed which scans through the dataset looking for violators. Using ideas presented in D the violator is made a support vector. Blocking points are identified and pruned away by using the ideas presented in E. The algorithm stops when all points are classified within an error bound i.e. yi f ( xi ) > 1 − ε i . The outline of our algorithm is as

⑥

following: Step1: Divide the training samples into two classes: P (positive) and N (negative); Step2: Cluster the two classes of k-means, and then there are 2k centers, the information of all sample can be compressed into 2k centers; Step3: Choose support vector={closest pair centers from opposite classes}; Step4: while there are violating points do Find a violated center Candidate SV = candidate SV violator if any

α p < 0 due

to addition of c to S then

candidate SV = candidate SV \ p repeat till all such points are pruned end if end while

The above is the algorithm we develop; use the algorithm to three publicly available datasets, the result of experiments validate our algorithm’s useful.

Clustering-Based Geometric Support Vector Machines

215

5 Experiment and Result Compare our algorithm with SVB on 3 publicly available datasets [10]. All experiments were run on a 3.06 GHz, Intel Pentium4 HT machine with 512MB RAM running Windows 2000. The code was written in C as well as in MATLAB. We uniformly used a value of 0.001 for the error bound, i.e. we stop the algorithm when yi f ( xi ) > 0.999 ∀i . The results are reported in tables. Use the WPBC dataset from the UCI Machine Learning repository [10]. This dataset consists of 683 data points, each having a dimension of 9. We used the Gaussian kernel with C = 100 , σ 2 = 4.0, k = 100,90,80, 70 . We do our experiment and reproduce our results in table 1. Table 1. CBGSVM VS SVM on WPBC Database kernel

Algorithm CBGSVM

Training samples 140

Support vectors

Accuracy

Time

92

88.2%

168.5(s)

CBGSVM

160

96

89.1%

172.0(s)

CBGSVM

180

102

92.9%

176.5(s)

CBGSVM

200

106

93.4%

181.5(s)

SVM

683

276

95.6%

432.5(s)

Use the Adult-1 dataset from the UCI Machine Learning repository[10]. This is a sparse dataset which consists of 1605 data points, each having a dimension of 123. We used the Gaussian kernel with C = 100 , σ = 10.0, k = 200,180,160,140 . We do our experiment and reproduce our results in table 2. 2

Table 2. CBGSVM VS SVM on Adult-1 Database kernel

Algorithm

Training samples

Support vectors

Accuracy

Time

CBGSVM

280

201

85.6%

610.5(s)

CBGSVM

320

206

87.4%

618.0(s)

CBGSVM

360

213

90.6%

624.0(s)

CBGSVM

400

238

92.3%

664.5(s)

SVM

1605

689

93.8%

1884.5(s)

Use the Adult-7 dataset from the UCI Machine Learning repository is a sparse data set of 16, 100 points, each having a dimension of 123[10]. We used the Gaussian kernel with C = 5 , σ 2 = 10.0, k = 2000,1800,1600,1400 . We do our experiment and reproduce our results in table 3.

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J. Chen and F. Pan Table 3. CBGSVM VS SVM on Adult-7 Database kernel

Algorithm

Training samples

Support vectors

Accuracy

Time

CBGSVM

2800

2378

83.1%

1056.5(s)

CBGSVM

3200

2389

84.3%

1082.0(s)

CBGSVM

3600

2416

87.5%

1146.5(s)

CBGSVM

4000

2571

89.3%

1231.5(s)

SVM

16100

10236

91.4%

3732.5(s)

As can be seen from the above tables, the CBGSVM algorithm outperforms the standard SVM on each one of datasets tested. The number of support vectors found by CBGSVM and SVM is different. The number of CBGSVM is much less than SVM. Because k-mean clustering compress the information of samples of cluster into the center, the number of center is decided by the k . The accuracy of CBGSVM and SVM is almost the same, but the training time is greatly diminished. From the above tables, we can see the training time of CBGSVM is 3 times less than the SVM. But there is one problem, k is chose arbitrary, from the three tables, we can see the bigger k that lead to more accuracy, and it takes more time to train, so we should develop the method to choose suitable k , this k will lead to consider the time and accuracy at the same time, this is the work we will do in the future.

6 Conclusions The new algorithm is proposed in this paper that is efficient, intuitive and fast. We show that the algorithm significantly outperforms the standard SVM in terms of the number of kernel computations. Because we use the k-mean clustering to compress the information of samples, and the approach build the support vector, set our algorithm does not suffer from numerical instabilities and round off errors that plague other numerical algorithms for the SVM problem. Our algorithm currently does not use any kind of kernel cache to reuse kernel computations. We are currently investigating methods to speed up the algorithm using some efficient caching scheme. K-mean clustering is used to cut the samples greatly, the reduction of sample accelerates the training process, and the training time is also greatly diminished. We also observe that the memory utilization of the algorithm is governed by the R matrix which 2

scales as O( S ) . Hence, our algorithm does not scale well for those problems where the R matrix cannot be held in main memory. But, this is not a serious limitation, for example on a machine with 512 MB of main memory we can store the R matrix corresponding to as many as 20,000 support vectors. We are investigating methods to store a compact representation of R in order to reduce this memory overhead. It can be observed that the addition of a vector to the support vector set is entirely reversible. Using this property Gert Cauwenberghs et. al. [7] have calculated the leave one out error. We propose to use similar techniques to calculate the leave one out error based on the order in which the data points were added to support vector set.

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References 1. Vapnik, V.N.: The nature of statistical learning theory. Springer, Heidelberg (1995) 2. Cf, http://www.clopinet.com/isabelle/Projects/SVM/appllist.html 3. Osuna, E., Freund, R., Girosi, F.: Training support vector machines: An application to face detection. In: Proceedings of the 1997 Conference on Computer Vision and Pattern Recognition (CVPR 1997), Puerto Rico, pp. 17–19 (1997) 4. Joachims, T.: Making large-scale support vector machine learning practical. In: Advances in Kernel Methods: Support Vector Machines, pp. 169–184. MIT Press, Cambridge (1999) 5. Platt, J.: Fast training of support vector machines using sequ- ential minimal optimization. In: Advances in Kernel Methods: Support Vector Machines, pp. 185–208. MIT Press, Cambridge (1999) 6. Danny, R.: Direct SVM: A simple support vector machine perceptron. Journal of VLSI Signal Processing Systems (2001) 7. Gert, C., Tomaso, P.: Incremental and decremental support vector machine learning. In: Advances in Neural Information Processing Systems (NIPS*2000), vol. 13. MIT Pres, Cambridge (2001) 8. Zhang, T., Ramakrishnan, R., Livny, M.B.: An efficient data clustering method for very large databases. In: Proc. ACM SIGMOOD Int. Conf., Management of Data, pp. 103–114 (1996) 9. Han, J.W., Kamber, M.: Data Mining: Concept and Techniques. Morgan Kanfmann, San Mateo (2000) 10. Blake, C.L., Merz, C.J.: UCI repository of machine learning databases (1998) 11. Vishwanathan, S.V.N., Smola, A., Narasimba, N., Murty, S.: A simple SVM algorithm. In: ICML, Washington, DC, USA (2003) 12. Danny, R.: DirectSVM: A fast and simple support vector machine perceptron. In: Proceedings of IEEE International Workshop on Neural Networks for Signal Processing, Sydney, Australia (2000) 13. Jain, A., Murty, M., Flynn, P.: Data clustering: A review. ACM Comput. Surv. 31, 264–323 (1999) 14. Fayyad, U., Piatetsky-Shapiro, G., Smyth, P., Uthurusamy, R.: Advances in Knowledge Discovery and Data Mining. MIT Press, Cambridge (1996) 15. Zhang, T., Ramakrishnan, R., Livny, M.: BIRCH: A new data clustering algorithm and its applications. In: Data Mining and Knowledge Discovery, vol. 1 (1997) 16. Brachman, R., Khabaza, T., Kloesgen, W., Piatetsky-Shapiro, G., Simoudis, E.: Industrial Applications of Data Mining and Knowledge Discovery. Communications of ACM 39 (1996) 17. Bradley, P.S., Usama, F., Cory, R.: Scaling clustering algorithms to large databases. In: Proc. of the 4th Int. Conf. on Knowledge Discovery and Data Mining (KDD 1998), pp. 9–15. AAAI Press, Menlo Park (August 1998) 18. Selim, S.Z., Ismail, M.A.: K-Means-Type Algorithms: A Generalized Convergence Theorem and Characterization of Local Optimality. IEEE Trans. on Pattern Analysis and Machine Intelligence PAMI-6 (1984) 19. Keerthi, S.S., Shevade, S.K., Bhattachayya, C., Murth, K.R.K.: Improvements to platt’s smo algorithm for SVM classifier design. Neural Computation 13, 637–649 (2001)

A Fuzzy-PID Depth Control Method with Overshoot Suppression for Underwater Vehicle Zhijie Tang, Luojun, and Qingbo He School of Mechatronics Engineering and Automation, Shanghai University, Shanghai, China, 200072 [email protected]

Abstract. This paper presents an underwater vehicle depth fuzzy-PID control method based on overshoot prediction. The underwater vehicle in the shallow waters is affected by the inevitable surge. In order to achieve reliable and stable depth control, this paper realizes the depth and overshoot forecasts by calculating quadratic equation with depth error acceleration and depth error change rate to derive the overshoot time possibility. With this time possibility and depth error, the fuzzy controller calculates the PID controller parameters, and then the underwater vehicle completes the fast and non-overshoot depth control. The simulation results show that the method is effective and feasible. Keywords: Fuzz y-PID, Overshoot Suppression, Depth control, underwater vehicle.

1 Introduction Underwater vehicles because of its flexible are very suitable for the completion of underwater monitoring, detection, testing and other tasks. The motion equation of underwater robot is six degrees of freedom. There are the coupling effects between each degree of freedom. This brought great difficulties to the control system design. When underwater vehicles working in shallow water, the inevitable surge will impact them, resulting in errors in the depth of control, affecting its normal operation. When the underwater vehicles control in a complex environment, improving the response speed and overshoot suppression is the most important [1]. In particular, for vertical control, overshoot means that the locating speed of the underwater vehicles will be relatively large; it would appear the phenomena of underwater robots damage at the bottom of the lake, which is to be avoided when designing the controller. In the current depth control methods of underwater vehicles, there are PID control, sliding mode control and neural network control[2-6]. Simple PID control, there is parameter configuration contradictory between response speed and overshoot control. Sliding mode control could easily lead to system jitter and affect control accuracy. Neural network control for the underwater vehicles in disturb surge, need for online learning and revise the operating parameters, will

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 218–224, 2010. © Springer-Verlag Berlin Heidelberg 2010

A Fuzzy-PID Depth Control Method with Overshoot Suppression

219

lead the lag phenomenon of the underwater vehicles response, can not meet the requirements of rapid response. This article established the motion equations of underwater vehicles. With error acceleration and error change rate, the paper completes the overshoot forecast. This becomes the Fuzzy-PID controller input reference. Finally, this paper designs the fuzzy rules to achieve the PID drive control parameters. Thus, no overshoot and fast response depth control for underwater vehicles is achieved. Simulation results show that the method is effective, feasible and fast.

2 Mathematical Model of Vertical Movement Combination of rigid body motion and fluid mechanics principles, underwater robot motion model can be achieved. The dynamics of a 6-degree-of-freedom underwater vehicle can be described in the following general form as formula (1) [7]:

Ma + C (a )a + D(a )a + g ( x) = τ

(1)

x = J ( x)q Where a =[u v w p q r]T is the body-fixed velocity vector in surge, sway, heave, roll, pitch and yaw; x=[x y z Φ θ Ψ]T is the corresponding earth-fixed vector; M is a 6×6 symmetric positive definite inertia matrix; C is a 6x6 matrix of Coriolis and centripetal terms; D is a 6×6 dissipative matrix of hydrodynamic damping terms; g(x) represents the restoring forces and moments and J(q) is the transform matrix relating the body-fixed reference frame to the inertial reference frame; τ is the input torque vector.

3 Design of Fast Fuzzy PID Control Mechanism The system block diagram of the underwater depth fuzzy PID controller (FC-PID) is shown in Figure 1.

Fig. 1. The system diagram of FC-PID

220

Z. Tang, Luojun, and Q. He

As shown, the system increased the overshoot prediction module of underwater robots in the future to respond to the current state to predict. 3.1 Overshoot Predictor The error acceleration is the secondary variable of the error change detection. The error acceleration directly reflects the error change speed and the response of the control system. When we get the acceleration and velocity of the error change, then we can calculate the final error over time shown in formula (2).

SE =

1 ecc ⋅ t 2 + ec ⋅ t = e 2

(2)

Where ecc is the error acceleration, the ec is the error change rate, e is the initial error. Form the formula (2) we set that the final error is zero, and then we can get the overshoot prediction time(OT) by calculating quadratic equation of time t. With this OT and error (E), we set the fuzzy rule to change the PID Parameter.

，

3.2 Fuzzy PID Parameter Adjustments As we all know, PID controller law is shown in formula (3).

u = Kp ⋅ e + Ki ⋅ ∫ e ⋅ dt + Kd ⋅ e

(3)

Where Kp, Ki and Kp are proportional, derivative and integral gains respectively. Considering the control purpose, we use the Mamdani-type fuzzy logic controller to adaptively tune these gains based on several sets of fuzzy rules. The proportional gain Kp has the effect of reducing the rise time and the steadystate error. The larger it is the faster will the system response. The proportional gains of high value will have the system perform robustly. However, usually low gains are preferred in practice in order to avoid an oscillatory response. Then if the OT is big then we increase the Kp, when OT decrease, we decrease the Kp. The fuzzy rule of Kp is shown in Table 1. Similarly, The proportional gain Ki has the effect of eliminating the static error of the system. The larger it is the smaller will be the static system error. However, the integral gains of high value will easily cause overshoot. The fuzzy rule of Ki is shown in Table 2. Similarly, the derivative gain Kd has the effect of dynamic smoothing of the system. The larger it is the larger will be the dynamic correction features. The derivative gains of high value will have the system overshoot suppression perform. However, the derivative gains of high value will easily cause low system response. The fuzzy rule of Kd is shown in Table 3.

A Fuzzy-PID Depth Control Method with Overshoot Suppression Table 1. Rule base for Kp

N

Kp B Z O

P

P OT

M

S

S P P

P

P

P

P

P

P

P M

P M

P B

P S

S

M

B

P

P

P

P B

S

S

S

B

P

P

P

P M

S

S

M

B

Z

P

P

P S

O

S

M

B

P

P

P

O

S

S

B

B

P

P

E Z

N S

S

M

M

N

P B

P B

P B

Table 2. Rule base for Ki

N

Ki B Z O

P

P OT

M

M

S P P

Z

P

P

P

P

P

P S

P S

P S

P M

M

M

M

P

P

P

P B

B

B

B

S

P

P

P

P M

B

B

M

S

P

P

P

P S

B

B

S

O

P

P

Z

O

B

M

O

B

P

P

E Z

N S

B

S

M

N

Z O

P S

Z O

Table 3. Rule base for Kd

N

Kd B Z O

P

P OT

P

Z

P P

P

P

Z O

Z O

Z O

P S

S

S P

S

P

P

P

P B

M

M

M Z

O

P

P

P

P M

B

B

S Z

O

P

P

Z

P S

B

M

O Z

O

P

P

Z O

O

B

S

E Z

N S

M

O

M B

M

S

S

N

Z O

Z O

Z O

221

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4 Simulation Results With Simplified treatment, we get the underwater vehicle depth control transmission function as formula (4).

G (s) =

0.2466 s + 0.1251 3 s + 1.8101s 2 + 0.9412 s + 0.1603

(4)

We get a set of parameters of the quick response PID controller (Q-PID) that Kp is equal to 16, and Ki is equal to 1 and Kd is equal to 0.5. Then we get another set of parameters of the non-overshoot PID controller (N-PID) that Kp is equal to 3, and Ki is equal to 2 and Kd is equal to 0.5. Then we get the 5 meter depth control simulation results shown in figure 2. The PID parameter turning of FC-PID is shown in figure 3. The overshoot prediction OT is shown in figure 4. From the simulation, we can see that FC-PID can get quicker response than N-PID and FC-PID can get non-overshoot than Q-PID.

Fig. 2. Response of 5 meter set depth control using Q-PID, N-PID and FC-PID

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Fig. 3. PID parameter turning of FC-PID

Fig. 4. Overshoot prediction OT

5 Conclusions This paper presents a new underwater vehicle depth fuzzy-PID control method based on overshoot prediction. For the further research, the overshoot prediction OT should be smoothed. Form the simulation we can conclude that this method can effectively realize the depth control without overshoot.

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Acknowledgement. The authors would like to acknowledge Shanghai Education Commission Foundation for providing financial support for this work under grant

No. 09zz091.

References 1. Silvia, M.Z., Giuseppe, C.: Remotely operated vehicle depth control. Control Engineering Practice 11, 453–459 (2003) 2. Lee, P.M., Hong, S.W., Lim, Y.K., Lee, C.-M., Jeon, B.H., Park, J.W.: Discrete-time quasisliding mode control of an autonomous underwater vehicle. IEEE Journal of Oceanic Engineering, 388–395 (1999) 3. Shi, X.C., Xiong, H.S., Wang, C.G., Chang, Z.H.: A New Model of Fuzzy CMAC Network with Application to the Motion Control of AUV. In: Proceedings of the IEEE International Conference on Mechatronics & Automation, Niagara Falls, Canada, pp. 2173–2178 (2005) 4. Song, F., Smith, S.M.: Design of sliding mode fuzzy controllers for an autonomous underwater vehicle without system model. In: MTS/IEEE Oceans, pp. 835–840 (2000) 5. Bin, X., Norimitsu, S., Shunmugham, R.P., Fred, P.: A Fuzzy Controller for Underwater Vehicle-Manipulator Systems. In: MTS/IEEE Oceans, pp. 1110–1115 (2005) 6. Lin, C.K.: Adaptive Critic Control of Autonomous Underwater Vehicles Using Neural Networks. In: Proceedings of the Sixth International Conference on Intelligent Systems Design and Applications, pp. 122–127 (2006) 7. Fossen, T.I., Sagatun, S.I.: Adaptive control of nonlinear systems:A case study of underwater robotic systems. Journal of Robotic Systems, 339–342 (1991)

Local Class Boundaries for Support Vector Machine Guihua Wen, Caihui Zhou, Jia Wei, and Lijun Jiang South China University of Technology,Guangzhou 510641, China [email protected]

Abstract. The support vector machine (SVM) has proved effective in classification.However,SVM easily becomes intractable in its memory and time requirements to deal with the large data, and also can not nicely deal with noisy, sparse, and imbalanced data. To overcome these issues, this paper presents a new local support vector machine that first finds k nearest neighbors from each class respectively for the query sample and then SVM is trained locally on all these selected nearest neighbors to perform the classification. This approach is efficient,simple and easy to implement. The conducted experiments on challenging benchmark data sets validate the proposed approach in terms of classification accuracy and robustness.

1

Introduction

The SVM algorithms proposed by Vapnik are a well-known class of data mining algorithms based on the statistical learning theory[1]. Benchmarking studies reveal that, in general, the SVM performs best among current classification techniques[2] due to its ability to capture nonlinearities[3]. However, SVM easily becomes intractable in its memory and time requirements to deal with the large data[4]. It has also been empirically investigated that when the data such as text data are largely unbalanced, that is, when the positive and negative labeled data are in disproportion, the classification quality of standard SVM deteriorates[5]. Another popular approach is the k -nearest neighbor (KNN) that is simple and often results in good classifications in experiments with different data sets[6,7]. However, it often suffers from high variation caused by finite sampling, so that various attempts have been made to remedy this situation, such as fuzzy set theory and evidential reasoning are applied to make strategy[8]. Since two approaches have distinguished advantages and disadvantages, it is natural to combine them together to perform the classification by making full use of both superiorities. One way is to find k nearest neighbors for the query sample and then apply SVM on these k nearest neighbors to classify the query sample[9], such as SVM-KNN[10]. The key of this approach lies in the finding of k nearest neighbors that should be consistent with human intuition. When the training data is imbalanced, the existing approaches are challenging to find the true neighbors to the query sample. The imbalance problem will be met when the data in one class heavily outnumber the data in another class, the class boundary K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 225–233, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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can be skewed towards the class with fewer data samples[16,17,18]. Under such cases, the classification performance easily decreases[19]. Currently the sampling strategies can be applied to solve this problem. However, the removal of training samples in large categories may lose some important information and always sacrifices the classification performance in some cases. As an alternative, decision rule can be modified by assigning a big weight for neighbors from small class, and assigning a little weight for neighbors contained in large category,instead of balancing the training data[19]. This problem can be also solved to some extent by fuzzifying the training set. With this manner, training data are no longer treated equally but treated differently according to their relative importance[20]. However, these approaches can not deal with imbalanced problem well as where the k nearest neighbors can not be selected reasonably. The class with more training samples has more probability than one with few to be selected as nearest neighbors. The problem is that they may not be much representative while class with few samples have not describe its topology structure clearly.To overcome the problem,this paper presents a new local support vector machine approach that first finds k nearest neighbors from each class respectively for the query sample and then SVM is trained on all these selected nearest neighbors to perform the classification for the query sample. This approach not only produces better classification accuracies, but also reveal better robustness.

2

Proposed Approach

The proposed approach is based on SVM-KNN[10], but with new approach to finding the nearest neighbors. 2.1

Support Vector Machines

SVM seeks to find the optimal separating hyperplane between classes to perform the classification, which focuses on the training cases that lie at the edge of the class distributions, the support vectors, with the other training cases effectively discarded[9]. Thus, not only is an optimal hyperplane fitted, but also the approach may be expected to yield high accuracy with small training sets. The solution to the problem is only dependent on a subset of training data points which are referred to as support vectors. Using only support vectors, the same solution can be obtained so that the classification time can be reduced. The principle of SVM can be illustrated as Fig.1 from reference[9]. Let training data Y = {(x1 , y1 ), (x2 , y2 ), · · · , (xn , yn )} where xi ∈ Rm is an m-dimensional vector representing the sample, and yi is the class label of xi ∈ {+1, −1}. The support vector machine separates samples of the two classes by an optimal hyperplane. It takes the form y(x) = sign[wT ϕ(x) + b]

(1)

where ϕ is the nonlinear function that maps the input space to a high dimensional feature space. In this feature space, the hyperplane can be constructed

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Fig. 1. Illustration of SVM

through wT ϕ(x) + b = 0 to discriminate the two classes. By minimizing wT w, the margin between two classes is maximized. However, the classifier is actually never evaluated in this form because it is hard for us to define the ϕ nonlinear function. We take the other way to reach the goal. We define the convex optimization problem as follows: T

min Υ (w, b, ξ) = w w + C

w,b,ξ

N 

ξi

(2)

i=1

subject to yi [wT ϕ(xi ) + b] ≥ 1 − ξi

(3)

ξi ≥ 0

(4)

where the variables ξi are slack variables which are needed to allow misclassifications in the set of inequalities (such as due to overlapping distributions).The constant C should be considered as a tuning parameter in the algorithm. The first part of the objective function tries to maximize the margin between both classes in the feature space, whereas the second part minimizes the misclassification error. The Lagrangian can be applied to solve this optimization problem and then leads to the following classifier: N  ai yi K(xi , x) + b] y(x) = sign[

(5)

i=1

where K(xi , x) = ϕ(xi )T ϕ(x) is taken with a positive definite kernel satisfying the Mercer theorem. The Lagrangian multipliers ai are then determined by means of the following optimization problem(dual problem): max − ai

N N  1  yi yj K(xi , x)ai aj + ai 2 i,j=1 i=1

(6)

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subject to N 

ai yi = 0, 0 ≤ ai ≤ C

(7)

i=1

The entire classifier construction problem now simplifies to a convex quadratic programming problem in ai . Note that one does not have to calculate w nor ϕ(xi ) in order to determine the decision surface. Thus, no explicit construction of the nonlinear mapping ϕ(xi ) is needed. Instead, the kernel function K will be used. A variety of kernel functions can be used, typical examples include Gaussian kernel and polynomial kernel. Gaussian Kernel K(x, y) = exp[−

x − y2 ], σ ≥ 0 2σ 2

(8)

Polynomial Kernel K(x, y) = [x · y + 1]d , d ∈ R

(9)

For low-noise problems, many of the ai will typically be equal to zero (sparseness property). The training observations corresponding to nonzero ai are called support vectors and are located close to the decision boundary. 2.2

Proposed LCB-SVM

To deal with large learning tasks with many training examples, SVM easily becomes intractable in its memory and time requirements. Fortunately, SVM often performs better than the other classification approaches on smaller data. Accordingly, it is natural to design the local SVM to reach the better performance on larger data,such as SVM-KNN[10], illustrated as Fig.2. point to be classified

SVM Train k=3 model

SVM

SVM Classify

class label of the red point

Fig. 2. Local support vector machine which defines the local region by k nearest neighbors

However, SVM-KNN can not work well on the sparse, noise, or imbalanced data, as in such cases the classification boundary may be heavily deformed and in turn threaten the classification results. Therefore, we first construct the balanced local class boundaries for each query sample to train the SVM and then classify this query sample, where local class boundaries are defined by selecting k nearest neighbors from each class respectively, as illustrated as Fig.3. This approach is denoted as LCB-SVM.

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point to be classified

SVM Train

k=3

model

SVM

SVM Classify

class label of the red point

Fig. 3. Local support vector machine which defines the local region by local class boundaries

Algorithm LCB-SVM(X,q,k ) /* q be the query sample,X be the training samples, and k be the neighborhood size defining the local region for SVM*/ Step 1. Let X j be the training sample subset from class ωj where j ∈ {1, 2, · · · , nc }, nc be the number of classes, and σ be the permutation of index of samples in X j , we find k nearest neighbors set from X j as follows, denoted as Ω j (q, k): Ω j (q, k) = {xσ(i) ∈ X j |d(q, xσ(i) ) ≤ d(q, xσ(i+1) )} where d(q, xσ(1) ) ≤ d(q, xσ(i) ), xσ(i) ∈ X j ,1 ≤ i ≤ k and d be Euclidean distance. Step 2. Define the local region by all selected nearest neighbors as follows Ω(q, k) =

nc 

Ω j (q, k)

j=1

Step 3. Train SVM on the local region defined by Ω(q, k) to get SVM classifier. Step 4. Apply trained SVM classifier to classify the query sample q. This algorithm becomes tractable on the large data. A comparison in time complexity is summarized in Table.1, where n is the number of training examples, #SV the number of support vectors,nc is the number of classes in training examples, and k is the number of nearest neighbors. Table 1. Time complexity comparison No. SVM SVM-KNN LCB-SVM Training O(n2 ) none none Testing O(#SV ) O(n + k2 ) O(n + (nc ∗ k)2 )

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LCB-SVM works well when k takes smaller value. To the other extreme, such as k = n, the local region defined by local class boundaries for each sample is the whole training data. In this case, the testing time increases much. Generally, LCB-SVM outperforms KNN and SVM-KNN, and remains efficient when the problem involves in noisy and imbalanced data, or becomes intractable for SVM.

3

Experimental Results

The experiments are conducted to validate the proposed ideas and approach by comparing with KNN, CAP[11],SVM[12], SVM-KNN[10,9], and LCB-SVM. All compared methods are implemented using matlab software tool, where LIBSVM software is used for SVM[12]. 3.1

Experimental Setting

The experiments involve in three parameters(C, γ, k), while only RBF kernel is utilized[13]. For each problem, we estimate the generalized accuracy on training data sets using different kernel parameter γ, cost parameter C, nearest neighbor size k: γ = [24 , 23 , 22 , ..., 2−10 ], C = [212 , 211 , ..., 2−2 ] , k = 3 : 3 : 30. More concretely,we tried parameter pairs of γ × C for SVM, γ × C × k for SVMKNN and LCB-SVM. Parameters are chosen respectively according to the best performance. Though this approach is completely suboptimal for choosing the optimal values of the free parameters, it still validates the proposed LCB-SVM if it can perform better than the other approaches on these parameters. Each training set is partitioned into training and testing subsets, usually in the ratio of 70-30%[14]. Ten such partitions are generated randomly for the experiments. On each partition, the compared algorithms are trained and tested for each pair of parameters, respectively, and then the best performance is reported. Subsequently the average performance over these ten partitions is calculated and compared. Experimental data sets are taken from UCI repository of machine learning databases[15], where the records with missing values and non-numeric attribute are all removed. Most of these real data may be noisy, sparse, and imbalanced. It can be observed from Table.2 that LCB-SVM outperforms SVM-KNN by the average accuracy 2.17%, while has almost the same standard deviation of the accuracy as that of SVM-KNN. Although LCB-SVM only lightly outperforms SVM by the average accuracy 0.08%, it can work on the large data whereas SVM can not. These results do indicate the significant value of the proposed idea and the classifier. Generally every method has its strengths and weaknesses. It is necessary to apply a measure to evaluate the robustness of the different methods. We take the usually used measure to quantify the robustness by computing the ratio bm of the error rate em of method m and the smallest error rate over all methods being compared in a particular dataset: bm = em / min1≤k≤10 ek [6]. Thus, the best method m∗ for that problem has bm =1, and all other methods have larger values bm > 1. The larger the value of bm , the worse the performance of the method.

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Table 2. Average values and standard deviations of the accuracies of compared methods on real data sets(%) Data wine dermatology diabetes ionosphere glass optdigits segmentation yeast yaleface iris

Size 178 358 768 351 214 1797 210 1484 165 150 avg

KNN 72.50±4.88 87.54±2.46 74.69±1.48 85.19±2.32 65.73±7.82 98.61±0.26 76.66±4.03 59.36±1.97 60.88±7.85 96.27±3.66 77.74±3.67

CAP 72.30± 91.41± 74.04± 89.90± 66.88± 98.93± 78.73± 57.34± 68.66± 95.81± 79.40±

5.52 2.24 1.50 2.13 5.39 0.19 3.36 2.02 6.14 2.85 3.14

SVM 90.00± 96.03± 76.13± 94.80± 68.85± 98.95± 86.50± 59.52± 69.33± 95.81± 83.59±

5.34 1.65 1.56 2.08 4.70 0.34 2.72 1.67 6.35 4.07 3.05

SVM-KNN 86.34± 4.29 93.30± 2.10 74.65± 1.34 90.38± 1.50 67.70± 4.44 99.02± 0.31 81.74± 5.08 58.66± 2.10 66.44± 5.78 96.74± 3.13 81.50± 3.01

LCB-SVM 90.00± 5.03 96.13± 1.29 75.95± 1.39 95.67± 2.27 68.52± 6.63 99.00± 0.30 86.50± 4.24 58.86± 1.43 69.55± 3.92 96.51± 4.13 83.67± 3.06

This means that the distribution of bm will be a good indicator reflecting its robustness.We calculate em for each method in terms of the average error rate of the ten best results with respect to parameters on each data set. Fig.4 shows the distribution of bm for each method over the ten real data sets, which is drawn using matlab function: boxplot. The box area represents the lower and upper

3.5

3

Values

2.5

2

1.5

1 KNN

CAP

SVM

SVM−KNN

LCB−SVM

Fig. 4. Average performance distribution of the compared classifiers

quartiles of the distribution that are separated by the median. The outer vertical lines show the entire range of values for the distribution.Clearly, the spread for LCB-SVM is much narrower and closer to one. This result demonstrates that it obtains the most robust performance over these data sets. Although these results are data-specific and sensitive to how the classifiers were parameterized, LCBSVM is of consistent behaviors on showing better performance than SVM-KNN. These do indicate the value of proposed approaches and related classifiers.

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Conclusion and Future Work

Although SVM has remarkable generalization performance, it can not nicely deal with the large data as well as the noisy, sparse, and imbalanced data. This paper proposes a new approach to define the local region where SVM can be applied to locally perform for classification. This approach uses local class boundaries to define the local region, so that the imbalanced problem can be solved. Furthermore,it obviously improves the performance just with small additional cost from SVM-KNN. Although our approach only lightly outperforms SVM, it can work on the large data whereas SVM can not, as well as with the most robust performance. This approach is simple and easy to be implemented, so that it can be expected to have wider applications.

Acknowledgements This work was supported by China National Science Foundation under Grants 60973083, The Fundamental Research Funds for the Central Universities,SCUT, Guangdong Science and Technology project under Grants 2007B030803006,and Hubei Science and Technology project under Grants 2005AA101C17.

References 1. Vapnik, V.N.: Statistical Learning Theory. John Wiley and Sons, Chichester (1998) 2. Baesens, B., Van Gestel, T., Viaene, S., Stepanova, M., Suykens, J., Vanthienen, J.: Benchmarking State-of-the-Art Classification Algorithms for Credit Scoring. J. Operational Research Soc. 54(6), 627–635 (2003) 3. Martens, D., Baesens, B., Van Gestel, T.: Decompositional Rule Extraction from Support Vector Machines by Active Learning. IEEE Trans. Pattern Anal. Machine Intell 21(2), 178–192 (2009) 4. Joachims, T.: Making Large-Scale SVM Learning Practical. Advances in Kernel Methods C Support Vector Learning (1999) 5. Lewis, D.D., Yang, Y., Rose, T., Li, F.: RCV1: A New Benchmark Collection for Text Categorization Research. Journal of Machine Learning Research 5, 361–397 (2004) 6. Domeniconi, C., Peng, J., Gunopulos, D.: Locally adaptive metric nearest-neighbor classification. IEEE Trans. Pattern Anal. Machine Intell. 24(9), 1281–1285 (2002) 7. Karl, S.N., Truong, Q.N.: An Adaptable k-Nearest Neighbors Algorithm for MMSE Image Interpolation. IEEE Trans. Image Processing. 18(9), 1976–1987 (2009) 8. Zhu, H., Basir, O.: An Adaptive Fuzzy Evidential Nearest Neighbor Formulation for Classifying Remote Sensing Images. IEEE Transaction on Geoscience and Remote Sensing 43(8), 1874–1889 (2005) 9. Blanzieri, E., Melgan, F.: Nearest neighbor classification of remote sensing images with the maximal margin principle. IEEE Trans. Geoscience and Remote Sensing. 46(6), 1804–1811 (2008) 10. Zhang, H., Berg, A.C., Maire, M., Malik, J.: SVM-KNN: Discriminative Nearest Neighbor Classification for Visual Category Recognition. In: Proceeding of Conference on Computer Vision and Pattern Recognition (CVPR 2006), pp. 2126–2136. IEEE Computer Society, New York (2006)

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11. Hotta, S., Kiyasu, S.: Pattern recognition using average patterns of categorical k-nearest neighbors. In: Proc. 17th ICPR, pp. 412–415 (2004) 12. Chang, C.-C., Lin, C.-J.: LIBSVM: A library for support vector machines (2004), http://www.csie.ntu.edu.tw/~ cjlin/libsvm 13. Hsu, E.C.-W., Lin, C.-J.: A Comparison of Methods for Multiclass Support Vector Machines. IEEE Transaction on Neural Network 13(2), 415–425 (2002) 14. Lam, W., Han, Y.: Automatic Textual Document Categorization Based on Generalized Instance Sets and a Metamodel. IEEE Trans. Pattern Anal. Mach. Intell. 25(5), 628–633 (2003) 15. Asuncion, A., Newman, D.J.: UCI Machine Learning Repository. University of California, School of Information and Computer Science, Irvine, CA (2007), http://www.ics.uci.edu/~ mlearn/MLRepository.html 16. Veropoulos, K., Campbell, C., Cristianini, N.: Controlling the sensitivity of support vector machines. In: Proc. Int. Joint Conf. Artif. Intell. (IJCAI 1999), Stockholm, Sweden, pp. 55–60 (1999) 17. Akbani, R., Kwek, S., Japkowicz, N.: Applying support vector machines to imbalanced datasets. In: Boulicaut, J.-F., Esposito, F., Giannotti, F., Pedreschi, D. (eds.) ECML 2004. LNCS (LNAI), vol. 3201, pp. 39–50. Springer, Heidelberg (2004) 18. Wu, G., Cheng, E.: Class-boundary alignment for imbalanced dataset learning. In: Proc. ICML 2003 Workshop Learn. Imbalanced Data Sets II, Washington, DC, pp. 49–56 (2003) 19. Tan, S.: Neighbor-weighted K-nearest neighbor for unbalanced text corpus. Expert Systems with Applications 28, 667–671 (2005) 20. Liu, Y.-H., Chen, Y.-T.: Face Recognition Using Total Margin-Based Adaptive Fuzzy Support Vector Machines. IEEE Transaction on Neural Network 18(1), 178–192 (2007)

Research on Detection and Material Identification of Particles in the Aerospace Power Shujuan Wang1, Rui Chen1, Long Zhang1, and Shicheng Wang2 1

School of Electrical Engineering and Automation, Harbin Institute of Technology, 150001, Harbin, P.R. China [email protected] 2 Army Aviation Institute of PLA, 101123, Beijing, P.R. China

Abstract. The aerospace power is widely used in the aerospace system. Its reliability directly affects the safety of the whole system. However, the particles generated in the production process usually cause failures to the aerospace power. In this paper, a novel automatic detection method for particles in the aerospace power is proposed based on Particle Impact Noise Detection (PIND) test. Firstly, stochastic resonance algorithm is presented to detect the existence of tiny particles. Secondly, in order to obtain the sources of particles, wavelet packet transform is used to extract energy distribution vectors of different material particles, and Learning Vector Quantization (LVQ) network is brought in for material identification of particles. Finally, the results indicate that the accuracy meets the requirements of practical application. Keywords: Particles detection, Stochastic resonance, Material identification.

1 Introduction The aerospace power is widely used to supply power for the aerospace system, so its reliability directly affects the performance of the whole system. However, due to the complex production process of aerospace power, it’s easy to assemble particles into it, consequently bringing great harm to the aerospace equipment. Therefore, most aerospace powers have to make delivery inspection to check whether there are particles in them. Currently, particles generated during the production process in the aerospace power mainly include tin granular, copper shot, iron filings and wire skin.

Fig. 1. Structure of the particle detection system for the aerospace power

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 234–240, 2010. © Springer-Verlag Berlin Heidelberg 2010

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Referring to PIND method [1], a turntable is adopted to vibrate a tested aerospace power, and a data acquisition card is used to collect the audio signal generated by the process of particles impacting the cavity of the test component. The audio signal is analyzed by computer software in order to achieve particle detection and material identification. The main structure of the novel detection system is shown in Fig. 1.

2 Signal Analysis and Processing The signal from the acoustic sensor includes system noise and particle signal. The particle signal is a series of oscillation attenuation pulse generated by particle impact, and the particle signal frequency is between 30 kHz and 140 kHz, especially around 80 kHz. The detection method of big particle signal adopts wavelet analysis and threshold [1]. But this method has low accuracy for tiny particle detection because the useful information of the particle signal is partly eliminated during the de-noising processing. So the stochastic resonance method is presented to detect tiny particles. 2.1 Application of Stochastic Resonance Stochastic resonance means that, in some nonlinear system with a weak lowfrequency input signal, increasing the system noise increases the signal noise ratio (SNR) of output rather than deteriorating the output signal, which is the synergy of the system noise, the weak signal and the nonlinear system. A typical nonlinear bistable stochastic resonance model is given [3]

x = ax − bx 3 + s (t ) + N (t ) a > 0, b > 0 .

(1)

Where x is the system output, a and b are structure parameters of nonlinear system, s (t ) is the input signal, N (t ) is Gaussian white noise and meets E[ N (t )] = 0 ,

E[ N (t ) N (t + τ )] = 2 Dδ (t − τ ) , and D is the noise intensity. This equation essentially describes the mass point driven by both external forces and noise does over-damped motion in double potential wells. When there is no system noise or input signal, the potential function is

a b V ( x) = − x 2 + x 4 . 2 4

(2)

The barrier height of bi-stable system is ΔV = a 2 / 4b , and its two steady state points are xm = ± a / b where the potential function reaches the minimum. When a = 1, b = 1 , the potential well graph of bi-stable system is shown in Fig. 2. When there is input signal and system noise, the mass point is likely to jump between the two potential wells; meanwhile, the jump frequency is synchronous with the weak signal where stochastic resonance happens. Runge-Kutta algorithm is adopted to solve the nonlinear equation (1). Stochastic resonance is only effective for weak low-frequency signal, but the particle signal is a

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high-frequency signal around 80 kHz, so the sub-sample method is used to collect particle signal to meet the requirements of stochastic resonance [4], namely, the sampling rate is no less than 50 times of particle signal frequency.

Fig. 2. Potential well graph of bi-stable system

2.2 Tiny Particle Detection Based on Stochastic Resonance

In the paper, the tiny particle signal, the system noise and nonlinear bi-stable system constitute three key elements of stochastic resonance, regarding that the system noise could not be decreased in practical engineering application, so this paper will adjust a and b to adapt to the tiny particle signal in order to produce stochastic resonance. And the values of a and b will be obtained by simulation. The simulation process is as follows: the first step is that the system noise as shown in Fig. 3 is taken as the only input signal of the nonlinear bi-stable system, and the second step is that the tiny particle signal shown in Fig. 4 is added into input signal located between 0.4 s and 0.6 s. Adjusting a and b to assure that the nonlinear system couldn’t generate state transition in the first step, but could generate stochastic resonance in the second step.

Fig. 3. Signal of system noise

Fig. 4. Weak signal of tiny particle

Through simulation, the structure parameters of the bi-stable system a = 2, b = 20000 are obtained and its two steady state points are xm = ± a / b = ±0.01 . The output

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processed by stochastic resonance algorithm is shown in Fig. 5. The system generate state transition at the location where exists the tiny particle signal. To verify whether the simulation results a and b are available in practical particle detection, stochastic resonance algorithm is used to process the tiny particle signal in Fig. 4, and the output is shown in Fig. 6. The existence of particles can be easily judged by detecting the state transition.

Fig. 5. Simulation output

Fig. 6. Practical output

3 Material Identification of Particles Material identification includes extraction of the eigenvectors and identification by neutral network. Its accuracy is determined by both eigenvectors describing the pattern characteristic and the identification method whether possessing excellent identification performance. The particles in the relay are classified by linear method [2], but many nonlinear factors are ignored, which leads to low accuracy. In this paper, neutral network algorithm is proposed to improve the accuracy of material identification of particles because it takes into account the nonlinear factors [5]. 3.1 Extraction of the Eigenvector

The signals of different material particles have different energy, so energy distribution vectors are used as the eigenvectors of particles and are extracted by wavelet packet transform that is suitable for processing signals with wide energy distribution, because it can not only deal with the low frequency scale factors, but also dispose high-frequency wavelet coefficients. The particle signal x( N ) is decomposed into m layers wavelet packet. And 2m wavelet tree nodes are obtained, then they are reconstructed to get original signal component in the corresponding frequency subspace [6]. Wavelet packet transform meets the Pasaweier theorems so there is 2m

x( N ) = ∑ WPTi ( N ) . i =1

Here WPTi ( N ) is the i th reconstructed signal.

(3)

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In this paper energy distribution vector e is defined in (4), we can conclude e has no unit and is characterized as (5) and (6). n

e(i ) =

∑ WPT

i

j =1

2

( j)

n

∑ x ( j)

1 ≤ i ≤ 2m .

(4)

.

(5)

2

j =1

0 ≤ e (i ) ≤ 1 2m

∑ e(i) = 1 i =1

(6)

.

By the definition, the more dimensions of the energy distribution vector lead to the higher frequency resolution and the more comprehensive information about particle material. Experimental results show that the outcome of 3 layers wavelet packet decomposition is mostly close to that of more than 3 layers, so considering declining the amount of computation the paper adopts 3 layers whose energy distribution vector is 8 dimensions. At the same time, DB4 wavelet is adopted according to particle signal characteristic. The energy distribution vectors of typical particles are shown in Table 1. Table 1. Energy distribution vectors of typical particles

Particles

e1

e2

e3

e4

e5

e6

e7

e8

Iron

0.0984 0.4102 0.0524 0.2715 0.0124 0.0267 0.1016 0.0268

Tin

0.0168 0.1423 0.0771 0.1406 0.1084 0.2161 0.1681 0.1306

Copper Wire Skin

0.0099 0.0737 0.0698 0.0798 0.1733 0.2854 0.1677 0.1403 0.1880 0.3967 0.0222 0.3098 0.0091 0.0174 0.0401 0.0167

3.2 Material Identification Using LVQ Network

LVQ network is adopted to identify the material. Six representative training samples are respectively provided for each of the four kinds of particles, and three of them are actual test samples and the other three are added with 5% noise in order to improve the abilities of network generalization, so there are totally twenty-four samples. The specific steps are as follows. (1) The energy distribution vectors of typical samples are used as the input of LVQ network while expected outputs representing copper, iron, tin and wire skin particles are 1, 2, 3 and 4, respectively. (2) The number of hidden layer neuron should be larger than 100 in order to acquire better identification results, so 200 neurons are selected to make the network flexible. (3) Using Matlab

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neural network toolbox to build and train the LVQ network, and the network converges when the number of the step reaches 252 and the training error reaches 0.

4 Experiments and Conclusion 4.1 Experimental Results and Analysis

In the experiment, four kinds of particles are put into the aerospace powers, and the weight of each kind of particles increases from 0.5 mg to 2.5 mg in steps of around 0.1 mg, meanwhile, each kind of weight of particles is provided four kinds of shape including square, triangle, circle and polygon. So there are 336 test samples in total. The above 336 test samples and 20 samples without any particles are experimented to detect the existence of particles and identify the particle material. The detection accuracy is above 90%. The reason of the detection error is that the particles are not effectively activated due to electrostatic adsorption. The material identification results are shown in Table 2, and the accuracy of material identification is above 80 %. Different weight and shape lead to the discrepancy of the energy distribution vector; hence, it affects the accuracy of material identification, which is the reason of the error. Table 2. Experimental results of material identification

Copper

84

Num of Correct Identification 71

Iron

84

72

12

85%

Tin Wire Skin

84

70

14

83%

84

68

16

80%

Material

Total Num

Num of Error Identification 13

Accuracy 84%

4.2 Conclusion

In this paper, a novel method of automatic detection material identification of particles in the aerospace power is proposed, and some valuable conclusions are generalized as follows: Stochastic resonance algorithm is put forward to process tiny particle signal, and the accuracy of particle detection is improved. (2) Energy distribution vectors are adopted as eigenvectors of different material particles, and LVQ network is brought in for material identification according to eigenvectors. (3) An automatic detection system is developed for the experiments detecting and identifying the material of particles in the aerospace power, and the experimental results indicate the detection accuracy is above 90% and the accuracy of material identification is above 80%, which completely meet the requirements of practical application.

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References 1. Gao, H.L., Zhang, H., Wang, S.J.: Research on Auto-detection for Remainder Particles of Aerospace Relay based on Wavelet Analysis. Chinese Journal of Aeronautics 20(1), 74–80 (2007) 2. Wang, S.J., Gao, H.L., Zhai, G.F.: On Feature Extraction of Remnant Particles of Aerospace Relays. Chinese Journal of Aeronautics 20(6), 253–259 (2007) 3. Harmer, G.P., Davis, B.R., Abbott, D.: A Review of Stochastic Resonance: Circuits and Measurement. IEEE Transactions on Instrumentation and Measurement 51(2), 209–309 (2002) 4. Leng, Y.G., Wang, T.Y., Qin, X.D.: Power Spectrum Research of Twice Samplings to Resonance Response in a Bistable System. ACTA Physica Sinica, 53(3), 717–723 (2004) (in Chinese) 5. Li, D.Q., Pedrycz, W., Pizzi, N.L.: Fuzzy Wavelet Packet based Feature Extraction Method and Its Application to Biomedical Signal Classification. IEEE Transactions on Biomedical Engineering 52(6), 1132–1139 (2005) 6. Wang, B.C., Omatu, S., Abe, T.: Identification of the Defective Transmission Devices Using the Wavelet Transform. IEEE Transactions on Pattern Analysis and Machine Intelligence 27(6), 919–928 (2005)

The Key Theorem of Learning Theory Based on Sugeno Measure and Fuzzy Random Samples Minghu Ha1, Chao Wang2, and Witold Pedrycz3 1

College of Mathematics and Computer Sciences, Hebei University, Baoding, 071002, P.R. China 2 College of Physics Science & Technology, Hebei University, Baoding, 071002, P.R. China 3 Department of Electrical and Computer Engineering, University of Alberta, Edmonton, T6G2V4, Canada. and Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland {Minghuha,wangchaohbu}@yahoo.com.cn, [email protected]

Abstract. Statistical Learning Theory is one of the well-developed theories to deal with learning problems about small samples, and it has become an important conceptual and algorithmic vehicle of machine learning. The theory is based on the concepts of probability measure and random samples. Given this, it becomes difficult to take advantage of the theory when dealing with learning problems based on Sugeno measure and fuzzy random samples which we encounter in real-world problems. It is well known that Sugeno measure and fuzzy random samples are interesting and important extensions of the concepts of probability measure and random samples, respectively. This motivates us to discuss the Statistical Learning Theory based on Sugeno measure and fuzzy random samples. Firstly, some definitions of the distribution function and the expectation of fuzzy random variables based on Sugeno measure are given, and the law of large numbers of fuzzy random variables based on Sugeno measure is proved. Secondly, the expected risk functional, the empirical risk functional and the principle of empirical risk minimization based on Sugeno measure and fuzzy random samples are introduced. Finally, the key theorem of learning theory based on Sugeno measure and fuzzy random samples is proved, which will play an important role in the systematic and comprehensive development of the Statistical Learning Theory based on Sugeno measure and fuzzy random samples. Keywords: Sugeno measure, Fuzzy random variables, The principle of empirical risk minimization, Key theorem.

1

Introduction

Statistical Learning Theory (SLT) was proposed in 1960s, and has been fully established in 1990s by Vapnik [12-14] et al. The theory offers a sound theoretical framework for statistical learning realized in presence of small data samples. Its fundamental idea is to make the learning machines compatible with the small samples and improve their generalization abilities to the highest possible extent. The study of SLT has become a new hotspot [2-6]. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 241–249, 2010. © Springer-Verlag Berlin Heidelberg 2010

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The SLT is based on the concepts of probability measure and random samples. Given this, it becomes difficult to take advantage of the theory when dealing with statistical learning problems based on Sugeno measure and fuzzy random samples which we encounter in real-world problems. It is well known that Sugeno measure [8, 11] and fuzzy random samples [9, 10] are interesting and important extensions of the concepts of probability measure and random samples, respectively. In this study, we develop the SLT based on Sugeno measure and fuzzy random samples, and prove the key theorem of learning theory based on Sugeno measure and fuzzy random samples, which will play an important role in the systematic and comprehensive development of the SLT based on Sugeno measure and fuzzy random samples. The organization of this paper is structured as follows. In Section 2, some notions pertaining to Sugeno measure, random set and fuzzy random variables are reviewed. The expectation of fuzzy random variables based on Sugeno measure is defined, and then the law of large numbers of fuzzy random variables based on Sugeno measure is proved. In Section 3, some notions of the SLT based on Sugeno measure and fuzzy random samples are introduced, the key theorem of learning theory based on Sugeno measure and fuzzy random samples is discussed.

2 Preliminaries Throughout this paper, we assume that Ω is a nonempty set, F is a nonempty class of subsets of Ω , and μ is a nonnegative, extended real-valued set function. Let K( R m ) be the family of all nonempty compact subsets of Euclidean space R m . Sugeno [11] introduced the concept of λ -fuzzy measure that satisfies λ -rule. Let us recall that satisfies the σ - λ -rule (on F ) iff there exists ⎛ ⎞ 1 , ∞ ⎟ ∪ {0} λ ∈⎜− ⎝ sup μ ⎠

where sup μ = sup E∈F μ ( E ) , such that

∞ ⎧1 ⎛ ∞ ⎞ ⎪ λ {∏ i =1[1 + λ ⋅ μ ( Ei )] − 1}, λ ≠ 0 μ ⎜ ∪ Ei ⎟ = ⎨ ⎝ i =1 ⎠ ⎪ ∞ μ ( E ), λ =0 i ⎩∑ i =1 for any disjoint sequence { E1 , E2 , , En } of set in F whose union is also in F .

When F is a σ -algebra, μ is called Sugeno measure ( g λ -fuzzy measure) iff it sat-

isfies σ - λ -rule on F and μ ( Ω ) = 1 . In this paper, we denote Sugeno measure by g λ , while F is a σ -algebra. g λ random variable was introduced by Ha, et al [2], subsequently the distribution function and expectation of g λ random variables were discussed. Let ξ : Ω → R be a real-valued function. ξ is called a g λ random variable if for any given real number x , we have

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{ω : ξ (ω ) ≤ x} ∈ F . The distribution function of ξ is defined by Fgλ ( x ) = g λ ({ξ ≤ x}) , ∀x ∈ R . If ∫

∞

−∞

∞

x dFgλ ( x) < ∞ , ∫ xdFgλ ( x) is called the expectation of ξ , denoted by E gλ (ξ ) . −∞

A random set is a Borel measurable function F : Ω → K( R m ) , i.e.

F −1 ( K ) = {ω ∈ Ω : F (ω ) ∩ K ≠ ∅} ∈ F , ∀K ∈ K( R m ) . The definition of expectation of random set in a probability space was introduced by Aumann [1] in 1965. We define the distribution function and expectation of random set in Sugeno measure space [7]. Some basic notions of fuzzy set introduced by Zadeh [15] are reviewed as follows. A fuzzy set is a function u : R m → [ 0,1] . We denote by u r = { x ∈ R m : u ( x ) ≥ r} , 0 ≤ r ≤ 1

as r − level set. We denote the support of u by Suppu = { x ∈ R m : u ( x ) > 0} .

Let F ( R m ) denote the family of all fuzzy sets u : R m → [ 0,1] which satisfies the fol-

lowing conditions: (1) The 1-level set u1 = { x ∈ R m : u ( x ) = 1} ≠ ∅ . (2) u : R m → [ 0,1] is upper semicontinuous, i.e. for each 0 ≤ r ≤ 1 , the r − level set ur = { x ∈ R m : u ( x ) ≥ r} is a closed subset of R m . (3) The support set Suppu = cl { x ∈ R m : u ( x ) > 0} is compact. We say a fuzzy set u is convex iff u ( λ x + (1 − λ ) y ) ≥ min {u ( x ) , u ( y )} , ∀x, y ∈ R , r ∈ ( 0,1] . m m Let F c ( R ) be the family of all convex fuzzy sets in F ( R ) . For any fuzzy

set u ∈ F ( R m ) , the convex hull cou is defined as

cou = sup {r ∈ ( 0,1] : x ∈ cour } .

A fuzzy random variable [9] (fuzzy set-valued random variable) is a mapping F : Ω → F ( R m ) , such that Fr (ω ) = {x ∈ R m : F (ω ) ( x ) ≥ r} is a random set, for

any 0 ≤ r ≤ 1 . A convex fuzzy random variable is a mapping F : Ω → Fc ( R m ) . Properties of fuzzy random variables have been discussed quite intensively in the past [9, 10], we continue along this line by presenting fuzzy random variables based on Sugeno measure and discussing their main features. A sequence of fuzzy random variables {F k : k = 1, 2, , n} is called to be independent (identical) based on Sugeno measure if for any r ∈ ( 0,1] the sequence of random sets {F kr : k = 1, 2, , n} is independent (identical) based on Sugeno measure. A sequence of fuzzy random variables {F k : k = 1, 2, , n} is called to converge to a fuzzy random variable F , if d H ( F k (ω ), F (ω )) → 0, a.e.g λ as k → ∞ .

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The expectation of a fuzzy random variable F based on Sugeno measure is defined as

Egλ ( F )( x) = sup(r ∈ (0,1] : x ∈ Egλ ( Fr )) . Therefore, we have ( Egλ ( F )) r = Egλ ( Fr ) . In order to discuss SLT problems based on Sugeno measure and fuzzy random samples, the law of large numbers of fuzzy random variables is of interest. Lemma 2.1. If ∀ω ∈ Ω , we have F (ω ) ∈ Fc ( R ) , then the following assertions hold (1) Fr (ω ) = ⎡⎣ Fr− ( ω ) , Fr+ (ω ) ⎤⎦ ; (2) Fr− (ω ) and Fr+ (ω ) are both g λ random variables. Proof. For any F (ω ) ∈ Fc ( R ) , it is a closed fuzzy number. In this way (1) has been

proved. It is also clear that Fr− (ω ) and Fr+ (ω ) are both g λ random variables—we can

demonstrate this by using the definitions of random sets and fuzzy random variables.

{

Corollary 2.2. Let F 1 , F 2 ,

}

, F n be i.i.d. (independent and identically distributed)

convex fuzzy random variables based on Sugeno measure. Then {F 1r− (ω ), F r2− (ω ) ,

, F nr− (ω )}

{F 1r + (ω ) , F r2+ (ω ) ,

, F nr+ (ω )}

are i.i.d. g λ random variables sequences accordingly Theorem 2.3. Let {F 1 , F 2 ,

, F n } be an i.i.d. fuzzy random variables based on

Sugeno measure such that Egλ SuppF 1 < ∞ . Then

1 n gλ → E (coF 1 ) . ∑ F k ⎯⎯⎯ n →∞ n k =1 Proof. In virtue of Lemma 2.1 and Corollary 2.2, we can arrive at the following

co( F k )λ = [co( F k )λ− , co( F k )λ + ] . And {co( F k )λ− , k = 1, 2,

n} , {co( F k )λ+ , k = 1, 2,

n} are both independent and iden-

tical distributed fuzzy variables. So we can get the conclusion easily by use of the law of large numbers of g λ random variables [2].

3 The Key Theorem of Learning Theory Based on Sugeno Measure and Fuzzy Random Samples Any loss function of fuzzy random samples is denoted by Q( z , α ), α ∈ Λ , and the corresponding risk functional based on Sugeno measure and fuzzy random samples is defined by

The Key Theorem of Learning Theory Based on Sugeno Measure

245

RF (α ) = Egλ (coQ ( z , α )) . Generally, we consider the problem of minimizing the risk functional on the basis of empirical data. It was shown that different problems such as pattern recognition, regression estimation, and density estimation can be reduced to this scheme by specifying a loss function in the risk. However, we cannot minimize the functional directly since the distribution function of fuzzy random samples based on Sugeno measure that defines the fuzzy risk functional is unknown. So the principle of empirical risk minimization based on Sugeno measure and fuzzy random samples has to be introduced. Instead of minimizing the risk functional RF (α ) = Egλ (coQ ( z , α )) ,

(1)

we minimize the functional

RFemp (α ) =

(2)

1 l ∑ Q ( zi , α ) l i =1

which is referred to as the empirical risk functional based on Sugeno measure and fuzzy random samples. The empirical risk functional (2) is constructed on the basis of data z1 , z2 , , zl obtained according to the distribution function of fuzzy random variables. Let the minimum of the risk functional (1) be attained at Q ( z , α 0 ) and let the minimum of the empirical risk functional (2) be attained at Q ( z , α l ) . We consider the function Q ( z , α l ) as an approximation to the function Q ( z , α 0 ) . This principle guiding the process of solving the risk minimization problem is called the empirical risk minimization principle based on Sugeno measure and fuzzy random samples. We say that the method of minimizing empirical risk is strictly consistent for the set of loss functions {Q ( z , α ) , α ∈ Λ} and the corresponding distribution function of fuzzy random samples if for any nonempty subset Λ (c) = {α : RF (α ) ≥ c}, c ∈ [0, ∞) ,

the convergence holds true gλ inf RF (α ) ⎯⎯⎯ → inf RFemp (α ) . n →∞

α ∈Λ ( c )

α ∈Λ ( c )

Corollary 3.1. If the method of empirical risk minimization is strictly consistent, the following convergence holds true gλ

RF (α l ) → inf RFemp (α ) . l →∞ α ∈Λ ( c )

Proof. Denote inf RF (α ) = RF (α 0 ) = a . ∀ε > 0 , we consider subset Λ (a + ε ) of Λ , α ∈Λ

such that Λ (a + ε ) = {α : RF (α ) ≥ a + ε } .We choose a ε , such that Λ (a + ε ) is not empty. Then the convergence gλ

inf

α ∈Λ ( a + ε )

RFemp (α ) → inf

l →∞ α ∈Λ ( a + ε )

RF (α )

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holds. Hence, we have lim g λ { inf

α ∈Λ ( a + ε )

l →∞

RFemp (α ) > inf

α ∈Λ ( a + ε )

We also know that the inequality

inf

α ∈Λ ( a + ε )

ε

RF (α ) − } = 1 . 2

RF (α ) ≥ a + ε

is valid. Then we obtain the equality

ε

RFemp (α ) ≥ a + } = 1 . α ∈Λ ( a + ε ) 2

lim g λ { inf l →∞

(3)

For any given c ∈ [ 0, ∞ ) , the convergence gλ

inf RFemp (α ) → inf RF (α ) = a

α ∈Λ ( c )

l →∞ α ∈Λ ( c )

is satisfied, then we have obtain the equality

ε

lim g λ {inf RFemp (α ) ≥ a + } = 0 . l →∞ α ∈Λ 2

(4)

The equalities (3) and (4) imply lim g λ {α l ∈ Λ (a + ε )} = 0 . l →∞

On the other hand, since α l ∉ Λ (a + ε ) , the inequality holds true a ≤ RF (α l ) ≤ a + ε Therefore, we have gλ

RF (α l ) → inf RF (α ) = a . l →∞ α ∈Λ

Remark 3.2. Strict consistency implies that the empirical risk minimization converges to the risk minimization while l is large enough. It means that the loss function of the empirical risk minimization give a closed value of the possible risk. We say the empirical risk uniform one-sided to the expected risk for the set of functions {Q ( z , α ) , α ∈ Λ} and the corresponding distribution function of fuzzy random

samples, if for any ε > 0 ,

lim g λ {sup ( RF (α ) − RFemp (α ) ) > ε } = 0 l →∞

α ∈Λ

is valid. In the view of the above definitions, the key theorem of learning theory based on Sugeno measure and fuzzy random samples is given, that transforms the question of consistency to the issue of existence of uniform one-sided convergence. Theorem 3.3. Assume that there exist constants a and b such that for all fuzzy random variables {Q ( z , α ) , α ∈ Λ} and for the corresponding distribution function of fuzzy random samples, the inequality holds true a < RF (α ) < b, α ∈ Λ .

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Then the following two statements are equivalent. a) For the corresponding distribution function, the empirical risk minimization method is strictly consistent on the set of functions {Q ( z , α ) , α ∈ Λ} . b) For the corresponding distribution function of fuzzy random samples, the uniform one-sided convergence holds true, i.e., lim g λ {sup( RF (α ) − RFemp (α )) > ε } = 0 . l →∞

α ∈Λ

We first show the necessity part of the proof. Suppose that ERM is valid. Then for any set Λ(c) = {α : RF (α ) ≥ c, α ∈ Λ} ≠ ∅ ( ∀c ∈ (−∞, ∞) ), we have gλ inf RFemp (α ) ⎯⎯⎯ → inf RF (α ) . n →∞

α ∈Λ ( c )

α ∈Λ ( c )

∀ε > 0 , we consider a finite sequence of numbers a1 , a2 ,… , an , such that a i +1 −ai
ε by A . Suppose that A takes place, α ∈Λ

* * then there will be α ∈ Λ , such that RF (α ) − RFemp (α ) > ε . From α * , we find k such *

that α * ∈ Λ (ak ) and RF (α * ) − ak < chosen set Λ (ak ) the inequality

ε 2

. In the view of inf RF (α ) ≥ ak , then for the α ∈Λ ( ak )

RF (α * ) − inf R(α ) < α ∈Λ ( ak )

ε 2

holds. Therefore for the chosen α and Λ (ak ) , the following inequalities hold: *

α ∈Λ ( a k )

R F (α ) −

ε

> R Fem p (α * ) ≥ inf R Fem p (α ) α ∈Λ ( a k ) 2 That is, the event Bk occurs, so does B , i.e. g λ ( A) < g λ ( B) → 0 . Then inf

l →∞

gλ {sup( RF (α ) − RFemp (α )) > ε } → 0 . α ∈Λ

l →∞

Now let us consider the sufficiency part of the proof. Suppose that the uniform oneRF (α ) − inf RFemp (α ) > ε } . side convergence holds true. Denote A = { αinf ∈Λ ( c ) α ∈Λ ( c ) Then the event A is the union of the two events A = A1 ∪ A2 , A1 ∩ A2 = ∅ , where

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A1 = { inf RF (α ) < inf RFemp (α ) − ε α ∈Λ ( c )

α ∈Λ ( c )

},

A2 = { inf RF (α ) > inf RFemp (α ) + ε } . α ∈Λ ( c )

α ∈Λ ( c )

Suppose that the event A1 takes place. We will find function Q( z , α * ), α * ∈ Λ(c) , such that RF (α * ) < inf RF (α ) + α ∈Λ ( c )

RF (α * ) +

ε

dH (

2

. Then we have

< inf RF (α ) + ε < inf RFemp (α ) < RFemp (α * ) , α ∈Λ ( c )

2

i.e. RFemp (α * ) − RF (α * ) >

ε

ε 2

α ∈Λ ( c )

. In the view of the inequality

1 l ε ∑ Q( zi ,α * ), E (coQ( z,α * )) ≥ RFemp (α * ) − RF (α * ) > 2 l i =1

By the use of the law of large numbers of fuzzy random variables based on Sugeno measure, we have ⎧ g λ ( A1 ) ≤ g λ ⎨ d H ⎩

⎛1 l ⎞ ε⎫ * * 0. ⎜ ∑ Q( zi , α ), E ( coQ ( z , α ) ) ⎟ > ⎬ l→ l ⎝ i =1 ⎠ 2 ⎭ →∞

On the other hand, if A2 occurs, there will exist a function Q( z , α ** ), α ** ∈ Λ (c) such that RFemp (α ** ) +

ε 2

< inf RFemp (α ) + ε < inf RF (α ) < RF (α ** ) . Then we have α ∈Λ ( c )

α ∈Λ ( c )

RFemp (α k (0) ) − RF (α k (0) ) ≥ ε 2 . Because θ λ g λ ( A ) is a probability measure, so we have

g λ ( A) = θ −1λ

(θλ

g λ ( A ) ) = θ −1λ (θ λ g λ ( A1 ) + θ λ g λ ( A2 )) → 0 . l →∞

Hence, the proof has been completed.

4 Conclusions Sugeno measure as a representative of non-additive measures comes as an interesting extension of the probability measure. Fuzzy random variables are the generalization of random variables. In this paper, we have discussed the SLT based on Sugeno measure and fuzzy random samples, provided the key theorem of learning theory based on Sugeno measure and fuzzy random samples, which constitute the generalization of corresponding results of the SLT based on probability measure and random samples. They are also the generalization of the SLT based on Sugeno measure and random samples. Based on Sugeno measure and fuzzy random samples, further research may focus on the bounds on the rate of uniform convergence of learning process based on the VC dimension, structural risk minimization principle, and the construction of support vector machines.

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Acknowledgments. This work was supported by the National Natural Science Foundation of China (No. 60773062), the Natural Science Foundation of Hebei Province of China (No. F2008000633), and the Key Scientific Research Project of Education Department of Hebei Province of China(No.2005001D).

References 1. Aumann, R.J.: Integrals of set-valued functions. Journal of Mathematical Analysis and Applications 12, 1–12 (1965) 2. Ha, M.H., Li, Y., Li, J., Tian, D.Z.: The key theorem and the bounds on the rate of uniform convergence of learning theory on Sugeno measure space. Science in China, Series E, Information Sciences 36, 398–410 (2006) 3. Ha, M.H., Pedrycz, W., Chen, J.Q., Zheng, L.F.: Some theoretical results of learning theory based on random sets in set-valued probability space. Kybernetes 38, 635–657 (2009) 4. Ha, M.H., Pedrycz, W., Zheng, L.F.: The theoretical fundamentals of learning theory based on fuzzy complex random samples. Fuzzy Sets and Systems 160, 2429–2441 (2009) 5. Ha, M.H., Tian, J.: The theoretical foundations of statistical learning theory based on fuzzy number samples. Information Sciences 178, 3240–3246 (2008) 6. Ha, M.H., Wang, C., Zhang, Z.M., Tian, D.Z.: Uncertainty of Statistical Learning Theory. Science Press, Beijing (2010) 7. Ha, M.H., Wang, C.: Some properties of random sets on Sugeno measure space. In: Proceedings of the Seventh International Conference on Machine Learning and Cybernetics, vol. 1, pp. 1583–1587 (2008) 8. Ha, M.H., Yang, L.Z., Wu, C.X.: Generalized Fuzzy Set-Valued Measure Theory. Science Press, Beijing (2009) 9. Li, S., Ogura, Y.: Strong laws of large numbers for independent fuzzy set-valued random variables. Fuzzy Sets and Systems 157, 2569–2578 (2006) 10. Puri, M.L., Ralescu, D.A.: Fuzzy random variables. Journal of Mathematical Analysis and Applications 114, 406–422 (1986) 11. Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D. Thesis, Tokyo Institute of Technology, Tokyo (1974) 12. Vapnik, V.N.: An overview of statistical learning theory. IEEE Transactions on Neural Networks 10, 988–999 (1999) 13. Vapnik, V.N.: Statistical Learning Theory. Wiley-Interscience Publication, New York (1998) 14. Vapnik, V.N.: The Nature of Statistical Learning Theory. Springer, New York (1995) 15. Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)

Recognition of Fire Detection Based on Neural Network Yang Banghua, Dong Zheng, Zhang Yonghuai, and Zheng Xiaoming Shanghai Key Laboratory of Power Station Automation Technology, Department of Automation, College of Mechatronics Engineering and Automation, Shanghai University, Shanghai, 200072, China [email protected]

Abstract. Aiming to the fire detection, a fire detection system based on temperature and pyroelectric infrared sensors is designed in this paper. According to the National Fire Detection Standard, a great number of test data are acquired. A model based on Levenberg-Marquardt Back Propagation (LM-BP) neutral network is established to recognize the fire status using the acquired data. Among the data, 200 groups of samples are used to train the established LM-BP networks while 1500 groups of samples test the LM-BP model. A 90% recognition rate is obtained by the LM-BP model. Compared with the other neutral networks such as Radial Basis Function (RBF) network, the LM-BP neural network has a significantly higher recognition rate (90%) than the RBF net (70%). The initial results show that the LM-BP recognition method has a favourable performance, which provides an effective way for fire detection. Keywords: fire detection; BP neural network; RBF network.

1 Introduction The fire is an extremely complex combustion process. It not only deprives great estates and financial losses, but also threatens human’s security. The fire detection technology is an effective method for the fire preventing[1]. At present, a variety of fire detection sensors are mainly used to detect fire resultant, such as smoke, carbon monoxide and temperature. Firstly, the physical and chemical parameters mentioned above are acquired and preprocessed. And then a fire detection system is used to obtain the fire data. Finally the fire is predicted effectively through the result of the algorithm recognition. How to enhance the accuracy and the response time of the fire detection and how to reduce the failure and misinformation are the keys of following research. The accuracy of fire detection involves many factors. The algorithm identification is the most important factor in the same hardware conditions. With the development of fire detection, fire detectors and algorithms researches have made great progress. Various new and excellent fire detectors are used to make raw data reliable, which lay a solid foundation for the recognition algorithm. The current fire detection and K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 250–258, 2010. © Springer-Verlag Berlin Heidelberg 2010

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recognition algorithms are as follows: 1) threshold method: the method uses a single sensor signal processing, in which the fixed threshold and the change rate are the two main detection algorithms[2]. The fixed threshold algorithm will make a comparison between the fire signal amplitude and a pre-set threshold signal. When the signal amplitude exceeds the threshold, the fire alarm signal will produce an output. The change rate algorithm judges the increasing rate of temperature signal and indicates whether the fire has gone to the exothermic phase. 2) The duration algorithm: when a fire occurs, there is a clear change of the detector signals and the change will maintain some time[3]. The fire detection signal on fire condition exceeds a fixed threshold longer than on no fire condition. The changes in trend and duration over time are used the signal feature to detect a fire, which increases the accuracy of fire detection. These two methods have advantages and disadvantages. The first method is simple and easy to realize, but the selection of the threshold depends on the human experience and repeated experiments. Its adaptability to the environment and its anti-interference capability are poor, therefore it has a high false alarm rate. The second method takes into account the duration that the fire signal exceeds the threshold and use digital filters to realize the algorithm. Its anti-interference capability has been improved, but the construction of filters and parameters’ selection are complicated. To further enhance the effectiveness in the fire detection, some artificial neural network (ANN) intelligent algorithms have been widely used in the field of fire detection in recent years. The ANN is composed by a large number of interconnected artificial neurons and used to simulate the brain structure and function of the system. The neural network has many different types, such as the Back Propagation (BP) network, Radial Basis Function (RBF) network, Elman network, self organizing network, Hopfield network, Boltzmann networks, Support Vector Machine (SVM) network, which is not less than 20 kinds of algorithms[4]. A fire detection system based on temperature and pyroelectric infrared sensors is designed in this paper. According to the National Fire Detection Standard about fire detections, a great number of test data are acquired. A model based on LevenbergMarquardt BP(LM-BP) neutral network is established to recognize the fire status using the acquired data. Among the data, 200 groups of samples are used to train the established BP networks while 1500 groups of samples are used to test it.

2 Introduction to Fire Detection System 2.1 System Architecture The fire detection system block diagram is shown in Fig.1. It consists five parts: temperature and infrared sensors, amplifying and filtering circuit, A/D converter, microprocessor recognition processing and data output. The temperature and infrared sensors are used for fire detection. The temperature sensor can also detect the target and ambient temperature. The infrared sensor operates around the 4.3μm region and the detection angle is 120°. The two output

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signals of composite detectors largely reflect the major changes in the environment when the fire occurs. Coupled with the fire combustion, the temperature of the surrounding environment will be gradually increased, while the infrared flame radiation will change much instantly. Through the detection and corresponding data fusion by two sensors, the possibility of misstatement and omission can be reduced. The amplifying part amplifies the output signal of the composite sensors appropriately and the corresponding filtering part removes signal noise and glitches to make the signal smoother and easier to identify. The A/D conversion part converts the analog signal into digital signals, which are analyzed in the microprocessor through corresponding signal processing algorithms. Finally, the data output part displays the results of the monolithic processing with the fire alarm light.

Fig. 1. Fire Detection System Block Diagram

The microprocessor signal recognition is the core in fire detection system and it relates to accuracy of fire detection. This paper focuses on the algorithm of the fire detection in order to enhance the reliability and validity of the fire detection system. 2.2 The Data Acquisition

℃

Based on the established detection system, in the outdoor test environment (27 ), under the National Fire Detection Standards, the specific data acquisition process is as follows: 1 Pour 2000g anhydrous ethanol (concentration 95%) into 2mm thick, 33cm × 33cm bottom size, 5cm height of steel containers; 2 Light the fire with the way of flame ignition; 3 The detector system is put at the 12 meters away from the fire, and the data from detector system are recorded about 30 seconds during the fire combustion. After 30 seconds, a shutter is used to block the detector. The block time is 5 seconds and the corresponding 5 seconds data also are recorded, the data from the 5 seconds corresponds to no fire combustion. Then remove the shutter and record 30 seconds data from fire combustion. The shutter is again used to block the fire and the 5 seconds are recorded. The process is repeated many times and the whole test data are recorded; 4) The detector system is put at the 17 meters away from the fire, repeat the same experiment as 3) and record the test data;

） ） ）

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）

5 The detector system is put at the 25 meters away from the fire, repeat the same experiment as 3) and record the test data; Through this process, thousands of groups of test data are acquired, which contain data from 12, 17 and 25 meters between detector system and fire. The data lay a foundation for the neural network constructed below.

3 Fire Signal Processing Based on Neural Network 3.1 Principle of Neural Network The Artificial Neural Network (ANN) theory was born only just over half a century from the late 1940s. However, it has many advantages such as distributed storage, parallel information processing and self-learning ability, etc. It need not be described in the object modeling and can also describe the nonlinear systems and uncertain systems well. It has been widely applied in the field of automatic control, fault diagnosis and testing, pattern recognition and classification so far. In particular, the BP network based on back propagation algorithm can realize arbitrary accuracy to continuous function approaching. In the practical application of artificial neural networks, the BP neural network has been adopted around 80 - 90%[5]. The BP network is a forward neural network with three or more layers which has a one-way communication, including input, output and hidden layers. Upper and lower layers have full connections, and each layer has no connection between neurons. The traditional BP algorithm uses back-propagation algorithm to adjust training parameters for network weights and bias repeatedly and make the output vector and expect vector as close as possible. When the output layer error sum squares is less than the specified error, the training is completed and the network weights and bias are preserved. Essentially this is a nonlinear optimization problem. And by the negative gradient descent algorithm, this learning method has the slow convergence and is easy to the local minimum. Therefore the improved BP network training algorithm is adopted in this paper. The LM-BP algorithm is the most widely used optimization algorithm. It outperforms simple gradient descent and other conjugate gradient methods in a wide variety of problems. The LM-BP algorithm is the combination of the gradient descent algorithm and the GaussNewton algorithm. The problem for which the LM algorithm provides a solution is called Nonlinear Least Squares Minimization. Because of adopting the approximate second-order derivative information, the LM algorithm is faster than the gradient descent algorithm. And LM algorithm is also superior to the variable learning rate method of BP algorithm. The specific learning process and steps are shown in reference[6]. Besides the BP neural network, the RBF neural network is often used in pratical application. The RBF network is based on a strong interpolation theory and it learns fast and is suitable for real time control. But the number of its network hidden nodes is difficult to determine and its center vector and standardized constants are hard to

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find, which affect the precision of network learning. Although BP network has a slow convergence speed compared to RBF network its learning algorithm belongs to the global approach and has a better generalization ability, better fault tolerance and better recognition effect. The BP and RBF network are constructed in this paper. Considering the speed and accuracy comprehensively, the neural network algorithm which is more suitable for the detection system and has been chosen finally. 3.2 Design of BP Network Model A forward BP network with three layers is adopted in this paper, including input, output and hidden layers. The input layer links the network with the external environment. The hidden layer makes non-linear transform between input spaces with hidden spaces, which usually has higher dimension. The output layer provides the response signal for its activation signal. Fig.2 shows the BP network with three layers. The BP with three input nodes, seven hidden nodes and one output node is adopted in this network.

Fig. 2. BP Neural Network Structure with Three Layers

This three layers BP neural network is to realize the nonlinear relation from the infrared radiation changes to the fire status. Therefore inputs are temperature changes and infrared radiation variation and the output is fire probability. According to the above principles the input layer is designed to three nodes (X1, X2, X3), respectively, the target temperature(X1), ambient temperature(X2), pyroelectric infrared (X3) The output layer is designed to one node(Y) which is whether there is fire or not (1 is yes, 0 no). The number of hidden units has direct relationship with the network performance. Too many hidden units will result in a long learning time and can not obtain the best error. At the same time, it has poor fault tolerance and can not identify unknown samples. Base on the following formula:

n1 = m + n + l (l ∈ [1,10]) , Where m is the number of output units, n is the number of input units, n1 is the number of hidden units.

(1)

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n

∑ C n > k , Where, k is the number of samples. i =0

i

255

(2)

i

The selected hidden layer cell number ranges from 7 to 13 according to above two forma. Through the training, learning and comparison of the output error of test samples, 10 hidden nodes are selected because of good convergence and little error. The hidden layer transfer function adopts ‘tansig’ function and the output layer transfer function adopts ‘logsig’ function while ‘trainlm’ is selected as training function. To enable more efficient network training, the neural network input and output data are normalized. The input and output data is mapped to [0, 1] range. 3.3 Fire Prediction Based on BP Using the BP model constructed and related parameters chosen above, the test data of target temperature, ambient temperature, infrared pyroelectric are used as input X1, X2, X3 respectively and the target output Y is whether the fire occurs. Sorting the experimental data which are acquired by the experiment 2.2 and 1700 groups of samples are obtained including three kinds of data in 12 17 25 meters between detector system and fire under the National Fire Detection Standards.. Among the data, there are 200 groups of training samples while 1500 groups of testing samples. The main parameters of the training are as follows: Training epochs: 200; Training goal: 0; Learning rate: 0.1. 200 groups of samples are trained and the training convergence curve is shown in Fig 3.

、 、

Fig. 3. Training Convergence Curve

It can be seen from Fig.3 that the network can forecast the fire correctly after 200 times training. Since the network training is successful, the test samples are used to test the network. Test results on 1500 groups of samples show that misinformation is only 4 groups and a 90% recognition rate is obtained by the BP model. The BP neural network error curve is shown in Fig.4.

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Fig. 4. BP Neural Network Error Curve

3.4 Fire Prediction Based on RBF To further test the validity of BP network, a RBF network is built to train and simulate fire data. The RBF neural network consists of two layers. The input nodes transfer input signal to the hidden layer. The hidden nodes are constructed by radial effect function like Gaussian function. Output layer nodes are usually simple linear functions. According to the RBF model input layer is designed to three nodes-the target temperature, ambient temperature, pyroelectric infrared respectively. The output layer is designed to one node which is the fire probability. The model is based on selforganizing learning algorithm. Sorting the experimental data which are acquired by the experiment 2.2, 1700 groups of samples are obtained including three kinds of data in 12 17 25 meters between detector system and fire under the National Fire Detection Standards. Among the data, there are 200 groups of training samples while 1500 groups of testing samples. 200 groups of samples are trained and obtain the training convergence curve as shown in Fig.5.

，

，

、 、

Fig. 5. Training Convergence Curve

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It can be seen from the Fig.5 that after 11 times’ training, the network achieves training objectives. Then the test samples are used to test the network. Test results on 1500 groups of training samples show that a 70% recognition rate is obtained by the RBF model. The RBF neural network error curve is shown in Fig.6. The RBF network has a faster convergence speed compared with BP network but the relatively recognition rate is low. Comparing the identification results of the BP and RBF network, it can be seen that although BP network has a slow convergence speed compared to RBF network it has a less relative error and a better recognition effect. Because the RBF network has a worse generalization than BP network when the goal is not continuous. And in this fire detection system, the goal is whether there is fire or not (1 is yes, 0 no).

Fig. 6. RBF Neural Network Error Curve

4 Conclusion A fire detection system based on temperature and pyroelectric infrared sensors is designed in this paper. A LM-BP neural network fire detection model is constructed. Through test data acquisition under National Fire Detection Standard and comparison results between the BP and RBF network detection model, the preliminary simulation results show that the BP network can obtain a high fire identification rate which is more than 90%. The BP recognition method has a better performance compared with RBF network, which provides an effective way for fire detection.

Acknowledgments This project is supported by National Natural Science Foundation of China (60975079), Shanghai University, "11th Five-Year Plan" 211 Construction Project, Shanghai Key Laboratory of Power Station Automation Technology (08DZ2272400).

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References 1. James, A.M.: Using Multiple Sensors for Discriminating Fire Detection. J. FL. USA, 150–164 (1999) 2. Okayama, Y.: A Primitive Study of a Fire Detection Method Controlled by Artificial Neural Net. J. Fire Safety Journal 17, 535–553 (1991) 3. Pfister, G.: Multisensor fire detection: A new trend rapidly becomes state of art. J. Fire Technology 33, 115–139 (1997) 4. James, A.M., Thomas, J.: Analysis of signature patterns for discriminating fire detection with multiple sensors. J. Fire Technology 31, 120–136 (1998) 5. Moghavvemi, M., Seng, L.C.: Pyroelectric infrared sensor for intruder detection. J. Analog and Digital Techniques in Electrical Engineering, 656–659 (2004) 6. Thuilldar, M.: New method for reducing the number of false alarms in fire detection systems. J. Fire Technology 31, 250–268 (1994)

The Design of Predictive Fuzzy-PID Controller in Temperature Control System of Electrical Heating Furnace Ying-hong Duan College of Electronic Information and Automation, Tianjin University of Science &Technology, Tianjin 300222, China

Abstract. An electrical heating furnace temperature control system is characterized by large inertia, pure time-delay and parameters time-varying, which needs a long time to control with conventional control methods. It is hard to meet the technical requirements. An improved Smith predictive fuzzy-PID composite control method is therefore presented. A mathematic model of the electrical heating furnace temperature control system is established and the structure of the improved Smith predictive fuzzy PID controller and the method of generating fuzzy control rules are introduced. The simulation result shows that the control system may reduce the overshoot, shorten the time to stabilize, improve control accuracy, and work well for the electrical heating furnace system. Keywords: temperature control, fuzzy-PID control, simulation, Smithpredictor, electrical heating furnace.

1 Introduction Electrical furnaces are widely used in metallurgy, chemical industry, machinery and other modern industries, and their performances decide the quality of relevant products. They may be considered to have a pivotal position in the national economy. But their temperature control is featured by one-way heating, large inertia, large time delay, time-varying characteristics, and so on. [1] And the pure delay time and the inertia time are difficult to identify, so using conventional control methods, such as PID regulator, take longer time to stablize and can not meet the technical requirements properly. Therefore, this article presents a new controller based on improved Smith fuzzy PID Algorithm for electrical heating furnace temperature control system, that is, selfadaptive fuzzy-tuning PID controller instead of ordinary PID controller, to make the controller parameters of the controlled object to self-adapt to overcome the adverse effects of object model parameters changing, to improve the system response speed, and to improve control accuracy; using the improved Smith predictor control instead of conventional Smith control to realize the dynamic compensation of the hysteresis. The controller can estimate the process of the basic dynamic characteristics in a pure delay phenomenon under disturbance in advance, so that regulators act ahead of time to reduce the overshoot and adjustment time. Simulation results show that the K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 259–265, 2010. © Springer-Verlag Berlin Heidelberg 2010

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improved fuzzy PID controller of Smith estimated system can reduce the adjustment time, and make the system free of overshoot with higher control accuracy, and effective control of electrical heating furnace may be achieved.

2 Establishment of a Control System Model For convenience, it is often to consider the dynamic characteristics of furnace temperature as a linear system in the process control. The approximation is feasible in many cases that using one or two inertia link in series as a pure delay part to express. [2] It can be that the electric heating furnace is a large volume lagged behind, with a certain amount of pure delay (volume lag is much longer than the delay time). If the controller object, that is electric furnace, is rated 8 kilowatts, powered by 220V single-direction AC power, we use KS200A/800VA TRIAC zero trigger control. The transfer function of the electric heating furnace measured from the band soaring characteristic curve is as follows:

G p (S) =

K p *e −τs Tp + 1

=

2.8e −40s . 178S + 1

(1)

3 Design of Improved-Type Predictive Fuzzy PID Controller 3.1 Smith Predictor and Its Improved Version

In order to deal with the problem that PID control is difficult to address the complete elimination of the effects of pure delay time constant, this article introduces Smith predictor as the compensation control. The principle of Smith predictor is to estimate the basic dynamic process characteristics in advance under disturbance, and then is to compensate by the predictor, aiming to make the overshoot of delayed τ reflected in the regulator, to make the compensated equivalent transfer function no longer with time delay, to allow regulators work ahead of time, to reduce the overshoot and speed up the adjustment process, and to shorten the adjustment time. The most significant drawbacks of Smith predictive control are the following. The first is that the control quality will deteriorate and even diverge when there is a margin of error between the estimated model and the real model; the second is that the control is very sensitive to enable the poor robustness under external disturbances. Therefore, this system uses the improved version of Smith predictor proposed by C.C.Hang. The improved Smith predictor structure is shown in Figure 1. In the figure G (S) is the transfer function after the object is removed with delay − τs

part ( e ), and K is its gain; Gm(s)e (-τms) .is the Smith predictor transfer function, and Km is its gain; GC0 (s ) is the main regulator transfer function, and GC (s) is the secondary regulator transfer function. The improved Smith predictor transfer function of the main feedback channel is not 1, but

G f (s) =

G C (S)G m (s) 1 + G C (s)G m (s)

(2)

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R(s) _

GC0(s)

G(S)e(-τS)

Gm(s)

Gm(s)e(-τms)

261

Y(s)

Ym(s _

)

Gc(s) Gm(s) 1+Gc(s)Gm(s) Fig. 1. Improved Smith Predictor Schematic Diagram

When GC (s) is the fuzzy PID regulator and Gm (s) is the first-order inertia link, making the integral regulator time constant equal to the model time constant, so that the primary feedback channel transfer function Gf (s) could be reduced to

G f (s) =

1 tf s + 1

(3)

tf = Tm / (KCKm) is the filter time constant, Tm is the predictor time constant, KC is the GC (s) gain. The obtained closed-loop system transfer function is: GC ( S )G ( s)e( −τ s ) Y ( s) = R (s) 1 + G ( s)G (s ) + G ( s)(G ( s)e( −τ s ) − G ( s)e ( − tm s ) ) C m C m

1 t f s +1

(4)

Formula (4) is the same as the conventional Smith predictor control program, and the improved program has no effect on the system. Actually, this improvement program is that introducing a one-order inertia link 1/(tfs+1) in the main feedback channel. tf can be adjusted to change the closed-loop system characteristic equation roots, thereby changing the performance of the control system. When tf is 0, the system is equivalent to the conventional Smith Predictor Control System [3]. 3.2 Improved Smith Estimated Fuzzy PID Controller

This improved design of Smith predictive fuzzy PID controller is mainly used to improve the system dynamic performance, and to shorten the adjustment time. The controller thorough fuzzy self-tuning PID parameters may improve the system response speed, improve accuracy, and maintain a system of control targets. Then using the modified Smith predictor may achieve the dynamic hysteresis compensation control strategy. Among them, fuzzy adaptive PID parameter tuning methods are the main regulator, which makes the corresponding decision by fuzzy inference and on-line tuning of PID parameters Kp, Ki, Kd, to obtain a satisfactory control effect, according to the input signal (ie, the error E) the size, direction and trend of such features. The improved Smith, the auxiliary regulator regulator, is mainly used to eliminate the system inertia and delay. Its system block diagram is shown in Figure 2:

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R(s) _

Fuzzy PID control

G(S)exp(-τs)

Y(s)

Ym(s Gm(s)exp(-τms) ) _

Gm(s) 1 (tfs+1)

Fig. 2. Improved Fuzzy PID Controller Smith Predictor Block Diagram

G (S) is the control object transfer function; Gm (s) is the Smith predictor. Master regulator fuzzy self-tuning PID controller. It based on these features such as the input signals (the error e), sizes, direction and change trend. Through the corresponding decision made by FUZZY reasoning and on-line tuning of PID parameters Kp, KI, KD, the system can obtain satisfactory control effect. Its functions are to reduce the overshoot and response time to make the system fast access to steady-state and strong adaptability of time-delay, time-varying. According to the features of this control object system, the article selects twodimensional fuzzy controller, the bias and the bias rate of change of E as the EC language input variables, and the PID parameters Kp, Ki, Kd as output. The values of E and EC were chosen as the language of negative big (NB), negative middle (NM), negative small (NS), zero (Z), positive small (PS), positive middle (PM), positive big (PB) 7 files, the range of fuzzy control is (-3, -2, -1, 0, 1, 2, 3). For convenient of realizing, input bias E, input and output error rate of change of EC membership functions are used a linear function. Designed using MATLAB fuzzy deviation, E, EC, Kp, Ki, Kd of the discourse of fuzzy linguistic variables corresponding to the membership function are shown in Figure 3: NB NM

-3

NS

-2

-1

u Z

0

PS

PM

1

2

PB

／ ／／

E/EC Kp Ki Kd 3

／

Fig. 3. E/EC Kp/Ki/Kd Curve of Membership Grade Functions

The discrete PID regulator expression:

u (k ) = K p e(k ) + K i T ∑ e ( j ) + K d Δ e ( k ) / T

(5)

T is the sampling period, e (k), Δe (k) are the input, which are known quantities; while the unknown quantities are the controller parameters Kp, Ki, Kd; Ki = Kp / Ti, Kd = Kp / Td. Ti is the integration time constant, Td is the differential time constant.

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From formula (6) we can see PID controller structure and algorithms have been identified, and the quality of control system depends largely on control parameter selection. By summarizing the engineering practical experience in the past, we conclude the setting principles of Kp, Ki, Kd for different E and Ec: When E is large, in order to have a good tracking performance, the system should take a larger Kp smaller Kd, while, in order to avoid a large overshoot, integral action should be limited, usually taking Ki = 0; when E and Ec are appropriate, in order to have a smaller overshoot, Kp should be made smaller, and Kd values impact greatly on the system, so Kd should be made smaller and Ki should be appropriate; when E is small, in order to have good stability and performance, Kp, and Ki should be made bigger, and in order to avoid oscillation near the set value, anti-jamming performance should be taken into account, when Ec is large, Kd can be made small; When Ec is small, Kd can be made bigger. [4] [5] Based on the above analysis and linguistic variables set, one can conclude the selfadjusting control rules of Kp, Ki, Kd (see Table 1, 2, 3) The above fuzzy control rules can also be written in the form of conditional statements. According to the rate of change error E, the deviation to quantify the value of the entire EC and fuzzy relationship, through the synthesis of fuzzy inference rules of operation, the amount of the corresponding changes in fuzzy sets can be drawn, that is a fuzzy subset reflecting the control language of a combination of different values. Table 1. Control Rule of Kp

E EC NB NM NS Z PS PM PB

NB kp PB PB PM PM PS Z Z

NM

NS

Z

PS

PM

PB

PB PB PM PS NS Z NS

PM PM NS PS Z NS NS

PM PM PS Z NS NM NM

PS PS Z NS NS NM NM

PS Z NS NM NM NM NB

Z Z NM NM NM NB NB

Table 2. Control Rule of Ki

E EC NB NM NS Z PS PM PB

NB

NM

NS

Z

PS

PM

PB

NB NB NM NS NS Z Z

NB NM NM NS Z PS PS

NM NM NS Z PS PM PM

NM NM Z PS PS PM PB

Z Z PS PS PB PB PB

Z Z PS PM PM PB PB

Ki NB NB NM NM NS Z Z

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E EC NB NM NS Z PS PM PB

NB

N M Kd

PS NS NB

PS NS NB

NB NB PM PS

PM PM NS Z

PS

PM

PB

Z NS NS

Z Z Z

PB PS PS

PB PM PM

NS NS NS Z

Z Z Z Z

PS PS PS PB

PM PS PS PB

NS

Z

Z NS B M PM NS NS Z

Fuzzy reasoning method is Mamdani- type. The method of taking AND is min, the method of OR is max, the method of Implication is min, the method of Aggregation is max, the method of Defuzzification is the focus of the averaging to find the corresponding amount of control. The auxiliary regulator uses an improved version of previously mentioned Smith Predictor Controller for the purpose of reducing the overshoot and speeding up the adjustment process to further shorten the adjustment time.

4 Simulation Results For comparative analysis and to validate the effectiveness of the design of an improved predictive fuzzy PID controller, this article uses an improved predictive Fuzzy PID controller and conventional PID controller to do simulation test. Two kinds of controllers are both of the above-mentioned electrical heating furnace mathematical models, when the unit step r (t) = 80 input under the two kinds of controller's output waveform is as shown in Figure 4:

Xi (Temperature/ C) Fig. 4. Controller’s Output Waveform

Xj (Time/s)

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Table 4. Dynamic Parameter List of Two Controllers

Overshoot PID controller Improved predictive fuzzy PID controller

3.2℃ （4%） 0

Steady-state error

Accuracy

Adjustment time (seconds)

0.255

99.681%

2014

-0.0094

99.988%

465

From the data of Table 4 and the simulation of Chart 4, we can see that both methods can achieve the requirements setting temperature. Comparing to the traditional PID controller, the qualities of improved estimate of Fuzzy PID controller have improved significantly, like overshoot, settling time and precision performance indicators. Especially the improved controller does not have overshoot. The adjustability of system has reached a short time, steady-state error (only 0.0094) and other expectations. This is the same as theory, like improving control accuracy, improving response time and suppressing the overshoot. It also shows that the improved Fuzzy PID controller has no overshoot, fast response and high precision characteristics, and that using the controller to improve electric heating furnace temperature control system is feasible.

5

Conclusion

According to changing the value of error and bias to determine the size of the intelligent PID parameters, and based on fuzzy control rules, improved Fuzzy PID controller combines the ordinary PID , fuzzy control, and improved combination of Smith predictor to make the system has good response speed and accuracy degrees, and no overshoot in the heating process. The controller software is easily available. In the practical application of the programming, the temperature curve is divided into two sections: The first is that the system takes full power output to increase temperature rapidly, when the temperature falls below 85% of given temperature; The second is the system add improve estimated fuzzy PID-type controller to realize intelligent heat up, to reduce the overshoot in the increasing temperature process, and to obtain a more satisfactory process control curve, when the temperature rises to 90% of the given temperature.

References 1. Liang, F., Ma, X.J.: A kind of online interpolation fuzzy controller is applied in temperature control system. J. Manufacturing Automation. 6, 21–22 (2006) 2. Liu, A.M., Cao, A.H., Jin, J.Y.: Smith predictor base on adaptive temperature control system. J. Science & Technology Information 11, 9–15 (2008) 3. Chen, Y., Yang, Q.W.: Design and Simulation of Fuzzy Smith Intelligent Temperature Controller. J. Control Engineering of China 4, 422–429 (2007) 4. Wen, D.D.: Temperature Control System of Furnace Based on Fuzzy Control Algorithm. J. Industrial Furnace 5, 30–33 (2007) 5. Li, H.L.: Fuzzy self-adaptive PID control for temperature of resistance furnace. J. Journal of JILIN Institute of Chemical Technology 2, 66–67 (2007)

Stability Analysis of an Impulsive Cohen-Grossberg-Type BAM Neural Networks with Time-Varying Delays and Diffusion Terms Qiming Liu, Rui Xu, and Yanke Du Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiahzuang 050003, China [email protected]

Abstract. An impulsive Cohen-Grossberg-type BAM neural network with time-varying delays and diffusion terms is investigated. By using suitable Lypunov functional and the properties of M-matrix, sufficient conditions to guarantee the uniqueness and global exponential stability of the equilibrium solution of such networks are established. Keywords: BAM neural networks, time delay, impulses, diffusion, global stability.

1 Introduction In 1988, Kosko proposed bidirectional associate memory (BAM) neural networks which has good application perspective in the area of pattern recognition, signal and image process [1]. Later, Gopalsamy and He investigated BAM models with axonal signal transmission delays which has obtained significant advances in many fields such as pattern recognition, automatic control, etc. [2]. In the implementation of neural networks, time delays are unavoidably encountered due to the finite speeds of the switching and transmission of signals in a neural network. Various sufficient conditions, either delay-dependent or delay-independent, have been proposed to guarantee the stability for various neural networks with delays, for example, see [3-8]. On the other hand, in real world, many evolutionary processes are characterized by abrupt changes at certain time. These changes are called to be impulsive phenomena, which are included in many neural networks such as Hopfield neural networks, bidirectional neural networks and recurrent neural networks and can affect dynamical behaviors of the systems just as time delays, many results are presented in recent years[7-12]. At the same time, the diffusion phenomena could not be ignored in the factual operations in neural networks and electric circuits once electrons transport in a non-uniform electromagnetic field. Hence, it is essential to consider the state variables are varying with the time and space variables. The study on the stability of reaction–diffusion neural networks, for instance, see [11-17], and references therein. Motivated by above discussion, the objective of this paper is to study the global exponential stability of a class of impulsive Cohen-Grossberg-type BAM (CGBAM) neural networks with time-varying delays and reaction-diffusion terms by using K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 266–276, 2010. © Springer-Verlag Berlin Heidelberg 2010

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suitable Lypunov functional and the properties of M-matrix, without assuming the boundedness of the activation functions, as needed in many other papers. The rest of this paper is organized as follows: Model description and preliminaries are given in Section 2. The sufficient conditions which guarantee the uniqueness and global exponential stability of the equilibrium solution for CGBAM neural networks are given in Section 3. An example is given in Section 4 to demonstrate the main results.

2 Model Description and Preliminaries Consider the following impulsive CGBAM neural networks with distributed delays and reaction-diffusion terms. m ∂ui (t , x ) = Di Δui (t , x) − ai (ui (t , x))[bi (ui (t , x)) − ∑ cij f j (v j (t , x )) ∂t j =1 m

− ∑ d ij f j (v j (t − τ ij (t ), x )) − I i ], t ≠ tk ,

(1a )

j =1

ui (tk+ , x) = ui (t k− , x) + Eik (ui (tk− , x)), t = tk , k ∈ N , ∂v j (t , x) ∂t

(1b) n

= D j Δv j (t , x ) − a j (v j (t , x ))[b j (v j (t , x)) − ∑ c ji g i (ui (t , x)) i =1

n

− ∑ d ji g i (uI (t − τ ji (t ), x )) − I j ], t ≠ tk ,

(1c)

i =1

v j (tk+ , x) = v j (t k− , x ) + E jk (v j (tk− , x )), t = tk , k ∈ N

for 1 ≤ i ≤ n, 1 ≤ j ≤ m and t > 0 ,

where N = {1, 2,

}

(1d )

is a set of positive integers,

xq (t ), 1 ≤ q ≤ l are space variables, ui and vi denote the state variable of the ith neuron from the neural field FX and the jth neuron from the neural field FY at time t, respectively; Di ≥ 0 and D j ≥ 0 correspond to the transmission diffusion coefficient along the ith neurth from FX and the jth neuron from FY , respectively; denote by Δui = ∑ q =1 l

∂v 2j (t , x) ∂u 2i (t , x) l Δ v = and ∑ q =1 ∂x 2 the Laplace operators; f j (v j (t, x)) j ∂xq2 q

and gi (ui (t , x)) denote the signal functions of the ith neuron from FX and the jth neuron from FY at time t and in space X, respectively; I i and I j denote inputs of the ith neuron from the neural field FX and the jth neuron from the neural field FY at the time t, respectively; ai (ui (t , x)) and a j (v j (t , x)) represent amplification functions; bi (ui (t , x)) and b j (v j (t , x)) are appropriately behaved functions; cij , dij , c ji , d ji are the connection weights, respectively; τ ij (t ) and τ ji (t ) are time-vary delays; tk is called −

impulsive −

moment, + k

and + k

satisfy

0 < t1 < t2
0 in R l . Denote by ∂ / ∂l the outward normal derivative. The boundary conditions and initial conditions are given by ∂ui ∂l

= 0, ∂Ω

∂v j ∂l

=0

t ≥ 0, i = 1, 2,

for

, n, j = 1, 2,

m,

(2a)

∂Ω

ui ( s, x) = ϕi ( s, x), v j ( s, x) = φ j ( s, x), − σ ≤ s ≤ 0, where σ = max{τ ,τٛ},τ =

max τ ij ,τٛ=

1≤ i ≤ n ,1≤ j ≤ m

(2b)

max τٛji .

1≤ i ≤ n ,1≤ j ≤ m

If impulsive operator Eik = 0 and E jk = 0 , the system (1) may reduce to the following system with boundary conditions and initial conditions (2): m ∂ui (t , x ) = Di Δui (t , x ) − ai (ui (t , x ))[bi (ui (t , x )) − ∑ cij f j (v j (t , x )) ∂t j =1 m

− ∑ d ij f j (v j (t − τ ij (t ), x )) − I i ], j =1

∂v j (t , x ) ∂t

ٛ = D j Δv j (t , x ) − aٛj (v j (t , x ))[b j (v j (t , x )) − ٛ

n

ٛ

n

∑ i =1

cٛji gi (ui (t , x ))

(3)

ٛ

− ∑ d ji gi (u j (t − τٛji (t ), x )) − I j ]. i =1

System (3) is called the continuous system of the impulsive system (1). For convenience, we introduce some notation and recall some basic definitions. Δ

PC[ J × Ω , R n + m ] = {ψ (t , x) : J × Ω → R n + m | ψ (t , x)

is

a

continuous

at

t ≠ tk , ψ (t , x) = ψ (tk , x) and ψ (t , x) exists for tk , tk ∈ J , k ∈ N } , where J ∈ R is a interval. + k

Δ

− k

PC[Ω ] = {η (t , x) : (−σ , 0] × Ω → R n + m | η (t , x)

η ( s + , x) = η ( s, x) for s ∈ (−σ , 0] ,

η ( s − , x) exists for s ∈ (−σ , 0] , η ( s − , x) = η ( s, x) for all but at most a finite number of points s ∈ (−σ , 0]. . Δ

PC = {η (t ) : (−σ , 0] → R n + m |

η ( s + ) = η ( s) for s ∈ (−σ , 0] , η ( s − ) exists for

s ∈ (−σ , 0] , η ( s − ) = η ( s ) for all but at most a finite number of points s ∈ (−σ , 0]. Throughout this paper, we assume that

Stability Analysis of an Impulsive Cohen-Grossberg-Type BAM Neural Networks

269

(H1) Function ai (ui ) and aٛj ( v j ) are bounded, positive and continuous, i.e. there exist ٛ

ٛ

constants αi , β i , αٛj and β j such that 0 < α i ≤ ai (ui ) ≤ βi and 0 < αٛj ≤ aٛj (u j ) ≤ β j . (H2) Function bi (ui ) and b j (v j ) are monotone increasing , i.e. there exists positive numbers bi and b j such that

ٛ

ٛ

b j (ξ1 ) − b j (ξ 2 ) ٛ bi (ξ1 ) − bi (ξ 2 ) ≥ bi , ≥ bj ξ1 − ξ2 ξ1 − ξ 2 for all ξ1 ≠ ξ 2 , 1 ≤ i ≤ n, 1 ≤ j ≤ m. (H3) For activation function f j (⋅) and gi (⋅) satisfy Lipschitz conditions. That is, there exists constant L j and Li such that

ٛ

| f j (ξ1 ) − f j (ξ 2 ) |≤ L j | ξ1 − ξ 2 | , gi (ξ1 ) − gi (ξ2 ) |≤ Li | ξ1 − ξ 2 | for any ξ1 , ξ 2 ∈ R , 1 ≤ i ≤ n, 1 ≤ j ≤ m. ٛ

ٛ

(H4) Let hik = ui + Eik and h jk = vi + E jk are Lipschitz continuous, i.e. there exist

nonnegative number γ ik and γٛjk such that

| hik (ξ1 ) − hik (ξ 2 ) |≤ γ ik | ξ1 − ξ 2 |, | h jk (ξ1 ) − h jk (ξ 2 ) |≤ γ jk | ξ1 − ξ 2 | for any ξ1 , ξ 2 ∈ R , 1 ≤ i ≤ n, 1 ≤ j ≤ m. Definition 1. Function (u (t , x), v(t , x))T ([−σ , +∞) × Ω → R n + m ) is said to be the solution of (1), if the following conditions are satisfied. (i) For fixed x, (u (t , x), v(t , x))T is piecewise continuous with first king discontinuity

at the points tk , k ∈ N . Moreover, (u (t , x), v(t , x))T is right continuous at each discontinuity point. (ii) (u (t , x), v(t , x))T satisfies (1a) and (1c) for t ≥ 0 and u ( s, x) = ϕ ( s, x), v( s, x) = φ ( s, x), − σ ≤ s ≤ 0, where u (t , x) = (u1 (t , x), u2 (t , x), v(t , x) = (v1 (t , x), v2 (t , x),

, un (t , x)),

, vm (t , x)).

Especially, a point (u * , v* )T is called an equilibrium point of system (1), if (u (t , x), v(t , x))T = (u * , v* )T is a solution of the system. (H5) If (u * , v* )T is an equilibrium of continuous system (3), the impulsive jumps satٛ

isfy Eik (ui * ) = 0 and E jk (v j * ) = 0 [9-11]. By (H5), we know that if (u * , v* )T is an equilibrium of continuous system (3), (u * , v* )T is also the equilibrium of impulsive system (1). For (u (t , x), v(t , x))T ∈ R n + m , define || (u (t , x), v(t , x))T || = ∑ i =1 || ui (t , x) ||2 + ∑ j =1 || v j (t , x) ||2 , where n

m

1

1

|| ui (t , x) ||2 = [ ∫ | ui (t , x) |2 dx] 2 , || v j (t , x) ||2 = [ ∫ | v j (t , x) |2 dx]2 , 1 ≤ i ≤ n,1 ≤ j ≤ m. Ω

Ω

then PC[Ω ] is a Banach space with respect to || (u (t , x ), v (t , x ))T || .

270

Q. Liu, R. Xu, and Y. Du

Definition 2. A equilibrium point (u * , v* )T of system (1) is said to be globally exponentially stable, there exist positive constants λ and M > 0 such that || (u (t , x) − u* , v(t , x) − v* )T ||≤ M || (ϕ (t , x) − u * , φ (t , x) − v* )T ||σ e − λ t where

|| (ϕ (t , x) − u* , φ (t , x) − v* )T ||σ = sup

−σ ≤ s ≤ 0

∑

n i =1

|| ϕi (s, x) − ui* ||2 + sup

−σ ≤ s ≤ 0

∑

m j =1

|| φ j (s, x) − v j * ||2 .

Definition 3. A real matrix A = (aij ) n× n is said to be a nonsingular M-matrix if

aij ≤ 0 (i, j = 1, 2,

, n, i ≠ j ) and all successive principle minors of A are positive.

Lemma 1. A matrix with non-positive off-diagonal elements A = ( aij ) n× n is an M-

matrix if and if only there exists a vector p = ( pi )n×1 such that pA > 0. Lemma 2. [11] Let τ > 0, a < b ≤ +∞ . Suppose that v(t ) = (v1 (t ), v2 (t ),

, vn (t ))T ∈

C[[a, b), R n ] satisfy the following differential inequality: + ⎪⎧ D v(t ) ≤ Pv(t ) + (Q ⊗ V (t ))en , t ∈ [a, b), ⎨ s ∈[−τ , 0], ⎪⎩v(a + s ) ∈ PC ,

where P = ( pij ) n× n and pij ≥ 0 for i ≠ j , Q = (qij ) n× n ≥ 0, V (t ) = (v(t − τ ij (t )))n× n . ⊗ means Hadamard product. If the initial condition satisfies v(t ) ≤ κξ e− λt , κ ≥ 0, t ∈ [a − τ , 0], ξ = (ξ1 , ξ2 , , ξn )T > 0 and the positive number λ is determined by the following inequality: λτ [λ E + P + (Q ⊗ ε (t )]ξ < 0, ε (t ) = (e ij ) n×n

Then v(t ) ≤ κξ e− λ (t − a ) for t ∈ [a, b).

3 Existence and Global Exponential Stability of Equilibria Lemma 3. Under assumptions (H1)-(H3) and (H5), system (1) has a unique equilibrium point if M is anonsingular M-matrix, and

⎛ 2B − H M =⎜ ٛ ⎝ −F

−F ⎞ ٛ ٛ 2 B − H ⎟⎠

(4)

in which

B = diag (b1 , b2 ,

, bn ), H = diag (h1 , h2 ,

, hn ), hi = ∑ j =1 L j | cij + dij |, m

ٛ ٛ ٛ ٛ ٛ ٛ ٛ ٛ n ٛ ٛ ٛ B = diag (b1 , b2 ,..., bm ), H = diag ( h1 , h2 ,..., hn ), hi = ∑ i =1 Li | cٛji + d ji |, ٛ ٛ ٛ ٛ ٛ F = ( Fij )n ×m , Fij = L j | cij + d ij |, F = ( Fij )m×n , Fij = Li | cٛji + d ji | .

Proof. Assume (u * , v* )T = (u1* , u2* , , un* , v1* , v2* , , vm* ) is an equilibrium point of (1). By (H5), we know it must be an equilibrium point of (3), which satisfies the following equation owing to (H1).

Stability Analysis of an Impulsive Cohen-Grossberg-Type BAM Neural Networks

271

m

bi (ui* ) − ∑ ( cij + d ij ) f j ( v*j ) − I i* = 0, j =1

ٛ

ٛ

n

ٛ

b j ( v *j ) − ∑ (cٛji + d ji ) gi (ui* ) − I *j = 0 i =1

From Theorem 3.1 in [10], it is easy to know that the proposition holds. Theorem 1. Under assumptions (H1)-(H5), system (1) has a unique equilibrium point which is globally exponentially stable with convergence rate (λ − η ) / 2 , if the following conditions hold. (H6) There exists a positive number λ such A is a M-matrix, and

−D ⎛ 2A − C − λE ⎞ Α=⎜ ٛ ٛ ٛ 2 A − C − λ E ⎟⎠ −D ⎝

(5)

in which ٛ ٛ ٛ ٛ A = diag (α1b1 , α 2 b2 ,..., α n bn ), A = diag (αٛ1b1 , αٛ2 b2 ,..., αٛm bm ), C = diag ( β1 ∑ j =1 L j (| c1 j | + | d1 j |), β 2 ∑ j =1 L j (| c2 j | + | d 2 j |), ..., β n ∑ j =1 L j (| cnj | + | d nj |)), m

ٛ

ٛ

m

ٛ

ٛ

ٛ

m

ٛ

ٛ

ٛ

ٛ

ٛ

C = diag ( β1 ∑ i =1 Li (| cٛ1i | + | d1i |), β1 ∑ i =1 Li (| cٛ2i | + | d 2i |),..., β m ∑ i =1 Li (| cٛmi | + | d mi |)), n

D = ( Dij ) m×n , Dij = L j β i (| cij | + | d ij | e

n

λτ ij

ٛ

ٛ

n

ٛ ٛ

ٛ

ٛ

), D = ( Dij ) m×n , Dij = Li β j (| cٛji | + | d ji | e

λτٛji

).

⎧ ln ηk ⎫ (H7) η = supk∈N ⎨ {1, γ ik2 , γٛ2jk }. ⎬ < λ , ηk = 1≤i ≤max n ,1≤ j ≤ m t − t ⎩ k k −1 ⎭

Proof. Since (5) holds implies (4) hold, we obtain from lemma 3 that system (1) has a unique equilibrium point (u * , v* )T , let ui (t , x) = ui (t , x) − ui* , v j (t , x) = v j (t , x) − v*j . It is easy to see that system (1) can be transformed into the following system m ∂ui (t , x ) = Di Δui (t , x ) − ai (ui (t , x ))[bi (ui (t , x )) − ∑ cij f j (v j (t , x )) ∂t j =1 m

− ∑ d ij f j (v j (t − τ ij (t ), x ))], t ≠ tk , j =1

+ k

ui (t , x ) = ui (t k , x ) + Eik (ui (tk− , x )), t = tk , k ∈ N , ∂v j (t , x ) ∂t

−

ٛ = D j Δv j (t , x ) − aٛj (v j (t , x ))[b j (v j (t , x )) − ٛ

n

−∑ i =1

+ k

n

∑ i =1

(6) cٛji gi (ui (t , x ))

ٛ

d ji gi (ui (t − τٛji (t ), x ))], t ≠ tk , ٛ

v j (t , x ) = v j (t k− , x ) + E jk (v j (tk− , x )), t = tk , k ∈ N for t > 0 , 1 ≤ i ≤ n, 1 ≤ j ≤ m, where ai (ui (t , x )) = ai (ui (t , x ) + ui* ), aٛj (v j (t , x )) = aٛj (v j (t , x ) + v*j ),

272

Q. Liu, R. Xu, and Y. Du ٛ

ٛ

ٛ

bi (ui (t , x )) = bi (ui (t , x ) + ui* ) − bi (ui* ), b j ( v j (t , x )) = b j ( v j (t , x ) + v *j ) − b j ( v *j ) , f j (v j (t , x)) = f j (v j (t , x) + v*j ) − f j (v*j ), gi (ui (t , x) = gi (ui (t , x) + ui* ) − gi (ui* ) , ٛ

ٛ

Eik (ui (tk− , x )) = Eik (ui (tk− , x ) + ui* ), E jk (v j (tk− , x )) = E jk (v j (tk− , x ) + v *j ) . The boundary conditions and initial conditions of system (6) are given by ∂ui ∂l

= 0, ∂Ω

∂v j ∂l

= 0 for t ≥ 0, i = 1, 2,

, n, j = 1, 2,

m,

∂Ω

ui ( s, x) = ϕi ( s, x) − ui* , v j ( s, x) = φ j ( s, x) − v*j , − σ ≤ s ≤ 0 Multiply both sides of the first equation (6) by ui (t , x) and integrate over Ω , we get l 1 d (ui (t , x))2 dx = ∑ ∫ ui (t , x) Di Δu (t , x)dx ∫ Ω 2 dt Ω q =1

− ∫ ui (t , x)ai (ui (t , x))bi (ui (t , x))dx Ω

m

+ ∑ cij ∫ ui (t , x )ai (ui (t , x)) f j (v j (t , x))dx j =1

(7)

Ω

m

+ ∑ dij ∫ ui (t , x)ai (ui (t , x)) f j (v j (t − τ ij (t ), x))dx j =1

Ω

Since system (6) follows Neumann boundary conditions, from Green's formula we get

∫Ω u (t , x)Δu (t , x)dx = − D ∫Ω | ∇u (t , x) | dx ≤ 0 2

i

i

i

i

(8)

By using Holder inequality, together with condition (H1)-(H3) and (8), from (7) we obtain 1 d || ui (t , x) ||22 ≤ −α i bi || ui (t , x) ||22 2 dt m

+ ∑ β i | cij | L j || ui (t , x ) ||2 ⋅ || v j (t , x) ||2 j =1 m

+ ∑ β i | dij | L j || ui (t , x) ||2 ⋅ || v j (t − τ ij (t ), x) ||2 . j =1

According to || ui (t , x) ||2 ⋅ || v j (t , x ) ||2 ≤

1 (|| ui (t , x) ||22 + || v j (t , x) ||22 ) 2

and || ui (t , x) ||2 ⋅ || v j (t − τ ij (t ), x) ||2 ≤

1 (|| ui (t , x) ||22 + || v j (t − τ ij (t ), x) ||22 ) 2

(9)

Stability Analysis of an Impulsive Cohen-Grossberg-Type BAM Neural Networks

273

We have from (9) that m d || ui (t , x) ||22 ≤ (−2α i bi + β i ∑ L j (| cij |+ | dij |) || ui (t , x) ||22 dt j =1 m

+ βi ∑ L j | cij | || v j (t , x) ||22

(10)

j =1 m

+ βi ∑ L j | dij | || v j (t − τ ij (t ), x) ||22 . j =1

Similarly, we have from (6) that ٛ ٛ n ٛ ٛ d || v j (t , x ) ||22 ≤ ( −2αٛj b j + β j ∑ Li (| cٛji |+ | d ji |) || v j (t , x ) ||22 dt i =1 ٛ ٛ + β j ∑ Li | cٛji | || ui (t , x ) ||22 n

(11)

i =1

ٛ

ٛ

n

ٛ

+ β j ∑ Li | d ji | || ui (t − τٛji (t ), x ) ||22 . i =1

Let v = (|| u1 ||22 ,|| u2 ||22 , that

,|| un ||22 ,|| v1 ||22 ,|| v2 ||22 ,

,|| vm ||22 ) , we have from (10) and (11)

dv(t ) ≤ Pv(t ) + (Q ⊗ V (t ))en , t ≥ 0, dt

(12)

where

⎛ −2 A + C P=P=⎜ ٛ ⎝ D1

⎞

D1

⎛O ⎝ D2

ٛ⎟ , Q = ⎜ ٛ

ٛ

−2 A + C ⎠

D2 ⎞ ⎟, O⎠

ٛ

A, C , A and C see (5). ٛ ٛ ٛ ٛ ٛ D1 = ( D1ij ) m×n , D1ij = L j β i | cij |, D1 = ( D1ij )m×n , D1ij = Li β j | cٛji | . ٛ

ٛ

ٛ

ٛ ٛ

ٛ

D2 = ( D2ij )m×n , D2ij = L j βi | d ij |, D2 = ( D2ij )m×n , D2 ij = Li β j | d ji | . From (5), we have ⎛ 2A − C − λE (λ E + P + (Q ⊗ ε (λ )) = − ⎜ ٛ −D ⎝

⎛ O ⎞ = − A , ε (λ ) = ⎜ λτٛ ⎟ ji ⎜ 2A − C − λE ⎠ ⎝ ( e ) m ×n ٛ

−D ٛ

According to Lemma 1, there exists a vector ξ = (ξ1 , ξ 2 , (λ E + P + (Q ⊗ ε (λ ))ξ < 0. Let

κ=

|| (ϕ (t , x) − u * , φ (t , x) − v* )T ||σ . min1≤ i ≤ m + n {ξi }

(e

λτ ij

) n×m ⎞ ⎟ O ⎠⎟

, ξ n + m )T such that (13)

274

Q. Liu, R. Xu, and Y. Du

Note | ui (tk ) ||22 ≤ γ ik2 || u1 (tk− ) ||22 , || v j (tk ) ||22 ≤ γٛ2jk || v j (tk− ) ||22 . Similar to proof of Theorem 1 in [11], applying mathematical induction, we have from Lemma 2 that

v(t ) ≤ κη0η1

(14)

ηk −1ξ e−λt , tk −1 ≤ t < tk , k ∈ N .

Furthermore, we have from (14) and (H7) that (15)

v(t ) ≤ κη e− ( λ −η )t , tk −1 ≤ t < tk , k ∈ N . This implies that

|| ui (tk ) ||2 ≤ κξi e

−

λ −η 2

t

, i = 1, 2,

, n, t ≥ 0

and

|| v j (tk ) ||2 ≤ κξn + j e

−

λ −η 2

t

, j = 1, 2,

, m, t ≥ 0.

Hence || (u (t , x) − u* , v(t , x) − v* )T ||≤ M || (ϕ (t , x) − u* , φ (t , x) − v* )T ||σ e for t ≥ 0, where κ =

∑

n+m i =1

−

λ −η 2

t

κξi

. min1≤ i ≤ m + n {ξi } It implies the equilibrium point of system (1) is globally exponentially stable with convergence rate (λ − η ) / 2 . This completes the proof. Corollary 1. Under assumptions (H1)-(H5), system (1) has a unique equilibrium which is globally exponentially stable if the following conditions hold. (H ′6 ) Matrix A ′ is a nonsingular M-matrix, and

− D′ ⎞ ⎛2A− C A′ = ⎜ ٛ ٛ⎟ ٛ ⎝ − D′ 2 A − C ⎠ in which ٛ ٛ ٛ ٛ ٛ ٛ D ′ = ( Dij′ )m×n , Dij′ = L j β i (| cij | + | d ij |), D ′ = ( Dij′ )m×n , Dij′ = Li β j (| cٛji | + | d ji |). ٛ

A, C , C see (6). (H ′7 ) max(γ ik2 , γٛ2jk ) ≤ 1, i = 1, 2,..., n, j = 1, 2,..., m. Proof. Since A′ is a M-matrix, it is easy to know from (13) that there is a suitable small positive number λ such that (H6) holds. (H ′7 ) implies η k = 1 , it follows that η = 0 , which means (H7) holds too. From Theorem 3.2, Corollary 1 holds.

4 One Example In this section, we will give one example to illustrate our results. Consider the following system

Stability Analysis of an Impulsive Cohen-Grossberg-Type BAM Neural Networks

275

∂u1 (t , x ) = D1Δu1 (t , x ) − (| sin t | +1)[3ui (t , x ) − 0.5 f1 (v1 (t , x )) − 0.5 f 2 (v2 (t , x )) ∂t − 0.6 f1 ( v1 (t − 1, x )) − 0.9 f 2 ( v2 (t − 1, x ))], t ≠ tk , u1 (tk+ , x ) = u1 (t k− , x ) + 0.6ui (tk− , x ), t = tk , k ∈ N , ∂v1 (t , x ) ٛ = D1Δv1 (t , x ) − (| sin t | +1)[3v1 (t , x ) − 0.2 g1 (u1 (t , x )) + 0.4 g1 (u1 (t − 1, x ))], t ≠ tk , ∂t ∂v2 (t , x ) ٛ = D2 Δv2 (t , x ) − 2[2v2 (t , x ) − 0.3g1 (u1 (t , x )) − 0.3g1 (u1 (t − (2 + 0.25sin t ), x )) − 1], t ≠ tk , ∂t v1 (tk+ , x ) = v1 (t k− , x ) + e0.001k v1 (tk− , x )), t = tk , k ∈ N ,

(16)

v2 (tk+ , x ) = v2 (t k− , x ) + e0.002 k v2 (tk− , x )), t = tk , k ∈ N ,

where f i ( y) = tanh( y), i = 1, 2; g1 ( y ) = tanh( y ) and tk = tk −1 + 0.005k. By simple computation, one can obtain that there exists λ = 0.4208 such that ⎛ 7.945756711046097 -4.443763264760875 -6.1656448971413124807 ⎞ ⎜ ⎟ A = ⎜ -2.695842176507250 1.745756711046097 0 ⎟ ⎜ -1.747921088253625 ⎟ 0 9.945756711046096 ⎝ ⎠

is a nonsingular M-matrix and λ = 1.0542 > η = 0.4 . By theorem 1 we know system (16) has a equilibrium point, this equilibrium point is (0,0,0), and the equilibrium is globally exponentially stable with convergence rate 0.3271.

References 1. Kosto, B.: Bi-directional associative memories. IEEE Trans. Syst. Man Cybern. 18, 49–60 (1988) 2. Gopalsamy, K., He, X.Z.: Delay-independent stability in bi-directional associative memory neural networks. IEEE Trans. Neural Networks. 5, 998–1002 (1994) 3. Mohamad, S.: Global exponential stability in continuous-time and discrete-time delay bidirectional neural networks. Physica D 159, 233–251 (2001) 4. Gopalsamy, K.: Stability of artificial neural networks with impulses. Appl. Math. Comput. 154, 783–813 (2004) 5. Wang, L., Zhou, Q.: Global exponential stability of BAM neural networks with timevarying delays and diffusion terms. Physics Letters A 37, 83–90 (2007) 6. Xiang, H., Cao, J.: Exponential stability of periodic solution to Cohen-Grossberg-type BAM networks with time-varying delays. Neurocomputing 72, 1702–1711 (2009) 7. Bai, C.: Stability analysis of Cohen-Grossberg BAM neural networks with delays and impulses. Chaos, Solitons Fract 35, 263–267 (2008) 8. Xia, Y., Huang, Z., Han, M.: Exponential p-stability of delayed Cohen-Grossberg-type BAM neural networks with impulses. Chaos, Solitons Fract 38, 806–818 (2008) 9. Li, K.: Stability analysis for impulsive Cohen-Grossberg neural networks with timevarying delays and distributed delays. Nonlinear Analysis: Real World Appl. 10, 2784–2798 (2009) 10. Zhou, Q., Wang, L.: Impulsive effects on stability of Cohen-Grossberg-Type bidirectional associative memory neural networks with delays. Nonlinear Anal.: Real World Appl. 10, 2531–2540 (2009)

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11. Li, Z., Li, K.: Stability analysis of impulsive Cohen-Grossberg neural networks with timevarying delays and reaction-diffusion terms. Neurocomputing 72, 231–240 (2008) 12. Pan, J., Liu, X., Zhong, S.: Stability criteria for impulsive reaction-diffusion CohenGrossberg neural networks with time-varying delays. Math. Comput. Modelling (2010), doi:10.1016/j.mcm.2009, 12.004 13. Song, Q., Cao, J.: Global exponential stability and existence of periodic solutions in BAM networks with delays and reaction-diffusion terms. Chaos Soliton. Fract. 23, 421–430 (2005) 14. Song, Q., Cao, J., Zhao, Z.: Periodic solutions and its exponential stability of reactiondiffusion recurrent neural networks with continuously distributed delays. Nonlinear Anal.: Real World Appl. 7, 65–80 (2006) 15. Zhao, Z., Song, Q., Zhang, J.: Exponential periodicity and stability of neural networks with reaction-diffusion terms and both variable and unbounded delays. Comput. Math. Appl. 51, 475–486 (2006) 16. Lou, X., Cui, B.: Boundedness and exponential stability for nonautonomous cellular neural networks with reaction-diffusion terms. Chaos Soliton. Fract. 33, 653–662 (2007) 17. Allegretto, W., Papini, D.: Stability for delayed reaction-diffusion neural networks. Phys. Lett. A 360, 669–680 (2007) 18. Zhou, Q., Wan, L., Sun, J.: Exponential stability of reaction-diffusion generalized CohenGrossberg neural networks with time-varying delays. Chaos Soliton. Fract. 32, 1713–1719 (2007)

Characterizing Multiplex Social Dynamics with Autonomy Oriented Computing Lailei Huang and Jiming Liu Department of Computer Science, Hong Kong Baptist University {llhuang,jiming}@comp.hkbu.edu.hk

Abstract. Multiplexity, or the fact that people interact through different social contexts with different purposes, is one of the most fundamental characters of social interactions. However, it is rarely addressed in both empirical study and theoretical modeling. In this paper, we intend to address this gap by proposing an autonomy-oriented model that captures the multiplexity of social interactions in two specific contexts: (1) cultural interaction and (2) social resource sharing. The experimental results demonstrate that the positive feedback in local cultural interaction together with the autonomous behavior in resource sharing can clearly capture the emergent multiplex effects, i.e. the nonlinear effect of resource utilization performance corresponds to different level of cultural diversity and cultural support tendency. Therefore, our method offers an alternative view to characterize social behavior and social interactions.

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Introduction

Due to the newly availability of large scale, longitudinal datasets of social interactions from technologies like social web (Facebook, Twitter, etc.), reality mining [1], computational sociology has becoming a promising field to reveal deep laws in human society. Watts recently lists three fundamental issues to be further studied [2]: (1) the dynamic feature of social networks, (2) the interplay of social network structure and collective social dynamics and (3) the multiplex characteristics of social interactions. Consistent with this call, a great number of attention has been devoted to study the interplay of social network dynamics and social behavior dynamics [3] [4] [5]. However, the multiplex features of social interactions are still, as far as we known, rarely studied in the literature. In this paper, our purpose is to clarify and study the multiplex aspects of social interactions. Here, multiplexity means a social tie can serve as the channel of social interactions with different roles and purposes [6] at the same time and the potential coupled result. To be concrete, we focus our study of the multiplex effects on two specific social interaction contexts: (1) cultural interaction [7] and (2) social resource sharing [8]. Picklert et al. [9] have demonstrated that similarities among people can foster mutual support. Lizardo [10] points out that cultural taste helps constructing social relations, establishing networks of trusting relations and facilitating resource mobilization. Based on these observations, we intend to build a model to capture the inherent relationship between the two dynamic processes. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 277–287, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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Many studies have been done in characterizing cultural dynamics [3] [7] [11] and social resource sharing [4] [5]. The Axelrod model [7] was proposed to study the persistence of cultural diversity under the mutual reinforcement of homophily and social influence. Other studies [3] [11] have extended this model to complex social networks and incorporated network dynamics. In this paper, we extend the Axelrod model [7] to the multiplex social interaction contexts. Social resource means valued goods (i.e. information, expertise) that are embedded in the personal network and accessible through direct neighbors or further referrals [8]. Several recent works has characterized the social interaction with certain types of resources. For example, Rosvall and Sneppen [5] have studied the emergent of heterogeneous social groups. In this model, information is the resource that drives the entities’ behavior. Qiu and Liu [4] studies the formation of scale-free dynamic trust network under a self-organization mechanism. In their work, the entities’ ability is a kind of social resource. Different from the above studies, this work focuses on characterizing the multiplex social interactions in cultural dynamics and social resource sharing. We adopt the methodology of Autonomy-Oriented Computing (AOC) [12] [13] [14] to achieve this goal by explicitly emphasizing the process of self-organization and local autonomy design. The autonomy captures the need of a social entity to make decisions in different social contexts. The self-organization process is realized to capture the emergence of cultural diversities. Experimental results show that the multiplex effects, i.e. the interplay of cultural diversity, local autonomous behavior and resource utilization efficiency, can be clearly captured. The rest of the paper is organized as follows: Section 2 gives a detailed statement in characterizing the multiplex social dynamic processes. Section 3 formulates cultural and social resource sharing dynamic processes. Experimental results and discussions are presented in section 4. Section 5 draws the conclusion and points out the future directions.

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Problem Statement

Human society is an open, dynamic complex system in which social entities (e.g. people, organizations, etc) have different coupling relationships (e.g. trust relationship [4]) with each other formed through social interactions. In this paper, we consider the multiplexity as a kind of meta coupling. More specifically, what we want to characterize and observe is not the coupling formation in a single type of social interaction like those studied in [3] [4] [5] but the interplay of multiple social interaction processes. To characterize the multiplex effects, we should first specify the multiple social interaction contexts in consideration. In this paper, we formulate cultural and social resource sharing dynamics. In the former part we use the feature vector based definition of culture which follows [7]. On the social resource sharing part, we take resources as the attributes of the owners e.g. different information, knowledge known by entities. Therefore, we also use the vector based definition.

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Definition 1. The culture vector Vculture = [σ1 , σ2 , . . . , σFc ] has Fc number of cultural features. Each feature stands for cultural related individual characters such as musical choices, clothing preferences, reading interests, etc. Each feature has q possible trait values i.e. σi ∈ {0, 1, . . . , q − 1} to represent the difference of entities on that feature. Definition 2. Social resource embedded in the social network is denoted as ri , or the ith resource which stands for job information, legal expertise, etc. The resource vector Vresource = [r1 , . . . , rRc ],ri ∈ {0, 1} where Rc is a parameter representing the number of resource types in the artificial society. The above two definitions specify the meaning of culture and social resource at the individual level. However, in order to characterize the multiplex effects at the global level, we need to introduce two measurements: cultural diversity and resource utilization efficiency. Definition 3. The culture diversity Cdiv = Smax /N denotes the average size of the largest cultural region normalized by the the number of entities N . Here, a cultural region denotes a community of connected entities on the lattice with the identical values on all cultural features [7]. The resource utilization efficiency is characterized by the average resource access ratio defined below: i (t) Definition 4. The resource access ratio of an entity i at time t or Racc is the number of resources the entity can access proportional to the total number of resources in the artificial society. The average  resource access ratio is the i i on all entities, i.e. Racc (t) = N1 · N average of Racc i Racc (t).

Given the above definitions related to cultural and social resource interactions, the research questions to be explored in this paper can be specified as: 1. At the individual level, how to characterize cultural and resource sharing interactions? More importantly, how to design the local autonomous behavior to couple the two social interaction processes together? 2. At the global level, how does the resource utilization efficiency change in the resource sharing dynamics? More importantly, how does the cultural diversity defined above, based on the designed local autonomy in individuals, affect the resource utilization efficiency?

3 3.1

Modeling the Multiplex Social Dynamics Basic Ideas

In order to formalize the multiplex social dynamics, we introduce an autonomyoriented computing (AOC) based model in this section. AOC is an unconventional computing paradigm which puts emphasis on self-organization, self-organized computability encoded in the local autonomy of entities to model complex system

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or solve computationally-hard problems [12]. On topics related to social dynamics modeling, it has successfully been used to characterize the human web surfing behaviors [15], social network dynamics in service based social interactions [16]. Taking the same perspective, in this work, we model individuals in multiplex social interactions as autonomous entities. Definition 5. A social entity e represents an individual involved in social interactions, i.e. communicating mutual interests, sharing social resources. It can formally be defined as a tuple, i.e. ei =< id, attribute, rules >, where id denotes the identifier, attribute = {attrcul , attrres } represents the attribute of e related to culture and resource sharing dynamics respectively. rules = {rulecul , ruleres } denotes the different behavioral rules related to the two dynamic processes. Based on this building-block, we emphasize two aspects of AOC: positive feedback and autonomous behavior. Firstly, self-organization process is reflected in the positive feedback which captures the mutual enhancement of homophilic interaction and social influence in cultural dynamics (section 3.2). Specifically, the interaction probability between two entities is proportional to the the accumulated cultural similarity which is further improved after a successful cultural interaction. This mechanism leads to different levels of cultural diversity. Secondly, to capture the inherent relationship between the two dynamic processes [9] [10], we introduce the local autonomous behavior of entities in social resource sharing (section 3.3). We design different reply strategies for a resource owner in social resource sharing, i.e. reply behavior based on cultural similarity with the enquirer. Moreover, we use a probability parameter to capitare the tendency of such behavior. The two aspects play the key role in characterizing the global multiplex effects. Below, we introduce each of the two aspects and the details of the related dynamic processes. 3.2

Positive Feedback and Cultural Dynamics

The basic idea of the positive feedback in this model is: on the one hand the more similar two entities on the cultural tastes the more likely they will interact (homophily principle); on the other hand the more interactions happen between two entities, the more similar they are likely to become (peer social influence). An illustrative example is shown in Figure 1. This mechanism was first proposed in the Axelrod model [7]. Computationally, it can be realized based on the definition of cultural similarity. And the detailed cultural dynamic process is as follows: Definition 6. The culture similarity sim(ei , ej ) denotes the overlap of the  c cultural feature values of ei and ej , i.e. sim(ei , ej ) = F1c · F f =1 δ(σif , σjf ). 1. At each time step, an entity ei uniformly randomly picks a parter ej in its neighborhood. 2. ei calculates and uses the similarity sim(ei , ej ) as the interaction probability. 3. If sim(ei , ej ) equals 1 or 0 goto step 5 (i.e. they are identical or totally dissimilar). Otherwise, ei generates a random number rnd uniformly, if rnd < sim(ei , ej ) goto step 4, otherwise goto step 5.

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4. ei picks a cultural feature f that differs from ej ’s (if exists), i.e. σif = σjf and set σif ← σjf . 5. Nothing happens between ei and ej . In this procedure, step 1 stands for the parter selection behavior; step 2 to 5 compose the homophilic conf orm behavior. Therefore, the mutual reinforcement between cultural similarity and interaction probability is implemented. 3.3

Autonomous Behavior and Resource Sharing Dynamics

In AOC, behavioral rules necessarily define the local autonomy of entities [12] which decide how certain behaviors are internally or externally triggered at different circumstances. In this work, we implement an intuitive local autonomy design based on the entities’ behavioral rule in resource sharing. First, we introduce the overall process of social resource sharing. An example is illustrated in Figure 2. The detailed dynamic process is described as follows:

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1. At each time step, an entity ei picks a resource ri that it does not hold (i.e. ei .Vresource .ri = 0) randomly as the need and generates an enquiry message i EMi with the maximal valid time Tmax . 2. ei picks a neighbor ej uniformly randomly in its neighborhood as the enquiring partner and sends the message. 3. If ej receives an enquiring message, it first checks the need ri in ej .Vresource . If it holds the resource, ej replies to the source entity ei with the reply rule res Rreply . Otherwise, it uniformly randomly picks a neighbor ek (not involved in the enquiry session) and transfers the message. 4. At each time step, ei checks whether it receives the feedback on EMi , if it i fails to get the feedback within Tmax steps, this enquiry fails; otherwise if it gets the feedback, it changes its state on the resource ri from unknown to known i.e. ei .Vresource .ri ← 1. In this resource sharing process, step 1 and 2 denote the enquiry behavior, step 3 denotes the inf ormation handling and step 4 reflects the enquiry checking res noted in behavior. The autonomous behavioral rule used in this model is Rreply AW step 3. The two extreme cases are always reply, denoted as Rreply and cultural SM which means the resource holder ej replies the similarity based reply or Rreply enquiry by considering the cultural similarity with the source entity ei . We introduce a parameter α, noted as cultural support tendency, to model the internal tendency (probability) that ej will trigger the reply behavior based on AW α=0 the cultural similarity to the interaction partner. Thus, we have Rreply = Rreply SM α=1 and Rreply = Rreply . We may use the α to adjust the autonomous reply behavior of an entity in social resource sharing. 3.4

The Main Algorithm

Algorithm 1 summarizes the procedure of the cultural and social resource sharing dynamics. Two major assumptions are needed to be further discussed. The first assumption is about initialization. As an initial effort, we use two dimensional lattice as the underlying social network structure in the current model, just as in [7]. Complex network structure and network dynamics will be studied in the next step. Besides, although not confined to, in the current work, we set the number of resource type Rc = 1 (definition 2). Moreover, we introduce a parameter η to reflect the abundance of the resource in the artificial society, i.e. initially there are N · η entities holding the resource. The second crucial assumption is the ordering of the two dynamic processes because it affects how they are coupled together through entities’ local autonomous behavior. The two possibilities are: (1) serial ordering and (2) parallel ordering. The former means the resource sharing dynamics happens after the convergence of cultural dynamics. The later means two dynamic processes happen simultaneously. In the current model, we implement the serial version. Under this scenario, we assume that the environment is relatively coherent and stable which means within the social communities entities quickly reach convergence on different

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Input: Lattice size lattsize, cultural vector parameters < Fc , q >, resource type number Rc , resource abundance η and cultural support tendency α. Build the underlying lattice network G with size lattsize × lattsize; for each node v in G do Build an entity e = create entity(v, Fc , q, Rc ) end Uniformly randomly pick η · N entities as the initial resource holders; // Cultural dynamics phase Convg = true do while F lagculture for each entity ei do ei .partner selection(); ei .homophilic conf orm(); end end // Social resource sharing phase Convg while F lagresource = true do for each entity ei do ei .enquiry(); ei .inf ormation handling(α); ei .enquiry checking(); end end

Algorithm 1. The main simulation algorithm cultural tastes. On the other hand, entities have daily resource needs which are continuously generated and propagated through the social network. We note this is only one possibility. In future work we plan to test the effect of parallel ordering and the co-evolution of the two dynamic processes.

4

Experimentation

In the previous section, we have presented an intuitive model to characterize the individual level interactions in both social dynamic processes as well as a coupling mechanism based on the reply behavior strategies in resource sharing. The purpose of this section is to demonstrate the multiplex effects based on the proposed model. We want to address the following two issues: – to observe the separated evolution of cultural dynamics and resource sharing dynamics by testing the cultural diversity Cdiv (definition 3) and resource utilization efficiency Racc (definition 4). – to discover the multiplex effects at the global level when two dynamic processes are coupled by the local autonomous behavior of social entities. To achieve these objectives, we design three experiments to capture: (1) cultural evolution; (2) resource utilization and (3) local behavior based multiplex mechanism. Table 1 summarizes the model parameters in use.

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The number of social entities. Cultural feature number (definition 1). Value range of one cultural feature (definition 1). Number of social resource type (definition 2). Cultural support tendency (section 3.3). The ratio of entities initially chosen as resource owners. Max enquiry message valid cycles.

Evolution of Cultural Diversity

This experiment aims to observe how cultural regions evolve in cultural dynamics. The target measurement is cultural diversity Cdiv . To observe the evolution of cultural diversity, we set Fc = 3 and change q ∈ {0, 1, . . . , 40}. We use N ∈ {100, 400, 900} for sensitive comparison. Each data value is the average result on 30 simulation runs. Figure 3 demonstrates that for fixed cultural feature value Fc , slightly change of the trait range or q leads to a dramatic change of the size of cultural regions - the phase transition also observed in [7]. A small q value leads to the monocultural region which we call homogenous culture while a large q value leads to the highly cultural diversity which we call heterogeneous culture. The state of the society with critical q value (e.g. q = 10) is named medium diverse culture. 4.2

Resource Utilization Efficiency

In this experiment, we intend to observe the global performance of resource sharing dynamics. The target measurement is the average resource access ratio Racc . To achieve this goal, we use N = 1600, Rc = 1, η = 1%. We set α = 0 (always reply) and α = 1 (similarity reply). Besides, we use Tmax ∼ N oraml(2, 5)1 . In the case with coupled cultural dynamics, we set Fc = 3, q = 15. Figure 4(a) shows that all entities can access the resource finally under the experimental setting. Figure 4(b) demonstrates that the accessability to the resource is inhibited under the cultural similarity based reply rule, i.e. there are only 4.5% of the population can access the resource. The observation in Figure 4(a) is not very realistic in the sense that in reality, some entities may not be able to access the resource (e.g. job information) no matter how many enquiries are locally sent. Therefore it is more realistic when cultural similarity based behavior is considered as is shown in in Figure 4(b). We conjecture that the inhibition is due to two reasons. First, because of the serial ordering of the dynamics processes, the resource sharing dynamics happens on cultural regions that have already been formed. Second, the resource is initially 1

Slightly change on the mean and variance values do not affect the results.

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randomly distributed, therefore entities in one cultural region cannot access the resource located in the other regions because of the cultural boundary. In summary, the experiment demonstrates the idea that cultural similarity based reply behavior can generate inhibition on the resource sharing dynamics. It can be taken as an indication of a multiplex effect. However, to fully reveal the multiplex effects, two issues - the different levels of cultural diversity and different tendency of cultural support - are needed to be further clarified. 4.3

The Multiplex Effects

The purpose of this experiment is to couple the observation of the previous two experiments together. More specifically, we want to characterize the multiplex effects based on different level of cultural diversities in cultural dynamics and different tendency of reply behavior in resource sharing. We set N = 400, Fc = 3 and change q from 5 to 15 in the cultural dynamics part. In the resource sharing

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dynamics, we adopt Rc = 1, η = 1% and vary α from 0 to 1. Again, each simulation result is based on 30 replications. Figure 5(a) demonstrates the effect of different level of cultural diversities on the resource utilization efficiency under the cultural similarity based reply (α = 1). More than 90% entities can access the resource in the artificial society when q is smaller than 9. However, when q increases to 14 the proportion rapidly decreases to less than 10% - a clearly demonstration of the multiplex effects. Figure 5(b) illustrates the different tendency of cultural similarity based support. The result shows that entities’ behavioral tendency have little effect on the resource utilization if the resource sharing is based on the homogenous culture (q = 5). On the other hand, if the culture is highly heterogeneous (q = 15), then the more an entity interacts based on cultural similarity, the more likely the overall resource utilization is inhibited. The nonlinear effect is also shown when q is around critical value (q = 9, 10, 11), i.e. the results diverge significantly with a small increase in q and constantly increase of cultural support tendency.

5

Conclusion and Future Work

In this paper, we are interested in characterizing the multiplex effects in cultural and social resource sharing dynamics. Specifically, we implement an autonomyoriented model to capture entities’ local interactions in each social dynamic process. We implement the positive feedback mechanism that characterizes the mutual reinforcement of cultural similarity and interaction probability. Besides, different reply strategies are designed to reflect the autonomy of entities in resource sharing interactions. The experimental results show that the global multiplex effects can emerge from the local interactions. Specifically, under the different cultural support tendency in entities’ resource sharing behavior, the resource utilization efficiency is nonlinearly influenced by the state of cultural

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diversity. To further improve the comprehensiveness of the current work, in the next steps we focus on (1) studying the parallel ordering of dynamic processes and (2) exploring the role of network dynamics in the multiplex social interactions.

References 1. Eagle, N., Pentland, A.S.: Reality mining: sensing complex social systems. Personal and Ubiquitous Computing 10, 255–268 (2006) 2. Watts, D.J.: A twenty-first century science. Nature 445, 489 (2007) 3. Centola, D., Gonzalez-Avella, J.C., Eguiluz, V.M., Miguel, M.S.: Homophily cultural drift and the co-evolution of cultural groups. Journal of Conflict Resolution 51(6), 905–929 (2007) 4. Qiu, H., Liu, J., Zhong, N.: A dynamic trust network for autonomy-oriented partner finding. In: 5th Internal Conference on Active Media Technology, pp. 323–334 (2009) 5. Rosvall, M., Sneppen, K.: Reinforced communication and social navigation generate groups in model networks. Phys. Rev. E. 79, 026111 (2009) 6. Verbrugge, L.M.: Multiplexity in adult friendships. Social Forces 57(4), 1286–1309 (1979) 7. Axelrod, R.: The dissemination of culture. Journal of Conflict Resolution 41(2), 203–226 (1997) 8. Lai, G., Lin, N., Leung, S.Y.: Network resources, contact resources, and status attainment. Social Networks 20, 159–178 (1998) 9. Plickert, G., Cote, R.R., Wellman, B.: It’s not who you know, it’s how you know them: who exchanges what with whom. Scoial Networks 29(3), 405–429 (2007) 10. Lizardo, O.: How cultural tastes shape personal networks. American Sociological Review 71(5), 778–807 (2006) 11. Klemm, K., Eguiluz, V.M., Toral, R., Miguel, M.S.: Nonequilibrium transitions in complex networks: a model of social interaction. Phys. Rev. E. 67, 026120 (2003) 12. Liu, J.: Autonomy-oriented computing (AOC): the nature and implications of a paradigm for self-organized computing(keynote talk). In: Proceeding of the 4th International Conference on Natural Computation and the 5th International Conference on Fuzzy Systems and Knowledge Discovery, pp. 3–11 (2008) 13. Liu, J., Jin, X., Tsui, K.C.: Autonomy oriented computing (AOC): formulating computational systems with autonomous components. IEEE Transactions on Systems, Man, and Cybernetics, Part A 35(6), 879–902 (2005) 14. Liu, J., Jin, X., Tsui, K.C.: Autonomy oriented computing: from problem solving to complex systems modeling. Springer, Heidelberg (2005) 15. Liu, J., Zhang, S., Yang, J.: Characterizing web usage regularities with information foraging agents. IEEE Transactions on Knowledge and Data Engineering 16, 566– 584 (2004) 16. Zhang, S., Liu, J.: Autonomy-oriented social networks modeling: discovering the dynamics of emergent structure and performance. International Journal of Pattern Recognition and Artificial Intelligence 21(4), 611–638 (2007)

A Computational Method for Groundwater Flow through Industrial Waste by Use of Digital Color Image Takako Yoshii1 and Hideyuki Koshigoe2,, 2

1 Graduate School of Engineering, Chiba University Faculty of Engineering, Department of Urban Environment Systems, Chiba University, 1-33 Yayoi, Chiba, 263-8522, Japan [email protected]

Abstract. In this article we propose a computational method coupled with the digital color image and discuss its application to the numerical simulation of the groundwater flow through industrial wastes. The solution method combines finite difference approximations based on fictitious domain method (also called domain embedding method) and pixels which are arranged in two dimensional grid in the color image. The object is the groundwater flow through the industrial waste deposits which were dumped in Teshima Island. The mathematical model of the groundwater flow is formulated by Darcy’s law, the water permeability and transmission conditions on geologic boundary surfaces. The result of numerical simulations for two dimensional groundwater flow through industrial wastes is presented.

1

Introduction

From 1978 to 1990, the large-scale industrial wastes were illegally dumped to a part of Teshima Island in Japan. The area, the weight and the height of the industrial waste heaped in the field are about 7 ha, 500,000 tonnes and 16 m, respectively. The environmental report showed that groundwater in the alluvium and granite layers was contaminated at levels above the environmental water criteria and contaminated groundwater was leaching to the sea from the north coast of Teshima ([7]). From the experience of Teshima, we study the groundwater flow in the geological profile of industrial wastes and soil layers which consist of the shredderdust, cinder, clay, sandy soil, fractured granite and fresh granite layers([6]). Numerical model for mathematical one is formulated by fictitious domain method based on distribution theoretic approach. The characteristic of the numerical computation is to construct the approximation of each layer by pixel of digital color image, which is the smallest unit of image and forms a small square. We then propose a finite difference method coupled with the pixel information, and simulate the groundwater flow through industrial wastes.  

Corresponding author. This work was supported by JSPS KAKENHI (22540113).

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 288–296, 2010. c Springer-Verlag Berlin Heidelberg 2010 

A Computational Method for Groundwater Flow through Industrial Waste

2 2.1

289

Formulation of a Mathematical Model Steady Groundwater Flow

Before proceeding to the mathematical model, we state the piezometric head, the Darcy’s law and the governing equation of a groundwater flow. (A) The piezometric head u is defined by u=z+

p . ρg

(1)

Here u means a sum of an elevation and pressure head. The piezometric head is given in units of meter[m]. (B) The Darcy’s law is defined by v s = −k∇u

(2)

where v s is called the Darcy’s velocity and it is described by the permeability k and the hydraulic gradient ∇u. (C) The governing equation of a groundwater flow has a form S0

∂u = −∇ · v s ∂t

(3)

where S0 is a specific storage. In this paper we consider a steady state flow in (3) since the velocity of infiltration in Teshima is 1[mm/day] and slow ([6]). Therefore, we get a basic governing equation of the steady groundwater flow from (1)-(3); div(k∇u) = 0.

(4)

Remark 1. The permeability is ability of material to allow water to move through it, expressed in terms of [cm/s]. In this article we use the permeability and the geological profile as shown in Table 1 and Figure 1, respectively([6]). Table 1. Permeability of earth materials and notations material notation of layer permeability[cm/s] notation of permeability shredderdust Ω1 6.72 × 10−4 k1 cinder Ω3 1.30 × 10−6 k3 clay Ω3 1.00 × 10−6 k3 sandy soil Ω2 1.22 × 10−3 k2 fractured granite Ω5 2.23 × 10−3 k5 fresh granite Ω4 5.31 × 10−5 k4

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Fig. 1. Geological profile

2.2

Mathematical Model of the Groundwater Flow in Teshima

Now let’s show the mathematical model of the groundwater flow in Teshima. Using the geological profile (cf. Fig.1) and its permeability {ki } (cf. Table 1), we define the function k(x, y) as follows. k(x, y) =

5 

ki χΩi (x, y)

i=1



where

1 if (x, y) ∈ Ωi (i = 1, 2, 3, 4, 5), 0 otherwise. Then the following mathematical model is deduced from (4). Mathematical model Find u ∈ H 1 (Ω) satisfying ⎧ div( k(x, y)∇u) = 0 in D (Ω) ⎪ ⎨ u = 0 on Γ1 , u = g2 on Γ2 ⎪ ⎩ ∂u = 0 on Γ , ∂u = 0 on Γ . χΩi (x, y) =

3

∂n

∂n

4

Here D (Ω) means the Schwarz distribution and g2 is the piezometric head on Γ2 as shown in Figure 2. Here g2 is the datum investigated by the borehole logging ([6]). Ω = Ω1 ∪ Ω2 ∪ Ω3 ∪ Ω4 ∪ Ω5 and n is the outer normal vector. Remark 2. It follows from the mathematical model that [k

∂u ]=0 ∂n

(5)

where [ ] denotes the jump on geologic boundary surfaces. In fact, denoting ui the piezometric head in each layer Ωi , δΓi,j Dirac Delta

A Computational Method for Groundwater Flow through Industrial Waste

291

¯i ∩ ∂ Ω ¯j ) and n is the outer normal vector on Γi,j , the function on Γi,j (= ∂ Ω Schwarz distribution shows that 5     ∂ui ∂uj  div k(x, y)∇u = − kj δΓi,j + k(x, y)Δu ki ∂n ∂n i,j=1

=0

in D (Ω).

Hence we get (5), i.e., ∂ui ∂uj = kj on Γi,j . ∂n ∂n Therefore the mathematical model is also rewritten as follows: Find ui ∈ H 1 (Ωi ) (i = 1, 2, 3, 4, 5) satisfying. ki

Δui = 0 ui = uj ∂ui ∂u = kj ∂nj ki ∂n ui = 0 ui = g2 ∂ui = 0 ki ∂n ∂ui = 0 ki ∂n

in Ωi ( i = 1, 2, 3, 4, 5 ) ¯i ∩ ∂ Ω ¯ j ( i = j ) on ∂ Ω

(6) (7)

¯i ∩ ∂ Ω ¯ j ( i = j ) on ∂ Ω

(8)

on Γ1 ( i = 1, 2, 3 ) on Γ2 ( i = 1, 4, 5 ) on Γ3 ( i = 4 ) on Γ4 ( i = 1, 3 )

Fig. 2. Original profile (Ωi ) and boundary (Γi )

Remark 3. The equations (7) and (8) on geologic boundary surfaces are a type of transmission conditions which was introduced by J.L.Lions ([2],[5]).

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Computational Method by Use of Digital Color Image

In this section, we present a numerical model coupled with a digital color image, a fictitious domain formulation for the mathematical model and our computational method which combines the regular mesh in finite difference approximations with pixel of image. 3.1

Digital Approximation of Geological Profile by Pixels

The original profile (Fig.1, [6]) is converted into a digital imageDIn this digital image, based on the geological layers, color is assigned to each pixel as shown in Fig.3. In the obtained color image, it is noted that aspect ratio of the original profile, 1 width : 1 height, is changed into 2,010 pixel width : 510 pixel height, and both width and height in each pixel represents 10 cm in the original profile.

Fig. 3. Six color image (2010 pixels×510 pixels) Table 2. Color corresponding to layer layer notation color of layer shredderdust Ω1 blue cinder Ω3 green clay Ω3 green sandy soil Ω2 yellow fractured granite Ω5 pink fresh granite Ω4 brown air Ω6 white

To show the digital approximation of geological profile, we proceed in three steps. Step 1: NTSC value A digital color image pixel is just a RGB data value. Each pixel’s color sample has three numerical RGB components (Red, Green, Blue) to represent the color. These three RGB components are three 8-bit numbers (Rn, Gn, Bn) for each pixel. We define for each pixel, the following NTSC value: a = 0.3×Rn + 0.59×Gn + 0.11×Bn.

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Step 2: Cell value In our computation, we equate a regular grid point with a 10 × 10 pixels which forms a 10 pixels width and 10 pixels height, which we call cell. Using the NTSC value (a) of each pixel, we also define the cell value (b): b =the total of NTSC value of pixels included in a cell. Therefore the maximum of cell value is 25, 500. Step 3: Digital approximation of the geological profile i } of {Ωi } as follows. Using cell values, we define the digital approximation {Ω 1 Ω  Ω2 3 Ω 4 Ω  Ω5 6 Ω

=o =o =o =o =o =o

cell cell cell cell cell cell

, , , , , ,

cell cell cell cell cell cell

value value value value value value

= = = = = =

11, 400 21, 900 12, 200 10, 600 19, 200 25, 500

In addition, we let the cell of geologic boundary surfaces belong to its lower layer. 3.2

Fictitious Domain Formulation for Computation

i In the previous section, we introduced the cell values and decided the domain Ω       which approximated Ωi digitally. Here D = Ω1 ∪ Ω2 ∪ Ω3 ∪ Ω4 ∪ Ω5 ∪ Ω6 , Ω ⊂ D and D is called the fictitious domain mathematically. i , we define the following function  k(x, y) such that Using domain Ω  k(x, y) =

6 

ki χΩ i (x, y)

i=1

Fig. 4. Digital fictitious domain

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where { k1 , k2 , k3 , k4 , k5 } is the permeability and k6 is a new parameter in 6 , which is the domain occupied with air. We then propose our numerical Ω model for the mathematical model and call it a fictitious domain formulation for computation. Numerical model Find u ∈ H 1 (Ω) satisfying ⎧ k(x, y)∇u) = 0 in D (D) ⎨ div(  u = 0 on Γ1 , u = g2 on Γ2 ⎩ ∂u ∂u ∂n = 0 on Γ3 , ∂n = 0 on Γ5 Remark 4. Historically Lions([5]) and kawarada([3]) showed that Neumann’s boundary condition is approximated by the fictitious domain method. From the point of view, we approximate the Neumann’s condition on Γ4 in Mathematical model by choosing a small number as k6 .

4

Numerical Computation

The numerical computation of Numerical model is based on a kind of finite difference approximations defined on ”cells” and a regular mesh size corresponds to a 10 pixels width. 4.1

Data of Numerical Model

The piezometric head g2 [m] on Γ2 and the permeability ki [m/s] (i = 1, 2, 3, 4, 5) in the geological profile are given as follows([6]). g2 = 10.7 [m] , k1 = 6.72 × 10−4 [m/s] k2 = 12.2 × 10−4 [m/s] k3 = 0.013 × 10−4 [m/s]k4 = 0.531 × 10−4 [m/s] k5 = 22.3 × 10−4 [m/s] From the viewpoint of the fictitious domain method which we stated in 3.2, we choose the parameter k6 such that k6 = 10−9 [m/s]. 4.2

Numerical Calculation

Using the data and the finite difference algorithm for Numerical model , we get the numerical solution ( or piezometric head ) uh . From uh and the Darcy’s law (2), the groundwater flow through the geological profile is shown below Fig.5. The result of numerical simulations shows that groundwater flows from the mountain site to the north coast in Teshima island and this calculation gives a good information about the piezometric head all over the domain Ω.

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295

Fig. 5. Direction and strength of groundwater flow

Fig. 6. Our numerical computation with borehole logging at 

4.3

Comparison of Piezometric Heads

In 4.2, we use the piezometric head g2 of the fresh granite layer as the boundary condition on Γ2 and get the piezometric head uh in Ω. On the other hand, the piezometric head of the sandy soil was reported([6]). Its drilling sites marks  on Fig.6 where 100 [m] from the sea side and 22 [m] from the top of the shredderdust layer. We now compare two piezometric heads at the point  and have the following. The piezometric head by the borehole logging is 3.11 [m]and the piezometric head by the numerical calculation is 3.57 [m]. These data are almost same. Hence this computational method may be useful to predict the groundwater flow in Ω through industrial wastes.

5

Conclusion

We showed the numerical method based on the pixel information of the digital color image. Approximating layers of the geological profile by pixels, this numerical method is very simple and useful to predict the various groundwater flow from landfill in our life.

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References 1. Fujita, H., Kawahara, H., Kawarada, H.: Distribution theoretic approach to fictitious domain method for Neumann problems. East-West J. Numer. Mathe. 3(2), 111–126 (1995) 2. Glowinski, R., Lions, J.L., Tremolieres, R.: Numerical Analysis of Variational Inequalities. In: Studies in Mathematics and its Applications, vol. 8. North-Holland, Amsterdam (1981) 3. Kawarada, H.: Free Boundary Problem, Theory and Numerical Method. Tokyo University Press (1989) (in Japanese) 4. Koshigoe, H., Shiraishi, T., Ehara, M.: Distribution algorithm in finite difference method and its application to a 2D simulation of temperature inversion. Journal of Computational and Applied Mathematics 232, 102–108 (2009) 5. Lions, J.L.: Perturbations Singulieres dans les Problems aux Limites et en Control Optimal. Lecture Notes in Mathematics, vol. 323. Springer, Heidelberg (1973) 6. MIC: Investigation result: Environmental dispute coordination in the Teshima Island industrial waste case (September 1995) (in Japanese) 7. Takatsuki, H.: The Teshima Island industrial waste case and its process towards resolution. J. Mater Cycles Waste Manag. 5, 26–30 (2003)

A Genetic Algorithm for Solving Patient- Priority- Based Elective Surgery Scheduling Problem* Yu Wang1, Jiafu Tang1, and Gang Qu2 1

Dept of Systems Engineering, College of Information Science & Engineering, Key Lab of Integrated Automation of Process Industry, Northeastern University, Shenyang, 110819, China 2 Hospital Affiliated with Dalian University, Dalian, 116000, China [email protected], [email protected]

Abstract. Surgery generates the largest cost and revenue in the hospital. The quality of operation directly affects the level of patient satisfaction and economic benefit. This paper focuses on partitioning patients into different priorities according to the state of illness, an optimization model with the aim of maximizing customer satisfaction is established under the consideration of a three-dimensional parameter constraint related patients, operating rooms and medical staffs. An Genetic algorithm is proposed with two-dimensional 0-1 encoding for solving the surgery scheduling problem with the data derived from an upper first-class hospital, the experimental results show the efficiency of the model and algorithm. Keywords: Surgery Scheduling; Genetic Algorithm; Patient Priority.

1 Introduction With the reformation of the health care system being in progress, the community has paid increasing attention on medical service industry. The improvements of the service quality, efficiency and patient satisfaction have become important issues which should be addressed by researches. Since the operating theater is the hospital’s largest cost item and revenue center [1], it has a major impact on the efficiency and performance of the hospital as a whole. There are two challenges should be addressed 1. The confliction between patient priority and preference. 2. The scarcity of costly resources [2]. Therefore, surgical case scheduling allocates hospital resources to individual surgical case and decides on the time to perform the surgeries [3], which has plagued the staff in the operating theater for a long time, becoming the bottleneck of the department. Over the past years, several investigations on surgical scheduling have evolved by researches. Two major patient classes are usually sorted in the literature, namely elective and non-elective patients. The former class represents patients for whom the surgery can be well planned in advance, whereas the latter class groups patients for *

This research is supported by the National Science Foundation of China (70625001 and 70721001), the Fundamental Research Funds for the central Universities of MOE (N090204001).

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 297–304, 2010. © Springer-Verlag Berlin Heidelberg 2010

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whom a surgery is unexpected and hence needs to be performed urgently[4]. Adan and Vissers [5] consider elective patients in their research. They formulate a mix integer programming model to identify the mix of patients that have to be admitted to the hospital in order to obtain the target utilization of different resources. Wullink et al. [6] use discrete-event simulation to examine whether it is preferred to reserve a dedicated operating room or to reserve some capacity in order to improve the responsiveness to emergencies. In this paper patients were divided into different priorities according to the state of illness. Emergencies have the highest priority so that they should be operated on as soon as possible with the purpose of saving their life, body organ or limb function. In additional to patient priority, the inherent influence factors in surgeries such as the daily maximum workload of surgeon, the degree of patient satisfaction and the resource constraints are considered in the proposed model. A genetic algorithm is proposed to solve the problem. A satisfactory solution is obtained, which showed that the presented model and algorithm are effective.

2 Problem Description and Modeling An operation scheduling flowchart is shown in Fig.1. Firstly, the inputs which consist of the character of patients, medical staffs and the limitation of operating room resources are integrated. Then the high-performance solutions are selected according to the results of the objective function. Finally, the surgery scheduling should be improved continuously based on the feedback of the output of the system. According to the operation scheduling flowchart shown in Fig.1, the problem can be described as follows: There are totally N surgical groups in one hospital. Each surgical group’s daily maximum workload is Fi hour. The relationship between surgical

Patient Priority Severity Index Surgery Length Surgery Category

Patient Disease Type

Visiting day School day Maximum Workload

Medical Staff Information

Surgery Team

Technique Category

Integrate Inputs

Monitor Objective Function

Professional Title Quantity

Opening Hours

Operating Room Resource

Feedback

Professional Equipment

Fig. 1. An operation scheduling flowchart

Select Excellent Solutions

A Genetic Algorithm for Solving Patient- Priority- Based Elective Surgery

299

group and patient has been given, Oij = 1 denotes that the surgical group i is arranged to make operations on patient j, otherwise O ij = 0 .Each patient’s operation date is scheduled under the consideration of his surgeon’s visiting day, that is to say, operations shouldn’t be arranged on surgeons’ visiting day. Because the quantity of different resources is limited such as the opening time of operating theatre, at most M patients can be performed surgery each day. The total number of patients who make appointments to be operated on is S. Each patient has his own latest operation time cj , surgical length d j and priority aj (the bigger of aj , the higher level of priority for patient). Aik denotes that whether or not surgical group i can perform surgery on day k, Aik =1 denotes that the Day k is not the surgical group i’s visiting day, that is to say, the Surgeon i can make operations on the Day k, otherwise Aik=0. xjk is the decision variable, x jk = 1 denotes that patient j will be performed surgery on day k, otherwise x jk = 0 . u j ( x j ) is the satisfaction function for patient j. The scheduling cycle is T days. The objective function is to maximizing the degree of satisfaction for patients under the consideration of patient priority. The operation scheduling model proposed in this paper is described as follows. S

max G1 = ∑ a j u j ( x j )

(1)

j =1

S

s.t. ∑ Oij x jk d j ≤ Fi

∀i = 1, 2,..., N ; k = 1, 2,..., T

(2)

j =1

kx jk ≤ c j

∀k = 1, 2..., T ; j = 1, 2,..., S

S

∑x j =1

T

∑x k =1

jk

jk

≤M

=1

(3)

∀k = 1, 2..., T

(4)

∀j = 1, 2,..., S

(5)

T

u j (x j ) =

c j − ∑ kx jk + 1 k =1

cj

x jk ∈ {0,1} Oij x jk ≤ Aik

∀j = 1, 2,..., S ; x j = ( x j 0 ,...x jk ..., x j ,(T −1) )

(6)

∀j = 1, 2,..., S ; k = 1, 2,..., T

(7)

∀i = 1, 2,..., N ; j = 1, 2,..., S ; k = 1, 2,...T

(8)

The objective function (1) is to maximize patient satisfaction under the consideration of patient priority. Constraint (2) ensures that the maximum of each group’s working time cannot exceed Fi hour per day. Constraint (3) insures that each

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patient’s operating time should be scheduled earlier than his latest operating time. Constraint (4) insures that the quantity of operations scheduled can’t be up to M each day. Constraint (5) ensures that each patient can be performed surgery only once. Formula (6) is the satisfaction function of patient j, the value of this function is above 0, that is to say, it doesn’t have the condition of complete dissatisfaction as long as patients have been arranged surgery. When patient j is placed to be performed surgery on the first day of the planned cycle, his degree of satisfaction achieves the maximum T

value 1. y j = ∑ kx jk is the day when patient j is assigned to be operated on. Conk =1

straint (7) makes sure that surgeons are arranged to make operations only on the non-visiting day. Constraint (8) denotes that the decision variable xjk is binary. Constraints (3), (5) and (7) are hard constraints which must be met during the surgical scheduling process while constraints (2) and (4) are soft constraints which contribute to a more reasonable, humanized schedule.

3 Genetic Algorithm Based on Two-Dimensional 0-1 Encoding Genetic algorithm is an effective technique for probability search. Compared with other methods, genetic algorithm has obvious character of global search and robustness of solving problems. Therefore, genetic algorithm can be used to solve some problems which are impossible or too difficult to deal with by other optimization methods. Genetic algorithm has many promising features, one of which is not directly to make iterative operation on variables, but that making use of variable encoding to find the global optimal solution through genetic manipulation imposed on individuals. It is recommended to use binary encoding when applied mode theorem for analyzing code scheme by Holland [7]. To give full play to the advantages of genetic algorithm based on multi-dimensional 0-1 encoding, this paper selects multi-dimensional 0-1 encoded genetic algorithm to solve the operation scheduling problem. 3.1 Algorithm Realization 1) Encoding

In the algorithm, individual uses the binary coding scheme of a two dimensional array. The row in a chromosome represents the serial number of patients and the column in a chromosome represents the day of the planning cycle. Then the value located in the row j, column k equals 1 meaning that the patient j is arranged surgery on the day k. Each individual represents a possible result of surgery scheduling. Generate initial population meeting the hard constraints (3), (5) and (7). 2) Determination of Fitness Function To meet the soft constraints (2) and (4), the penalty factors p and q are introduced in the fitness function, so that the survival chance of individuals who don’t meet the soft constraints can be sharply reduced. When constraints (2) and (4) are met, p and q

A Genetic Algorithm for Solving Patient- Priority- Based Elective Surgery

equal 0, otherwise p and q equal a positive number. ness function is expressed as follows.

Oij x jk ≤ Aik

max f ( x ) =

301

G1 is the objective function. Fit-

∀i = 1, 2,..., N ; j = 1, 2,..., S ; k = 1, 2,...T

(9)

G1 S

S

j =1

j =1

1 + p ( ∑ Oij x jk d j − Fi ) + q ( ∑ x jk − M )

.

(10)

3) Selection

Roulette wheel selection is chosen, where the average fitness of each chromosome is calculated depending on the total fitness of the whole population. The chromosomes are randomly selected proportional to their average fitness. 4) Mutation

In order To introduce variations into the individuals, the mutation operator is applied in the proposed genetic algorithm, which avoids convergence to local optima in the solution space. Firstly generate a random number r between 0 and 1. Then chromosomes are selected to mutate when the random number is smaller than r. At last select the mutation position randomly and change the chosen gene. 5) Crossover

To keep the quantity of 1 in each row is no more than 1 in the two-dimensional array, that is, to ensure the solution is feasible, it would be desirable to swap the chosen row generated randomly. Firstly, randomly select pairs of individuals from the mating pool. Then generate a random number which is smaller than the number of rows. At last swap the genes of the parents whose row numbers are bigger than the random number. 6) Correction

Traverse the offspring chromosomes and explicitly converse the genes which cannot satisfy the hard constraints (3), (5) and (7).

4 Experimental Results and Analysis In this section the performance of the proposed algorithm would be evaluated. We experiment on the surgery reservation data which comes from an upper first-class hospital located in Liaoning Province during the period of 2010-01-4 to 2010-01-8. Data as shown in the following: The total number of patients waiting for surgery is 39. There are 10 surgical groups. The maximum workload of each surgical group is 8 hours per day. Each day at most 10 operations can be made. The priority of patient j is

302

Y. Wang, J. Tang, and G. Qu Table 1. The available date for surgical group

Aik

2010-1-4

2010-1-5

2010-1-6

2010-1-7

2010-1-8

1

1

1

0

1

1

2

0

1

1

1

1

3

1

1

1

0

1

4

1

0

1

1

1

5

1

1

1

1

0

6

1

1

0

1

1

7

1

1

1

0

1

8

0

1

1

1

1

9

1

1

1

1

0

10

1

0

1

1

1

Table 2. The basic data of patients j

1

2

3

4

5

6

7

8

9

10

11

12

13

aj

1

1

1

1

2

2

2

1

1

1

1

1

3

cj (d)

4

4

4

4

4

4

5

5

5

5

5

5

5

dj (h)

3

5

6

2

3

2

3

2

5

6

1

5

2

i

5

2

8

4

6

1

10

4

2

6

4

5

1

j

14

15

16

17

18

19

20

21

22

23

24

25

26

aj

1

1

1

1

1

2

1

3

1

3

1

1

1

cj(d)

5

5

5

5

5

5

5

5

5

5

5

5

5

dj(h)

4

2

2

4

4

3

5

3

3

3

3

2

7

i

9

7

2

3

3

7

2

9

6

5

1

9

6

j

27

28

29

30

31

32

33

34

35

36

37

38

39

aj

1

1

1

2

2

1

1

3

1

1

1

1

1

cj(d)

5

5

5

5

5

5

5

5

5

5

5

5

5

dj(h)

1

5

5

5

3

3

2

3

1

3

1

2

2

i

10

2

7

4

9

8

10

4

10

9

9

4

9

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303

Table 3. The surgical date of each patient Num

Date

Num

Date

Num

Date

Num

Date

1

10-1-6

11

10-1-8

21

10-1-5

31

10-1-4

2

10-1-6

12

10-1-5

22

10-1-4

32

10-1-7

3

10-1-6

13

10-1-4

23

10-1-7

33

10-1-7

4

10-1-7

14

10-1-7

24

10-1-7

34

10-1-6

5

10-1-4

15

10-1-6

25

10-1-7

35

10-1-4

6

10-1-5

16

10-1-7

26

10-1-8

36

10-1-5

7

10-1-4

17

10-1-5

27

10-1-7

37

10-1-6

8

10-1-8

18

10-1-8

28

10-1-5

38

10-1-8

9

10-1-8

19

10-1-5

29

10-1-6

39

10-1-4

10

10-1-5

20

10-1-7

30

10-1-4

Fig. 2. The number of operations per day

represented as a j , the latest operation time of patient j is represented as cj (day), the length of the operation performed on patient j is estimated dj hour, the available time for surgical group is shown in Table 1, and the basic data of patients is shown in Table 2. We obtain Figure 2 and 3 through the analysis of the scheduling results. As shown in Figure 2, the number of operations which have been arranged is no more than 10 per day. As shown in Figure 3, we find that the working hours for each surgical group are less than 8 hours per day. Fig.2 and Fig.3 indicate that the soft constraints have been satisfied. The results in Tables 3, Fig. 2 and 3 validate that the operation scheduling model and genetic algorithm proposed in this paper are feasible in the investigated cases.

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Fig. 3. Working hours per day for each surgical group

Compared with traditional manual scheduling methods, this method is more humanized, automatic and intelligent. It has greatly reduced the workload of staffs. The restriction on the length of working hours per day avoids surgeons to be overtired, which indirectly reduces the probability of medical accidents.

5 Conclusion This paper proposes an operation scheduling model and solves the problem with genetic algorithm. The result of the experiment demonstrates the feasibility and validity of this method. The emergency scheduling hasn’t been deeply referred to in this paper. The arrival time of emergency patients is random, which greatly increases uncertainty and difficulty of the problem. More attention will be paid on random conditions of surgery scheduling problem in future research.

References 1. Health Care Financial Management Association: Achieving operating room efficiency through process integration. Technical Report (2005) 2. Glauberman, S., Mintzberg, H.: Managing the care of health and the cure of disease – Part I: Differentiation. Health Care Management Review 26, 56–69 (2001) 3. Pham, D.N., Klinkert, A.: Surgical case scheduling as a generalized job shop scheduling problem. European Journal of Operational Research 185, 1011–1025 (2008) 4. Brecht, C., Erik, D., Jeroen, B.: Operating room planning and scheduling: A literature review. European Journal of Operational Research 201, 921–932 (2010) 5. Adan, I.J.B.F., Vissers, J.M.H.: Patient mix optimization in hospital admission planning: A case study. International Journal of Operations and Production Management 22, 445–461 (2002) 6. Wullink, G., Van, M., Houdenhoven, E.W., Hans, J.M.: Closing emergency operation rooms improves efficiency. Journal of Medical System 31, 543–546 (2007) 7. Holland, J.H.: Adaptation in nature and artificial system. Ann Arbor University of System. University of Michigan Press (1975)

A Neighborhood Correlated Empirical Weighted Algorithm for Fictitious Play Hongshu Wang, Chunyan Yu, and Liqiao Wu College of Mathematics & Computer Science, FuZhou University, Fuzhou, Fujian, China, 350108 [email protected], [email protected], [email protected]

Abstract. Fictitious play is a widely used learning model in games. In the fictitious play, players compute their best replies to opponents’ decisions. The empirical weighted fictitious play is an improved algorithm of the traditional fictitious play. This paper describes two disadvantages of the empirical weighted fictitious play. The first disadvantage is that distribution of the player’s own strategies may be important to make a strategy as times goes. The second is that all pairs of players selected from all players ignore their neighborhood information during playing games. This paper proposes a novel neighborhood correlated empirical weighted algorithm which adopts players' own strategies and their neighborhood information. The comparison experiment results demonstrate that the neighborhood correlated empirical weighted algorithm can achieve a better convergence value. Keywords: Learning model, Fictitious play, Empirical weight, Neighborhood information.

1 Introduction Fictitious play is an iterative procedure in which players compute their best replies at each step based on the assumption that opponents’ decisions follow a probability distribution in agreement with the historical frequency of their past decisions [1]. In this process, players believe that their distribution of opponents’ strategies is stationary but unknown. Fictitious play has been proved to converge to Nash equilibrium [2]. Fictitious play and its variants have been widely used in many parts, such as simultaneous auctions [3], the interesting class of anonymous games [4], and so on. Fictitious play has also aroused interest as an optimization heuristic [5]. In an evolutionary environment, fictitious play also plays well [6]. Fictitious play has three basic forms. The first is traditional fictitious play. The second is stochastic fictitious play. The third is empirical weighted fictitious play. In the stochastic fictitious play, players’ random behavior can prevent players from being “manipulated” by clever deterministic rules. Hence, the player adopts the strategy stochastically instead of adopting pure strategy. The stochastic fictitious play is proved to be convergence [7, 8, and 9]. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 305–311, 2010. © Springer-Verlag Berlin Heidelberg 2010

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In traditional fictitious play players suppose the weights of the strategies are the same. In empirical weighted fictitious play, players may be more interesting in the resent facts than the olds, they may find the olds is out and give them less weight. However, in traditional empirical weighted fictitious play, the player just considers the opponents’ strategies. As the time goes, his self strategies will maybe become more important because some new ideas in the process may be emerged. Unfortunately, in traditional empirical weighted fictitious play, the player doesn’t consider this point. Secondly when there are a lot of players among which stochastically to select two players to play games, the player in traditional fictitious play can’t use his neighborhood information which consists of other players’ strategies space. However, other players' strategy space may be useful to the player. Hence, this paper proposes an improving algorithm for empirical weighted fictitious play, and observes the results of players' learning condition with two experiments simulated on the swarm platform. The paper is structured as follows. In section 2, we introduce traditional empirical weighted fictitious play. In Section 3, a neighborhood correlated empirical weighted algorithm is proposed. Several interesting examples of the model are discussed in Section 4. In section 5, the main results of the paper are summarized.

2 The Traditional Empirical Weighted Fictitious Play In the case of a two-player simultaneous-move game, assuming S represents finite strategy spaces, and u represents payoff function. The player i has an exogenous initial weight function k0i : S0 − i → R + . The records should be set with a different weight, for in the learning process the resent records are always better than the former. Defining the weight of the record is φ τ −1 (0 ≤ φ ≤ 1) at date τ . Then at the t

period, this weight function kt i ( s − i ) is updated like that: t −1

kt i ( s − i ) =

I ( st − i , s − i ) + ∑ φ k I ( sk − i , s − i ) k =1 t −1

1 + ∑φ k

(1)

k =1

−i −i ⎪⎧1 if st = s where I ( st − i , s − i ) = ⎨ ⎪⎩0 others

The probability that player i assigns to player −i playing s − i at date t is given by ⎧ kt i ( s − i ) if s − i ∈ Si − i ⎪ i −i i −i ( ) k s γ t ( s ) = ⎨ ∑ s∈Si− i t ⎪ others ⎩0

(2)

The player i at date t calculates that period’s expected payoff ut i ( s − i ) with their prediction γ t i ( s − i ) and the payoff matrix taking the following form:

A Neighborhood Correlated Empirical Weighted Algorithm for Fictitious Play

ut i ( s − i ) = ∑ s − i ∈S − i γ t i ( s − i )ut i ( s i , s − i )

307

(3)

i

In the end the player i selects the action to maximize that period’s expected payoff with ut i ( s − i ) .

3 The NCEWFP Algorithm The NCEWFP (neighborhood correlated empirical weighted algorithm for fictitious play) algorithm is an improving method of the traditional empirical weighted fictitious play. The NCEWFP algorithm considers three factors: opponent’s strategy, his own strategy, neighborhood information. Its definition: FRL = aFP + (1 − a) R − > rL

(4)

The FRL means the player’s strategy which is selected with rules; FP represents the traditional empirical weighted fictitious play, and a is the probability of FP ; R means the player make a strategy by calculating his frequency of his strategies, and 1 − a is the probability of R ; L represents the neighborhood information, and r is the weight of L ; the arrow means here reference, that is, at first the strategy made by former compares with the neighborhood information, if the strategy is the same one, then return the strategy. Otherwise, with probability r it will return the strategy which is made by the neighborhood information. In the case of r = 0 , the formula (4) degenerates into extension of the traditional empirical weighted fictitious play, and can be used in only 2-player situation. Otherwise it can be used in the two-player which maybe selected in lots of players. In the case of a = 0 , the formula (4) degenerates into reinforcement learning model. In the case of a = 1 , the model degenerates into traditional fictitious play. The NCEWFP algorithm is extension of the traditional fictitious play. It can help players to make better actions and be more adaptive in the complex situations. 3.1 The Steps to Make a Strategy for the Player with NCEWFP Algorithm

Assuming Pi represents player i . P− i represents the opponent of the player i . Si represents a strategy selected by Pi . S − i represents a strategy selected by P− i . ui represents a payoff of Pi . 1) player Pi selects a strategy Si1 based on opponent’s strategy according to the traditional empirical weighted fictitious play; 2) Pi makes a strategy Si2 based on his own strategy according to the traditional empirical weighted fictitious play; a) Pi is supposed to be wise to assume that his opponent would select a strategy

S−2i

to get a

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maximum payoff by the payoff matrix with Si2 that is supposed to be selected by Pi . At this situation, the opponent would get the payoff u− i , and the player would get the payoff ui ; b)

Pi would select a strategy Si2 to get a maximum payoff by the payoff matrix with supposed

to

be

selected

S−2i

P− i .

by

that is At

this

situation, Pi got the payoff ui ; c) if ui > ui , the player finally adopts the strategy

Si2 with the probability p(0 ≤ p ≤ 1) , otherwise he

3) Pi

adopts the strategy Si2 . Here, probability p is introduced to avoid the fact that the player always assumes twice, maybe sometime once guess is better. gets the strategy Si3 with formula

Si3 = a( Si1 ) + (1 − a ) Si2 (0 ≤ a ≤ 1) with probability a . 4) Pi

calculates the strategy

Sil in his neighborhood

k =n

with the formula

Sil = max(∑ sak ) (a ∈ A) where

n is the

k =0

number of players in his vision and the A is the set of the players in his vision; 5) the player Pi , following the crowd with probability

r , select his player

will

Otherwise,

strategy as follow:

make the

the player

action

with

adopts

If Si3 = Sil , the strategy

Si3 .

Sil

with

strategy

probability r (0 ≤ r ≤ 1) .

4 The Game Model in the Experiment A game model is composed of the players and the environment which the players live in. The players are autonomy, and they are freedom to make actions without a center organization. The environment has twofold meaning. The one is that the players' environment is composed of other players. The other one is that the environment is the space the players live in. With a two-tuple the game model can be described as Mod =< Env, A > . The Env represents the environment and A represents the set of the players. The environment the players live in is complex. In real world, the players' communication is impacted with lots of factors such as education, culture, the space. In the real world, there are mountains, rivers, cities, etc. To model simply, the Env in the

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model can be modeled with n × n grids. The players can randomly walk and play games without constraints in the grids. For each player a ∈ A , it can be modeled by a seventuple a =< Pos, S , u , M ,ψ ,ϑ , vis > . The Pos represents the current position of the player. The S represents the player's space of strategies. The u is the payoff function of a , and its expression is u : S a × S − a → R . The M is the player's memory set which consist of the player's every action in the game. The ψ represents how wise the player is! The ϑ represents the player is a man who follows the crowd whether or not. The vis represents the player's vision, and in his vision he could select a player to play game. To test the NCEWFP algorithm which is effective or not, the paper adopts some payoff matrixes with different game model. The payoff matrixes just like that in the table 1. The value in the brackets is Nash equilibrium in table 1. Table 1. The payoff matrix with different game model Type of Game Prisoner's Dilemma (0) Hawk Dove Game (0.714) Stag Hunt Game (1)

0.7, 1, 0, 0.2, 1, 0.3,

Payoff Matrix 0.7 0, 1 0 0.2, 0.2 0 1, 0.2 1 0, 0 1 0, 0.3 0 0.3, 0.3

In the experiment, the environment is 50 × 50 grids, in which there are 50 players whose position is set randomly. The player at each period walks randomly and chooses a player as the opponent to play games in his vision. At each period the player must just play only once. In the game he makes a strategy with the NCEWFP algorithm, and puts the payoff and his strategy into his memory set. There are two experiments to show the NCEWFP algorithm is effective in a different way. The first experiment shows players' neighborhood information can help players converge to a better value. The second shows the NCEWFP algorithm is much better than traditional empirical weighted fictitious play. The table 2 shows the different experiment results with the player's neighborhood information and without his neighborhood information in the NCEWFP algorithm. Table 2. The NCEWFP algorithm with and without neighborhood information in Hawk-Dove game The 500th period with neighborhood 0.6643016437084 information without neighborhood 0.6646794434832 information

The 1500th period

The 3000th period

0.68706186205923

0.69162191764788

0.66870364960522

0.67241030850638

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The experiment result shows the player's neighborhood information is very effective. At the 3000th period that is already stable, the convergence value with neighborhood information is over the data without neighborhood information. With the data, it can be seen that at the 500th period in the game, the information is still incomplete, and then players' neighborhood information is less to help players to make a better strategy. As the time goes, the information is more and more, and players will make better strategies with their memory sets. At that time the neighborhood information can help players always make optimalizing choice. The table 3 is the experiment with the traditional empirical weighted fictitious play and the NCEWFP model. It shows the difference in the two ways. Table 3. The convergence value at stable with different algorithm Type Of Game Prisoner's Dilemma (The 3000th period) Hawk Dove Game The 3000th period) Stag Hunt Game (The 3000th period)

Traditional Empirical Weighted The NCEWFP AlgoFictitious Play rithm 3.469368870002722E-4 1.977068264046457E-4 0.6347911938748314

0.6916219176478893

0.9989036424609946

0.9992565218025018

Take Hawk-Dove game for example from the table 3, the convergence value is more or less 0.6347911938748314 with traditional empirical weighted fictitious play, whenever how long the period runs! But if the players in that game adopt the NCEWFP algorithm, the convergence value can be raised up to 0.6916219176478893. With the contrast data, it can be demonstrated the NCEWFP algorithm is better than traditional empirical weighted fictitious play.

5 Conclusion Fictitious play is widely used in games as a learning model. Based on the empirical weighted fictitious play, this paper proposes a NCEWFP algorithm. In the NCEWFP algorithm players not only compute the best replies to opponents’ decisions, but also compute the best replies to their decisions. At the last step of the NCEWFP algorithm, players make decisions by comparing the decisions with the neighborhood information. By the three key steps, the NCEWFP algorithm becomes more effective than the traditional weighed fictitious play. And then, with the two contrast experiment results, the NCEWFP algorithm is demonstrated to achieve a better convergence value than the traditional weighed fictitious play at the same period. Acknowledgments. This work was supported in part by a grant from the National Natural Science Foundation of China (No.60805042) and a grant form Program for New Century Excellent Talents in Fujian Province University (No.XSJRC2007-04).

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References 1. Brown, G.W.: Iterative solution of games by fictitious play. In: Koopmans, T. (ed.) Activity Analysis of Production and Allocation, pp. 347–376. Wiley, New York (1951) 2. Drew, F., David, K.L.: The theory of learning in games. MIT Press, Cambridge (1995) 3. Enrico, H., Gerding, Z.R., Andrew, B., Edith, E., Nicholas, R.J.: Approximating mixed Nash equilibria using smooth fictitious play in simultaneous auctions. In: 7th International Joint Conference on Autonomous Agents and Multiagent Systems (2008) 4. Zinovi, R., Enrico, G., Maria, P., Nicholas, R.J.: Generalised fictitious play for a continuum of anonymous players. In: 21st International Joint Conference on Artifical Intelligence, pp. 245–250. Morgan Kaufmann Publishers Inc., Pasadena (2009) 5. Lambert, T., Epelman, M., Smith, R.: A fictitious play approach to large-scale optimization. In: Operations Research, vol. 53, pp. 477–489. INFORMS (2005) 6. Michal, R., Robert, M.S.: Fictitious play in an evolutionary environment. In: Games and Economic Behavior, vol. 68, pp. 303–324. Elsevier Inc., Amsterdam (2010) 7. Josef, H., William, H.S.: On the Global Convergence of Stochastic Fictitious Play (2002) 8. Noah, W.: Stability and Long Run Equilibrium in Stochastic Fictitious Play. Princeton University, Princeton (2002) 9. Drew, F., Satoru, T.: Heterogeneous beliefs and local information in stochastic fictitious play. In: Games and Economic Behavior. Elsevier Inc., Amsterdam (2008)

Application of BP Neural Network in Exhaust Emission Estimatation of CAPS Linqing Wang and Jiafu Tang College of System Engineering, Northeastern University, Shenyang 110819, China [email protected], [email protected]

Abstract. An approach is put forward in this paper to introduce environment factors into transportation system optimization management of enterprises which supply a CAPS. The method of estimatate the exhaust emission of vehicles in CAPS is presented and involves four steps. The designing BP NN(Backprogapation Neural Network), in step 1, is the main content and is for forcasting the pickup demand of CAPS as the basic study of the whole method. In part 3 the process of estimation exhaust emission of vehicles in CAPS is prensented. The case study of pickup demand forecasting in CAPS is given. According to the case study, the structure of the BP NN is proved to be correct and reasonable and can be used to forecast the pickup demand of CAPS as long as there is enough samples to train the designing network. Moreover, the further studies is proposed at last. Keywords: exhaust emission estimation, pickup demand forecasting, city airport pickup service, backpropagation neural network.

1 Introduction Economic profits of enterprises are optimized in many studies by modeling VRP(vehicle routing problem) with the target of the minmum travel distance, the minmum travel cost or the minimun travel time and then by computing the optimal paths of vehicles with the specific algorithms. Jiafu Tang, Zhendong Pan and Jun Zhang[1] proposed a WVRP(Weighted Vehicle Routing Problem), in which the demand of each retailer is considered when routing schedule is made. The retailers with larger demand have priority to be visited earlier. Finally, a genetic algorithm named PB-GA using a special partition method is proposed to solve this model. Gang Dong, Jiafu Tang, Kin Keung Lai and Yuan Kong[2] addressed a vehicle routing and scheduling problem arising in CAPS. The problem is formulated under the framework of VRPTW(Vehicle Routing Problem with Time Windows), with the objective of minimizing the total operational costs, i.e. fixed start-up costs and variable traveling costs. A 0–1 mixed integer programming model is presented, in which service quality is factored in constraints by introducing passenger satisfaction degree functions that limit time deviations between actual and desired delivery times. These studies are reflection of refined management and are results of introducing economic factors, K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 312–320, 2010. © Springer-Verlag Berlin Heidelberg 2010

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which are practical, into mathematical models, which are abstract and theoretical, and are popular in the current academia. The original interior process of logistics or transportations that are rough is optimized. Consequently tremendous economic profits are realized. In this sence, it is a problem deserving of further study that whether the environmental benefits could also be realized with introduing environment factors into transportation system optimization management of enterprises. It has been hightlight in the book “Introduction of Logistics” that green transportation means decreasing traffic congestion and reducing environmental pollotion. It is to control or reduce the use of high-polluting vehicles on the traffic that predicing vehicle emissions and developing emission standards[3]. Zhigao Zhu, Tiezhu Li and Wenquan Li[4] suggests a method to control the quantity of traffic pollutants emission through adjusting and controlling the composition of vehicle type, then traffic volume, and indirectly vehicle’s running speed. In their paper, the pollutants, carbon monoxide (CO), nitrogen oxides (NOx) and hydrocarbon (HC) are considered as evaluation factors. On the Basis of the relation between the emission factor and vehicle’s velocity and the relation amongst three parameters (Volume, Speed, Density) of traffic flow, they developed the model that calculates the total quantity of pollutants emission from traffic based on vehicles’ speed, proportion of vehicle type, and traffic volume. In the circumstance of our research, exhaust emission of vehicles is taken into account while caculating the optimal paths of vehicles that will be used to pickup passengers to the airport. The quantitative of vehicle exhaust emission per day in CAPS, which is discharged in the process of completing the pickup tasks of the day , is estimated by forecasting the total pickup demand of one city per day and then by comuputing how much exhaust emission will be produced by vehicles to finish the pickup tasks of the day, whose amount equals to the forecasting value above, in the optimal paths. Therefore the environment impact of vehicles in the pickup service can be supervised with the predicted value and can also be used as the background research for further studies. The method to forecasting exhaust emission estimation of vehicles in CAPS is proposed in this paper. The designing BP neural network is presented with its detailed structure and is trained and proved by a case according to our 2008 survey data that count the pickup demand several residential areas in Shenyang.

2 BP NN Model for Pickup Demand Forecasting of CAPS The first step is to define our problem of forecasting pickup demand in CAPS with BP NN. For supervised networks, this means designing the structure, size and properties of a set of input vectors and a set of associated desired output vectors called target vectors[5]. As in this problem,the values of CAPS pickup demand are defined to be the output values and the relative properties values of the CAPS pickup demand are defined to be the input.These relative properties are determined by our 2008 survey data and list as follows: S=F(C,X1,X2,X3,X4,E).

(1)

where S is the measured pickup demand values of CAPS in one residential area, C is the measured quantitative value of the residential area. X1, X2, X3 and X4 are four 0-1 variables to describe the convenience of surrounding traffic of the residential area:

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• It means the residential area is not near any subway when X1 equals to 0 and it is on the contrary when X1 equals to 1. • It means the residential area is not near any railway station when X2 equals to 0 and it is on the contrary when X2 equals to 1. • It means the residential area doesn’t have convenient bus communications when X3 equals to 0 and it is on the contrary when X3 equals to 1. • It means it is hard to take taxis around the residential area when X4 equals to 0 and it is on the contrary when X4 equals to 1. E is the evaluatation value of the average monthly incomes of people in the residential area. If E equals to 1, then the average monthly incomes value of people in the residential area is between 0 RMB and 1000 RMB. If E equals to 2, then the value is between 1000 RMB and 2000 RMB. And so on, if E equals to 8, then the value is between 7000 RMB and 8000 RMB. The two-layer feed-forward network, which is also the most common network used with backpropagation, is chosen to be the framework of our neural network model for the purpose that master the complex relationship between C,X1,X2,X3,X4,E and S by learning from the survey data. Two-layer feed-forward networks can potentially represent any input-output relationship with a finite number of discontinuities, assuming that there are enough neurons in the hidden layer[5]. Function tansig, as shown in Fig.1.(a)., is selected to be the transfer function for hidden layers and function purelin, as shown in Fig.1.(b)., is designed to be the transfer function for the output layer.

α

α

+1

n

0

n

-1 α = tan sig ( n ) = (a)

2 −1 1 + e −2 n

α = purelin ( n ) = n (b)

Fig. 1. Transfer functions of the desining BP NN

With the two transfer functions, the network is allown to learn both nonliner and linear relationships between input and output vectors above[5]. In summary, the structure of the designing BP NN with the two-layer tansig/purelin network in our problem is shown next. As shown in Fig.2., 10 neurons are arranged as the hidden layer in BP network. The learning course of network includes two courses, one is the input data transmitting forward directed and another is the error transmitting backward directed. In the

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Forward Input

C X1 X2 X3 X4 E 6

Hidden Layer

p1 6 ×1

Output Layer

IW 1,1 10 × 5

⊕

α1

n1

5 ×1

LW 2 ,1

1

b2

5 ×1

1

b1 10 ×1

5

α 1 = tan sig ( IW1,1 p1 + b1 )

1× 5

1× 1

α2

1× 1

⊕

n2

S

1×1

1

α 2 = purelin ( LW2,1α 1 + b 2 )

Backward Fig. 2. Two-layer tansig/purelin BP Feedforward Network

transmitting forward direction, the input data goes to the hidden layer from input layer and goes to the output layer[6]. If the output of output layer is different with the target, expected output result, then the output mean square error will be calculated, the mean square error will be transmitted backward directed then the weights between the neurons of the two layers will be modified in order to make the mean square error decreases. The key of the BP NN is the error transmitting backward and is also the essential process of the BP NN to learn the relationships between C,X1,X2,X3,X4,E and S as a positive reinforce to correct the network behavior. The course is accomplished through minimize the mean square error between the actual output of network and the target. Weight and bias values are updated according to Levenberg-Marquardt optimization[6]. Through the BP network training with some known samples, satisfy the accuracy requirement, then the interconnect weighing between every nodes in the input vectors are ascertained. Thus, the trained network can identify and predict the unknown sample.

3 Approach to Estimated Exhaust Emission of Vehicles in CAPS The issue of mesuring the vehicle exhaust emission is a very complicated one and involves studies in many disciplines like chemistry, automobile dynamics and many others. Only the approach to estimated exhaust emission of vehicles in CAPS is given in this paper but not how to measure it. The approach is shown as the following flowchart Fig.3. The process of step 1, as shown above, is exactly what we presented in part 2. The content in step 3 and 4 is one of our core research and can be settled with any algorithm developed. The studies in step 2 are in progress now.

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Fig. 3. Flowchart of the approach to estimated exhaust emission of vehicles in CAPS

4 Case Study In this part, the case study of the designing BP NN in part 2 is given. The 2008 residential areas pickup demand survey data is applied to for this BP NN model. During

Fig. 4. Performance plot chart of the train

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the training of the BP NN, 256 groups survey data is selected to be the input and target vectors, 64 groups of that are from residential areaS with 0~300 families and 64 groups of that are from residential areaS with 300~600 families and the two 64 groups remaining are from residential areaS with 600~900 and 900~1200 respectively. The train is accomplished with the neural network toolbox of MATLAB R2010a. One results of the train process is listed below. In Fig.4., the performance plot has three lines, because the 256 input and targets vectors are randomly divided into three sets. 60% of the vectors are used to train the network. 20% of the vectors are used to validate how well the network generalized. The last 20% of the vectors provide an independent test of network generalization to data that the network has never seen. Training on the training vectors continues as long as the training reduces the network's error on the validation vectors. After the network memorizes the training set, training is stopped. This technique automatically

Fig. 5. Regression chart of the train

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avoids the problem of overfitting, which plagues many optimization and learning algorithms[5]. The performance plot here is reasonable, because the test set error and the validation set error have similar characteristics, and it doesn't appear that any significant overfitting has occurred. In Fig.5., the upper-left chart, the upper-right chart and the lower-left chart respectively show how well the outputs of train vectors, validate vactors and test vactors track the targets. The lower-right char show the comprehensive performance of the network. All the R-values are around 0.9, which presents the result is reseasonable well. The following data in table below, which is unknown for the BP NN, is input to demonstrate the forcasting function of the trained BP NN. Table 1. Input data and the forecasting result

C X1 X 2 X 3 X 4 E S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

1180 1076 978 1170 905 968 1115 929 1114 972 1024 987 945 993 1090 906 1087 960 1019 1040 1140 966 1015 1172 937 913 1107 989 1199 955 1067 1077

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

12 32 33 47 88 82 79 86 7 24 41 45 68 69 80 85 10 30 34 45 87 88 89 96 6 22 38 46 62 70 81 88

C X1 X 2 X 3 X 4 E S 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

1040 1039 942 966 1117 1199 1067 1093 1103 1186 1010 1041 938 924 1063 1102 1153 1080 918 1122 946 925 914 924 964 1016 986 945 1090 934 1121 1167

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

30 46 50 52 86 88 89 90 12 26 50 56 70 73 88 90 13 33 37 50 90 78 79 86 7 23 28 43 61 60 82 85

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As shown in Table 1., the column “C”,”X1”,”X2”,”X3”,”X4” and ”E” are the input data and the column “S” is the corresponding pickup demand forecasting value given by the BP NN. The sum value of the column “S” is the total pickup demand forecasting value in one day of the above 64 residential areas in the table. The analyses are carried out as below from the forcasting results: • Pickup demand per residential area increases with amount of families in the residential area; • While others are the same, pickup demand in residential areas where hard to call taxis is larger than those where easy to call taxis; • While others are the same, pickup demand in residential areas that is far from the railway station is larger than those near the railway station; • While others are the same, pickup demand variations are slight with the convenience of the bus communications arround; • All in all, the residential areas with families those have high average monthly incomes have more pickup demand. It also can be concluded from the above results that the BP NN masters the relationship between C,X1,X2,X3,X4,E and S basiclly and that the BP NN realizes the expected forecasting function.

5 Conclusions The pickup demand forecasting of CAPS can be completed effectively by the designing BP NN when the network is trained. Thus, the structure of the BP NN is correct and reasonable and can be used to forecast the pickup demand of CAPS as long as there is enough samples to train the network.In part 3, the process of estimate the exhaust emission of vehicles in CAPS according to the pickup demand forecasting value of CAPS is prensented. Since the study of step 2 of the total process in part 3 has not yet finished, only the case study of pickup demand forecasting of CAPS is given but not the whole case study. The distribution forcasting of the pickup demand, including the demand locations and timewindows, would be our next study. In this paper, a method based on BP NN to forcast the pickup demand in CAPS is designed. Combined the forcasting value with VRP, companies who provide a CAPS could realized prediction, assessment and management of vehicles’ exhaust emissions in the pickup transportation. Thus, it helps companies to achieve a coordinated operation management of the economy, resource and environment.

References 1. Pan, Z., Tang, J., Han, Y.: Vehicle routing problem with weight coefficients. Jou. Man. Sci. 10, 23–29 (2007) 2. Gang, D., Tang, J., Kei, K.L., Kong, Y.: An exact algorithm for vehicle routing and scheduling problem of free pickup and delivery service in flight ticket sales companies based on set-partitioning model. Jou. Int. Man. Online-first (2009)

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3. Qi, Z.: Introduction of Logistics. Tsinghua University Press, Beijing (2005) 4. Zhu, Z., Li, T., Li, W.: Study on Optimal Total Link Vehicle Pollutants Emission Quantity. Jou. Tra. Sys. Eng. and Inf. Tec. 8, 80–84 (2008) 5. Brain, R.H., Ronald, L.L., Jonathan, M.R.: A Guide to MATLAB for Beginners and Experienced Users. Cambridge University Press, Cambridge (2003) 6. Tian, J., Gao, M., Zhang, F.: Research and Application of Urban Logistics Demand Forecast Based on High Speed and Precise Genetic Algorithm Neural Network. In: Proc. Int. Workshop Educ. Technol. Comput. Sci. ETCS, pp. 853–856. Inst. of Elec. and Elec. Eng. Computer Society, United States (2009)

Dynamic Behavior in a Delayed Bioeconomic Model with Stochastic Fluctuations Yue Zhang1 , Qingling Zhang1 , and Tiezhi Zhang2 1

Institute of Systems Science, Northeastern University, 110004, Shenyang, Liaoning, China zhangyue [email protected] 2 Institute of Design, Northeastern University, 110013, Shenyang, Liaoning, China

Abstract. The dynamic behavior of a bioeconomic model with time delay is investigated within stochastically fluctuating environment. Local stability and Hopf bifurcation condition are described on the delayed model system within deterministic environment. It reveals the sensitivity of the bioeconomic model dynamics on time delay. A phenomenon of Hopf bifurcation occurs as the time delay increases through a certain threshold. Subsequently, a stochastic model is discussed, which is established by incorporating white noise terms to the above deterministic delayed model system. With the help of numerical simulation, it can be shown that the frequency and amplitude of oscillation for the population density is enhanced as environmental driving forces increase. Keywords: Bioeconomic model system; Time delay; Hopf bifurcation; Stochastic fluctuating environment.

1

Introduction

In recently, as the growing human needs for more food and more energy, mankind is facing the dual problems of resource shortages and environmental degradation. Consequently, there is considerable interest in the use of bioeconomic modeling to gain insight in the scientific management of renewable resources. Much analysis and modeling has been devoted to the problem of harvesting a renewable resources. In paper[1], the authors study the asymptotic behavior of a singlespecies model with stage structure and harvesting for the constant, variable and periodic harvesting effort. In paper[2], the exploited model with periodic birth pulse and pulse fishing occur at different fixed time are proposed and the conditions of equilibrium stability, bifurcations and chaos behavior are obtained. In papers[3,4,5], the authors discuss the optimal management strategy of renewable resources, like maximum sustainable yield, maximum harvest and maximum current profit. In papers[6,7], the dynamic behavior of bioeconomic models in stochastically fluctuating environment are discussed. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 321–332, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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In paper[8], the authors establish the following model of predator-prey with stage-structure ⎧ ⎨ x˙ 1 = αx2 − r1 x1 − βx1 − ηx21 − β1 x1 x3 (1) x˙ 2 = βx1 − r2 x2 ⎩ x˙ 3 = x3 (−r + kβ1 x1 − η1 x3 ) where x1 and x2 represent the densities of the immature and mature of the prey population, respectively. x3 represents the density of the predator population. x˙ i = dxi /dt˜ and α, r1 , β, η, β1 , r2 , r, η1 , k are positive constants. α is the birth rate of the immature population, r1 is the death rate of the immature, β is the transformation rate of mature, the immature population is density restrictive (cf. the term ηx21 ), r2 is the death rate of the mature population, r is the death rate of the predator, the predators compete among themselves for food and hence the terms −η1 x23 . The second species is a predator of the immature population of the first species (cf. the terms β1 x1 x3 and kβ1 x1 x3 ). k is a digesting constant. With the introduction of the following non-dimensional variables y1 = βr21 x1 , y2 = kββ 1 x2 , y3 = ηr21 x3 , dt˜ = r12 dt, system (1) is transformed into the following non-dimensional form ⎧ ⎨ y˙ 1 = ay2 − by1 − cy12 − dy1 y3 y˙ 2 = y1 − y2 (2) ⎩ y˙ 3 = y3 (−e + y1 − y3 ) i where a = αβ , b = r1r+β , c = βη1 , d = βη11 , e = rr2 , y˙ i = dy dt . r22 2 Now assuming that the mature species are subjected to harvesting, system (2) can be written as ⎧ ⎨ y˙ 1 = ay2 − by1 − cy12 − dy1 y3 (3) y˙ 2 = y1 − y2 − qEy2 ⎩ y˙ 3 = y3 (−e + y1 − y3 )

where qEy2 is the harvesting yield, q is the catchability coefficient and E is the harvesting effort. In paper [8], the authors discussed the problem of the optimal harvesting yield of system (3). They obtained the optimal harvesting policy and the threshold of the harvesting for sustainable development. In fact, the reproduction of predator after predating the immature population is not instantaneous but will be mediated by some discrete time lag required for gestation of predator. Therefore, researching some delayed models are more realistic and there have obtained some interesting results[1,10,14]. In addition, it is common that resources science and management operate in fluctuating environment. Recently, the stochastic modelling of ecological population systems are getting more attention from several scientists[6,7,11,12]. To these papers, it is seldom to investigate the dynamic behavior of a delayed bioeconomic model within stochastically fluctuating environment. In this paper, the dynamic behavior of the bioeconomic model system (3) under stochastic fluctuations with time delay is discussed. The rest of the paper

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is organized as follows: in section 2, the local stability and Hopf bifurcation condition are described on the delayed bioeconomic model. In section 3, the effect of fluctuating environment on the bioeconomic model is discussed with time delay. Numerical simulation results are presented on the bioeconomic model in section 4. Finally, concluding remarks are given in section 5.

2

Basic Model with Time Delay

In this section, the time delay required for gestation of predator is considered and the following delayed differential equations corresponding to the model system (3) is used to describe this case. ⎧ ⎨ y˙ 1 = ay2 − by1 − cy12 − dy1 y3 y˙ 2 = y1 − y2 − qEy2 (4) ⎩ y˙ 3 = y1 (t − τ )y3 (t − τ ) − ey3 − y32 where τ is the dimensionless parameter expressing the gestation period for predator population. In paper [8], Zhang et al. study the optimal harvesting policy on the model system (4) with τ = 0. In paper [10], M. Bandyopadhyay et al. discuss the stability on the above system when qEy2 = 0. Here, we will investigate the dynamic behavior of the bioeconomic model system (4) according to the time delay. It is easy to obtain that the model system (4) has three equilibrium points within the positive quadrant if a > (b + ce)(1 + qE), namely P0 (0, 0, 0), a−b(1+qE) ∗ ∗ ∗ P1 ( a−b(1+qE) c(1+qE) , c(1+qE)2 , 0) and the interior equilibrium point P2 (y1 , y2 , y3 ), where 1 [1 + (de − b)(1 + qE)], y1∗ = (c + d)(1 + qE) y2∗ =

1 [1 + (de − b)(1 + qE)], (c + d)(1 + qE)2

y3∗ =

1 [a − (b + ce)(1 + qE)]. (c + d)(1 + qE)

The stability switch due to the variation of the time delay τ around the interior equilibrium point P2 in the model system (4) will be discussed. Furthermore, a phenomenon of Hopf-bifurcation will be exhibited. The characteristic equation to the interior equilibrium point P2 of the model system (4) is as following ⎞ −a dy1∗ λ + b + 2cy1∗ + dy3∗ ⎠=0 −1 λ + 1 + qE 0 det ⎝ 0 λ + e + 2y3∗ − y1∗ exp(−λτ ) −y3∗ exp(−λτ ) ⎛

⇒ M (λ) + N (λ)exp(−λτ ) = 0

(5)

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where M (λ) = λ3 + M1 λ2 + M2 λ + M3 , N (λ) = N1 λ2 + N2 λ + N3 , a + (1 + c)y1∗ + y3∗ > 0, M1 = 1 + qE + 1 + qE a a ]y1∗ + cy1∗ y3∗ + cy1∗2 + [1 + qE + ]y ∗ > 0, M2 = [(1 + c)(1 + qE) + 1 + qE 1 + qE 3 M3 = c(1 + qE)y1∗ (y1∗ + y3∗ ) > 0, N1 = −y1∗ < 0, a + cy1∗ − dy3∗ ], 1 + qE N3 = −(1 + qE)y1∗ (cy1∗ − dy3∗ ). N2 = −y1∗ [1 + qE +

It will be investigated that the time delay plays a crucial role in the dynamic behavior of the model system (4). For this purpose, it is assumed that there exist a critical value τ = τ0 > 0 and a positive quantity ω0 such that the characteristic equation (5) will have a pair of purely imaginary roots λ = ±iω0 . Then substituting λ = iω0 in equation (5) and separating real and imaginary parts, two transcendental equations can be obtained as follows: M1 ω 2 − M3 = (N3 − N1 ω 2 )cos(ωτ ) + N2 ωsin(ωτ )

(6)

ω 3 − M2 ω = N2 ωcos(ωτ ) − (N3 − N1 ω 2 )sin(ωτ ).

(7)

Squaring and adding (6) and (7), it can be calculated that (N3 − N1 ω 2 )2 + N22 ω 2 = (M1 ω 2 − M3 )2 + (ω 3 − M2 ω)2 ⇒ ω 6 + (M12 − 2M2 − N12 )ω 4 + (M22 − N22 − 2M1 M3 + 2N1 N3 )ω 2 + M32 − N32 = 0.

(8)

It is easy to obtain that if the following conditions: a(3c − d) + (bd − 3bc + 3cde − c2 e)(1 + qE) < 0 a − (b + ce)(1 + qE) > 0 hold, there exists a positive ω0 satisfying Eq.(8). The characteristic equation (5) has a pair of purely imaginary roots of the form ±iω0 . From (6) and (7), it can be obtained that τ0 corresponding to ω0 is given by τ0 =

1 (M1 ω 2 − M3 )(N3 − N1 ω 2 ) + N2 ω 2 (ω 2 − M2 ) arccos[ ]. ω0 (N3 − N1 ω 2 )2 + (N2 ω)2

(9)

Now checking the sign of d(Reλ) at τ = τ0 which is given by (9) to verify transverdτ sality condition for Hopf-bifurcation[13], by the following result[14]: Sign{

d(Reλ) dλ }λ=iω0 ,τ =τ0 = Sign{Re( )−1 }λ=iω0 ,τ =τ0 . dτ dτ

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It is easy to obtain that dλ −1 ) }λ=iω0 ,τ =τ0 dτ 1 2ω 6 + (M12 − 2M2 − N12 )ω04 + (N32 − M32 ) = 2 Sign[ 0 ]. ω0 (N3 − N1 ω02 )2 + (N2 ω0 )2

Θ = Sign{Re(

According to the values of Mi and Ni , i = 1, 2, 3, it can be obtained that M12 − 2M2 − N12 > 0. Furthermore, based on the assumption that Eq.(8) has a positive real root, it follows that N32 − M32 > 0. Thus Sign{ d(Reλ) dτ }λ=iω0 ,τ =τ0 > 0 and the following theorem is obtained. Theorem 1. If a(3c−d)+(bd−3bc+3cde−c2e)(1+qE) < 0 and a−(b+ce)(1+ qE) > 0, the model system (4) undergoes a Hopf-bifurcation around the interior equilibrium point P2 for τ = τ0 . Furthermore, P2 remains asymptotically stable for τ < τ0 and P2 is unstable for τ < τ0 . These results can be exhibited through Fig.1-3.

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3

The Model under Stochastic Fluctuations

In this section, the effect of stochastically fluctuating environment on the model system (4) is investigated. Here, the Gaussian white noise is incorporated into each equations of the model system (4). It is assumed that the presence of

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fluctuating driving forces on the deterministic growth of these population at time t. Thus, the stochastic version corresponding to the delayed model system (4) within the fluctuating environment takes the following form: ⎧ ⎨ y˙ 1 = ay2 − by1 − cy12 − dy1 y3 + σ1 ξ1 (t) y˙ 2 = y1 − y2 − qEy2 + σ2 ξ2 (t) (10) ⎩ y˙ 3 = y1 (t − τ )y3 (t − τ ) − ey3 − y32 + σ3 ξ3 (t) where σi , i = 1, 2, 3 are real constants expressing the environmental driving forces and the perturbed terms ξi (t), i = 1, 2, 3 are mutually independent Gaussian white noises characterized by ξi (t) = 0 and ξi (t)ξj (t1 ) = δij δ(t − t1 ), i, j = 1, 2, 3. · represents the ensemble average due to the effect of fluctuating environment, δij is the Kronecker delta expressing the spectral density of the white noise and δ is the Dirac delta function with t and t1 being the distinct times. The linearization of system (10) around P2∗ is as follows ⎧ ⎨ y˙ 1 = α11 y1 + α12 y2 + α13 y3 + σ1 ξ1 (t) y˙ 2 = y1 + α22 y2 + σ2 ξ2 (t) (11) ⎩ y˙ 3 = α33 y3 + β31 y1 (t − τ ) + β33 y3 (t − τ ) + σ3 ξ3 (t) a where α11 = −cy1∗ − 1+qE , α12 = a, α13 = −dy1∗ , α22 = −1−qE, α33 = −e−2y3∗ , β31 = y3∗ , β33 = y1∗ . Taking Fourier transform of equations in (11), the following system can be obtained  = A(s)Y (s) ξ(s) (12)

 = (σ1 ξ1 (s), σ2 ξ2 (s), σ3 ξ3 (s))T , Y (s) = ( where ξ(s) y1 (s), y2 (s), y3 (s))T and ⎛ ⎞ is − α11 −α12 −α13 ⎠ A(s) = ⎝ −1 is − α22 0 0 is − β33 exp(−isτ ) − α33 ⎛ −β31 exp(−isτ ⎞) A11 A12 A13 = ⎝ A21 A22 A23 ⎠ A31 A32 A33 Then the following equation can be obtained

where

 Y (s) = A−1 (s)ξ(s)

(13)

⎞ ⎛ ∗ ∗ ∗ ⎞ k11 k12 k13 A11 A21 A31 1 ⎝ A∗12 A∗22 A∗32 ⎠ A−1 (s) = ⎝ k21 k22 k23 ⎠ = det(A(s)) k31 k32 k33 A∗13 A∗23 A∗33

(14)

⎛

A∗ij , i, j = 1, 2, 3 are the algebraic cofactor of Aji . Thus yi (s) =

3  j=1

kij σj ξj (s), i = 1, 2, 3.

(15)

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If the function Z(t) has zero mean value, the fluctuation intensity of the components in the frequency band s and s + ds is SZ ds where the spectral density SZ (s) is formally defined[9] 2  |Z(s)|  . T →∞ T

SZ (s)ds = lim

(16)

Hence  2 |ξ(s)| T →∞ T  T2  T2 1 ξ(t)ξ(t )exp(is(t − t ))dtdt . = lim T →∞ T − T − T 2 2

Sξ (s)ds = lim

(17)

Therefore, from (15) and (17), it is obtained that

Syi (s) =

3 

σj2 |kij |2 Sξj (s), i = 1, 2, 3

(18)

j=1

because ξi (t) = 0 and ξi (t)ξj (t1 ) = δij δ(t − t1 ), i = 1, 2, 3. Furthermore, Sξi (s) = 1, i = 1, 2, 3, so the fluctuation intensity in yi is written by[9] 

∞

σy2i =

1 2π

=

1 2π

=

3  1  ∞ 2 σ |kij |2 ds, 2π j=1 −∞ j

Syi (s)ds −∞ 3  ∞  σj2 |kij |2 Sξj (s)ds −∞ j=1

(19) i = 1, 2, 3.

After some calculation, the fluctuation intensity yi is given by σy2i =

 3 A∗ji 1  2 ∞ |2 ds. σj | 2π j=1 −∞ det(A(s))

(20)

The expressions of the fluctuation intensity of population are concerned with the environmental driving forces σi , i = 1, 2, 3. It will increase as the increase of the environmental driving force. With the help of numerical simulation, it can be exhibited that the intensities of population fluctuations from their steady state value increase gradually as the increase of delayed parameter τ and environmental driving forces σi , i = 1, 2, 3. These results can be exhibited through Fig.4-6.

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4

Numerical Simulation

In this section, some numerical simulation results will be presented on the following stochastic delayed differential equation model system for different values of the delayed parameter τ . For system (10), a = 25.8779, b = 4.3133, c = 0.4667, d = 1.8750, e = 0.7778 when α = 7.889, r1 = 1.225, β = 2.657, η = 0.7, β1 = 1.5, r2 = 0.9, r = 0.7, k = 1, η1 = 0.8 (the set of parameter values from [10]). Furthermore, let q = 0.2, E = 1. For the model system (10) with τ = 0 and σi = 0, i = 1, 2, 3, one can easily verify that the condition of theorem 1 holds for the set of parametric values. Consequently, numerical simulations exhibit that the length of gestation delay τ affects the dynamic behavior. The interior equilibrium point P2 is stable for the delayed time τ < τ0 ≈ 12.8381 (see Fig.1). It looses its stability behavior as τ passes through its critical value τ0 ≈ 12.8381. Furthermore, the limit cycle is presented as depicted in Fig.2. The model system (10) undergoes a Hopf bifurcation around the interior equilibrium point P2 . Fig.3 depicts the unstable population distribution of the species for τ = 15.8381. Following, Fig.4-Fig.6 exhibit the stochastically stable orbits for τ = 4.8381, limit cycle period orbits with oscillation for τ = 12.8381 and unstable orbits with oscillation for τ = 15.8381 on the model system with environment driving force σi = 1, respectively. It is clear that the magnitude of environmental driving forces play a crucial role to determine the magnitude of oscillation (as the magnitude of delay parameter is same in both cases).

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331

Conclusion

In this paper, the effect of environmental fluctuations and gestation delay on a bioeconomic model system is investigated. The present work is the extension of earlier works by Zhang[8] and M. Bandyopadhyay[10] through incorporating the time delayed parameter and white noise terms. The above analysis shows that the sensitivity of the bioeconomic model dynamics on maturation time delay and environmental fluctuation. The gestation delay with larger magnitude has ability to drive the system from stable to unstable within the same fluctuating environment. In the same time, the frequency and amplitude of oscillation for the population density is enhanced as the environmental driving forces increase. These indicate that the magnitude of gestation delay plays a crucial role to determine the stability or instability and the magnitude of environmental driving forces plays a crucial role to determine the magnitude of oscillation of the harvesting population model system within fluctuating environment.

Acknowledgments This work is supported by Postdoctoral Foundation of China(20080441097) and the Fundamental Research Funds for the Central Universities(N090305005). The authors gratefully acknowledge the reviewers for their comments and suggestions that greatly improved the presentation of this work.

References 1. Song, X.Y., Chen, L.S.: Modelling and analysis of a single-species system with stage structure and harvesting. Mathematical and Computer Modelling 36, 67–82 (2002) 2. Gao, S.J., Chen, L.S., Sun, L.H.: Optimal pulse fishing policy in stage-structured models with birth pulses. Chaos, Solitons and Fractals 25, 1209–1219 (2005) 3. Claire, W.A., Anders, S.: Marine reserves: a bio-economic model with asymmetric density dependent migration. Ecological Economics 57, 466–476 (2006) 4. James, N.S., James, E.W.: Optimal spatial management of renewable resources: matching policy scope to ecosystem scale. Journal of Environmental Economics and Management 50, 23–46 (2005) 5. Jerry, M., Raissi, N.: The optimal strategy for a bioeconomical model of a harvesting renewable resource problem. Mathematical and Computer Modelling 36, 1293–1306 (2002) 6. Luis, H.R.A., Larry, A.S.: Optimal harvesting of stochastically fluctuating populations. Journal of Mathematical Biology 37, 155–177 (1998) 7. Tapan, K.K.: Influence of environmental noises on the Gompertz model of two species fishery. Ecological Modelling 173, 283–293 (2004) 8. Zhang, X.A., Chen, L.S., Neumann, A.U.: The stage-structured predatorprey model and optimal harvesting policy. Mathematical Biosciences 168, 201–210 (2000)

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9. Nisbet, R.M., Gurney, W.S.C.: Modelling Fluctuating Populations. Wiley Interscience, New York (1982) 10. Bandyopadhyay, M., Banerjee, S.: A stage-structured prey-predator model with discrete time delay. Applied Mathematics and Computation 182, 1385–1398 (2006) 11. Saha, T., Bandyopadhyay, M.: Dynamical analysis of a delayed ratio-dependent prey-predator model within fluctuating environment. Applied Mathematics and Computation 196, 458–478 (2008) 12. Bandyopadhyay, M., Saha, T., Pal, R.: Deterministic and stochastic analysis of a delayed allelopathic phytoplankton model within fluctuating environment. Nonlinear Analysis: Hybrid Systems 2, 958–970 (2008) 13. Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Application of Hopfbifurcation. Cambridge University Press, Cambridge (1981) 14. Cooke, K.L., Grossman, Z.: Discrete delay, distributed delay and stability switches. Journal of Mathematical Analysis and Applications 86, 592–627 (1982)

A Feature Points Matching Method Based on Window Unique Property of Pseudo-Random Coded Image Hui Chen, Shiwei Ma*, Hao Zhang, Zhonghua Hao, and Junfeng Qian School of Mechatronic Engineering & Automation, Shanghai University, Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200072, China [email protected]

Abstract. Proposed a method for object feature points matching used in active machine vision technique. By projecting pseudo-random coded structured light onto the object to add code information, its feature points can be easily identified exclusively taking advantage of the window unique property of pseudo-random array. Then, the 3D coordinates of object both on camera plane and code plane can be obtained by decoding process, which will provide the foundation for further 3D reconstruction. Result of simulation shows that this method is easy to operate, calculation simple and of high matching precision for object feature points matching. Keywords: Machine Vision, Coded Structured Light, Feature Points Matching, Window Unique Property.

1 Introduction Research on mechatronics has made great progress over recent years. In particular, industrial robots and automatic machines equipped with machine vision have played an important role in our society and have contributed significantly in making it productive and efficient[1]. In this field, automatic 3D reconstruction of object surface is becoming increasingly important commercially, with applications in fields such as rapid prototyping, teleshopping, virtual reality, robot navigation and manufacturing. The existing 3D measurement techniques may be broadly divided into passive vision method and active ones. Stereovision is one of the most widely known passive vision methods, in which the 3D coordinates of an object point can be obtained by triangulation from two cameras. However, the approach to determine the position of same point on the surface of object within two images is quite difficult. In the active vision method, it should be noticed that, for each object point, its projection on the optical sensor image planes should be known first. In fact, in order to obtain the coordinates of a given 3D point, the given projections have to be recovered from the same object point. This is known as correspondence problem and can be considerably alleviated by active vision method such as structured light. Active vision *

Corresponding author.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 333–341, 2010. © Springer-Verlag Berlin Heidelberg 2010

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technique often uses scanning structured light methods to obtain object surface information. It has a good performance and robust, and can accurately return densely sampled range data without much scene texture. And it is computationally efficient and inexpensive especially considering the declining prices of computer-controlled projectors and cameras[2]. Among structured light methods, coded structured light method was developed from multi-line structured light method. By using coded structured light, it can realize precise location of light line and light point only by fewer steps. It firstly encodes structured light, then takes advantage of different code strategy which will give every pixel priori code information, finally the code information is decoded to retrieve object’s surface information[3]. However, in coded structured light method, the problem of how to accurately match each feature point within two or more frames needs to be resolved. In recent years, there develops a coded structured light method which uses a projector to project pseudo-random coded pattern image onto the surface of object[5,7]. This method has merits such as low cost, high measurement matching accuracy, free of scanning device with high accuracy, easy to operate, and easy to change encoding image so as to adapt different kinds of code model. Since pseudo-random code has socalled window unique property[4], one can employ it to characterize 3D scene space surface which covered by a frame of image. As a result, all feature points of the space surface can be identified exclusively and matched precisely in two 3D coordinate systems, which are camera plane and code plane respectively, through pseudo-random code projected method. In our work, in order to realize accurate feature matching in 3D reconstruction, the character of this method is investigated, and the related theory study and simulated experiment results are given in this paper.

2 The Window Unique Property of Pseudo-Random Coded Image If a sequence can be predetermined and duplicated, meanwhile it has random property in statistics, this sequence is named pseudo-random sequence. For a given sequence a0 a1a2 = 000100110101111 , suppose that a consecutive sub-sequence with length 2 m − 1 is a pseudo-random sequence, and a window of width m is moving along the sequence. Hence, each of the sub-sequence will be different to others in a same cycle in the view of the moving window. This is the well-behaved window property of pseudo-random sequence[4]. However, in practical application of pseudorandom coded structured light 3D active vision technique, one should use pseudorandom array (image) other than pseudo-random sequence to build the coded template. 1 2 Given that the cycle of a pseudo-random ( a0 a1a2 an −1 ) is n = 2 k ∗k − 1 , and m = k1 × k 2 , where k1 , k 2 is the window parameters of pseudo-random array. Hence, the size of pseudo-random array is n1 × n2 , where

n1 = 2 k1 − 1

(1)

n2 = n / n1

(2)

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The process of converting pseudo-random sequence into pseudo-random array is shown as an example in Fig.1. The first step is to build a table with size n1 × n2 , and fill out its left-top corner with the first value of pseudo-random sequence. The second step is to fill out the table one by one along its main diagonal line, and then jump to the left boundary of next row or the top boundary of next column when encountering the boundary of table. The previous two steps should be done repeatedly until all the values are filled in the table. As a result, the k1 × k 2 coded window in the left-top corner of pseudo-random array can determine the whole pseudo-random array.

0

1 c

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d

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1 1

e 0

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1 d

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Fig. 1. Sketch of converting pseudo-random sequence into pseudo-random array

The other very important property of the pseudo-random array is that, a sub-array

k1 × k 2 in the window can be identified uniquely when a coded window with size k1 × k 2 is moving in an array with size n1 × n2 . As what is known for any pseudorandom array, the position of a sub-array in the whole array can be determined as long as the sub-array of a window is known.

3 Method of Feature Points Extraction The accuracy of extraction for feature points will profoundly influence their matching in 3D reconstruction. Among existing methods of feature point extraction, the Harris operator[6] has advantages of easy calculation, high operation speed and precision. According to the method of multicolor corner detection based on Harris operator[7], first let the auto-correlation matrix M at each pixel point ( x, y ) as,

⎛A C⎞ ⎟⎟ M = G (σ ) ⊗ ⎜⎜ ⎝C D⎠

(3)

where G (σ ) is a Gauss template, and A = R x2 + G x2 + B x2 C = R x R y + G x G y + Bx B y D=

R y2

+ G y2

+

(4)

B y2

where R, G and B represent three-channel component figures of the multicolor image respectively, and there has

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R x = ∂R / ∂x, G x = ∂G / ∂x, B x = ∂B / ∂x R y = ∂R / ∂y , G y = ∂G / ∂y , B y = ∂B / ∂y

(5)

R ( x, y ) = det( M ) − k (trace( M )) 2 where k is constant. Based on the above equations, a two-dimension array which saved the measurement function value in each image can be gotten readily. When the valve of which is greater than the threshold setting beforehand, the candidate points which are lower than the threshold will be removed, and the image points will be considered as the feature points (corner points). In order to obtain higher precision, the sub-pixel location of corner is implemented by using quadratic curve fitting to pixel gray level around the feature point and solving its extreme value as[8]

R( x, y ) = ax 2 + bxy + cy 2 + dx + ey + f

(6)

where ( a, b, c, d , e, f ) are unknown coefficients. Extreme value point of the fitting regions should satisfy the following equations:

⎧ ∂R ( x, y ) = 2ax + by + d = 0 ⎪⎪ ∂x ⎨ ∂R ( x, y ) ⎪ = bx + 2cy + e = 0 ⎪⎩ ∂y

(7)

By solving this equation, the offset relative to the integer pixel point ( x0 , y0 ) can be easily obtained.

4 Method of Feature Points Matching During the process of reconstructing from 2D to 3D, the primary challenge is to obtain the correspondence between 2D coordinates of feature points in pixel image and those in code projected template. However, when the feature points coordinates on camera plane is ready, the correspondence problem will reduce to the problem of calculating the feature points’ coordinates on projection plane. The process of feature matching is shown in Fig.2. Defining the coordinates of point as P( x p , y p ) in projector plane, by using the window unique property of pseudo-random coded image, the position of sub-array in window will correspond to the whole coded array. Every feature points on 3D scene surface can be identified uniquely. Consequently, in order to get feature points’ coordinates on projection plane, one need only to calculate the sub-array of feature points in detecting window. In order to determine the position of detecting window in case of its sub-array was known, a table of number index should be built firstly, based on the color pseudorandom coded pattern related to sub-array and position in the window. The method is to convert different color squires in pseudo-random coded image into different

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Fig. 2. The process of feature points matching

numbers. For example, 0 stands for red, 1 green, 2 blue and 3 yellow. Then, the coordinates P( x p , y p ) can be simply obtained by retrieving the position of sub-array in this number index table. The reflection of projected pattern from scene can be detected as the corner coordinates, by using some better Harris operators. Since the coordinates P( xc , y c ) in camera plane are known according to the equation (7), the objective will become the work to solve all the correspondence ϕ between the transition camera coordinates and the projector coordinates, where

⎧⎪⎛ X c1 , Yc1 ⎞ ⎛ X c 2 , Yc 2 ⎞ ⎛ X c 3 , Yc 3 ⎞

⎛ X ci , Yci ⎞⎫⎪

⎟，⎜ ⎟，⎜ ⎟，… ⎜ ⎟⎬ ϕ = ⎨⎜⎜ ⎪⎩⎝ X p1 , Y p1 ⎟⎠ ⎜⎝ X p 2 , Y p 2 ⎟⎠ ⎜⎝ X p 3 , Y p3 ⎟⎠ ⎜⎝ X pi , Y pi ⎟⎠⎪⎭

(8)

As a result, the process of feature points matching based on window unique property of pseudo-random coded image will be equivalent to a path in 2D grid, as illustrated in Fig.2(b). The grid’s horizontal axis and vertical axis represent the feature points’ coordinates in different plane.

5 Simulation Results and Analysis According to the above theory, we have developed an experimental system for testing the algorithm by using 3D human face model as a target. The camera used in this system is a CCD (ARTRAY,ARTCAM-150PШ), whose effective pixel is 1392(H)×1040(V) and the size of effective pixel is 7.60mm×6.20mm. A Seiko lens (TAMRON 53513) with focus 16.0mm is used. And, a LCD projector (HITACHI -HCP600X.) with a resolution of 1024*768 pixels is used. In order for a more comparable emulation result to human face model in experiment, we design different pseudo-random coded pattern. The discrete square pseudorandom coded image is shown in Fig.3a, and the circle coded image in Fig.3b.

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A2

A2

A2

A2

A2

A2

Fig. 3a. Discrete square color code model

Fig. 3b. Circle color code model

Fig. 4. Feature points matching image of human face model projected by square coded window

Under natural daylight source, the discrete square and circle color coded model image are projected to a human face model, and reflected image information is acquired. The feature points matching experiment image of human face model projected by square coded window, as shown in Fig.4. The feature points’ coordinates on the projection plane can be obtained by the multi-color corner detection method. Then an index table is built up based on pseudo-random coded model, which contains window sub-array and window position. If the sub-array in detected window is known, we can attain the position where the detection window is by checking index table.

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At the same time, the coordinates of the six feature points in this window are confirmed. The matching results, as shown in table 1. The coordinate valves as shown in the table are the coordinates of the four corners in the first small square at the upper left of the window on camera plane and projector plane. They are correspondence one to one just as the data can be seen. Using the same window matching method, all other feature points matching can be finished in the image. If the pseudo-random coded detection window is bigger, the difficulty of image processing will increase. But if the window is smaller, the detection range of the image will decrease. In our experiment, the size of window is 3× 2 , which is chosen as it is the most suitable one by testing different code scheme for many times. Projecting pseudo-random circle coded pattern to test, the matching results are shown in Fig.5 and table 2. Table 1. Feature points matching results projected by discrete square coded Plane

Camera plane coordinate Projector plane coordinate

Coordinate1

Coordinate2

Coordinate3

Coordinate4

（18.400183, （16.297035, （36.427584, （35.773310, 20.501544） 88.681428） 70.004595） 19.477080） （180, 140） （180, 150） （190, 150） （190, 140）

Fig. 5. Feature points matching image of human face model projected by circle coded window

Finally, the 3D human model reconstruction is carried out by using the above experiment results. One of a reconstruction results is shown in Fig.7. It can be found that, the 3D human face shape can be roughly restored. However, since the features on real human surface are remarkable, not all of the surface information can be recovered by projecting circle coded image onto it. But, in the case of the discrete square coded image, the surface information can be well recovered, and the whole feature points’ information can be obtained.

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Plane Camera plane coordinate Projector plane coordinate

Coordinate 1 (15.721415, 19.255567)

Coordinate 2 (27.436432, 50.326923)

(160,70)

(160,80)

Coordinate 3 (21.072250, 87.435316)

Coordinate 4 (55.968193, 83.488125)

Coordinate 5 (51.748094, 49.541112)

Coordinate 6 (50.746084, 17.386764)

(160

(170

(170

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90)

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80)

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Fig. 6. The 3D reconstruction result of human face model

6 Conclusion By taking use of the window unique property of pseudo-random coded image, the problem of feature points matching during 3D reconstruction of coded structured light machine vision can be resolved. The research results indicate that different coded pattern projecting to object surface has different effect, which will influence the matching precision. By adopting square code projection mode, the precision is higher, but the processing program is more complicated than that of circle code mode. By adopting circle code projection mode, though the processing program is simple, but it is difficult to locate the round centre of object’s surface. Acknowledgments. This work was supported by Shanghai Science and Technology Research Foundation No.09dz2201200, No.09dz2273400, 08DZ2272400

References 1. Ejiri, M.: Robotics and machine vision for the future-an industrial view. In: Proceedings of IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Como, vol. 2, pp. 917–922 (2001) 2. Liao, J.R., Cai, L.L.: A Calibration Method for Uncoupling Projector and Camera of a Structured Light System. In: IEEE/ASME International Conference on Advanced Intelligent Mechatronics, AIM 2008, pp. 770–774 (2008)

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3. Kong, L.F., Chen, S.P., Guo, F.T.: A Correspondence Algorithm Based on Coded Structured Light for 3D Reconstruction. In: Computer Application Federation of China Instrument and Control Society, CACIS, Zhejiang, pp. 91–95 (2007) 4. MacWilliams, F.J., Sloane, N.J.A.: Pseudo random sequences and arrays. Proceedings of the IEEE 64, 1715–1729 (1976) 5. Zhang, Y.B., Lu, R.S., Fei, Y.T.: Multicolor pseudo-random array pattern projector and its calibration. Journal of Harbin Institute of Technology 36, 59–68 (2004) 6. Harris, C., Stephens, M.: A Combined Corner and Edge Detector. In: Proceeding of 4th Alvey Vision Conference, Manchester, pp. 147–151 (1988) 7. Chen, H., Ma, S.W., Xu, Z.Y.: A novel method of corner detection for multicolor pseudorandom encoded image. In: 8th International Conference on Electronic Measurement and Instruments, ICEMI, Xi’an, pp. 2894–2897 (2007) 8. Liang, Z.M.: Sub-pixels corner detection for camera calibration. J. Transaction of the China Welding Institution. 27, 102–105 (2006) 9. Zhou, J., Chen, F.I., Gu, J.W.: A Novel Algorithm for Detecting Singular Points from Fingerprint Images. IEEE Transactions on Pattern Analysis and Machine Intelligence 31, 1239–1250 (2009) 10. Zhang, L., Curless, B., Seitz, S.M.: Rapid shape acquisition using color structured light and multi-pass dynamic programming. In: Proceedings of First International Symposium on 3D Data Processing Visualization and Transmission, pp. 24–36 (2002) 11. Olaf, H.H., Szymon, R.: Stripe boundary codes for real-time structured-light range scanning of moving objects. In: Proceedings of Eighth IEEE International Conference on Computer Vision, ICCV 2001, vol. 2, pp. 359–366 (2001)

A Reconstruction Method for Electrical Impedance Tomography Using Particle Swarm Optimization Min-you Chen, Gang Hu, Wei He, Yan-li Yang, and Jin-qian Zhai State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, Chongqing 400044, China

Abstract. The inverse problem of Electrical Impedance Tomography (EIT), especially for open EIT which involves less measurement, is a non-linear ill-posed problem. In this paper, a novel method based on Particle Swarm Optimization (PSO) is proposed to solve the open EIT inverse problem. This method combines a modified Newton–Raphson algorithm, a conductivity-based clustering algorithm, with an adaptive PSO algorithm to enhance optimal search capability and improve the quality of the reconstructed image. The results of numerical simulations show that the proposed method has a faster convergence to optimal solution and higher spatial resolution on a reconstructed image than a Newton–Raphson type algorithm. Keywords: Open Electrical Impedance Tomography, Reconstruction Algorithm, Clustering, Particle Swarm Optimization.

1 Introduction The electrical impedance tomography (EIT) problem is the inverse problem of determining the spatially varying electrical conductivity of an inaccessible region through currents injected into the region and the resulting voltages measured on the surface. As the conductivity of biological tissue contain a large amount of information that reflects the physiological state and function of tissues and organs, EIT uses this electrical characteristic information to conduct noninvasive imaging. As a functional imaging modality, EIT is very suitable for early detection, diagnosis, prediction and evaluation after healing for nerve diseases and malignant tumors [1], due to its low cost and noninvasive properties. As we know, the inverse problem of EIT is a non-linear ill-posed problem. While numerous ad hoc reconstruction methods have been tried for EIT, the commonly used approach is using one of a family of regularized Newton-type methods, the Newton's One-Step Error Reconstructor (NOSER) algorithm being a well known example [2]. With a pre-computed gradient matrix, the NOSER can produce a reconstruction image quickly. However, a linearized solution will only be accurate when the true conductivity is close to the initial estimate; also, the process is very sensitive to modeling and measurement errors. From a mathematical point of view, the EIT inverse problem consists of finding the coordinates of a point in m-dimensional hyperspace, where m is the number of K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 342–350, 2010. © Springer-Verlag Berlin Heidelberg 2010

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discrete elements whose union constitutes the imaging area. Recently, some researchers have put forward some imaging methods based on using search algorithms, such as GA [3] and PSO [4]. These search algorithms appear to be of value in EIT reconstruction since no assumption on function continuity needs to be made. Usually the search algorithm can improve spatial resolution at the cost of expensive computing time and resources. After the one-step iteration of the NOSER algorithm, absolute values of meshes cannot be determined but some useful information on the objective images can be obtained. In this paper, a clustering-based PSO reconstruction method for OEIT was proposed, which used the result of NOSER to initialize the conductivity distribution and reduce the dimension of problem space by clustering the meshes with a similarity factor. With the exploitation of prior knowledge, convergence of the algorithm was accelerated and the quality of imaging was improved.

2 The Mathematical Model of OEIT 2.1 The Principle of Open Electrical Impedance Tomography To solve the problems of the traditional closed EIT [5], an open EIT (OEIT) model using a fixed electrode array is proposed. By increasing information available from the local area, one can enhance the imaging precision and the quality of the reconstructed image. The OEIT model has the potential to overcome the problems of closed EIT in clinical application, such as the inconvenience of application and inaccurate positioning of the electrodes. A 2-dimensional OEIT model is shown in Fig.1. It is composed of the open field, Ω, and the open boundary, ∂Ω. It is not necessary to solve the boundary problem of the whole open field. On the one hand, the imaging area is the local area beneath the electrodes, so an appropriate area should be chosen as the research area. On the other hand, the injected currents are mainly distributed around the shallow area beneath the surface of the object, the further away from the electrodes, the lower the current density. So a virtual boundary, ∂Ω2, composed of BC, CD and DA as shown in Fig.1 can be defined. The field, which is surrounded by a virtual boundary, can be treated as the open EIT field model and the open EIT problem transformed into an approximate closed EIT problem [6].

Fig. 1. The OEIT model

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Fig. 2. The FE model for OEIT

2.2 The FE Model of OEIT In this paper, the FE model for OEIT is selected empirically [6], as shown in Fig.2, which has 864 elements and 463 nodes. The eight evenly spaced electrodes array put on the top boundary. The parameters H and W are related to the width (D) of the electrode arrays: H=D, W=1.3D. With NOSER, the simulation results showed that better resolution and positioning accuracy could be obtained when one object was at high sensitivity region (the region within the broken line in Fig.2) below the surface. 2.3 The Mathematical Model of the Forward Problem Many models have been proposed in order to study the boundary problem of electrical impedance tomography. In this paper, a simple model is used to quickly solve the forward problem at the cost of losing some information about the measurement. With this model, the effect of the contact impedance is ignored by treating the voltages as zero on injection electrodes. The potential distribution function, ϕ, and conductivity distribution function, σ, satisfy the Laplace equation:

∇[σ ( x, y ) ⋅ ∇ϕ ( x, y )] = 0 ( x, y ) ∈ Ω

(1)

The boundary conditions are:

σ ( x, y ) and

∂ϕ ( x, y ) = j ( x, y ) ( x, y ) ∈ ∂Ω1 , ∂n

ϕ ( x , y ) = f ( x, y )

( x, y ) ∈ ∂Ω 2 .

(2)

(3)

Equations (2) and (3) are Neumann boundary conditions for Equation (1), where f is the measuring voltage on the boundary, j is the current density on the boundary, ∂Ω1 and ∂Ω2 are the boundaries of Ω and n is the unit outward normal vector to the boundary surface. Computation of the potential ϕ(x, y) for the given conductivity distribution σ(x, y) and the boundary conditions is called the forward problem. The numerical solution for the forward problem can be obtained using the finite element method (FEM). In this paper, the FE model, shown in Fig.2, is used to solve the forward problem.

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2.4 The Inverse Problem The inverse problem, also known as the image reconstruction, consists of reconstructing the conductivity distribution, σ(x,y) from potential measured on the boundary of the object. Mathematically, the EIT reconstruction problem is a nonlinear ill-posed inverse problem. Many different approaches to solve the reconstruction problem have been proposed. One of the most common is the minimization of the squared norm of the difference between the measured boundary voltages and the calculated boundary voltages, which is usually presented as minimizing the objective function, E(σ) , with respect to σ, where N

N

E (σ ) = ∑∑ (Vij − U ij (σ ))2 ,

(4)

i =1 j =1

where Vij are the measured boundary voltages and Uij are the calculated boundary voltages with respect to conductivity σ. The EIT reconstruction problem can be treated as finding the optimal conductivity ïdistribution, σ, such that the error function, E, is minimum. In this study, a clustering-based PSO algorithm is used to solve EIT reconstruction problem.

3 Image Reconstruction Using Clustering-Based PSO 3.1 PSO Algorithm The particle swarm optimization (PSO) is a relatively new intelligence optimization method proposed by Kennedy and Eberhart [7], which was inspired by the choreographic behavior of flocks. The PSO algorithm has been found to be successful in a wide variety of optimization tasks [8]-[10]. In this paper, the modified PSO algorithm suggested in [10] is introduced to solve the EIT inverse problem. The modified particle swarm algorithm is described below: The particle’s position is modified according to the following equations:

vi (t + 1) = wvi (t ) + α [r1 ( pi − xi (t )) + r2 ( p g − xi (t ))] + vm (t )

(5)

xi (t + 1) = vi (t + 1) + xi (t ) .

(6)

The second term in Equation (5) can be viewed as an acceleration term, which depends on the distances between the current position, xi, the personal best, pi, and the global best, pg. The acceleration factor, α, is defined as follows:

α = α 0 + t / Nt ,

t = 1,2,..., N t ,

(7)

where Nt denotes the number of iterations, t represents the current generation, and the suggested range for α0 is [0.5, 1].

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Furthermore, instead of using a linearly-decreasing inertia weight, a random number is used to improve the performance of the PSO in some benchmark functions. The inertia weight at every generation is changed via the following formula:

w = w0 + r ( w1 − w0 )

(8)

where w0∈[0, 1], w1> w0 are positive constants, and r is a random number uniformly distributed in [0, 1]. The suggested range for w0 is [0, 0.5], which makes the weight, w, randomly varying between w0 and w1. In this way, a uniformly distributed random weight combination can be obtained at every iteration. The third term vm(t) in Equation (5) is a mutation operator, which is set proportionally to the maximum allowable velocity, vmax. When the historic optimal position, pi of the particle swarm is not improving with increasing generations, it indicates that the whole swarm being trapped in a local optimum from which it becomes impossible to escape. Then, a particle is selected randomly and then a random perturbation (mutation step size) is added to a randomly selected modulus of the velocity vector of that particle by a mutation probability. The mutation term is described as follows:

vm = sign ( 2 rand − 1) ⋅ β ⋅ vmax ,

(9)

where β ∈ [0, 1] is a constant, rand is a random number uniformly distributed in [0,

⎧ 1 if x ≥ 0 , which is used to ⎩− 1 if x ≺ 0

1], and the sign function is defined as sign( x) = ⎨ decide the particle’s moving direction.

3.2 Image Reconstruction Using Clustering Based PSO The EIT inverse problem is usually presented as minimizing the objective function relative to some sort of residual between the measured and expected potential values. The EIT inverse problem searches for the parameters in a high-dimensional space which is determined by the elements of the FE model. In the implementation of PSO for the EIT inverse problem, the position vector, x, represents m-dimensional conductivity values, where m is the number of elements. The ith member of a population’s position is represented as xi=[ xi1, xi2, …, xim]=[σi1, σi2,…, σim], where σi1 is the conductivity value of the 1st element and so on. In EIT imaging, it normally requires a large number of elements in the FE model (e.g. m=864 in our FE model) to obtain satisfactory resolution in the imaging region. In such a high-dimensional space, it is very difficult for the PSO algorithm to search for optimal solutions. In real life, the existence of biological tissue is always classified. Similar tissues have similar electrical properties. If all the elements can be reasonably classified into several clusters under certain similarity measures without losing key information, it will reduce the dimensions of the problem space significantly and improve the convergence of the PSO algorithm. To increase the feasibility of the PSO in EIT reconstruction, we propose a similarity measure to control the clustering, which includes two factors, one is concerned with the conductivity value and the other is concerned

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with the adjacency of space location of elements. These two factors can be treated as two distance measures, represented by VD and SD, defined as follows:

VD = σ ei − σ cj ,

(10)

SD = ( xei − xcj ) 2 + ( yei − ycj ) 2 ,

(11)

where σei is the conductivity of the ith element, σcj is the conductivity of the prototype of the jth cluster, xei and yei are the central coordinates of the ith element, xcj and ycj are the coordinates of the central point of the jth cluster. The conductivity of a cluster is defined as follows:

σ cj = ∑ σ i / G j ,

(12)

where σi is the conductivity value of an element belonging to cluster j, and Gj is the number of elements belonging to cluster j. The central coordinates of a cluster are defined as follows:

xcj = ∑ xi / G j ; ycj = ∑ yi / G j ,

(13)

where xi and yi are the central coordinates of an element belonging to cluster j and Gj is the number of elements in cluster j. Using the similarity measures VD and SD, a self-organizing network based clustering algorithm [11] is introduced to generate the element clusters. With different thresholds for VD and SD, different partitions of element clusters will be produced. After the clustering process, the dimensions of the solution space of the PSO algorithm can be greatly reduced from m to N, where N is the number of clusters which is much smaller than the number of elements m. Thus, the population of the PSO can be generated in an N-dimensional space, the position of an individual in the population is represented as xi=[ xi1, xi2, …, xiN]=[σi1, σi2,…, σiN] , where σi denotes the conductivity distribution. Instead of random initialization, one particle in the initial population is initialized by the conductivity distribution [σi1, σi2,…, σiN], which are calculated through the NOSER algorithm. This clustering-based PSO algorithm is used for EIT image reconstruction. The image reconstruction process can be described as following steps: Step 1. Set the threshold of VD and SD; set the number of clusters equal to one. Step 2. Use a self-organizing network to generate element clusters using the given VD, SD values. Step 3. Initialize the first particle using the conductivity values calculated through the NOSER algorithm and the others initialized randomly. Step 4. Search for an optimal solution (conductivity distribution) using the PSO algorithm presented in Equations (5) and (6). Step 5. Store the optimal solution as the final conductivity distribution.

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3.3 Simulation Study To test the effectiveness of the proposed reconstruction method, a simulation experiment for EIT image reconstruction was conducted. Background conductivity is set to 6 S/m and the target conductivity to 30 S/m in elements 316, 317, 318 and 319, shown in Fig.3. This is simulates the insertion of a high conductivity material into a low conductivity background material. The aim of the simulation is to identify the location of the inserted high conductivity material using the clustering-based PSO method. The measurement method of EIT is neighboring method. This involves using the 64 voltages which are calculated by the FEM and adding some disturbing errors (the signal-to-noise ratio was 30DB) to locate the target (illustrated in black in Fig.3). To illustrate the effect of the clustering algorithm on the convergence of the PSO, different clustering results produced for different threshold values of VD and SD were used for comparison. First, we let threshold of VD=0.01 and SD=0.3. All the elements shown in Fig.4 were clustered into 91 clusters using the self-organizing network. Then we set VD=0.05 and SD=0.4. After self-organizing clustering, 22 clusters were generated. The convergence performance of the PSO algorithm with different clustering results is shown in Fig.4. It can be seen that the speed of convergence of the PSO with the clustering process (the line with dot markers and the line with diamond markers) is faster than the PSO algorithm without clustering (the line with triangle markers). But without sufficient clusters, it is easy for the algorithm to be trapped in a local optimum (see the line with diamond markers for iterations > 6). It is also shown that the PSO with 91 clusters (the line with dot markers) achieved a better solution than the PSO with 22 clusters (the line with diamond markers), in which some useful information might be lost. It is therefore noted that the selection of the threshold for VD and SD is important to the performance of the PSO algorithm.

Fig. 3. The FE model used in computer simulation

Fig. 4. The convergent performance of different threshold for VD and SD

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(c)

Fig. 5. (a) The reconstruction image using NOSER. (b) The reconstruction image using PSO (c) The reconstruction image using clustering-based PSO.

The comparative study on image reconstruction using NOSER only, PSO only and the proposed clustering based PSO with NOSER initialization was conducted, and shown in Fig.5 (a), (b) and (c), respectively. In these figures, the conductivity value was represented by brightness, i.e. the objective with higher conductivity corresponds to brighter region. The ideal result should be that a bright region appears at the position where the high conductivity material was inserted (the black region in Fig.3) and the other region is dark. From figure 5(b) we cannot confirm the position of the inserted objective with high conductivity, while the image reconstruction result shown in figure 5(c) is better than figure 5(a) and (b), in which the bright region is closer to the position of the inserted objective. In order to test the repeatability of the proposed method, the image reconstruction has been run 10 times independently with random initial conditions and adding randomly distributed noise to the calculated voltages. From Fig.5, it is clearly shown that the clustering-based PSO can improve the reconstruction image on the accuracy of solution compared with NOSER, and produce a better reconstruction image within limited iterations (100 times) compared with the PSO algorithm alone.

4 Conclusions and Discussion In this paper, an EIT image reconstruction method based on clustering-PSO using information provided by NOSER was developed for the solution of the EIT inverse problem. Although PSO is presently unsuitable for real-time tomographic applications, the exploitation of prior knowledge has the potential to produce better reconstructions. The proposed clustering-based PSO-EIT method is able to finding optimal solution of reconstruction that can improve imaging resolution. Further research should be done to determine the optimal partition of the clusters. The number of clusters is dependent on the similarity measures of VD and SD. In a different imaging situation, the number of clusters would be different. The method of selecting the most suitable values for VD and SD is still needed.

Acknowledgements The authors gratefully acknowledge the support from National “111” Project of China (B08036), and thank Professor P. J. Fleming (University of Sheffield, UK) for his valuable help on paper presentation.

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References 1. Brown, B.H.: Electrical impedance tomography. Journal of Medical Engineering & Technology 27(3), 97–108 (2003) 2. Lionheart, W.R.B.: EIT reconstruction algorithms: pitfalls, challenges and recent developments. Physiol. Meas 25, 125–142 (2004) 3. Olmi, R., Bini, M., Priori, S.: A Genetic Algorithm Approach to Image Reconstruction in Electrical Impedance Tomography. IEEE Transactions on Evolutionary Computation 4(1), 83–88 (2000) 4. Ijaz, U.Z., Khambampati, A.K., Kim, M.C., et al.: Particle swarm optimization technique for elliptic region boundary estimation in electrical impedance tomography. In: AIP Conf. Proc., vol. 914, pp. 896–901 (2007) 5. He, C.H.H., He, W., Huang, S., Xu, Z.H.: The research of the Open EIT theory, simulation and early experiment. In: Advances in Chinese Biomedical Engineering, vol. 1654 (2007) 6. Chen, M.Y., Zhang, X.J., Luo, C.Y., He, W.: Modeling and Simulation Based on Open Electrical Impedance Tomography. Journal of Chongqing University 32(7), 731–735 (2009) 7. Kennedy, J., Eberhart, R.: Particle Swarm Optimization. Neural Networks. In: Proc. IEEE Inter. Conf. on Neural Networks, Perth, pp. 1942–1948 (1995) 8. Zhang, L.H., Hu, S.: A New Approach to Improve Particle Swarm Optimization. In: Cantú-Paz, E., Foster, J.A., Deb, K., Davis, L., Roy, R., O’Reilly, U.-M., Beyer, H.-G., Kendall, G., Wilson, S.W., Harman, M., Wegener, J., Dasgupta, D., Potter, M.A., Schultz, A., Dowsland, K.A., Jonoska, N., Miller, J., Standish, R.K. (eds.) GECCO 2003. LNCS, vol. 2723, pp. 134–139. Springer, Heidelberg (2003) 9. Mahfouf, M., Chen, M.Y., Linkens, D.A.: Adaptive Weighted Particle Swarm Optimisation for Multi-objective Optimal Design of Alloy Steels. In: Yao, X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervós, J.J., Bullinaria, J.A., Rowe, J.E., Tiňo, P., Kabán, A., Schwefel, H.-P. (eds.) PPSN 2004. LNCS, vol. 3242, pp. 762–771. Springer, Heidelberg (2004) 10. Chen, M.Y., Wu, C.S., Fleming, P.J.: An evolutionary particle swarm algorithm for multiobjective optimization. In: Processing of the 7th World Congress on Intelligent Control and Automation, pp. 3269–3274. IEEE Press, Los Alamitos (2008) 11. Linkens, D.A., Chen, M.Y.: Hierarchical Fuzzy Clustering Based on Self-organising Networks. In: Proceedings of World Congress on Computational Intelligence (WCCI 1998), vol. 2, pp. 1406–1410. IEEE, Piscataway (1998)

VLSI Implementation of Sub-pixel Interpolator for AVS Encoder Chen Guanghua1, Wang Anqi1, Hu Dengji1, Ma Shiwei1, and Zeng Weimin2 1

School of Mechatronics Engineering and Automation, Shanghai Key Laboratory of Power Station Automation Technology, Shanghai University, Shanghai 200072, P. R. China 2 Key Laboratory of Advanced Display and System Applications, Ministry of Education & Microelectronic Research and Development Center, Shanghai University, Shanghai 200072, P. R. China [email protected]

Abstract. Interpolation is the main bottleneck in AVS real-time high definition video encoder for its high memory bandwidth and large calculation complexity caused by the new coding features of variable block size and 4-tap filter. In this paper, a high performance VLSI architecture of interpolation supporting AVS Baseline@L4 is presented. Vertical redundant data reuse, horizontal redundant data reuse and sub-pixel data reuse schemes are presented to reduce memory bandwidth and processing cycle. The separated 1-D interpolation filters are used to improve throughput and hardware utilization. The proposed design is implemented on FPGA with operating frequency of 150MHz and can support 1080p (1920×1080)/30fps AVS real-time encoder. It is a useful intellectual property design for real-time high definition video application. Keywords: AVS, interpolation, data reuse, separated 1-D architecture.

1 Introduction AVS[1] standard is the Chinese video compression standard. Being adopted many new coding features and functionalities, AVS achieves more than 50% coding gains over MPEG-2[2] and similar performance with lower cost compared with H .264[3]. In AVS real-time high definition video encoder, the difficulties of interpolation hardware design arise from variable block-sizes (VBS) and quarter-pixel precision. VBS provides more accurate prediction compared with traditional fixed block-size. A macro block (MB) with VBS can be divided into four kinds of block modes including 16×16, 16×8, 8×16 and 8×8. As a result, a larger amount of frame memory accesses are required due to the smaller block size (e.g. 8×8), and the hardware implementation becomes difficult. The quarter-pixel precision interpolation improves the coding efficiency over a half-pixel precision case, as the precision of motion vector (MV) is one of the important factors that affect prediction accuracy. However, it increases memory bandwidth and calculation complexity considerably for its 4-tap filter. So the interpolation unit becomes one of the most data intensive parts of the AVS encoder especially in high definition video application. The design of interpolation hardware K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 351–359, 2010. © Springer-Verlag Berlin Heidelberg 2010

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architecture for high definition video application encoder is a tough work. Recently, some interpolation architectures have been proposed[5]-[7]. However, these architectures based on former standards usually do not suit for the block sizes and 4-tap filter specified in the AVS standard. The reduction of memory access and calculation time is two important issues in our design. In this paper, three data reuse techniques are presented for the reduction of memory bandwidth and computation complexity. And we proposed a novel VLSI architecture based on separate 1-D approach to improve throughput and hardware utilization. Using these techniques above, we successfully design an interpolation hardware circuit. This paper is organized as follows. The interpolation algorithm applied in AVS is described in Section 2. Section 3 explains redundancy reduction approach. Section 4 presents the details of the proposed design. The last section concludes the paper.

2 Algorithm Description In AVS, a separate 2-D interpolation method named two steps four taps interpolation (TSFT)[8] is adopted. The position of integer pixels and sub pixels is illustrated in Fig.1. According to TSFT, the luminance prediction values at half-pixel locations b, h are interpolated by applying a 4-tap filter with tap values (-1/8, 5/8, 5/8, -1/8), and j is interpolated by applying the same 4-tap filter on the values at half-pixel location b or h. The prediction values at quarter-pixel locations a, c, i, k, d, n, f and q are achieved by applying another 4-tap filter with tap values (1/16, 7/16, 7/16, 1/16). The prediction values at quarter-pixel positions e, g, p and r are achieved by applying a bi-linear filter on the values at half-pixel position j and the one at integer positions D, E, H and I respectively. Under these features, quarter pixel precision interpolation of an X×Y luminance block requires (X+5)×(Y +5) integer pixels.

C

ee

bb

gg

G

A

aa

ff

hh

D d h n H

K

B

a b c E e f g i j k m p q r s I

dd

F cc J

L

Fig. 1. Position of integer/half/quarter pixel

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3 Redundancy Reduction Schemes As described in Section 2, interpolation of an AVS encoder have very high memory bandwidth and computation requirement. Furthermore, we have to take the hardware implementation complexity into account in our design. So we must make a trade off among memory bandwidth, processing cycle and hardware utilization carefully. In AVS, 8×8 block is the smallest processing element for ME. Every sub-block can be divided into several 8×8 blocks with the different MV. For example, a 16×16 block can be partitioned into four 8×8 blocks to utilize the 8-pixel interpolation unit in figure 2(b). The hardware usage is 100%, however, some redundant data in the shadowed regions have to be accessed from off-chip memory twice and calculated twice accordingly. Obviously it increases the memory bandwidth and processing cycle. On the contrary, if a 16-pixel interpolation unit is used, all the horizontal redundant data are reused, which avoids all the redundant computations. However, the hardware can not be fully utilized especially for small blocks, and a great deal of hardware resource is needed. To solve this problem efficiently, we propose an 8-pixel interpolation unit hardware architecture with the higher hardware utilization and the smaller bandwidth. Therefore, we concentrate on designing an 8 pixel interpolation unit for 8×8 block and reusing it in all block types. The redundant data appear in the adjacent 8×8 blocks as showed in Fig.2 (b). Bandwidth and processing cycle are wasted accessing these data. These data can be classified into two types, vertical redundant data (VRD) and horizontal redundant data (HRD) which can be reused by VRD/HRD reuse techniques respectively.

Fig. 2. Sub-block and its interpolation window

Firstly, we introduce the VRD reuse approach in detail. The traditional straightforward memory access scheme is to process every 8×8 block, as the general case always load a 14×14 integer pixels for interpolation, as shown in Fig.2 (a). In order to minimize the bandwidth and computation operation, a VRD reuse technique is presented and the correspondingly hardware architecture will be given in section 4. In Fig.2 (b), the two vertically adjacent 8×8 blocks can be treated as a 8×16 block and their interpolation window is 14×22 block instead of two 14×14 blocks. As a result,

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Table 1. VRD reuse of different sub-blocks block type

block number

cycle/block

reduced bandwidth/cycle

16×16 16×8 8×16 8×8

1 2 2 4

22×2 14×2 22×1 14×1

21 0 21 0

％ ％

21% of memory bandwidth and cycle are saved for a 16×16 block. Indeed, the improvement varies with the height of each sub-block as show in Tab.1, 11% of the memory bandwidth and cycles is reduced in average. As for HRD reuse technique, an additional 22×48 bit on-chip memory called HRD reuse memory is required to save overlapped HRD of horizontal adjacent 8×8 blocks. This scheme can further provide 11% bandwidth reduction in average for the VRD reuse scheme. Actually, sub-blocks with different width get different improvements as show in Tab.2. The on-chip memory is referred to as reuse memory in the following section. By using these two kinds of data reuse techniques, 20% bandwidth and 11% cycle are reduced altogether. After half-pixel ME, quarter-pixel data are required for quarter-pixel ME. According to half-pixel MV, the required quarter-pixel data are interpolated with the adjacent integer pixels and half pixels as shown in Fig.1. A sub-pixel data reuse scheme is proposed to avoid interpolating necessary half pixels repeatedly and reading integer pixels which have been accessed in half-pixel interpolation. As a result, a sub-pixel reuse memory (42×88 bit) is needed for saving these useful b/h type half pixels and another one (42×80 bit) is also used for buffering integer pixels and j type half pixels. Therefore, off-chip memory access is not necessary during quarter-pixel interpolation operation, and about 50% memory bandwidth can be reduced. Moreover, 50% calculating operation can be cut down because of the remove of redundant half-pixel interpolation operation. Table 2. HRD reuse of different sub-blocks block type

block number

access data/block

reduced bandwidth

16×16 16×8 8×16 8×8

1 2 2 4

22×22 14×22 14×22 14×14

21 21 0 0

％ ％

4 Proposed Architecture Figure.3 shows the block diagram for the proposed design of interpolation hardware architecture. The input data include integer MV acquired in ME and searching area data from off-chip memory which is inputted row by row. HRD reuse memory is used to buffer HRD for half-pixel interpolation. Besides, a sub-pixel reuse memory is used

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to buffer useful intermediate integer/half pixels for quarter-pixel interpolation. And the architecture employs a finite state machine (FSM) to generate control signals for all components. Firstly, FSM translates the integer MV for address generation of the off-chip memory. Then data in interpolation window are loaded consecutively without bubble cycle. Input buffer is in charge of selecting the data loaded from off-chip memory and HRD reuse memory. Finally, after the generation of half pixels in half interpolation unit, some horizontal overlapped integer pixels are saved into HRD reuse memory for the next half-pixel interpolation, and all of half pixels are written into sub-pixel reuse memory with some of the integer pixels for quarter-pixel interpolation. Quarter interpolation unit uses the integer/half pixels for generating all the necessary quarter pixels around the half or integer pixel according to half MV acquired in ME. The proposed interpolation hardware architecture is based on a separated 1-D approach of which concept is separating 2-D FIR filter into vertical and horizontal 1-D FIR filter [5]. The 2D 4-tap filter can be decomposed into two 1D 4-tap filters. The detail architecture of 8-pixel half-pixel interpolation unit in our design is shown as fig. 4 (To make it clear, the architecture in this figure is shortened as a 4pixel interpolation unit). It is composed of two types of 4-tap FIR filter (H_HFIR, H_VFIR) and register array. H_HFIR is a horizontal 4-tap FIR filter using horizontal integer pixels to generate horizontal half pixels, while H_VFIR is a vertical 4-tap FIR filter utilizing integer pixels/horizontal half pixels to generate vertical and diagonal half pixels respectively. The adder tree based 4-tap FIR filter is shown in Fig.5(a). And register array buffers all the integer pixels and horizontal half pixels for H_VFIR.

Fig. 3. The proposed interpolation architecture

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The half-pixel interpolation unit has good data reusability, because register array can keep intermediate data for the vertical data reuse during the interpolation of vertically adjacent blocks. Ten adjacent integer pixels in off-chip frame memory must be able to access arbitrarily for 7 H_HFIRs which generate 7 horizontal half pixels. The 7 half pixels with adjacent 6 integer ones are buffered and shifted downward in register array. The vertical and diagonal half pixels are generated by filtering the data in buffers with 13 H_VFIR. After the generation of half-pixel MVs which are acquired in ME, Quarter interpolation unit determines which data should be fetched from sub-pixel reuse memory. For example, if MV points to horizontal half pixel, 20 integer pixels, 16 horizontal and 25 vertical with 20 diagonal half pixels are read for quarter-pixel interpolation as shown in Fig.6. When the half-pixel MV points to different positions, different data are required as show in Table.3. The quarter-pixel interpolation unit employs another 4-tap FIR filter which is different from the one used in half-pixel interpolation unit. Besides bi-linear filter is utilized for quarter pixels generation. The 2-D 4-tap quarter-pixel FIR filter can be partitioned into two 1D 4-tap filters (Q_HFIR, Q_VFIR) with register array which delay registers is used to balance output cycle, as shown in Fig.7. The adder tree based 4-tap FIR filter is shown in Fig.5 (b). Q_HFIR is a horizontal 4-tap FIR filter using integer/half pixels for generating horizontal quarter pixels. Q_VFIR is a vertical 4-tap FIR filter used to generate vertical quarter pixels. And the bi-linear filter is in charge of generating diagonal quarter pixels. Register array buffers keeps all useful pixels for Q_VFIR and bi-linear filter.

Fig. 4. Architecture of half interpolation unit

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Fig. 5. Adder tree based 4-tap FIR

Fig. 6. Integer/half pixels needed for quarter-pixel interpolation

Table 3. Half-pixel best matching candidate and its needed data Half-pixel MV

integer pixels

integer pixel horizontal half pixel vertical half pixel diagonal half pixel

32 5×4

horizontal pixels 5×4 32

half vertical half pixels 4×5 0

diagonal half pixels 5×5 4×5

altogether 97 72

4×5

0

32

5×4

72

5×5

4×5

5×4

32

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Fig. 7. Architecture of Quarter Interpolation Unit

Finally, a testbench is established under the environment of Modelsim which can fulfill the RTL simulation and verification of the module. The design is implemented on the Virtex4 XC4VLX200 field programmable gate array with operating frequency of 150MHz. In order to fulfil 1080p@30 high definition video real-time processing, interpolation should be accomplished in less than 600 cycles. The proposed design only needs 200 cycles to interpolate, and it meets the real-time requirement of the AVS high-resolution encoder, and it is a useful intellectual property design for realtime high definition video application.

5 Conclusions In this paper, a high performance VLSI architecture of interpolation supporting AVS Baseline@L4 is presented. The proposed architecture is attractive for low memory bandwidth, high throughput and high hardware utilization. The design mainly involves three types of data reuse techniques and the separated 1-D interpolation approach. From the data reuse analysis, it can efficiently reduce 60-80% memory bandwidth and 30% processing cycles. The design is implemented on FPGA and meets the requirement of the AVS high definition video real-time encoder. So the design is an efficient hardware solution for real-time HDTV encoder.

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Acknowledgments The research is supported by Shanghai University, "11th Five-Year Plan" 211 Construction Project and Postgraduate Innovation Fund of Shanghai University.

References 1. GB/T 20090.2—2006, Information technology - Advanced coding of audio and video - Part 2: Video (2006) 2. Information technology- General coding of moving picture and associated audio information: Video. ITU Recommendation H.262|ISO/IEC 13818-2 (MPEG-2) Standard draft (March 1994) 3. Advanced video coding for generic audiovisual services. ITU-T Recommendation H.264 | ISO/IEC 14496-10 AVC Standard draft (2005) 4. Gao, W., Wang, Q., Ma, S.W.: Digital Audio Video Coding Standard of AVS. ZTE Communications 4(3), 6–9, 19 (2006) 5. Lappalainen, V., Hallapuro, A., Hamalainen, T.D.: Complexity of optimized H.26L video decoder implementation. IEEE Transactions on Circuits and Systems for Video Technology 13, 717–725 (2003) 6. Junhao, Z., Wen, G., David, W., Don, X.: A Novel VLSI Architecture of Motion Compensation for Multiple Standards. IEEE Transactions on Consumer Electronics 54, 687–694 (2008) 7. Liang, L., John, V.M., Sakir, S.: Subpixel Interpolation Architecture for Multistandard Video Motion Estimation. IEEE Transactions on Circuits and Systems for Video Technology 19, 1897–1901 (2009) 8. Ronggang, W., Chao, H.: Sub-pixel motion compensation interpolation filter in AVS. In: IEEE Int. Conf. Multimedia and Expo. (ICME 2004), Taipei, vol. 1, pp. 93–96 (June 2004)

Optimization of Refinery Hydrogen Network Yunqiang Jiao and Hongye Su State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Yuquan Campus, Zhejiang University, Hangzhou, 310027, P. R. China [email protected]

Abstract. Tighter environmental regulations and more heavy-end upgrading in the petroleum industry lead to increased demand for hydrogen in oil refineries. In this paper, the method proposed to optimize the refinery hydrogen network is based upon mathematical optimization of a superstructure whose objective function is the minimizing total annual cost (TAC) within the hydrogen network in a refinery. The constraints of flowrate, pressure, purifiers, impurities, and compressors were considered. The superstructure considers all the feasible connections and then subjects this to mixed-integer nonlinear programming (MINLP). This approach makes best use of resources and can provide significant environmental and economic benefits. A refinery system from China is used to illustrate the applicability of the approach. Keywords: Optimization; Refinery; Superstructure; Hydrogen management.

1 Introduction Refineries consume hydrogen in large amounts for removing sulfur and nitrogen compounds and producing lighter fuels. Hydrogen availability has become a very important issue because refiners are facing challenges of stricter regulation and increasing demand for transport fuel. The optimization and the improvement of existing hydrogen system is a necessity [1-3]. Refiners are interested in the lower-cost alternative of optimizing and revamping the hydrogen distribution network. The complexity of the problem creates a need for systematic methods and tools for hydrogen management. Two main methods are practiced for an efficient hydrogen management system: graphical [3] and mathematical approach [4]. Graphical approach considers solely the purity and flow rates of streams, while the pressure of sources and sinks also has to be considered. This method will give a theoretical solution which is not necessarily applicable in a real system. This approach will find the minimum use of hydrogen, while the commercial and environmental aspects have to be considered as well. Thus the targets generated may be too optimistic and unachievable in a real design [3]. Superstructure method recently used in the optimization of hydrogen network is a mathematical optimization method, like many other optimization method, which K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 360–367, 2010. © Springer-Verlag Berlin Heidelberg 2010

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includes objective function and constraint conditions and finally obtain the optimum result at the restrictions of some constraint conditions through some optimization solution methods. Superstructure method is better than graphical method when used to optimize of hydrogen network. Superstructure method has many advantages. Firstly, it can consider more constraints than graphical method, such as pressure, compressors and purifiers. In refinery, these costs account for a important part of all refinery hydrogen system, so it is necessary to consider compressors and purifiers in the optimization of hydrogen network. Secondly, it is very easy to add constraint conditions and change the objective function. Therefore, it is also a flexible and strict method to optimize the refinery hydrogen system. It is these superiorities that superstructure method is used more and more frequently in the optimization of hydrogen network. In this paper, we aim to present an improved mathematical approach to the optimization of refinery hydrogen network. Moreover, We not only account for pressure constraints, the existing compressors, purifiers and impurities, but also the necessity of adding or decreasing new compressors and purifiers. Meanwhile, the optimization model is converted into a mixed-integer nonlinear programming (MINLP). The final suitable optimization design is then decided by an optimization procedure. Comparing these methods mentioned in previous literatures, this method is simple, efficient, and easy to be implemented. Finally, a practical case study will be employed to illustrate the applicability of the approach.

2 Refinery Hydrogen Network In a refinery, there are many processes that either consume hydrogen, for example, hydrotreating, hydrocracking, isomerization, purification processes, and lubricant plants, or produce hydrogen, such as the catalytic reforming process. All these processes compose a hydrogen network. A source is a stream that makes hydrogen available to the network Hydrogen sources are the products of hydrogen-producing processes, the offgases of hydrogen-consuming processes, or the imported hydrogen (or fresh hydrogen). Therefore, except for the imported hydrogen, hydrogen sources are usually the outlet streams of different processes. A sink is a stream that takes hydrogen from the hydrogen network with fixed flow rate and purity requirements. Hydrogen sinks are the inlet streams of the various hydrogen-consuming processes such as hydrotreaters and hydrocrackers.

3 Mathematical Model To establish the Superstructure model, many factors that should be first considered are hydrogen supply devices, hydrogen consumption devices, compressors, purifiers and pipe network, then the sources and sinks are determined. The all possible connections from sources to sinks should be also established, and finally the optimization model is transformed into a mixed-integer nonlinear programming (MINLP). In this paper, we utilize the superstructure method to solve the optimization problem of refinery hydrogen network.

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Fig. 1. Schematic diagram of the superstructure for hydrogen distribution network

3.1 Superstructure of Hydrogen Distribution Network Fig.1 shows a simplified superstructure schematic diagram for a hydrogen distribution network. In this model, assuming that any source can supply hydrogen to sink, it means that all the possible connections from sources to sinks should be also considered. 3.2 Objective Function Considering the trade-offs between operating cost and investment cost, the aim of the optimum design should be the minimum total annual cost (TAC) that includes operation costs and annualized capital costs,

⎛ ⎞ TAC = CH 2 + C power − C fuel + Af ⎜ ∑ Ccomp + ∑ C purifier + ∑ C pipe ⎟ purifier pipe ⎝ comp ⎠

(1)

Where: CH2, Cpower and Cfuel are the cost of hydrogen, power and fuel respectively. Ccomp, Cpurifier and Cpipe are the capital costs of compressors, purifiers and pipe respectively. 3.3 Hydrogen Sink Restrictions The sink constraints are described as follows,

∑F

≥ Fk

(2)

y j ≥ Fk yk

(3)

j ,k

j

∑F

j ,k

j

Where: Fk is the hydrogen demand of sink, Nm3·h-1; yj is the hydrogen purity of source; yk is the hydrogen demand purity of sink. 3.4 Hydrogen Source Constraints The amount of gas available from each source must equal the total amount sent to the sinks.

∑F k

j ,k

≤ Fj

(4)

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Considering general binary variable Xj,k, the relationships between Xj,k and Fj,k are stated as,

X j ,k = 1 ⇔ Fj ,k > 0

∀( j , k ) ∈ O

(5)

X j ,k = 0 ⇔ Fj ,k = 0

∀( j , k ) ∈ O

(6)

Where: O is a set of matches that covers every possible match between source j and sink k. Therefore, based on the preceding binary variable Xj,k, the flowrate and pressure constraint is shown as the following,

Fj , k − X j ,k Fj ≤ 0

(7)

F j , k + (1 − X j , k ) Fj ≥ 0

(8)

Pj − Pj , k (1 − X j ,k ) ≤ Pk

(9)

3.5 Purifiers Constraints Purifiers such as membrane separation and PSA, can be modeled as one sink (inlet stream) and two sources (the product stream along with the residue stream). The flowrate balance and hydrogen balance for the purifiers are simply given by [2],

∑F

= ∑ FPp,i + ∑ FPr,i = FP ,i

(10)

⋅ y j = ∑ FPp,i ⋅ yPSAp + ∑ FPr,i ⋅ yPSAr

(11)

j

∑F j

j,P

j,P

i

i

i

i

Where: Fj,p is the flowrate from source j to purifier, Nm3·h-1; Fpp,j is the flowrate of purge from the outlet of purifier, Nm3·h-1; Frp,j is the flowrate of residue from the outlet of purifier, Nm3·h-1. Moreover, the following constraint should be applied to the purifier,

y rP ≤ yP ,i ≤ y Pp

(12)

FP ,i ≤ FP ,i ,max

(13)

Where: ypp and yrp are the purity of purge and residue from the outlet of purifier. 3.6 Impurities Constraints Catalyst of hydrogenation has high requirement for impurities. Therefore, it is necessary to analyze the impurities of sources and the effect to catalyst. Impurities constraints are stated as flows,

X j ,k = 0 ∀( j , k ) ∈ ON

(14)

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X j ,i + X i , k ≤ 1

∀i = I , ( j , k ) ∈ ON

(15)

Where: ON is the set of forbidden blends between sources and sinks on the condition of impurities constraints. 3.7 Compressors Constraints The amount of gas fed to the compressor must be equal to the amount that leaves it as well as its gas purity. Flowrate balance,

∑F

= ∑ Fi ,comp

(16)

ycomp = ∑ Fi ,comp yi

(17)

comp , j

i

Hydrogen balance,

i

∑F

comp , j

i

i

The amount of gas fed to one compressor must never exceed its maximum capacity.

∑F

i , comp

≤Fmax,comp

(18)

i

4 Model Simplification and Model Solution It is necessary to reduce the characteristic of variables before starting optimization procedure [5]. Some variables such as pressure, temperature and purity of some streams, sources and sinks must be set to their operating value. Assuming that the inlet and outlet pressures of the compressors, the inlet and outlet purity of the purifiers are fixed at their design values. These variables should be eliminated from the superstructure as well. The nature of variables must be determined and the unrealistic variables should be removed from the system formulation before starting the optimization mathematical procedure [6]. According to the above model simplification and assumption, this superstructure mode can be transformed into a MINLP problem. This MINLP problem can be solved with Lingo 8.0.

5 Case Study The case study is from a real refinery system in China. Fig. 2 shows the existing hydrogen system in the refinery. There are eight consumers which are four hydrotreating units (HT1-HT4) and two hydrocracking units (HC1, HC2). Hydrogen producers include a continuous catalytic reforming unit (CCR), a semiregenerated catalytic reformer (SCR), a fertilizer plant (FER) and two hydrogen plants. Two PSA plants are also presented to purify the hydrogen product of SCR and CCR respectively in the refinery.

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Fig. 2. The existing network in the case study

Applying the above superstructure optimization strategy, the new network structure is shown in Fig. 3. No additional purifier and compressor are needed, in other words, only piping investment is required. Comparing Fig. 3 with Fig. 2, one of the hydrogen plants as well as four compressors is shut down. Combining these modifications with restructuring, 13% reduction of hydrogen production and a saving of 7.17 million $/year could be achieved.

6 Conclusion An improved mathematical approach to the optimization of refinery hydrogen networks was presented. The method is based on setting up a superstructure that includes the feasible connections and then subjects this to mixed-integer nonlinear programming (MINLP). The constraints of flowrate, pressure, purifiers, impurities, and compressors were considered. Based on this optimization, 13% reduction of hydrogen production and a saving of 7.17 million $/year could be achieved. This saving could be achieved

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Fig. 3. Optimized hydrogen network

without any new equipment addition to the plant and just by using the superstructure optimization technology. Further saving can be attained while considering the utilization of off-gases from all the sinks. Acknowledgment. Financial support provided by the National High Technology Research and Development Program of China (2008AA042902), National High Technology Research and Development Program of China (2009AA04Z162) and Program of Introducing Talents of Discipline to University (B07031) are gratefully acknowledged.

References 1. Towler, G.P., Mann, R., Serriere, A.-L., Gabaude, C.M.D.: Refinery hydrogen management: cost analysis of chemically-integrated facilities. J. Industrial & Engineering Chemistry Research 35, 2378–2388 (1996)

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2. Liu, F., Zhang, N.: Strategy of purifier selection and integration in hydrogen networks. J. Chemical Engineering Research & Design 82, 1315–1330 (2004) 3. Alves, J.J., Towler, G.P.: Analysis of refinery hydrogen distribution systems. J. Industrial & Engineering Chemistry Research 41, 5759–5769 (2002) 4. Hallale, N., Liu, F.: Refinery hydrogen management for clean fuels production. J. Advances in Environmental Research 6, 81–98 (2001) 5. Khajehpour, M., Farhadi, F., Pishvaie, M.R.: Reduced superstructure solution of MINLP problem in refinery hydrogen management. J. International Journal of Hydrogen Energy 34, 9233–9238 (2009) 6. Fonseca, A., Vitor, S., Bento, H., Tavares, M.L.C., Pinto, G., Gomes, L.A.C.N.: Hydrogen distribution network optimization: a refinery case study. J. Journal of Cleaner Production 16, 1755–1763 (2008)

Overview: A Simulation Based Metaheuristic Optimization Approach to Optimal Power Dispatch Related to a Smart Electric Grid Stephan Hutterer1 , Franz Auinger1 , Michael Affenzeller2 , and Gerald Steinmaurer3 1

Upper Austria University of Applied Sciences {stephan.hutterer,f.auinger}@fh-wels.at 2 Josef Ressel Center Heureka! [email protected] 3 Austrian Solar Innovation Center [email protected]

Abstract. The implementation of intelligent power grids, in form of smart grids, introduces new challenges to the optimal dispatch of power. Thus, optimization problems need to be solved that become more and more complex in terms of multiple objectives and an increasing number of control parameters. In this paper, a simulation based optimization approach is introduced that uses metaheuristic algorithms for minimizing several objective functions according to operational constraints of the electric power system. The main idea is the application of simulation for computing the fitness- values subject to the solution generated by a metaheuristic optimization algorithm. Concerning the satisfaction of constraints, the central concept is the use of a penalty function as a measure of violation of constraints, which is added to the cost function and thus minimized simultaneously. The corresponding optimization problem is specified with respect to the emerging requirements of future smart electric grids.

1

Introduction

In coming decades, the electrical power grid, faced with decentralization, liberalization of the energy market, and an increasing demand for high-quality and reliable electricity, is becoming more and more stressed [1]. The change to a smart electric grid is seen to be a promising approach to match these upcoming needs. Concerning the term ”smart electric grid”, there exist multiple definitions in literature all partially varying in the features that the power grids should implement. Among all these, the standard is the usage of sensors and communications technologies to enable more efficient use of energy, provide improved reliability, and enable consumer access to a wider range of services [2]. Therefore, a core feature will be the integrated intelligence, which allows more efficient energy generation and distribution through better information. The term ”efficient” thereby concerns on the one hand to a decrease in overall energy consumption, on the other hand to an increase of the reliability of electrical power supply while, at the same time, improving environmental friendliness. A further concept that K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 368–378, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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accompanies the smart grid development is to integrate the future requirements of e- mobility into the power grid. Hence, battery vehicles could act as controllable loads and distributed energy resources. With this so called vehicle- togrid (V2G) concept, plugged-in batteries may provide capacity during off- peak periods and can act as generating devices when power-demand is high [3]. This smart electrical grid requires new technologies for power system operation. Here, optimization is an essential problem which is the main background of this paper. With the emerging smart electrical grid and the V2G concept, the power flow optimization problem related to later is getting more and more complex. The reasons for this increasing complexity are obvious: the progressive decentralization of comparatively smaller generation units and the hype of environmentally friendly zero- emission generators like photovoltaic plants or wind turbines that cause stochastic delivery of electric power, complicate the supply- side situation in the power grid drastically. On the other end of the grid, the consumer side, the possibility of demand side management (DSM) leads to additional control parameters in the optimization problem. Furthermore the usage of battery vehicles as controllable loads and generating devices simultaneously asks for determination whether a battery should be assigned to the grid or not. This determination introduces a further optimization problem, the so called unit commitment problem [4]. Referring to the objective functions that have to be minimized in order to satisfy the criterion ”efficient”, there exists not only a single objective like the overall energy consumption defined in the original ED. Additional objectives like grid reliability, low emissions and costumer side cost minimization have to be taken into account, yielding in a multi-objective optimization problem. The rest of this paper is organized as follows: Section II presents an overview of the economic dispatch respectively the optimal power flow (OPF) problem with introduction of typical methods of solution, also referring to the unit commitment problem. Special focus is on metaheuristic approaches. Section III goes more into detail with emerging demands of future smart electrical grids. Also the formulation of the corresponding optimization problem for efficient power distribution is discussed. In section IV, the conceptual framework of the simulation- based optimization approach using metaheuristic algorithms is presented subject to the problem defined in section III. Section V provides some concluding remarks. Here, the special abilities of this simulation- based approach with metaheuristics are highlighted, as well as central criteria that will decide it’s applicability.

2

Power System Optimization

Economic Dispatch The optimization of power dispatch in order to match objectives like economical efficiency is one of the most important optimization problems in power system research. Conventional solutions to the ED respectively OPF are the usage of Lagrange Multipliers and the Karush- Khun- Tucker conditions as well as Linear Programming [4]. Further investigations have been made by using sequential quadratic programming as reported in [5], an approximate dynamic programming approach was presented in [7] in the context of future smart grids.

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Referring to the fact that ED respectively OPF try to minimize some objective function by scheduling outputs of committed generators, the unit commitment (UC) problem addresses the decision whether to assign a supply unit or not [4]. Thus, from a mathematical point of view, UC is a discrete optimization problem, while ED is a continuous one. With the mentioned conventional nonmetaheuristic techniques, the two problems are handled separately [4]. Here the ability of metaheuristics like EAs to handle continuous as well as discrete control variables simultaneously in order to minimize some cost function is a major advantage of these methods. Metaheuristic Approaches Due to disadvantages of the mentioned conventional methods for solving the ED, like for instance the difficulties/inabilities when handling multiple objectives, several metaheuristic approaches have been investigated in the literature. For example, [6] showed the applicability of Particle Swarm Optimization and Differential Evolution for reactive power control, which is a problem similar to OPF. As reported in [8], [9], [10], hybrid approaches with Artificial Neural Networks (ANN) have been introduced with a view to decrease the calculation runtime for ED, which is important for its capability for online- control. Subject to multiobjective optimization, [11], [12], [13] presented several Evolutionary Algorithms (EA) techniques, for instance in form of Genetic Programming (GP), in power system research. For representing the strength of these techniques, [11] solved the ED applied to the standard ”IEEE 30-bus” test case1 . Several Multiobjective Evolutionary Algorithms like Strength Pareto Evolutionary Algorithm (SPEA), Nondominated Sorted Genetic Algorithm (NSGA) and Niched Pareto Genetic Algorithm (NPGA) have been compared to the results achieved with conventional techniques like linear programming. Also worked out by [13], the obtained best solutions by EAs are matchable or even better than those of conventional techniques. A special ability of these evolution-based methods is the usage of a population of solutions, instead of a single solution. So, the final population captures multiple optimal solutions. The claim to not only optimize a single objective function like the overall energy usage is obvious. Thus, additional objectives like grid stability and environmentally friendliness are getting more important. The so called environmental/economic dispatch (EED) is an exemplary task in multiobjective power system optimization, also investigated by [11] in the context of Multiobjective Evolutionary Algorithms.

3

Smart Grid Characteristics and the Resulting Optimization Problem

3.1

General Optimization Problem

Originally, the electrical power system dispatch optimization problem can be formulated as an aggregation of objective and constraint functions as follows (the subsequent formulations are based on explanations in [4]): 1

http://www.ee.washington.edu/research/pstca/

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Minimize some cost function: Fi (PG ), subject to the constraints:



g(P G ) = 0;



h(P G ) ≤ 0,

where Fi (PG ) may cover a single objective function like the total energy- consumption of the system or can represent a set of objective functions for i = 1,...,n in the multiobjective case. The optimization problem is constrained by a set of equality and inequality constraints, where the equality constraints represent for J  instance the power balance P G − P load − P loss = 0 with PG as the vector of j=1

power outputs of all generators defined as P G = [P G1 , ..., P Gj ]. Ploss , as the sum of power losses over all transmission lines L in the system, can be calculated depending on PG and by solving the load flow equations, using Newton- Raphson or Gauss- Seidel method [4]. Pload is the given load power that has to be satisfied. The set of inequality constraints includes lower and upper bounds for variables to assure stable operation, for instance the generation capacity of each generator, that is restricted to a range: P Gj min ≤ P Gj ≤ P Gj max , for j = 1, ..., J, and the upper limit of the power flow through transmission lines: S l ≤ S l max , for l = 1, ..., L, with J being the total number of generators and L being the total number of transmission lines. Coming from the so defined original problem in power system optimization, it’s obvious that future smart electric grids will come up with characteristics that ask for a new definition of the problem which than should be able to cover a much higher functionality as we see thereinafter. 3.2

Smart Grid Characteristics

Subsequently, those features are explained that have main impact to the above mentioned optimization problem. As noted in the introduction of this paper, there exist muliple definitions about the smart electric grid that are partially varying in the features that the power grids should realize. Apart from that, many of those features only exist in a fictitious manner and are far away from practical implementation. Therefore, the focus is laid on core aspects of smart electric grids. Reliability: As proposed before and building a central yield of [1], a major requirement to future power grids will be increased reliability of power supply due to the fact that so many critical infrastructures like communications and finance depend on reliable electrical power supply. But just this reliability is getting more and more stressed because of decentralization, liberalization of the energy market, stochastic behaviour of zero- emission plants and complex interdependencies in the grid. So, a central feature of smart electric grids should be the ability of autonomous and fast self- healing after some kind of black- out,

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or at least, the intelligent avoidance of critical operation points. Caused by the complex terminology of power grid stability, security and reliability, here, reliability generally is understood as the ability to supply adequate electric service over a period of time. Additional Availability and Usage of Electrical Storage: Complementary to large- scale storage devices that are already established in electric power grids in different types, the progressive development of small- scale storages like lithium- ion or sodium sulphur (NaS) batteries establish new possibilities for the modern grid in regard to cost aspects, energy- and power- densities or duration of load cycles of storage devices. Especially these small-scale devices substantiate the way to the idea of distributed energy storage, being a major characteristic of smart grids. There exists a large spectrum of possible applications like the ability of storing energy in multiple locations closer to the end- use consumer to increase gridreliability. Another aspect of storage devices is the ability of time- shift generated energy from the moment it was generated to a later point when it is needed or when transmission capacity is available. The probably most important point when talking about distributed energy storage is the V2G- concept. Here, the massive amount of plugged- in batteries from electric vehicles can provide enough storage capacity as well for large- scale applications like ancillary services or regulation of the stochastic behaviour of renewables [14], as for end- user aspects in domestic, commercial or industrial buildings [14]. Load Control: Coming from the end- user side in the smart electric grid, another important characteristic is the usage of automated meter reading (AMR) for generation and load control [15], which is a basic technology in terms of increasing reliability in the power grid, being a major aim of smart grids as mentioned above. Further advanced metering systems should enable the possibility to control load demand. Therefore, the end- use customer accepts the risk of delay or refusal of energy- use by the utility in order to receive some discount [16], which enables the so called load shedding. 3.3

Impacts to the Defined Optimization Problem

The original optimization problem can now be reformulated with detail to the defined smart- grid characteristics. The major change affects the cost function which has to be extended in terms of load control respectively load shedding and the ability of using electricity from electric storages. The new cost function is expressed as follows: M inimize : Fi (PG , Pshed , Psto )

(1)

where Pshed = [Pshed1 , ..., PshedLB ] is the vector of shed loads over all load buses, and Psto = [Psto1 , ..., PstoST ] is the vector of loads taken from or given to storages.

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The following considerations are based on the problem formulation investigated in [15], motivated by power dispatch in a distributed generation case. Now, the extended cost function related to the original optimization problem formulated at the beginning of this chapter can be stated as follows: Minimize: F (PG ) +

LB  i=1

αi Pshedi + F (Psto ),

where F(PG ) is the resulting financial cost value of the generated power over all generators. αi is the penalty cost paid by the distribution company for shedding the load at bus i, respectively the financial reward to the end- use customer; Pshedi is the power shed by the distribution company at bus i, which can be substituted by the term (PL oldi − PL newi ), where PL oldi is the power at load bus i before load control and PL newi is the power at load bus i after load control; F (Psto ) is the resulting financial cost value over all storages that are charged or discharged. Additionally, new constraints will have to be defined in order to realize different criterions that the new parameters have to satisfy, for instance the maximum power when charging / discharging a battery. The mathematical appearence of these new constraints will mainly be the same as those described above. Principally, the definition of constraints in this paper shows their general mathematical formulation to demonstrate their position in the general optimization problem. Some concrete examples of constraints have been stated, without intending to be exhaustive. Surely, numbers of polymorphic constraints exist in this context, depending on the concrete optimization scenario. For example, multiple constraints have been defined in [17] to come over power grid stability like constraints corresponding to contingency, coupling or spinning reserves considerations. In terms of environmental aims, additional constraints referring to maximum emissions have to be defined, also stated in [17]. When talking about smart electric grid aspects, especially the increasing usage of distributed storage, also including the afore mentioned V2G- concept, requires additional constraints. For instance inequality constraints will be required to prevent loss of battery life in case of V2G, or to ensure charging- targets of the electric vehicle’s battery as described in [18]. It’s obvious, that the complexity of the advanced optimization problem is growing rapidly respectively to the original one. Additionally, it will be hard to define a meaningful cost function for different batteries or other electrical storages. So, in sum, the optimization problem respectively the corresponding objective function is getting more and more difficult to define analytically in order to match new requirements of the intelligent power grid. But, what about the mentioned demand for increasing reliability? Principally, security aspects are implemented as constraints to the optimization problem. Wouldn’t it make sense to realize reliability concerns of a grid generally as an objective function? Maybe, another approach can be investigated, that is mainly independent from the exact mathematical formulation of the optimization problem.

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A Simulation- Based Metaheuristic Approach

Supported by the extensive advances in computing power and memory over the last two decades, the research community around simulation-based optimization is able to deal with more and more practical tasks. In [19], Law and McComas defined the aim of this approach as following: ”The goal of an ”optimization” package is to orchestrate the simulation of a sequence of system configurations [each configuration corresponds to particular settings of the decision variables (factors)] so that a system configuration is eventually obtained that provides an optimal or near optimal solution. Furthermore, it is hoped that this ”optimal” solution can be reached by simulating only a small percentage of the configurations that would be required by exhaustive enumeration.” Additionally to increasing computational conveniences, improvements in the fields of usable generic optimization techniques like metaheuristics speeded up developments and practical applications of this approach. 4.1

General Principle

Principally, the aim of this process is to optimize values of defined control variables of a system, where the performance of the system, which basically represents the value of the cost function, is measured through simulation. Here, simulation and optimization are indeed two independent software- tools which build together the simulation-based approach. Thus, the optimization package generates the parameter vector ϕ as possible solution due to a specific algorithm, considering ideally all constraints defined before. The simulation tool then uses this vector as input for the simulation-run and generates some performance measure of the system depending on these parameters. This measure is used to quantify the fitness of the solution which consequently influences the candidate solution generated next by the optimization package. Vector ϕ therefore contains all decision variables. In the case of the optimization problem defined above, ϕ is formulated as: ϕ = [PG , Psto , PL new ] where each of the elements of ϕ is itself a vector, with a length different from the others. Thus, it would be better to generate the three different vectors PG , Psto , PL new and deliver them to the simulation tool independently. 4.2

Evaluation of the Fitness Function

Independent of the used metaheuristics, it’s necessary to formulate some performance- measure for the generated solution in order to influence the way in which the next solutions will be created. Here, the big advantage of the simulation-based approach is that it’s not necessary for the optimization package to implement any exact mathematical formulation of some cost function concerning its constraints. For instance, if the objective is the minimization of transmission losses, without simulation tool it would be necessary to solve the load-flow equations, which would further need to implement mathematical solvers like the Newton-Raphson method. In the case of simulation-based optimization, this mathematical burden is sourced out to the simulation tool.

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Thus, the big advantage of the approach is the flexibility in defining cost functions respectively performance measures without analytical definition of the underlying system. So, metrics for evaluating the cost function of some optimization objective can be created easily for every possible analysis that can be done by the simulation tool, for example usage of available transfer capability, total energy consumption and transmission losses or reliability of the grid, just to mention a few. Regarding power dispatch optimization in the smart electric grid, this flexibility is essential. But, the constraints to the optimization problems have also to be taken into account, which is not always that easy as we see further but can be realized due to the introduction of soft constraints. 4.3

Generation of Candidate Solutions and Their Feasibility

If the parameter vector created by any metaheuristic algorithm is denoted as ϕ , the following statement must be satisfied concerning the feasibility of solutions: ϕ ∈ Φ, where Φ is the feasible region of solutions. Here, the feasible region of solutions is defined as the set of admissible solutions such that all equality and inequality constraints are satisfied, but can be violated to a user defined degree in the case of soft constraints. Considering trivial constraints like lower or upper bounds for parameter- values, it should be relatively easy to fulfil feasibility. For example when optimizing a production layout, where the size of some buffer is constrained by an upper bound, this will not cause any difficulties. In case of power system optimization, for bounds to the upper and lower generation limits of any plant, the same holds. Now, why should the satisfiability of constraints lead to problems? When taking a look at the following constraint as defined above, the equality constraint that represents power balance, it will be shown that it’s not alway that easy: J  PG − Pload − Ploss = 0. This equation expresses the fact that when the metaj=1

heuristic algorithm produces a candidate solution consisting of the power outputs of all controllable generators, the sum of all these power values must be equal to the sum of load power plus transmission losses. In this case, the problem occurs that the power loss itself is a function of the generated power, so Ploss = f (PG ). Thus, it’s obvious that finding a candidate solution while satisfying power balance per se is a more or less complex problem which needs computational effort and time. Hence, for a whole optimization run, where hundreds or thousands of candidate solutions have to be generated, the runtime complexity would grow rapidly because of such nontrivial constraints. Now the question arises how to overcome this problem? 4.4

Introduction of a Penalty Function

In operations research, the use of a penalty function if nonlinear constraints have to be handled is a valid approach. The idea is that the constrained optimization problem is approximated through a non-constrained one by adding some penalty

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term to the cost function if any restriction is violated. The effect of this method is that the cost function per se and the penalty term are minimized simultaneously. Thus, when finding the optimal solution, the penalty term converges to zero, i. e. the optimal solution does not hurt any constraints. The resulting optimization formulation to our problem trivially follows: M inimize : F(PG , Pshed , Psto ) + r ∗ p(PG , Pshed , Psto )

(2)

subject to all trivial constraints. p(PG , Pshed , Psto ) denotes the penalty function that punishes the cost- function afterwards if any nontrivial constraint is violated; r is a constant multiplier to the penalty function for determining its weight relative to the cost function. Principally, the value of the penalty function should be high compared to the value of the cost function to guarantee that the found optimal solution does not hurt any constraint. Additionally, it has to be ensured that when the optimization run stops (not important which stopping- criteria is used), the penalty term already converged to zero in order to make sure that the found solution is feasible. Summing up, the constrained optimization problem is approximated through a partly constrained one (trivial constraints are still included) by adding some penalty function that punishes the violation of nontrivial constraints. It’s obvious, that the violation itself is determined by the simulation run. Concerning the used software, in the context of this paper two packages are used, namely NEPLAN (www.neplan.ch) being the simulation software, and HeuristicLab [20] [21] forming a general and open metaheuristic optimization environment2 .

5

Conclusion

In this paper, an overview is presented on how future electric power dispatch problems can be formulated and solved by a simulation- based metaheuristic optimization approach. Sure, there are several open issues concerning the objectives that have to be handled and about their formulation, caused by the fictitious characteristic of the smart grid paradigm, but the simulation-based approach should be able to provide enough flexibility for ad-hoc definition and optimization of different and partly complex cost- functions. Here, a special strength of this approach is seen in exactly this flexibility, which enables the evaluation of different power system scenarios with more or less detailed implementation of smart electric grid features. Thus, multiple test cases can be evaluated without the burden and effort of exact mathematical problem formulations. Referring to the scope of this paper, the authors are not focusing on the exact metaheuristic strategies that will be applied. Due to the nature of the HeuristicLab framework, the opportunity of testing and evaluating the suitability of different algorithms is given. Main criteria will be the convergence of the algorithm respectively the runtime of the optimization and its ability to find 2

www.heuristiclab.com, gagp2009.heuristiclab.com [22]

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good solutions where the successful treatment of the penalty function will be essential. The formulation of the penalty function itself and finding a meaningful weight relative to the cost function value will demand much operational effort. The application of metaheuristic algorithms contrary to deterministic methods in terms of the simulation-based approach is well-founded due to their successful usage in related works. Additionally, some advantages already mentioned in chapter II, like the ability of handling local minima or the capability of multiobjective optimization of population-based methods further motivate this approach. The additional possibility of handling continuous and discrete decision variables simultaneously, which enables the integrated solution to the unit commitment problem, is not treated here. But as it is a central feature of the application of metaheuristics, it will surely be part of further investigations. ¨ This project was supported by the program Regionale Wettbewerbsf¨ahigkeit OO 2010-2013, which is financed by the European Regional Development Fund and the Government of Upper Austria.

References 1. Amin, S.M., Wollenberg, B.F.: Toward a Smart Grid: Power Delivery for the 21st Century. IEEE Power and Energy Magazine 3, 34–41 (2005) 2. Potter, C.W., Archambault, A., Westrick, K.: Building a Smarter Grid Through Better Renewable Energy Information. Power Systems Conference and Exposition (2009) 3. Guille, C., Gross, G.: A Conceptual Framework for the Vehicle-To-Grid (V2G) Implementation. Energy Policy (2009) 4. Wood, A.J., Wollenberg, B.: Power Generation, Operation, and Control, 2nd edn. Wiley Interscience, Hoboken (1996) 5. Mo, N., Zou, Z.Y., Chan, K.W., Pong, T.Y.G.: Transient stability constrained optimal power flow using particle swarm optimization. IET Generation Transmission and Distribution 1(3), 476–483 (2007) 6. Bakare, G. A., Krost, G., Venayagomoorthy, G. K., Aliyu, U. O.: Comparative Application of Differential Evolution and Particle Swarm Techniques to Reactive Power and Voltage Control. In: International Conference on Intelligent Systems Applications to Power Systems (2007) 7. Werbos, P.J.: Putting More Brain-Like Intelligence into the Electric Power Grid: What We Need and How to Do It. In: Proceedings of the International Joint Conference on Neural Networks, IJCNN 2009 (2009) 8. Panta, S., Premrudeepreechacharn, S., Nuchprayoon, S., Dechthummarong, C.: Optimal Economic Dispatch for Power Generation Using Artificial Neural Network. In: 8th International Power Engineering Conference (2007) 9. Mohammadi, A., Mohammad, H., Kheirizad, I.: Online Solving of Economic Dispatch Problem Using Neural Network Approach And Comparing It With Classical Method. In: International Conference on Emerging Technologies (2006) 10. Tangpatiphan, K., Yokoyama, A.: Adaptive Evolutionary Programming With Neural Network for Transient Stability Constrained Optimal Power Flow. In: 15th International Conference on Intelligent Applications to Power Systems (2009)

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11. Abido, M.A.: Multiobjective Evolutionary Algorithms for Electric Power Dispatch Problem. IEEE Transactions on Evolutionary Computation 10(3) (2006) 12. Chan, K.Y., Ling, S.H., Chan, K.W., Lu, H.H.C., Pong, T.Y.G.: Solving MultiContingency Transient Stability Constrained Optimal Power Flow Problems with an Improved GA. In: Proceedings IEEE Congress on Evolutionary Computation, pp. 2901–2908 (2007) 13. Calderon, F., Fuerte-Esquivel, C.R., Flores, J.J., Silva, J.C.: A Constraint-Handling Genetic Algorithm to Power Economic Dispatch. In: Gelbukh, A., Morales, E.F. (eds.) MICAI 2008. LNCS (LNAI), vol. 5317, pp. 371–381. Springer, Heidelberg (2008) 14. Kempton, W., Tomic, J.: Vehicle-to-grid power implementation: From stabilizing the grid to supporting large- scale renewable energy. Article in press, Science Direct, Journal of Power Sources (2005) 15. Bruno, S., et al.: Load control through smart-metering on distribution networks. In: IEEE Bucharest Power Tech Conference (2009) 16. Hirst, D.: Settlement issues for advanced metering with retail competition. In: CIRED Seminar: SmartGrids for Distribution Paper (2008) 17. Momoh, J.A.: Electric Power System Applications of Optimization, 2nd edn. CRC Press, Boca Raton (2009) 18. Han, S., Han, S., Sezaki, K.: Development of an Optimal Vehicle-to-Grid Aggregator for Frequency Regulation. IEEE Transactions on Smart Grid 1(1) (June 2010) 19. Law, A.M., McComas, M.G.: Simulation-Based Optimization. In: Proceedings of the 2002 Winter Simulation Conference, San Diego, CA, USA (2002) 20. Wagner, S., Affenzeller, M.: HeuristicLab: A Generic and Extensible Optimization Environment. In: Adaptive and Natural Computing Algorithms. Springer Computer Science, pp. 538–541. Springer, Heidelberg (2005), http://www.heuristiclab.com 21. Beham, A., Affenzeller, M., Wagner, S., Kronberger, G.K.: Simulation Optimization with HeuristicLab. In: Proceedings of the 20th European Modelling and Simulation Symposium (EMSS 2008), Campora San Giovanni, Italy (2008) 22. Affenzeller, M., Winkler, S., Wagner, S., Beham, A.: Genetic Algorithms and Genetic Programming. In: Modern Concepts and Practical Applications. Chapman and Hall, Boca Raton (2009)

Speed Control for a Permanent Magnet Synchronous Motor with an Adaptive Self-Tuning Uncertainties Observer Da Lu1 , Kang Li1 , and Guangzhou Zhao2 1

School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, BT9 5AH, UK 2 College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China

Abstract. This paper presents a robust speed control method for a permanent magnet synchronous motor (PMSM) drive. The controller designed from conventional field-oriented vector control method with constant parameters will result in an unsatisfactory performance of a PMSM due to dynamic uncertainties such as changes in load and inertia. In this paper an adaptive self-tuning (ST) observer that estimates dynamic uncertainties on-line has been developed, where output is fed forward as compensation to the PMSM torque. The stability of the observer is studied using the discrete-time Lyapunov theory. A performance comparison of the proposed controller with the constant parameter based conventional field-oriented vector controller is presented though a simulation study, which illustrates the robustness of the proposed controller for PMSMs. Keywords: Permanant magnet synchronous motor, Self-tuning observer, Speed control, Field-oriented vector control.

1

Introduction

In the last decade, the permanent magnet synchronous motor (PMSM) has received widespread appeal in robotics, machine tools, and other high performance industrial servo applications due to some desirable features such as high air-gap flux density, high efficiency, and robustness [11]. However, It is not easy to control a PMSM because of its nonlinearity. This has drawn many efforts from the control experts. There are two most popular control types for a PMSM: direct torque control (DTC)[17, 16, 15] and field-oriented vector control (FOC)[1, 14, 13]. Based on the output of the hysteresis controllers for the stator flux linkage and the torque, DTC controls the stator flux linkage and the torque directly, not though the control of the stator current. The advantage of DTC is its computational efficiency, fast response and good robustness. However, using hysteresis controllers lead to a variable switching frequency, the current and torque ripple, and the requirement of a high sampling frequency. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 379–389, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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FOC employs a position sensor to drive a rotating reference frame transformation, which transforms the ac values to dc quantities. Then it utilizes synchronous frame proportional-integral (PI) current controllers with an active state-decoupling scheme to achieve the separate control of the stator flux linkage and the torque. But one of the main disadvantages is that the parameter of a PMSM needs to be known exactly to guarantee a good performance [7]. This is however not realistic in industrial application for several reasons. Firstly, PMSMs may be produced by different manufacturers, so it is not always possible to know the precise values of the parameters of a PMSM are not possible to always be known precisely enough. Secondly, some parameters such as the stator resistance, phase inductance, the inertia and external load may vary with the working condition [5, 2, 10]. Some different FOC control methods have been proposed. Sliding mode control (SMC) can achieve robustness [1, 3], but it also creates the chattering problem, which leads to a high-frequency oscillation of the torque and noise during the operation. Fuzz logic control (FLC) can also enhance the robustness of system [14]. However, the expert experience is needed to design the rule base, input and output membership functions, which makes the procedure very difficult to achieve. Artificial neural network (ANN) is another important method [13, 10]. It can efficiently map inputs to outputs without an exact system model. The shortage of ANN is that environmental disturbances and structural uncertainties may cause the instability of the overall system and a not fast enough online learning procedure. Adaptive control has been the subject of active research for a long time. Some successful applications in PMSMs control have been reported. So far, most works concentrate on parameters estimation, then the speed controller is designed depend on the estimation results [8, 4, 12? ]. The drawback of these strategies is that they are not able to tackle unconsidered uncertainties. Paper [6] has proposed a new adaptive control algorithm for a PMSM, which has a simple structure with high noise immunity. It collects all the parameter changes and external disturbances as one parameter that represents lump uncertainties. As a result, it can copy with expected disturbances as well as unconsidered uncertainties. However, it did not consider the changing of the moment of inertia and external load, which can heavily deteriorate the performance of speed controller. In this paper, a robust speed control method for a PMSM is proposed. First, to equivalently place lump disturbances outside the PMSM torque control, a selftuning (ST) observer is proposed. Second, dynamic uncertainties are estimated by minimizing the estimation error using a gradient algorithm. Thirdly, the output of the observer is used to feed forward as compensation to the changing of parameters. The organization of this paper is as follows. In Section 2, the PMSM model will be formally described, which covers the description of the used motels with uncertainties. In Section 3 the design process of the ST speed controller is explained and the algorithm stability is analyzed. The validity and usefulness of the

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proposed control scheme are proven by simulation results in Section 4. Finally, Section 5 concludes this paper.

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PMSM Model Description

The PMSM stator voltage models with uncertainties in the rotor reference frame can be described as vq = Rs iq + Lq

diq + ωr (Ld id + ψf ) + ηq dt

(1)

did (2) − ωr Lq iq + ηd dt where vq and vd are the d− and q−axis voltages, id and iq the d-current and q-axis stator current, Ld and Lq the d− and q−axis stator inductance, Rs the stator phase resistance, ψf the permanent magnet flux linkage, ωr the electric velocity, ηq and ηd the lump of d−and q−axis voltage models uncertainties as given by diq + ωr (ΔLd id + Δψf ) + εq (3) ηq = ΔRs iq + ΔLq dt vd = Rs iq + Ld

did − ωr ΔLq iq + εq (4) dt where ΔRs is the stator resistor disturbances, ΔLq andΔLd the d− and q−axis inductance disturbance, Δf the permanent magnet flux linkage disturbance, εd and εq the d− and q− axis unstructured uncertainties due to unmodeled dynamics. The PMSM torque model with uncertainties in the rotor reference frame is given as 3P (ψf iq − iq id (Lq − Ld )) + ηT e1 (5) Te = 2 where Te is the electromagnetic torque, P the number of poles in PMSM, ηT e1 the lump of PMSM dynamics uncertainties given by ηd = ΔRs id + ΔLd

3P (Δψf iq − iq id (ΔLq − ΔLd )) + εT e1 (6) 2 where Δψf is the permanent magnet flux linkage disturbance, εT e1 is the unstructured uncertainties due to unmodeled dynamics. The PMSM dynamic model is ηT e1 =

B J dωr + ωr + ηT e2 (7) P dt P where J is the inertia, B the frictional coefficient, ηT e2 the lump of PMSM dynamics uncertainties given by Te =

ηT e2 =

ΔB ΔJ dωr + ωr + TL + εT e2 P dt P

(8)

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where ΔJ is the moment of inertia disturbances, ΔB the frictional coefficient disturbances, TL the load torque, εT e2 the PMSM dynamics unstructured uncertainties due to unmodeled dynamics.

3

The Proposed Control Scheme

Fig.1 shows the block diagram of the proposed self-tuning control scheme. The system adopts the popular two-level hierarchical control structure, where the higher level uses speed controller to determine the torque from (7) and the lower level completes the d− and q− axis voltage computation through current controllers from (1) and (2). The current observers and controllers have been proposed in Mohamed(2007), so this paper focuses on the speed controller design. An adaptive ST observer runs in parallel with the actual PMSM. When the error between the estimated speed and actual speed exceeds a specific threshold, the observer is self-tuning to estimate the lump uncertainties. Then the estimation is fed forward to compensate the changs in the parameters and external disturbances. 3.1

Design of the Self-Tuning Observer

For a nonsalient pole PMSM, Ld and Lq are equal, and Te is proportional to iq as in (5). PMSM is assumed to be nonsalient in this paper for simplicity. Actually, if the PMSM is salient, the only modification is to introduce a term containing id into the uncertainties. In discreet time, the (5) and (7) can be reformulated by replacing all continuous quantities by their finite difference. Using the discrete time models, the speed update equation is: ωr (n) = Xωr (n − 1) + Y iq (n − 1) + W ηT e (n − 1) where X = 1 − BΔt J , Y = sample interval.

3ψf ΔtP 2 , 2J

W =

P Δt J ,

(9)

ηT e = ηT e1 − ηT e2 , and Δt is the

Fig. 1. Block diagram of the proposed field-oriented vector control

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The observer is similar to equation (9) with the estimated electric velocity ω ˆ r (n) and the estimated lump of uncertainties ηˆT e (n − 1) to replace the actual electric velocity ωr (n) and lump of uncertainties ηT e (n − 1) given by: ω ˆ r (n) = Xωr (n − 1) + Y iq (n − 1) + W ηˆT e (n − 1)

(10)

The error function is given by: E(n) =

1 2 e (n) 2

(11)

where e(n) = ω(n) − ω ˆ (n)

(12)

For the above given error function, there are many different parameter estimation methods such as [9]: the stochastic approximation, instrumental variables, extended Kalman filtering, maximum likelihood, and deepest descent algorithm. This paper adopts the deepest descent algorithm [9] due to its simplicity and effectiveness for dynamic system applications. To minimize (11) with respect to ηˆT e (n−1), we can form the recursive scheme: ηˆT e (n) = ηˆT e (n − 1) + γW e(n)

(13)

where γ > 0 is the adaptive gain. As the observer has only one parameter and the calculation of adaptive law (13) is quite simple, the update procedure costs relatively short time to complete. If the γ is selected carefully, the sample interval for the PMSM speed loop is large enough to calculate the recursive process given in (13) several times. It guarantees the convergence of the algorithm in every sample interval. So the lump uncertainties tracking objective has been achieved under parameter changes and external disturbance. 3.2

Stability Analysis of the Self-Tuning Observer

The adaptive gain γ should be a carefully choice as an inappropriate value may lead to instability of the observer. In this paper the range of γ is derived through a Lyapunov function, which guarantees the stability of the self-tuning observer. Define the Lyapunov function V (n) as V (n) =

e2 (n) 2

(14)

Differentiating (14) gives: 1 2 1 e (n + 1) − e2 (n) 2 2 The change in e(n) can be easily derived through (9), (10) and (13) : ΔV (n) = V (n + 1) − V (n) =

Δe(n) = e(n + 1) − e(n) = 1 − γW 2 e(n)

(15)

(16)

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The change in δV (n) is calculated as: 1 γW 2 (γW 2 − 2)e2 (n) (17) 2 Since V (n) > 0, to make ΔV < 0, the simplest way is to make the item in the bracket negative. Then the following results can be obtained. ΔV (n) =

2 (18) W2 Therefore a γ selected within the range given in (18) guarantees the stability of the observer. 0 0 is predetermined input power level. Signal-to-noise constraint: The closed loop system is described from (1)-(3) x(t) ˙ = (A − BK)x(t) + Bn(t).

(5)

If the system is stable, then the power spectral density of un (t) as [9] can be rewritten 2 Sun (ω) = |T (s)| Sn (ω) where T (s), s = jw is the continuous-time closed loop transfer function from n(t) to un (t). And from [9] un (t)pow has the following definition  ∞ 1 un (t)pow = Su (ω)dω . 2π −∞ n

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Thus (4) becomes 2

2

P > un pow = T H2 Φ

∞ 2 1 1/2 is H2 norm of T (s). where T H2 = ( 2π −∞ |T (jw)| dω) It is obviously that (4) is equivalent to the form of SNR constraint 2

T H2
0, if there exists P¯ > 0, Z > 0 and K   ¯ + P¯ AT − (B K) ¯ T B AP¯ − B K 0. Kx(t) KP −1 K T We readily obtain the following from (11) and (12). un (t)L∞ < γ 2 n(t)L2 Then, by Definition 1 continuous-time system (1) is mean-square stable with H2 performance γ and by solving the optimal problem ρ∗ = min ρ s.t.(10) − (12)

(15)

the optimal value ρ∗ can be obtained. To illustrate the developed theory the following example is given. Example 1: If the parameters of continuous-time system (1) are chosen   p 0 A= 1 , p ∈ {1, 2, 3, 4, 5}, i = 1, 2, 0 p2  i  1 p1 = p2 , B = 1 then by solving (15) we can get Table 1 which indicates the impact of unstable poles on the minimal SNR required for stabilization. Where ◦ denotes the case of p1 = p2 which is not in the calculation range of this example. Remark 2. As it can be seen in Table 1, the minimal SNR for stabilization satm  Re{pi }. This demonstrates the same result in [9]. isfying the formula ρ∗ = 2 i=1

But our approach is not only to make theoretical analysis by Lyapunov method but also to compute numerical solution by lmi-toolbox in Matlab software. Table 1. The relation of unstable poles and the minimal SNR ρ∗ p2 =1 p2 =2 p2 =3 p2 =4 p2 =5

p1 =1 ◦ 6 8 10 12

p1 =2 6 ◦ 10 12 14

p1 =3 8 10 ◦ 14 16

p1 =4 10 12 14 ◦ 18

p1 =5 12 14 16 18 ◦

A Novel Approach to System Stabilization over Constrained Channels

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Conclusions

This paper presents a LMI approach to the problem of stabilization over a channel with the SNR constraint. The two advantage of our approach is the numerical computation and simulation can be easily realized. The numerical examples are presented to support explicitly this conclusion. Notations: Denotes by C+ open right halves of the complex plane. The expectation operator is denoted by E. The size of signal is measured by L2 and ∞ L∞ norm, n(t)2L∞ = sup E{nT (t)n(t)}, and n(t)2L2 = 0 nT (t)n(t)dt for continuous-time.

Acknowledgment This work was supported by the National Science Foundation of China under Grant 60834002,60774059,60904016, The Excellent Discipline Head Plan Project of Shanghai under Grant 08XD14018, The graduate student innovation fund of Shanghai University under Grand SHUCX101016, Shanghai Key Laboratory of Power Station Automation Technology and The Mechatronics Engineering Innovation Group project from Shanghai Education Commission.

References 1. Yang, T.C., Yu, H., Fei, M.R.: Networked control systems: a historical review and current research topics. Measurement and Control 38(1), 12–17 (2005) 2. Hespanha, J., Naghshtabrizi, P., Xu, Y.: A survey of recent results in networked control systems. Proceedings of the IEEE 95(1), 138–162 (2007) 3. Lin, C., Wang, Z.D., Yang, F.W.: Observer-based networked control for continuoustime systems with random sensor delays. Automatica 45(2), 578–584 (2009) 4. Schenato, L.: To Zero or to Hold Control Inputs With Lossy Links. IEEE Transactions on Automatic Control 54(5), 1093–1099 (2009) 5. Rojas, A.J., Braslavsky, J.H., Middleton, R.H.: Control over a Bandwidth Limited Signal to Noise Ratio constrained Communication Channel. In: Proceedings of the 44th IEEE Conference on Decision and Control, and The European Control Conference 2005 Seville, Spain, pp. 197–202 (2005) 6. Tian, E., Yue, D., Zhao, X.: Quantised control design for networked control systems. IET Control Theory Appl. 1(6), 1693–1699 (2007) 7. Minero, P., Franceschetti, M., Nair, G.N.: Data rate theorem for stabilization over time-varying feedback channels. IEEE Transactions on Automatic Control 54(2), 243–255 (2009) 8. Braslavsky, J.H., Middleton, R.H., Freudenberg, J.S.: Feedback stabilization over signal-to-noise ratio constrained channels. Technical Report (2003) 9. Braslavsky, J.H., Middleton, R.H., Freudenberg, J.S.: Feedback stabilization over signal-to-noise ratio constrained channels. IEEE Transactions on Automatic Control 52(8), 1391–1403 (2007) 10. Gao, H.J., Lam, J.: Robust energy-to-peak filter design for stochastic time-delay systems. Systems and Control Letters 55(2), 101–111 (2006)

An Efficient Algorithm for Grid-Based Robotic Path Planning Based on Priority Sorting of Direction Vectors Aolei Yang, Qun Niu, Wanqing Zhao, Kang Li, and George W. Irwin Intelligent Systems and Control Group School of Electronics, Electrical Engineering and Computer Science Queen’s University of Belfast, Belfast, BT9 5AH, UK

Abstract. This paper presents an efficient grid-based robotic path planning algorithm. This method is motivated by the engineering requirement in practical embedded systems where the hardware resource is always limited. The main target of this algorithm is to reduce the searching time and to achieve the minimum number of movements. In order to assess the performance, the classical A* algorithm is also developed as a reference point to verify the effectiveness and determine the performance of the proposed algorithm. The comparison results confirm that the proposed approach considerably shortens the searching time by nearly half and produces smoother paths with less jagged segments than A* algorithm. Keywords: path planning, grid-based map, priority sorting, time-efficient algorithm.

1

Introduction

Path planning is considered as one of the most important tasks in building robotic systems and has been researched for several decades. The path planning problem which has been proved to be a PSPACE hard problem [1] is characterized by the ability to search or find a feasible collision-free path from a start location to a goal location. Many applications rely on path planning such as: intelligent wheelchairs, computer games, computer animals, and robot guidance [2, 3]. Many published paper have addressed the path planning problem. Potentialfield algorithms are efficient for high-dimensional systems under complex constraints. Its main idea is to construct an attractive potential at the goal location and repulsive potentials on the obstacles. The path is then generated by following the gradient of a weighted sum of potentials [4]. Sampling-based algorithms are currently considered as good choice for motion planning in high-dimensional spaces. These are based on uniform sampling which considers the whole map environment as uniformly complex and thus the overall sampling density will be equivalent to the density needed by the most complex region. The result is that every region in the space has the same computational complexity [5]. Lowdimensional path planning problems can be solved with grid-based algorithms K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 456–466, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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that overlay a grid on top of the map. Several approaches to such grid-based path planning have been proposed. Breadth-first search (BFS), a graph search algorithm, can be used to solve grid-based path planning problems. This method begins at the root node and explores all the neighbouring nodes. Then for each of those nearest nodes, it explores their unexplored neighbour nodes, and so on, until it finds the goal [6, 7]. The A* algorithm is a classical method and along with its variants has been widely applied. It uses a heuristic idea to focus the search towards the goal position. Using an edge cost and a heuristic based on the Euclidean distance, the A* algorithm can search the shortest paths [8]. Although there are many algorithms available to solve the path planning problem, these methods are not always suitable in practical engineering applications because of the limitations on runtime and on the resources available in embedded systems. For the classical A* algorithm, its scalability is limited by its memory requirements. It stores all explored nodes of a search graph in the memory, using an Open List to store nodes on the search frontier and a Closed List to store already-expanded nodes. One of the slowest parts of the A* algorithm is to find the grid square on Open List because it needs to check each item on the list to make sure the lowest F cost item is found. Depending on the size of the grid map, dozens, hundreds or even thousands of nodes may have to be searched at any given time. Needless to say, repeatedly searching through these lists can slow down the application significantly. Therefore, from the perspective of a real-time engineering application, the algorithm designed should be simple, efficient, consume minimal resources and be easy to implement. This paper mainly addresses the grid-based robotic path planning problem discussed above. A practical and efficient approach is proposed for generating a path with the minimum number of steps. It is termed as the Direction Priority Sequential Selection (DPSS) method as it is based on the Goal Direction Vector (GDV). In order to assess the algorithm performance, the classical A* algorithm is also included to verify the effectiveness of the proposed technique and to determine its performance. Results show that the proposed approach considerably reduces the search time by nearly 50% and produces smoother paths with less jagged segments than the A* alternative. This paper is organised as follows: Section II presents a detailed description of the direction priority sequential selection algorithm. Section III reports results from comparative simulation studies. Section IV contains the conclusion and future work.

2 2.1

Direction Priority Sequential Selection Algorithm Preliminaries

Consider the three 7 × 7 grid-based maps shown in Fig.1, a “0” means free space and a “1” for an obstacle. This paper assumes that nodes are placed in centres of grid squares and each node is an eight-connected mapping. For simplicity, each square of the map is considered as a node and this node can be represented using the coordinates of the cell, for example, Node (1, 1).

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Fig. 1. Three examples for illustrating the process of sorting neighbour node

As for the algorithm, there are a few considerations which should be taken into account. The first is the rule about cutting corners movement. In the implementation of algorithm, a diagonal movement is not allowed if any of two adjacent diagonal nodes are filled with obstacles. The second consideration is about the “Length” between adjacent nodes and it can also be considered as a “Cost” of moving from one node to another.√There is an assumption that the length between diagonal adjacent nodes is of 2 units and the length of vertical and horizontal adjacent nodes is of 1 unit. The third consideration is to introduce a few definitions: “steps”, “current node”, “tail node”, “Goal Direction Vector (GDV)” and “Neighbour Direction Vector (NDV)”. A single movement of the robot between neighbouring nodes is called as one step. In the next of this paper, “Steps” stands for the number of step in a path. A “current node” is one which is currently under search. In the user-defined data structure (Link List) introduced in this paper, a current node is pointed to by a head pointer. As for the “tail node”, it is the last node of the link list which is pointed to by a tail pointer. The Goal Direction Vector(GDV) and the Neighbour Direction Vector(NDV) are shown in Fig. 1. The GDV is represented by a dashed arrow which is the direction vector from the current robot location to the destination node. The NDV is shown by a solid arrow line which is the direction vector from the current location to every valid neighbouring node. Here, a few rules for sorting neighbouring nodes for the new DPSS method are given below: 1. The angles between the GDV and the NDV are used to define the priority of every NDV for the current node. The closer the angle between the GDV and the NDV, the higher the priority of the corresponding neighbouring node. 2. If the GDV of the current node is parallel to the horizontal or vertical directions, the NDV which parallels the GDV holds the highest priority and the reverse direction of the GDV is assigned the lowest priority. The remaining NDVs of the nodes are also allocated according to the angle between the GDV and the NDV. However, there are always three pairs of NDVs having the same angles, the solution is to randomly specify the priority of NDVs.

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Coordinate of Location (x, y) Simple Node Structure Index of Parent Node in Path Link List

Fig. 2. User-defined data structure

To illustrate clearly the DPSS method, Fig. 1 shows the application of the rules introduced above. Robots (Circle R stands for current node) traverse the grid map and reach the goal location (Oval D stands for goal node). Note that the number near NDV represents the sorting priority, the smaller the value, the higher its priority. In Fig. 1(a), eight neighbouring nodes of the current node are labelled “0”, which mean these are valid nodes for the next search process. Fig. 1(b) shows that six neighbouring nodes of the current node are valid. The two cases in Fig. 1(a), (b) can adopt rule 1 to get the priority for all valid NDVs. Actually every NDV corresponds to a neighbouring node, so these nodes can be sorted according to NDV priority and can be stored into a user-defined link list. For the case of Fig. 1(c), rule 2 can be used to achieve the sorting priority. Considering Fig. 2, a simple node data structure is designed to store the search path information sequentially. The “coordinate of location” field is filled with the coordinate position in the map, and the “index of parent node” field is filled by the index of the parent node of the current one in user-defined link list. Fig. 4 displays the using of this data structure. 2.2

Introduction of DPSS Algorithm

The DPSS algorithm consists primarily of two functions: a Node Sorting Search (NSS) and Path Fin Cutting (PFC). The Node Sorting Search function is based on a direction priority sorting and can be used to achieve a feasible path with minimum steps which is called as raw path. Path Fin Cutting is used to optimize the raw path. Since the focus of the NSS function is on the steps rather than the length/cost, it is necessary to develop some measures to generate smoother paths with less jagged segments and to improves the quality of the raw path. Consider the problem of traversing the grid map as shown in Fig. 3(a). The NSS function would search and identify all nodes that surround the current one. After a node is searched, it is considered as a visited node. All these visited nodes are recorded in order into a user-defined link list according to the rules of sorting mentioned above. The question arises as to how to know a node is a visited node in the next search operation? A widely used method that is used by the A* algorithm is to search an “Open List” or a “Closed List”. However, repeatedly searching through a list is a very inefficient method if thousands of nodes are stored there. The method used in the implementation of the new DPSS algorithm is to build a mirror image of the grid-based map. If a node in the map has been visited, a corresponding position in the mirror image can be recorded. When judging the state of a node, it is then just necessary to read the value in the corresponding position of the mirror image.

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Fig. 3. The process of operating of NSS function

Fig. 3 illustrates the detailed operation of the NSS algorithm for solving a simple path planning problem. Node(2, 1) is the start location and Node(1, 4) is the destination. The grey squares indicate the obstacles on the grid map and the light grey squares represent nodes which have already been visited in a previous searching operation. The mirror image keeps track of previous visited nodes by marking them temporally in the workspace, as if they were obstacles. Consider the search process shown in Fig. 3, where the NSS function has performed eight searches. Fig. 4 illustrates the process of evolution about the link list structure corresponding to Fig. 3. The head and tail arrows in Fig. 4 indicate the user-defined pointers which are used to indicate the head and tail of the link list. The head pointer is always considered as the current robot location which can also be called the focus node. The tail pointer points to the last node of the link list as the NSS function runs continuously. The method always searches the node pointed by head pointer and stores the valid traversal node in the position pointed by the tail pointer. To simplify matters, the search operation is stopped when the goal node is found for the first time or when finding the target node fails. If the goal node is found, the method just starts here, and works backwards moving from one node to its parent, following the parent pointers. This will eventually track back to the start node, and produce the resulting path. As depicted in Fig. 4(h), a feasible path can then be easily achieved as follows: N ode(2, 1) → N ode(2, 2) → N ode(1, 3) → N ode(1, 4). Since the main target of the DPSS algorithm is to decrease the search time and to achieve the minimal step number of a path, it is possible that there are different path lengths with the same step number. So it is then necessary to design the Path Fin Cutting function to optimize the raw path in order to produce smoother paths with less jagged path segments.

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Fig. 5 show a segment of a feasible path. Consider the √ path from Node A to Node E. The step number is equal to 4, the length is 4 × 2 units, and further there are no obstacles on the straight line N ode A → N ode X → N ode Y → N ode Z → N ode E. It is obvious that the path N ode A → N ode B → N ode C → N ode D → N ode E can be replaced by the path N ode A → N ode X → N ode Y → N ode Z → N ode E which has smaller length (4 × 1 units) but the same number steps. Similarly, the path N ode E → N ode F → N ode G → N ode H → N ode I can also be replaced by a path N ode E → N ode O → N ode P → N ode Q → N ode I.

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To implement the algorithm, C++ is used to develop the Graphical User Interface (GUI) to display the trajectory. All simulation tests were performed on Intel Centrino Due2300 processor with 1G RAM and Windows XP. In order to assess the performance, the A* algorithm was used for comparison purposes. The performances of the two algorithms were compared with respect

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(a) A* algorithm

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Fig. 6. Comparison on manually generated maps for Test 1

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to run times, length and steps of the generated path. Note that the “runtime” displayed on the GUI is the average time spent for 100 tests running in the same environment. In the simulation, various types of tests were carried out on manually generated maps, random maps with a uniform distribution, and small to large maps. In all cases, a black circle stands for the start node, a white circle stands for the goal node, and white squares were using for free spaces and black squares for obstacles. 3.1

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In the manually generated maps which consisted of a 20 × 20 grid of cells, Fig. 6 (Test 1) and Fig. 7 (Test 2) clearly compare the DPSS and A* algorithms under

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the following conditions: the same map, the same goal node, but with different start nodes. As depicted in Fig. 6 (a) and the Fig. 7 (a), the A* is not optimal and the DPSS gave better results. Table 1 compares the time consumption, and the path lengths and steps. The Fig. 6, Fig. 7 and Table 1 together confirm that the performance of the DPSS technique is better than the A* algorithm. The time spent by the DPSS algorithm was nearly 50% less that from the A* method in all tests. Considering Test 1, the DPSS method achieved smaller steps and shorter lengths than A*, while in Test 2 the DPSS length was shorter than that of the A* for the same number of steps. Table 1. Comparison of two algorithms Map Type Case Start Node Goal Node Parameter DPSS Algorithm A*Algorithm Time (ms) 0.066 0.114 The same Test 1 (11, 9) (17, 10) Length 32.07 32.656 map Steps 30 31 generated Time (ms) 0.066 0.143 manually Test 2 (3, 9) (17, 10) Length 26.414 27.242 Steps 26 26

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Similarly to [9, 10], several different densities of obstacles can be chosen to assess the performance of the new algorithm in different environments. In this paper, nine different densities of obstacles were selected, viz. 0%, 5%, 10%, 15%, 20%, 25%, 30%, 35%, 40% of filled squares in the whole grid. Through many tests, it was found that if the density of obstacles was greater than 40%, there was almost no chance of a feasible path being found. Fig. 8(a), (b) present the path solutions from the DPSS algorithm under the following conditions: 10% and 25% density of obstacles, the same map of size (50 × 50), and the same start and goal location. The obstacles were filled randomly in the map space. A detailed comparison is provided in Table 2. As shown in Table 2, the two algorithms produced exactly the same paths under the same density of filled obstacles. This is a precondition for comparing the times taken. Some valuable conclusions can be reached: 1. The average time spent by the new DPSS is almost 50.4% shorter than that of the A* algorithm in the tests performed. 2. The time spent by the DPSS first increases, and then decreases as the density of obstacles increases. By contrast, the time spent by the A* algorithm decreases monotonously with increasing density of obstacles. 3. The density of obstacles has less impact on the performance of the new DPSS technique than that of A* method.

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(a) Path solution with 10% obstacles filled

(b) Path solution with 25% obstacles filled

Fig. 8. Path solution of DPSS algorithm with 10% and 25% density of obstacles filled randomly Table 2. Algorithm comparison with the gradual change of density of obstacle filled Test DPSS Algorithm A* Algorithm Density environment Time(ms) Length Steps Time(ms) Length Steps 0% 0.38 63.63 45 1.01 63.63 45 Map size: 5% 0.42 67.146 51 0.98 67.146 51 50*50 10% 0.46 70.662 57 0.96 70.662 57 Start node: 15% 0.46 73.248 60 0.94 73.248 60 (2,2) 20% 0.47 75.592 64 0.92 75.592 64 Goal node: 25% 0.48 75.936 66 0.88 75.936 66 (47,47) 30% 0.44 79.694 71 0.83 79.694 71 35% 0.39 94.866 87 0.72 94.866 87 40% 0.26 106.038 99 0.46 106.038 99

Same path Yes Yes Yes Yes Yes Yes Yes Yes Yes

Shorten Average time shorten 62.4% 57.1% 52.1% 51.1% 48.9% 50.4% 45.5% 47.0% 45.8% 43.5%

Table 3. Algorithm comparison with the gradual change of map size Map Start Goal DPSS Algorithm A* Algorithm Density Dimension node node Time(ms) Length Steps Time(ms) Length Steps 10% 0.065 22.382 17 0.15 22.382 17 20*20 (2,2) (17,17) 25% 0.075 27.898 25 0.13 27.898 25 10% 0.29 54.178 43 0.61 54.178 43 40*40 (2,2) (37,37) 25% 0.3 62.28 54 0.57 62.038 55 10% 0.66 83.63 65 1.43 83.63 65 60*60 (2,2) (57,57) 25% 0.67 88.904 74 1.28 88.904 74 10% 1.17 114.49 88 2.63 113.66 88 80*80 (2,2) (77,77) 25% 1.22 124.216 106 2.56 124.216 106 10% 1.84 142.534 109 4.56 142.534 109 100*100 (2,2) (97,97) 25% 1.91 157.184 134 3.77 157.184 134

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Comparison of Performances on Large Maps

In addition to assessing the performance with small and medium sized maps, the large maps were also examined. Table 3 illustrates the change in performance with increasing map size from 20 × 20 to 100 × 100. Several conclusions can be drawn: 1. The DPSS algorithm shortens the running time by nearly half in the same environment (the same map, same start and goal locations). 2. On large maps (100 × 100), the time consumption is shorteded by 59.6% and 49.3% corresponding to 10% and 25% density of obstacles respectively. 3. The DPSS algorithm can always achieve the smallest number of steps in a path for all the tests, although the path length for the DPSS is larger than that of the A* algorithm under certain conditions, as shown in bold in Table 3 (map of size 40 × 40 and 25% density).

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Conclusions and Future Work

A novel DPSS algorithm has been proposed and applied to grid-based path planning problems in this paper. It was compared with the well known A* algorithm which is extensively used in practical systems to verify its effectiveness and to determine its performance. A number of simulation tests showed that the proposed approach considerably reduces the search time and generates smoother paths with less jagged segments. The execution time of the DPSS was shorter by nearly 50% in all tests. The path step number was always smaller than that of A* alternative, although sometimes the smallest path length was not also achieved. Compared with the A* algorithm, the DPSS one only needs to design one Link List, whose size is proportional to the number of grid squares. Moreover, there is no necessity to search “Open List” and “Close List” frequently. These are the key factors in ensuring that the DPSS algorithm is efficient and can also reduce the memory requirements during search. The proposed algorithm can be further improved in the future. Here, to improve the search speed, the search was stopped when the goal node was found for the first time. If achieving the minimum path length is the main target, it is essential to continue the search operation until a path with minimum length is found. Further, this algorithm can be further extended to dynamically adapt to the environment by the changing maps and changing the goal location. Acknowledgments. This work was supported by the China Scholarship Council and EPSRC under UK-China Science Bridge grant EP/G042594/1.

References [1] Reif, J.H.: Complexity of the mover’s problem and generalizations. In: Proceeding of the 20th IEEE Symposium on Foundations of Computer Science, pp. 421–427. IEEE, New York (1979)

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[2] Burgard, W., Cremers, A.B., Fox, D., Hahnel, D., Lakemeyer, G., Schulz, D., Steiner, W., Thrun, S.: The interactive museum tourguide robot. In: Proceeding of the Fifteenth National Conference on Artificial Intelligence, Madison, Wisconsin, United States, pp. 11–18 (1998) [3] Arai, T., Pagello, E., Parker, L.E.: Advanced in multi-robot systems. IEEE Transactions on Robotics and Automation 18(5), 655–661 (2002) [4] Al-Sultanl, K.S., Aliyul, M.D.S.: A new potential field-based algorithm for path planning. Journal of Intelligent and Robotic Systems (17), 265–282 (1996) [5] Taha, T., Valls Miro, J., Dissanayake, G.: Sampling based time efficient path planning algorithm for mobile platforms. In: Proceeding of the 2006 International Conference on Man-Machine Systems (ICoMMS 2006), Langkawi, Malaysia (2006) [6] Zhou, R., Hansen, E.A.: Breadth-first heuristic search. Artificial Intelligence 170(4), 385–408 (2006) [7] Vacariu, L., Roman, F., Timar, M., Stanciu, T., Banabic, R., Cret, O.: Mobile robot path planning implementation in software and hardware. In: Proceedings of the 6th WSEAS International Conference on Signal Processing, Robotics and Automation, pp. 140–145 (2007) [8] Hart, P., Nilsson, N., Raphael, B.: A formal basis for the heuristic determination of minimum cost paths. IEEE Transactions on Systems Science and Cybernetics (2), 100–107 (1968) [9] Nash, A., Daniel, K., Koenig, S., Felner, A.: Any-angle path planning on grids. In: Proceedings of the AAAI Conference on Artificial Intelligence (AAAI), pp. 1177–1183 (2007) ˇ sl´ [10] Siˇ ak, D., Volf, P., Pˇechouˇcek, M.: Accelerated a* trajectory planning: Grid-based path planning comparison. In: Proceedings of ICAPS 2009 Workshop on Planning and Plan Execution for Real-World Systems, Greece, pp. 74–81 (2009)

A Novel Method for Modeling and Analysis of MeanderLine-Coil Surface Wave EMATs Shujuan Wang, Lei Kang, Zhichao Li, Guofu Zhai, and Long Zhang School of Electrical Engineering and Automation, Harbin Institute of Technology, 150001, Harbin, P. R. China [email protected]

Abstract. A novel 3-D model for meander-line-coil surface wave electromagnetic acoustic transducers (EMATs) operating on the Lorentz principle is established by combining numerical calculations and analytical solutions. Simulation and analysis find that surface waves generated by the Lorentz force due to dynamic magnetic field is more sensitive to flaws, and the force due to the dynamic magnetic field generates surface waves more efficiently than that due to the static one when the excitation current exceeds 528.9 A. The accuracy of the established model is verified by experiment. Keywords: surface waves, EMATs, meander-line coil, finite element method.

1 Introduction Electromagnetic acoustic transducers (EMATs) are a kind of non-contact ultrasonic transmitting and receiving devices, which have gradually become a popular ultrasonic nondestructive testing technique due to advantages such as being free of couplant, having no requirement of preconditioning, etc [1]. Surface waves are widely used to detect surface and sub-surface flaws as they are sensitive to such flaws with low attenuation [2]. Combining advantages of both electromagnetic acoustic technique and surface waves, surface wave EMATs can successfully detect surface defects in hostile environments including elevated temperatures, on-line inspection, etc [3-12]. To study the mechanism of surface wave EMATs, researchers have developed many EMAT models [3-10]. However, these models failed to consider the influence of dynamic magnetic field generated by excitation current during the transduction process of EMATs, which inevitably leads to the inaccuracy of the models. Subsequently, [11] and [12] built a new model for spiral-coil surface wave EMATs with a consideration of the influence of dynamic magnetic field, and found that Lorentz force due to the dynamic magnetic field played a more significant role in transmitting surface wave than that due to the static one when the excitation current is about 300 A. However, models for meander-line-coil surface wave EMATs containing the influence of the dynamic magnetic field have been hardly reported. Comparing with the spiral-coil EMATs, the meander-line-coil EMATs possess higher directivity, and they are more widely used in real applications of surface wave detection. Consequently, this paper establishes a novel 3-D model for the meander-line-coil EMATs and analyzes EMATs’ characteristics based on the model. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 467–474, 2010. © Springer-Verlag Berlin Heidelberg 2010

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2 Lorentz Principle of EMATs and Governing Equations Electromagnetic acoustic transducers consist of a coil, a static magnetic field and a sample under test. To generate surface waves with high energy density and high directivity, a meander-line coil and a vertical or a horizontal magnetic field are generally used. The Lorentz mechanism of surface wave EMAT with a meander-line coil on an aluminum plate is shown in Fig. 1. An alternating current density J0 in the coil generates a dynamic flux density Bd as well as an eddy current density JE in the plate. The eddy current experiences Lorentz forces under the interaction with both the dynamic magnetic field Bd from the excitation current and the external static magnetic field Bs from the permanent magnet. The forces will generate vibrations under the coil. The spacing intervals between neighboring conductors of the coil are halfwavelength of the surface waves, so the vibrations under each conductor interfere constructively and thus generate surface waves in the plate. The Cartesian coordinate system is shown in Fig. 1, and the origin O locates on the aluminum plate surface.

Fig. 1. Lorentz mechanism of surface wave EMAT with a meander-line coil

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(1) ~ (6) are a typical quasi-static field problem, which can be solved by finite element method (FEM) [3-9]. The displacement of surface waves consists of an in-plane displacement and an out-of-plane displacement, and the former one is made up of a radial component and a tangential component. Surface waves generated by surface point forces are shown in Fig. 2. Fz is the out-of-plane point force, Fr is the in-plane point force; A1, A2 are the working points of Fz and Fr respectively, B1, B2 are the oscillating points with a distance r from A1, A2 respectively; ξzz and ξrz are the out-of-plane displacement and the in-plane radial displacement at B1 generated by Fz respectively; ξzr, ξrr and ξfr are the out-of-plane displacement, the in-plane radial displacement and the in-plane tangential displacement at B2 generated by Fr respectively; φ is the angle between r and Fr. Surface acoustic waves generated by the out-of-plane and in-plane point forces were given for a large distance R by Nakano [13],

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where vL, vT and vR are velocities of longitudinal, transverse and surface waves respectively; in aluminum plate, vL=6380 m/s; vT=3140 m/s; vR=2930 m/s; ω = 2π f , f is frequency of the force; K L = ω vL ; K T = ω vT ; K R = ω vR ; η1 = K R2 − K L2 ; η 2 = K R2 − K T2 ; μ is Lamé constant, μ = ρ vT2 ; ρ is aluminum density, A1 = ( jK T ) / 2μπ , F '( K R ) = ∂ ⎡ (2 K R2 − K T2 ) 2 − 4 K R2 K R2 − K L2 K R2 − K T2 ⎤ / ∂K R . ⎣ ⎦ Assuming the Lorentz force in the aluminum plate is decomposed into N point force components, and the ultrasonic displacement resulted from the ith point force component FLi is upi, the total displacement by a meander-line-coil EMAT is: N

up (t ; X P , YP , ZP ) = ∑ upi (t ; X P , YP , ZP ; FLi ) . i =1

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3 Modeling of Meander-Line-Coil Surface Wave EMATs A 3-D physical model of a commonly used meander-line-coil surface wave EMAT is established using FEM software, as shown in Fig. 3 (a). When the magnet is set invisible, the coil and the aluminum plate are shown in Fig. 3 (b). In this model, the coil consists of 12 conductor wires. Each conductor has a length of 35 mm, a width of 0.5 mm and a thickness of 0.05 mm. The lift-off distance between the coil and the plate is 0.1 mm. The excitation current is a 500 kHz tone burst signal with amplitude of 50 A. The size of the magnet is 60×60×25 mm3. It has a remnant magnetism of 1.21 T and a maximum magnetic energy product of 279 kJ/m3. To reduce calculation, only the part of the plate where electrical-acoustic energy conversion takes place is calculated and its size is 90×90×1.58 mm3. The resistivity of aluminum is 2.6×10-8 Ω/m. For clarity, this model is called the “whole model”. The frequency of Lorentz forces fd is twice that of fs. According to (7) ~ (11), surface-wave components generated by fd and fs should be calculated separately. Then, surface waves generated by the EMAT as a whole can be acquired by calculating the sum of the two components. However, fd and fs are mixed together in the whole model. To study their influence individually, a physical model only consists of a meander-line coil and an aluminum plate is also built as shown in Fig. 3 (b). As there is no static magnetic field in this model, the Lorentz force is completely generated by the interaction of the eddy current with the dynamic magnetic field. For clarity, this model is called the “dynamic magnetic field model” (DMF model). The parameters in the DMF model are the same as that in the whole model. Finite element models are obtained by dividing the physical models into numerous tetrahedron elements as shown in Fig. 4. To increase the accuracy of the models, the region under the coil is further meshed with more elements. fd and fL can be obtained by solving the finite element models, and fs can be acquired by equation (5). To calculate surface waves on the aluminum plate, the plate is also divided into many small ring elements, as shown in Fig. 5. Surface wave displacements on each element generated by Lorentz forces can be calculated according to (7) ~ (12).

(a) Whole model

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Fig. 3. 3-D Physical model of a commonly used meander-line-coil surface wave EMAT

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Fig. 4. 3-D finite element models of a commonly used meander-line-coil surface wave EMAT

Fig. 5. Ring-element division of aluminum plate for calculating surface waves

4 Simulation and Analysis Assuming fsX, fsY, fsZ and fdX, fdY, fdZ are components of fs and fd in X, Y, Z directions respectively, the distribution of the six components are shown in Fig. 6 (a) ~ (f). Comparing fs with fd shows that fs is about 9.32 times larger than fd, and both of the forces chiefly distribute along the outline of the coil; fd mainly acts as a repulsive force between the coil and the plate; the frequency of fd is twice that of the excitation current; while the frequency of fs is the same as that of the excitation current. Fig. 7 shows the amplitude distribution of surface wave on the aluminum plate transmitted by the meander-line-coil EMAT, which illustrates that the EMAT can generate surface waves with high directivity.

Fig. 6. Distribution of Lorentz force density fd (a ~ c) and fs (d ~ f)

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Fig. 7. Distribution of surface waves on aluminum plate surface 1.5 1.0 0.5 0 -0.5 -1.0 -1.5 0

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Surface wave displacements generated by fs and fd at the point (0, 300, 0) are shown in Fig. 8. Fig. 8 indicates that the wave amplitude due to fd is about 1/5 of that due to fs; which means fs generates surface waves more efficiently than fd when the peak-to-peak (p-p) amplitude of excitation current is 100 A. Moreover, the frequency of displacements due to fd is twice that due to fs, so surface waves due to fd have higher sensitivity to flaws. As surface waves with two frequencies are excited during the inspection process, sensitivity and resolution of EMATs are enhanced. It was reported that for a spiral coil EMAT, fd generates surface waves more efficiently than fs when the excitation current is 300 A [12]. So it is interesting to study the influence of Lorentz forces due to the static and the dynamic magnetic fields for the meander-line-coil EMAT at different currents, as shown in Fig. 9. Fig. 9 illustrates that the amplitude of surface waves due to fs is equal to that due to fd when the current is 528.9 A. Surface waves due to fd increases more quickly with the increment of the current, and fd plays a more important role when the current exceeds 528.9 A. So ignoring the contribution of the dynamic field may lead to the inaccuracy of the model. 20 16

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Fig. 9. Amplitude of surface-wave displacement at different excitation currents

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5 Experiment A specially designed EMAT is used to detect the surface waves excited by the modeled meander-line-coil EMAT. The amplitude of the receiving signal of the special EMAT is proportional to the out-of-plane velocity of surface waves when the magnetic field is arranged horizontally (Br), as shown in Fig. 10 (a) [10]. The sensor coil is enwound using enamel wires, with 3 mm high, 3 mm long, and 20 turns; the horizontal magnetic field is about 0.63 T. Fig. 10 (b) shows the photo of the special EMAT. The dimensions of the plate are 500×160×30 mm3. The transmitting EMAT locates at one end of the plate, and the receiving EMAT is moved along the sound beam axis of the surface wave sound field by a 3-D Cartesian coordinate robot. After normalization and curve fitting, the measured variation of the out-of-plane vibration velocity along the sound beam axis is shown by the solid line in Fig. 11, and the calculated one based on the established model is illustrated by the dashed line in Fig. 11. Fig. 11 indicates that the measured out-of-plane velocity agrees well with the calculated one, which verifies the accuracy of the established model.

(a) Schematic diagram

(b) photo of the special EMAT

Fig. 10. Detection of surface waves by a specially designed EMAT

Fig. 11. Comparison between measured and calculated velocity along sound beam axis (normalized)

6 Conclusions A novel 3-D model for meander-line-coil surface wave EMATs operating on the Lorentz principle has been established. Lorentz forces of the EMATs due to both static and dynamic magnetic field have been calculated respectively by finite element method, and the surface wave displacements generated by the Lorentz forces are obtained respectively by an analytical method. Simulation and analysis shows that the frequency of surface waves due to the dynamic magnetic field is twice that due to the

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static one. Therefore, surface waves with two frequencies are excited by EMATs, which enhances the sensitivity and resolution of the transducers. Surface wave generated by the Lorentz force due to dynamic magnetic field increases more quickly with the increment of the excitation current than that due to the static one, and when the current exceeds 528.9 A, the Lorentz force due to the dynamic field plays a more significant role in the excitation of surface waves.

References 1. Hirao, M., Ogi, H.: EMATs for Science and Industry: Noncontacting Ultrasonic Measurements. Kluwer Academic Publishers, Boston (2003) 2. Viktorov, I.A.: Rayleigh and Lamb Waves: Physical Theory and Application. Plenum Press, New York (1967) 3. Thompson, R.B.: A Model for the Electromagnetic Generation and Detection of Rayleigh and Lamb Waves. IEEE Transactions on Sonics and Ultrasonics. SU 20(4), 340–346 (1973) 4. Ludwig, R.: Numerical Implementation and Model Predictions of a Unified Conservation Law Description of the Electromagnetic Acoustic Transduction Process. IEEE Transactions on Ulrtasonics, Ferroelectrics, and Frequency Control 39(4), 481–488 (1992) 5. Ludwig, R., You, Z., Palanisamy, R.: Numerical Simulations of an Electromagnetic Acoustic Transducer-Receiver System for NDT Applications. IEEE Transactions on Magnetics 29(3), 2081–2089 (1993) 6. Ludwig, R., Dai, X.W.: Numerical Simulation of Electromagnetic Acoustic Transducer in the Time Domain. J. Appl. Phys. 69(1), 89–98 (1991) 7. Shapoorabadi, R.J., Sinclair, A.N., Konrad, A.: Finite Element Determination of the Absolute Magnitude of an Ultrasonic Pulse Produced by an EMAT. In: IEEE Ultrasonics Symposium, pp. 737–741 (2000) 8. Shapoorabadi, R.J., Sinclair, A.N., Konrad, A.: Improved Finite Element Method for EMAT Analysis and Design. IEEE Transactions on Magnetics 37(4), 2821–2823 (2001) 9. Shapoorabadi, R.J., Sinclair, A.N., Konrad, A.: The Governing Electrodynamic Equations of Electromagnetic Acoustic Transducers. J. Appl. Phys. 10E102, 1–3 (2005) 10. Kawashima, K.: Electromagnetic Acoustic Wave Source and Measurement and Calculation of Vertical and Horizontal Displacements of Surface Waves. IEEE Transactions on Sonics and Ultrasonics SU-32(4), 514–522 (1985) 11. Jian, X., Dixon, S., Edwards, R.S.: Ultrasonic Field Modeling for Arbitrary Non-Contact Transducer Source. In: Third Intl. Conf. on Experimental Mechanics and Third Conf. of the Asian Committee on Experimental Mechanics, vol. 5852, pp. 515–519 (2005) 12. Jian, X., Dixon, S., Grattan, K.T.V., Edwards, R.S.: A Model for Pulsed Rayleigh Wave and Optimal EMAT Design. Sensors and Actuators A 128, 296–304 (2006) 13. Nakano, H.: Some Problems Concerning the Propagations of the Disturbance in and on Semi-Infinite Elastic Solid. Geophys. Mag. 2, 189–348 (1930)

The Design of Neuron-PID Controller for a Class of Networked Control System under Data Rate Constraints Lixiong Li, Rui Ming, and Minrui Fei School of Mechanical Engineering and Automation, Shanghai University, Shanghai Key Laboratory of Power Station Automation Technology, Shanghai, 200072, P.R. China [email protected]

Abstract. In networked control system where control loop is closed over communication network, limited data rate may deteriorate control performance even destabilize the control system. In this paper, for a second-order control system closed over typical control networks, performance analysis with data rate constraints is conducted and Critical Data Rate (CDR) is explored. Furthermore, under the circumstances that CDR is available, Neuron-PID controller with Smith predictor is employed to eliminate the negative influence cause by communication network. Finally, case study is provided and acceptable control performance can be guaranteed with this design methodology. Keywords: Networked Control System, Critical Data Rate, Performance of Control, Neuron-PID, Smith Predictor.

1 Introduction Communication networks are frequently employed as transmission medium in control systems. However, in networked control system (NCS), the inserted communication networks inevitably introduce some non-determinacy and then degrade the control performance [1]. In terms of control theory, it is a popular assumption that limitations in the communication links, such as limited data rate do not affect performance significantly [2, 3]. Therefore, the selection of data rate is usually ignored during the analysis and design of the NCS for control engineers. In this paper, network performance (Quality of Service, QoS) and control performance (Quality of Performance, QoP) analysis under data rate constraints [3-6] are conducted when the network is at low network loads, and the Critical Data Rate (CDR) can be acquired. Then the design of controller can be implemented based on CDR information. Our main purpose is to explore a new design road to controller in NCS under limited data rate.

2 Performance Analysis under Data Rate Constraints In this section, the negative influence on network performance and control performance induced by limited data rate communication links is explored. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 475–483, 2010. © Springer-Verlag Berlin Heidelberg 2010

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2.1 Network Delay Network performance refers to the service quality of a communications as seen by the customer (node or computer in NCS). There are many different ways to measure the performance of a network, as each network is different in nature and design. In terms of Networked Control System, network-induce delay is a frequently-used and key metric of network performance. The delay on the network channel, Ttx is one of the important components in network-induced delay (see literature [7] for details). Ttx includes the total transmission time of a message and the propagation time of the network, and can be described as:

Ttx = Tsend + Tprop Tsend =

L R

(1) (2)

where Tsend represents the frame sending time, Tprop represents the propagation time. In expression (2), L is the length of message size, R is the data rate. The propagation delay Tprop is far smaller than Tsend and can be ignored in this paper. When the network is at low loads, Ttx or Tsend is dominant over other parts of the network-induced delay, upon that inverse relation is expected between data rate and delay. 2.2 Performance Simulation A preliminary case study on the network performance and control performance is performed for three typical control networks: Ethernet, ControlNet and DeviceNet. For the network performance, assuming length of frame is 512bits, 10 nodes with equal periods and data sizes are employed by simulation and the simulation results are respectively shown in Figure 2 (see dashed lines). It is clear that lower data rate induces longer delay; rough inverse relation is found between data rate and delay when networks are at low network loads. For control performance, the plant to be controlled is a DC servo motor, which can be described by the following second-order transfer function:

G (s ) =

1000 s(s + 1)

In this case, discrete PID strategy is employed for the controller, and control performance simulation under different data rates is conducted. The average ITAE is acquired and comparison of performance is implemented, as shown in Figure 1 (see solid lines). Similar to the delays, lower data rate induces poorer ITAE performance; and significant inverse relation can be found between data rate and ITAE when networks are at low network loads.

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(a) Ethernet

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(b) ControlNet

(c) DeviceNet Fig. 1. Performance of NCS over Control Networks (at low network loads)

2.3 Critical Data Rate Critical Data Rate (CDR) is defined as an important data rate, where lower data rate might induce significant performance deterioration [8]. In some cases, it can be regarded as the Minimum Acceptable Data Rate for NCS. Here The Critical Data Rate can be acquired approximately using the method in literature [8], as the midpoint of Point A and B. In Figure 2, Point A responds to the rate point where the value of ΔJ/ΔR approaches zero, and Point B responds to the rate point where the value of ΔJ/ΔR approaches infinity. The Critical Data Rate for the simulation in §2.2 can be acquired using this method, and each CDR can be illustrated in Figure 4. The CDR is at 193.5kbps for Ethernet, 194.0kbps for ControlNet and 132.4kbps for DeviceNet (see Figure 3 for details). As a result, in this low network loads case, it is sufficient to guarantee acceptable control performance if the data rate is larger than 200kbps.

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Fig. 2. Critical Data Rate

(a) Ethernet

(b) ControlNet

(c) DeviceNet Fig. 3. Critical Data Rate of NCS over Control Networks (at low network loads)

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3 Neuron-PID Control Algorithm In this section, Neuron-PID controller, plus Smith predictor with buffer technology is implemented, as the data rate of control network is chosen near the Critical Data Rate mentioned above. 3.1 Buffer Technology Smith predictor can be used to solve the time-delay system problem, by transferring the delay outside the control loop. However, for the NCS with time-varying networkinduced delays, accurate models of the plant and delays should be provided. The Smith predictor may become an impossible approach when the delays are randomly time-varying [9]. Randomly time-varying delay may make the Smith predictor a failure. In this regards, buffers should be installed to provide fixed delays. Our intent here is to present strategies to solve this problem and improve overall control performance with buffer technology. We assume that both the network-induced delays τsc and τca are bounded, and it is always possible to reduce the problem of time-varying delay to a time-invariant one by introducing suitable additional delay such that the sum of the delays experienced by a signal on the communication links is always constant. We call this way the maximum delay policy. This policy can be implemented by introducing suitable buffers at both the input of controller and identifier as illustrated in Figure 4.

Fig. 4. Buffers in NCS

3.2 Neuron PID Controller As the controller is running under data rate constraints in NCS, a robust control algorithm should be presented to eliminate the network non-determinacy. In this paper, neuron-PID control algorithm is employed, where a single neuron is adopted to implement PID control strategy with self-learning parameters. The inputs of the neuron can be described as:

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x1 = e(k ) − e(k − 1) x2 = e(k )

(3)

x3 = e(k ) − 2e(k − 1) + e(k − 2 ) The output of neuron can be described as the controller output (see Figure 5):

Δ u (k ) = K (w1 ⋅ x1 + w 2 ⋅ x 2 + w 3 ⋅ x 3 ) 3

u (k ) = u ( k − 1) + K ∑ wi′ (k ) xi (k )

(4) (5)

i =1

3

wi′ (k ) = wi (k ) / ∑ wi (k )

(6)

i =1

where wi (i = 1,2,3) represents weight coefficient of the neuron controller, K is coefficient of proportionality, and then K

P

= Kw 1 , K I = Kw 2 , K D = Kw 3

Proportional Gain KP, Integral Gain KI and Derivative Gain KD can be tuned through Hebb learning strategy (see formula (7)), and better control performance can be expected.

w1 ( k ) = w1 (k − 1) + η P z ( k )u (k )(e(k ) + Δe(k )) w2 (k ) = w2 ( k − 1) + η I z (k )u (k )(e( k ) + Δe( k )) w3 ( k ) = w3 ( k − 1) + η D z ( k )u (k )(e(k ) + Δe( k )) where Δ e (k ) = (e (k ) − e (k − 1 )), z (k ) = e (k ) .

rin (k )

x1 x2 x3

w1 w2 w3

∑

Δu (k )

u (k )

z−1

Fig. 5. Neuron-PID Controller

(7)

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3.3 Neuron-PID -Smith Algorithm An early control methodology for time delayed plants is the Smith predictor, in which the plant model is utilized to predict the non-delayed output of the plant and move the delay out of the control loop. Recent Smith predictor based tele-operation control architectures have used linear or fixed-parameter dynamic approximations of the slave/environment at the master for environment contact prediction. It is thus imperative in applications of the Smith predictor compensation for networked control. As control over a network using delay measurement and the Smith predictor based approach is now emerging, in this paper, Neuron PID controller along with Smith predictor is employed. With the installation of buffers in Figure 5 the total end-to-end delay can be got as:

τ max = τ camax + τ scmax Then this scheme of Neuron-PID controller with Smith predictor can be realized, and the diagram of control system is shown in Figure 6.

e−τ

e−τ

max

max ca

s

s

e−τ

max sc

s

Fig. 6. Neuron PID Controller with Smith Predictor

4 Case Study In this case study, the plant to be controlled is just the one used in §2.2, and Ethernet is used as communication network, step signal used as reference input. As mentioned above, the Critical Data Rate for Ethernet is 193.5kbps. In this regard, data rate is selected as 200kbps. Here the performance margin is so small that a robust control strategy should be employed. Under data rate constraint, PID, PID-Smith and neuron-PID-Smith control algorithms are respectively used to conduct simulation. The simulation results are shown in Figure 7. Neuron-PID-Smith enjoys the best performance among these three algorithms. The comparison of ITAE performance can be found in Table 1.

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Fig. 7. System output

Table 1. Control Performance of Different Control Algorithms Combination

Combination

PID

PID-Smith

Neuron-PID-Smith

ITAE

0.0054

0.0022

0.0017

5 Conclusions and Future Work In this paper, performance analysis of NCS under data rate constraints is investigated for three typical control networks at low network loads. With the Critical Data Rate explored by performance analysis, a neuron-PID controller with Smith predictor is implemented for a second-order system. Our future efforts will focus on network scheduling algorithms, such as RM and EDF, to present a co-design approach combined with network scheduling and control algorithm.

Acknowledgement The work was supported by National Science Foundation of China under Grant No.60774059.

References 1. Murray, R.M., Astrom, K.J., Boyd, S.P., Brockett, R.W., Stein, G.: Future Directions in Control in an Information-rich World. IEEE Control Systems Magazine 23, 20–33 (2003) 2. Baillieul, J., Antsaklis, P.J.: Control and communication challenges in networked real-time systems. Proceedings of the IEEE 95, 9–27 (2007)

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3. Nair, G.N., Fagnani, F., Zampieri, S., Evans, R.J.: Feedback control under data rate constraints: an overview. Proceeding of the IEEE 95, 108–137 (2007) 4. Nair, G.N., Evans, R.: Stabilization with data-rate-limited feedback: Tightest attainable bounds. Syst. Control Lett. 41, 49–76 (2000) 5. Nair, G.N., Evans, R.: Exponential stability of finite-dimensional linear systems with limited data rates. Automatica 39, 585–593 (2003) 6. Tatikonda, T., Mitter, S.: Control under communication constraints. IEEE Transactions on Automatic Control 7, 1056–1067 (2004) 7. Lian, F.L., Moyne, J.R., Tilbury, D.M.: Performance Evaluation of Control Networks: Ethernet, ControlNet, and DeviceNet. IEEE Control Systems Magazine 21, 66–83 (2001) 8. Li, L., Fei, M.R., Ding, J.: Performance analysis of a class of networked control systems under data rate constraints. Journal of Shanghai University (English Edition) 13, 356–358 (2009) 9. Chen, C.H., Lin, C.L., Hwang, T.S.: Stability of Networked Control Systems with TimeVarying Delays. IEEE Communications Letters 11, 270–272 (2007)

Stochastic Optimization of Two-Stage Multi-item Inventory System with Hybrid Genetic Algorithm Yuli Zhang1, Shiji Song1, Cheng Wu1, and Wenjun Yin2 1

Department of Automation, Tsinghua University, Beijing 100084 2 IBM China Research Lab, Beijing 100084 {yl-zhang08,shijis}@mails.tsinghua.edu.cn

Abstract. This paper considers a two-stage, multi-item inventory system with stochastic demand. First we propose two types of exact stochastic optimization models to minimize the long-run average system cost under installation and echelon (r, nQ) policy. Second we provide an effective hybrid genetic algorithm (HGA) based on the property of the optimization problem. In the proposed HGA, a heuristic search technique, based on the tradeoff between inventory cost and setup cost, is introduced. The long-run average cost of each solution in the model is estimated by Monte Carlo method. At last, computation tests indicate that when variance of stochastic demand increases, echelon policy outperforms installation policy and the proposed heuristic search technique greatly enhances the search capacity of HGA. Keywords: Two-stage inventory; Stochastic optimization; Heuristic search; Hybrid Genetic Algorithm; Monte Carlo method.

1 Introduction Multi-stage inventory control problem have been studied for decades. The inventory system can be categorized into series system, assembly system, distribution system, tree system and general system. Clark and Scarf [1] firstly study the serial inventory system and propose echelon inventory police. Then many researchers extend this model to general arborescent structure (e.g. Bessler and Veinott [2]) and assembly systems (e.g. Axsater and Roling [3]). For assembly and distribution inventory system, many useful results exist. For example, it is proved that so-called zero-inventory policies always dominate others and stationary-interval policies also perform well [4]. However, general inventory system is more complex. Many scholars have studied inventory models with uncertainties. Giimiis and Guneri [5] give a recent literature review of the multi-echelon inventory management with uncertainty. Axsater and Zhang [6] consider a distribution system and compare the echelon and installation policy. Inventory model with stochastic lead-time is investigated by Mohebbi and Posner [7] and Chiang and Monahan [8]. This paper considers a two-stage, multi-item inventory system with stochastic demand. The complexity of this problem mainly comes from two aspects: first this two-stage inventory has a general many-to-many supply-demand relationship; second, the uncertainty makes it difficult to evaluate the performance of different policies. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 484–492, 2010. © Springer-Verlag Berlin Heidelberg 2010

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In this paper, first we propose two types of exact stochastic optimization models under installation and echelon (r, nQ) policy. Then a hybrid genetic algorithm has been designed to minimize the long-term average cost. In the HGA, the initial r and Q are randomly generated based on the near-optimal solution of deterministic problem; the heuristic search technique takes account into the balance between the inventory cost and the setup cost for each node respectively; Monte Carlo method are used to evaluate the long-term average cost. At last, by the numerical experiments, we find that echelon (r, nQ) policy outperforms installation policy when the variance of stochastic demand is increasing and the proposed heuristic search technique greatly enhances the search capacity of the HGA. In section 2, we provide notations. In section 3 stochastic optimization models are established. In section 4, HGA is proposed. In section 5 numerical experiments are carried out and in section 6 the conclusions of this paper are summarized.

2 Notation and Preliminary We assume that items stocked in different warehouses are different and in order to produce one unit of product, factories must order certain amount of items from the upstream warehouse. Each factory or warehouse is referred to as a node on stage 1 or 2. We consider two types of cost: one is inventory cost which contains holding cost and shortage cost; another is setup cost. The basic parameters of this model are denoted as follows , : number of node on stage 1 and stage 2, , : node on stage or item on stage , 1,2, per unit per unit time, , , , : holding cost and shortage cost of item , : demand at node 1, at period , , ∑ to 2, , and , , , : setup cost of an order from 1, , , , : setup cost of an order from 2, to outside supplier, , to 1, and from outside supplier to 2, , , , , : leadtime from 2, . , : the quantity of item m comsumed in producing a unit of item 1, Inventory order position (IOP) contains inventory on hand, outstanding orders in transit, outstanding orders backlogged and minus backorders; inventory position (IP) is equal to IOP minus the outstanding orders backlogged while inventory level (IL) contains only inventory on hand and minus backorders. Installation inventory terms focus only on the node while echelon terms contain its inventory and its downstream inventory. Note that for node on stage 1 all forms of installation inventory is always equal to the corresponding echelon inventory. For inventories on stage 1 and 2, these terms are denoted as follows , ,

, ,

, , ,

,

, ,

, , ,

, ,

,

, , , , ,

,

, of item 1, at node 1, at beginning of , , of item 1, at node 1, at end of period , , of item 2, at node 1, at beginning of t, , of item 2, at node 1, at end of period t, , at node 2, at beginning of , , at node 2, at end of ,

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, , , at node 2, at beginning of , , , , at node 2, at end of . The decision variables under installation and echelon policy are denoted as follows and at node , , , , , , and order quantity at node , , , , , , : and order quantity at node , at t, , 0, , 0, , 0 0 , , : , , , , 1, , 1, , 0 0 ,

,

3 Stochastic Optimization Model In this section, we establish the stochastic optimization model under installation and echelon (r, nQ) policy. First, we give the following assumptions,

··Demand is backordered and not lost in case of shortage. ··Lead-time is deterministic constant. ··No initial backorders at beginning on all stage. ··No outstanding order backlogged at nodes on stage 2.

Then we introduce intermediate variables to make the analysis simply. backorders of item 1, at node 1, at period , backorders of item 2, at node 1, at period , , , backorders of item 2, at node 2, for node 1, at , , , outstanding orders in transit from 2, to 1, at period . , , To calculate IOP, we have the following equations ,

, ,

1

, ,

, ,

∑

, ,

,

/

, ,

1

, ,

0,

,

,

,

,

(1) (2)

,

, where

,

,

τ

, ,

is (3)

,

,

,

,

,

Now we define the intermediate inventory on hand, which occurs after the replenishment and before the demand occurs as follows , ,

, ,

,

1

, ,

, ,

, ,

⁄

,

(4) (5)

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Now, we define the fulfilled demand at t as follows min

,

,

1

,

1

,

, ,

,

,

1

,

, ,

,

(8) , ,

(9)

(10)

, ,

, ,

,

(7)

0 0

, ,

,

,

,

0, x,

where

(6)

,

,

, ,

, ,

,

,

,

⁄

,

(11)

,

.

(12)

Then we introduce the intermediate variable inventory on hand after just replenishment at 2, , ,

,

,

,

τ

,

(13) (14)

,

1

,

,

∑

1

,

at .

(15)

,

,

Then, we define the FIFO function as ,

,

,

, ,

, ,

,

,

∑

,

,

,

,

1

, ,

(16) (17)

, ,

At last, we obtain the objective cost function as follows ∑ ∑

where ∑

∑

,

, ,

,

∑

∑

, , ,

, ,

, ,

, ,

(18) ,

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∑

,

∑

,

,

,

,

Since the difference between those two policies lies in the calculation of inventory, so model under echelon policy can be obtained by following simply modification. ,

, ,

1

, ,

,

∑

1

,

∑

,

,

τ

,

(19) ,

(20)

,

4 Hybrid Genetic Algorithm Although Van Slyke and Roger Wets [9] propose a well-known L-shaped method for stochastic two-stage linear recourse problems, there is still no general efficient algorithm for stochastic nonlinear programming problems. HGA has been developed to makes use of the heuristic information [10]. GA has been applied to inventory problem. See Moon et al. [11]. The improvements of the HGA lie in the initialization, heuristic search technique and evaluating cost by Monte Carlo method, which will be discussed in the following. 4.1 Initialization via Deterministic Model and Monte Carlo Method Consider the corresponding deterministic two-stage multi-item inventory system and disregard the leadtime. The long term run cost can be approximated as ,

∑

min

,

∑ ,

,

,

,

,

,

,

,

,

,

,

,

∑

,

,

,

,

,

,

,

,

,

,

(21)

, ,

∑

,

, ,

,

,

According to Economic Order Quantity when backorder is allowed [12], we have

,

,

⁄

,

,

,

,

,

,

, ,

,

,

(22)

Let vector r, Q denotes a solution and let D denote the demand. Assume the probability density function of D is f (D) and the cost function is C=p(r, Q, D).Then the objective function to minimize the expectation of the total cost can be obtained as min

, ,

(23)

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We use Monte Carlo method to estimate the average cost as following, min

, ,

∑

, ,

(24)

4.2 Problem-Based Heuristic Search Considering the deterministic optimization model in subsection 4.1, the order quantity Q has a direct impact on the inventory cost. There is a balance between the inventory cost and setup cost when we change Q [4]. For the reorder point r, it seems that the increase of r will lead to increase of holding cost and decrease of shortage cost since r can be viewed as an approach to deal with the uncertainty during the leadtime. For single stage inventory, the above analysis is correct; but in multi-stage inventory, it is not necessarily true. However, as the following numerical experiments show, if the target cost are relative large, adjusting parameters will almost decrease the total cost. See Fig 2.

Fig. 2. Numerical experiments for heuristic rules

For parameter r, the installation holding cost and shortage cost keep a consistent increase or decrease and the setup cost is unchangeable when r varies; however, when parameter Q varies, the installation setup cost and main trend of holding cost (denoted as “ * ” in the table 1) keep a consistent change and shortage cost change irregularly. From the above analysis and the computational results from Fig 2, we summarize the heuristic rules in table 1. Golden section search approach is an effective line search methods for strictly unimodal objective function. See Bertsekas [13] for detail. Table 1. Heuristic search rules Cost item CA high CH high CB high

R Decrease Increase

Q Increase Decrease*

The effectiveness of this heuristic search method will be validated in the next section. For detail of the selection, mutation and crossover operation, see [14].

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5 Numerical Experiments In this section, numerical experiments are carried out to compare the proposed two types of stochastic optimization models and to validate the effectiveness of the HGA by cost-time comparison with the general GA in deterministic model and stochastic model. The HGA is implemented by Java and runs on Intel (R) Xeon (R) CPU E5410. First, we consider the comparison of those two models under different stochastic demand variances. In the test example, M = N = 2 and T =40. The parameters in the model are given in table 2 and 3. Table 2. Deterministic parameters Demand node (1,1) (1,1) (1,2) (1,2) (2,1) (2,2) (1,1)

Supply node (2,1) (2,2) (2,1) (2,2) Outside Outside (2,1)

Leadtime 2 2 3 4 1 2 2

,

1 2 3 2 1 1 1

Table 3. Stochastic parameters parameters h1( j ), j=1,2 h2( j ), j=1,2 a1(i,j), i,j=1,2 A2( j ), j=1,2 p1( j ), j=1,2

Range(uniform distribution) [4,5] [2,3] [30,40] [60,70] [10,20]

To compare those two policies, HGA, with population size=100, max iteration=50 mutation possibility=0.4, crossover possibility=0.3 is implemented. We assume the stochastic demand satisfies truncated normal distribution with average vector (6, 8). The following Fig 3. (a)-(b) are obtained by applying Echelon and Installation policy to 10 different problems. Numerical examples show that when the uncertainty increases, the echelon (r, nQ) policy outperforms the installation (r, nQ) policy. Second, we compare the HGA with the general GA in two ways. By applying HGA and GA to deterministic model, we obtain Fig 4. Fig 4(a) shows that HGA can find better solution. To test the efficiency of HGA, the Fig a (b) is obtained by keeping both HGA and GA run the same time span. It shows that HGA still outperforms GA in the same run time for deterministic model. The computational results for stochastic optimization model are demonstrated at Fig 5(a) and 5(b). It shows that for stochastic optimization model, the HGA is still an effective and efficient improved algorithm for general GA. In summary, we find that when the variance of customer demand increases, echelon (r, nQ) performs better than installation (r, nQ) policy and the proposed HGA is

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Fig. 3 (a). Var(D)=(0.3, 0.4)

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Fig. 3 (b). Var(D)=(1.5, 2.0)

Fig. 4(a). Comparison for same iteration Fig. 4(b). Comparison with same run time

Fig. 5(a). Comparison for same iteration Fig. 5(b). Comparison with same run time

effective to obtain better near-optimal solution and more efficient than general GA for this two-stage multi-item inventory control problem.

6 Conclusions This paper considers two-stage, multi-item inventory problem with stochastic demand. First, we present two types of exact stochastic optimization models under installation and echelon (r, nQ) policy. Those exact stochastic models provide basis for further research and application. Second, we propose a HGA based on problem property to solve the proposed models. By analyzing the deterministic inventory system and numerical computation, we propose a set of effective heuristic rules. Then a heuristic search approach based on the balance between the inventory cost and setup cost has been designed and is embodied into the general GA. The initialization has been

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improved and evaluation operations are implemented by Monte Carlo simulation. Numerical experiments show that: when the demand variance increase, the echelon (r, nQ) policy outperforms the installation policy; for both the deterministic model and stochastic model, the proposed HGA is not only effective to find better solution than GA, but also more efficient when they run the same time.

Acknowledgements The paper is supported by UK-China Bridge in Sustainable Energy and Built Environment (EP/G042594/1), Distinguished Visiting Research Fellow Award of Royal Academy of Engineering of UK, NSFC (No.60874071) and RFDP (No.20090002110035).

References 1. Clark, A.J., Scarf, H.: Optimal Policies for a Multi-echelon Inventory Problem. Management Science 50(12), 1782–1790 (2004) 2. Bsssler, S.A., Veinott, A.F.: Optimal Policy for a Dynamic Multi-echelon Inventory Model. Naval Research Logistic Quarterly 13(4), 355–399 (2006) 3. Axsater, S., Rosling, K.: Installation vs. Echelon Stock Policies for Multi-level Inventory Control. Management Science 39, 1274–1280 (1993) 4. Zipkin, P.H.: Foundations of Inventory Management. McGraw-Hill, New York (2000) 5. Giimiis, A.T., Guneri, A.F.: Multi-echelon Inventory Management in Supply Chains with Uncertain Demand and Lead Times: Literature Review from an Operational Research Perspective. In: Proceedings of the Institution of Mechanical Engineers, Part B, pp. 1553–1570. Professional Engineering, Publishing, London (2007) 6. Axsater, S., Zhang, W.-F.: A Joint Replenishment Policy for Multi-echelon Inventory Control. Int. J. Prod. Econ. 59, 243–250 (1999) 7. Mohebbi, E., Posner, M.: Continuous-review Inventory System with Lost Sales and Variable Leadtime. Naval Research Logistic 45(3), 259–278 (1998) 8. Chiang, W., Monahan, G.E.: Managing Inventories in a Two-echelon Dual-channel Supply Chain. European Journal of Operational Research 162(2), 325–341 (2005) 9. Van Slyke, R.M.: Roger Wets.: L-Shaped Linear Programs with Application to Optimal Control and Stochastic Programming. Journal on Applied Mathematics 17(4), 638–663 (1969) 10. Pan, Z., Kang, L., Chen, Y.: Evolutionary Computation. Tsinghua University Press, Beijing (1998) 11. Moon, I.K., Cha, B.C., Bae, H.C.: Hybrid Genetic Algorithm for Group Technology Economic Lot Scheduling Problem. International Journal of Production Research 44(21), 4551–4568 (2006) 12. Axsater, S.: Inventory Control. Kluwer Academic Publishers, Boston (2000) 13. Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999) 14. Zhang, Y., Song, S., Wu, C., Yin, W.: Multi-echelon Inventory Management with Uncertain Demand via Improved Real-Coded Genetic Algorithm. In: Proceedings of the International Symposium on Intelligent Information Systems and Applications, pp. 231–236. Academy Publisher, Oulu (2009)

Iterative Learning Control Based on Integrated Dynamic Quadratic Criterion for Batch Processes Li Jia1, Jiping Shi1, Dashuai Cheng1, Luming Cao1, Min-Sen Chiu2 1 Shanghai Key Laboratory of Power Station Automation Technology, Department of Automation, College of Mechatronics Engineering and Automation, Shanghai University, Shanghai 200072, China 2 Faculty of Engineering, National University of Singapore, Singapore

Abstract. An integrated neuro-fuzzy model and dynamic R-parameter based quadratic criterion-iterative learning control is proposed in this paper. Quadratic criterion-iterative learning control with dynamic parameters is used to improve the performance of iterative learning control. As a result, the proposed method can avoid the problem of initialization of the optimization controller parameters, which are usually resorted to trial and error procedure in the existing iterative algorithms. Lastly, example is used to illustrate the performance and applicability of the proposed method. Keywords: batch processes; iterative learning control; quadratic criterion; dynamic parameter.

1 Introduction Recently, batch processes have been used increasingly in the production of low volume and high value added products, such as special polymers, special chemicals, pharmaceuticals, and heat treatment processes for metallic or ceramic products [1-10]. For the purpose of deriving the maximum benefit from batch processes, it is important to optimize the operation policy of batch processes. Therefore, optimal control of batch processes is very significant. The key of optimal control depends on obtaining an accurate model of batch processes, which can provide accurate predictions. Usually, developing first principle models of batch processes is time consuming and effort demanding. For example, Terwiesch et al. pointed out that the effort for building a useful kinetic model that is still just valid for a limited range of operating conditions often exceeds one man year [2]. This may not be feasible for batch processes where frequent changes in product specifications occur and a type of product is usually manufactured for a limited time period in response to the dynamic market demand [3]. Motivated by the previous works, a neuro-fuzzy model-based iterative learning optimization control is proposed in this paper. The dynamic R-parameter based quadratic criterion is employed in this work to improve the performance. Moreover, rigorous theory analysis indicates that the zero-error -tracking would be possible under K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 493–498, 2010. © Springer-Verlag Berlin Heidelberg 2010

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R-adjustment control strategy. That is to say, the changes of the control policy converge to zero with zero-error -tracking as the batch index number tends to infinite, which are normally obtained only on the basis of the simulation results in the previous works.

2 Integrated Neuro-Fuzzy Model for Batch Processes Suppose that define that

t f is the batch end time, which is divided into T equal intervals. And

U k = [uk (1),

k-th batch and

, uk (T )]T is a vector of control policy variables during

Yk = [ yk (1),

, yk (T )]T is a vector of product quality variables

during k-th batch, where k represents the batch. y ∈ R n and u ∈ R m are, respectively, the product quality and control action variables. In this paper, neuro-fuzzy Model (NFM) is employed to build the nonlinear relationship between uk and yk , namely

yˆk (t +1) = NFM[ yk (t), yk (t −1), , yk (t − ny +1), uk (t), uk (t −1) , uk (t − nu +1)] where

(1)

n y and nu are integers related to the model order.

In this work, only the product quality at the end of a batch, y d (t f

) is concerned so

the proposed NFM is used to predict the product quality at the end of the k-th batch,

yˆ k ( t f

)

U k = [uk (0),

for the giving initializations and input sequence of

k -th batch

, uk (T − 1)]T , noted by yˆ k ( t f ) = f NFM (U k )

(2)

f NFM is the nonlinear function related to NFM. In batch process, there are three axes, namely t , k and x . In the direction of axes k , there are K batches of data vectors, in which one vector is with the dimensions of n y + nu , and in the direction of axes t , it is divided into T equal intervals. Then the data are unfolded up from the axes of t and thus the number of the data is K × T (noted by KT ).

where

Therefore, Eq 1 can be rewritten to

yˆ (kt + 1) = NFM [ x ( kt )]

(3)

where

x(kt ) = [ y (kt ), y (kt − 1), and

, ykt (t − ny + 1), u (kt ), u (kt − 1),

，

kt represents the data point kt = 1, 2,

, KT .

, ukt (t − nu + 1)]T

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3 Dynamic R-Parameter Based Quadratic Criterion-Iterative Learning Control for Batch Processes In this work, k is the batch index and t is the discrete batch time, uk and yk are, respectively, the input variable of the product qualities and product quality variable of batch processes, U k is the input sequence of a batch, yˆ k (t ) is the predicted values of product quality variable at time t of the k-th batch,

yk +1 (t f ) is the modified predictions

of the neuro-fuzzy model at the end of the k+1-th batch and NFM model is employed to predict the values of product quality variable at the end of the k+1-th batch

yˆ k +1 (t f ) .

In this study, based on the information from previous batch, the optimization controller can find an updating mechanism for the input sequence U k +1 of the new batch using improved iterative optimal control law derived by the rigorous mathematic proof method discussed shortly. At the next batch, this procedure is repeated to let the product qualities at the batch end convergence asymptotically towards the desired product qualities y d (t f ) . Owing to the model-plant mismatches, the process output may not be the same as the one predicted by the model. The offset between the measured output and the model prediction is termed as model prediction error defined by

eˆkf = ykf − yˆ kf

(4)

Since batch processes are repetitive in nature, the model prediction at the end of the k-th batch, yˆ k +1 (t f ) , can be corrected by

ykf+1 = yˆ kf+1 + Where

α ≥0

α k +1

eˆk f

(5)

is a bias correction parameter, and eˆk (t f

) the average model prediction

errors of all the previous batches described as

eˆk f =

1 k f 1 k f f ∑ eˆi = ∑ ( yi − yˆi k i =1 k i =1

)

(6)

Observe that the objective is to design a learning algorithm to implement the control trajectory U k so that the product qualities at the batch end converges asymptotically towards the desired product qualities. Thus, instead of using the whole reference sequences for product qualities, which is less important for batch processes product quality control, only the desired product quality yd Here the following quadratic objective is employed:

( t ) at the end of a batch is used. f

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min J (U k +1 , k + 1) = ekf+1

U k +1∈Θ U

where ek +1 = f

2

Q

+ ΔU k +1

2

(7)

Rk

ydf − ykf+1 is the difference between desired product qualities ydf and

modified predictions of the neuro-fuzzy model at the end of the k+1th batch

ykf+1 .

The updating function of optimization controller is defined as

ΔU k +1 = U k +1 − U k

(8)

u low ≤ uk +1 (t ) ≤ u up

(9)

u low and u up are the lower and upper bounds of the input trajectory, Q and R are both weighting matrices and defined as Q = q × IT ， R k = rk × IT where rk and q where

are all positive scalar. Theoretically, the increase of r and q by same multiple does not affect on the optimal solution of optimization problem (abbreviated as OPTP) described by eq.6 to eq.11. As a result, it is equal to adjust r or q at the same time. Thus, in our work, r is defined as dynamic parameter while q is a constant. The convergence and tracking performance of the proposed dynamic parameters based quadratic criterion-iterative learning control are analyzed as following theorem. Theorem 1. Considering the optimization problem described by eq. 6 to eq.11 with the adaptive updating law eq.15, if the limitation of the dynamic parameter R k satisfies

R k ≠ 0 , then lim ΔU k = 0 and lim ΔWk = 0 hold. the condition of lim k →∞ k →∞ k →∞ Proof: the proof is omitted. Theorem 2. The tracking error under OPTP* described by eq.12 and eq.13 is bounded-tracking for arbitrary initial control profiles U k0 . Moreover, if the condition that the function

g (i ) = e(i )

2 Q

is derivable and the optimization solution is not in

the boundary, it is zero-tracking for arbitrary initial control profiles U k0 . Proof: the proof is omitted.

4 Example In this section we implement the proposed method to control product quality of a continuous stirred tank reactor (CSTR), in which a first-order irreversible exothermic k1 k2 reaction A ⎯⎯ → B ⎯⎯ → C takes place [11-13]. This process can be described by the following dynamic equations

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x1 = −4000 exp(−2500 / T ) x12 x2 = 4000 exp(−2500 / T ) x12 − 6.2 × 105 exp(−5000 / T ) x2 where T denotes reactor temperature, x1 and x2 are, respectively, the reactant concentration. In this simulation, the reactor temperature is first divided into 10 equal intervals and normalized by using Td = (T − Tmin ) / (Tmax − Tmin ) , in which Tmin and Tmax are respectively 298 by 0 ≤ Td

（K）and 398（K）. T is the control variable which is bounded d

≤ 1 , and x2 (t ) is the output variable. The control objective is to manipulate the reactor temperature Td to control concentration of B at the end of the batch x2 (t f ) .

Fig.1 represents the control trajectory of the system while the relevant output is showed in Fig.2. The product quality can achieve 0.6086 while the tracking error decreasing to 1.4000×10-3. 0.7 1th batch 3th batch 10th batch 20th batch

0.6

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0.1

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0.4

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1

Fig. 2. Output trajectory

5 Conclusion Considering the potentials of iterative learning control as a framework for industrial batch process control and optimization, an integrated neuro-fuzzy model and dynamic

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R-parameter based quadratic criterion-iterative learning control is proposed in this paper. Moreover, we make the first attempt to give rigorous description and proof to verify that a perfect tracking performance can be obtained, which are normally obtained only on the basis of the simulation results in the previous works. Then examples illustrate the performance and applicability of the proposed method. Acknowledgement. Supported by Research Fund for the Doctoral Program of Higher Education of China (20093108120013), Shanghai Science Technology commission (08160512100, 08160705900, 09JC1406300 and 09DZ2273400), Grant from Shanghai Municipal Education commission (09YZ08) and Shanghai University, "11th Five-Year Plan" 211 Construction Project.

References 1. Bonvin, D.: Optimal operation of batch reactors: a personal view. Journal of Process Control 8, 355–368 (1998) 2. Terwiesch, P., Argarwal, M., Rippin, D.W.T.: Batch unit optimization with imperfect modeling: a survey. J. Process Control. 4, 238–258 (1994) 3. Zhang, J.: Batch-to-batch optimal control of a batch polymerisation process based on stacked neural network models. Chemical Engineering Science 63, 1273–1281 (2008) 4. Cybenko, G.: Approximation by superposition of a sigmoidal function. Math. Control Signal Systems. 2, 303–314 (1989) 5. Girosi, F., Poggio, T.: Networks and the best approximation property. Biological Cybernetics 63, 169–179 (1990) 6. Park, J., Sandberg, I.W.: Universal approximation using radial-basis-function networks. Neural Comput. 3(2), 246–257 (1991) 7. Su, H.T., McAvoy, T.J., Werbos, P.: Long-term prediction of chemical process using recurrent neural networks: a parallel training approach. Ind. Eng. Chem. Res. 31, 1338–1352 (1992) 8. Tian, Y., Zhang, J., Morris, A.J.: Modeling and optimal control of a batch polymerization reactor using a hybrid stacked recurrent neural network model. Ind. Eng. Chem. Res. 40, 4525–4535 (2001) 9. Tian, Y., Zhang, J., Morris, A.J.: Optimal control of a batch emulsion copolymerisation reactor based on recurrent neural network models. Chemical Engineering and Processing 41, 531–538 (2002) 10. Xiong, Z., Zhang, J.: Modelling and optimal control of fed-batch processes using a novel control affine feedforward neural network. Neurocomputing 61, 317–337 (2004) 11. Xiong, Z.H., Zhang, J.: Product quality trajectory tracking of batch processes using iterative learning control based on time-varying perturbation model. Ind. Eng. Chem. Res. 42, 6802–6814 (2003) 12. Lu, N., Gao, F.: Stage-based process analysis and quality prediction for batch processes. Industrial and Engineering Chemistry Research 44(10), 3547–3555 (2005) 13. Ray, W.H.: Advanced Process Control. McGraw-Hill, New York (1981)

Impedance Measurement Method Based on DFT Xin Wang Institute of Measurement and Process Control, Jiangnan University, Lihu Road 1800, 214122 Wuxi, China [email protected]

Abstract. A principle of the impedance measurement based on DFT is proposed, and an implementation of the principle is designed. The principle raises the requirement of a 2-channel simultaneous sampling. In order to satisfy the requirement with a single A/D converter, both a hardware solution and a software solution are proposed, and the latter, which in fact is a kind of software compensation, is preferred. A larger measurement range requires a larger dynamic range of the data acquisition, and thus a floating-point A/D converter is designed to substitute for a monolithic fixed-point A/D converter, because the dynamic range of a FP-ADC is much larger than that of a fixed-point one. As a digital measurement technique, the method takes full advantage of the powerful arithmetic and processing ability of a digital signal processor. Simulation experiments on two extreme cases are studied to verify the performance of the measurement system and the result curves are plotted. Keywords: impedance measurement, DFT, floating-point A/D converter, DSP.

1 Introduction The impedance measurement is of great importance in the electrical engineering. Measuring the impedances of the key components in a system precisely helps estimate the state of the system, which is especially true for the active components in the devices operating at radio frequency and microwave frequency [1]. Recently, impedance measurement has found its places in many other fields, such as materials science [2], biochemistry [3], and medicine [4]. At present, conventional impedance measurement techniques typically rely on manually-adjusted or auto-balancing bridge networks, vector-voltmeter systems, or resonant circuits [5].

2 Measurement Principle In Fig. 1, U is a sinusoidal voltage source, ER is the voltage across the reference resistance R, EZ is the voltage across the unknown impedance Z, and I is the current through R and Z. Remarkably, a principle using the relative relation between ER and EZ to measure Z does not impose strict requirements on the precision and stability of the power source. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 499–504, 2010. © Springer-Verlag Berlin Heidelberg 2010

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I R

ER

Z

EZ

U

Fig. 1. Diagram of the measurement principle

Consider the problem in the time-domain, where, u (t ) = U m cos(ω t + ϕ ) ,

eR ( t ) =

R

U m cos (ω t + ϕ − θ )

(1)

U m cos (ω t + ϕ − θ + φ )

(2)

R+Z

And Z

eZ ( t ) =

R+Z

where Z is the modulus of Z φ is the angle of Z and θ is the angle of (R+Z). Theoretically, one set of simultaneous values of eR(t) and eZ(t) are enough for calculating Z and φ provided that ω is known. However, using only one set of simultaneous values of eR(t) and eZ(t) can lead to a notable error. Now, consider the principle of the impedance measurement based on the DFT. If eR(t) and eZ(t) are uniformly and simultaneously sampled at a sample rate of Nω ( 2π ) in a signal cycle to produce two sequences of length N, respectively, denoted as eR(nTS) and eZ(nTS), where TS = 2π /( N ω ) and n = 0,1, 2, ⋅ ⋅ ⋅, N − 1 , then

R ⎛ 2π ⎞ U m cos ⎜ n + ω t0 + ϕ − θ ⎟ R+Z ⎝ N ⎠

(3)

⎛ 2π ⎞ U m cos ⎜ n + ω t0 + ϕ + θ − φ ⎟ R+Z ⎝ N ⎠

(4)

eR ( nTS ) = eZ ( nTS ) =

Z

where t0 is the starting time of the data acquisition. Construct aR, bR, aZ and bZ from the DFT of eR(nTS) and eZ(nTS), that is,

aR =

⎡ e nT cos ⎛ 2π n ⎞⎤ ∑ ⎢⎣ ( ) ⎜⎝ N ⎟⎠⎦⎥ N 2

N −1

R

n =0

S

(5)

Impedance Measurement Method Based on DFT

bR =

501

⎡e nT sin ⎛ 2π n ⎞⎤ ∑ ⎢⎣ ( ) ⎜⎝ N ⎟⎠⎦⎥ N

(6)

⎡ e nT cos ⎛ 2π n ⎞⎤ ∑ ⎢⎣ ( ) ⎜⎝ N ⎟⎠⎦⎥ N

(7)

⎡ e nT sin ⎛ 2π n ⎞⎤ ∑ ⎢⎣ ( ) ⎜⎝ N ⎟⎠⎦⎥ N

(8)

2

N −1

S

R

n =0

aZ =

2

N −1

S

Z

n=0

bZ =

2

N −1

S

Z

n =0

It has been proved that aZ + bZ

2

a R + bR

2

2

Z =

2

(9)

R

⎛ aZ bR − aR bZ ⎞ ⎟ ⎝ aZ aR + bZ bR ⎠

φ = tan ⎜ −1

(10)

Above is the fundamental measurement principle [6]. It can be easily verified that the equations (9) and (10) is still tenable if k*eR(t)+dR and k*eZ(t)+dZ (where dR, dZ, and k are constants) are substituted for eR(t) and eZ(t) provided that N is even. As can be see thereafter, an even N helps simplify the implementation and cancel some errors.

3 Implementation of the Principle Fig. 2 presents the block diagram of an implementation of the measurement principle above. IA1 and IA2 are two instrumentation amplifiers, and S/H1 and S/H2 are two sample-and-hold circuits.

U

R

+In Vo IA1Ref -In

S/H1 FSR/2

Z

Ref +In IA2 Vo -In

S/H2

2 to 1 Switch

FloatingPoint A/D Converter

DSP 16

Fig. 2. Block diagram of an implementation of the measurement principle

IA1 and IA2 are of the same gain. The FSR is the full scale range of the floatingpoint A/D converter (FP-ADC). Such an offset of FSR/2 is added to convert the bipolar voltages to unipolar ones, which will simplify the FP-ADC to a unipolar one. In order to realize the simultaneous data acquisition, S/H1 and S/H2 are routed to the

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same logic input. Moreover, a 2-to-1 switch is employed to accomplish the 2-channel data acquisition with a single A/D converter. At the final stage, the outputs of the A/D conversion are received and processed by a digital signal processor (DSP) to obtain the modulus and angle of Z. Alternately, another solution is a kind of software compensation. In Fig, 2, if S/H1 and S/H2 are omitted, and the outputs of IA1 and IA2 are directly connected to the inputs of the 2-to-1 switch, the data from each channel are not simultaneous but have a fixed interval. Actually, if the global sample rate of the FP-ADC is ω N π , the effective sample frequency of eR(t) and eZ(t) is 1 TS = ω N ( 2π ) . In this case, the data obtained are not eR(nTS) and eZ(nTS), but eR(nTS) and eZ((n+1/2)TS), n = 0,1, 2, ⋅ ⋅ ⋅, N − 1 . Calculating aZ and bZ using eZ((n+1/2)TS) instead of eZ(nTS) in equations (7) and (8), the modulus obtained from equation (9) is the same as before, but the angle φ ′ obtained from equation (10) has a fixed difference from the actual angle φ . Obviously, φ = φ ′ + π N . Such a solution simplifies the implementation of the system, brings down the costs and saves room for the system. The most key and difficult module in the system is the FP-ADC. The bother constructing a FP-ADC instead of employing a monolithic fixed-point A/D converter is due to the fact that experiments show that a 12-bit fixed-point A/D converter doesn’t have dynamic range large enough to handle some exceptions precisely. In fact, the quantization range of a FP-ADC is several orders of magnitude higher than that of a fixed-point analog-to-digital converter [7]. Fig. 3 is a simplified diagram of the FP-ADC, which has both large dynamic range and high throughput rate. Firstly, S/H_1 holds the input signal. Meanwhile, the programmable gain amplifier (PGA) in conjunction with the comparator circuit determines which binary quantum the input falls within, relative to the full scale range. Once the PGA has settled to the appropriate level, then S/H_2 can hold the amplified signal while the 12-bit ADC performs its conversion routine. Finally, the output data is presented as a 16-bit word, the lower 12 bits from the A/D converter form the mantissa and the upper 4 bits from the digital signal used to set the gain form the exponent. The FP-ADC achieves its high throughput rate by overlapping the acquisition time of S/H_1 and the settling time of PGA with the conversion time of the A/D converter.

Vin

S/H_1

+5V 10k 5k 2.5k 1.25k 1.25k

Vin A0

+5V Vcc 1In+ 1Out 1In2In+ 2Out 2InLM399 3In+ 3Out 3In4In+ 4Out 4InGND

Decoder Circuit

PGA Vo A1

A2

S/H_2 Vin

D11 D10 D9 D8 D7 12-bit D6 ADC D5 D4 D3 D2 D1 D0

Fig. 3. Block diagram of the floating-point A/D converter

Mantissa

Exponent

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4 Simulation Experiments In order to verify the performance of the measurement system fully but in a simple way, simulation experiments on two extreme cases are studied. In the simulation, the primary error of the system, the quantization error of the A/D conversion, is considered. In fact, many other errors can be cancelled with an even N. Assume that Um=2.4V, f = 1000Hz, FSR = 5V, R = 100Ω , and N = 160.

2

x 10

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Fig. 4. The simulation result curves with a pure resistance

Firstly, take a pure resistance as the measurand and assign a set of logarithmically equally spaced values to it respectively. The value of the angle of Z and the relative error of the modulus of Z are computed and then plotted in Fig. 4, where Z is the pure resistance. Secondly, take a pure capacitance as the measurand and assign a set of logarithmically equally spaced values to it respectively. The value of the angle of Z and the relative error of the modulus of Z are computed and then plotted in Fig. 5, where Z is the pure capacitance. It can be seen that the angle and the modulus hold different error-tolerable ranges. The effective measurement range is the range where the errors of both the angle and the modulus are tolerable. Change the reference resistance R and repeat the steps above. As is determined by the measurement principle, the width of the effective measurement range logarithmically holds the same, but the position of the effective measurement range is shifted.

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Fig. 5. The simulation result curves with a pure capacitance

5 Conclusion In an effective measurement range, the system offers a considerable accuracy. It is the measurement range that the system need improve. The measurement range of the system is relevant to the value of reference resistance R provided that ω is fixed. The FP-ADC extends the measurement range of the system effectively. Adjusting R manually, however, is required for a larger range.

References 1. García-Beneytez, J.M., Vinai, F., Brunetti, L., García-Miquel, H., Vázquez, M.: Study of magneto impedance effect in the microwave frequency range for soft magnetic wires and microwires. Sensors and Actuators A: Physical 81, 78–81 (2000) 2. Mertens, S.F.L., Temmerman, E.: Study of zinc passivation in chromium (VI)-containing electrolytes with short-term impedance measurements. Corrosion Science 43, 301–316 (2001) 3. Bordi, F., Cametti, C., Gliozzi, A.: Impedance measurements of self-assembled lipid bilayer membranes on the tip of an electrode. Bioelectrochemistry 57, 39–46 (2002) 4. Oelze, M.L., Frizzell, L.A., Zachary, J.F., O’Brien, W.D.: Frequency-dependent impedance measurements of rat lung as a function of inflation. Ultrasound in Medicine and Biology 29, S111 (2003) 5. Carullo, A., Parvis, M., Vallan, A., Callegaro, L.: Automatic Compensation System for Impedance Measurement. IEEE Transactions on Instrumentation and Measurement 52, 1239– 1242 (2003) 6. Zhang, J.Q., Sun, J.W., Liu, T.M.: A Fully Digital Ratio Impedance Measurement device. Electrical Measurement Techniques 2, 23–25 (1996) 7. Groza, V.Z.: High-Resolution Floating-Point ADC. IEEE Transactions on Instrumentation and Measurement 50, 1822–1829 (2001)

A 3D-Shape Reconstruction Method Based on Coded Structured Light and Projected Ray Intersecting Location Hui Chen, Shiwei Ma*, Bo Sun, Zhonghua Hao, and Liusun Fu School of Mechatronic Engineering & Automation, Shanghai University, Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200072, China [email protected]

Abstract. This paper proposes a 3D-shape reconstruction method based on coded structured light. Projecting color pseudo-random coded pattern on surface of object and using the thought of ray intersecting location, 3D shape reconstruction can be implemented by seeking the intersection point of the projection ray and the imaging ray. The former is projected ray of a feature point at projector side, and the later is image ray of the same feature point received by camera. Experimental results show that this method is effective. Keywords: 3D-shape reconstruction, coded structured light, ray intersecting location.

1 Introduction 3D-shape reconstruction is the main purpose of human vision and is also the most important research and application direction in computer vision. 3D reconstruction technology of non-texture surface objects based on coded structured light has advantages of the inherent wide range, large measuring range, non-contact, high accuracy and easy implementation. So it is paid more and more attention in industry and has been applied to automatic processing, electronics, automobile, textile and other manufacturing industries. Such a vision system consists of a projector and a camera component generally. The projector will project coded structured light onto surface of object, then camera collect object image information from another angle. After image processing, the 3D surface data is obtained and finally 3D reconstruction is achieved[1]. This method can implement 3D reconstruction by means of only one image and have many merits such as smaller data quantity and high calculation speed, comparing with the traditional binocular vision method. Projector is often used as a delivery vehicle of coded structured light in this method. However, it does not participate in subsequent reconstruction substantive. In this paper, *

Corresponding author.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part II, LNCS 6329, pp. 505–513, 2010. © Springer-Verlag Berlin Heidelberg 2010

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in view of the projection characters of projector, an illumination system is built up based on multicolor pseudo-random coded structured light. Projector is considered as a reverse camera during the reconstruction of spatial coordinates of feature points. Based on the thought of ray intersecting location, the 3D reconstruction is implemented by seeking the intersection of the projection ray and the imaging ray. The former is a projected ray of a feature point at projector side, and the later is an imaging ray of the same feature point received by camera.

2 Basic Principle of Ray Intersecting Location Binocular stereo vision is the main channel to get 3D information of scene by human eyes. There are two methods to reconstruct the 3D object shape and location in binocular stereo vision generally[2]. One is that two images of the surrounding scene are obtained from two cameras at different angles at the same time, and the other is that two images of the surrounding scene are gained from one camera at different angles and time. The visual model is shown in Fig.1. For any point p on the surface of an object, there is an imaging point p1 on the image plane of camera C1 . Therefore, the point P is on the half line which is located by camera C1 's optical center point and the point P1 . However, the 3D position of point P can not be determined by one imaging rays. So if we use two cameras C1 and C2 to observe the point P at the same time, it can be determined, which the point p1 on the camera C1 and the point P2 on the camera C2 are the imaging point of P on the same object.

P

p1

p2

Fig. 1. Schematic diagram of binocular vision model

Because of the space point P both on the ray C1 P1 and C2 P2 , the point P is the intersection point of two rays, and its 3D position can be uniquely determined: the 3D coordinates of point P can be reconstructed by calculating the intersection point of two imaging rays. The number of such imaging rays is the same as the cameras that grabbing images interactively. And all the rays should go through the same point. Therefore, we can locate space point based on the principle that all the rays must be intersected at the same point. This is the ray intersecting location principle of the binocular vision.

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3 Active Vision System Model Based on Projection Ray Intersecting Location This study will apply ray intersecting location theory to active vision system of coded multicolor pseudo-random structured light projection. There are two rays, which one is the projection ray from projector to the feature point and the other is the imaging ray of the same feature point collected by camera. 3D reconstruction is implemented by calculating the intersection of two rays. An active vision system model based on projection of coded structured light is shown in Fig 2. The spatial coded feature point P is available when the point P1 is projected on object surface along the direction from light center point C P to it. As the direction of projection ray C P P1 and the imaging ray are inversed between each other, the projector can be regarded as a reverse camera. A number of projection rays are derived from the projection center C P and various feature points at horizontal and vertical section of color-coded plane. Typically, the projector's position and focal length will be fixed, once the visual system has been calibrated. So the relative space position between these projection rays is fixed. In other words, the position of the projection rays beginning with the point of light center C P and over the point p1 will remain unchanged in the space coordinates.

Fig. 2. Schematic diagram of active vision system model based on projected structured light

According to the invariance of projection ray, the direction vectors of projection rays that corresponded by horizontal and vertical section feature points on coded plane can serve as parameters of projector. These parameters can locate feature points. In addition, based on the principle of camera imaging geometry, feature point P is on the ray that beginning at the camera optical center Cc and going through the point p2 . In terms of the thought of ray intersecting location, by calculating the intersection point of the projection ray C P P1 and the imaging ray P2Cc , where the two rays correspond to the same point, the 3D coordinate of point P can be gotten.

4 Implementation of 3D Reconstruction An active vision system based on multicolor coded of pseudo-random structured light consists of a LCD projector and a colorful planar array CCD camera. Based on

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pseudo-random array theory, the coded image including several particular colors can be identified uniquely. And 3D reconstruction can be achieved by only one image. Multicolor coded image of pseudo-random array is projected onto 3D scene by projector, and camera collects colorful coded image modulated by surface shape of object at the cross position. Supposed that projector and camera satisfy the ideal pinhole projection model and camera's internal and external parameters have been calibrated by the linear model, the corresponding relationship between 2D image coordinates and 3D image coordinates can be found. For the camera, the homogeneous coordinates mi and M i of the number i feature point are located on image plane and corresponding space respectively. The relationship between them is as follows:

ki mi = Ac [ Rc | Tc ] M i

(1)

mi = (ui , vi ,1) T , M i = ( X wi , Ywi , Z wi ,1) T Among them,

Ac is internal parameters matrix and [ Rc Tc ] is external parameters in

the projection matrix of camera. Through these parameters, the imaging ray at camera side is available. The matrix form of equation (1) is:

⎛ ui ⎞ ⎛ w11 w12 ⎜ ⎟ ⎜ ki ⎜ vi ⎟ = ⎜ w21 w21 ⎜ 1 ⎟ ⎜w ⎝ ⎠ ⎝ 31 w32

Eliminating the coefficient

w13 w23 w33

⎛ ⎞ ⎜ ⎟ w14 ⎞ ⎜ X wi ⎟ ⎟ w24 ⎟ ⎜ Ywi ⎟ ⎜ ⎟ w34 ⎟⎠ ⎜ Z wi ⎟ ⎜1 ⎟ ⎝ ⎠

(2)

ki in equation (2), we get:

⎧(ui w31 − w11) Xw + (ui w32 − w12 )Yw + (ui w33 − w13)Zw = w14 −ui w34 ⎨ ⎩(vi w31 − w21) Xw + (vi w32 − w22 )Yw + (vi w33 − w23)Zw = w24 − vi w34

(3)

As known by the geometrical relationship, the two equations are both plane equation in equation (3). By solving the simultaneous equations, one ray in space can be determined, which is the right another imaging ray we need to locate space coded feature point. Therefore, image the coded feature point coordinates of object surface should be substituted into equation (3) in reconstruction. The direction vector of imaging rays that correspond to coded feature points can be obtained by carrying out outer product operation of the two plane normal vectors. In practice, since there are noise and calibration parameter error in the process of camera imaging, the projection ray and imaging ray obtained do not intersect at one point probably. The traditional method of solving such problem is to calculate optimal solution of intersection in the least square sense, considering the optimal solution as 3D

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coordinates of space point, but this method is complex in calculation and has bigger error. In this paper, according to the geometry sense of two rays, we regard the two rays above as a pair of noncoplanar lines, and then the problem of rays intersection location is changed to problem of seeking the public vertical line in the noncoplanar lines, regarding the midpoint coordinates of the vertical line as the spatial coordinates of coded feature point. Its principle is given as Fig.3. L1

p1

l1

M1

M ( x, y , z ) L2

p2

l2

M2

Fig. 3. Model of the noncoplanar lines

As shown in Fig.3, assume that the direction vectors of the rays L1 and L2 are

l1 and l2 respectively, P1 and P2 are two known points on the two rays, M 1 and M 2 are foot points of the public vertical line with the two rays, M 2 M 2 is the public vertical line of two noncoplanar lines, then 1 / 2( M 1 + M 2 ) is the midpoint coordinate of the vertical line. As the public vertical line is perpendicular to the two rays respectively, we have,

⎧⎪ M 1M 2 ⋅ p1M 1 = 0 ⎨ ⎪⎩ M 1M 2 ⋅ p1M 2 = 0

(4)

⎧(( p1 + t1 ⋅ l1 ) − ( p2 + t2 ⋅ l2 )) ⋅ l1 = 0 ⎨ ⎩(( p1 + t1 ⋅ l1 ) − ( p2 + t2 ⋅ l2 )) ⋅ l2 = 0

(5)

That is

Solve this equation, obtained:

⎧ ( p2 − p1 ) ⋅ l1 ⋅ l2 − ( p2 − p1 ) ⋅ l2 ⋅ (l1 ⋅ l2 ) ⎪t1 = l1 ⋅ l2 − (l1 ⋅ l2 )2 ⎪ ⎨ ⎪t = ( p2 − p1 ) ⋅ l1 ⋅ (l1 ⋅ l2 ) − ( p2 − p1 ) ⋅ l2 ⋅ l1 ⎪2 l1 ⋅ l2 − (l1 ⋅ l2 ) 2 ⎩

(6)

After substituting position coefficient t1 and t 2 into ray equation and calculating the 3D coordinates of two foot points M 1 and M 2 , the coordinate of space point is considered as 1 / 2( M 1 + M 2 ) approximately .It can be proved that adopting the midpoint of public vertical line of the noncoplanar lines to approximate space point has smallest error in geometry[10].

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The coded feature points coordinates in projected coded image are extracted firstly to match feature points based on the window unique property of pseudo-random coded image[1]. The corresponding position of coded feature points in projection template is identified, then we can find out projection rays derived from the feature points and achieve direction vector and reference point coordinates of the feature point. After that, combining with the internal and external parameters matrixes which have obtained at the camera pre-calibration phase, the direction vector of imaging rays at camera side can be calculated. Finally, using the constraint condition that the camera light center point is on all the imaging rays, the point coordinate reconstruction is achieved based on ray intersecting location in the European 3D space.

5 Experimental Results and Analysis The experimental system of this paper is shown in Fig.4. A LCD projector (HITACHI -HCP-600X.) has been used, with a resolution of 1024*768 pixels and brightness of 2,500 lumens, whose resolution is 1024 × 768 pixels and contrast ratio is 500:1.

Fig. 4. Photograph of the experiment setup

By analyzing the obtained experimental results in literature [4] and [5], it indicates that projecting discrete square pattern is easy to extract and match feature points, and has higher accuracy. In this paper, the pseudo-random coded color image is square discrete codes whose size is 14×14. Each square vertex represents one coded feature point in this image. In vertical and horizontal direction, there are 28×28 coded feature points in all, which correspond to 28×28 projection rays. The calibration target is a 7×7 circle center target, and the distances of circles centre are 20mm by measuring. Three different positions ( z =-10cm、 z =0 and z =10cm) of the circle center target can be available by moving it parallelly. Combining with the distances of circles centre obtained, the center of circle target 3D coordinates at different position is available. The 28×28 projection rays at vertical and horizontal direction could also be obtained by fitting the imaging rays based on the principle of cross-ratio invariance[8]. Taking these imaging rays as constant parameters, coded feature points in scene can be located accurately during the process of reconstruction.

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The camera used in this system is a CCD (ARTRAY,ARTCAM-150PШ), whose effective pixel is 1392(H)×1040(V) and the size of effective pixel is 7.60mm×6.20mm. A Seiko lens (TAMRON 53513), the focus of which is 16.0mm. By previous calibrating, the camera's internal and external parameters matrixes are as follows: 2.2 641.6 0 ⎞ ⎛ 3545.4 ⎜ ⎟ Ac = ⎜ 0 3513.2 573.5 0 ⎟ ⎜ 0 0 1 0 ⎟⎠ ⎝ ⎛ ⎞ ⎜ 0.9997 0.0064 −0.022 −35.8 ⎟ ⎜ ⎟ [ Rc | Tc ] = ⎜ −0.0066 0.9999 −0.0091 −91.7 ⎟ ⎜ ⎟ ⎜ −0.022 −0.0092 −0.9997 1204.3 ⎟ ⎜0 ⎟ 0 0 1 ⎝ ⎠

Two panels are fixed up and made into a flat clamp, and then a cube is close to the two panels. One purpose of this experiment is to restore the angle θ between two cube adjacent surfaces, and the other purpose is to restore the distances ( L1 and L2 ) between the cube’s two visible surfaces and the corresponding folder planes. It is shown in Fig.5.

θ

Fig. 5. Schematic diagram of measurement model of the cube and the flat folder

Fig. 6. Obtained image coverd by quadrature code

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Fig. 7. Reconstruction result

Single coded covered image which obtained by experiment is shown in Fig.6. The first step is to extract the coordinates of all the feature points in image and match these coordinates with coded template. The second step is to export all the parameters of the projection rays which corresponded by feature points and getting direction vectors of imaging rays by these parameters. Finally, the facial reconstruction of the model is implemented based on the principle of ray intersection location. Reconstruction result is shown in Fig.7. These 3D points obtained in reconstruction are classified to determine which plane they belong to. Then fitting out the planes of scenes by the least squares algorithm, the spatial location parameters will be obtained naturally. Thereby, the angle θ between two cube adjacent surfaces and the distances ( L1 and L2 ) of two visible surfaces of the cube to the relative folder planes could also be calculated. Table 1 is the results of 3D reconstruction and error analysis. It can be seen from table 1 that the results of 3D reconstruction have a higher accuracy. Table 1. Reconstruction results and error

Reconstruction value Measured value error

angle θ (°)

distance L1 (mm)

distance L2 (mm)

90.32 90.62 0.33%

13.840 13.774 0.50%

13.835 13.904 0.49%

In 3D reconstruction technique which based on projection of coded structured light, the main factors affecting the accuracy include template pattern selection of the coded structured light, feature point’s detection accuracy, pre-calibration accuracy of projection system and so on. If more precision target, better-quality optical lens and more reasonable method are adopted to improve the precision of pre-calibration, it should laid a more reliable foundation for the subsequent 3D reconstruction. In addition, the reconstruction precision can be improved further if the false corner points are eliminated and the missing corner points are added, when detecting the feature points.

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6 Conclusions The proposed 3D reconstruction method requires only calibrating camera parameters and projection rays, but not the internal and external parameters of the projector, by regarding projector as a reverse camera and treating the projection and imaging rays as a pair of noncoplanar lines. Based on the thought of ray intersecting location, the spatial coordinates of coded feature points can be reconstructed by calculating the public vertical center coordinate of the noncoplanar lines, and 3D shape and angle information of the object in reality can be available. Since only one coded image is needed in this method, it is expected that 3D reconstruction in dynamic scenes can be realized. Since this experiment is processing on the condition that the projector and camera according to the ideal pinhole model, it doesn't consider the nonlinear model of them. If lens has distortion, the accuracy of reconstruction should be affected. Therefore, the affection of lens distortion on the 3D reconstruction needs further study. Acknowledgments. This work was supported by Shanghai Science and Technology Research Foundation No.09dz2201200, No.09dz2273400, 08DZ2272400.

References 1. Zhang, G.J., Tian, X.: Structured light 3D vision and its industry application. Journal of Beijing University of Aeronautics and Astronautics 22(6), 650–654 (1996) 2. Qi, X.: Research on key technology of 3D reconstruction system based on Binocular stereo vision. Zhongnan University (2007) 3. Jessie, F., Williams, M., Neil, J.S.: Pseudo random sequences and arrays. Proceedings of the IEEE 64(12), 1715–1729 (1976) 4. Zhang, Y.B.: Research on Uncalibrated 3D Euclidean Reconstruction Based on Single Encoded Image. Hefei University of Technology (2004) 5. Fu, L.Q.: The Integration of 3D Reconstruction System Based on Single Encoded Image. Hefei University of Technology (2007) 6. Fofi, D., Salvi, J., Mouaddib, E.M.: Euclidean reconstruction by means of an uncalibrated structured light sensor. In: 4th Ibero-American Symposium on Pattern Recognition (SIARP 2000), Lisboa, Portugal, pp. 159–170 (2000) 7. Zhang, Y.B., Chen, X.H., Lu, R.S.: Uncalibrated Euclidean 3D Reconstruction Based on Stratification Algorithm. Chinese Journal of Scientific Instrument 26(7), 710–714 (2005) 8. Faugeras, O.: Stratification of Three-dimensional Vision: Projective Affine and Metric Representations. Journal of the Optical Society of America A: Optics, Image Science, and Vision 12(3), 465–484 (1995) 9. Ma, S.W., Chen, H., Xu, Z.Y., et al.: Research on Corner Detection of Multicolor Pseudo-random Encoded Image. Acta Metrologica Sinica 29(2), 110–113 (2008) 10. Kanatani, K.: Computational Projective Geometry. CVGIP: Image Understanding 54(3), 333–348 (1991) 11. Zhang, W.Z., Chen, G.D., Gao, M.T.: The Computational Methods Based on Invariance of Cross Ratio. Computer Simulation 24(9), 225–227 (2007) 12. Fu, Q., Wang, C.P.: New Algorithm for Three-dimensional Construction of Point Object. Science Technology and Engineering 8(3), 643–647 (2008)

2 3

[email protected]

M E = y(t) − yˆ(t) NPPE  [

N 

(ˆ y (t) − y(t))2 /

t=1

N 

(3) y(t)2 ]1/2 × 100%

(4)

t=1

M E = yi − yˆi

(3)

n n   NPPE  [ (ˆ yi − yi )2 / yi2 ]1/2 × 100%

(4)

i=1

i=1

_______________________________________________ The original online version for this chapter can be found at http://dx.doi.org/10.1007/978-3-642-15597-0_2

_______________________________________________

Author Index

Abeykoon, Chamil II-9, II-E1 Affenzeller, Michael II-368 Alla, Ahmed N. Abd II-190 Allard, Bruno I-416 An, Bo I-256 Anqi, Wang II-351 Auinger, Franz II-368 Banghua, Yang

Fan, Guo-zhi I-486 Fan, Jian I-147 Fan, Xin-Nan I-304 Fan, Zhao-yong I-104 Fei, Minrui I-1, I-62, I-87, I-147, I-221, I-456, II-40, II-49, II-450, II-475 Fu, Liusun II-505 Fu, Xiping II-49

II-250

Cai, Zongjun I-36 Cao, Guitao I-313 Cao, Longhan II-93 Cao, Lu II-58 Cao, Luming I-36, I-112, II-493 Chen, Haifeng I-389 Chen, Hui II-333, II-505 Chen, Jindong II-207 Chen, Jing I-136 Chen, LiXue II-138 Chen, Min-you I-104, II-342 Chen, Qigong I-87 Chen, Rui II-234 Chen, Wang II-148 Chen, Xiaoming II-120 Chen, Xisong I-409 Chen, Yong I-486 Cheng, Dashuai I-36, I-112, II-493 Cheng, Liangcheng I-45 Chiu, Min-Sen I-36, I-112, II-493 Dai, Rui II-93 Deng, An-zhong I-486 Deng, Jing I-7, II-9, II-40, II-E1 Deng, Weihua I-1, II-450 Dengji, Hu II-351 Di, Yijuan I-456 Ding, Guosheng I-244 Ding, Yan-Qiong I-304 Dong, Yongfeng I-324 Du, Dajun I-62, I-290 Du, Yanke II-266 Duan, Ying-hong II-259

Ganji, Behnam I-434 Gao, Lixin I-157 Gao, Weijun I-361 Gao, Yanxia I-416 Ge, Bo I-506 Geng, Zhiqiang II-84 Gou, Xiaowei II-69 Gu, Hong I-52 Gu, Li-Ping I-304 Guanghua, Chen II-351 Guo, Chunhua I-282, I-333 Guo, Fengjuan I-205 Guo, Huaicheng I-120 Guo, Shuibao I-416 Guo, Xingzhong II-433 Guo, Yi-nan I-213 Hadow, Mohammoud M. II-190 Hailong, Xue II-400 Ha, Minghu II-241 Han, Yongming II-84 Hao, Zhonghua II-333, II-505 He, Qingbo II-218 He, Ran I-466 He, Tian-chi II-1 He, Wei II-342 Hong, Moo-Kyoung I-380 Hou, Weiyan I-129 Hu, Gang II-342 Hu, Huosheng II-450 Huang, Bin I-497 Huang, Hong II-390 Huang, Lailei II-277 Huang, Mingming I-466 Huang, Quanzhen I-95

516

Author Index

Huang, Zaitang II-166 Hutterer, Stephan II-368 Irwin, George W.

II-40, II-456

Jia, Fang II-158 Jia, Li I-36, I-112, II-69, II-493 Jiang, Enyu I-95 Jiang, Jing-ping I-77 Jiang, Lijun II-225 Jiang, Ming I-87 Jiang, Minghui I-497 Jiang, Xuelin I-185 Jiao, Yunqiang II-360 Jing, Chunwei I-233 Jingqi, Fu II-128 Kang, Lei II-467 Karim, Sazali P. Abdul II-190 Kelly, Adrian L. II-9, II-E1 Kim, Jung Woo I-372 Kong, Zhaowen I-185 Koshigoe, Hideyuki II-288 Kouzani, Abbas Z. I-434 Lang, Zi-qiang I-104 Lee, Hong-Hee I-372, I-380 Liao, Ju-cheng I-104 Li, Bo II-148 Li, Jianwei I-324 Li, Kang I-1, I-7, I-62, I-270, I-399, II-9, II-40, II-379, II-456, II-E1 Li, Lixiong I-350, II-475 Li, Nan I-196 Li, Qi I-409 Li, Ran II-199 Li, Sheng-bo I-486 Li, Shihua I-409 Li, Xiang II-180 Li, Xin II-180 Li, Xue I-290 Li, Yaqin I-45 Li, Yonghong I-26 Li, Yongzhong I-233 Li, Zhichao II-467 Liang, Chaozu I-185 Lin, Huang I-297 Linchuan, Li II-400 Lin-Shi, Xuefang I-416 Littler, Tim II-421

Liu, Enhai I-324 Liu, Fei II-442 Liu, Jiming II-277, II-410 Liu, Li-lan I-166, II-101 Liu, Qiming II-266 Liu, Shuang I-297 Liu, Xiaoli II-93 Liu, Xueqin I-7 Liu, Yi II-390 Lu, Da II-379 Lu, Huacai II-433 Luojun, II-218 Luo, Yanping I-157 Ma, Shiwei I-185, II-69, II-333, II-505 Martin, Peter J. II-9, II-E1 McAfee, Marion I-7, II-9, II-E1 McSwiggan, Daniel II-421 Meng, Ying II-433 Menhas, Muhammad Ilyas II-49 Ming, Rui II-475 Niu, Qun Niu, Yue

II-21, II-456 I-416

Pan, Feng II-207 Pang, Shunan II-30 Pedrycz, Witold II-241 Pei, Jianxia I-290 Peng, LingXi II-138 Qi, Jie II-30 Qian, Junfeng II-333 Qin, Xuewen II-166 Qin, Zhaohui I-129 Qu, Fu-Zhen II-148 Qu, Gang II-297 Ren, Hongbo

I-361

Shao, Yong I-95 Shi, Benyun II-410 Shi, Jiping I-112, II-493 Shi, Lukui I-324 Shi, Xiaoyun II-84 Shi, Yan-jun II-148 Shiwei, Ma II-351 Shu, Yunxing I-506 Shu, Zhi-song I-166, II-101 Song, Shiji I-399, II-484

Author Index Song, Yang I-1, I-129 Steinmaurer, Gerald II-368 Su, Hongye II-360 Su, Zhou I-425 Sun, Bo II-505 Sun, Hao I-304 Sun, Meng I-313 Sun, Shijie I-244 Sun, Sizhou II-433 Sun, Xin I-456 Sun, Xue-hua I-166, II-101 Tan, Weiming II-166 Tang, Jiafu II-297, II-312 Tang, LiangGui I-256 Tang, Zhijie II-218 Tao, Jili II-120 Tegoshi, Yoshiaki I-425 Tian, Shuai I-166 Trinh, H.M. I-434 Wang, Binbin I-389 Wang, Chao II-241 Wang, Fang I-157 Wang, Haikuan I-129 Wang, Hongbo I-196 Wang, Hongshu II-305 Wang, Jia II-128 Wang, Jianguo II-69 Wang, Jiyi I-297 Wang, Junsong I-69 Wang, Ling II-49, II-69 Wang, Lingzhi II-110 Wang, Linqing II-312 Wang, Nam Sun I-45 Wang, Qiang II-58 Wang, Shentao II-93 Wang, Shicheng II-234 Wang, Shujuan II-234, II-467 Wang, Tongqing I-282, I-333 Wang, Xihong I-147 Wang, Xin II-499 Wang, Xiukun I-466 Wang, Xiuping I-136 Wang, Yu II-297 Wei, Jia II-225 Wei, Lisheng I-87 Wei, Pi-guang I-477 Weimin, Zeng II-351 Wen, Guangrui I-16

Wen, Guihua II-225 Wu, Cheng I-270, I-399, II-484 Wu, Jue II-138 Wu, Jun II-158 Wu, Liqiao II-305 Wu, Mingliang II-93 Wu, Qiong I-361 Xiao, Da-wei I-213 Xiaoming, Zheng II-250 Xiaozhi, Gao II-400 Xia, Qingjun I-205 Xu, Jing I-233 Xu, Lijun I-221 Xu, Li-Zhong I-304 Xu, Rui II-266 Xu, Zhi-yu I-176 Yang, Aolei II-456 Yang, Ating I-445 Yang, Bo II-75 Yang, Huizhong I-45 Yang, Jun I-409 Yang, Lei II-138 Yang, Mei I-213 Yang, Nanhai I-466 Yang, Qingxin I-324 Yang, T.C. I-221, I-350, I-456 Yang, Yan-li II-342 Ye, Longhao I-120 Yin, Wenjun II-484 Yin, Xueyan II-442 Yonghuai, Zhang II-250 Yoshii, Takako II-288 Yuan, Ruixi I-69 Yu, Chunyan II-305 Yu, Shuxia I-120 Yu, Tao I-166, II-101 Yu, Wei I-221 Yu, Xiao-ming I-77 Yu, Yajuan I-120 Yun, Shiwei I-506 Zeng, Xiangqiang I-95 Zhai, Guofu II-467 Zhai, Jin-qian I-104, II-342 Zhang, An I-205, II-58 Zhang, Changjiang II-75 Zhang, Chao II-390 Zhang, Dexing II-180

517

518 Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang,

Author Index Hao I-270, II-333 Hong-yu I-486 Hui II-390 Jie I-176 Jun I-52 Long II-234, II-467 Qian I-389 Qingling II-321 Tiezhi II-321 Ting I-497 Xianxia I-313 Xining I-16 Xue-Wu I-304 Yaozhong I-205 Yuanyuan II-84 Yue II-321 Yukui I-399 Yuli II-484 Zhihua I-425

Zhao, Guangzhou I-52, I-196, II-199, II-379 Zhao, Liang I-342 Zhao, Lindu I-26, I-445 Zhao, Ming I-16 Zhao, Ming-sheng II-1 Zhao, Shian II-110 Zhao, Wanqing II-456 Zheng, Dong II-250 Zheng, Xiaoming II-180 Zhou, Caihui II-225 Zhou, Fang II-21 Zhou, Hao I-477 Zhou, Taijin II-21 Zhou, Weisheng I-361 Zhuang, Chun-long I-486 Zhu, Peng II-1 Zhu, Wei II-390 Zhu, Wen-hua I-477 Zhu, Yong II-120