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研究生：	黃志霖 Huang, Chih-Lin
論文名稱：	邱氏濾波系統之非線性常微分方程組解算電路設計 Design of the Nonlinear ODEs Solver in Yau Filtering System
指導教授：	賴源泰 Lai, Yen-Tai
學位類別：	碩士 Master
系所名稱：	電機資訊學院 - 電機工程學系 Department of Electrical Engineering
論文出版年：	2003
畢業學年度：	91
語文別：	英文
論文頁數：	59
中文關鍵詞：	邱氏濾波器、卡門巴希濾波器
外文關鍵詞：	Yau filter, Kalman-Bucy filter
相關次數：	點閱：70 下載：1
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卡門和巴希提出了一個在現代工業界被廣泛使用的重要濾波器。但是，卡門巴希濾波器並不是完美的。它最弱的地方是僅能被用在線性的動態系統中。近幾年，邱成棟提出了一個相當成功的邱氏濾波器。
在邱氏濾波器中，最特別的是，將DMZ方程式化簡成一組線性的常微分方程式和一組非線性的常微分方程式。最主要的優點是以這一組非線性的常微分方程式來代替偏微分方程式，如果狀態變數的維度非常大時，可以有效地增加計算的彈性，因此這個方法可以用硬體電路來實現。
在本論文中，我們提出一個非線性常微分方程式的解算電路架構。以傳統的Runge-Kutta的方法設計，並且用Avant! 0.35um的製程技術實現此電路，實驗結果證明此電路設計方法適合使用於超大型積體電路技術上。

Kalman and Bucy published a historically important paper on filtering that is high influential to the modern industry. Despite its usefulness, however, the Kalman-Bucy filter is not perfect. The weakness is that it restricts only to those linear dynamical system. More recently, Stephen S. –T. Yau apply the new direct method successfully to Yau filtering systems which include both Kalman-Bucy filter and Benes filter as special cases.
In the Yau filtering system, more specifically, the solution of robust DMZ equation is reduced to the on-line of solution of a linear system of ODEs and off-line solution of a nonlinear system of ODEs. The major advantage is that we can use a simple system of nonlinear ODEs to compute the solution of Kolmogorov equation which makes the computation feasible even if the state dimension is fairly large. Therefore the method can be implemented in the hardware.
In this thesis, we present architecture for solving the nonlinear ODEs in the Yau filtering system. The classical Runge-Kutta method used to solve the nonlinear ODEs problem and implement it with Avant! 0.35um cell library. Experimental results show the effectiveness of the architecture.

ABSTRACT
CONTENTS
LIST OF FIGURES
LIST OF TABLES
CHAPTER 1 INTRODUCTION… … … … … … … … … 1
1.1 Historical Perspective … … … … … … … 1
1.2 The Motivation… … … … … … … … … …   2
CHAPTER 2 The Yau Filtering system… ….. … … … 4
2.1 General Information On Yau Filtering System… …  4
2.1.1 What Is a Yau Filter… … … … … … … … …  4
2.1.2 How It Came to Be Called a Filter… … … … …  5
2.1.3 What It Is Used For… … … … … … … … … … 5
2.2 The Yau Filtering System… … … … … … … … …  7
2.2.1 Basic Filtering Problem… … … … … … … …  7
2.2.2 Explicit Solution of DMZ Equation in terms of Solutions of ODEs.. 10
CHAPTER 3 Runge-Kutta Method And IEEE Floating-Point
Standard… … … … … … … … … … … … … … 15
3.1 Runge-Kutta Method… … … … … … … … … …  15
3.2 The ANSI/IEEE Floating-Point Standard… … … … 20
3.2.1 Introduction… … … … … … … … … … … … 20
3.2.2 Basic Floating-Point Algorithm… … … … … … 23
3.2.3 Conversions… … … … … … … … … … … … … 24
3.2.4 Rounding Schemes … … … … … … … … … … …  25
CHAPTER 4 Floating-Point Operations… … … … … … 29
4.1 Floating-Point Adder/Subtractor… … … … … … 29
4.2 Floating-Point Adder Algorithm… … … … … …  32
4.2.1 Notation… … … … … … … … … … … … … 32
4.2.2 Floating-Point Algorithm… … … … … … … …  33
4.3 Optimization Techniques… … … … … … … … …  34
4.3.1 Pre- and Postshifting… … … … … … … …  34
4.3.2 Leading Zeros Counting … … … … … … … … 37
4.4 Rounding … … … … … … … … … . … … … 39
4.5 Floating-Point Multiplier… … … … . . … … 41
CHAPTER 5 Design of the Nonlinear ODEs Solver … … 44
5.1 Vectorization Nonlinear ODEs … … … … … … 44
5.2 The Architecture of the Nonlinear ODEs Solver… … 49
5.3 Experimental Results … … … … … … … … … … 50
CHAPTER 6 Conclusions … … … … … … … … … … …  56
REFERENCES… … … … … … … … … … … … … … …  57
                                    

[1] S. S. –T. Yau and Yen-Tai Lai, “Explicit solution of DMZ equation in nonlinear filtering via solution of ODEs”, IEEE transactions on automatic control, vol. 48, NO.3, pp.505-508, Mar 2003.
[2] Yen-Tai Lai, S. S. –T Yau, and Ping-Hua Chen, “Design of The Ordinary Differential Equation Solver in The Yau Filtering System”, Proceeding of the American Control Conference Anchorage AK, pp.5144-5149, May 2002.
[3] S. S. –T. Yau and G-Q Hu, “Finite dimensional filters with nonlinear drift X: explicit solution of DMZ equation”, IEEE Transactions on Automatic Control, vol. 46, NO.1, pp. 142-148, January 2001.
[4] Martin S. Schmookler and Kevin J. Nowka, “Leading zero anticipation and detection – A comparison of methods”, Proceedings of the 15th IEEE symposium on computer arithmetic, Vail CO. USA, pp.7-12, June 2001.
[5] Peter-Michael Seidel and Guy Even, “On the design of fast IEEE floating-point adders”, Proceedings of the 15th IEEE symposium on computer arithmetic, Vail CO. USA, pp184-194, June 2001.
[6] Asger Munk Nielsen, David W. Matula, C. N. Lyu, and Guy Even, “An IEEE compliant floating-point adder that conforms with the pipelined packet-forwarding paradigm”, IEEE transactions on computers, vol.48, NO.1, pp.33-47, January 2000.
[7] Guy Even and Peter-Michael Seidel, “A comparison of three rounding algorithms for IEEE floating-point multiplication”, IEEE transactions on computer, vol. 49, NO.7, pp.638-650, July 2000.
[8] Yen-Tai Lai, Ping-Hua Chen, and Ming-Feng Wu, “A scheme of the implementation of Yau filtering system”, Proceedings of the 10th VLSI Design/CAD Symposium,pp.73-76, August 1999.
[9] Javier D. Bruguera and Tomas Lang, “Leading-one prediction with concurrent position correction”, IEEE transactions on computer, vol.48, NO.10, pp.1083-1097, October 1999.
[10] Stephen S. T. Yau and Amid Rasoulian, “Classification of four-dimensional estimation algebras”, IEEE Transactions on Automatic Control, vol. 44, NO. 12, pp.2312-2318, December 1999.
[11] Xi Wu and S. S. –T Yau, “Explicit solution to generalized time-varying Kolmogorov equation”, Proceedings of the 38th Conference on Decision & Control, Phoenix Arizona USA, pp.1897-1902, December 1999.
[12] S. S. –T. Yau and Guo-Qing Hu, “Recent results on classification of finite dimensional maximal rank estimation algebras with state space dimension up to 5”, Proceedings of the 37th conference on decision & control, Tampa Florida USA, pp.1311-1312, December 1998.
[13] S. S. –T. Yau, Xi Wu, and Ke-Pao Lin, “Recent advance in computing Kolmogorov equation arising from nonlinear filtering”, Proceedings of the 37th conference on decision & control, Tampa Florida USA, pp.2904-2905, December 1998.
[14] S. T. Yau. and S.S.-T. Yau, “Finite-dimensional filters with nonlinear drift Ⅲ: Duncan-Mortensen-Zakai equation with arbitrary initial condition for the linear filtering system and the Benes filtering system”, IEEE Transactions on Aerospace and Electronic System, vol. 33, NO. 4, pp. 1277-1294, October 1997.
[15] Zhigang Liang, S. S. –T. Yau, and S. T. Yau, “Finite-dimensional filters with nonlinear drift V: solution to Kolmogorov equation arising from linear filtering with non-Gaussian initial condition”, IEEE Transactions on Aerospace and Electronic System, vol. 33, NO. 4, pp. 1295-1307, October 1997.
[16] Chi-Wah Leung, Stephen S. –T. Yau, and Jie Chen, “Explicit construction of finite-dimensional nonlinear filters with state space dimension 4”, Proceedings of the 36th Conference on Decision & Control, San Diego, California USA, pp.4462-4466, December 1997.
[17] Guo-Qing Hu and Stephen S. –T. Yau, “Explicit solution of DMZ equation”, Proceedings of the 36th Conference on Decision & Control, San Diego, California USA, pp.4455-4459, December 1997.
[18] S.S.-T. Yau and Guo-Qing Hu, “Direct method without the Riccati equation for a linear filtering system with arbitrary initial condition”, Proceedings the 13th Triennial World Congress, San Francisco USA, pp. 469-474, 1996.
[19] Hon-Wing Cheng and S.S.-T Yau, “Parallel ODE-solvers for Kalman-Bucy filter with Arbitrary Initial Condition”, Proceedings of the 35th Conference on Decision & Control, Kobe, Japan, pp.4138-4145, December 1996.
[20] S.S.-T Yau and Guo-Qing Hu, “New direct method for Yau filtering system with arbitrary initial condition”, Proceedings of the 35th Conference on Decision and Control Kobe, Japan, pp.2539-2544, December 1996.
[21] Stephen S. T. Yau and Amid Rasoulian, “Construction of new finite dimensional nonlinear filters”, Proceedings on the 34th Conference on Decision & Control, New Orleans LA, pp.4002-4005, December 1995.
[22] Jie Chen, Stephen S. –T. Yau, and Chi-Wah Leung, “Explicit construction of finite-dimensional nonlinear filters with state space dimension 3”, Proceedings on the 34th Conference on Decision & Control, New Orleans LA, pp.4030-4034, December 1995.
[23] Jie Chen, “On Ubiquity of Yau Filters”, Proceedings of the American Control Conference Baltimore, Maryland, pp. 252-254, June 1994.
[24] Hon-Wing Cheng and S.S.-T Yau, “Computing the exponential of matrices”, Proceedings of the American Control Conference Baltimore, Maryland, pp. 3543-3547, June 1994.
[25] S.S.-T Yau and S.T. Yau, “New direct method for Kalman-Bucy Filtering system with arbitrary initial condition”, Proceedings of the 33rd Conference on Decision & Control, Lake Buena Vista Florida, pp.1221-1225, December 1994.
[26] Rui-Tao Dong, W.S. Wong and S.S.-T Yau, “Some PDE problems from estimation algebras”, SIAM Jour. On Control and Optimization, pp.143-147, April 1993.
[27] S.S.-T. Yau and Wei Lin Chiou, “Explicit construction of the finite dimension nonlinear filters with state space dimension 2”, Proceedings of the 32nd conference on Decision and Control, San Antonio, Texas, pp. 710-713, December 1993.
[28] S. S. –T. Yau and Chi-Wah Leung, “Recent result on classification of finite dimensional maximal rank estimation algebras with state space dimension 3”, Proceedings of the 31st Conference on Decision & Control, Tucson Arizona, pp.2247-2250, December 1992.
[29] S. S. -T Yau and S. -T Yau, “Explicit fundamental solution to Kolmogorov equation”, Proceedings of the 31st Conference on Decision & Control, Tucson Arizona, pp.1508-1511, December 1992.
[30] S. S. –T. Yau and Wen-Lin Chiou, “Recent results on classification of finite dimensional estimation algebras: dimension of state space≦2”, Proceedings of the 30th Conference on Decision & Control, Brighton England, pp.2758-2760, December 1991.
[31] S.S.-T Yau. “Recent results on nonlinear filtering: new class of finite dimensional filters”, Proceedings of the 29th Conference on Decision & Control, Honolulu Hawaii, pp. 231-233, December 1990.
[32] Luen-Fai Tam, Wing Shing Wong, and S. S. –T. Yau, “Recent results on finite dimensional exact estimation algebras”, Proceedings of the 28th Conference on Decision & Control, Tampa Florida, pp.2574-2575, December 1989.

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校外：2003-07-15公開

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