ABSTRACT XUE, XIANGZHONG. Electronic System Optimization Design via GP-Based Surrogate Modeling. (Under the direction of Dr. Paul D. Franzon.) For an electronic system with a given circuit topology, the designer’s goal is usually to automatically size the device and components to achieve globally optimal performance while at the same satisfying the predefined specifications. This goal is motivated by a human desire for optimality and perfection. This research project improves upon current optimization strategies. Many types of convex programs and convex fitting techniques are introduced and compared, and the evolution of Geometric Program (GP)-based optimization approaches is investigated through a literature review. By these means, it is shown that a monomial-based GP can achieve optimal performance and accuracy only for long-channel devices, and that a piecewise linear (PWL)-based GP works well only for short-channel, narrow devices without many data fitted. Based on known GP optimizer and convex PWL fitting techniques, an innovative surrogate modeling and optimization algorithm is proposed to further improve performance accuracy iteratively for a wide transistor with a short channel. The new surrogate strategy, which comprises a fine model and a coarse model, can automatically size the device to create a reusable system model for designing electronic systems and noticeably improving prediction accuracy, particularly when compared to the pure, GP-based optimization method. To verify the effectiveness and viability of the proposed surrogate strategy, a widely used two-stage optimization design is employed, entailing an operational amplifier (op-amp) and LC-tuned oscillator. In addition, an involved analysis and simulation demonstrate that the optimal results of both coarse and fine models in the proposed surrogate strategy may gradually converge to each other iteratively while achieving over 10% improvement in performance accuracy compared to the previous, PWL-based GP algorithm. As a result, the proposed surrogated modeling and optimization algorithm can serve as an efficient Computer Aided Design (CAD) tool, with the capability of dramatically improving performance for Integrated Circuit (IC) design. © Copyright 2012 by Xiangzhong Xue All Rights Reserved Electronic System Optimization Design via GP-Based Surrogate Modeling by Xiangzhong Xue A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Electrical Engineering Raleigh, North Carolina 2012 APPROVED BY: _______________________________ ______________________________ Dr. Paul D. Franzon Dr. Michael B. Steer Committee Chair ________________________________ ________________________________ Dr. W. Rhett Davis Dr. Harvey Charlton DEDICATION To my wife, son, and all other family members for a lifetime of support and encouragement. ii BIOGRAPHY Xiangzhong Xue was born on July 17, 1976 in Chongqing, China. In 1997, he received a Bachelor of Science degree from Chongqing University in China. He later moved to the United States and studied at North Carolina Agricultural and Technical State University in Greensboro, North Carolina, graduating with an Master of Science from the Department of Electrical and Computer Engineering in August 2003. Continuing his doctoral work at North Carolina State University in Raleigh, North Carolina, he graduated in May of 2012 with a Ph.D. in Electrical and Electronic Engineering. iii ACKNOWLEDGMENTS This dissertation would not have been possible without the great help and encouragement of my advisor, Dr. Paul D. Franzon, whom I most sincerely and deeply thank for all he has done to help me in my studies and my life. I will always be grateful, too, for the guidance I received from Dr. Michael B. Steer, Dr. Rhett Davis, and Dr. Harvey Charlton. A special warm thanks to my wife, Xiuqiong Liu; my son, Ziang Xue; and all of my friends for their continued support, care, and assistance. An expression of sincere gratitude is extended to all my roommates, who have provided me with generosity, help, and friendship through my time in Raleigh. I could not mark the occasion without acknowledging them. Finally, in appreciation for a lifetime of support and encouragement, I would like to thank my parents, Bangquan Xue and Xifang Xiang, as well as the rest of my family in China. iv TABLE OF CONTENTS List of Tables ................................................................................................................... viii List of Figures ..................................................................................................................... ix Chapter 1 Introduction .......................................................................................................... 1 1.1 Background and Motivation ................................................................................... 1 1.2 Research Objectives ............................................................................................... 3 1.3 Research Contributions .......................................................................................... 3 1.4 Dissertation Organization ....................................................................................... 6 Chapter 2 Convex Program .................................................................................................. 8 2.1 Overview and Induction ......................................................................................... 8 2.2 General Mathematical Optimization Problem .......................................................... 9 2.3 Conic Programs ................................................................................................... 10 2.4 Convex Programs ................................................................................................. 11 2.4.1 Convex Set and Convex Function .............................................................. 13 2.4.2 Convexity Verification .............................................................................. 14 2.4.3 Theoretical Properties ................................................................................. 18 2.4.4 Numerical Solution Algorithms ................................................................. 20 2.4.5 Applications ............................................................................................... 20 Chapter 3 Geometric Program ........................................................................................... 22 3.1 Evolution of Geometric Programming .................................................................. 22 3.2 Standard Geometric Program ............................................................................... 25 3.3 Relevant Terminology for the Geometric Program ............................................... 26 3.3.1 Monomial Functions and Examples ........................................................... 26 3.3.2 Posynomial Functions and Examples ......................................................... 27 3.3.3 Inverse Posynomial Functions and Examples ............................................. 28 3.3.4 Generalized Posynomial Functions and Examples ..................................... 29 3.3.5 Signomial Functions and Examples ........................................................... 30 3.4 Transformation Methods for the Geometric Program ........................................... 30 3.4.1 GP-compatible Algebraic Transformation ................................................. 31 3.4.2 Fractional Powers of Posynomials ............................................................. 33 3.4.3 Maximum of Posynomials ......................................................................... 35 3.4.4 Function Composition ............................................................................... 36 3.4.5 Additive Log Terms .................................................................................. 39 3.4.6 Generalized Posynomial Equality Constraints ............................................ 40 3.4.7 Application Example: Geometric Program in Convex Form ...................... 44 3.5 Optimization Sensitivity Analysis ........................................................................ 47 3.5.1 Tradeoff Analysis ...................................................................................... 47 3.5.2 Optimization Sensitivity Theory ................................................................ 49 v 3.6 Convex Approximation and Fitting ....................................................................... 52 3.6.1 Convex Approximation and Fitting Theory ................................................ 53 3.6.2 Monomial Fitting ...................................................................................... 57 3.6.3 Extended Monomial Fitting ....................................................................... 59 3.6.4 Max-monomial Fitting ............................................................................... 60 3.6.5 Posynomial Fitting .................................................................................... 63 3.6.6 Convex Piecewise-Linear (PWL) Fitting ................................................... 64 3.7 Methods for Solving Geometric Programs ............................................................ 66 3.8 Generalized GP and Relevant Solution Methods .................................................. 69 3.9 Signomial Problem and Relevant Solution Methods ............................................