Advanced Model-Order Reduction Techniques for Large-Scale Dynamical Systems by Seyed-Behzad Nouri, B.Sc., M.A.Sc. A thesis submitted to the Faculty of Graduate and Postdoctoral Affairs in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering Ottawa-Carleton Institute for Electrical and Computer Engineering Department of Electronics Carleton University Ottawa, Ontario, Canada © 2014 Seyed-Behzad Nouri Abstract Model Order Reduction (MOR) has proven to be a powerful and necessary tool for various applications such as circuit simulation. In the context of MOR, there are some unaddressed issues that prevent its efficient application, such as “reduction of multiport networks” and “optimal order estimation” for both linear and nonlinear circuits. This thesis presents the solutions for these obstacles to ensure successful model reduction of large-scale linear and nonlinear systems. This thesis proposes a novel algorithm for creating efficient reduced-order macromodels from multiport linear systems (e.g. massively coupled interconnect structures). The new algorithm addresses the difficulties associated with the reduction of networks with large numbers of input/output terminals, that often result in large and dense reduced-order mod- els. The application of the proposed reduction algorithm leads to reduced-order models that are sparse and block-diagonal in nature. It does not assume any correlation between the responses at ports; and thereby overcomes the accuracy degradation that is normally as- sociated with the existing (Singular Value Decomposition based) terminal reduction tech- niques. Estimating an optimal order for the reduced linear models is of crucial importance to ensure accurate and efficient transient behavior. Order determination is known to be a challenging task and is often based on heuristics. Guided by geometrical considerations, a novel and efficient algorithm is presented to determine the minimum sufficient order that ensures the ii accuracy and efficiency of the reduced linear models. The optimum order estimation for nonlinear MOR is extremely important. This is mainly due to the fact that, the nonlinear functions in circuit equations should be computed in the original size within the iterations of the transient analysis. As a result, ensuring both ac- curacy and efficiency becomes a cumbersome task. In response to this reality, an efficient algorithm for nonlinear order determination is presented. This is achieved by adopting the geometrical approach to nonlinear systems, to ensure the accuracy and efficiency in tran- sient analysis. Both linear and nonlinear optimal order estimation methods are not dependent on any spe- cific order reduction algorithm and can work in conjunction with any intended reduced modeling technique. iii Dedicated: To the living memories of my father, who lived by example to inspire and motivate his students and children. I also dedicate this to my mother, my understanding wife, and my wonderful sons Ali and Ryan for their endless love, support, and encour- agements. iv Acknowledgments First and foremost, I would sincerely like to express my gratitude to my supervisor, Pro- fessor Michel Nakhla. Without his guidance, this thesis would have been impossible. I appreciate his insight into numerous aspects of numerical simulation and circuit theory, as well as his enthusiasm, wisdom, care and attention. I have learned from him many aspects of science and life. Working with him was truly an invaluable experience. I am also sincerely grateful to to my co-supervisor, Professor Ram Achar, for his helpful suggestions and guidance, which was crucial in many stages of the research for this thesis. Most of all I wish to thank him for his motivation and encouragements. I would like to thank my current and past fellow colleagues in our Computer-Aided Design group for keeping a spirit of collaboration and mutual respect. They were always readily available for some friendly deliberations that made my graduate life enjoyable. I will always fondly remember their support and friendship. I am thankful towards the staff of the Department of Electronics at Carleton University for having been so helpful, supportive, and resourceful. Last but not least, I give special thanks to my family for all their unconditional love, encouragement, and support. I am eternally indebted to my wife and both my sons for their unconditional, invaluable and relentless support, encouragement, patience and respect. I would like to thank Mrs Zandi for all her understandings and gracious friendship with my v family. My final thoughts are with my parents to whom I am forever grateful. I cherish the memories of my late father with great respect. Words cannot express my admiration for the endless kindness, dedication, and sacrifices that my parents have made for their children. I believe that I could not have achieved this without their unlimited sacrifice. This is for them. Thank you all sincerely, vi Table of Contents Abstract ii Acknowledgments v Table of Contents vii List of Tables xiii List of Figures xiv List of Acronyms xx List of Symbols xxii Introduction 1 1 Background and Preliminaries 6 1.1DynamicalSystems.............................. 6 1.2LinearSystems................................ 7 1.2.1 Important Property of Linear Systems ................ 9 1.2.2 Mathematical Modeling of Linear Systems . ............ 10 vii 1.3NonlinearSystems.............................. 13 1.3.1 Solutions of Nonlinear Systems . ................ 15 1.3.2 Linear versus Nonlinear . .................... 16 1.4 Mathematical Modeling of Electrical Networks . ............ 16 1.5OverviewofFormulationofCircuitDynamics................ 18 1.5.1 MNA Formulation of Linear Circuits ................ 19 1.5.2 MNA Formulation of Nonlinear Circuits . ............ 20 2 Model Order Reduction - Basic Concepts 25 2.1Motivation................................... 26 2.2 The General Idea of Model Order Reduction . ................ 26 2.3 Model Accuracy Measures . ........................ 28 2.3.1 Error in Frequency Domain . .................... 31 2.4 Model Complexity Measures . ........................ 32 2.5 Main Requirements for Model Reduction Algorithms ............ 33 2.6 Essential Characteristic of Physical Systems . ................ 34 2.6.1 Stability of Dynamical Systems . ................ 34 2.6.2 Internal Stability . ........................ 35 2.6.3 External Stability . ........................ 38 2.6.4 Passivity of a Dynamical Model . ................ 38 2.7TheNeedforMORforElectricalCircuits.................. 39 3 Model Order Reduction for Linear Dynamical Systems 40 viii 3.1PhysicalPropertiesofLinearDynamicalSystems.............. 41 3.1.1 Stability of Linear Systems . .................... 41 3.1.2 Passivity of Linear Systems . .................... 46 3.2LinearOrderReductionAlgorithms..................... 49 3.3 Polynomial Approximations of Transfer Functions . ............ 50 3.3.1 AWE Based on Explicit Moment Matching . ............ 52 3.4 Projection-Based Methods . ........................ 53 3.4.1 General Krylov-Subspace Methods . ................ 56 3.4.2 Truncated Balance Realization (TBR) ................ 58 3.4.3 Proper Orthogonal Decomposition (POD) Methods . ........ 64 3.5 Non-Projection Based MOR Methods .................... 67 3.5.1 Hankel Optimal Model Reduction . ................ 67 3.5.2 Singular Perturbation . ........................ 67 3.5.3 Transfer Function Fitting Method . ................ 68 3.6 Other Alternative Methods . ........................ 76 4 Model Order Reduction for Nonlinear Dynamical Systems 77 4.1PhysicalPropertiesofNonlinearDynamicalSystems............ 78 4.1.1 Lipschitz Continuity . ........................ 79 4.1.2 Existence and Uniqueness of Solutions . ............ 80 4.1.3 Stability of Nonlinear Systems .................... 81 4.2NonlinearOrderReductionAlgorithms................... 84 4.2.1 Projection framework for Nonlinear MOR - Challenges . 84 ix 4.2.2 Nonlinear Reduction Based on Taylor Series ............ 86 4.2.3 Piecewise Trajectory based Model Order Reduction . 91 4.2.4 Proper Orthogonal Decomposition (POD) Methods . ........ 95 4.2.5 Empirical Balanced Truncation . ................ 98 4.2.6 Summary . ............................100 5 Reduced Macromodels of Massively Coupled Interconnect Structures via Clustering 101 5.1 Introduction . ................................101 5.2 Background and Preliminaries ........................104 5.2.1 Formulation of Circuit Equations . ................105 5.2.2 Model-Order Reduction via Projection ................106 5.3 Development of the Proposed Algorithm . ................107 5.3.1 Formulation of Submodels Based on Clustering . ........108 5.3.2 Formulation of the Reduced Model Based on Submodels . 110 5.4 Properties of the Proposed Algorithm ....................114 5.4.1 Preservation of Moments . ....................114 5.4.2 Stability ................................115 5.4.3 Passivity . ............................116 5.4.4 Guideline for Clustering to Improve Passivity ............123 5.5NumericalExamples.............................125 5.5.1 Example I . ............................126 5.5.2 Example II . ............................130 x 6 Optimum Order Estimation of Reduced Linear Macromodels 136 6.1 Introduction . ................................136 6.2 Development of the Proposed Algorithm . ................137 6.2.1 Preliminaries . ............................137 6.2.2 Geometrical Framework for the Projection