DOCSLIB.ORG

Advanced Model-Order Reduction Techniques for Large-Scale Dynamical Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Advanced Model-Order Reduction Techniques for Large-Scale Dynamical Systems 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
Load more

Recommended publications
  • Differential Analysis of Nonlinear Systems: Revisiting the Pendulum Example
    Differential analysis of nonlinear systems: revisiting the pendulum example F. Forni, R. Sepulchre Abstract— Differential analysis aims at inferring global prop- where k 0 is the damping coefficient and u is the torque erties of nonlinear behaviors from the local analysis of the input. The≥ specific aim of the paper is therefore to understand linearized dynamics. The paper motivates and illustrates the as much as possible of the global behavior of model (1) from use of differential analysis on the nonlinear pendulum model, its linearized dynamics ((δ#; δv) T ) an archetype example of nonlinear behavior. Special emphasis is 2 (#;v)X put on recent work by the authors in this area, which includes a δ#_ 0 1 δ# 0 differential Lyapunov framework for contraction analysis [24], = + (2) δv_ cos(#) k δv δu and the concept of differential positivity [25]. − − =:A(#;k) I. INTRODUCTION where any solution| (δ#{z( ); δv( ))}lives at each time instant The purpose of this tutorial paper is to revisit the role · · t in the tangent space T , where (#( ); v( )) is a of linearization in nonlinear systems analysis and to present (#(t);v(t)) solution to (1). X · · recent developments of this differential approach to systems The nonlinear pendulum model is an archetype example and control theory. Linearization is often considered as a of nonlinear systems analysis. As a control system, it is one synonym of local analysis, whereas nonlinear systems anal- of the simplest examples of nonlinear mechanical models ysis aims at a global understanding of the system behavior. and many of its properties extend to more complex electro- The focus of the paper is therefore on system properties that mechanical models such as models of robots, spacecrafts, allow to address non-local questions through the local-in- or electrical motors.
    [Show full text]
  • Far-From-Equilibrium Attractors and Nonlinear Dynamical Systems Approach to the Gubser Flow
    PHYSICAL REVIEW D 97, 044041 (2018) Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow † ‡ Alireza Behtash,1,2,* C. N. Cruz-Camacho,3, and M. Martinez1, 1Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA 2Kavli Institute for Theoretical Physics University of California, Santa Barbara, California 93106, USA 3Universidad Nacional de Colombia, Sede Bogotá, Facultad de Ciencias, Departamento de Física, Grupo de Física Nuclear, Carrera 45 No 26-85, Edificio Uriel Guti´errez, Bogotá D.C. C.P. 1101, Colombia (Received 19 November 2017; published 26 February 2018) The nonequilibrium attractors of systems undergoing Gubser flow within relativistic kinetic theory are studied. In doing so we employ well-established methods of nonlinear dynamical systems which rely on finding the fixed points, investigating the structure of the flow diagrams of the evolution equations, and characterizing the basin of attraction using a Lyapunov function near the stable fixed points. We obtain the attractors of anisotropic hydrodynamics, Israel-Stewart (IS) and transient fluid (DNMR) theories and show that they are indeed nonplanar and the basin of attraction is essentially three dimensional. The attractors of each hydrodynamical model are compared with the one obtained from the exact Gubser solution of the Boltzmann equation within the relaxation time approximation. We observe that the anisotropic hydro- dynamics is able to match up to high numerical accuracy the attractor of the exact solution while the second-order hydrodynamical theories fail to describe it. We show that the IS and DNMR asymptotic series expansions diverge and use resurgence techniques to perform the resummation of these divergences.
    [Show full text]
  • Feedback Control of Sector-Bound Nonlinear Systems with Applications
    Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2013 Feedback control of sector-bound nonlinear systems with applications to aeroengine control Luis Donaldo Alvergue Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Electrical and Computer Engineering Commons Recommended Citation Alvergue, Luis Donaldo, "Feedback control of sector-bound nonlinear systems with applications to aeroengine control" (2013). LSU Doctoral Dissertations. 3358. https://digitalcommons.lsu.edu/gradschool_dissertations/3358 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. FEEDBACK CONTROL OF SECTOR-BOUND NONLINEAR SYSTEMS WITH APPLICATIONS TO AEROENGINE CONTROL A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Division of Electrical and Computer Engineering by Luis Donaldo Alvergue B.S., McNeese State University, USA, 2004 M.S., Louisiana State University, USA, 2008 May 2012 Acknowledgments I would like to thank my advisor, Professor Guoxiang Gu, for his goodwill, patience, encouragement, and guidance by example that have made this dissertation possible. His support has been essential and I have been very fortunate to have him as my main advisor. I'd also like to thank Professor Sumanta Acharya for his trust in me that this project could be completed and for the generous financial support that I received through the IGERT fellowship.
    [Show full text]
  • Barrier Lyapunov Function-Based Adaptive Back-Stepping Control for Electronic Throttle Control System
    mathematics Article Barrier Lyapunov Function-Based Adaptive Back-Stepping Control for Electronic Throttle Control System Dapeng Wang 1,2,* , Shaogang Liu 1 , Youguo He 3 and Jie Shen 4 1 College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, China; [email protected] 2 713 Research Institute of China Shipbuilding Industry Corporation, Zhengzhou 450015, China 3 Automotive Engineering Research Institute, Jiangsu University, Zhenjiang 212013, China; [email protected] 4 Department of Computer and Information Science, University of Michigan-Dearborn, Dearborn, MI 48128, USA; [email protected] * Correspondence: [email protected] Abstract: This paper presents an adaptive constraint control approach for Electronic Throttle Control System (ETCS) with asymmetric throttle angle constraints. The adaptive constraint control method, which is based on barrier Lyapunov function (BLF), is designed not only to track the desired throttle angle but also to guarantee no violation on the throttle angle constraints. An ETC mathematic model with complex non-linear system is considered and the asymmetric barrier Lyapunov function (ABLF) is introduced into the design of the controller. Based on Lyapunov stability theory, it can be concluded that the proposed controller can guarantee the stability of the whole system and uniformly converge the state error to track the desired throttle angle. The results of simulations show that the proposed controller can ensure that there is no violation on the throttle angle constraints. Keywords: electronic throttle control; constraint control; barrier Lyapunov function; throttle opening Citation: Wang, D.; Liu, S.; He, Y.; angle; adaptive back-stepping control Shen, J. Barrier Lyapunov Function-Based Adaptive Back-Stepping Control for Electronic Throttle Control System.
    [Show full text]
  • Dynamic Lyapunov Functions✩
    Automatica 49 (2013) 1058–1067 Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Dynamic Lyapunov functionsI Mario Sassano a,1, Alessandro Astolfi a,b a Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma ``Tor Vergata'', Via del Politecnico 1, 00133, Rome, Italy b Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK article info a b s t r a c t Article history: Lyapunov functions are a fundamental tool to investigate stability properties of equilibrium points of Received 16 September 2011 linear and nonlinear systems. The existence of Lyapunov functions for asymptotically stable equilibrium Received in revised form points is guaranteed by converse Lyapunov theorems. Nevertheless the actual computation (of the analytic 11 November 2012 expression) of the function may be difficult. Herein we propose an approach to avoid the computation Accepted 14 November 2012 of an explicit solution of the Lyapunov partial differential inequality, introducing the concept of Dynamic Available online 23 February 2013 Lyapunov function. These functions allow to study stability properties of equilibrium points, similarly to standard Lyapunov functions. In the former, however, a positive definite function is combined with a Keywords: Lyapunov methods dynamical system that render Dynamic Lyapunov functions easier to construct than Lyapunov functions. Nonlinear systems Moreover families of standard Lyapunov functions can be obtained from the knowledge of a Dynamic Partial differential equations Lyapunov function by rendering invariant a desired submanifold of the extended state-space. The invariance condition is given in terms of a system of partial differential equations similar to the Lyapunov pde.
    [Show full text]