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Passive Network Synthesis An Approach to Classification A. Morelli M. C. Smith Advances in Design and Control Passive Network Synthesis Advances in Design and Control SIAM’s Advances in Design and Control series consists of texts and monographs dealing with all areas of design and control and their applications. Topics of interest include shape optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control. The series focuses on the mathematical and computational aspects of engineering design and control that are usable in a wide variety of scientiic and engineering disciplines. Editor-in-Chief f Ralph C. Smith, North Carolina State University Editorial Board Stephen L. Campbell, North Carolina State University Jennifer Mueller, Colorado State University Michel C. Delfour, University of Montreal Michael Ross, Naval Postgraduate School Fariba Fahroo, Air Force Ofice of Scientiic Research John Singler, Missouri University of Science and J. William Helton, University of California, San Diego Technology Kirsten Morris, University off Waterloo f Stefan Volkwein, Universität Konstanz Series Volumes Morelli, A. and Smith, M. C., Passive Network Synthesis: An Approach to Classiication Özbay, Hitay, Gümü soy, Suat, Kashima, Kenji, and Yamamoto, Yutaka, Frequency Domain Techniques for H∞ Control f of Distributed Parameter Systems Khalil, Hassan K., Hişgh-Gain Observers in Nonlinear Feedback Control Bauso, Dario, Game Theory with Engineering Applications Corless, M., King, C., Shorten, R., and Wirth, F., AIMD Dynamics and Distributed Resource Allocation Walker, Shawn W., The Shapes of Things: A Practical Guide to Differential Geometry and the Shape Derivative Michiels, Wim and Niculescu, Silviu-Iulian, Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach, Second Edition Narang-Siddarth, Anshu and Valasek, John, Nonlinear Time Scale Systems in Standard and Nonstandard Forms: Analysis and Control Bekiaris-Liberis, Nikolaos and Krstic, Miroslav, Nonlinear Control Under Nonconstant Delays Osmolovskii, Nikolai P. and Maurer, Helmut, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Suficient Optimality Conditions in Calculus of Variations and Optimal Control Biegler, Lorenz T., Campbell, Stephen L., and Mehrmann, Volker, eds., Control and Optimization with Differential-Algebraic f Constraints Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition Hovakimyan, Naira and Cao, Chengyu, L1 Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation Speyer, Jason L. and Jacobson, David H., Primer on Optimal Control Theory Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second Edition Shima, Tal and Rasmussen, Steven, eds., UAV Cooperative Decision and Control: Challenges and Practical Approaches Speyer, Jason L. and Chung, Walter H., Stochastic Processes, Estimation, and Control Krstic, Miroslav and Smyshlyaev, Andrey, Boundary Control of PDEs: A Course on Backstepping Designs Ito, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and Applications Xue, Dingyü, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design with MATLAB Hanson, Floyd B., Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation Michiels, Wim and Niculescu, Silviu-Iulian, Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach Ioannou, Petros and Fidan, Bar , Adaptive Control Tutorial Bhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix Problems Robinett III, Rush D., Wilson, Daıvᶊid G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic Programming for Optimization of Dynamical Systems Huang, J., Nonlinear Output Regulation: Theory and Applications Haslinger, J. and Mäkinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation, and Computation Antoulas, Athanasios C., Approximation of Large-Scale Dynamical Systems Gunzburger, Max D., Perspectives in Flow Control and Optimization Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and Optimization Betts, John T., Practical Methods for Optimal Control Using Nonlinear Programming El Ghaoui, Laurent and Niculescu, Silviu-Iulian, eds., Advances in Linear Matrix Inequality Methods in Control Helton, J. William and James, Matthew R., Extending H∞ Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives Passive Network Synthesis fi An Approach to Classi cation A. Morelli University of Cambridge Cambridge, United Kingdom M. C. Smith University of Cambridge Cambridge, United Kingdom Society for Industrial and Applied Mathematics Philadelphia Copyright © 2019 by the Society for Industrial and Applied Mathematics 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. Publications Director Kivmars H. Bowling Executive Editor Elizabeth Greenspan Developmental Editor Gina Rinelli Harris Managing Editor Kelly Thomas Production Editor David Riegelhaupt Copy Editor Susan Fleshman Production Manager Donna Witzleben Production Coordinator Cally A. Shrader Compositor Cheryl Hufnagle Graphic Designer Doug Smock Library of Congress Cataloging-in-Publication Data Names: Morelli, A. (Alessandro), 1987- author. | Smith, Malcolm C. (Malcolm Clive), 1957- author. Title: Passive network synthesis : an approach to classiication / A. Morelli, M. C. Smith, University of Cambridge, Cambridge, United Kingdom. Description: Philadelphia : Society for Industrial and Apfplied Mathematics, [2019] | Series: Advances in design and control ; 33 | Includes bibliographical references and index. Identiiers: LCCN 2019005292 (print) | LCCN 2019017504 (ebook) | ISBN 9781611975826 | ISBN 9781611975819 (print) Subjecfts: LCSH: System analysis. Classiication: LCC T57.6 (ebook) | LCC T57.6 .M64 2019 (print) | DDC 003--dc23 LC recford available at https://lccn.loc.gov/2019005292 is a registered trademark. Contents 1 Introduction 1 1.1 Outline ..............................................................................................3 1.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Classicalresultsofnetworksynthesis 5 2.1 Preliminariesofelectricalnetworks . . . . . . . . . . . . . . . . 5 2.2 FosterandCauercanonicalforms . . . . . . . . . . . . . . . . . 7 2.3 Positive-realfunctionsandpassivity . . . . . . . . . . . . . . . . 9 2.4 TheFosterpreambleandBrunecycle . . . . . . . . . . . . . . . 11 2.5 TheBott–Duffinconstructionanditssimplifications . . . . . . . . 13 2.6 Darlingtonsynthesis .............................................................................15 2.7 Reactanceextraction .............................................................................17 3 Moderndevelopmentsofnetworksynthesis 19 3.1 Regularpositive-realfunctionsandtheLadenheimcatalogue . . . 19 3.2 Reichert’stheorem 22 3.3 Algebraiccriteriaforcircuitrealizations . . . . . . . . . . . . . . 24 3.4 Thebehavioralapproachtopassivity . . . . . . . . . . . . . . . . 26 4 Mechanicalnetworks 29 4.1 Networkanalogiesandtheinerter . . . . . . . . . . . . . . . . . . 29 4.2 Applicationsofmechanicalnetworksynthesis . . . . . . . . . . . 31 5 Theenumerativeapproachtonetworksynthesis 41 5.1 DefinitionandderivationoftheLadenheimcatalogue . . . . . . . 42 5.2 Approachtoclassification . . . . . . . . . . . . . . . . . . . . . . 43 5.3 Classicalequivalences . . . . . . . . . . . . . . . . . . . . . . . 46 6 StructureoftheLadenheimcatalogue 47 6.1 Cataloguesubfamilystructurewithorbitsandequivalences . . . . 48 6.2 One-,two-,andthree-elementnetworks. . . . . . . . . . . . . . . 52 6.3 Four-elementnetworks 52 6.4 Five-elementnetworks 54 6.5 Summaryofrealizabilityconditions . . . . . . . . . . . . . . . . 56 6.6 Realizabilityregionsforfive-elementnetworks . . . . . . . . . . 62 7 Mainresultsanddiscussion 67 7.1 Cauer–Fostertransformation . . . . . . . . . . . . . . . . . . . . 67 7.2 FormalresultsontheLadenheimcatalogue . . . . . . . . . . . . . 70 v vi Contents 7.3 Smallestgeneratingsetofthecatalogue. . . . . . . . . . . . . . . 73 7.4 RemarksonKalman’s2011Berkeleyseminar . . . . . . . . . . . 74 7.5 Anoteond-invarianceofRLCnetworks . . . . . . . . . . . . . . 75 7.6 Six-elementnetworkswithfourresistors . . . . . . . . . . . . . . 78 8 Conclusions 83 A Realizationtheorems 85 A.1 EquivalenceclassIV1 ..........................................................................85 B A.2 EquivalenceclassV1 ..........................................................................88 A A.3 EquivalenceclassV1 ..........................................................................90 B A.4 EquivalenceclassV1 ..........................................................................95 C A.5 EquivalenceclassV1 ..........................................................................97 D A.6 EquivalenceclassV1 ........................................................................102 E A.7 EquivalenceclassV1 ........................................................................104 F A.8 EquivalenceclassV1 ........................................................................107 G A.9 EquivalenceclassV1 ........................................................................116 H A.10 EquivalenceclassV ...........................................................................120 I B Basicgraphs 125 C TheLadenheimnetworks(numericalorder) 129 D TheLadenheimnetworks(subfamilyorder) 135 Bibliography 145 Index 153 Chapter 1 Introduction Thecentralgoalofnetworksynthesisistodeviseanetworkwhichrealizesaprescribed behavior.Implicitinthistaskisthecharacterizationofthebehaviorsthatareinprinciple realizablewithcertainspecifiedcomponentsandthosethatarenot. Inpassivenetwork synthesisthespecifiedcomponentsarethestandard(passive)electricalelementssuchas theresistor,capacitor,andinductorortheirmechanicalequivalents. Incontrasttosyn- thesis,networkanalysisstudiesthebehaviorofagivencircuitwhichobeystherelevant component laws and the interconnection rules, e.g., Kirchhoff’s law. Thus, synthesis implicitlycontainsanalysisandhasahigherlevelofcomplexity. Thefirstmajorresultofelectricalnetworksynthesis,andarguablythebirthofnet- work synthesis, is the 1924 publication of the reactance theorem of Foster [25]. The theorem gives a necessary and sufficient condition that specifies the impedance func- tion of an arbitrary network comprising inductors, capacitors, and transformers only. An explicit construction is given for a network realizing any impedance of the pre- scribedclass. WiththistheoremFosterestablishednetworksynthesisasanexemplarof anengineeringtheorythatsoughttodescriberigorouslytheentitieswhicharepossible torealizephysicallyinagivendomain. Networksynthesisdevelopedrapidlyinthefollowingdecades. Bythe1960sacor- pusofresultshadbeenestablishedwhichisnowconsideredclassical. Foster’stheorem was generalized to two-element-kind networks by Cauer [10]. Brune [8] introduced the important class of positive-real functions, which are those functions that are ca- pableofbeingrealizedasatwo-terminalnetworkwitharbitrary(linear)passivecom- ponents. Brune provided an explicit construction method for a general positive-real functionwhichmadeuseofresistors,inductors,capacitors,andtransformers,asdida lateronebyDarlington[19]. AremarkabletheoremofBottandDuffin[7]showedthat transformerswereunnecessary. Networksynthesisflourishedasanactiveresearchtopicinthefirsthalfofthetwen- tieth century due to the broad scope it offered and due to the practical motivation of developing useful results for analogue filter design. Research gradually petered out in the 1960s with the advent of integrated circuits. Nevertheless, the classical theory continuedtoexertconsiderableinfluencein relatedfieldssuchascontrolandsystems andisstillfeltstronglytoday. Linearsystemstheory,dissipativity,theparametrization ofstabilizingcontrollers,conservationlaws,systemsongraphs,etc.,alloweadebtto networksynthesis. 1 2 Chapter1. Introduction Inthelastdecadetherehasbeenaresurgenceofinterestinnetworksynthesismo- tivated in part by the introduction of the inerter, a new fundamental element for mechanical control, and independently due to the advocacy of Kalman [51]. From both perspectivestheclassicaltheoryremainedincompleteinregardtotransformerlesssyn- thesis. ThisishighlightedintheBott–Duffintheorem. Theconstructionisdescribedin asingle-pagepaperintheJournalofAppliedPhysics,yetthesharpnessoftheresultis instarkcontrasttothelackofintuitivenessoftheproof. Inhisexpositionofthecon- struction[33,Chapter10]Guilleminwritesthatitisdifficult“tomakethesuccessionof steps appear natural and meaningful except that they do ultimately lead to the desired result.” ThemostunusualaspectoftheBott–Duffinproofisthatthenumberofenergystor- ageelements(inductorsandcapacitors)intherealizationappearsexcessivecompared tothedegreeoftheimpedancefunction. Therewereanumberofindependentattempts tosimplifytheconstruction[23,62,68]withmarginalsuccess. Fosterconcededin[27] thatnofurthersimplificationsseemedlikelybysimilarmethodsandissuedachallenge “to derive a much better method of synthesis.” No further progress was achieved for halfacentury. RecentworkhasestablishedthattheBott–Duffinconstructionisthesimplestpos- sibleamongseries-parallelnetworksforcertainbiquadraticimpedancefunctions[42], andthattheknownsimplificationscannotbeimprovedingeneral[39],apartfromsome newly discovered circuits of equal complexity. At one level Foster’s challenge is an- sweredinthenegative. Butatanother,muchstillremainstobebetterunderstood,since agreatmanypositive-realfunctionscanberealizedinamuchsimplermanner. The present state of knowledge brings an earlier approach to RLC synthesis into the foreground again: the enumerative approach. An example of this is the Master’s thesisofLadenheim[56], astudentofFosteratthePolytechnicInstituteofBrooklyn. Ladenheim determines the class of all essentially distinct two-terminal electrical networks comprising at most two reactive elements (capacitors or inductors) and at most three resistors—nowknown asthe “Ladenheim catalogue.” Until recent yearsLaden- heim’s dissertation appears to have been virtually unknown. A single citation in [28] independently led to two publications: one by Jiang and Smith [47] and the other by Kalman[51]. IntheLadenheimcataloguetheimpedanceforeachnetworkiscomputed, andthe inverseprocessisperformed;i.e.,giventheimpedance,anexpressionforeachelement of the network is stated. However, there are no derivations in Ladenheim’s work and hiscatalogueleavesseveralimportantquestionsunanswered,themostimportantbeing theabsenceofconditionsontheimpedancecoefficientswhichensurepositivityofthe network parameters. Furthermore, Ladenheim only considers what will be referred to hereasgenericnetworks,yetthatconceptisonlyimplicitinthethesis. The purpose of this monograph is to provide an outline of the main (classical and modern)contributionstothefieldofpassivenetworksynthesisandtopresentnewre- searchintothefieldontheenumerativeapproachandtheclassificationofnetworksof restrictedcomplexity. Inparticular,weprovideanewanalysisandclassificationofthe Ladenheimcatalogue,buildingonrecentwork,toobtainanimprovedunderstandingof thestructureandrealizationpoweroftheclasswithinthebiquadraticpositive-realfunc- tions.Themotivationistwofold.Withtheintroductionoftheinerter,recentapplications of network synthesis in the mechanical domain have highlighted the need for a better understandingofthemost“economical”waytorealizeagivenpassiveimpedance. The Ladenheimclass,comprisingatmosttworeactiveelements,isthesimplestclassthatis nontrivialandisthereforeidealforanin-depthanalysis.Morebroadly,thefundamental 1.1. Outline 3 natureofthetheoremofBottandDuffinaloneinvitesfurtherstudy. Recentresultshave establishedthatRLCrealizationislessstructuredthanrealizationoflineardynamical systems, and this does suggest that Foster’s challenge should be modified to a search foranimprovedgeneralunderstandingofRLCsynthesis. 1.1 Outline Fundamental notions of two-terminal networks are reviewed in Chapter 2, and classi- calresultsfrompassivenetworksynthesisaredescribed. Moderndevelopmentsinthe field are summarized in Chapter 3, where relevant results on the classification of bi- quadratics are introduced. The inerter device, the mechanical-electrical analogy, and some applications of passive network synthesis to mechanical networks are then de- scribedinChapter4. Oneofthemaincontributionsofthisworkisacompleteanalysis of the Ladenheim catalogue. The formal derivation of the catalogue is described in Chapter 5, where the group action and equivalence relations, which will be our main toolsfortheclassificationofnetworks,areintroduced. Foreachequivalenceclass,the set of impedances that can be realized is derived in explicit form as a semi-algebraic set. Realizabilityconditions,expressedintermsofnecessaryandsufficientconditions, aregiveninChapter6foreachequivalenceclass. Basedonsuchderivations,themain results are formalized in Chapter 7, where a new notion of generic network is intro- ducedandobservationsaremadeonthecompletesetofequivalences, onthesmallest generatingsetfortheclass,andonKalman’slatestwork. Theclassofsix-elementnet- workswithfourresistorsisalsoanalyzedinChapter7. Finally,conclusionsaredrawn in Chapter 8. A series of appendices which are useful in the study of the catalogue areprovided,includingtherealizationtheoremsforallthefive-elementnetworksinthe catalogue. 1.2 Acknowledgments The Ladenheim catalogue was a central part of discussions at the four workshops on mathematical aspects of network synthesis initiated by Uwe Helmke which were held alternately in Würzburg and Cambridge from 2010 to 2016, and it is from these roots thatthepresentmonographhasemerged. ThemonographowesaspecialdebttoUwe Helmke and two of the participants of the early workshops, Rudolf Kalman and Jan Willems,allthreeofwhomhavesadlypassedawaysince2010. The authors are most grateful to Tim Hughes, Jason Jiang, Rodolphe Sepulchre, and Paolo Rapisarda for their helpful comments on the manuscript. The first author acknowledges the support of The MathWorks for its funding of The MathWorks stu- dentshipinEngineeringattheUniversityofCambridge.