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Generalized Sylvester equations. Unified parametric solutions PDF

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Mathematics D u Generalized Sylvester equations (GSEs) are applied in many fields, including a applied mathematics, systems and control, and signal processing. Generalized n Guang-Ren Duan Sylvester Equations: Unified Parametric Solutions presents a unified parametric approach for solving various types of GSEs. G In an extremely neat and elegant matrix form, the book provides a single unified parametric solution formula for all the types of GSEs, which further e reduces to a specific clear vector form when the parameter matrix F in the Generalized n equations is a Jordan matrix. Particularly, when the parameter matrix F is diagonal, the reduced vector form becomes extremely simple. e r The first chapter introduces several types of GSEs and gives a brief overview a Sylvester Equations of solutions to GSEs. The following two chapters then show the importance l i of GSEs using four typical control design applications and discuss the z F-coprimeness of a pair of polynomial matrices. The next several chapters deal e with parametric solutions to GSEs. The final two chapters present analytical Unified Parametric Solutions d solutions to normal Sylvester equations (NSEs), including the well-known continuous- and discrete-time Lyapunov equations. An appendix provides the S proofs of some theorems. y l Features v • Covers several types of very general GSEs, including GSEs with e arbitrary orders, arbitrary dimensions, unknown parameter matrices, s t and GSEs without controllability or regularizability assumption e • Proposes a whole set of highly unified parametric general solutions to r the various types of GSEs, which are in very simple and neat analytical E closed-forms and are complete in the sense of providing all the degrees of freedom q • Provides numerically simple and reliable unified procedures using u matrix elementary transformations or singular value decompositions a for solving the relevant polynomial matrices based on which general t solutions to the GSEs can be immediately constructed i o n s K23460 6000 Broken Sound Parkway, NW ISBN: 978-1-4822-4396-3 Suite 300, Boca Raton, FL 33487 90000 711 Third Avenue New York, NY 10017 an informa business 2 Park Square, Milton Park www.crcpress.com Abingdon, Oxon OX14 4RN, UK 9 781482 243963 w w w.c r c p r e s s .co m K23460 cvr mech.indd 1 4/20/15 8:42 AM Generalized Sylvester Equations Unified Parametric Solutions Generalized Sylvester Equations Unified Parametric Solutions Guang-Ren Duan MATLAB® and Simulink® are trademarks of The MathWorks, Inc. and are used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® and Simulink® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® and Simulink® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20150506 International Standard Book Number-13: 978-1-4822-4398-7 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, includ- ing photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface........................................................................... xv Notations....................................................................... xxi Abbreviations .................................................................xxiii 1 Introduction ................................................................. 1 1.1 ThreeTypesofLinearModels............................................ 1 1.1.1 First-OrderSystems............................................... 1 1.1.2 Second-OrderSystems............................................ 2 1.1.3 Higher-OrderSystems............................................ 5 1.2 ExamplesofPracticalSystems............................................. 8 1.2.1 CircuitSystem..................................................... 8 1.2.2 MultiagentKinematicSystems.................................. 10 1.2.3 ConstrainedLinearMechanicalSystems........................ 13 1.2.3.1 MatrixSecond-OrderForm............................ 14 1.2.3.2 MatrixFirst-OrderForm............................... 14 1.2.4 Flexible-JointRobots............................................ 16 1.3 SylvesterFamily.......................................................... 19 1.3.1 First-OrderGSEs................................................ 19 1.3.1.1 HomogeneousGSEs................................... 19 1.3.1.2 NonhomogeneousGSEs............................... 20 1.3.2 Second-OrderGSEs............................................. 20 1.3.3 Higher-OrderGSEs.............................................. 21 1.3.4 NSEs............................................................. 22 1.4 AnOverview:WorkbyOtherResearchers.............................. 24 1.4.1 GSEsRelatedtoNormalLinearSystems........................ 25 1.4.1.1 NumericalSolutions................................... 25 1.4.1.2 ParametricSolutions................................... 27 1.4.1.3 SolutionswithApplications............................ 27 1.4.2 GSEsRelatedtoDescriptorLinearSystems..................... 28 1.4.2.1 SolutionstotheEquations............................. 28 1.4.2.2 SolutionswithApplications............................ 29 v vi ■ Contents 1.4.3 OtherTypesofEquations........................................ 30 1.4.3.1 NonhomogeneousFirst-OrderGSEs.................. 30 1.4.3.2 Second-andHigher-OrderGSEs...................... 30 1.5 AbouttheBook.......................................................... 31 1.5.1 Purposes.......................................................... 31 1.5.1.1 ShowingApplicationsofGSEs........................ 31 1.5.1.2 HighlightingtheSylvesterParametricApproaches..... 32 1.5.1.3 PresentingSolutionstoGSEs.......................... 32 1.5.2 Structure......................................................... 33 1.5.3 BasicFormulas................................................... 35 1.5.3.1 FormulaSetIfortheCaseofArbitraryF............. 35 1.5.3.2 FormulaSetIIfortheCaseofJordanMatrixF....... 36 1.5.3.3 FormulaSetIIIfortheCaseofDiagonalF........... 36 1.5.4 Features.......................................................... 37 1.5.4.1 GeneralSuitability..................................... 37 1.5.4.2 HighUnification....................................... 38 1.5.4.3 CompletenessinDegreesofFreedom................. 39 1.5.4.4 NeatnessandSimplicity............................... 39 1.5.4.5 NumericalSimplicityandReliability.................. 39 2 ApplicationHighlightsofGSEs........................................... 41 2.1 ESAandObserverDesigns.............................................. 42 2.1.1 GeneralizedPole/ESA............................................ 42 2.1.1.1 StateFeedbackCase.................................... 42 2.1.1.2 OutputFeedbackCase................................. 43 2.1.1.3 CommentsandRemarks............................... 44 2.1.2 ObserverDesign................................................. 45 2.1.2.1 LuenbergerObservers.................................. 45 2.1.2.2 PIObservers........................................... 46 2.1.2.3 CommentsandRemarks............................... 47 2.2 ModelReferenceTrackingandDisturbanceDecoupling............... 48 2.2.1 ModelReferenceTracking....................................... 48 2.2.2 DisturbanceRejection........................................... 50 2.3 SylvesterParametricControlApproaches............................... 52 2.3.1 GeneralProcedure............................................... 52 2.3.2 MainSteps....................................................... 53 2.3.2.1 SolvingGSEs........................................... 53 2.3.2.2 ControllerParametrization............................ 54 2.3.2.3 SpecificationParametrization.......................... 54 2.3.2.4 ParameterOptimization............................... 54 2.4 NotesandReferences.................................................... 55 2.4.1 ProblemofESAinHigher-OrderSystems...................... 55 Contents ■ vii 2.4.2 Author’sWorkonSylvesterParametricApproaches............. 57 2.4.2.1 ESA.................................................... 58 2.4.2.2 ObserverDesign....................................... 59 2.4.2.3 FaultDetection........................................ 59 2.4.2.4 DisturbanceDecoupling............................... 60 2.4.2.5 RobustPoleAssignment............................... 60 3 F-Coprimeness ............................................................ 63 3.1 ControllabilityandRegularizability..................................... 63 3.1.1 First-OrderSystems.............................................. 63 3.1.1.1 ControllabilityandStabilizability..................... 63 3.1.1.2 RegularityandRegularizability ....................... 64 3.1.2 Higher-OrderSystems........................................... 65 3.1.2.1 ControllabilityandStabilizability ..................... 65 3.1.2.2 RegularityandRegularizability........................ 68 3.2 Coprimeness ............................................................ 72 3.2.1 ExistingConcepts................................................ 72 3.2.2 GeneralizedConcepts............................................ 73 3.2.3 CoprimenessofA(s)andB(s)................................... 74 3.2.3.1 IrregularizableCase.................................... 74 3.2.3.2 RegularizableCase..................................... 75 3.3 EquivalentConditions................................................... 76 3.3.1 SFR.............................................................. 77 3.3.2 GeneralizedRCF................................................. 78 3.3.3 DPE.............................................................. 81 3.3.4 UnifiedProcedure................................................ 82 3.4 RegularizableCase....................................................... 83 3.4.1 F-LeftCoprimewithRankn.................................... 84 3.4.1.1 EquivalentConditions................................. 84 3.4.1.2 UnifiedProcedure...................................... 85 3.4.2 Controllability................................................... 89 3.4.2.1 EquivalentConditions................................. 89 3.4.2.2 UnifiedProcedure...................................... 91 3.5 Examples................................................................. 93 3.5.1 First-OrderSystems.............................................. 94 3.5.2 Second-OrderSystems........................................... 98 3.5.3 Higher-OrderSystems.......................................... 100 3.6 NumericalSolutionBasedonSVD.................................... 102 3.6.1 ProblemDescription........................................... 102 3.6.2 MainSteps...................................................... 103 3.6.2.1 DataGenerationviaSVD............................ 103 3.6.2.2 PolynomialRecovering............................... 104 3.6.3 NumericalSolution............................................. 106 viii ■ Contents 3.7 NotesandReferences................................................... 110 3.7.1 CoprimeFactorizations......................................... 111 3.7.2 UnifiedProcedure.............................................. 112 3.7.3 NumericalAlgorithmUsingSVD.............................. 112 4 HomogeneousGSEs ...................................................... 115 4.1 SylvesterMappings..................................................... 115 4.1.1 DefinitionandOperations..................................... 115 4.1.2 RepresentationofGSEs........................................ 117 4.2 First-OrderGSEs....................................................... 119 4.2.1 GeneralSolution............................................... 119 4.2.2 Example........................................................ 123 4.3 Second-OrderGSEs.................................................... 124 4.3.1 GeneralSolution............................................... 125 4.3.2 Example........................................................ 129 4.4 Higher-OrderGSEs.................................................... 131 4.4.1 GeneralSolution............................................... 132 4.4.2 Example........................................................ 135 4.5 CaseofF BeinginJordanForm....................................... 137 4.5.1 GeneralSolution............................................... 138 4.5.2 Example........................................................ 142 4.6 CaseofF BeingDiagonal.............................................. 144 4.6.1 CaseofUndeterminedF....................................... 145 4.6.2 CaseofDeterminedF.......................................... 148 4.7 Examples............................................................... 152 4.7.1 First-OrderSystems............................................ 152 4.7.2 Second-OrderSystems......................................... 155 4.7.3 Higher-OrderSystems.......................................... 157 4.8 NotesandReferences................................................... 158 4.8.1 GSEsAssociatedwithNormalLinearSystems................. 159 4.8.2 GSEsAssociatedwithDescriptorLinearSystems.............. 160 4.8.3 Second-OrderGSEs............................................ 162 4.8.4 Higher-OrderGSEs............................................ 163 4.8.5 OtherRelatedResults.......................................... 164 4.8.5.1 NumericalSolutionsofGSEs........................ 164 4.8.5.2 Sylvester-ConjugateMatrixEquations............... 164 5 NonhomogeneousGSEs .................................................. 165 5.1 SolutionBasedonRCFandDPE...................................... 166 5.1.1 ParticularSolution.............................................. 166 5.1.2 GeneralSolution............................................... 171 5.1.3 SolutiontoGSE(1.70)......................................... 172 Contents ■ ix 5.2 Condition(5.11)....................................................... 174 (cid:2) 5.2.1 SolutionofR ................................................... 175 5.2.2 RegularizableCase.............................................. 177 5.3 SolutionBasedonSFR................................................. 179 5.3.1 ParticularSolution.............................................. 180 5.3.2 GeneralSolution............................................... 183 5.4 ControllableCase...................................................... 184 5.4.1 Results.......................................................... 184 5.4.2 Examples........................................................ 187 5.5 CaseofF BeinginJordanForm....................................... 192 5.5.1 SolutionBasedonRCFandDPE.............................. 193 5.5.2 SolutionBasedonSFR......................................... 195 5.5.3 Example........................................................ 199 5.6 CaseofF BeingDiagonal.............................................. 201 5.6.1 SolutionBasedonRCFandDPE.............................. 202 5.6.2 SolutionBasedonSFR......................................... 204 5.6.3 Example........................................................ 207 5.7 CaseofF BeingDiagonallyKnown................................... 208 5.7.1 SFRandSVD.................................................. 209 5.7.1.1 DiscretizedRCFandDPE........................... 210 5.7.1.2 DiscretizedFormofCondition(5.11)............... 210 5.7.2 SolutionBasedonSVD........................................ 211 5.8 Examples............................................................... 215 5.8.1 First-OrderSystems............................................ 215 5.8.2 Second-OrderSystems......................................... 217 5.8.3 Higher-OrderSystems.......................................... 219 5.9 NotesandReferences................................................... 220 5.9.1 First-OrderGSEs............................................... 221 5.9.2 Second-andHigher-OrderGSEs.............................. 222 6 FullyActuatedGSEs ...................................................... 225 6.1 FullyActuatedGSEs................................................... 225 6.1.1 FullyActuatedSystems......................................... 225 6.1.2 FullyActuatedGSEs........................................... 227 6.1.3 Examples........................................................ 229 6.2 HomogeneousGSEs:ForwardSolutions.............................. 232 6.2.1 GeneralSolutions............................................... 232 6.2.1.1 CaseofF BeingArbitrary............................ 232 6.2.1.2 CaseofF BeinginJordanForm..................... 233 6.2.1.3 CaseofF BeingDiagonal............................ 235 6.2.1.4 StandardFullyActuatedGSEs....................... 236 6.2.2 TypeofSecond-OrderGSEs................................... 237

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