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. 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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