Generalized Sylvester equations. Unified parametric solutions

457 Pages·2015·2.364 MB·English

by  Duan, Guangren R.

Mathematics D
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Generalized Sylvester equations (GSEs) are applied in many fields, including
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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.
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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
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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
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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
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• 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
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o
n
s
K23460
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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
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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