Control of Nonlinear Dynamical Systems: Methods and Applications

396 Pages·2008·4.33 MB·english

by  Felix L. Chernousko

Communications and Control Engineering
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· ·
Felix L. Chernousko Igor M. Ananievski
Sergey A. Reshmin
Control of Nonlinear
Dynamical Systems
Methods and Applications
123
F.L.Chernousko I.M.Ananievski
RussianAcademyofSciences RussianAcademyofSciences
InstituteforProblemsinMechanics InstituteforProblemsinMechanics
VernadskyAve.101-1 VernadskyAve.101-1
Moscow Moscow
Russia119526 Russia119526
[email protected] [email protected]
S.A.Reshmin
RussianAcademyofSciences
InstituteforProblemsinMechanics
VernadskyAve.101-1
Moscow
Russia119526
[email protected]
ISBN:978-3-540-70782-0 e-ISBN:978-3-540-70784-4
DOI:10.1007/978-3-540-70784-4
CommunicationsandControlEngineeringISSN:0178-5354
LibraryofCongressControlNumber:2008932362
(cid:2)c 2008Springer-VerlagBerlinHeidelberg
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Preface
Thisbookisdevotedtonewmethodsofcontrolforcomplexdynamicalsystemsand
dealswithnonlinearcontrolsystemshavingseveraldegreesoffreedom,subjected
tounknowndisturbances,andcontaininguncertainparameters.Variousconstraints
areimposedoncontrolinputsandstatevariablesortheircombinations.
Thebookcontainsanintroductiontothetheoryofoptimalcontrolandthetheory
ofstabilityofmotion,andalsoadescriptionofsomeknownmethodsbasedonthese
theories.
Majorattentionisgiventonewmethodsofcontroldevelopedbytheauthorsover
thelast15years.Mechanicalandelectromechanicalsystemsdescribedbynonlinear
Lagrange’sequationsareconsidered.Generalmethodsareproposedforaneffective
construction of the required control, often in an explicit form. The book contains
various techniques including the decomposition of nonlinear control systems with
many degrees of freedom, piecewise linear feedback control based on Lyapunov’s
functions,methodswhichelaborateandextendtheapproachesoftheconventional
controltheory,optimalcontrol,differentialgames,andthetheoryofstability.
The distinctive feature of the methods developed in the book is that the con-
trols obtained satisfy the imposed constraints and steer the dynamical system to a
prescribedterminalstateinfinitetime.Explicitupperestimatesforthetimeofthe
processaregiven.Inallcases,thecontrolalgorithmsandtheestimatesobtainedare
strictlyproven.
The methods are illustrated by a number of control problems for various en-
gineeringsystems:roboticmanipulators,pendularsystems,electromechanicalsys-
tems,electricmotors,multibodysystemswithdryfriction,etc.Theefficiencyofthe
proposedapproachesisdemonstratedbycomputersimulations.
The authors hope that the monograph will be a useful contribution to the sci-
entificliteratureonthetheory andmethods ofcontrolfordynamical systems.The
v
vi Preface
bookcouldbeofinterestforscientistsandengineersinthefieldofappliedmathe-
matics,mechanics,theoryofcontrolanditsapplications,andalsoforstudentsand
postgraduates.
Moscow, FelixL.Chernousko
April2008 IgorM.Ananievski
SergeyA.Reshmin
Contents
Introduction ....................................................... 1
1 Optimalcontrol................................................ 11
1.1 Statementoftheoptimalcontrolproblem....................... 11
1.2 Themaximumprinciple ..................................... 16
1.3 Open-loopandfeedbackcontrol .............................. 21
1.4 Examples ................................................. 23
2 Methodofdecomposition(thefirstapproach) ..................... 31
2.1 Problemstatementandgameapproach......................... 31
2.1.1 Controlledmechanicalsystem ......................... 31
2.1.2 Simplifyingassumptions .............................. 32
2.1.3 Decomposition ...................................... 35
2.1.4 Gameproblem ...................................... 36
2.2 Controlofthesubsystemandfeedbackcontroldesign ............ 37
2.2.1 Optimalcontrolforthesubsystem ...................... 37
2.2.2 Simplifiedcontrolforthesubsystem .................... 42
2.2.3 Comparativeanalysisoftheresults ..................... 45
2.2.4 Controlfortheinitialsystem........................... 52
2.3 Weakcouplingbetweendegreesoffreedom .................... 54
2.3.1 Modificationofthedecompositionmethod............... 54
2.3.2 Analysisofthecontrolledmotions...................... 56
2.3.3 Determinationoftheparameters........................ 59
2.3.4 Caseofzeroinitialvelocities .......................... 61
2.4 Nonlineardamping ......................................... 67
2.4.1 Subsystemwithnonlineardamping ..................... 67
2.4.2 Controlforthenonlinearsubsystem..................... 69
2.4.3 Simplifiedcontrolforthesubsystemandcomparative
analysis ............................................ 74
2.5 Applicationsandnumericalexamples.......................... 82
2.5.1 Applicationtorobotics................................ 82
vii
viii Contents
2.5.2 Feedbackcontroldesignandmodellingofmotionsfor
two-linkmanipulatorwithdirectdrives.................. 86
2.5.3 Modellingofmotionsofthree-linkmanipulator........... 92
3 Methodofdecomposition(thesecondapproach)...................103
3.1 Problemstatementandgameapproach.........................103
3.1.1 Controlledmechanicalsystem .........................103
3.1.2 Statementoftheproblem..............................105
3.1.3 Controlintheabsenceofexternalforces.................106
3.1.4 Decomposition ......................................108
3.2 Feedbackcontroldesignanditsgeneralizations .................112
3.2.1 Feedbackcontroldesign ..............................112
3.2.2 Controlinthegeneralcase ............................114
3.2.3 Extensiontothecaseofnonzeroterminalvelocity.........117
3.2.4 Trackingcontrolformechanicalsystem .................124
3.3 Applicationstorobots.......................................131
3.3.1 Symbolicgenerationofequationsformultibodysystems ...131
3.3.2 Modellingofcontrolforatwo-linkmechanism(withthree
degreesoffreedom) ..................................136
3.3.3 Modellingoftrackingcontrolforatwo-linkmechanism
(withtwodegreesoffreedom) .........................144
4 StabilitybasedcontrolforLagrangianmechanicalsystems .........147
4.1 Scleronomicandrheonomicmechanicalsystems ................147
4.2 Lyapunovstabilityofequilibrium .............................151
4.3 Lyapunov’sdirectmethodforautonomoussystems ..............151
4.4 Lyapunov’sdirectmethodfornonautonomoussystems ...........153
4.5 Stabilizationofmechanicalsystems ...........................153
4.6 ModificationofLyapunov’sdirectmethod......................155
5 Piecewiselinearcontrolformechanicalsystemsunderuncertainty...157
5.1 Piecewiselinearcontrolforscleronomicsystems ................157
5.1.1 Problemstatement ...................................157
5.1.2 Descriptionofthecontrolalgorithm ....................159
5.1.3 Justificationofthealgorithm...........................161
5.1.4 Estimationofthetimeofmotion .......................166
5.1.5 Sufficient condition for steering the system to the
prescribedstate......................................168
5.2 Applicationstomechanicalsystems ...........................170
5.2.1 Controlofatwo-linkmanipulator ......................170
5.2.2 Controlofatwo-masssystemwithunknownparameters ...173
5.2.3 Thefirststageofthemotion ...........................178
5.2.4 Thesecondstageofthemotion.........................182
5.2.5 System“aloadonacart” .............................186
5.2.6 System“apendulumonacart”.........................187
Contents ix
5.2.7 Computersimulationresults ...........................197
5.3 Piecewiselinearcontrolforrheonomicsystems .................199
5.3.1 Problemstatement ...................................199
5.3.2 Controlalgorithmforrheonomicsystems ................200
5.3.3 Justificationofthecontrol.............................201
5.3.4 Resultsofsimulation .................................210
6 Continuous feedback control for mechanical systems under
uncertainty....................................................213
6.1 Feedbackcontrolforscleronomicsystemwithagivenmatrixof
inertia ....................................................213
6.1.1 Problemstatement ...................................213
6.1.2 Controlfunction .....................................214
6.1.3 Justificationofthecontrol.............................217
6.1.4 Sufficientconditionforcontrollability...................222
6.1.5 Computersimulationresults ...........................223
6.2 Controlofascleronomicsystemwithanunknownmatrixofinertia 229
6.2.1 Problemstatement ...................................229
6.2.2 Computersimulationofthemotionofatwo-linkmanipulator234
6.3 Controlofrheonomicsystemsunderuncertainty.................237
6.3.1 Problemstatement ...................................237
6.3.2 Computersimulationresults ...........................241
7 Controlindistributed-parametersystems.........................245
7.1 Systemoflinearoscillators ..................................245
7.1.1 Equationsofmotion..................................245
7.1.2 Decomposition ......................................246
7.1.3 Time-optimalcontrolproblem .........................248
7.1.4 Upperboundfortheoptimaltime.......................249
7.2 Distributed-parametersystems................................252
7.2.1 Statement of the control problem for a distributed-
parametersystem ....................................252
7.2.2 Decomposition ......................................254
7.2.3 First-orderequationintime............................257
7.2.4 Second-orderequationintime .........................258
7.2.5 Analysisoftheconstraintsandconstructionofthecontrol ..259
7.3 Solvabilityconditions .......................................263
7.3.1 Theone-dimensionalproblems.........................263
7.3.2 Controlofbeamoscillations ...........................266
7.3.3 Thetwo-dimensionalandthree-dimensionalproblems .....267
7.3.4 Solvabilityconditionsinthegeneralcase ................270