Self-Oscillations in Dynamic Systems: A New Methodology via Two-Relay Controllers

163 Pages·2015·3.46 MB·English

by  Luis T. Aguilar

Systems & Control: Foundations & Applications
Luis T. Aguilar
Igor Boiko
Leonid Fridman
Rafael Iriarte
Self-
Oscillations
in Dynamic
Systems
A New Methodology via Two-Relay
Controllers

Systems & Control: Foundations & Applications
SeriesEditor
Tamer Bas¸ar, University of Illinois at Urbana-Champaign, Urbana, IL,
USA
EditorialBoard
KarlJohanÅström,LundUniversityofTechnology,Lund,Sweden
Han-FuChen,AcademiaSinica,Beijing,China
BillHelton,UniversityofCalifornia,SanDiego,CA,USA
AlbertoIsidori,SapienzaUniversityofRome,Rome,Italy
MiroslavKrstic,UniversityofCalifornia,SanDiego,CA,USA
H.VincentPoor,PrincetonUniversity,Princeton,NJ,USA
MeteSoner,ETHZürich,Zürich,Switzerland;
SwissFinanceInstitute,Zürich,Switzerland
RobertoTempo, CNR-IEIIT,PolitecnicodiTorino,Italy
Moreinformationaboutthisseriesathttp://www.springer.com/series/4895
Luis T. Aguilar (cid:129) Igor Boiko (cid:129) Leonid Fridman
Rafael Iriarte
Self-Oscillations in Dynamic
Systems
A New Methodology via Two-Relay
Controllers
LuisT.Aguilar IgorBoiko
CITEDI DepartmentofElectricalEngineering
InstitutoPolitecnicoNacional ThePetroleumInstitute
Tijuana,BC,Mexico AbuDhabi,UAE
LeonidFridman RafaelIriarte
DivisiondeIngenieriaElectrica DivisiondeIngenieriaElectrica
UniversidadNacional UniversidadNacional
AutonomadeMexico AutonomadeMexico
MexicoCity,DF,Mexico MexicoCity,DF,Mexico
ISSN2324-9749 ISSN2324-9757 (electronic)
Systems&Control:Foundations&Applications
ISBN978-3-319-23302-4 ISBN978-3-319-23303-1 (eBook)
DOI10.1007/978-3-319-23303-1
LibraryofCongressControlNumber:2015953028
MathematicsSubjectClassification(2010):93B12,93C10,93C80,34D20,34C25
SpringerChamHeidelbergNewYorkDordrechtLondon
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Wededicatethisbookwithloveand
gratitudeto
Luis’swifeErica,his sonGabriel,andhis
daughterElisa,
Igor’swifeNatasha,
Leonid’swifeMillie,
Rafael’sfamily:hissonRafa,hisdaughter
Erika,andhiswifeJudith

Preface
This book began with a question asked by Professor Luis T. Aguilar to Professor
Leonid Fridman in 2005: How can I generate oscillations of low frequency and
particularamplitudeusingvariablestructurecontrol?
Coincidentally, this question was asked in the right place and at the right
time, because in 2005, Professors Igor Boiko and Leonid Fridman completed
their research on the second-order sliding mode control algorithms in frequency
domain [13, 14, 16–18], resulting in the possibility of calculating the amplitude
andfrequencyofchatteringinsystemswithsecond-orderslidingmodecontrollers.
They discovered that describing functions (DF) of the second-order sliding mode
controlalgorithmscouldshiftthepointcharacterizingtheoscillatorymoderesulting
from chattering to the second and third quadrantsof the complex plane. With the
discoveryofthisproperty,astraightforwardlogicalconclusioncouldbemadethat
the problem of generation of self-oscillations (SO) with desired amplitude and
frequencycouldbedefinedasaninverseproblemwithrespecttotheonepreviously
studied. Motivated by this question, Professor Rafael Iriarte found his subject of
researchinthisareatoo.
Usually, the DF of a single-valued nonlinearity is located on the negative part
of the real axisof the complexplane. So forthe designof SO in such a situation,
onlydynamiccompensatorscan be employed,butthe possibilityof compensators
toshapetheNyquistplotoftheplantisverylimited.
Therefore,theideathatthecontrolleritselfcouldbedesignedinsuchawaythat
itsDF (negativereciprocaloftheDF) mightbeplacedin anydesiredpointofthe
complexplanewasconceived.Thisideaservesasthebasisforthemainsubjectof
thisbook.
In this book,the two-relaycontroller(TRC) is proposed,which is intendedfor
the generation of SO in dynamic systems. A remarkable feature of this controller
is the possibility, with a simple changeof controllergains, for one to producethe
DF in every phase angle between 0 and 360ı, which corresponds to the crossing
of the Nyquist plot of the plant and the negative reciprocal of the DF of the
controller in any desired point. This point would define the SO produced in the
systemcontainingtheplantandthecontroller.ThedesignproceduresforTRCare
vii
viii Preface
proposedusingthreedifferentmethodologiesbasedonthefollowing:DF,Poincaré
maps, and locus of a perturbed relay system (LPRS) method. Three strategies
of robustification of generated SO are also proposed. The theoretical results are
illustratedbyexperimentsonSO generationin fourunderactuatedsystems:wheel
pendulum, Furuta pendulum, three-link robot, and three-degrees-of-freedom (3-
DOF) helicopter. The experiments are recorded with available video recordings
presentedinthefollowingweblinks:
– https://www.youtube.com/watch?v=t_1DcUdwFGE
– https://www.youtube.com/watch?v=MwXVQXlbJMQ
Acknowledgments
We wish to thank our friends and colleagues Professors Leonid Freidovich, Luis
Martínez-Salamero,andIvánCastellanosandDrs.AlejandraFerreiraandAntonio
Estrada.
TheworkwaspartiallysupportedbyCONACYT undergrant132125,SIP-IPN
Grants, PAPIIT under Grant IN113613 and IN112915, DGAPA PASPA Program,
andRIFPProjectNo.12310ofthePetroleumInstitute.
Tijuana,Mexico LuisT.Aguilar
AbuDhabi,UAE IgorBoiko
MexicoCity,Mexico LeonidFridman
June2015 RafaelIriarte
Notations and Acronyms
TRC Two-relaycontrol
SO Self-oscillation
DF Describingfunction
SOSM Second-orderslidingmodes
HOSM High-orderslidingmodes
LPRS Locusofaperturbedrelaysystem
DOF Degree(s)offreedom
AOS Asymptoticorbitalstability
R Thesetofrealnumbers
Rn Thesetofalln-dimensionalvectorwithrealnumbers
Rm(cid:2)n Thesetofallm(cid:2)nmatriceswithrealelements
C Thesetofcomplexnumbers
j Imaginaryunit
d.p;S/ DistancebetweenthepointpandthesetS(infx2Sjp(cid:3)xj)
q2Rn Jointpositionvector
qP 2Rn Thetimederivativeofthejointpositionvector
c1;c2 Two-relaycontrollergains
A1 Amplitudeoftheoscillation
! Frequencyoftheoscillation
˝ Particularordesiredvalueoffrequencyoftheoscillation
t Time
N.A;!/ Describing function depending on the amplitude and frequency of the
oscillation
s FrequencydomaincomplexvariablesDj!
W.s/ Transferfunction
(cid:2) Ratioofthetwo-relaycontrollergains
T Periodofasignal
(cid:3) Actuatedstatesvector
(cid:4) Unactuatedstatesvector
(cid:3)?; (cid:4)? FixedpointofPoincarémap
1
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