Design and Implementation of Piecewise-Affine Observers for Nonlinear Systems

166 Pages·2014·7.71 MB·English

Design and Implementation of Piecewise-Affine
Observers for Nonlinear Systems
AZITAMALEK
AThesis
in
TheDepartment
of
ElectricalandComputerEngineering
PresentedinPartialFulfillmentoftheRequirements
fortheDegreeofMasterofAppliedScience(ElectricalandComputerEngineering)at
ConcordiaUniversity
Montre´al,Que´bec,Canada
November2013
(cid:2)c AZITAMALEK,2013
CONCORDIAUNIVERSITY
SchoolofGraduateStudies
Thisistocertifythatthethesisproposalprepared
By: AZITAMALEK
Entitled: Design and Implementation of Piecewise-Affine Observers for Non-
linearSystems
andsubmittedinpartialfulfilmentoftherequirementsforthedegreeof
MasterofAppliedScience(ElectricalandComputerEngineering)
complies with the regulations of this University and meets the accepted standards with
respecttooriginalityandquality.
Signedbythefinalexaminingcommittee:
Dr. M.Z.Kabir,Chair
Dr. Y.M.Zhang,ExternalExaminer
Dr. A.Aghdam,Examiner
Dr. Sx. HashtrudiZad,Examiner
Dr. K.Khorasani,Supervisor
Dr. L.Rodrigues,Supervisor
Approvedby
Dr. W.E.Lynch,Chair
DepartmentofElectricalandComputerEngineering
Dr. C.W.Trueman
InterimDean,FacultyofEngineeringandComputerScience
ABSTRACT
DesignandImplementationofPiecewise-AffineObserversforNonlinearSystems
AZITAMALEK
This thesis is divided into two main parts. The contribution of the first part is to de-
sign a continuous-time Piecewise-Affine (PWA) observer for a class of nonlinear systems.
It is shown that the state estimation error is ultimately bounded. The bound on the state
estimation error depends on the PWA approximation error. Moreover, it is shown that the
state estimation error is still convergent and ultimately bounded when the output of the
system is only available at sampling instants. The proof of convergence is presented in
two parts: conditions dependent on the sampling time and conditions independent of the
sampling time. In addition, ultimate boundedness of the state estimation error is proven
in the presence of norm bounded measurement noise. It is shown that the bound on the
state estimation error is dependent on the sampling time, PWA approximation error and
the bound on the norm of the noise. The proposed approach for observer design leads to a
convexoptimizationwhichcanbesolvedefficientlyusingavailablesoftwarepackages.
The contribution of the second part is to implement the proposed PWA observer on
a real setup of a wheeled mobile robot (WMR) available at the Hybrid Control Systems
(HYCONS) Laboratory of Concordia University. Although some researchers have applied
differenttypesofobserverstoexperimentalapplications,practicalimplementationofPWA
observershasnotbeengivenmuchattentionbyresearchers. Inthisthesisforthefirsttimea
PWAobserverisappliedtotheWMR.TheWMRisanexampleofanonlinearsystemwith
iii
a sampled output in the presence of measurement noise. The results of the experimental
implementationvalidatetheproposedtheoreticalresultsinthefirstpart.
iv
“Ifweknewwhatitwasweweredoing,itwouldnotbecalledresearch,wouldit? ”
—AlbertEinstein
v
ACKNOWLEDGEMENTS
Firstandforemost,Iwouldliketothankmysupervisors,Dr. LuisRodriguesandDr.
Khashayar Khorasani. This thesis could not have been accomplished without their patient
guidance, encouragement and support. I learnt from them; not only the scientific matters,
but also lots of knowledge which are very helpful in all aspects of my life. I am very
grateful to my supervisors for giving me this opportunity to come to Concordia University
andjointheirresearchgroup.
I would like to convey my gratitude to all my committee members for devoting their
valuabletimeinevaluatingmywork. Moreover,Imustthanktheprofessors,administrative
andthetechnicalstaffofthedepartmentwhohaveplayedanimportantroleinmysuccess.
I would like to thank all my HYCONS friends Behzad, Miad, Sina, Camilo, Hadi,
Jamila, Amin, Tiago, Arthur, Javier, Qasim, Jesus, Manuel and Ram with whom I spent
great moments during this period of my life. I would like to thank Miad in particular,
which spent lots of time answering all my questions with patience. I also would like to
thankFarzadforbeingsuchagoodandsupportivefriendduringthepastfewyears.
Last, but by no means least, I would like to thank my parents for all of their uncon-
ditionallove,helpandsupportwhichcannotbeputintowords. Also,Iwouldliketothank
mylovelybrotherandsisterforalwaysbeingthereforme.
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Tomyparents.
vii
TABLE OF CONTENTS
ListofFigures x
ListofTables xvii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 LiteratureSurvey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 LinearObservers . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 NonlinearObservers . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Piecewise-AffineObservers . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 Sampled-DataObservers . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.5 ExperimentalImplementationofObservers . . . . . . . . . . . . . 12
1.3 ObjectivesandContributions . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 StructureoftheThesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 PreliminariesandPrerequisites 16
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 ReviewofPiecewise-AffineSystems . . . . . . . . . . . . . . . . . . . . . 16
2.3 ReviewofPiecewise-AffineObserverDesign . . . . . . . . . . . . . . . . 19
2.4 BoundednessandUltimateBoundedness . . . . . . . . . . . . . . . . . . . 22
2.5 NonlinearObservers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Piecewise-AffineObserverDesignforaClassofNonlinearSystems 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Piecewise-Affine Observer Design for a Class of Nonlinear Continuous-
TimeSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
viii
3.2.1 StabilityoftheStateEstimationErrorfortheNonlinearContinuous-
TimeSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Piecewise-Affine Observer Design for a Class of Nonlinear Sampled-Data
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1 Stability of the State Estimation Error for the Nonlinear Sampled-
DataSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Piecewise-Affine Observer Design for a Class of Nonlinear Sampled-Data
SystemsinthePresenceofNormBoundedMeasurementNoise . . . . . . . 50
3.5 NumericalExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 WheeledMobileRobotExperimentalResults 90
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 WheeledMobileRobotModeling . . . . . . . . . . . . . . . . . . . . . . 91
4.3 WirelessCommunication,ElectronicsandSensors . . . . . . . . . . . . . 93
4.4 ImplementationoftheContinuous-TimePiecewise-AffineObserveronthe
WheeledMobileRobot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5 ConclusionsandFutureResearch 124
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Appendix 148
ix
LIST OF FIGURES
2.1 PWAapproximationofy=x2 forx∈[−1,1]. . . . . . . . . . . . . . . . . 18
2.2 PWAobserverschematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Coveringcirclewithminimumarea(x∈R,xˆ∈R ). . . . . . . . . . . . . . 22
i j
2.4 Interconnectedobservers(takenfrom[1]). . . . . . . . . . . . . . . . . . . 34
3.1 PWAObserverDesignforaClassofNonlinearSampled-DataSystems. . . 50
3.2 WMRschematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 PWAapproximationof“sinψ”. . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Estimationandestimationerroroftheposition“y”ofthenonlinearcontinuous-
timesystem,usingPWAobserver. . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Estimation and estimation error of the heading angle “ψ” of the nonlinear
continuous-timesystem,usingPWAobserver. . . . . . . . . . . . . . . . . 57
3.6 Estimation and estimation error of the heading angle rate “R” of the non-
linearcontinuous-timesystem,usingPWAobserver. . . . . . . . . . . . . . 58
3.7 PWAregionsinwhichtheobserverisoperating. . . . . . . . . . . . . . . . 59
3.8 Estimationandestimationerroroftheposition“y”ofthenonlinearsampled-
datasystem(T =0.2s),usingPWAobserver. . . . . . . . . . . . . . . . . 60
3.9 Estimation and estimation error of the heading angle “ψ” of the nonlinear
sampled-datasystem(T =0.2s),usingPWAobserver. . . . . . . . . . . . 60
3.10 Estimation and estimation error of heading angle rate “R” of the nonlinear
sampled-datasystem(T =0.2s),usingPWAobserver. . . . . . . . . . . . 61
3.11 State estimation errors for the nonlinear sampled-data system in the pres-
enceofnormboundedwhiteGaussianmeasurementnoise,usingPWAob-
server. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
x