Pseudo Euler-Lagrange and Piecewise Affine Control Applied to

164 Pages·2013·3.82 MB·English

Pseudo Euler-Lagrange and Piecewise Affine Control
Applied to Surge and Stall in Axial Compressors
Seyed Mohamad AminSeyedahmady Zavieh
AThesis
in
TheDepartment
of
ElectricalandComputerEngineering
PresentedinPartialFulfillmentoftheRequirements
fortheDegreeofMasterofAppliedScience(ElectricalandComputerEngineering)at
ConcordiaUniversity
Montre´al,Que´bec,Canada
February2013
(cid:13)c SeyedMohamadAminSeyedahmadyZavieh,2013
CONCORDIAUNIVERSITY
SchoolofGraduateStudies
Thisistocertifythatthethesisproposalprepared
By: SeyedMohamadAminSeyedahmadyZavieh
Entitled: Pseudo Euler-Lagrange and Piecewise Affine Control Applied to
SurgeandStallinAxialCompressors
andsubmittedinpartialfulfilmentoftherequirementsforthedegreeof
MasterofAppliedScience(ElectricalandComputerEngineering)
complies with the regulations of this University and meets the accepted standards with
respecttooriginalityandquality.
Signedbythefinalexaminingcommittee:
Dr. XiupuZhang,Chair
Dr. WahidGhaly,Examiner,External
Dr. AmirAghdam,Examiner
Dr. MustafaMehmetAli,Examiner
Dr. KhashayarKhorasani,Supervisor
Dr. LuisRodrigues,Supervisor
Approvedby
Dr. W.E.Lynch,Chair
DepartmentofElectricalandComputerEngineering
Dr. RobinA.L.Drew
Dean,FacultyofEngineeringandComputerScience
ABSTRACT
PseudoEuler-LagrangeandPiecewiseAffineControlAppliedtoSurgeandStallinAxial
Compressors
SeyedMohamadAminSeyedahmadyZavieh
Thisthesisaddressesthecontroloftheaxialcompressorsurgeandstallphenomena
usingPseudoEuler-LagrangeandPiecewiseAffine(PWA)controllersynthesistechniques.
These phenomena are considered as major gas turbine compressor instabilities that may
result in failures such as the engine flame-out or severe mechanical damages caused by
high blade vibration. The common approach towards the detection of the rotating stall
and surge is to install various types of pressure sensors, hot wires and velocity probes.
The inception of the rotating stall and surge is recognized by the presence of pressure
fluctuation and velocity disturbances in the gas stream that are obtained through sensors.
The necessary measure is then taken by applying proper stall and surge stabilizing con-
trol actions. The Lyapunov stability of pseudo Euler-Lagrange systems in the literature
is extended to include additional nonlinear terms. Although Lyapunov stability theory is
considered as the cornerstone of analysis of nonlinear systems, the generalization of this
energy-based method poses a drawback that makes obtaining a Lyapunov function a dif-
ficult task. Therefore, proposing a method for generating a Lyapunov function for the
control synthesis problem of a class of nonlinear systems is of potential importance. A
systematicLyapunov-basedcontrollersynthesistechniqueforaclassofsecondordersys-
tems is addressed in this thesis. It is shown, in terms of stability characteristics, that the
iii
proposed technique provides a more robust solution to the compressor surge suppression
problem as compared to the feedback linearization and the backstepping methods. The
second contribution is a proposed new PWA approximation algorithm. Such an approx-
imation is very important in reducing the complexity of nonlinear systems models while
keeping the global validity of the models. The proposed method builds upon previous
work on piecewise affine (PWA) approximation methods, which can be used to approxi-
mate continuous functions of n-variables by a PWA function. Having computed the PWA
model of the stall and surge equations, the suppression problem is then solved by using
PWA synthesis techniques. The proposed solution is shown to have higher damping char-
acteristicsascomparedtothebacksteppingnonlinearmethod.
iv
“Ateverystepfromtheconceptionofarationalvisiontotheformulationofthetheory,
faithisnecessary: faithinthevisionasarationallyvalidaimtopursue,faithinthe
hypothesisasalikelyandplausibleproposition,andfaithinthefinaltheory,atleastuntil
ageneralconsensusaboutitsvalidityhasbeenreached.”
—ErichFromm,TheArtofLoving
v
Dedicatedto
MyParents
vi
ACKNOWLEDGEMENTS
First of all, I extend my deep gratitude to my supervisors Dr. Luis Rodrigues and
Dr. KhashayarKhorasaniforawakeningmyinterestinthisparticularfield. Thisresearch,
as one of my important life and academic goals, could not be accomplished without the
invaluable and technical guidance and also generous support of them. I must thank Dr.
W. Ghaly, Dr. A. Aghdam and Dr. M. Mehmet Ali as the defense examining committee
members for their constructive comments, as well as the administrative and the technical
staffofthedepartmentfortheirunstintingsupportandhelp.
I would like to heartily thank all my HYCONS family members and friends Miad
Moarref,HadiKarimi,SinaKaynama,CamiloOssa,MehdiAbedinpour,BehzadSamadi,
Mohsen Zamani, Azita Malek, Gavin Kenneally, Qasim Bhojani and Sahar Sedaghati,
with whom I spent great moments during this period of my life. In particular, my profuse
thanksgoes toMiad Moarref, whoI distracteda lotwithmy questionsand who answered
themunstintingly.
And of course, I owe my father Saeed, my mother Guiti and my sister Sarah a great
deptofeternalgratitudefortheircontinuoussupportandpassionatelove. Ialsowouldlike
tothankSulmasforherhelp,understandingandpatience.
AminZavieh
February2013
vii
TABLE OF CONTENTS
ListofFigures xi
ListofTables xiv
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 OnTheProposedMethodologies . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 PseudoEuler-LagrangeSystems . . . . . . . . . . . . . . . . . . 4
1.2.2 PiecewiseAffineSystems . . . . . . . . . . . . . . . . . . . . . 5
1.3 ContributionsoftheThesis . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 StructureoftheThesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 GasTurbinesCompressors 10
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 AxialCompressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 RotatingStallandSurge . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 BasicConcept . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 ABriefSurveyontheCompressorInstabilityProblem . . . . . . 14
2.3.3 CompressorModel . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.4 SimplifiedMoore-GreitzerModel . . . . . . . . . . . . . . . . . 21
2.3.5 UncertaintyinaParameter . . . . . . . . . . . . . . . . . . . . . 23
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 LyapunovStabilityofPseudoEulerLagrangeSystems 26
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 ReviewofPseudoEuler-LagrangeSystems . . . . . . . . . . . . . . . . 27
3.3 StabilityAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
viii
3.4 ControllerSynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 CompressorSurgeStabilization . . . . . . . . . . . . . . . . . . . . . . 33
3.5.1 Input-OutputFeedbackLinearizationTechnique . . . . . . . . . 33
3.5.2 BacksteppingTechnique . . . . . . . . . . . . . . . . . . . . . . 36
3.5.3 PseudoEuler-LagrangeTechnique . . . . . . . . . . . . . . . . . 41
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Intersection-basedPiecewiseAffineApproximationofFunctionsofn-Variables 52
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 ReviewofPWASystems . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 ApproximationTheoryForFunctionsofOne-Variable . . . . . . . . . . 54
4.3.1 TheIPWAAlgorithms . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 ApproximationTheoryForFunctionsofn-Variables . . . . . . . . . . . 69
4.4.1 TheIPWAAlgorithms . . . . . . . . . . . . . . . . . . . . . . . 97
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5 AxialCompressorSurgeandStallSuppression 106
5.1 PseudoEuler-Lagrange(PEL)Approach . . . . . . . . . . . . . . . . . . 107
5.1.1 PseudoEuler-LagrangeControlSynthesisProblem . . . . . . . . 107
5.1.2 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 PiecewiseAffineApproach . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.1 IPWAApproximationandtheModelingProblem . . . . . . . . . 117
5.2.2 PWAControlSynthesisProblem . . . . . . . . . . . . . . . . . . 120
5.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6 ConclusionsandFutureResearch 124
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.1.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
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6.2 FutureResearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
x