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 ix 6.2 FutureResearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 x