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Switching linear parameter-varying electronic throttle control for automotive engines Zandi Nia, Arman 2015

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Switching Linear Parameter-VaryingElectronic Throttle Control ForAutomotive EnginesbyArman Zandi NiaB.A.Sc., The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2015c© Arman Zandi Nia 2015AbstractModern automotive engines are mechanically mature technologies that havebeen significantly optimized to reduce harmful emissions, reduce fuel con-sumption, and improve driveablilty. Spark-ignition (SI) engines must op-erate under a stoichiometric air-to-fuel ratio that allows the three-way cat-alytic converter to effectively mitigate harmful tail-pipe emissions. Elec-tronic throttle control (ETC) is the primary means of regulating the amountof charge-air entering the cylinders. This research studies the applicationof switching linear parameter-varying (LPV) feedback control techniques tothe ETC problem.Electronic throttle valves are highly nonlinear systems due to their pack-aging, cost, and reliability constraints. The plant’s dynamics vary consider-ably with respect to slip-stick friction, limp-home springs, and unmodeleddisturbances. The complete plant model encapsulates the aforementionednonlinear dynamics as a linear model that changes affinely with parameter-varying slip-stick friction and voltage fluctuation. The ETC synthesis prob-lem is formulated as a linear matrix inequality as a non-convex optimizationproblem and solved via iterative methods.In previous literature pertaining to LPV controllers, large variationsthroughout the electronic throttle valve’s operating region have led to con-servative results. In this research, to reduce conservatism, the operatingregion has been partitioned into smaller subregions. Switching events be-tween subregions are based on the scheduling-parameters of the LPV system,which are related to slip-stick friction and voltage fluctuation.The devised switching LPV controller is tuned and validated and its per-formance is experimentally compared with popular control solutions to theETC problem. The baseline controllers include classic proportional-integral-iiAbstractderivative control, sliding-mode control, and gain-scheduling LPV control.Experimental results reveal that the switching LPV controller outperformsthe baseline controllers throughout all the prescribed operating regions.iiiPrefaceThis dissertation is original intellectual property of the author, Arman ZandiNia under supervision of Dr. Ryozo Nagamune. This work has been com-pleted in the Control Engineering Laboratory at the University of BritishColumbia. The results presented are going to be submitted for publications.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . 21.3 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . 42 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Electronic Throttle Controller . . . . . . . . . . . . . . . . . 52.1.1 Proportional-Integral-Derivative Control . . . . . . . 62.1.2 Sliding-Mode Control . . . . . . . . . . . . . . . . . . 72.1.3 Linear Parameter-Varying Control . . . . . . . . . . . 72.2 Controller Design Theory . . . . . . . . . . . . . . . . . . . . 82.2.1 Sliding-Mode Control Theory . . . . . . . . . . . . . 8vTable of Contents2.2.2 Linear Parameter-Varying Control Theory . . . . . . 92.2.3 Switching LPV Control Theory . . . . . . . . . . . . 102.3 Research Objectives and Methodology . . . . . . . . . . . . . 113 Electronic Throttle Control System . . . . . . . . . . . . . . 133.1 System Description . . . . . . . . . . . . . . . . . . . . . . . 133.2 Electronic Throttle Valve Modeling . . . . . . . . . . . . . . 143.2.1 Lumped-Element Model . . . . . . . . . . . . . . . . 143.2.2 Limp-Home Spring . . . . . . . . . . . . . . . . . . . 173.2.3 Slip-Stick Friction . . . . . . . . . . . . . . . . . . . . 183.2.4 Exogenous Disturbance . . . . . . . . . . . . . . . . . 193.2.5 Control-Oriented Model . . . . . . . . . . . . . . . . . 203.3 Control Strategy and Performance Metrics . . . . . . . . . . 213.3.1 Control Objectives . . . . . . . . . . . . . . . . . . . 223.3.2 Experimental Settings and Performance Specifications 224 ETC Structure and Design Method . . . . . . . . . . . . . . 254.1 Sliding-Mode Controller . . . . . . . . . . . . . . . . . . . . . 264.2 LPV Gain-Scheduling Controller . . . . . . . . . . . . . . . . 274.2.1 LPV Control Structure . . . . . . . . . . . . . . . . . 284.2.2 Limp-Home Spring Compensation . . . . . . . . . . . 294.2.3 LPV Model of plant . . . . . . . . . . . . . . . . . . . 304.2.4 Scheduling-Parameters . . . . . . . . . . . . . . . . . 334.2.5 Reduced-order Observer . . . . . . . . . . . . . . . . 335 Switching Gain-Scheduling LPV Controller . . . . . . . . . 365.1 Hysteresis Switching . . . . . . . . . . . . . . . . . . . . . . . 375.2 Switching gain-scheduling LPV Synthesis . . . . . . . . . . . 375.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 396 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 436.1 Experimental Test Bench . . . . . . . . . . . . . . . . . . . . 436.2 Parameter Identification . . . . . . . . . . . . . . . . . . . . 446.2.1 Stalled Motor Test . . . . . . . . . . . . . . . . . . . 44viTable of Contents6.2.2 Motor Constant Test . . . . . . . . . . . . . . . . . . 456.2.3 Static Load Test . . . . . . . . . . . . . . . . . . . . . 466.2.4 Summary of System Identification . . . . . . . . . . . 486.3 Experimental Cases . . . . . . . . . . . . . . . . . . . . . . . 486.3.1 Large Opening and Closing . . . . . . . . . . . . . . . 486.3.2 Small Opening and Closing . . . . . . . . . . . . . . . 496.3.3 Limp-Home Crossing . . . . . . . . . . . . . . . . . . 496.3.4 Voltage Fluctuations . . . . . . . . . . . . . . . . . . 507 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.1 Summary of Research . . . . . . . . . . . . . . . . . . . . . . 577.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 587.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60AppendixA Controller Parameters . . . . . . . . . . . . . . . . . . . . . . . 66A.1 PID Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 66A.2 SMC Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 66A.3 LPV Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 67A.4 Switching LPV Parameters . . . . . . . . . . . . . . . . . . . 68viiList of Tables5.1 Performance bound comparison between multi-region con-trollers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.1 Table of the identified parameter values . . . . . . . . . . . . 486.2 Performance of PID, SMC, LPV and S-LPV for case 1 . . . . 496.3 Performance of PID, SMC, LPV and S-LPV for case 2 . . . . 506.4 Performance of PID, SMC, LPV and S-LPV for case 3 . . . . 506.5 Performance of PID, SMC, LPV and S-LPV for case 4 . . . . 51A.1 Table of parameters for the baseline PID controller . . . . . . 66A.2 Table of parameters for the baseline SMC controller . . . . . 67viiiList of Figures1.1 Typical electronic throttle valve found in automotive engines 21.2 Air and fuel path in automotive engines . . . . . . . . . . . . 33.1 Electronic accelerator system . . . . . . . . . . . . . . . . . . 143.2 Lumped-element model of electronic throttle valve. Elementsare DC motor, reduction gear, and throttle plate. . . . . . . . 153.3 Reduction gear set of the electronic throttle valve . . . . . . . 163.4 Torque induced by return springs as a function of throttleposition θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Torque induced by slip-stick friction as a function of angularvelocity θ˙ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 Experimental trajectories for ETC . . . . . . . . . . . . . . . 234.1 Block diagram of a typical control system . . . . . . . . . . . 254.2 LPV control structure applied to the ETC plant . . . . . . . 284.3 ETC system block diagram . . . . . . . . . . . . . . . . . . . 304.4 LFT form of ETC system . . . . . . . . . . . . . . . . . . . . 314.5 Structure of reduced-order state-observer . . . . . . . . . . . 345.1 Hysteresis switching for two adjacent regions . . . . . . . . . 375.2 Lyapunov function of adjacent subregions during switching . 385.3 Iterative optimization algorithm for finding switching LPVcontrollers for r-subregions . . . . . . . . . . . . . . . . . . . 405.4 Switching subregions of the controller as a function of scheduling-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41ixList of Figures5.5 Visual comparison gains between one-region LPV and nine-region LPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.1 Experimental test bench of ETC system . . . . . . . . . . . . 446.2 Measured current as a function of slowly varied armature volt-age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.3 Experimentally measured angular velocity of intermediate re-duction gear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.4 Static load test showing the throttle valve angle as the motortorque is linearly increased (red) and subsequently linearlydecreased (blue) . . . . . . . . . . . . . . . . . . . . . . . . . 476.5 Experimental result of large opening and closing of the throt-tle plate position with PID (dotted line), SMC (dash-dottedline), LPV (thin line) and S-LPV (thick line) . . . . . . . . . 526.6 Experimental result of small opening and closing of the throt-tle plate position with PID (dotted line), SMC (dash-dottedline), LPV (thin line) and S-LPV (thick line) . . . . . . . . . 536.7 Experimental result of LHC of the throttle plate position withPID (dotted line), SMC (dash-dotted line), LPV (thin line)and S-LPV (thick line) . . . . . . . . . . . . . . . . . . . . . . 546.8 Experimental result of voltage increase with PID (dotted line),SMC (dash-dotted line), LPV (thin line) and S-LPV (thickline) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.9 Experimental result of voltage decrease with PID (dottedline), SMC (dash-dotted line), LPV (thin line) and S-LPV(thick line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56xGlossaryADC Analog to DigitalAFR Air-to-Fuel RatioECU Electronic Control UnitEMF Electromotive ForceETC Electronic Throttle ControlIFT Iterative Feedback TuningLFT Linear Fractional TransformationLMI Linear Matrix InequalityLTI Linear Time-InvariantLH Limp-HomeLHC Limp-Home CrossingLOC Large Opening & ClosingLPV Linear Parameter-VaryingPID Proportional-Integral-DerivativePWM Pulse-Width ModulationSMC Sliding-Mode ControlSI Spark-ignitionxiGlossarySOC Small Opening & ClosingTPS Throttle Position SensorTWC Three-Way Calalytic ConverterVF Voltage FluctuationxiiAcknowledgementsFirstly, I would like to express my sincerest gratitude to my supervisorDr. Ryozo Nagamune for the unfaltering support, for his patience, motiva-tion, and friendship. I could not imagine having a better supervisor andmentor and I am forever grateful for this opportunity.Besides my supervisor, I would like to thank the rest of my thesis com-mittee: Dr. Patrick Kirchen and Dr. Bhushan Gopaluni.I am indebted to my colleagues Dillon Melamed and Jeffrey Homer.They continuously inspired me, pushed me, distracted me, and aided methroughout my research. The oft stressful and difficult moments were easierbecause of you two.I would also like to thank my fellow CEL labmates, in particular MasihHanifzadegan, Masashi Karasawa, Pan Zhao, and Alireza Alizadegan whomhave been pillars of support and knowledge. To my friends, who I wasn’table to see for months at a time, thank you for bearing with me.Finally, I would like to thank my parents for always supporting me,loving me, and providing me with every opportunity to succeed. Thereare no words to express my appreciation, I owe them everything. Thankyou for being patient with me. To my lovely sister, I thank you for yourunderstanding and encouragement throughout my academic career.xiiiDedicationTo my parents,Hossein Zandi Nia and Najmeh Sajjadi,Without whom none of my success would be possible.xivChapter 1Introduction1.1 MotivationModern automotive engines are mechanically mature technologies that havebeen finely tuned and optimized over the past hundred years to reduce harm-ful emissions, reduce fuel consumption and increase power. However, inter-nal combustion engines still have potential for substantial improvements inperformance with the help of advanced control systems. With the ever-increasing complexities in automotive systems and strict environmental reg-ulations, model-based control system design will have a pivotal role in ad-dressing these issues [1], [2], [3].Engine pollutant emissions would exceed the typical limits set by mostregulating bodies if it were not for exhaust after-treatment systems. Thecrucial element for running a clean spark-ignition engine is the three-way cat-alytic converter (TWC) which converts the pollutant emissions to innocuousemissions. However, the conversion efficiency depends on the precise controlof the air-to-fuel ratio (AFR) such that the TWC operates within a verynarrow band where the AFR ratio is stoichiometric [4], [1], [5]. AFR controlhas had a significant impact in the adoption of advanced control strategiesin automotive engines. By precisely controlling the AFR, especially duringtransient operation, one can effectively allow the TWC to treat emissions,prevent engine knock and reduce fuel consumption [6]. AFR is adjusted bycontrolling the fuel-injectors and the electronic throttle valve.In conventional engines, the throttle plate is physically linked to theaccelerator pedal which is directly controlled by the driver, and thus, en-gine conditions do not impact the resulting throttle angle. Succeeding itsentirely mechanically operated predecessor, ETC uses a drive-by-wire sys-11.2. Problem Statementtem to regulate the throttle plate position through a DC motor, as can beseen in Figure 1.1 [6]. Throttle position solely determines the quantity ofcharge-air flowing into the cylinders, which makes ETC a key facilitator intorque-based engine control where the acceleration pedal corresponds to adesired torque. The ETC system is becoming a standard feature of modernengines [7], [8].DCMotorReduction GearsLimp-HomeSpringThrottlePlateFigure 1.1: Typical electronic throttle valve found in automotive engines1.2 Problem StatementThis research pertains to the normally aspirated spark-ignition internal com-bustion engine as seen in Figure 1.2. Air is drawn in through the electronicthrottle valve into the intake manifold reservoir where the amount of charge-air is regulated by the position of the throttle valve. The charge-air is thendistributed among the cylinders in the engine. Fuel is then injected intoeach air stream prior to entering each cylinder. Once the combustion stroke21.2. Problem Statementhas completed by igniting the air and fuel mixture, the exhaust streams re-combine and enter an exhaust reservoir where they eventually flow into theTWC for treatment [9], [1]. The amount of air and fuel is controlled by theelectronic control unit (ECU) which generates a desired trajectory for theposition of the throttle plate and the amount of fuel to be injected based onthe driver’s requested power demands while taking into consideration engineoperation mode information, emission constraints, safety factors, and so on[10], [11], [12]. This research does not consider the ECU’s generation of thedesired throttle position trajectory.Figure 1.2: Air and fuel path in automotive enginesThis current research concerns the control of an electronic throttle valvethat regulates the charge-air quantity. The control of the throttle valve is nota trivial task. The treatment of the nonlinear characteristics of the systemsuch as slip-stick friction, limp-home (LH) spring set, gear backlash, batteryvoltage fluctuations and the torque induced by the intake air flow signifi-cantly affect system performance [13], [11], [14]. In practice, it has beenshown that the electronic throttle should handle fast transient trajectorieswithout overshoot, be robust to parameter variations and disturbances yetmaintain high static precision [10].31.3. Organization of Thesis1.3 Organization of ThesisThis thesis is organized as follows. In Chapter 2, a detailed review of previ-ous literature concerning ETC and advanced control techniques is presented.In Chapter 3, a detailed mathematical model of the electronic throttle valvesystem is derived and the control objectives are defined. Chapter 4 presentsthe control synthesis techniques of the baseline controllers used for compar-ison purposes. Chapter 5 presents the the synthesis of a switching gain-scheduling LPV controller, which is the primary contribution of this thesis.Chapter 6 presents the experimental test bench, methods used for param-eter identification, and reveals the experimental results of four designedcontrollers. Chapter 7 presents the conclusions of the thesis, and discussespotential future research directions.4Chapter 2Literature ReviewThis chapter provides a summary of the previous works to improve the per-formance characteristics of electronic throttle valves in automotive applica-tions. ETC has been an active area of research over the past two decadesdue to drive-by-wire technology becoming more popular among automotivemanufacturers.This literature review will focus on works addressing the ETC problem inautomotive engines in Section 2.1 and literature regarding controller designmethods which have been utilized for ETC in Section Electronic Throttle ControllerModern techniques that address the ETC problem include proporitional-integral-derivative control (PID) [15], [16], [17], [18], [10], sliding-mode con-trol (SMC) [19], [20], [21], [22], [23], [24] and linear parameter-varying con-trol (LPV) [25], [13], [26]. Electronic throttle valves are an exemplary ap-plication of advanced control due to its inherent nonlinear dynamics andparametric uncertainty which play a significant role in overall engine perfor-mance. Robust control techniques consider system variations in electronicthrottle valves as uncertainties and deal with them using time-invariantcontrollers with guaranteed performance. However, due to the considerablevariations in the electronic throttle valve’s dynamics, robust control tech-niques generate conservative controllers [9].52.1. Electronic Throttle Controller2.1.1 Proportional-Integral-Derivative ControlIn industry, PID control is typically applied to powertrain control systemsdue to its simplicity and satisfactory performance with coarse tuning inspecific regions of operation [27], [15], [28]. However, PID control can beinadequate in some applications and generally is not an optimal methodof control. Parameter uncertainty is common in electronic throttle valves,especially since reducing production costs typically means less stringent em-phasis on manufacturing tolerances; frictional effects and dependency onenvironmental factors or system wear are increased as a result [17]. Thefundamental issue with PID control is its time-invariance with constant pa-rameters, as well as its ignorance to previous knowledge of the electronicthrottle valve dynamics. When applied alone, PID controllers can have dif-ficulty in handling nonlinearities, do not change with respect to varyingsystem behavior, and can lag in response to large disturbances. PID con-trollers can be modified by employing iterative feedback tuning (IFT) [15],or adaptive self-tuning [17], [10], methods to estimate corresponding modelparameters. However, adaptation may not converge very quickly and couldresult in performance losses or loss of stability during sudden changes inoperating conditions. Furthermore, PID controllers can be modified withadditional nonlinear compensation techniques to mitigate the nonlinear ef-fects [10]; these techniques may also ensure the stability of the system [16].To account for the inherent time-varying nature of the electronic throt-tle valve, a gain-scheduled PID was implemented in [25] to optimize per-formance at fixed operating points. Gain-scheduling controllers are time-varying controllers that adjust based on system variations using informationabout the plant dynamics through a specified operating range. Typically,gain-scheduled techniques based on interpolation have the shortcoming ofbeing unable to theoretically guarantee stability and performance of thesystem. In summary, PID controllers are linear controllers and even withmodifications, they do not guarantee performance or stability throughoutall regions of operation.62.1. Electronic Throttle Controller2.1.2 Sliding-Mode ControlSMC is a nonlinear class of controller that is designed to be robust to para-metric uncertainty and/or unmodeled dynamics. It can effectively quantifythe resulting trade-offs between the tracking performance and unmodeleddynamics. Furthermore, SMC provides a systematic approach to attainingrobust stability in the face of modeling imprecision [29]. SMC is a com-mon approach to the ETC problem where the nonlinear dynamics of springtorques and friction are considered as bounded disturbances and parametricuncertainties [26], [20], [21], [19]. With SMC, convergence to a stable statecan be guaranteed within a finite time, however the robust performanceof SMC cannot be guaranteed. As unmodeled nonlinear dynamics can haveseverely adverse effects on tracking performance, a discontinuous control lawacross the sliding-mode manifold is employed to push the system dynamicsback to a stable state. Since in actuality, switching is imperfect, this resultsin chattering within the system. Furthermore, the selection of the slidinghyperplane is a non-trivial task. ETC is susceptible to immediate changes indesired tracking setpoints, and trajectory design plays an important factorin the performance of SMC.2.1.3 Linear Parameter-Varying ControlLPV systems are defined by sets of linear differential equations which canvary based on time-varying parameters. A physics-based LPV throttlevalve model was established in [26], [13], where the authors converted thehighly nonlinear system into an LPV system. Uncertainties such as slip-stick friction and voltage fluctuations are modeled as LPV measurable gain-scheduling parameters. The nonlinear LH spring dynamics are not includedin the LPV model as they are compensated by another feedback controller.The purpose of LPV synthesis is to design a controller with the same struc-ture as the LPV system to meet the desired specifications on stability andperformance over the bounded regions of operation [30]. Consequently, theauthors designed LPV gain-scheduling controllers via linear matrix inequal-ity (LMI) techniques based on the LPV model to guarantee the robust sys-72.2. Controller Design Theorytem stability and performance. However, the apparent issue with LPV con-troller design techniques is the inherent conservatism of the controllers. InLPV systems with a broad range of parameter variation, it is possible thata single Lyapunov function may not exist. If a Lyapanuv function does notexist, designers must sacrifice performance in some part of the operatingregion in order to attain an LPV controller throughout all regions [31], [32].2.2 Controller Design TheoryTheory pertaining to the development of the current research topic is brieflyreviewed in the following subsections. For comparative purposes, robustcontrol methods such as SMC and LPV-based control are investigated inSubsections 2.2.1 and 2.2.2 respectively. The switching LPV based controlfor electronic throttle valves, being the emphasis of this current research, ispresented in Subsection Sliding-Mode Control TheorySMC is a popular control technique that can be applied to nonlinear andtime-varying systems. Furthermore, SMC is an attractive option due toability to handle uncertain system dynamics and parameters. SMC is aclass of variable structured systems that has a discontinuous nature. Thisdiscontinuity is exhibited when the controller switches between two uniquelydifferent structures [29], [33]. This distinct switching mechanism introducesa new type of system motion, coined sliding-mode, which exists in the man-ifold [34]. When the system dynamics become confined to the sliding-modemanifold, reduced-order dynamics emerge from the original system model.These characteristics provide adequate performance in disturbance rejectionand insensitivity to parameter variations, and since the control law is notcontinuous, the system can converge within a finite time [35].The SMC structure does have its disadvantages, however. SMC is proneto the chattering phenomenon that is perceived as oscillations across thesliding-mode manifold. The switching of the SMC structure ideally happens82.2. Controller Design Theoryat an infinite frequency to force the system dynamics along the confinedmanifold. However, due to delays imposed by computations, hardware andphysical limitations of the system, it is not possible to change the controlinput infinitely fast. These dynamics are often neglected in the models usedfor control design. The potential high frequency chattering that can exciteunmodeled dynamics, cause energy loss, instability or damage the plant.Furthermore, SMC has been shown to handle bounded uncertainties butfailed in dealing with unmatched uncertainties [36]. Another drawback ofSMC is that the performance of the SMC depends heavily on the nontrivialdesign of the sliding-mode manifold. If not designed adequately, the controlscheme will lead to poor performance.In summary, the SMC scheme is used to effectively account for the im-precision or inaccuracies in the system model, the presence of unmodeleddynamics, and simplifies higher-order systems. The robust stability of theSMC scheme can be guaranteed with parametric and dynamic uncertaintiesassociated with the plant. However, there is a trade-off between model-ing accuracy and performance; the robust performance of the SMC is notguaranteed with respect to chattering, unmatched uncertainties, and thenontrivial design of the sliding-mode manifold.2.2.2 Linear Parameter-Varying Control TheoryLPV methods have been used in a wide array of applications, and to no sur-prise, many of the first papers utilizing LPV modeling and control were per-tinent to the aerospace industry [37], [38]. Gain-scheduling control is a com-mon class of nonlinear control that is effective for nonlinear systems but isan amalgamation of linear controllers. Classical synthesis of gain-schedulingcontrollers requires numerous iterations of design for all subsequent regionsof operation, and the control law uses scheduling or interpolation methodsthat do not guarantee adequate performance and robustness throughoutall trajectories [39], [40]. With the advent of LMI techniques, there ap-peared efficient schemes for synthesizing gain-scheduling controllers in theLPV framework. LPV controllers, by design, incorporate gain-scheduling92.2. Controller Design Theorywhile guaranteeing the robust performance and stability of the system [37],[39], [13], [41]. Furthermore, LPV models provide a compromise between lin-ear and nonlinear systems, particularly for analysis since there are a varietyof tools available for evaluating linear systems which can be easily adoptedduring controller design.To design an LPV controller, the system plant is converted into an LPVstructure such that the state-space matrices depend affinely on the time-varying parameters [42], [43], [39]. The LPV model has been developed in[26], [13]. The time-varying parameters associated with the LPV model mustbe measurable in real-time. Moreover, derived as a result of the quadraticstability of the system, conditions to solve the synthesis problem are reducedto a set of LMIs. The synthesis problem has now effectively been reducedto a optimization problem.In summary, the LPV framework permits the synthesis of a controllerthat accounts for the time-varying nature of the system without the defi-ciencies associated with classical gain-scheduling or gain-interpolation tech-niques. LPV control provides a systematic approach for synthesis, and itguarantees the robust performance and stability of a system for an entirerange of trajectories at once so long as there exists a feasible solution.2.2.3 Switching LPV Control TheoryLiterature pertaining to LPV synthesis has been primarily concerned withfinding a single Lyapunov function throughout a system’s entire operatingrange. This tends to be a problem because a Lyapunov function may notexist over a large operating range or may otherwise lead to conservative con-trol [44], [31]. A successful strategy to handle systems with large regions ofoperation is to design a controller for each possible model within a subregionand switch using supervisory switching logic [45]. Multicontroller design isan effective means to deal with high system variability but naive switchingcan introduce transient instability during switching events.Using switching LPV control, the intrinsic conservatism associated withsingle-region LPV controllers can be mitigated by designing numerous Lya-102.3. Research Objectives and Methodologypunov functions over various subregions of the operating range. As previ-ously mentioned, classical switching methods can introduce transient insta-bility so hysteresis switching techniques are used to ensure safe switchingconditions between the subregions [32], [45]. These conditions are intro-duced by designing LMI’s over the overlapping subregions to ensure thatat each switching event the Lyapunov function is increasing relative to theadjacent subregion. Switching controllers can be more aggressive than non-switching controllers, especially in regards to systems with a large operatingranges. Switching LPV control is an approach for improving the perfor-mance of current LPV control synthesis techniques and will be the primarycontroller investigated in this thesis.2.3 Research Objectives and MethodologyETC is a popular area of research because of its value in automotive appli-cations and its non-trivial control problem. Many researchers have proposedeffective control methods for addressing the ETC problem but often do notconsider the time-varying dynamics, robust stability and robust performanceof the electronic throttle valve throughout all regions of operation. The pri-mary objective of this thesis is to address the aforementioned shortcomingsof previous literature, and to further improve upon the performance of thecurrent methods in a novel way. To elaborate, the research objectives pre-sented in this thesis are to design a controller that:• addresses time-varying and nonlinear dynamics of the electronic throt-tle valve, specifically friction, LH spring, and voltage fluctuations,• guarantees robust performance and stability throughout the entire re-gion of operation via linear matrix inequality optimization, and• simplifies the approach to LPV control synthesis for ETC problem.To apply this class of controller, we must:• obtain an LPV model of the system,112.3. Research Objectives and Methodology• define the control objectives and performance measured based on thosemetrics,• propose a feedback structure,• apply existing switching gain-scheduling techniques, and• experimentally validate controller performance with existing solutionsin terms of settling time, overshoot and .12Chapter 3Electronic Throttle ControlSystemIn this chapter, a detailed characterization of the electronic throttle valve ispresented. A macro description of the system, mathematical modelling ofthe system, and consequent performance requirements associated with ETCare presented in Sections 3.1, 3.2, and 3.3, respectively.3.1 System DescriptionThe electronic throttle valve consists of a DC motor, a set of reduction gears,throttle valve plate, throttle position sensor (TPS) and two return springs.When the accelerator pedal of the automobile is actuated, the ECU cal-culates the required throttle position such that the appropriate amount ofcharge-air enters the intake manifold and consequently the engine cylindersfor combustion. However, the setpoint angle generated by the ECU is notsolely a function of the accelerator pedal position. Rather it considers anarray of parameters related to environmental conditions, safety, efficiencyand performance based on estimations of the driver’s desired torque. Thethrottle position plays a crucial role in the amount of air that enters theengine cylinders as it is the sole means of regulating the charge-air. Oncethe reference trajectory is generated by the ECU, it is compared to the mea-surement of the TPS which is fed back into the ECU, as seen in Figure 3.1.As a result, the ECU generates a voltage signal across the DC motor whichis controlled via pulsewidth-modulation (PWM). The motor shaft rotationis transferred to the throttle shaft through a set of reduction gears to achievethe desired throttle position.133.2. Electronic Throttle Valve ModelingFigure 3.1: Electronic accelerator systemIn spite of being relatively straight forward, the electronic throttle valveis a highly nonlinear system due to friction, LH spring, and voltage variation.The nonlinear characteristics of the electronic throttle valve add complexityfor which classical control methods will not yield desirable results. Thenonlinear dynamics of the electronic throttle valve can be, for the mostpart, attributed to the presence of the LH return spring, slip-stick friction,voltage fluctuations in the automobile batteries, and disturbance torquesinduced on the throttle plate due to in-line pressure variations.3.2 Electronic Throttle Valve ModelingIn this section, an accurate mathematical model of the electronic throttlevalve is derived as described in [13], [26], [15]. Furthermore, the accompa-nying subsections will cover the linear and nonlinear system dynamics ingreater breadth.3.2.1 Lumped-Element ModelLumped-element models are a common method of capturing the dynamiccharacteristics of multi-domain systems. These models are an amalgamation143.2. Electronic Throttle Valve Modelingof discrete entities that accurately represent a system’s distributed behaviorunder certain assumptions. The electronic throttle valve is represented asa lumped-element model in Figure 3.2. The differential equations used toFigure 3.2: Lumped-element model of electronic throttle valve. Elementsare DC motor, reduction gear, and throttle plate.model the electrical DC motor dynamics areVa = Ladiadt+Raia + ea, (3.1)where Va, La, ia, Ra, and ea are the battery voltage, equivalent armatureinductance, current, resistance, and back electromotive force (EMF), re-spectively. However, due to the small value of the armature inductance,armature lag is neglected [10], [13]. The mechanical dynamics of the DCmotor areJmθ¨m = Ta −Bmθ˙m − Tm, (3.2)where Jm and Bm are the equivalent motor inertia and damping, respec-tively. Ta is the torque produced by the armature circuit and Tm is thetorque that is transmitted into the reduction gears and ultimately the throt-tle valve shaft as depicted in Figure 3.3.The torque Ta generated by the armature is linearly proportional to thecurrent ia running through the armature circuitTa = Kmia, (3.3)where Km is the motor constant. Equivalently, the armature torque Ta can153.2. Electronic Throttle Valve ModelingFigure 3.3: Reduction gear set of the electronic throttle valvebe represented byTa =KmRa(Va −Kvωm) (3.4)With respect to the conservation of power, the load torque Tl induced onthe throttle valve shaft through the reduction gears isTl = NTm, (3.5)where N is the compound gear ratio. Naturally, the backlash inherent toreduction gear sets introduces nonlinear dynamics, however they are quitesmall and have negligible effects on the system [19], [13].Finally, the dynamics of the throttle plate are described asJlθ¨l = Tl − Tfriction − Tspring − Tdisturbance, (3.6)where Tfriction, Tspring and Tdisturbance illustrate the dynamics resulting fromfriction, LH spring and exogenous disturbances, respectively. The followingsubsections will develop mathematical models for the aforementioned dy-namics.163.2. Electronic Throttle Valve Modeling3.2.2 Limp-Home SpringThe inclusion of the LH springs within the throttle valve mechanism is aprescribed safety measure where the ECU and the electronic throttle valvecan no longer communicate with each other. In this safety-mode, the returnsprings in the throttle valve force the throttle plate to return to a non-zeroresting position, called the LH position θ0. During, what is often referredto as, ’limp-home-mode’ the throttle is set to an idle condition where thecharge-air is fast enough to allow the vehicle to maneuver itself to safety butnot fast enough to generate lots of power. This safety feature results in thenonlinear dynamic exhibited in Figure 3.4. The dynamic characteristics ofFigure 3.4: Torque induced by return springs as a function of throttle posi-tion θthe LH springs dynamics are manifested as [26], [10]Tspring ={TLH +Ks(θl − θ0) : θ0 < θl < θmax−TLH −Ks(θl − θ0) : θmin < θl < θ0,Tspring = TLH sgn(θl − θ0) +Ks(θl − θ0),(3.7)173.2. Electronic Throttle Valve Modelingwhere the magnitude of the LH spring torque at the LH position is TLH . Ifthere is any divergence of the throttle position θl from the from the neutralposition, the dynamics are that of a linear spring with a spring stiffness Ks.3.2.3 Slip-Stick FrictionFriction factors quite heavily in the performance of the electronic throttlevalve. The friction present in the system is comprised of a nonlinear slip-stick friction Tf and linear viscous damping Tb [26], [10], given byTfriction = Tf + Tb (3.8)In the case of electronic throttle valves where the coefficient of kinetic fric-tion is less than the coefficient of static friction, there will be a trend ofintermittent jerking rather than a smooth motion. This phenomenon canbe characterized by the contact surfaces ’sticking’ until the force reachesthat of the static friction. Once the force has overcome the static friction, itwill continue ’slipping’ with a small kinetic friction until the contact surfacesstick once again. The dynamics of the slip-stick friction is illustrated by thesolid line in Figure 3.5. The dashed line is an approximation of the slip-stickfriction which will be revisited later.Furthermore, elastic deformations and relatively large manufacturing tol-erances in electronic throttle valves are significant contributors to slip-stickbehaviour that have a crucial impact on the performance of the controlsystem. The slip-stick model is given byTf = Fc sgn θ˙l, (3.9)where Fc is the magnitude of slip-stick friction coefficient, and sgn is thediscontinuous sign function defined assgn(θ˙l) = +1 if θ˙l > 0sgn(θ˙l) = −1 if θ˙l < 0(3.10)Moreover, the dynamic friction component is commonly modeled by a vis-183.2. Electronic Throttle Valve ModelingFigure 3.5: Torque induced by slip-stick friction as a function of angularvelocity θ˙cous damping model given byTb = Blθ˙l, (3.11)where Bl is the viscous damping coefficient. The direction of the dampingforce is opposite to the direction of throttle plate motion and its magnitudedepends on the nature of the contacting surfaces and angular velocity θ˙l.3.2.4 Exogenous DisturbanceDuring engine operation, there are various dynamic disturbances that affectthe tracking performance of the electronic throttle valve, these disturbancescan be modeled asTdisturbance = Tp + Tv (3.12)In practice, there is a disturbance torque Tp on the throttle plate inducedby pressure variations due the air-flow dynamics of the engine. Tp will beconsidered as an unmodeled disturbance. Since battery voltage typically193.2. Electronic Throttle Valve Modelingvaries during operation, we introduce that as a disturbance torque Tv to thethrottle valve asTv = −(NKmRa)∆Va, (3.13)where ∆Va is the voltage variation percentage. In this thesis, we will beconsidering ∆Va as a step change.3.2.5 Control-Oriented ModelThe preceding nonlinear dynamics presented lead to the formulation of amore complex control problem. Thus, an emphasis was placed on obtaininga strong mathematical representation of the system. Now that the discreteelements of the electronic throttle valve have been characterized for thesystem presented in Section 3.2.1 - 3.2.3, the differential equation 3.6 canbe expressed asJlθ¨l = NTm − (Fc sgn θ˙l +Blθ˙l)− (TLH sgn(θl − θ0) +Ks(θl − θ0))−(Tp −(NKmRa)∆Va)(3.14)Combining (3.4), (3.2), and (3.14) the differential equation representing theelectronic throttle valve is restructuredJeq θ¨ =NKmVbRau(1 + ∆)−(Beq +N2KvKmRa)θ˙ −Ksθ− Fc sgn θ˙ − TLH sgn θ − Tp, (3.15)where Jeq = Jl + N2Jm, Beq = Bl + N2Bm and θ = (θl − θ0) are theequivalent inertia, equivalent damping, and relative throttle angle measuredby the TPS. The relationship θl = Nθm is used to convert the angularposition of the motor θm into the angular position of the throttle valve θlmeasured by the TPS. The differential equation (3.15) can be represented203.3. Control Strategy and Performance Metricsby the continuous-time state space model.x˙ = Ax+Bu+ Γ(θ, θ˙),y = Cx,(3.16)where the matrices are defined asA =[0 1−KsJeq − 1Jeq (Beq + N2KvKmRa)], (3.17a)B =[0NKmVb(1+∆)JeqRa], (3.17b)Γ(θ, θ˙)=[0−Fc sgn θ˙ − TLH sgn θ − Tp]1Jeq, (3.17c)C =[1 0](3.17d)The state x seen in (3.16) isx =[θθ˙], (3.18)and input u is the PWM signal from the ECU. The term Γ(θ, θ˙)containsthe aggregate nonlinear terms given in (3.15).3.3 Control Strategy and Performance MetricsThe model of the electronic throttle valve system has been derived in thepreceding sections. Utilizing this model, a set of controllers are designed inthe subsequent chapters, and their respective performances will be analyzed.This section outlines the control objectives for the electronic throttlevalve, the trajectories used during experimentation, and performance met-rics that will be used to quantify the experimental results.213.3. Control Strategy and Performance Metrics3.3.1 Control ObjectivesElectronic throttle control has a significant impact on the performance of au-tomobiles. Performance in this context is a measure of the automobile’s fuelefficiency, emission control, and driveability. Accurate control of the elec-tronic throttle valve can lead to reductions in fuel consumption, emissions,and increases in comfort. In this section, the controller design objectives aremotivated to meet those target performances.To reiterate on earlier chapters, the precise control of the AFR is crucialin meeting emission regulations. ETC is a method of regulating the amountof charge-air entering the cylinders. Thus the tracking error between thethrottle plate and its setpoint must be minimized throughout all trajecto-ries. Consequently, this implies that accurate tracking and fast responsetime are vital characteristics of the controller that affect the AFR. Fur-thermore, a common source of discomfort in automobiles is associated withsystem overshoots and oscillations and thus these should be minimized oreliminated [25]. Electronic throttle valves are complex systems with nonlin-earities present due to their packaging, cost, and reliability constraints. It istherefore important to achieve robust performance and stability throughoutall possible trajectories despite the nonlinearities present in the system.3.3.2 Experimental Settings and Performance SpecificationsWhile driving a vehicle, the ECU will generate a diverse array of trajectoriesfor the ETC system. To validate the robust performance and stability of theETC, four cases which represent the most common, or at least a subset of,trajectories that can be generated by the ECU are used for experimentation;the cases are illustrated in Figure 3.6. The cases are:Case 1) large opening & closing (LOC). The setpoint moves from the LHposition to 80◦,Case 2) small opening & closing (SOC). The setpoint moves in small amountsfrom 1◦ to 5◦,223.3. Control Strategy and Performance MetricsCase 3) limp-home crossing (LHC). The setpoint travels through the LHposition from 2◦ to 10◦, andCase 4) voltage variation. The battery voltage drops and increases from14V to 7V .Time (s)0 0.5 1 1.5Throttle Angle (°)020406080(a) Large Opening and ClosingTime (s)0 0.5 1 1.5Throttle Angle (°)01234567(b) Small Opening and ClosingTime (s)0 0.5 1 1.5Throttle Angle (°)024681012(c) Limp-Home CrossingTime (s)0 0.5 1 1.5Voltage (V)051015(d) Voltage FluctuationFigure 3.6: Experimental trajectories for ETCThe aforementioned cases emphasize the sudden changes from small tolarge opening angles (and vice-versa), operation through the nonlinear LHposition, and operation during nonlinear voltage fluctuations.233.3. Control Strategy and Performance MetricsThe performance metrics that will be used for comparing controllersduring experiments are principally concerned with the transient performanceof the ETC. These metrics are described as follows:• no steady-state error,• minimal tracking error during transient,• minimal or no overshoot,• settling time < 200ms [13], [10], and• robust to nonlinear parameter-variations in slip-stick friction, LH springs,and external disturbances.Tracking error during transient operation in this context will be a measureof the average error between the controller setpoint and throttle positionduring a prescribed range. The tracking error metric will be used to quantifyperformance for controllers that do not settle within the required time.24Chapter 4ETC Structure and DesignMethodIn any control problem, there is disparity between the actual system andthe mathematically modeled plant. These mismatches happen due to un-known external disturbances, estimated plant parameter discrepancies, orunmodeled system dynamics. Robust control techniques can bridge the gapbetween model and plant mismatches.The most fundamental feedback structure of a control system can beseen in Figure 4.1. The controller structure can dramatically vary in com-Figure 4.1: Block diagram of a typical control systemplexity based on the designer’s objectives. In this chapter, popular robustcontroller structures for the ETC problem are developed for comparison insubsequent chapters. In Section 4.1, the SMC structure is developed sincethe application of SMC is prevalent among ETC researchers. In Section 4.2a gain-scheduling LPV controller is developed.254.1. Sliding-Mode Controller4.1 Sliding-Mode ControllerIn this section, we propose a SMC scheme for the ETC problem. A sys-tematic approach to designing a feedback controller which is insensitive tobounded disturbances and bounded parameter variations is presented here.Consider the second-order nonlinear plant model given in (3.15), it is as-sumed to take the following formθ¨ =1Jeqf + b2uk, (4.1)wheref = −(Beq +N2KvKmRa)θ˙ −Ksθ − Fc sgn θ˙ − TLH sgn θ,b2 =NKmVb(1 + ∆)JeqRa,(4.2)and uk is the control input. Since the dynamics of f are nonlinear andtime-varying, it is estimated as fˆ . The estimation error of f is assumed tobe bounded by some defined function F = F (θ, θ˙) such that|fˆ − f | ≤ F (4.3)In order to have the control system track θ = r, the sliding-manifold S = 0is defined as a function of the tracking error eS = λe+ e˙, (4.4)wheree = θ − r (4.5)A sliding-mode manifold S must be selected such that the stability of theLyapunov condition12ddtS2 ≤ −η|S|, (4.6)264.2. LPV Gain-Scheduling Controlleris satisfied, where η is a strictly positive constant. Thus we haveS˙ =(1Jeqfˆ + b2uk)− r¨ + λe˙ (4.7)Where uk = uˆk +Kp sgn(S). The input uˆk is defined as the best estimationof the equivalent control for the uncertain system. In order to satisfy thesliding-mode manifold conditions, the discontinuous term Kp sgn(S) acrossthe manifold S = 0 is supplemented. Therefore it can be seen that the bestapproximation of a control law uˆk required to achieve S˙ = 0 isuˆk =1b2(− 1Jeqfˆ + r¨ − λe˙)−Kp sgn(S) (4.8)where sgn is the sign functionsgn(S) = +1 if s > 0sgn(S) = −1 if s < 0(4.9)By choosing a large enough Kp-gain, the stability of the Lyapunov conditionin (4.6) of the SMC is achieved, whereKp = F + η (4.10)In summary, the higher-order nonlinear problem has been reduced toa first-order system with an intuitive feedback scheme that ”pushes” thesystem dynamics onto the sliding-mode manifold when there is an error.The SMC parameters used during experimentation can be found in Ap-pendix A.2.4.2 LPV Gain-Scheduling ControllerElectronic throttle valve dynamics change at different operating points. Asa result, the linearized model will perform differently at every consequentoperating point. To accommodate this problem, well known gain-schedulingLPV methods presented in [39], [43], are applied to the electronic throttle274.2. LPV Gain-Scheduling Controllervalve system. In this section, the structure for the ETC system used for LPVgain-scheduling controller synthesis is presented. By defining the closed-loopstructure, the control problem is defined and the controllers are tuned forexperimentation.4.2.1 LPV Control StructureThe general feedback structure imposed is illustrated in Figure 4.2. Thefeedback structure consists of two feedback controllers Kcomp and KLPV , astate-observer and the plant model of the electronic throttle valve.Figure 4.2: LPV control structure applied to the ETC plantTo design an LPV controller, the nonlinear system (3.17) needs to be con-verted into an LPV system. As such, the nonlinear components of Γ(θ, θ˙)are treated in a number of ways. The feedback controller Kcomp will be usedto compensate the nonlinear components of the LH spring torque Tspring. Incontrast, the slip-stick friction and voltage fluctuation terms will be mod-284.2. LPV Gain-Scheduling Controllereled as scheduling-parameters of the LPV plant used in synthesizing theLPV feedback controller, KLPV . Since electronic throttle valves are onlyequipped with a TPS, some crucial state information (angular velocity ofthe throttle plate) required in real-time measurement of the scheduling-parameters is unavailable. To address this shortcoming, an state-observer isdesigned to implicitly measure the scheduling-parameters.4.2.2 Limp-Home Spring CompensationReferring to Figure 4.2, it can be seen that the control input u to the elec-tronic throttle valve is a combination of the control outputs of Kcomp andKLPV such thatu = uk + uo, (4.11)The control signal uo is used to compensate the nonlinear portion of theLH spring torque TLH sgn θ. In doing so, not only is the nonlinear termeffectively treated, it no longer needs to be modeled in the LPV plant usedin synthesizing an LPV controller. (3.15) becomesJeq θ¨ =NKmRaVb(1 + ∆)((uk + uo)− TLHRaNKmVb(1 + ∆)sgn θ)−(Beq +N2KvKmRa)θ˙ −Ksθ − Fc sgn θ˙ − Tp, (4.12)whereuo =TLHRaNKmVb(1 + ∆)sgn θ (4.13)The emerging differential equation is reduced toJeq θ¨ =NKmVb(1 + ∆)Rauk −(Beq +N2KvKmRa)θ˙ −Ksθ − Fc sgn θ˙ − Tp(4.14)The remaining nonlinear elements in (4.14) are treated as scheduling-parametersin the eventual LPV system further described in the following sections.294.2. LPV Gain-Scheduling ControllerFigure 4.3: ETC system block diagram4.2.3 LPV Model of plantThe distinguishing feature of LPV systems when compared to their LTIcounterparts is that they are nonstationary. LPV systems can be used torepresent LTI systems subject to a vector of scheduling-parametersΘ = diag{ρ1, ρ2, . . . , ρn} (4.15)Weighting functions defined by We and Wu generate a weighted error e˜ andweighted input u˜Σ, respectively. These weighted functions will be minimizedduring controller synthesis.For controller synthesis, the control system depicted in Figure 4.3 is con-verted into linear fractional transformation (LFT) form shown in Figure 4.4.The parameter-varying state-space matrices of the generalized plant G(Θ)take the formx˙ = A(Θ)x+B1(Θ)w +B2(Θ)u,z = C1(Θ)x+D11(Θ)w +D12(Θ)u,y = C2(Θ)x+D21(Θ)w +D22(Θ)u,(4.16)304.2. LPV Gain-Scheduling ControllerFigure 4.4: LFT form of ETC systemwhere state x, output y, and input u of the system arex =xPxIxDruPuIuD, y = e, and u = u˜Σ, (4.17)The state-space matrices of the augmented plant G(Θ) in (4.16) are definedas A(Θ) B1(Θ) B2(Θ)C1(Θ) D11(Θ) D12(Θ)C2(Θ) D21(Θ) D22(Θ) =A BCI BCD 0 B BDI BDD0 AI 0 0 0 BI 00 0 AD 0 0 0 BD−C 0 0 1 0 0 00 0 0 0 1 1 1−C 0 0 1 0 0 0,(4.18)314.2. LPV Gain-Scheduling Controllerand the exogenous inputs w and performance channel z arew = r, z =[e˜u˜Σ], (4.19)The closed-loop stability of an gain-scheduling LPV controller and L2-gain bound, denoted by γ, from the w to z, is guaranteed when the LMIconstraint M11 M12 M13∗ M22 M23∗ ∗ M33 < 0, (4.20)whereM11 = X˙ + (A+B2KC2)TX +X(A+B2KC2),M12 = X(B1 +B2KD21),M13 = (C1 +D12KC2)T ,M22 = −γI,M23 = (D11 +D12KD21)T ,M33 = −γI,(4.21)holds for all values of the scheduled-parameters Θ and Θ˙. The LMI variablesinclude a positive-definite Lyapunov function X(Θ), gain-scheduling LPVcontroller K(Θ), and L2-gain bound γ. This is a non-convex problem, wherethe L2-gain bound is chosen to be minimized by iteratively solving for X(Θ)and K(Θ), thereby generating a locally optimal controller. The parameter-dependent Lyapunov function is defined asX(Θ) = X0 +X1ρ1 +X2ρ2, . . . ,+Xnρn, (4.22)Similarly, the parameter-dependent, gain-scheduling controller K(Θ) is de-fined asK(Θ) = K0 +K1ρ1 +K2ρ2, . . . ,+Knρn (4.23)324.2. LPV Gain-Scheduling Controller4.2.4 Scheduling-ParametersThe scheduling-parameter vector Θ for the ETC problem contains approx-imations for the nonlinear elements appearing in (4.14). The scheduling-parameter vector is defined as Θ = diag{ρ1, ρ2}, for simplicity, where ρ1and ρ2 are the scheduling-parameters associated with slip-stick friction andvoltage fluctuation, respectively. The scheduling-parameters are presentedin this subsection.The slip-stick friction can be approximated asFc sgn θ˙ =∣∣∣∣Fcθ˙∣∣∣∣ θ˙ + ∆Fc = ρ1θ˙ + ∆Fc, (4.24)where the scheduling-parameter ρ1 is defined asρ1 ={Fcθ˙: |θ˙| ≥ θ˙minFcθ˙min: else, (4.25)It should be noted that ∆Fc is applied as an uncertain input to correct forthe error between the actual slip-stick friction function (solid line) and theapproximation (dashed line) seen in Figure 3.5.The variations in the battery voltage are illustrated byVa = Va(1 + ∆) = Va(1 + ρ2), (4.26)where the scheduling-parameter ρ2 is defined asρ2 = ∆ (4.27)4.2.5 Reduced-order ObserverIn this section, the design of a reduced-order state-observer is presented [46].A state-observer is a system that estimates the internal states of the actualsystem via implicit measurements of the input and output of the plant. Inmost practical applications, including ETC, not all states can be measureddirectly. Since the TPS measures the throttle position θ explicitly, a reduced-334.2. LPV Gain-Scheduling ControllerFigure 4.5: Structure of reduced-order state-observerorder state-observer is designed to estimate the angular velocity θ˙ which willbe used to calculate the scheduling-parameter ρ1, implicitly.The model used for state-observer design will be different than the modelused for LPV controller synthesis. The proposed state-observer uses a linear,non parameter-varying, state-space model with additional nonlinear controlinputs to compensate the nonlinear elements and accurately track the statesof the system. The structure of the state-observer is illustrated in Figure 4.5.The state-space model of the reduced-order state-observer isx˙ = Aox+Bouobs + Γ(θ, θ˙),y = Cox,(4.28)where the input to the state-observer isuobs = u− unl, (4.29)where uobs is the combination of the output of the controller u, and the344.2. LPV Gain-Scheduling Controllernonlinear compensation terms unl, defined asunl =Γ(θ, θ˙)Bo(4.30)The state-observer state-space model becomesx˙ = Aox+Bou,y = Cox,(4.31)which is an adequate linear approximation for the ETC plant. The state-space matrices in (4.31) are defined asAo =[0 1−KsJeq − 1Jeq (Beq + N2KvKmRa)]=[A˜11 A˜12A˜21 A˜22],Bo =[0NKmVbJeqRa]=[B˜1B˜2],Co =[1 0](4.32)Since throttle position θ can be measured directly, a reduced-order observeris used to estimate the unmeasured state, angular velocity θ˙. The equationfor the reduced-order state-observer isddtz(t) = (A˜22 − LuA˜12)z(t)+[(A˜21 − LuA˜11) + (A˜22 − LuA˜12)Lu]y(t)+ (B˜2 − LuB˜1)u(t),z(t) = zˆ2(t)− Luy(t),(4.33)where Lu is observer-gain. The estimated angular velocityˆ˙θ isˆ˙θ = zˆ2(t) + Lu (4.34)35Chapter 5Switching Gain-SchedulingLPV ControllerThe principle contribution of this thesis, a switching gain-scheduling LPVcontroller, is presented in this chapter. Solving the ETC problem by de-signing a switching gain-scheduling LPV controller accounts for the time-varying nature of the plant while reducing the conservatism apparent ingain-scheduling LPV controllers. The LPV framework introduced in the pre-ceding chapters is extended by converting it into a multicontroller synthesisproblem. Switching controllers reduce the inherent conservatism in single-region LPV controllers. The proposed concept is to partition the electronicthrottle valve’s operating region into smaller subregions. A parameter-dependent Lyapunov function is obtained for every partitioned subregionand is used to synthesize a gain-scheduling LPV controller for each subse-quent subregion. The controllers will be switched in real-time such that theactive controller is the one associated with the subregion the system’s cur-rent operating point is within. Hysteresis switching techniques are appliedalong with conditions that guarantee that robust switching occurs at eachrespective switching event. These conditions are imposed by guaranteeingan increase of the approaching Lyapunov function with respect to the Lya-punov function of the adjacent subregion. The resulting controller will fieldmore aggressive results while guaranteeing theoretical stability and robustperformance throughout the entire operating region.365.1. Hysteresis SwitchingFigure 5.1: Hysteresis switching for two adjacent regions5.1 Hysteresis SwitchingThe scheduling-parameter vector Θ is divided into subregions where any twoadjacent subsets are overlapped, illustrated in Figure 5.1. The switching-event depends on which subregion the scheduling-parameter ρn belongs toduring operation. To guarantee the switching performance, the imposedcondition is that the Lyapunov function of the active subregion increasesas it approaches the adjacent subregion; this condition is depicted in Fig-ure 5.2. To make it clearer, the Lyapunov function in the current subregionmust be greater than the Lyapunov function of the subregion it is entering.By adjusting the width of the overlapping subregions, one can make thiscondition easier to satisfy but the tradeoff would be a more conservativecontroller.5.2 Switching gain-scheduling LPV SynthesisConsider the augmented plant G(Θ) of the ETC system in LPV formx˙ = A(Θ)x+B1(Θ)w +B2(Θ)u,z = C1(Θ)x+D11(Θ)w +D12(Θ)u,y = C2(Θ)x+D21(Θ)w +D22(Θ)u,(5.1)375.2. Switching gain-scheduling LPV SynthesisFigure 5.2: Lyapunov function of adjacent subregions during switchingwhere the matrices of the augmented plant G(Θ) are given in (4.17) - (4.18).To synthesize the multicontroller problem, the LMI constraint that guaran-tees the closed-loop stability and L2-gain bound γ, from w to z, within eachsubregion, denoted as r(i, j) isM11 M12 M13∗ M22 M23∗ ∗ M33 < 0, (5.2)whereM11 = X˙(r) + (A+B2K(r)C2)TX(r) +X(r)(A+B2K(r)C2),M12 = X(r)(B1 +B2K(r)D21),M13 = (C1 +D12K(r)C2)T ,M22 = −γI,M23 = (D11 +D12K(r)D21)T ,M33 = −γI,, (5.3)must hold. The LMI conditions that guarantee robust switching-events atthe adjacent switching surface S(r), areX(i) ≥ X(j) Θ ∈ S(i,j),X(i) ≤ X(j) Θ ∈ S(j,i), (5.4)385.3. Simulation Resultssuch that the Lyapunov function X(r) is positive definite in each subregionX(i) ≥ 0 ∀Θ(i),X(j) ≥ 0 ∀Θ(j) (5.5)The LMI variables are the Lyapunov function X(r)(Θ), switching gain-scheduling LPV controller K(r)(Θ), and L2-gain bound γ. The parameter-dependent Lyapunov function for each subregion r is defined asX(r)(Θ) = X(r)0 +X(r)1 ρ1 +X(r)2 ρ2, . . . ,+X(r)n ρn, (5.6)Similarly, the parameter-dependent, switching gain-scheduling controller foreach subregion r is defined asK(r)(Θ) = K(r)0 +K(r)1 ρ1 +K(r)2 ρ2, . . . ,+K(r)n ρn (5.7)The proposed LMI conditions are non-convex, so an iterative method isadopted to find a locally optimal solution. The general structure of theoptimization algorithm is illustrated in Figure 5.3. Based on an initialLTI controller, the algorithm searches through the entire operating rangeof the constrained trajectories Θ to confirm the existence of a parameter-dependent Lyapunov function X0(Θ). Once a Lyupunov function is foundfor the designed operating range, the algorithm partitions the region intoa set of overlapping subregions to find unique Lyapunov functions X(r)(Θ).For each subsequent subregion, a switching gain-scheduling LPV controllerK(r)(Θ) is found such that the performance bound γ will be decreasing withevery additional subregion and thus improving the L2 gain of the closed-loop system. This ”X(Θ)−K(Θ)” iteration may continue until the L2-gainbound no longer has any visible changes.5.3 Simulation ResultsIn this section, the notion of decreasing performance bound γ with multi-ple subregions is validated. One single-region LPV and four multi-region,395.3. Simulation ResultsFigure 5.3: Iterative optimization algorithm for finding switching LPV con-trollers for r-subregionsswitching LPV controllers were synthesized as illustrated in Figure 5.4. Thedashed line represents the overlapping of the adjacent regions. Figure 5.4(a)is an example of a single-region LPV controller with no switching. Fig-ure 5.4(b) and Figure 5.4(c) are partitioned into two subregions, whereasFigure 5.4(d) and Figure 5.4(e) are partitioned into four and nine subre-gions, respectively. The advantage of increasing the number of subregionsis that by reducing the range of parameter variations, a less conservativeLyapunov function can be found, yielding more aggressive controllers. Con-sequently, the performance bound γ can be seen to be decreasing as seen inTable 5.1.Table 5.1: Performance bound comparison between multi-region controllersPerformance Bound ComparisonSubregions γ Comparision1 0.1533 -2 0.1473 96.1%2 0.1492 97.3%4 0.1414 92.2%9 0.1404 91.5%405.3. Simulation Results(a) 1 Region (b) 2 Region(c) 2 Region (d) 4 Region(e) 9 RegionFigure 5.4: Switching subregions of the controller as a function of scheduling-parameters415.3. Simulation ResultsFigure 5.5 visually illustrates the substantial difference in controller gainsK(r)n with additional subregions. It is clear that the smaller subregions willresult in less conservative results but the drawback is that more switchingsubregions would result in more computational difficulties.Region X [-]Region Y [-] 0P [-](a) K0P for one-region LPV controller3Region X [-]213Region Y [-]2100.511.52K 0P [-](b) K0P for nine-region LPV controllerFigure 5.5: Visual comparison gains between one-region LPV and nine-region LPV42Chapter 6Experimental ResultsA controller’s performance is only as good as the system model being adopted.System parameter identification tends to be one of the most crucial andchallenging tasks associated with system modeling, analysis and synthesis.Mechanical deformations over time and large manufacturing tolerances cul-minate into systems with high parametric and dynamic variances. In thischapter, the experimental test bench and system identification techniquesare presented in Sections 6.1 and 6.2, respectively. Then controllers are val-idated and their tracking performance is analyzed in Section 6.3. The con-trollers appearing in this section are classical PID, SMC, gain-schedulingLPV and switching gain-scheduling LPV. The experimental cases are de-signed to exemplify important operational scenarios, which include: largeopening and closing, small opening and closing, limp-home crossing, andvoltage fluctuations.6.1 Experimental Test BenchIn this section, the test bench for parameter identification and implemen-tation of the final controllers used for experimentation are explained. TheMotohawk ECU565-128 family of ECU was the platform on which all subse-quent controllers were implemented. The ECU provided a means to adjustthe PWM signal which in turn is output into an h-bridge driver. The h-bridge allows the polarity of the voltage to be changed which means thatthe electronic throttle valve can be driven in both directions. The analogto digital conversion (ADC) rate for the TPS is 1ms so the sampling timeof the ECU was set to Ts = 1ms. The electronic throttle valve used inthis research was a Denso 157. It is equipped with two TPS from which436.2. Parameter IdentificationFigure 6.1: Experimental test bench of ETC systemthe average value at each time-step was used for feedback control. Systeminformation such as sensor data, reference data and so on could be loggedand observed in real-time using a PC via accompanying MotoTune software.The test bench is illustrated in Figure Parameter IdentificationIn this section, a combination of estimation via laboratory experiments andparameter identification algorithms is used to determine the nominal param-eters of the mathematical system model. Frequency domain analysis was notused due to the high nonlinearity of the Γ(θ, θ˙)terms in (3.16) which havedifferent dynamics at different operating points. The coupling terms ap-parent in the state-space matrices make standard step and frequency sweepmethods unviable.6.2.1 Stalled Motor TestThe stalled motor test is applied to determine the equivalent armature re-sistance Ra. At steady state, stalling the rotation of the motor, the back446.2. Parameter IdentificationVoltage (V)0 1 2 3 4 5 6 7 8Current (A)00.511.522.53Measured CurrentBest FitFigure 6.2: Measured current as a function of slowly varied armature voltageEMF ea and inductance voltage VLa go to zeroVa = 0V La +Raia +0ea (6.1)Applying a slowly varying dc voltage Va across the armature terminals re-sults in a linearly proportional variation in the armature current ia depictedin Figure 6.2. The inverse slope of the line of best fit yields the nominalarmature resistance Ra of the electronic throttle valve.6.2.2 Motor Constant TestThe motor constant Km defines the ability of the motor to translate elec-trical power to mechanical power. Despite not addressing the thermal orviscous losses of the motor, the motor constant still provides a good nom-inal approximation of the motor’s performance. In this section, the motorconstant Km associated with the experimental throttle valve is identified.In this experiment, the motor is decoupled from the throttle shaft allow-ing the armature to freely rotate without the influence of the throttle valvemechanics. The motor constant Km will be identified by determining the456.2. Parameter Identificationarmature velocity constant Kv since Km = Kv. The model for back EMFis given byea = Kv θ˙m (6.2)where Kv and θ˙m are the armature velocity constant and armature angularvelocity, respectively. By applying a constant voltage Va across the armatureterminals, the armature velocity constant can be estimated byKv =1θ˙m(Va −Raia) (6.3)Allowing the armature shaft to rotate freely permits a steady-state measure-ment of the angular velocity θ˙m and current ia. It should be noted that theangular velocity was measured using video analysis software on the reduc-tion gear rotary motion; the measurements and best fit estimation of θ˙m areillustrated in Figure 6.3.Time (s)0 2 4 6 8 10 12Angular Velocity (rad/s)05101520Measured Angular VelocityBest FitFigure 6.3: Experimentally measured angular velocity of intermediate re-duction gear6.2.3 Static Load TestIn this section, the forces present during static operation in the electronicthrottle valve are estimated. This experiment will estimate the magnitudesof slip-stick friction Fc, limp-home spring torque TLH , and spring stiffness466.2. Parameter IdentificationKs. When the throttle plate is held statically, (3.14) is reduced toNTm = Jl0θ¨l + (Fc sgn 0θ˙l +Bl0θ˙l) + (TLH sgn(θl − θ0) +Ks(θl − θ0)) (6.4)To determine the aforementioned parameters, a linearly increasing andMotor Torque (N m)0.2 0.3 0.4 0.5 0.6 0.7 0.8Angle (rad) TcFigure 6.4: Static load test showing the throttle valve angle as the motortorque is linearly increased (red) and subsequently linearly decreased (blue)decreasing current to the electronic throttle valve and use the TPS to recordthe throttle position. This is effectively torque control and since we havedetermined the motor constant Km, we can plot the throttle position versusthe implicitly calculated torque as shown in Figure 6.4. The halfway markbetween the two rising and falling curves is approximately the preload torqueTLH . The distance from TLH to either curve is the ±Fc. Once the preloadtorque and friction have been overcome, the throttle shaft angle increaseslinearly, the associated slope is spring constant Ks.476.3. Experimental Cases6.2.4 Summary of System IdentificationThe parameters identified in the preceeding sections have been listed inTable 6.1.Table 6.1: Table of the identified parameter valuesParameter IdentificationParameter Symbol ValueResistance Ra 3.06 ΩInductance La 800 µHMotor Constant Km 0.0275N ·mAEMF Constant Kv 0.0275V ·sradSpring Constant Ks 0.084N ·mradLH Spring Torque TLH 0.58 N ·mSlip-stick Friction Fc 0.2 N ·mViscous Damping Beq 0.03325 N ·m · sradMoment of Inertia Jeq 0.0018 Kg ·m26.3 Experimental CasesIn this section, the experimental results for each unique case that the elec-tronic throttle valve operates in is reviewed and analyzed. The trajectorycases 1-4, and performance metrics that will be used in the subsequent sec-tions were previously discussed in Section 3.3.2. The experimental controllerparameters used for PID, SMC, LPV and switching LPV can be found inAppendix A.6.3.1 Large Opening and ClosingIn this section, the experimental results of the controllers are evaluated andcompared for case 1. The time-domain experimental results of large open-ing and closing of the electronic throttle valve are illustrated in Figure 6.5and performance analysis is given in Table 6.2. The performance valuesappearing in Table 6.2 - 6.5 are the averaged totals for opening and closing.486.3. Experimental CasesTable 6.2: Performance of PID, SMC, LPV and S-LPV for case 1Large Opening & Closing|Tracking error|(◦) Percent Overshoot (%) Settling Time (ms)PID 0.2637 0.7 91.0SMC 0.3155 0 123.0LPV 0.9309 2.2 91.2S-LPV 0.7296 1.5 83.0The settling-time for all the controllers tested for case 1 are all underthe prescribed 200ms, however it is clear that the PID, LPV and S-LPVresponses are improvements over the SMC. All controllers with the exceptionof SMC experienced small overshoots. An important point to note is thatthe S-LPV’s overshoot is 33% less than the LPV. All controllers exhibitedgood tracking performance.6.3.2 Small Opening and ClosingIn this section, the experimental results of the controllers are evaluated andcompared for case 2. The time-domain experimental results of small openingand closing of the electronic throttle valve are illustrated in Figure 6.6 andperformance analysis is given in Table 6.3.Similar to case 1, the settling time of all the controllers is under theprescribed required settling time, but the PID, LPV and S-LPV controllerscontinue to outperform the SMC. The settling-time of the S-LPV was im-proved by 23% over LPV, the best baseline controller for case 2. Further-more, the S-LPV had a 71%, and 83% reduction in overshoot compared tothe PID and LPV, respectively. The tracking error of the S-LPV is improvedby 32%, 70%, and 2% compared to PID, SMC, and LPV, respectively.6.3.3 Limp-Home CrossingIn this section, the experimental results of the controllers are evaluatedand compared for case 3. The time-domain experimental results of the LH496.3. Experimental CasesTable 6.3: Performance of PID, SMC, LPV and S-LPV for case 2Small Opening & Closing|Tracking error|(◦) Percent Overshoot (%) Settling Time (ms)PID 0.0753 4.5 > 200SMC 0.1688 0 163.0LPV 0.0475 7.7 62.6S-LPV 0.0511 1.3 47.7crossing of the electronic throttle valve are illustrated in Figure 6.7 andperformance analysis is given in Table 6.4.The settling-time of the S-LPV is 44% less than the next best baselinecontroller. S-LPV has minimal overshoot but more noticeably a 76% reduc-tion in overshoot compared to LPV. The PID had inadequate performanceas it failed to settle within the prescribed time frame. The SMC appears totemporarily delineate from its normal trajectory at the LH position whichmay be a result of unmatched uncertainties.Table 6.4: Performance of PID, SMC, LPV and S-LPV for case 3Limp-Home Crossing|Tracking error|(◦) Percent Overshoot (%) Settling Time (ms)PID 0.5027 0 > 200SMC 0.2423 0 92.0LPV 0.0993 8.5 30.0S-LPV 0.0952 2 Voltage FluctuationsIn this section, the experimental results of the controllers are evaluatedand compared for case 4. The time-domain experimental results of volt-age fluctuation of the electronic throttle valve are illustrated in Figure 6.9;the vertical black line represents the instance when the voltage fluctuatesbetween 13.2V and 7.2V . The performance analysis is given in Table 6.5.506.3. Experimental CasesTable 6.5: Performance of PID, SMC, LPV and S-LPV for case 4Voltage Fluctuation|Tracking error|(◦) Settling Time (ms)PID 0.3284 > 200SMC 0.0937 > 200LPV 0.0568 30.4S-LPV 0.0721 30.4The settling time of the S-LPV and LPV are both immediate whereas thePID and SMC do not appear settle and furthermore exhibit what appears toheavy chattering and instability. The tracking error of the S-LPV performed25% better than the next best controller during in voltage increase case andnearly identical in the voltage decrease case. Moreover, it is clear that thePID and SMC struggle to converge quickly to the setpoint in part due totheir LTI nature and unmatched dynamic uncertainty.516.3. Experimental CasesTime (s)0.45 0.5 0.55 0.6 0.65 0.7Throttle Angle (°)0102030405060708090SetpointPIDSMCLPVS-LPV(a) Experimental large openingTime (s)0.45 0.5 0.55 0.6 0.65 0.7Throttle Angle (°)0102030405060708090SetpointPIDSMCLPVS-LPV(b) Experimental large closingFigure 6.5: Experimental result of large opening and closing of the throttleplate position with PID (dotted line), SMC (dash-dotted line), LPV (thinline) and S-LPV (thick line)526.3. Experimental CasesTime (s)0.45 0.5 0.55 0.6 0.65 0.7Throttle Angle (°)01234567SetpointPIDSMCLPVS-LPV(a) Experimental small openingTime (s)0.45 0.5 0.55 0.6 0.65 0.7Throttle Angle (°)01234567SetpointPIDSMCLPVS-LPV(b) Experimental small closingFigure 6.6: Experimental result of small opening and closing of the throttleplate position with PID (dotted line), SMC (dash-dotted line), LPV (thinline) and S-LPV (thick line)536.3. Experimental CasesTime (s)0.45 0.5 0.55 0.6 0.65 0.7Throttle Angle (°)02468101214SetpointPIDSMCLPVS-LPV(a) Experimental LHC openingTime (s)0.45 0.5 0.55 0.6 0.65 0.7Throttle Angle (°)02468101214SetpointPIDSMCLPVS-LPV(b) Experimental LHC closingFigure 6.7: Experimental result of LHC of the throttle plate position withPID (dotted line), SMC (dash-dotted line), LPV (thin line) and S-LPV(thick line)546.3. Experimental CasesTime (s)0.45 0.5 0.55 0.6 0.65 0.7Throttle Angle (°)99.51010.511SetpointPIDSMCLPVS-LPV(a) Experimental voltage voltage fluctuationTime (s)0 0.5 1 1.5 2Voltage (V)7891011121314(b) Reference voltage trajectoryFigure 6.8: Experimental result of voltage increase with PID (dotted line),SMC (dash-dotted line), LPV (thin line) and S-LPV (thick line)556.3. Experimental CasesTime (s)0.45 0.5 0.55 0.6 0.65 0.7Throttle Angle (°)99.51010.511SetpointPIDSMCLPVS-LPV(a) Experimental voltage voltage fluctuationsTime (s)0 0.5 1 1.5 2Voltage (V)7891011121314(b) Reference voltage trajectoryFigure 6.9: Experimental result of voltage decrease with PID (dotted line),SMC (dash-dotted line), LPV (thin line) and S-LPV (thick line)56Chapter 7ConclusionThe significant achievements accomplished in this research are summarizedin Section 7.1. The contributions and future work related to the currentresearch are presented in Section 7.2, and 7.3, respectively.7.1 Summary of ResearchIn this research, a number of controllers were designed to handle the elec-tronic throttle valve’s time-varying dynamics, including PID, SMC, LPVand switching LPV which were used for comparison. The switching LPVsystem was designed to account for the parameter-varying nature of theplant and guarantee robust performance and stability throughout the elec-tronic throttle valve’s entire operating region.A complete model of the electronic throttle valve was derived and con-verted into an LPV system. The control structure was designed to handlethe nonlinearities and time-varying dynamics of the plant. The LPV modelwas defined using two time-varying scheduling-parameters which were ap-proximations of the slip-stick friction phenomenon, and voltage fluctuations.These parameters were assumed to be measurable in real-time.Four different control techniques were used to solve the ETC problem.These feedback controllers were designed to track the reference throttle po-sition as per request of the ECU. A novel feedback control scheme, usingswitching LPV synthesis techniques was presented. The principle of this con-cept was converting the single-region LPV controller into a multicontrollerproblem. By partitioning the entire operating region into subregions, moreaggressive controllers could be synthesized. A Lyapunov function is foundfor each subsequent subregion where the robust performance, stability, and577.2. Contributionsswitching is guaranteed.Experiments were done to demonstrate the switching LPV’s performanceto other baseline controllers. Four cases which represent an array of poten-tial trajectories the ETC system can undergo were tested. These caseswere large opening & closing, small opening & closing, limp-home crossing,and voltage fluctuations. These experiments showed how the parameter-dependent nature of the plant can affect system performance. The resultingoutcome illustrated that the switching LPV controller outperformed PID,SMC, and LPV methods in cases 1-4. It can be concluded that the switchinggain-scheduling LPV controller will in the worst case perform equivalently tothe single-region LPV controller but will likely improve when the operatingregions are large.7.2 ContributionsThe contributions of this research for ETC problem are:• Applied switching LPV techniques are applied to the ETC problem.The resulting switching LPV controller guarantees the L2 performanceover the entire operating range of the electronic throttle. Furthermore,the controller is less conservative than its predecessors.• Novel method that guarantees that the performance bound γ of multi-region, switching gain-scheduling LPV controller will always be lowerand hence perform better than a single-region LPV controller.7.3 Future WorkElectronic throttle control is a mature problem but can still be improved.Advanced control techniques can still lead to substantial improvements overthe current methods. Some of the future work that can be done in this areais summarized as follows587.3. Future WorkPressure Disturbance ModelingModeling the pressure disturbance on the throttle plate. The disturbanceon the throttle plate due to the in-line pressure variations can change sig-nificantly based on throttle angle. These dynamics may impact the overallperformance of the electronic throttle valve in practice.Experimental Validation of Pressure DisturbanceExperimentally validate effects of pressure disturbances on electronic throt-tle valves. This can be done via actual engine experimentation or approxi-mating the pressure with mechanical disturbances.Optimal Subregion DesignIn this research, the subregions were heuristically selected within the elec-tronic throttle valve’s operating region. Substantial improvements can bemade to systematically partition the operating region. 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Yang, Modern Control Engineering, Prentice-Hall En-glewood Cliffs, NJ, 1970.65Appendix AController ParametersA.1 PID ParametersThe PID controller used as a baseline controller for comparison in the ex-periments takes the formu(t) = Kpe(t) +Ki∫ t0e(t)δt+Kdddte(t) (A.1)where e(t) is the feedback error. The proportional gain Kp, integral gain Ki,and differential gainKd used during experimentation are shown in Table A.1.These parameters were tuned experimentally until a desirable performancelevel was achieved.Table A.1: Table of parameters for the baseline PID controllerPID ParametersValue (-)Kp 0.6Ki 0.1Kd 0.005A.2 SMC ParametersThe SMC controller used as a baseline controller for comparison in theexperiments takes the formu(t) =1b2(− 1Jeqfˆ(t) + r¨(t)− λe˙(t))−Kp sgn(S) (A.2)66A.3. LPV Parameterswhere e(t) is the feedback error, r¨ is the twice differentiated trajectory, andfˆ is the estimation of the nominal plant. Tuning parameters of the SMCusing during experimentation are presented in Table A.2.Table A.2: Table of parameters for the baseline SMC controllerPID ParametersValue (-)Kp 0.7λ 50A.3 LPV ParametersThe gain-scheduling LPV controller used as a baseline controller for com-parison in the experiments takes the formu(t) = diag (K0 +K1ρ1 +K2ρ2) e(t)∫ t0 e(t)δtddte(t) (A.3)where e(t) is the feedback error, ρ1 and ρ2 are the varying-parameters of theLPV system given in (4.16). The operating regions of ρ1 and ρ2 areρ1 ∈ [0.0062 32.26]ρ2 ∈ [−0.4 0.1](A.4)The controller gains gains K0, K1, and K2 areK0 =0.67010.19330.0038 , K1 = 0.17660.0008−0.0001 , K2 =−1.16970.1956−0.0016 , (A.5)67A.4. Switching LPV ParametersA.4 Switching LPV ParametersThe gain-scheduling switching LPV controller, which is the primary contri-bution of this thesis, used during experiments takes the formu(t) = diag(Kij0 +Kij1 ρ1 +Kij2 ρ2) e(t)∫ t0 e(t)δtddte(t) (A.6)where e(t) is the feedback error, ρ1 and ρ2 are the varying-parameters of theLPV system given in (4.16). In the switching LPV controller, the operatingregions of ρ1 and ρ2 are partitioned into smaller subregions. The operat-ing region of ρ1 and ρ2 have been partitioned i = 1, 2, 3 and j = 1, 2, 3subregions, respectively.ρ11 ∈ [0.0062 1.1], ρ21 ∈ [0.9 15.1], ρ31 ∈ [14.9 32.26]ρ12 ∈ [−0.4 − 0.295], ρ22 ∈ [−0.305 − 0.095], ρ32 ∈ [−0.105 0.1](A.7)The controller gains Kij0 , Kij1 , and Kij2 used in the experiment areK110 =0.41610.17830.0026 , K210 =0.66350.16130.0040 , K310 =0.70370.16580.0053 ,K120 = 1.08860.2985−0.0013 , K220 = 1.16290.2617−0.0004 , K320 =1.24930.23310.0009 ,K130 = 1.42140.2541−0.0011 , K230 = 1.59700.2562−0.0004 , K330 =1.75100.23160.0004 ,(A.8)68A.4. Switching LPV ParametersK111 = 0.81660.0801−0.0029 , K211 = 0.77620.0779−0.0036 , K311 = 0.74300.0743−0.0044 ,K121 = 0.2108−0.00090.0001 , K221 = 0.1890−0.00090.0001 , K321 = 0.1564−0.0007−0.0000 ,K131 = 0.1295−0.00160.0000 , K231 = 0.1362−0.0014−0.0000 , K331 = 0.1164−0.0012−0.0000 ,(A.9)K112 =−2.26000.0902−0.0002 , K212 =−1.36710.04040.0028 , K312 =−1.00490.03790.0080 ,K122 =−1.86410.2417−0.0039 , K222 =−1.49450.1732−0. , K322 =−1.07960.14010.0020 ,K132 =−3.09000.1505−0.0064 , K232 =−2.41210.1596−0.0068 , K332 =−1.50440.15360.0009 ,(A.10)69


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