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Air-fuel ratio control in spark ignition internal combustion engines using switching LPV techniques Postma, Marius 2010-12-24

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Air-Fuel Ratio Control inSpark Ignition Internal Combustion EnginesUsing Switching LPV TechniquesbyMarius PostmaB.A.Sc., University of British Columbia, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of Applied ScienceinTHE FACULTY OF APPLIED SCIENCE(Mechanical Engineering)The University Of British Columbia(Vancouver)December 2010c Marius Postma, 2010AbstractThe Three-Way Catalytic Converter (TWC) is a critical component for the mitiga-tion of tailpipe emissions of modern Internal Combustion (IC) engines. Becausethe TWC operates effectively only when a stoichiometric ratio of air and fuel iscombusted in the engine, accurate control of the air-fuel ratio is required. Totrack the desired ratio, a switching Linear Parameter Varying (LPV) air-fuel ratiofeedback controller, scheduled based on engine speed and air flow, and providingguaranteed L2 performance, is introduced. The controller measures the air-fuelratio in the exhaust flow using a Universal Exhaust Gas Oxygen (UEGO) sensorand adjusts the amount of fuel injected accordingly.A detailed model of the air-fuel ratio control problem is developed to demon-strate the non-linear and parameter-dependent nature of the plant, as well as thepresence of pure delays. The model’s dynamics vary considerably with enginespeed and air flow. A simplified model, widely used in literature and known as aFirst Order Plus Dead Time (FOPDT) model, is then derived. It effectively cap-tures the control problem using a model which is linear but parameter-varyingwith engine speed and air flow.iiLarge variation of the FOPDT model across the engine’s operating range hasled to conservative LPV controllers in previous literature. For this reason, the op-erating range is divided into smaller subregions, and an individual LPV controlleris designed for each subregion. The LPV controllers are then switched based onthe current engine speed and air flow and are collectively referred to as a switch-ing LPV controller. The controller design problem is expressed as a Linear MatrixInequality (LMI) convex optimization problem which can be efficiently solved us-ing available LMI techniques.Simulations are performed and the air-fuel ratio tracking performance of theswitching LPV controller is compared with that of conventional controllers in-cluding, H¥ and LPV, as well as a novel adaptive controller. The switching LPVcontroller achieves improved performance over the complete operating range ofthe engine.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Air-Fuel Ratio Controllers . . . . . . . . . . . . . . . . . 61.3.2 Controller Design Theory . . . . . . . . . . . . . . . . . . 12iv1.4 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . 162 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1 A Measure of Air-Fuel Ratio . . . . . . . . . . . . . . . . . . . . 182.2 Mean Value Modeling . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Detailed Model of Plant . . . . . . . . . . . . . . . . . . . . . . . 192.3.1 Air Flow Dynamics . . . . . . . . . . . . . . . . . . . . . 202.3.2 Wall Wetting Dynamics . . . . . . . . . . . . . . . . . . . 212.3.3 Four-Stroke Engine Cycle Delays . . . . . . . . . . . . . 222.3.4 Exhaust Mixing . . . . . . . . . . . . . . . . . . . . . . . 252.3.5 Exhaust Transport Delay . . . . . . . . . . . . . . . . . . 262.3.6 Sensor Dynamics . . . . . . . . . . . . . . . . . . . . . . 272.3.7 Complete Model . . . . . . . . . . . . . . . . . . . . . . 282.4 Reduction to First Order Plus Dead Time . . . . . . . . . . . . . . 292.4.1 Steady State Gain . . . . . . . . . . . . . . . . . . . . . . 312.4.2 Time Constant . . . . . . . . . . . . . . . . . . . . . . . . 322.4.3 Pure Delay . . . . . . . . . . . . . . . . . . . . . . . . . 322.5 Rational Plant Model . . . . . . . . . . . . . . . . . . . . . . . . 333 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 LPV Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.1 Linear Parameter Varying Form of the Plant . . . . . . . . 383.1.2 Scheduling Parameter . . . . . . . . . . . . . . . . . . . . 40v3.1.3 LPV Controller Design . . . . . . . . . . . . . . . . . . . 403.2 Switching LPV Controller . . . . . . . . . . . . . . . . . . . . . 423.2.1 Linear Parameter Varying Form of Plant . . . . . . . . . . 433.2.2 Scheduling Parameter . . . . . . . . . . . . . . . . . . . . 463.2.3 Switching LPV Controller Design . . . . . . . . . . . . . 473.3 H¥ Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4 Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . 534 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.1 LPV Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.1.1 Time-Invariant Simulations . . . . . . . . . . . . . . . . . 594.1.2 Time-Varying Simulations . . . . . . . . . . . . . . . . . 604.2 Switching LPV Controllers . . . . . . . . . . . . . . . . . . . . . 624.2.1 Time-Invariant Simulations . . . . . . . . . . . . . . . . . 634.2.2 Time-Varying Simulations . . . . . . . . . . . . . . . . . 654.3 Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.1 Time-Invariant Simulations . . . . . . . . . . . . . . . . . 694.3.2 Time-Varying Simulations . . . . . . . . . . . . . . . . . 715 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78viBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80A Switching LPV Controller Synthesis With Guaranteed L2 Perfor-mance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86A.1 Controller Design Problem . . . . . . . . . . . . . . . . . . . . . 87A.2 Switching Variable . . . . . . . . . . . . . . . . . . . . . . . . . 88A.3 Controller Design Method . . . . . . . . . . . . . . . . . . . . . 89A.3.1 Controller Performance . . . . . . . . . . . . . . . . . . . 90A.3.2 Practical Validity . . . . . . . . . . . . . . . . . . . . . . 91A.3.3 Switching Performance . . . . . . . . . . . . . . . . . . . 92A.3.4 Reduction to Finite Dimensional . . . . . . . . . . . . . . 93A.4 Recapitulative Procedure . . . . . . . . . . . . . . . . . . . . . . 95B Adaptive Predictive Control Algorithm using Laguerre Network . . 98B.1 Laguerre Network . . . . . . . . . . . . . . . . . . . . . . . . . . 99B.2 IMLLPC Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 100B.2.1 User-Defined Parameters . . . . . . . . . . . . . . . . . . 101B.2.2 Constant Matrices . . . . . . . . . . . . . . . . . . . . . . 101B.2.3 Recursive Estimation . . . . . . . . . . . . . . . . . . . . 102B.2.4 Control Move Calculation . . . . . . . . . . . . . . . . . 102viiList of TablesTable 2.1 Sensor Time Constant Values . . . . . . . . . . . . . . . . . . 27Table 3.1 Switching LPV Controller Design Weighting Functions . . . . . 45Table 3.2 Complexity and Computing Time of Switching LPV Controllers 51Table 3.3 Tuning Parameters of the Incremental Mode Linear LaguerrePredictive Control (IMLLPC) Algorithm . . . . . . . . . . . . . 54viiiList of FiguresFigure 1.1 Diagram of Engine Showing Air and Fuel Path . . . . . . . . 4Figure 2.1 Intake Manifold Model . . . . . . . . . . . . . . . . . . . . . 20Figure 2.2 Wall Wetting Model . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.3 Timing of a Four-Cylinder Four-Stroke Engine . . . . . . . . 24Figure 2.4 Response of a Four-Cylinder Engine to a Step Input . . . . . . 25Figure 2.5 Model of Exhaust Mixing . . . . . . . . . . . . . . . . . . . . 26Figure 2.6 Complete Detailed Spark Ignition (SI) IC Engine Model . . . . 28Figure 3.1 Comparison of LPV Controller Scheduling Structures . . . . . 37Figure 3.2 Closed-Loop System With LPV Controller . . . . . . . . . . . 38Figure 3.3 Closed-Loop System With Switching LPV Controller . . . . . 44Figure 3.4 Operating Space Divided Into R Subregions . . . . . . . . . . 49Figure 3.5 Comparison of H¥ Controller Scheduling With LPV Controller 52Figure 4.1 Comparison of Pure Delay and Pad´e Approximation . . . . . 57ixFigure 4.2 Disturbance rejection of H¥ Controller (Dashed), LPV Con-troller 1 (Thin), and LPV Controller 2 (Thick) at Fixed Oper-ating Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 4.3 Engine Speed and Air Flow Profiles . . . . . . . . . . . . . . 60Figure 4.4 Performance of H¥ Controller (Dashed), LPV Controller 1(Thin), and LPV Controller 2 (Thick) With Time-Varying PlantDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 4.5 Disturbance Rejection of H¥ Controller (Dashed) and S-LPV Con-troller 1 (Solid) . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 4.6 Disturbance Rejection of S-LPV Controller 2 (Dashed) andS-LPV Controller 3 (Solid) . . . . . . . . . . . . . . . . . . . 64Figure 4.7 Disturbance Rejection of S-LPV Controller 4 (Dashed) andS-LPV Controller 5 (Solid) . . . . . . . . . . . . . . . . . . . 64Figure 4.8 Air Flow and Engine Speed Profiles . . . . . . . . . . . . . . 66Figure 4.9 Engine Operating Point Trajectory . . . . . . . . . . . . . . . 67Figure 4.10 Time-Varying Switching Variable . . . . . . . . . . . . . . . 67Figure 4.11 Performance of H¥ Controller (Dashed), S-LPV Controller 1(Thin), and S-LPV Controller 4 (Thick) With Time-VaryingPlant Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 4.12 Adaptive Controller Reference Tracking for Fixed OperatingPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 4.13 Time-Varying FOPDT Parameters . . . . . . . . . . . . . . . . 71xFigure 4.14 Tracking Performance of S-LPV Controller 4 (Thin) and theAdaptive Controller (Thick) . . . . . . . . . . . . . . . . . . 72Figure A.1 An Example Parameter Trajectory Causing Switching BetweenTwo Subregions . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure B.1 Laguerre Network Model Structure . . . . . . . . . . . . . . . 99xiGlossaryECU Engine Control UnitFOPDT First Order Plus Dead TimeHEGO Heated Exhaust Gas OxygenIC Internal CombustionIMLLPC Incremental Mode Linear Laguerre Predictive ControlLMI Linear Matrix InequalityLNT Lean NOx TrapLPV Linear Parameter VaryingMVM Mean Value ModelingMAF Manifold Air FlowMAP Manifold Absolute PressureSI Spark IgnitionxiiTWC Three-Way Catalytic Converter or Three-Way CatalystUEGO Universal Exhaust Gas OxygenxiiiAcknowledgmentsI wish to expresses my sincere appreciation to my devoted supervisor and teacher,Dr. Ryozo Nagamune, for giving me the wonderful opportunity to grow intellectu-ally under his guidance. For his inspiration, encouragement, patience, friendship,and faith in me, I am forever indebted. I know few people who put their heartand soul into their work like Dr. Nagamune does, and I have enjoyed followinghis example. Furthermore, I am grateful for his support in the form of a ResearchAssistantship as well as the equipment necessary for conducting my research.1Special thanks go to both my committee members, Dr. Clarence de Silva andDr. Farrokh Sassani, whose knowledge and expertise I have benefitted from asearly as my undergraduate degree.Thanks go, also, to Dr. Greg Stewart at Honeywell for donating his personaltime and for taking an interest in the current research. His advice, feedback, andassistance has been of great value.I would like to thank my very dear friends and colleagues in the ControlEngineering Laboratory: Mr. Ehsan Azadi Yazdi, Mr. Masih Hanifzadegan, and1This work is supported through a Discovery Grant awarded to Dr. Ryozo Nagamune by theNatural Sciences and Engineering Research Council of Canada.xivMr. Mohammad Sepasi. The assistance, encouragement, and camaraderie theyhave provided during the period of my research is deeply valued and appreciated.To my other friends at UBC, Jeswin, Sina, and Susana, as well as Hamed andMahkameh, whom I have had the pleasure of studying with since the very begin-ning of my career, I extend my deepest gratitude for the support and friendship.To all my friends at home, I extend thanks for patience and understanding while Idisappeared into the laboratory for indefinite periods of time.Finally, I wish to acknowledge and thank my family. To my parents, who haveloved me, inspired me, given me so many opportunities and, through example,nurtured a faith in Christ, I owe nothing less than all my life. I have been blessedwith two beautiful sisters who have always encouraged me and, when necessary,endured me, and I thank and cherish them both.xvDedicationTo my Parents,an insufficient token of my appreciation of their unwavering love and faithfulnessxviChapter 1Introduction1.1 MotivationModern automotive engines have benefited from the use of the Three-Way Cat-alytic Converter (TWC) since it was introduced in production automotive vehiclesin 1981 as a response to increasingly stringent tailpipe emission standards. TheTWC is situated in the exhaust line of an automotive engine and improves thequality of exhaust gasses by reducing nitrogen oxides to nitrogen and oxygen, ox-idizing carbon monoxide to carbon dioxide, and oxidizing hydrocarbons to carbondioxide and water [16]. More generally, the TWC can be thought of as a buffer tosmall air and fuel flow variations in the exhaust gas as it is capable of rejectingsmall disturbances to the air-fuel ratio. The TWC achieves this function by storingoxygen on the catalytic surface when an excess is present and donating it back tothe exhaust gas when a deficiency exists.1The effective operation of the TWC relies heavily on two factors: Firstly, theoxygen storage level on the catalytic surface should be near half of capacity and,secondly, the air-fuel ratio of the incoming exhaust gasses should be close to stoi-chiometric. The air-fuel ratio of the exhaust gas is said to be stoichiometric whenthe correct proportion of air and fuel was combusted in the engine, yielding an ex-haust gas with neither an excess or deficiency of oxygen. Accurately maintainingthe desired air-fuel ratio in the engine is therefore essential to ensure good healthand operation of the TWC and, consequently, low tailpipe emissions.When the driver gives a throttle command by lowering the throttle pedal, thethrottle valve is opened either mechanically or by the Engine Control Unit (ECU)if the vehicle uses a drive-by-wire throttle system. The opening throttle valveallows more air to be drawn into the engine’s cylinders. In order to compensatefor the changing air flow and maintain a desired air-fuel ratio, the ECU adjusts thefuel flow by changing the duration of the fuel injection pulses.Conventional control strategies have relied heavily on static maps and feed-forward controllers to determine the correct fueling rate. These controllers main-tain the desired air-fuel ratio by estimating the air flow into the cylinders andsimply commanding a proportional fuel flow. A shortcoming of the feed-forwardcontroller architecture is that it does not compensate for the following situationsthat result in unknown disturbances on the actual air-fuel ratio achieved: Aging of the sensors used in the estimation of the air flow, causing a loss ofaccuracy.2 Inaccuracy of the modeled dynamics, stored in the ECU’s memory, usedto estimate the air flow into the cylinders due to engine aging and plantidentification deficiencies. Variations in the oxygen concentration of the air flow due to altitude, weatherconditions, and fuel canister purge cycles (which is when fuel vapors trappedinside the fuel tank are periodically released into the air flowing into the en-gine). Clogging or wear of the fuel injectors, causing the actual amount of fuelbeing injected to differ from the amount commanded by the ECU. Variations in the quality and composition of fuel.The introduction of the Universal Exhaust Gas Oxygen (UEGO) sensor hasmade it possible to measure the air-fuel ratio in the exhaust flow and rely on feed-back [4]. Using a feedback controller in conjunction with a feedforward controllerenables fast and accurate air-fuel ratio control. The feedback controller takes ameasurement of the air-fuel ratio in the exhaust flow and rejects disturbances tothe desired air-fuel ratio. The feed-forward controller estimates the air flow intothe cylinders.1.2 Problem StatementThe current research focuses specifically on the normally aspirated, port injected,Spark Ignition (SI) Internal Combustion (IC) engine which is commonly used inpersonal vehicles. Figure 1.1 shows the air and fuel paths through one cylinder of3Figure 1.1: Diagram of Engine Showing Air and Fuel Paththe engine. The driver varies the air flow allowed into the system by opening andclosing the throttle valve. In the intake manifold, the flow of air is divided into aseparate stream for each cylinder. Fuel is injected separately into each air streambefore it enters the cylinder, the amount injected being controlled by the ECUthrough varying the length of the injection pulses. After the air and fuel mixturehave completed the combustion stroke in each cylinder, the separate flows arecombined in the exhaust manifold before traveling to the TWC. A measurement ofthe air-fuel ratio is taken using UEGO sensors mounted in the exhaust line beforeand after the TWC.The air mass flow rate, which is controlled by the driver, is considered anuncontrolled input to the plant but is assumed to be available. Although this thesisdoes not discuss them, methods have been developed to measure or estimate theair flow into the cylinders of normally aspirated IC engines based on Manifold4Air Flow (MAF) and Manifold Absolute Pressure (MAP) sensor readings in e.g.[9], [14], [34], [35], and [36], and range from simple schemes like the speeddensity method to advanced observers that compensate for sensor and air flowdynamics. Because the fuel mass flow rate is closely proportional to the width ofthe fuel injection pulses which are controlled by the ECU, the rate of fuel injectedis considered as the controlled input to the plant. An air-fuel ratio measurementis available in the exhaust line both before and after the TWC. The method in thisthesis uses the air-fuel ratio measurement of the first UEGO sensor as the outputof the plant. Control schemes exist that use the measurement of the second UEGOsensor installed after the TWC to estimate the oxygen storage level in the TWC (seee.g. [3], [7], [10], [11], and [25]) but are beyond the scope of this thesis. Thesemethods may build on the control loop presented in this thesis as the inner of twocascaded loops.The closed-loop air-fuel ratio control problem is non-trivial and a poor fit forsimple linear control techniques for three reasons. Firstly, the discrete strokes ofthe IC engine cause a pure delay in the system, meaning that a rational transferfunction for the plant does not exist. Secondly, the engine dynamics are time-varying. The aforementioned pure delay, for example, varies considerably as afunction of engine speed and air flow. Finally, because the air-fuel ratio is a frac-tion consisting of the IC engine’s two inputs, air and fuel, it is impossible to ex-press the output, the air-fuel ratio, as a linear transfer function of both inputs, andthe control problem is therefore non-linear.51.3 Literature ReviewPreviously developed methods addressing the air-fuel ratio control problem in SIIC engines are discussed in Section 1.3.1. Literature concerning controller designtechniques influencing the current research is given in Section Air-Fuel Ratio ControllersCurrent methods that address the air-fuel control problem in SI IC engines includePI-Control [3], H¥ Robust Control [24], Kalman Filter [26], [37], Adaptive Con-trol [19], [31], [32], [41], [42], Linear Parameter Varying (LPV) Control [44], [45],[47], and others [22], [39]. Robust control methods treat the variations of the ICengine as uncertainty and employ time-invariant controllers with guaranteed per-formance. However, because the variation in plant dynamics is substantial overthe operating range of a normal engine, robust control methods yield undesirablyconservative controllers.Adaptive controllers are able to accommodate the time-varying engine dynam-ics. An attractive feature of adaptive controllers is that no a priori information ofthe relationship between the engine’s operating point and the plant dynamics isnecessary. However, adaptation is typically slow and may lose performance andeven stability during sudden changes in the engine operating point. A combinationbetween adaptation and gain-scheduling is therefore often used to adress a rapidlymoving operating point as is done in [26] and [42]. Adaptive methods generallylack a theoretical guarantee for stability and performance.Gain-scheduling controllers, on the other hand, are time-varying controllers6that adjust for the variations in the dynamics using a priori knowledge of theplant dynamics throughout the operating range. Gain-scheduling methods basedon interpolation also share the shortcoming that no theoretical guarantee of sta-bility and performance exists. Considerable advances have been made, however,and in [5], [6], [33], gain-scheduling methods are presented with guaranteed per-formance. The LPV methods in [47] and [48] use Lyapunov theory developed in[5] to create gain-scheduled controllers with guaranteed L2 performance. Thesemethods use a single parameter dependent Lyapunov function to guarantee per-formance over the operating range of the plant. Because the variation of the plantdynamics can be extensive across the full operating range of an SI IC engine, asingle Lyapunov function may not necessarily exist or, if it does exist, may leadto an unacceptably conservative controller.The following subsections take a more detailed look at literature that haveaddressed the air-fuel ratio control problem in SI IC engines and includes methodsbased on inverse dynamics, H¥, LPV-based gain-scheduling, and adaptive control.Similar approaches have been used in Diesel engines (see e.g. [1], [2], and [8])but are beyond the scope of this thesis.Detailed Model-Based and Inverse Dynamics ControlOne of the early papers on model-based IC engine air-fuel ratio control, [4], givesan introduction to the control problem. The author compares two control archi-tectures: Feed-forward control with emphasis on static maps, and a closed-loop,model-based controller which is obtained from inverting the dynamics of the en-7gine. The dynamics of the SI, IC engine are presented and include many non-linearrelationships and time-varying coefficients.Another model-based controller is presented in [29]. The method uses anobserver to estimate the air flow into the cylinders with great accuracy so that aproportional amount of fuel can be injected. Air-fuel ratio measurements fromoxygen sensors mounted in the exhaust line serve to improve the accuracy of theestimated observer model.In [27], a detailed non-linear model is used to capture the dynamics of the SIIC engine and includes air flow dynamics, wall wetting dynamics, engine inertia,process delays and UEGO sensor dynamics. A sliding mode controller is used toregulate the air-fuel ratio.A disadvantage of closed-loop controllers based on detailed engine models isthat their effectiveness heavily rely on accurate identification of the time-varyingengine parameters.H¥ ControlAn SI IC engine with drive-by-wire throttle is considered in [24]. A feedforwardcontroller is used to calculate the base fueling while an H¥ preview controllerprovides an adjustment based on feedback from the UEGO sensor mounted in theexhaust line. Because the ECU has control of the position of the throttle valve,and consequently the air flow, in a drive-by-wire configuration, predicted futureair flow values are available. The H¥ preview controller takes advantage of thisfact to remove the effect of delay in the closed loop plant.8Adaptive ControlAn adaptive controller which compensates for sensor aging is presented in [32].Because the time constant of a UEGO sensor may increase from about 50msecup to 1sec during its lifetime, the authors recommend using adaptation law in or-der to maintain performance of the closed loop controller including the sensor.The variation in sensor dynamics occurs very gradually over the life of the sen-sor, however, meaning that the adaptation rate may be very slow. The IC enginedynamics are assumed to be known and are modeled as a First Order Plus DeadTime (FOPDT). The time-varying nature of the engine dynamics is beyond thescope of the paper.Methods presented in [31], [41] and [19] use adaptation to compensate forthe time-varying engine dynamics. In each method the engine dynamics are sim-plified and modeled using an FOPDT transfer function. Adaptation is then usedto estimate some or all of the model parameters and a suitable controller is cal-culated in real-time. Feed-forward controllers are used to augment the adaptivecontrollers by compensating for the rapidly changing low frequency gain of theplant due to changes in air flow. An adaptive Smith predictor is used in [19] toremove the effect of the time-delay in the closed-loop.A structure similar to that of a Smith predictor is also used in [42] but, ratherthan using adaptation, the time-varying delay is computed using a priory knowl-edge and sensor measurements and is scheduled accordingly. The authors intro-duce an adaptive controller which uses adaptation in both the feed-forward con-troller and feedback loop for the regulation of both the air-fuel ratio and the TWC9oxygen storage level. A UEGO sensor mounted upstream of the TWC and a HeatedExhaust Gas Oxygen (HEGO) sensor installed downstream of the TWC are used toestimate the oxygen storage level inside the TWC. Adaptation is used to identifythe model of the IC engine including wall wetting, exhaust dynamics, and sensordynamics.Adaptive Kalman filters are used in [37], [25], and [26]. The method presentedin [37] uses an indirect model-reference adaptive controller but assumes that theUEGO sensor dynamics, as well as all system delays in the engine, are known andavailable in real-time from lookup tables. The parameter identification is onlyused to estimate the wall wetting dynamics of the engine. In [25], on the otherhand, the Kalman filter is used to estimate the time-varying delay and disturbancedynamics and, in [26], it is used to estimate the parameters of a FOPDT modelof the IC engine. While the low frequency gain and time constant of the plantdynamics in [26] are automatically estimated using the Kalman filter, the delay isexplicitly computed at each time-step and expressed in terms of sample periods.The authors present experimental data confirming the dependence of the delay onthe engine speed and air flow.LPV-Based ControlLPV-based control is used in [44] and [45] to track a reference air-fuel ratio ina lean burn IC engine. Unlike conventional IC engines, lean burn engines donot operate in the narrow band around stoichiometry but instead vary the air-fuelratio considerably in order to achieve the effective operation of both the TWC and10the Lean NOx Trap (LNT), creating a more challenging tracking problem. Themethod proposed in [44] uses input shaping on the reference air-fuel ratio in orderto improve transient tracking responses. Air flow and engine speed are measuredusing a hot wire MAF sensor and engine speed sensor, respectively, and are bothused to calculate the time-varying delay of the plant in real-time and schedule thecontroller accordingly.An air-fuel ratio controller for SI IC engines using LPV gain-scheduling tech-niques is presented in [47]. The controller takes feedback from the UEGO sensormounted in the exhaust line. The plant to be controlled, including engine and sen-sor dynamics, is modeled as an FOPDT which contains a time-varying delay andtime constant. A feed-forward controller calculates a base fueling rate which isthen adjusted by the feedback controller in a multiplicative manner. The controllerdesign method used is developed in [5] and uses Lyapunov theory to guaranteeperformance over the operating range of the engine. The method is revisited in[48] and a state-delayed LPV controller is introduced. Rather than using a Pad´eapproximation to model the delay in the plant for controller design, the delay isremoved from the dynamics and reintroduced into the LPV form of the plant ex-plicitly as an additional state. The authors of both papers assume that the enginedynamics, specifically the delay and time constant, are only affected by enginespeed. The effects of air flow are ignored. This omission causes poor system per-formance in some regions of the engine’s operating range and voids any guaranteeof stability that the method provides.111.3.2 Controller Design TheorySupporting theory for the development of the current research is presented in thefollowing subsections. The first two topics, H¥, and adaptive control, are exploredfor comparative purposes. The last topic, LPV-based control, is the emphasisof the current research and is applied to the air-fuel ratio control problem in anovel way. An overview of gain-scheduling approaches to non-linear control ispresented in [21] including methods of linearization and LPV formulations.H¥ ControlThe H¥ controller design method used in Section 3.3 is developed and presentedin [13] and benefits from existing Linear Matrix Inequality (LMI) techniques. Thecontroller design problem is formulated in terms of LMIs which yield a convexoptimization problem and can be efficiently solved using existing LMI techniques.An alternative but similar method is given in [33] and is capable of addressingmulti-objective H2 and H¥ problems.Adaptive ControlAn adaptive method using a series of orthonormal functions, known as a Laguerrenetwork, to model the plant dynamics is presented in [43]. The Laguerre net-work is a set of time-invariant or time-varying delay elements placed in series.The output of the network is given as a vector of coefficients, called the spectralgain, multiplied by the state vector of the network. The overall transfer functionof the network is very similar to a Pad´e approximation, making it a very suitable12model structure for plants containing a pure delay. The value of the spectral gainvector is estimated using a recursive least squares algorithm. The method is revis-ited in [46] to improve tracking performance and provide necessary and sufficientconditions for stability and performance.LPV-Based ControlThe LPV-based gain-scheduling techniques used in this research are first devel-oped in [6]. The plant is written in an LPV form where the plant state-spacematrices are assumed to depend affinely on a vector, q, called the scheduling pa-rameter. Methods for expressing non-linear plants as LPV systems are given in[18] and [30]. The scheduling parameter is assumed to be available in real-timeand is used to update the controller. The resulting controller is time-varying andautomatically scheduled within the bounds of q, known as the operating space.The controller design problem is formulated in terms of LMIs, yielding a convexoptimization problem. Given that the plant state-space matrices can be accuratelymodeled as affine functions of the scheduling parameter, the method provides aguarantee of H¥ performance for any operating point and for any movement ofthe scheduling parameter.In [5] the method is developed further and the requirement that the state-spacematrices of the plant vary affinely with the scheduling parameter is removed. Thisis done by sampling the operating space using a grid consisting of a finite num-ber of points. LMIs are used to define constraints at each point and are computedtogether in order to calculate the optimal parameter-dependent controller matri-13ces. A function is introduced so that it mimics the dependence of the Lyapunovfunction, and consequently of the controller matrices, on the scheduling parame-ter. This function then allows the assembly of the controller matrices at any pointwithin the operating space, including but not limited to the sampled points on thegrid. Furthermore, the controller design technique yields a less conservative con-troller because, unlike [6], a limit can be imposed on the rate of movement of thescheduling parameter within the operating range.The method is advanced further in [23], where switching is introduced. Theoperating space is divided into overlapping subregions, each subregion having acontroller developed using the the techniques in [5]. The overlap in subregionsis used to create hysteresis switching. Additional LMIs are introduced for pointsalong the switching surfaces and ensure that the value of the Lyapunov functionis decreasing at each switching event. Because each subregion can have its ownparameter-dependent Lyapunov function, the resulting switching controllers canbe much more aggressive than a single controller with a single parameter-varyingLyapunov function that covers the entire operating range.1.4 Research ObjectivesMultiple authors have proposed excellent methods for addressing the air-fuel ratiocontrol problem in SI IC engines, as discussed in the previous section. However,because authors have focussed on specific aspects of the air-fuel ratio problem,currently no single controller design technique exists which does: Address the time-varying and non-linear nature of the SI IC engine by ap-14propriately compensating for the engine’s operating point. Appropriately address the parameter-dependent pure delay in the plant dy-namics. Incorporate the design of a feed-forward controller or remove the need todesign it separately from the feedback controller. Guarantee stability and performance over the entire operating range of theengine.The overall objective of the current research is to apply advanced controller de-sign techniques to the air-fuel ratio control problem and thereby address the aboveshortcomings with a single air-fuel ratio controller design technique. The LPV-based controller design technique developed in [5] and [23] is used. A switchingLPV air-fuel ratio controller which is scheduled for both engine speed and air flowis designed and solved using LMI techniques. The proposed air-fuel ratio con-troller design technique will: Use a model of the SI IC engine which is realistic but simple enough forcontroller design purposes. Appropriately model the pure delay present in the engine dynamics. Treat non-linear engine dynamics as linear but parameter-dependent dynam-ics. Compensate for variations in the engine dynamics due to both air flow andengine speed.15 Remove the need for a separately designed feed-forward controller and in-stead use only the output of an air flow estimator. Track a reference air-fuel ratio and reject disturbances. Guarantee performance over the operating range of the engine using Lya-punov theory. Reduce the conservativeness of the resultant controller as compared to ex-isting methods. Be practically useful, meaning that all assumptions are valid for a real SIIC engine and only available or measurable information is used by the con-troller.Simulations are performed using a model of an SI IC engine to demonstratethe performance of the controller designed using the proposed technique.1.5 Organization of ThesisThis thesis is organized as follows. Chapter 2 presents the SI IC engine plantmodel in two parts. First a detailed model of the plant is developed. The modelserves to demonstrate its parameter-dependent and non-linear nature. A simplemodel, known as an FOPDT model, is then derived so that it can be used for con-troller design.In Chapter 3, the proposed switching LPV controller design technique is intro-duced. Two current air-fuel ratio controller design techniques, H¥, and LPV, are16also presented as well as a novel adaptive air-fuel ratio controller.The controllers are simulated in closed loop and the results are shown inChapter 4. Multiple controllers are developed using each technique in Chapter 3,resulting in a large number of controllers. Chapter 4 is therefore divided into threesections and the results are compared in meaningfully selected groups rather thansimultaneously.Conclusions drawn from the results are presented in Chapter 5 and the con-tributions of the current research are discussed. Outstanding and possible futuredevelopments are also briefly discussed.Controller design techniques are deferred to the Appendices. The switchingLPV controller design technique and the adaptive control algorithm are given inAppendix A and Appendix B, respectively.17Chapter 2Modeling2.1 A Measure of Air-Fuel RatioA measurement of the air-fuel ratio is commonly given by the variable l, whichis defined asl := 1Rstoich ˙mair˙mf uel; (2.1)where ˙m f uel and ˙mair are the fuel and air mass flows, respectively, and Rstoich isthe stoichiometric ratio which is approximately 14.7 parts air to one part fuel bymass. Equation (2.1) is referred to as the combustion equation in this thesis.When l < 1 the air-fuel mixture is said to be rich, meaning that insufficientoxygen was supplied for complete combustion to occur, thereby causing uncom-busted fuel to remain in the exhaust gas. Conversely, when l > 1 the mixture islean, meaning that insufficient fuel was supplied and that oxygen remains in theexhaust gas.182.2 Mean Value ModelingAlthough the four-stroke IC engine has discrete events making it suitable for dis-crete time modeling, it is common practice to model it in continuous time usingMean Value Modeling (MVM) [15]. Transient responses within each event are ig-nored by MVM, while the mean value over each event is captured well. This issufficient because the period of each discrete event is much smaller than the timescale of interest. MVM is an especially good fit for four-stroke engines with fourcylinders or more because of the mechanical design of these engines. The cyclesof the cylinders are shifted in phase such that, at any given time, at least one cylin-der is performing each of the four strokes: induction, compression, power, andexhaust. Because the engine is performing each of the four strokes at any giventime, it can be thought of as a continuous process where the strokes are occurringconcurrently and continuously. More importantly, air and fuel are continuouslyentering the engine and exhaust gas is continuously produced. The delay intro-duced by the engine will be discussed in Sections Detailed Model of PlantA detailed model of the air-fuel ratio dynamics in a normally aspirated SI IC en-gine is developed in this section. The air flow dynamics, fuel flow dynamics,exhaust mixing, system delays, and sensor dynamics are considered. The purposeof developing this model is to achieve a complete understanding of the plant dy-namics and to demonstrate the non-linear and parameter-dependent nature of theengine as well as the source of delays. The model is not used for controller design19or closed-loop simulations. Instead, a simplified but widely accepted model ispresented in Section Air Flow DynamicsFigure 2.1: Intake Manifold ModelDue to the compressibility of the air in the intake manifold, the air flow throughthe throttle valve is not necessarily the same as the air flowing into the cylinders,as depicted in Figure 2.1. Assuming the air flow is isentropic, ˙mat, the air flowthrough the throttle valve, which is measured by the MAF sensor, is given in [4] as˙mat = papRTaQ(a)Y(pmanpa); (2.2)where pa is the ambient air pressure, pman is the pressure inside the intake man-ifold and is measured by the MAP sensor, R is the ideal gas constant, Ta is theambient air temperature, and a is the angle of the throttle plate and determinesthe opening through the throttle valve. The functions Q(a) and Y(pmanpa ) are non-linear but static functions which limit the air flow into the intake manifold and areexperimentally determined.20The air flow entering the cylinders is˙mac = hvol(N; pman)pmanNVdnrRTman; (2.3)where hvol is the volumetric efficiency and is a function of engine speed, N, andmanifold pressure, pman. The cylinder’s volumetric displacement is denoted Vdand the number of crankshaft rotations per cycle is denoted by nr. For a four-stroke engine, nr = 2. If a measurement of the manifold temperature, Tman, is notavailable, it can reasonably be assumed to be the same as the ambient temperature,Tman Ta.The rate of change of the pressure in the manifold is given byd pmandt = K ( ˙mat ˙mac); (2.4)where the constant, K, is related to the size of the intake manifold.2.3.2 Wall Wetting DynamicsFigure 2.2: Wall Wetting ModelFuel is injected in the intake manifold, upstream of the intake valve, in liquid21form but is partially vaporized. Of the fuel injected, ˙m f in j, a fraction is depositedon the manifold walls as fuel puddles while the remaining fuel immediately entersthe cylinders [4]. Fuel in the puddles evaporates and also enters the cylinders. Letx f p denote the fraction of fuel injected that is deposited on the manifold walls.Then 1 x f p of the fuel injected goes directly to the cylinders. The rate of fuelevaporating is proportional to the mass of fuel in the puddles, m f p, and is dictatedby tf p, the evaporation time constant. The rate of change of the mass of fuel inthe puddles is given bydm f pdt = x f p ˙m f in j 1tf p m f p; (2.5)and the fuel flow into the cylinders is˙m f c =(1 x f p) ˙m f in j + 1tf pm f p: (2.6)2.3.3 Four-Stroke Engine Cycle DelaysAs stated in Section 2.2, the IC engine can be modeled as a continuous systemusing MVM, knowing that at any given time it is concurrently receiving fuel andair and expelling exhaust gas. It is important to consider, however, that the airand fuel mixture entering the engine at a given moment is not the same as thatleaving the engine because the mixture remains in the engine while a cycle iscompleted. This dwell time, which is a function of the engine speed, can beaccurately modeled as a pure delay, also known as dead time.22The time that fuel resides in the engine, Tf uel, given in seconds, isTf uel = 60 nr nin j exnstroke N; (2.7)where nin j ex denotes the number of strokes completed between the rising edgeof the fuel injection pulse and the start of the exhaust stroke. For a modern four-stroke engine, nr = 2 and denotes the number of crankshaft revolutions per cycle,while nstroke = 4 and denotes the number of stokes per complete cycle. The enginespeed, N, is measured in revolutions per minute.The dwell time for air and fuel are not necessarily equal. The air dwell time isgiven byTair = 60 nr nind exnstroke N; (2.8)where nind ex, the number of strokes from induction to exhaust, is always equalto three for a four-stroke engine. The values of nin j ex and nind ex are typicallynot equal because most port-injected IC engines inject fuel into the intake port oneor even up to three strokes prior to the intake valve opening during the inductionstroke.Figure 2.3 serves as an example and shows the timing for a 2:0L 1992 Pon-tiac Sunbird engine, which is a four-cylinder SI IC engine, over two completecycles. The figure displays four complete revolutions of the crankshaft, meaningtwo complete cycles. Stroke names, intake and exhaust valve positions, spark ig-nition, and fuel injection pulses are shown. This engine’s ECU injects fuel in twoequal pulses for each cycle. The decision of how much fuel to give is made before23Figure 2.3: Timing of a Four-Cylinder Four-Stroke Enginethe first pulse. The first pulse occurs three strokes before the intake valve opensduring the induction stroke. Fuel therefore remains in the engine for six strokes,from injection to exhaust, and therefore nin j ex = 6 as labeled for the first cylinderin the figure.242.3.4 Exhaust MixingIndividual flows from the cylinders are recombined in the exhaust manifold. Be-cause the cycles of the cylinders in an IC engine are shifted in phase, the cylindersexpel their contents into the exhaust manifold sequentially. A step change to theair-fuel ratio entering the engine will therefore result in a staircase-shaped out-put consisting of ncyl smaller steps where ncyl is the number of cylinders in theengine [20]. As an example, the output of a four-cylinder engine is shown in Fig-ure 2.4. Each step corresponds to one cylinder completing the exhaust stroke andexpelling its gasses. The duration of each step, Ts s, represents to the time passingfrom one cylinder completing the exhaust stroke until the next cylinder does so,and is related to the engine’s speed as follows:Ts s = 60 nrN ncyl(2.9)Figure 2.4: Response of a Four-Cylinder Engine to a Step Input25Recall that N is the engine speed in revolutions per minute and nr denotes thenumber of revolutions per complete engine cycle.Figure 2.5: Model of Exhaust MixingFigure 2.5 shows a block diagram of a model for the exhaust mixing. The air-fuel ratio of the mixture inside the engine is denoted by lcyl and the air fuel ratioin the exhaust manifold is denoted by lex. The staircase-shaped output is achievedusing delays of duration kTs s placed in parallel, where k = 0;1;:::;ncyl 1 .2.3.5 Exhaust Transport DelayAnother delay, Tex, arises from the time it takes the exhaust gas to travel from theexhaust manifold to the UEGO sensor mounted upstream of the TWC. Intuitively,Tex is inversely proportional to the exhaust gas volumetric flow rate. Because the26exhaust gas volumetric flow rate is proportional to the air mass flow rate for aconstant engine temperature, the exhaust gas transport delay can be expressed asa function of air mass flow rate,Tex = c˙mair: (2.10)The proportionality constant, c, in (2.10) is chosen such that Tex varies betweenthe values of 20msec and 500msec over the engine’s full operating range as rec-ommended in [20].2.3.6 Sensor DynamicsSensor dynamics are modeled as first order transfer functions (in [4], [11], [12], etc.)such that˜y = 1sty +1y; (2.11)where y is the measurand, i.e. the measured quantity, ˜y is the measurement, i.e. thequantity reported by the sensor, and ty is the time constant of the sensor. Typicaltime constants for the sensors used in the air-fuel ratio control problem are givenin Table 2.1.Table 2.1: Sensor Time Constant ValuesSensor Name Time ConstantMAF 10msec 60msec [17]MAP 3msec 20msec [17]UEGO 50msec [32]272.3.7 Complete ModelFigure 2.6 shows the completed model of the air-fuel ratio dynamics discussedthroughout Section 2.3. The inputs to the model are the throttle angle, a, and fuelinjection rate, ˙m f in j, and the outputs include the MAF and MAP sensor readings,˜˙mat and ˜pman respectively, and the measured air-fuel ratio, ˜lsensor. The modelFigure 2.6: Complete Detailed SI IC Engine Model28includes the air flow dynamics, given in (2.2), (2.3), and (2.4), which relate thethrottle angle to the airflow into the cylinders, ˙mac. Wall wetting dynamics definedby (2.5) and (2.6), relating the fuel flow injected, ˙m f in j, to the fuel flow into thecylinders, ˙m f c, are also included. The air and fuel dwell times inside the cylinders,presented in Section 2.3.3, are represented by e sTair and e sTf uel , respectively.The air-fuel ratio in the cylinders, lcyl, is a function of the air and fuel flowsentering the cylinders according to the combustion equation, (2.1). The exhaustmixing, which relates the air-fuel ratio in the cylinders to the air-fuel ratio of thecombined exhaust flow in the exhaust manifold, lex, is shown in Figure 2.5. Theexhaust gas transport delay, discussed in Section 2.3.5, is represented by e sTex.Finally, the first order sensor dynamics for the MAF, MAP, and UEGO sensors arealso included.2.4 Reduction to First Order Plus Dead TimeThe overall dynamics of the plant described in Section 2.3 include manifold airflow dynamics, wall wetting effects, combustion cycle and exhaust gas transportdelays, exhaust gas mixing, as well as sensor dynamics. However, the modeldescribing these dynamics precisely is non-linear and too complex for controllerdesign. The problem is aggravated by the fact that this model contains manyparameter-dependent values. Constants and parameter-dependent values are ex-perimentally identified and are unique to each engine. The identification of en-gine parameters is beyond the scope of the current research. The detailed modelis therefore not used for controller design or closed-loop simulations.29A model which accurately captures the overall dynamics of the plant, as wellas its dependence on the operating point, but is simple enough for controller designpurposes is therefore necessary. A popular option is to model the overal dynamicsof the IC engine as an FOPDT model. A controller which was designed for anFOPDT plant has been successfully implemented on an actual IC engine in [26].The FOPDT model is defined asgst+1 e sT; (2.12)which includes the following three time-varying parameters:g: The steady state gain of the system.t: The time constant of the first order component.T : The duration of the pure delay.This section uses the notation ˙mair and ˙m f uel to denote air flow and fuel flow,respectively. Note that ˙mair refers to the air flow into the engine’s cylinders,˙mair = ˙mac: (2.13)Methods have been developed to measure or estimate the air flow into the cylin-ders of normally aspirated IC engines based on MAF or MAP sensor readings andair flow dynamics in e.g. [9], [14], [34], [35], and [36]. Likewise, ˙m f uel refersto the fuel flow into the cylinders. Since the FOPDT model omits wall wetting30dynamics,˙m f uel = ˙m f in j = ˙m f c: (2.14)2.4.1 Steady State GainThe air-fuel ratio, l, is given as a non-linear function of air flow and fuel flowby the combustion equation (2.1). Notice that the uncontrolled input, air flow, en-ters (2.1) linearly while the controlled input, fuel flow, appears in the denominator.This non-linear relationship is udesirable for control purposes. A simple and com-monly used remedy is to define a new measure of air-fuel ratio, the equivalenceratio, as f = 1l . Now only the controlled input enters linearly:f := Rstoich ˙m f uel˙mair: (2.15)When f > 1 the air-fuel mixture is said to be rich and when f < 1 the mixture islean.By considering ˙mair as a time-varying parameter, rather than an input, the plantis expressed as a single-input-single-output system that can be characterized by apseudo-linear but parameter-dependent transfer function. Equation (2.15) thenrepresents the steady state gain of the transfer function of the plant as it relatesthe plant’s output, f, to its input, ˙m f uel, but ignores any dynamics or transientresponses of the engine. It can be rewritten asg = f˙mf uel= Rstoich˙mair: (2.16)31Equation (2.16) shows the steady state gain of the FOPDT model as a function ofthe air flow.2.4.2 Time ConstantIn Section 2.3.4, it was shown that a step in the air-fuel ratio inside the engineresults in a staircase-shaped response of the air-fuel ratio in the exhaust manifolddue to the fact that the cylinders sequentially expel their contents. The staircase-shaped output can be approximated as a first order response. The time constant isgiven in [20] ast = 60 nr(ncyl 1)N ncyl: (2.17)Recall that nr denotes the number of revolutions the crankshaft makes to completea cycle, ncyl denotes the number of cylinders in the engine, and N is the enginespeed in revolutions per minute. A measurement of the engine speed is readilyavailable on modern engines. The time constant’s dependence on the engine speedwas confirmed using experimental data in [26].2.4.3 Pure DelayBecause fuel flow is chosen as the input to the plant in Section 2.4.1, we areinterested in the delay in the fuel path. The total delay from fuel injection to theair-fuel ratio measurement, T , is therefore the summation of the time that fuelremains in the engine and the travel time to the sensor:32T = Tf uel + Tex;= 60 nr nin j exnstroke N+ c˙mair;(2.18)where Tf uel is given in (2.7) and Tex is given in (2.10). Notice that the total delaydepends on both the engine speed and air flow.2.5 Rational Plant ModelIn order to perform controller design using the method developed in [23], theFOPDT plant model which is presented in Section 2.4 needs to be expressed asa rational transfer function. The pure delay, e sT , can either be removed fromthe transfer function and be reintroduced using additional states in the state-spacerealization [48] or it can be approximated using a Pad´e approximation. The latteris used in this research. A modified Pad´e approximation with a second orderdenominator but first order numerator is employed because, as shown in [38], itexhibits better approximation. The plant transfer function is then given byf˙m f uel =gst+1 6 2sT6+4sT +(sT)2 ; (2.19)where g, t, and T are time-varying and depend on the engine speed and air flowas shown in (2.16), (2.17), and (2.18), respectively. When realized into the state-33space form, the plant is given as˙xp = Apxp +Bp ˙m f uel;f = Cpxp;(2.20)where xp is the plant state vector andAp =266664 1t6T 2 2T0 0 10  6T 2  4T377775; Bp =266664001377775;Cp = gt 0 0 :(2.21)34Chapter 3Controller DesignIn this chapter, the controller design technique is presented. Four techniques aregiven and include LPV control, switching LPV control, H¥ control, and adaptivecontrol. The LPV controller is presented first in Section 3.1 and is created in orderto validate the choice of scheduling parameter and compare the results with thatof an existing LPV-based gain-scheduling air-fuel ratio controller. The primarycontribution of the current research, an LPV-based gain-scheduling air-fuel ratiocontroller which also includes switching, is then described in Section 3.2. Finally,an H¥ and adaptive controller, both of which are designed for comparison, areexplained in Sections 3.3 and 3.4, respectively.As discussed in Chapter 1, the controller design objective is to track a ref-erence air-fuel ratio and reject disturbances. A reference air-fuel ratio other thanstoichiometric, meaning an equivalence ratio other than unity, may be desired dur-ing engine warmup or may be required by the TWC oxygen storage level controller35when a correction in storage level is necessary. Each controller designed thereforeincludes an integrator in order to improve the reference tracking performance.The controllers are designed for the simplified model of the SI IC engine givenin Section 2.4. Each controller attempts to track a reference equivalence ratio byadjusting ˙m f uel, the amount of fuel injected. Accurate measurements of the enginespeed, N, air flow, ˙mair, and equivalence ratio, f, are assumed to be available.3.1 LPV ControllerAn LPV-based gain-scheduled air-fuel ratio controller is designed in this section.1The controller design method uses LPV techniques presented in [5] and is similarto the air-fuel controller presented in [47]. The authors of [47] ignore the effect ofair flow on the dynamics of the SI IC engine and present an LPV-based controllerwhich is scheduled for engine speed only. As shown in Chapter 2, the air flowaffects both the delay, T , and the steady state gain, g, of the plant, however. Inorder to cope with the large variations in gain due to varying air flow, the authorsfix the gain in the plant transfer function of the plant model but use a feed-forwardcontroller to calculate a base fueling rate using a measurement of the airflow. TheLPV-based closed-loop controller then adjusts the fueling rate in a multiplicativemanner. Figure 3.1(a) shows the structure of this method where K represents thecontroller and P is the time-varying plant. This explicit multiplicative controllerscheduling based on air flow is not necessary for an LPV-based gain-schedulingcontroller which includes both air flow and engine speed in its scheduling param-1The controller presented in this section has been published in [28] and presented at the2010 ASME Dynamic Systems and Control Conference.36(a) Controller Scheduled for Engine Speed Only(b) Controller Scheduled for Air Flow and Engine SpeedFigure 3.1: Comparison of LPV Controller Scheduling Structureseter as shown in Figure 3.1(b).The closed-loop system used for controller design takes the form shown inFigure 3.2. The matrices Ap, Bp, Cp, and Dp are the state-space realization ofthe FOPDT plant given in Chapter 2 in (2.21) where Dp = 0. K represents theparameter-dependent controller. The integrator placed in series with the controllerimproves the tracking performance. The disturbance, d, is modeled on the output37Figure 3.2: Closed-Loop System With LPV Controllerof the plant. A weighting function defined by state-space matrices Ae, Be, Ce, andDe filters the error and outputs a weighed error, ˜e, which will be minimized by thecontroller design technique.3.1.1 Linear Parameter Varying Form of the PlantThe plant can be written in the LPV form as˙x = A(q)x+B1(q)w+B2(q)u;z = C1(q)x+D11(q)w+D12(q)u;y = C2(q)x+D21(q)w;(3.1)where q is a vector known as the scheduling parameter.The design method minimizes the L2 norm from the exogenous input, w, to theperformance channel, z. Since we wish to minimize the error due to disturbances38or reference changes, we choosew =264 dr375 and z = ˜e; (3.2)where ˜e is the weighed error. The weighting function placed on the error can beused to tune the controller during the design stage. The states, controller input,and controller output arex =266664xpxeR e377775; y =R e; and u; (3.3)respectively. For the plant shown in in Figure 3.2 expressed in LPV form, thestate-space matrices are given byA =266664Ap 0 0 BeCp Ae 0 Cp 0 0377775; B1 =2666640 0 Be Be 1 1377775; B2 =266664Bp BeDp Dp377775;C1 =  DeDp Ce 0 ; D11 =  De De ; D12 =  DeDp ;C2 = 0 0 1 ; D21 = 0 0 :(3.4)393.1.2 Scheduling ParameterAlthough most SI IC engines can run at up to 6000rpm and wide open throttle,they spend very little time in that range. The normal use operating range of mostSI IC engines is 800rpm to 3500rpm with an air flow of 10% to 50% of maxi-mum, wide open throttle. In order to be comparable with the previous literature,which consider only the normal operating range of the engine, the controller inthis section is also designed for only the normal operating range.The proposed gain-scheduling controller adjusts for both engine speed, N,and air flow, ˙mair. Noting that engine speed and air flow enter the parametersof the FOPDT plant in the denominator in (2.16), (2.17), and (2.18), we select thescheduling parameter, q, asq =264 q1q2375=264 1˙mair1N375: (3.5)3.1.3 LPV Controller DesignThe controller design method seeks to find a controller of the form˙xK = AK(q; ˙q)xK +BK(q; ˙q)y;u = CK(q; ˙q)xK +DK(q; ˙q)y;(3.6)for the LPV plant expressed in (3.1). The plant’s dependence on the schedulingparameter, q, should be mimicked by a function, denoted by r( ), which enters the40plant affinely. For the LPV plant described by (3.4) and the choice of schedulingparameter given in (3.5), r( ) can simply be selected as r1(q)= q1 and r2(q)=q2. The time-varying Lyapunov variables then also mimic the plant’s dependenceon q and are given byX = X0 +r1(q)X1 +r2(q)X2;Y = Y0;(3.7)which yields a practically valid solution. This implies that a measurement of ˙q isnot required to reconstruct the controller’s state space matrices in (3.6) from theLyapunov variables which are solved using a family of LMIs.For comparison, the LPV-based controller presented in [47] is also developed.The controller design technique is identical to that presented above but does notinclude the air flow, ˙mair, in the scheduling parameter, q. Therefore, q = 1N , whichmeans that r(q)=q. The Lyapunov function in (3.7) becomesX = X0 +r(q)X1;Y = Y0:(3.8)The varying steady state gain of the plant is addressed by multiplying the con-troller’s output by the air flow as shown in Figure 3.1(a).The LPV controllers developed in this section are referred to as LPV Con-troller 1 and 2 in the results section of this thesis (Chapter 4). LPV Controller 1is scheduled for only engine speed as discussed in the above paragraph while41LPV Controller 2 is scheduled for both engine speed and air flow.3.2 Switching LPV ControllerIn this section the primary contribution of the current research, a switching LPV-based gain-scheduling controller, is presented.2 The present method is a con-tinuation of that used in Section 3.1 and reduces conservatism by applying theswitching LPV controller design methods developed in [23] to the air-fuel ratiocontrol problem in order to compensate for time-varying engine speed and airflow. The proposed method divides the engine’s operating range into multiplesubregions. Each subregion has its own parameter dependent Lyapunov func-tion used to create an LPV controller for that subregion. The controllers are thenswitched in real-time so that the controller for the subregion that encompasses thecurrent operating point is always active. By overlapping the subregions, hysteresisis introduced to the switching, thereby making it possible to guarantee switchingperformance by ensuring that the Lyapunov function decreases for any switchingevent. As well as being more aggressive, the switching LPV controller developedin this paper is able to cover the entire operating range of the engine while stillproviding a theoretical guarantee of performance.The following five controllers are developed using the method presented inthis section. The controllers’ performance is compared in Chapter 4.2The Switching LPV-based gain-scheduling controller developed in this section has been sub-mitted for publication in the IEEE Transactions on Control Systems Technology.42S-LPV Controller 1: Single region LPV controller.S-LPV Controller 2: Switching LPV controller with two subregions(Switched along the axis of N).S-LPV Controller 3: Switching LPV controller with two subregions(Switched along the axis of ˙mair).S-LPV Controller 4: Switching LPV controller with four subregions.S-LPV Controller 5: Switching LPV controller with nine subregions.3.2.1 Linear Parameter Varying Form of PlantLike the LPV-based controller, the switching LPV-based gain-scheduling con-troller design process requires the open-loop plant expressed in an LPV form as˙x = A(q)x+B1(q)w+B2(q)u;z = C1(q)x+D11(q)w+D12(q)u;y = C2(q)x+D21(q)w;(3.9)where q is a time-varying vector known as the scheduling parameter and x is thestate vector of the LPV plant. A parameter dependent controller, also a functionof the scheduling parameter, will be found such that it minimizes the worst-caseL2-gain from the input, w, to the performance channel, z, and has the form˙xK = AK(q)xK +BK(q)y;u = CK(q)xK +DK(q)y;(3.10)43Figure 3.3: Closed-Loop System With Switching LPV Controllerwith controller states xK. The controller takes the measurement, y, and givesoutput, u, where y is the integral of the error as can be seen in Figure 3.3 and u isthe fuel flow, ˙m f uel. A disturbance, d, is modeled on the output of the plant. Theexogenous input and the performance channels arew =264 dr375 and z =264 ˜e˜u375: (3.11)Second order weighting functions We and Wu are included on the error signaland the controller output, respectively, to produce the performance channels, ˜e and˜u, and allow controller tuning. The weighting functions, Wn, where n :=fe;ug,are realized using the state-space form,˙xn = Anxn + Bnn;˜n = Cnxn + Dnn;(3.12)44with the state vector xn. The weighting function parameters are chosen suchthat low frequency errors and high frequency controller outputs are prevented.Table 3.1 shows values for the weighting functions used in this thesis.Table 3.1: Switching LPV Controller Design Weighting FunctionsWe WuLow Frequency Gain 10 103High Frequency Gain 104 10 1Crossover Frequency 5 103rad=sec 105rad=secThe states, controller input, and controller output of the LPV system arex =266666664xpxexuR e377777775; y =R e; and u; (3.13)45respectively. The state space matrices in (3.9) areA =266666664Ap 0 0 0 BeCp Ae 0 00 0 Au 0 Cp 0 0 0377777775; B1 =2666666640 0 Be Be0 0 1 1377777775; B2 =266666664Bp0Bu0377777775;C1 =264 0 Ce 0 00 0 Cu 0375; D11 =264  De De0 0375; D12 =264 0Du375;C2 = 0 0 0 1 ; D21 = 0 0 :(3.14)3.2.2 Scheduling ParameterAlthough not explicitly shown in (3.14), the plant is parameter-varying due to thedependence of Ap, Bp, and Cp on the parameters g, t and T which are, in turn,dependent on the engine speed and air flow as shown in Section 2.4. The enginespeed can vary from 800rpm to 6000rpm and the air flow can vary from 10% to100%. Air flow is normalized and given as a percentage of maximum flow in thisthesis. The engine speed and air flow ranges are collectively referred to as theengine’s operating range. Furthermore, the rate of change of the operating pointcan be up to 6000rpm per second for the engine speed and 100% per second for theair flow. Noting that air flow, ˙mair, and engine speed, N, enter (2.16), (2.17), and46(2.18) in the denominator, we select the scheduling parameter q asq =264 q1q2375=264 1˙mair1N375: (3.15)Although the LPV controller, presented in the previous section, is only designedfor the normal use operating range of the engine, the added performance of theswitching LPV controllers allows them to remain effective over the full operatingrange. From the limits on the operating range of the engine, we can calculatethe bounds on the scheduling parameter as well as the bounds on its derivative,denoted by Q and Qd, respectively:Q =n(q1;q2)j 1100% < q1 < 110%; 16000rpm < q2 < 1800rpmo;Qd =n( ˙q1; ˙q2)j 100%=sec(10%)2 < ˙q1 < 100%=sec(10%)2 ;  6000rpm=sec(800rpm)2 < ˙q2 < 6000rpm=sec(800rpm)2o:(3.16)3.2.3 Switching LPV Controller DesignA detailed review of the LPV controller design method used is given in Appendix A.The parameter space, Q, is divided into R overlapping subregions Q(r);r = 1;:::;R.A superscript in parenthesis refers to the subregion index in this thesis. Themethod uses a separate parameter dependent Lyapunov function to provide an47LPV controller for each subregion with guaranteed performance of the closed-loop plant within the limits Q(r) and Qd. The controllers are then switched de-pending on which subregion the scheduling parameter, q, belongs to at each pointin time. In order to guarantee switching performance, it must be ensured that theLyapunov function decreases at each switching event. In other words, when thescheduling parameter moves from one subregion to another, the Lyapunov func-tion defined within the subregion it is leaving must be greater than the Lyapunovfunction defined within the subregion it is entering. By increasing the overlap ofthe subregions, hysteresis is increased and this constraint becomes easier to meet.3The dependence of the Lyapunov function on the scheduling parameter, q, ismimicked by a set of functions, r( ), which are differentiable but not necessarilylinear functions of the scheduling parameter. The function r( ) is introduced andexplained in greater detail in [40]. For the plant given in (3.14), r( ) is simplyr( )=264 r1( )r2( )375=264 q1q2375 (3.17)and is used to define the LMI variables. For example, the parameter dependentLyapunov variables areX = X0;Y(r)(q) = Y(r)0 +r1(q)Y(r)1 +r2(q)Y(r)2 :(3.18)3Figure A.1 in Appendix A shows the use of subregion overlap to create hysteresis switching.48Note that X is a constant while Y(r)(q) is a function of the scheduling parameter.A practically valid controller (See Appendix A) can also be designed with X(r)(q)as a function of the scheduling parameter and Y held constant, but yields moreconservative results in our problem setting. Because no analytical method exists tochoose which Lyapunov variable to fix, both options must be tried and comparedin order to find the least conservative resultant controller.(a) R = 2 (S-LPV Controller 2) (b) R = 2 (S-LPV Controller 3)(c) R = 4 (S-LPV Controller 4) (d) R = 9 (S-LPV Controller 5)Figure 3.4: Operating Space Divided Into R Subregions49The five controllers developed using the LPV controller design method are re-ferred to as S-LPV Controller 1 to 5 throughout this thesis. S-LPV Controller 1uses one subregion to cover the entire operating range of the engine, Q(1) =Q,and therefore does not actually involve any switching. For S-LPV Controller 2and S-LPV Controller 3, the parameter space is divided into two subregions asshown in Figure 3.4(a) and Figure 3.4(b), respectively. The shaded areas denotethe overlap of the subregions. Recall that the parameter space is given in q, whichis actually the inverse of engine speed and air flow as shown in (3.15). However,graphical representations of the subregions in this thesis use the axes air flow andengine speed rather than q1 and q2 because their meanings are more intuitive.S-LPV Controller 4 and S-LPV Controller 5 are developed for four subregionsand nine subregions shown in Figure 3.4(c) and Figure 3.4(d), respectively.The benefit of increasing the number of subregions is that the parameter vari-ation within each region becomes smaller, consequently reducing the variation ofthe Lyapunov function and yielding a more aggressive controller for each sub-region. The drawback, however, is the added memory and time requirements tosolve and store the increasing number of LMI variables. Table 3.2 shows the num-ber of LMIs and LMI variables used for each switching LPV controller as well asthe time taken to solve for the LMIs yielding the controllers presented. Computa-tions were performed on the same computer and under similar conditions so thattimes can be meaningfully compared.50Table 3.2: Complexity and Computing Time of Switching LPV ControllersName No. of No. of LMI Time toSubregions LMI’s Variables SolveS-LPV Controller 1 1 32 17 1min 12secS-LPV Controller 2 2 68 32 1min 38secS-LPV Controller 3 2 68 32 1min 38secS-LPV Controller 4 4 144 62 7min 0secS-LPV Controller 5 9 336 137 58min 37sec3.3 H¥ ControllerTwo H¥ controllers are developed for comparison with the LPV and switching LPVcontrollers using the LPV plants defined by (3.4) and (3.14), respectively, withLMI techniques described in [13]. Unlike the LPV and switching LPV controllersdeveloped in the last section, the usual H¥ controller design method assumes atime-invariant plant and produces a single time-invariant controller.The H¥ controllers in this paper are modified to compensate for variations ing, the parameter dependent steady state gain of the system, using the followingmethod: First the values of t and T are sampled at a nominal engine speed and airflow and the steady state gain is given the value g = 1. A time-invariant H¥ con-troller for the plant with these values is then generated using the method in [13].A parameter dependent gain is placed on the output of the controller. This gainis equal to the inverse of the steady state gain of the time-varying plant and iscalculated in real-time. In this way, the parameter dependent gain on the outputof the controller cancels out the time-varying steady state gain of the plant.Figure 3.5(a) shows the closed-loop system with the parameter dependent51(a) Gain-Scheduled H¥ Controller(b) LPV or Switching LPV ControllerFigure 3.5: Comparison of H¥ Controller Scheduling With LPV Controllerplant, time-invariant controller, and parameter dependent gain. Figure 3.5(b)shows the equivalent closed-loop system with a parameter dependent LPV or switch-ing LPV controller for comparison. Since the gain-scheduled H¥ controller doesnot adjust for variations of the time constant, t, or pure delay, T , performancecannot be guaranteed at any operating point other than the nominal point that it isdesigned for.The first H¥ controller, developed for comparison with the LPV controllersof Section 3.1, is designed for a nominal operating point of 30% air flow and1500rpm engine speed. The controller is carefully tuned such that the closed loopwill remain stable over the normal operating range of 10% to 50% air flow and800rpm to 3500rpm engine speed because this is the operating range that the LPVcontrollers are designed for. It is important to keep in mind that, though the H¥52controller is tuned to work over the operating range, no guarantee of performanceexists. The second H¥ controller, which is developed for comparison with theswitching LPV controllers of Section 3.2, is designed for a nominal operatingpoint of 80% air flow and 4000rpm engine speed and is tuned to remain effectiveover the entire operating range of the engine.3.4 Adaptive ControllerAs an alternative to the gain-scheduling approach to the air-fuel ratio problem, anadaptive controller is also developed. The appeal of the adaptive approach is thatno a priori knowledge of the parameter variations of the plant is required, meaningthat the development of an accurate model of the plant is not necessary. Thecontroller identifies the plant dynamics in real-time using a recursive least squaresalgorithm and provides the appropriate control input, needing only measurementsof the plant’s input and output. Furthermore, the adaptive controller is able tocompensate not only for variations in the dynamics due to engine speed and airflow, but also for variations due to engine and sensor aging and environmentalfactors.An adaptive predictive controller using a Laguerre network is chosen for thisapplication.4 The controller design method is introduced in [43] and has had muchsuccess in controlling pH levels in an industrial bleach plant extraction stage aswell as other systems with large delays. The authors develop a model predictivecontroller which looks beyond the delay time in the plant. An attractive feature of4The Laguerre network is discussed in greater detail in Section B.1 in the Appendix.53using a Laguerre network to model the plant is that no assumptions about the plantorder or time delay are required, making it ideal for the air-fuel ratio control prob-lem since the dynamics of the plant are dominated by a time-varying delay. Themethod is improved in [46] where it is given the name Incremental Mode LinearLaguerre Predictive Control (IMLLPC) and is modified by inserting an integratingaction into the controller to provide better tracking performance. Necessary andsufficient conditions for stability and steady-state performance are also presentedbut are useful for analysis only and are not used during the design stage.The detailed control algorithm is reviewed in Appendix B and is not presentedhere for brevity. Table 3.3 shows values that act as the tuning parameters and areset by the designer before creating the controller. The values were found mostlythrough an iterative tuning procedure.Table 3.3: Tuning Parameters of the IMLLPC AlgorithmSymbol Description ValueTs Sampling time 0:1secp Time-scale 60N Order of Laguerre network 8d Prediction horizon 7M Control horizon 1Q Weighting matrix on error Id dR Weighting matrix on control input IM Ma Softening factor on reference model output 0.2l Forgetting factor for recurser 0.96In order to simulate the adaptive controller’s performance, a discrete-time54model of the plant is required. We use a state-space model of the formx[k+1] = Apx[k] + Bp ˙m f uel[k nT];f[k] = Cpx[k] + Dp ˙m f uel[k nT];(3.19)where nT is the delay time of the plant, T , expressed in number of sample peri-ods, nT = TT s, and Ap, Bp, Cp, and Dp are matrices of the discretized state-spacerealization of the transfer functionf˙m f uel =gst+1; (3.20)which is simply the first order component of the FOPDT model given in (2.19) inChapter 2. This model is then used to simulate the closed-loop of the system anditeratively tune the controller. The controller’s closed-loop simulation results areshown and discussed in Chapter 4.55Chapter 4Simulation ResultsIn this chapter the controllers are validated and their reference tracking and dis-turbance rejection performances are compared using simulations performed usingMATLAB and Simulink software. The chapter is divided into three sections. InSection 4.1, the LPV controllers are compared with each other as well as an H¥controller in order to support the decision to consider both engine speed and airflow in the scheduling parameter. In Section 4.2, the performance improvementsof the switching LPV controllers are shown. Finally, the performance of the adap-tive controller is compared with that of a switching LPV controller in Section 4.3.Each section is further divided into two subsections titled Time-Invariant Sim-ulations and Time-Varying Simulations. In the former, the dynamic model of theplant is sampled and held constant for the duration of each simulation, conse-quently producing a time-invariant plant. The time-invariant simulations are usedto compare the response time of the controllers at different points within the en-56Figure 4.1: Comparison of Pure Delay and Pad´e Approximationgine’s operating range. During the time-varying simulations, on the other hand,the engine speed and air flow follow realistic trajectories while the controllers areperforming reference tracking and disturbance rejection. The performance of thecontrollers is thereby validated for realistic operating conditions.Although the plant includes a Pad´e approximation during controller design, anon-rational pure delay is employed during validation. Figure 4.1 compares theeffect of a Pad´e approximation and a pure delay on the step response at an enginespeed of 800rpm, and an air flow of 50%. Both curves are acceptably similar,validating the choice of order of the Pad´e approximation given in Chapter 2.5.This holds true at all engine operating conditions.RemarkIn order to ensure stability of the controller, which was designed for a plant con-taining a Pad´e approximation, with a plant containing a pure delay, either of two57methods can be used.The first method is to place a low-pass filter on the output of the controller.The bandwidth of the filter should be equal to or less than the frequency at whichthe phase of the Pad´e approximation’s frequency response plot starts differingfrom that of the actual pure delay. Because the pure delay, T , is time-varying thisfrequency will vary also. Either a low-pass filter with a time-varying bandwidthor a time-invariant filter, with a bandwidth that corresponds to the worst case ofthe delay (which happens at 800rpm and 10% air flow) can be used. Addition ofthe filter effectively modifies the FOPDT plant’s frequency response so that it issimilar to that of an FOPDT approximated as a rational transfer function.An alternative method is to use the weighting function on the controller outputto tune the controller during controller design such that high frequency controlleroutputs are avoided. The controller should not output signals with any frequen-cies higher than the frequency where the phase of the Pad´e approximation startsdiffering from that of the pure delay.Both methods have been attempted and have yielded similar results. Resultsshown in this chapter are generated using aggressively tuned controllers with time-invariant low pass filters on their outputs.4.1 LPV ControllersThe H¥ controller developed in Section 3.3 and LPV Controller 1 and LPV Con-troller 2, both developed in Section 3.1, are compared in this section.1 The pur-1The results presented in this section have been published in [28] and presented at the2010 ASME Dynamic Systems and Control Conference.58(a) 800rpm and 50% Air Flow (b) 3500rpm and 50% Air Flow(c) 800rpm and 10% Air Flow (d) 3500rpm and 10% Air FlowFigure 4.2: Disturbance rejection of H¥ Controller (Dashed), LPV Con-troller 1 (Thin), and LPV Controller 2 (Thick) at Fixed OperatingPointspose of comparing these controllers is to show the performance improvementachieved by scheduling the controller for both engine speed and air flow.4.1.1 Time-Invariant SimulationsFigure 4.2 displays the system’s response to a 10% step disturbance applied attime t = 1sec while operating at each of the four extremes of the operating range.Figures 4.2(a) and 4.2(c) show that both LPV Controller 1 and LPV Controller 2have comparable results with the H¥ controller at low engine speed, when theopen-loop plant is at its slowest. At high engine speeds, however, the LPV con-trollers adjust for the faster plant, yielding a much faster closed-loop response59than that of the fixed controller as can be seen in Figures 4.2(b) and 4.2(d). Fig-ure 4.2(d) shows that LPV Controller 1, which does not compensate for variationsin air flow, is too aggressive for the plant at 10% air flow while Figure 4.2(b)shows it to be slow for the plant at 50% air flow. LPV Controller 2, which isscheduled for both air flow and engine speed, maintains acceptable responses forboth cases.4.1.2 Time-Varying Simulations0 10 20 30 40 50 60 70 80 9001000200030004000Time (sec)Engine Speed (RPM)(a) Engine Speed0 10 20 30 40 50 60 70 80 900102030405060Time (sec)Air Flow (%WOT)(b) Air FlowFigure 4.3: Engine Speed and Air Flow Profiles60(a) Disturbance Rejection(b) Reference TrackingFigure 4.4: Performance of H¥ Controller (Dashed), LPV Controller 1(Thin), and LPV Controller 2 (Thick) With Time-Varying Plant Dy-namicsIn order to demonstrate the system’s stability and performance while the en-gine operating point is changing, realistic profiles for engine speed and air floware used. Figure 4.3 shows the engine speed and air flow over a 90sec simulation.The profiles represent normal engine usage and encompass all four extremes of61the normal use operating range. In Figure 4.4(a) a disturbance having the shapeof a square wave with a period of 30sec and an amplitude of 10% of the referenceis applied to the system. A square reference signal varying from 1 to 1:1 witha period of 30sec is applied to the system in Figure 4.4(b). LPV Controller 2,compensated for both air flow and engine speed, shows some improvement overLPV Controller 1, and very significant improvement over the H¥ controller inboth reference tracking and disturbance rejection performance.4.2 Switching LPV ControllersThe switching LPV controllers developed in Section 3.2 as well as an H¥ controllerdeveloped in Section 3.3 are evaluated in this section.2 The purpose of comparingthese controller is to show the improvement in performance which can be achievedby dividing the operating space into subregions and to demonstrate that stabilityis maintained even for rapidly moving operating points.Recall that the naming convention of the switching LPV controllers is:S-LPV Controller 1: Single region LPV controller.S-LPV Controller 2: Switching LPV controller with two subregions(Switched along the axis of engine speed, N).S-LPV Controller 3: Switching LPV controller with two subregions(Switched along the axis of air flow, ˙mair).S-LPV Controller 4: Switching LPV controller with four subregions.S-LPV Controller 5: Switching LPV controller with nine subregions.2The results presented in this section have been submitted for publication in the IEEE Trans-actions on Control Systems Technology.624.2.1 Time-Invariant SimulationsThe controllers are tested at nine different points within the operating range ofthe engine in this section. Because the scheduling parameter, q, remains fixedthroughout the duration of each simulation, the parameter dependent plant isalso time-invariant during each simulation. A step disturbance is given at timet = 1sec and the response is plotted. The responses of the H¥ controller and ofS-LPV Controller 1 can be seen in Figure 4.5. At higher engine speed and airflow, the LPV controller takes advantage of the plant’s faster open-loop responseand yields improved closed-loop response while the H¥ controller’s response timeremains similar throughout the operating range. Both controllers have poor andoscillatory responses at low engine speed and air flow when the plant dynamicsFigure 4.5: Disturbance Rejection of H¥ Controller (Dashed) andS-LPV Controller 1 (Solid)63Figure 4.6: Disturbance Rejection of S-LPV Controller 2 (Dashed) andS-LPV Controller 3 (Solid)Figure 4.7: Disturbance Rejection of S-LPV Controller 4 (Dashed) andS-LPV Controller 5 (Solid)64are very slow.Figure 4.6 shows the closed-loop responses of S-LPV Controllers 2 and 3, bothhaving two subregions. S-LPV Controller 3 is switched along the air flow axis andyields improved disturbance rejection at all operating points. This demonstratesthe importance of carefully choosing the subregions for the switching controller.Intuitively, the subregions should be chosen such that the open-loop plant’s vari-ation due to the movement of the scheduling parameter is minimum within eachsubregion. An analytical method for finding the optimal subregions is beyond thescope of this research, however. Recall that the subregions used for the switch-ing LPV controllers are given in Figure 3.4 in Section 3.2 and have been chosenthrough trial and error.Figure 4.7 shows the disturbance rejection of S-LPV Controllers 4 and 5, hav-ing four and nine subregions, respectively. Due to the decreased parameter vari-ation within each subregion, these two controllers outperform the single regionLPV and two subregion switching LPV controllers at all operating points. The per-formance improvement of S-LPV Controller 5 compared to S-LPV Controller 4 issmall while it requires a considerable increase in memory and computation timeas shown in Table 3.2 in Section 3.2. S-LPV Controller 4, having four subregions,is therefore recommended as the controller of choice for this particular plant.4.2.2 Time-Varying SimulationsAn engine speed and air flow profile representing realistic engine use are em-ployed to demonstrate the system’s stability and performance while the engine650 10 20 30 40 50 60050100Time [sec]Air Flow [%]0 10 20 30 40 50 600200040006000Time [sec]Engine Speed [rpm]Figure 4.8: Air Flow and Engine Speed Profilesoperating point is varying. Figure 4.8 shows the engine speed and air flow overthe 60sec simulation. Idling, “revving”, high load, and engine breaking are allrepresented in the engine profiles. The resulting trajectory of the schedulingparameter within the operating space is depicted in Figure 4.9. The derivativeof the scheduling parameter remains within in the bounds Qd, set in (3.16) inSection 3.2. The H¥ controller, S-LPV Controller 1, and S-LPV Controller 4 arecompared using the time-varying simulation. The switching signal generated forS-LPV Controller 4 can be seen in Figure 4.10 and confirms that the parametertrajectory passes through all four subregions.660 1000 2000 3000 4000 5000 60000102030405060708090100Engine Speed [rpm]Air Flow [%]Figure 4.9: Engine Operating Point Trajectory0 10 20 30 40 50 601234Time [sec]Switching VariableFigure 4.10: Time-Varying Switching Variable67(a) Disturbance Rejection(b) Reference TrackingFigure 4.11: Performance of H¥ Controller (Dashed), S-LPV Controller 1(Thin), and S-LPV Controller 4 (Thick) With Time-Varying PlantDynamics68The disturbance rejection of the controllers with the time-varying plant is seenin Figure 4.11(a). A constant reference equivalence ratio, r = 1, is given whilea square-wave shaped disturbance with 10% amplitude and a period of 20sec isapplied. S-LPV Controller 4 provides reduced overshoot and improved settlingtime. The large overshoot at time t = 12sec and spike at time t = 47sec are dueto fast parameter movements at those times. Figure 4.11(b) shows the referencetracking performance of the controllers, given a square-wave shaped referencesignal, represented by the dotted line. Once again, S-LPV Controller 4 achievesreduced overshoot and improved settling time.4.3 Adaptive ControllerThe simulation results of the adaptive controller obtained in Section 3.4 are pre-sented in this section. Whereas the H¥, LPV, and switching LPV controllers aredeveloped in continuous time, the adaptive controller is developed and simulatedcompletely in discrete time domain. The discrete time state-space model of theFOPDT plant given in (3.19) is used for simulations and iterative tuning.4.3.1 Time-Invariant SimulationsIn order to demonstrate the adaptive controller’s adaptation capability, simulationsare first performed with the plant dynamics remaining fixed. At the start of thesimulation, at t = 0, the estimate vector, known a the spectral gain, is near zero. Anon-zero, initial estimate is selected because a zero-estimate results in divergence.An explanation of the adaptive controller design technique, including the Laguerre69Figure 4.12: Adaptive Controller Reference Tracking for Fixed OperatingPointsnetwork and the definition of the spectral gain, is given in Appendix B. As thesimulation progresses, the recursive least squares algorithm improves the estimateand the internal model of the plant becomes more accurate.Figure 4.12 shows the tracking performance of the controller at four constantoperating points of the engine. At low engine speed and air flow, when the plantdynamics are at their slowest, the adaptation rate appears to be very slow and thereference tracking performance only becomes acceptable around t = 40sec. Wheneither or both engine speed and air flow are high, on the other hand, the adaptationrate is fast, causing the controller to show acceptable reference tracking within thefirst 10sec. Recall that no a priory information of the plant dynamics is availableto the controller, meaning that it is completely ‘learning’ the dynamics of the plant700 10 20 30 40 50 600246Dead Time in Sample Periods  ActualRounded Up0 10 20 30 40 50 6000. Constant (sec)0 10 20 30 40 50 6000. State GainTime (sec)Figure 4.13: Time-Varying FOPDT Parametersduring the simulation.4.3.2 Time-Varying SimulationsEngine speed and air flow profiles representing realistic engine use are once againemployed to validate the system’s stability and performance while the engine op-erating point is varying. The engine speed and air flow profiles used are identicalto those used for the switching LPV controller in Section 4.2.2 and are shown in710 10 20 30 40 50 600.750.80.850.90.9511. [sec]Equivalence Ratio  Switching LPVAdaptiveReferenceFigure 4.14: Tracking Performance of S-LPV Controller 4 (Thin) and theAdaptive Controller (Thick)Figure 4.8. The resulting FOPDT parameter values, t, T , and g, can be seen inFigure 4.13. Because a discrete time model is used to simulate the FOPDT plant,the time delay must be expressed in terms of sample periods, Ts, meaning thatit cannot be continuously variable. In order to create a conservative model, thedelay is always rounded up. With the sampling interval of Ts = 0:1sec used inthese simulations, the resultant variation of the delay is between two and six sam-pling intervals as seen in Figure 4.13. Using a smaller sampling interval reducesrounding errors but increases the order of the discrete time model.72Figure 4.14 shows the tracking performance of the adaptive controller com-pared with that of the recommended switching LPV controller using four sub-regions (S-LPV Controller 4). A reference signal around unity with a period of20sec is given as represented by the dotted line. The adaptive controller exhibitsvery poor reference tracking performance between 10sec and 20sec into the sim-ulation, again between 30sec and 40sec, and once again around 50sec. At each ofthese times, the engine’s operating point is rapidly moving, as can be seen fromthe air flow and engine speed profiles in Figure 4.8, causing very fast changes inthe plant dynamics. Because the dynamics are changing fast, the adaptive con-troller is unable to keep up, fails to accurately track the reference, and may evenbecome unstable. Using a smaller sampling interval may improve the estimationrate and mitigate this problem but, as has already been stated, increases the or-der and the complexity of the model and adaptive controller. Further research isrequired to improve the adaptation rate and robustness of the adaptive controller.73Chapter 5Conclusions5.1 SummaryThe objective of this research was to develop an air-fuel ratio controller techniquefor the Spark Ignition (SI) Internal Combustion (IC) engine which appropriatelyaddresses the engine’s non-linear and parameter-varying nature as well as its timedelays and provides guaranteed closed-loop performance over the engine’s entireoperating range.A detailed model of the air-fuel ratio dynamics was developed in Chapter 2.The model demonstrated that the dynamics are non-linear and depend on enginespeed, air flow, and manifold pressure, which is related to air flow. Time delaysin the system were examined, and are also dependent on engine speed and airflow. The detailed model was not used for controller design purposes. Instead, awidely accepted simplified model of the engine dynamics, known as a First Order74Plus Dead Time (FOPDT) was used. The FOPDT model approximates the com-bined dynamics as a first order transfer function coupled in series with a puredelay. The model can be completely defined using only three time-varying pa-rameters: a steady state gain, g, first order time constant, t, and delay time, T .These three parameters were shown to be functions of the engine speed and airflow into the cylinders. An estimate of the air flow into the cylinders was assumedto be available. Methods that successfully estimate the air flow into the cylindershave been demonstrated in prior literature. By selecting air flow as a schedulingparameter, rather than an input to the plant, the engine model was expressed as apseudo-linear but parameter-dependent transfer function.Four different controller design techniques were used to develop air-fuel ra-tio feedback controllers to track reference air-fuel ratios and reject disturbancesin Chapter 3. A novel air-fuel ratio controller using switching Linear ParameterVarying (LPV) controller design techniques was introduced. In this method, theoperating range of the engine is divided into multiple subregions, and an LPV con-troller which guarantees performance within its respective subregion using Lya-punov stability theory is developed for each subregion. Switching stability is alsoguaranteed. The switching LPV controller takes both air flow and engine speed asthe scheduling parameter and provides a guarantee of performance over the oper-ating range of the engine. Three other design techniques, H¥ control, LPV-basedcontrol, and adaptive control, were also used to develop controllers for compara-tive purposes.Simulations performed in Chapter 4 were used to demonstrate the switching75LPV controller’s performance. Firstly, the results were used to validate the choiceof air flow and engine speed as the scheduling parameter. An LPV controllerscheduled for both engine speed and air flow was compared against an LPV con-troller which was compensated for only engine speed as well as an H¥ controller.The LPV controller scheduled for air flow and engine speed outperformed both theLPV controller scheduled for only engine speed and the H¥ controller. Secondly,switching LPV controllers using two, four, and nine subregions were comparedagainst an LPV controller and an H¥ controller. Simulations confirmed that usinga greater number of subregions, resulting in smaller parameter variations withineach subregion, reduced the controller’s conservativeness and therefore improvedthe closed-loop performance.Finally, the performance of a novel adaptive air-fuel ratio controller was com-pared with that of the recommended switching LPV controller. Simulations showedthat the adaptive controller could not adapt to the rapid changes in the engine’soperating point fast enough to be competitive with the switching-LPV controller.This is attributed to the fact that the switching LPV controller, like any gain-scheduled controller, uses a priory information on the parameter-dependent dy-namics of the plant to compensate for changes in the operating point instanta-neously while the adaptive controller needs to dynamically estimate the plant dy-namics in real-time.765.2 ContributionsThe main contribution of the current research is the development of a switchingLPV air-fuel ratio controller. Although the controller was developed specificallyfor a port injected, normally aspirated, SI IC engine in this research, it can beapplied to air-fuel ratio control problems in other engines, including forced induc-tion, lean burn, and direct injection, as well. The contributions of this thesis areoutlined below: A detailed model of the SI IC engine’s air and fuel dynamics is developedand confirms the engine’s non-linear nature and dependency on parameterssuch as air flow, engine speed, and manifold pressure. The delays that the four-stroke engine introduces to the air and fuel flowdynamics are considered in detail. Previous authors have assumed both theair and fuel dwell times to be equal to the time that it takes the engineto complete three strokes: induction, power, and exhaust. Because fuel isinjected between one and three strokes before the intake valve opens, thefuel dwell time is actually between four and six strokes. By taking air flow as a time-varying parameter rather than an input to theplant, the non-linear combustion equation is expressed in pseudo-linear butparameter-dependent form. The FOPDT model of the engine is improved by including a parameter-dependent low frequency gain. The low frequency gain is derived from77the combustion equation and is a function of the air flow. The effect of air flow on the delay in the FOPDT model is considered andcompensated for. The air-fuel ratio control problem is expressed in an LPV form and theswitching controller design technique developed in [5] and [23] is applied.The resultant switching-LPV controller has reduced conservativeness, com-pared to previous methods, and guarantees L2 performance over the entireoperating range of the engine.5.3 Future WorkThe air-fuel ratio control problem in IC engines is a mature problem. However,it is evident, even from related papers published in 2010 only, that the air-fuelratio continues to benefit from recent and increasingly advanced controller designtechniques. In future work, the research presented in this thesis can be expandedin four areas discussed below.State-Delayed Switching LPV ControllerAt the same time that the current research was improving on the LPV-based air-fuel ratio controller presented in [47] by applying switching LPV techniques, theauthors of [47] were improving the method and introducing a state-delayed LPV-based controller in [48]. The benefit of using a state-delayed LPV controller overthe LPV controller is that the delay in the plant does not need to be modeled by78a Pad´e approximation. Using a state-delayed LPV controller can also result inclosed-loop performance improvements. The same method used to express theLPV plant as a state-delayed LPV plant in [48] can be applied to the current re-search, and a state-delayed switching LPV controller can be produced.Optimal Selection of SubregionsIn Chapter 3, it is mentioned that the division of the engine’s operating space intosubregions was found through trial and error. Considerable improvements can bemade to the process of dividing the operating range into subregions. An optimalselection of subregions does exist and can be found analytically.Implementation on Physical IC EngineThe recommended switching LPV controller was simulated on a realistic FOPDTmodel of the SI IC engine. Once the required hardware is available, a real ICengine’s dynamics can be identified and the controller can be implemented.Improvement to Adaptive Controller PerformanceFinally, the novel adaptive air-fuel ratio controller developed in Chapter 3 as analternative to gain-scheduling lacked the performance to compete with the switch-ing LPV controller. Additional attention to the controller design and tuning methodmay greatly improve the controller’s adaptation rate and consequently its closed-loop performance.79Bibliography[1] D. Alberer, M. Hirsch, and L. del Re. A virtual references design approachfor diesel engine control optimization. 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Robust fuel-ing strategy for an SI engine modeled as an linear parameter varying time-delayed system. In Proceedings of the 2010 American Control Coference,Baltimore, Maryland, July 2010. !pages 7, 11, 33, 78, 7985Appendix ASwitching LPV Controller SynthesisWith Guaranteed L2 PerformanceIn this appendix, a method of designing a switching LPV controller with guar-anteed L2 performance that is utilized to design the gain-scheduled air-fuel ratiocontrollers throughout Section 3.2 is presented. The method is a combination ofresults in [5] and [23]. Although the body of the thesis addresses the developmentof a switching LPV controller specifically for air-fuel ratio regulation of SI IC en-gines, the method described in this appendix is not specific and can be applied toany plant which can be expressed in an LPV form.86A.1 Controller Design ProblemConsider a plant expressed in an LPV form as˙x = A(q)x+B1(q)w+B2(q)u;z = C1(q)x+D11(q)w+D12(q)u;y = C2(q)x+D21(q)w;(A.1)where x is the state vector, w is the disturbance input, u is the controlled input,z is the performance channel, and y is the measured output of the plant, and allthe matrices have compatible dimensions. The scheduling-parameter, q, is a col-umn vector containing measurable time-varying parameters that affect the plantdynamics. The bounds on q as well as ˙q, its rate of change, must be known andare given asq(t)2Q; 8t 0;˙q(t)2Qd; 8t 0;(A.2)where Q and Qd are hyperrectangles defining the space of the scheduling param-eters and their derivatives, respectively. The parameter space Q is divided into Roverlapping subregions Q(r); r = 1;:::;R. The superscript in parenthesis refers tothe index of a subregion. Variables with a superscript, r, are only defined withinthe subregion Q(r) while variables without it are shared throughout the parameterspace Q.87An LPV controller of the form˙xK = A(r)K (q; ˙q)xK +B(r)K (q; ˙q)y;u = C(r)K (q; ˙q)xK +D(r)K (q; ˙q)y;(A.3)is connected to the LPV plant. A set of controller matrices is designed for eachsubregion and is switched depending on the value of q as discussed below.A.2 Switching VariableAt any given time, the scheduling parameter, q, falls within one of the subregions,Q(r), of the parameter space, Q. We let Q(s) denote this subregion where s is apiecewise-constant function, s(t)2f1;2;:::;Rg;8t 0, and is called the switch-ing variable. Whenever q leaves the current subregion, Q(s), a switching eventoccurs and the value of s changes to the index of the subregion that q is enter-ing. Because switching only occurs when q leaves its current subregion, lettingsubregions overlap introduces hysteresis. Figure A.1 explains hysteresis switch-ing graphically and shows two arbitrary overlapping and neighboring subregionsincluding an example trajectory of q that crosses the two switching surfaces. Thepoints where switching events occur are marked with crosses. The overlap of thetwo subregions is shaded. At time t = 0, q falls within the subregion Q(i) andtherefore s = i. When the trajectory of q passes through the switching surface,S(i j), switching occurs and the value of s becomes j. Switching occurs again onlyonce the trajectory passes through the switching surface, S(ji), and s equals i fromthen on. Notice that switching from subregion i to j does not occur along the same88Figure A.1: An Example Parameter Trajectory Causing Switching BetweenTwo Subregionssurface where switching occurs from subregion j to i.A.3 Controller Design MethodA switching LPV controller of the form (A.3) is developed so that it guarantees in-ternal stability of the closed-loop as well as a bound of the L2-gain for the closed-loop with (A.1) and (A.3) from w, the disturbance signal, to z, the error signal, forany trajectory of q satisfying (A.2). The LPV controller matrices are computedby expressing the problem as an optimization problem using a family of LinearMatrix Inequality (LMI)s which are solved using available LMI techniques. Thefollowing subsections are organized as follows: The LMIs to guarantee perfor-mance within each subregion are first given in A.3.1. The controller’s dependenceon ˙q, the rate of change of the scheduling parameter, is then removed in A.3.2giving a practically valid controller. Additional LMIs are introduced to guaran-tee switching performance in A.3.3, and finally the problem is reduced to finitedimensions by gridding the parameter space in A.3.4.89A.3.1 Controller PerformanceWithin each subregion, closed-loop stability and an L2-gain bound, denoted byg, from the disturbance signal to the error signal are guaranteed when the LMIconstraints (A.4) and (A.5) hold at all values of q within the subregion Q(r) andthe rate of parameter change, ˙q, within Qd.266666664˙X(r)+X(r)A+ ˆB(r)K C2 +( )    ( ˆA(r)K )T +A+B2D(r)K C2  ˙Y(r)+AY(r)+B2 ˆC(r)K +( )   (X(r)B1 + ˆB(r)K D21)T (B1 +B2D(r)K D21)T  gI  C1 +D12D(r)K C2 C1Y(r)+D12 ˆC(r)K D11 +D12D(r)K D21  gI377777775<0;(A.4)264 X(r) II Y(r)375> 0: (A.5)The dependencies on the scheduling parameter, q, is hidden in (A.4) and (A.5)for brevity. All matrices are functions of q, however. The LMI variables includethe symmetric matrices X(r)(q) and Y(r)(q), known as the Lyapunov variables, aswell as ˆA(r)K (q), ˆB(r)K (q), ˆC(r)K (q), and D(r)K (q), the quadruple state space matrices.The bound of the L2-gain, g, can be chosen as a constant or can be included asan LMI variable and be minimized, thereby yielding an optimal controller. Matrixterms with an asterisk (*) should be completed such that the LMI matrix (A.4) issymmetric.90Once the LMI variables are obtained, the controller matrices in (A.3) can becomputed asA(r)K (q) = [N(r)(q)] 1[X(r)(q)˙Y(r)(q)+N(r)(q)[ ˙M(r)(q)]T + ˆA(r)K (q) X(r)(q)[A(q) B2(q)D(r)K (q)C2(q)]Y(r)(q) ˆB(r)K (q)C2(q)Y(r)(q) X(r)(q)B2(q) ˆC(r)K (q)][M(r)(q)];B(r)K (q) = [(N(r)(q)] 1[ ˆB(r)K (q) X(r)(q)B2(q)D(r)K (q)];C(r)K (q) = [ ˆC(r)K (q) D(r)K (q)C2(q)Y(r)(q)][M(r)(q)]T:(A.6)Computation of the matrices M(r)(q) and N(r)(q) is shown below.A.3.2 Practical ValidityThe controller shown in (A.3) is not a usual gain-scheduling controller because ofits dependence on ˙q, the rate of change of the scheduling parameter. Since a real-time measurement of ˙q is usually not available and can be difficult to compute,this dependence is undesirable. Controllers that do not rely on a measurementor estimate of ˙q are known as practically valid controllers [5]. The controller’sdependence on ˙q results from the presence of ˙X(r)(q) and ˙Y(r)(q) in (A.4) and of˙Y(r)(q) and ˙M(r)(q) in (A.6). These terms can be eliminated by fixing either oneof the Lyapunov variables, X(r)(q) or Y(r)(q). There is no analytical method forchoosing which variable should be constant and which should remain a functionof q. Both cases should be attempted and the resultant closed-loop performancecompared in order to find the choice that yields the least conservative controller.Depending on the choice of Lyapunov variables, the matrices M(r)(q) and91N(r)(q), used for controller reconstruction in (A.6), are computed as follows: Ifwe choose X(r)(q)= X0, a constant, then we haveM(r)(q) = X 10  Y(r)(q);N(r)(q) = X0:(A.7)Alternatively, for Y(r)(q)=Y0, we haveM(r)(q) = Y0;N(r)(q) = Y 10  X(r)(q):(A.8)A.3.3 Switching PerformanceThe LMI constraints in (A.4) and (A.5) guarantee closed-loop stability and L2 per-formance for the LPV plant and controller combination within each subregionQ(r). In order to ensure switching performance, the Lyapunov function must beforced to decrease at any switching event along the switching surface S(i j). Re-call that S(i j) denotes the switching surface from an arbitrary subregion Q(i) toa neighboring and overlapping subregion Q(j). This condition is satisfied wheneitherX(i)(q)  X(j)(q);Y(i)(q) (X(i)(q)) 1  Y(j)(q) (X(j)(q)) 1;(A.9)92or equivalently,Y(i)(q)  Y(j)(q);X(i)(q) (Y(i)(q)) 1  X(j)(q) (Y(j)(q)) 1;(A.10)hold for all values of q along the switching surface. A further explanation of(A.9) and (A.10), as well as the proof, is given in [23] although different notationis used. In order to synthesize a controller subject to the switching constraintsgiven above, the LMI family consisting of (A.4) and (A.5) is augmented withY(i)(q) Y(j)(q) 0 (A.11)if the Lyapunov variable X is chosen to remain constant, orX(i)(q) X(j)(q) 0 (A.12)if the Lyapunov variable Y is chosen to remain constant. While the LMIs (A.4)and (A.5) must hold for all values of the scheduling parameter, q, within eachsubregion, (A.11) and (A.12) must only hold true for values of q along the twoswitching surfaces, S(i j) and S(ji), for each neighboring subregion pair, Q(i) andQ(j).A.3.4 Reduction to Finite DimensionalBecause the LMIs (A.4) and (A.5) must hold for all values of the scheduling pa-rameter, q, within each subregion, and the LMI (A.11) or (A.12) must hold for all93values of q along the switching surfaces, there exists an infinite number of oper-ating points where the LMI variables need to be solved subject to the constraints.In order to make this family of LMIs solvable, it must therefore be reduced to afinite dimensional problem. This can be done by sampling each subregion witha finite number of points. Let G(r) denote a grid within the space of Q(r) andlet Gd denote a grid within the space of Qd. Since ˙q enters the LMIs linearlyin (A.4) and (A.5), the LMI constraints need to be checked only at its extremevalues. Gd therefore only needs to include the vertices of the space Qd. Further-more, let G(i j)S and G(ji)S denote grids along the switching surfaces between eachneighboring subregion pair. A finite family of LMIs can then be set up for all thesubregions, including LMIs (A.4) and (A.5) for all combinations of G(r) Gd andLMI (A.11) or (A.12) for each point in G(i j)S and in G(ji)S , and the LMI variablescan be calculated simultaneously.In order to calculate the controller matrices for any value of q within Q whilethe LMI variables are only solved at a finite number of points, the nature of theirparameter dependence needs to be known. A simple solution, proposed by [40],is to mimic the parameter dependence using a series of functions denoted by r( )such that ri( ); i = 1;:::;Nr are differentiable functions of q. Nr represents thenumber of individual functions required to capture the dependence on q and canbe larger than or equal to the number of elements in q. Copies of the function r( )94are then introduced into the LMI variables asˆA(r)K (q) = ˆA(r)K0 +Nr i=1ri(q) ˆA(r)Ki ;ˆB(r)K (q) = ˆB(r)K0 +Nr i=1ri(q) ˆB(r)Ki ;ˆC(r)K (q) = ˆC(r)K0 +Nr i=1ri(q) ˆC(r)Ki ;D(r)K (q) = D(r)K0 +Nr i=1ri(q)D(r)Ki ;(A.13)andX(r)(q) = X(r)0 +Nr i=1ri(q)X(r)i ;Y(r)(q) = Y(r)0 +Nr i=1ri(q)Y(r)i :(A.14)The LMI variables now become g and ˆA(r)K j; ˆB(r)K j; ˆC(r)K j; D(r)K j; X(r)j ; Y(r)j wherej = 0;1;:::;Nr and r = 1;2;:::;R. After the LMI variables are computed subject tothe LMI constraints, the quadruple state space matrices and the Lyapunov variablesare reconstructed as functions of q using (A.13) and (A.14) before the controllermatrices are calculated using (A.6).A.4 Recapitulative ProcedureThe overall procedure for finding the switching LPV controller can be describedas follows:1. Set up a finite system of LMIs:95(a) Define G(r), a grid within each parameter subspace Q(r) as well as Gd,the limits of the parameter’s rate of change.(b) Define G(i j)S , the grids along the switching surfaces between each pairof neighboring subregions.(c) Define the function r( ) so that it mimics the dependence on the schedul-ing parameter, q, and use it to define the quadruple state space matricesand the Lyapunov variables as functions of q in (A.13) and (A.14).(d) Create a family of LMIs such that an instance of (A.4) and (A.5) ispresent for each pair G(r) Gd of each subregion and an LMI (A.11)or (A.12) is present for each point in G(i j)S for each neighboring pair ofsubregions.2. Solve for variables subject to LMI constraints:(a) Minimize g subject to the LMI constraints and compute the values ofthe LMI variables.(b) Use the solution of the LMI variables to obtain the quadruple statespace data matrices and the Lyapunov variables as functions of q using(A.13) and (A.14).3. Check LMI constraints on a denser grid:(a) Define another grid in each subregion Q(r) such that it is denser thanG(r).96(b) Define another grid of each switching surface such that it is denserthan G(i j)S and G(ji)S .(c) Calculate the values of the LMI variables at each point of the densergrid using (A.13) and (A.14) and check that the LMI constraints holdat each point.(d) If Step 3.c fails, increase the grid density of G(r), G(i j)S , and G(ji)S andrestart Step 3.4. Compute controller matrices:(a) In real-time, calculate the controller matrices using (A.6).(b) Switch the controller matrices in real-time using the switching variables such thatAK(q) = A(s)K (q);BK(q) = B(s)K (q);CK(q) = C(s)K (q);DK(q) = D(s)K (q):(A.15)5. Perform Steps 1 through 4 twice, fixing either one of the two Lyapunovvariables X and Y each time, and choose the least conservative controller.97Appendix BAdaptive Predictive ControlAlgorithm using Laguerre NetworkIn this appendix, the algorithm used in Section 3.4 to create an adaptive air-fuelratio controller is presented. The method is introduced in [43] and is further de-veloped in [46] where it is given the title Incremental Mode Linear Laguerre Pre-dictive Control (IMLLPC). The method is suitable for any control problem wherethe plant dynamics are dominated by a delay. This appendix only serves as anoverview of the algorithm and does not show the derivation or any proofs associ-ated with the method.98B.1 Laguerre NetworkThe Laguerre function is defined as the seriesFi(t)=p2p ept(i 1)! di 1dti 1 ti 1 e2pt ; i = 1;2;:::;¥ (B.1)where p is a constant, called the time-scale and t2[0;¥) is the time variable. TheLaplace transform of the Laguerre Function isFi(s)= L[Fi(t)]=p2p(s p)i 1(s+ p)i ; i = 1;2;:::;¥ (B.2)Figure B.1 shows the structure of an open-loop stable system approximated by aLaquerre series with order N. The output of the model isYm(s)=N i=1CiFi(s)U(s)=N i=1Cili(s) (B.3)Figure B.1: Laguerre Network Model Structure99where C = [C1;:::;CN] and is called the Laguerre network’s spectral gain andL = [l1;:::;lN] is the Laguerre network’s state vector. The Laguerre network canbe expressed in a state-space form asL(k+1) = AL(k)+bu(k);Ym(k) = CT L(k);(B.4)with the model’s input, u(k), and output, Ym(k).B.2 IMLLPC AlgorithmThe IMLLPC algorithm in [46] makes use of the Laguerre network given in (B.4)as the model of the plant in a model reference adaptive approach. Rather thanusing the values of the input and output directly, IMLLPC uses incremental values,Du(k), DYm(k), and DL(k). The state-space form in (B.4) therefore becomesDL(k+1) = ADL(k)+bDu(k); (B.5)DYm(k) = CTDL(k): (B.6)By using incremental values, integration is effectively added to the resulting con-troller and reference tracking performance is improved. The IMLLPC algorithm isgiven below.100B.2.1 User-Defined ParametersThe following values are defined by the designer and serve as tuning parameters.For considerations and instructions on the selection of parameters, see [43] and[46].Ts Sampling timep Time-scaleN Order of Laguerre networkd Prediction horizonM Control horizonQ Weighting matrix on errorR Weighting matrix on control inputa Softening factor on reference model outputl Forgetting factor for recurserB.2.2 Constant MatricesThe A and b state-space matrices of the Laguerre network are constants and aregiven asA =266666664t1 0    0 (t1t2+t3)Ts t1    0... ... ... ...( 1)N 1tN 22 (t1t2+t3)Ts     (t1t2+t3)Ts t1377777775; (B.7)andbT = t4 ( t2Ts )t4    ( t2Ts )N 1t4 ; (B.8)101wheret1 = e pTs;t2 = Ts + 2p(e pTs 1);t3 =  Tse pTs 2p(e pTs 1);t4 = p2p1 t1p :(B.9)B.2.3 Recursive EstimationThe C matrix of the Laguerre network’s state-space form is estimated in a recur-sive loop such thatˆC(k)= ˆC(k 1)+ P(k 1)DL(k)l +DLT(k)P(k 1)DL(k)  Dy(k) ˆCT(k)DL(k 1) ;(B.10)and the covariance matrix is updated usingP(k)= 1l P(k 1) P(k 1)DL(k)DLT(k)P(k 1)l +DLT(k)P(k 1)DL(k) : (B.11)B.2.4 Control Move CalculationThe current output of the Laguerre network isym(k)= ˆCT(k) D(k); (B.12)102and the estimated future plant output isˆYp(k+1)= SHlDL(k)+Fym(k)+K[y(k) ym(k)]; (B.13)with y(k), the measured output of the plant. The matrices, S, Hl, F, and K areconstants and are defined asS =2666666641    0 0... 01 ...1 1    1377777775d dHl =266666664ˆCT(k)AˆCT(k)A2...ˆCT(k)Ad377777775d NF=26666666411...1377777775d 1K =26666666411...1377777775d 1(B.14)respectively.The controller outputs, including the current and future outputs up to k+M 1are calculated usingDUM(k)=(HTu ST QSHu +R) 1HTu ST Q Yr(k+1) ˆYp(k+1) ; (B.15)where Yr(k+1), the reference output is computed fromyr(k+i)=aiy(k)+(1 ai)w; i = 1;2;   ;d (B.16)103and w is the setpoint. Hu is a constant matrix:Hu =2666666666666664ˆCT(k)b 0    0ˆCT(k)Ab ˆCT(k)b 0... ... ... ...ˆCT(k)AM 1b ˆCT(k)AM 2b    ˆCT(k)b... ... ...ˆCT(k)Ad 1b ˆCT(k)Ad 2b    ˆCT(k)Ad Mb3777777777777775d M(B.17)The current control output is then simply extracted from the results of (B.15)using the equationDu(k)= DDUM(k); (B.18)whereD = 1 0    0 1 M: (B.19)Finally, the Laguerre network states are updated using (B.5).104


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