DISSERTATION. Ahmed Abad Al-Durra, B.S.E.C.E., M.S.E.C.E. Graduate Program in Electrical and Computer Engineering. The Ohio State University

Transcription

1 MODEL-BASED ESTIMATION FOR IN-CYLINDER PRESSURE OF ADVANCED COMBUSTION ENGINES DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ahmed Abad Al-Durra, B.S.E.C.E., M.S.E.C.E. Graduate Program in Electrical and Computer Engineering The Ohio State University 21 Dissertation Committee: Prof. Steve Yurkovich, Adviser Prof. Giorgio Rizzoni Prof. Yann Guezennec

2 c Copyright by Ahmed Abad Al-Durra 21

3 ABSTRACT Cylinder pressure is one of the most important parameters that characterize the combustion process in an internal combustion engine. Recent developments in engine control technologies suggest the use of cylinder pressure as a feedback signal for closed-loop combustion control. However, the sensors measuring in-cylinder pressure are typically subject to noise and offset issues, requiring signal processing methods to be applied to obtain a sufficiently accurate pressure trace. The signal conditioning implies a considerable computational burden, which ultimately limits the use of cylinder pressure sensing to laboratory testing, where the signal can be processed off-line. In order to enable closed-loop combustion control through cylinder pressure feedback, a real-time algorithm that extracts the pressure signal from the in-cylinder production grade sensor is proposed in this study. The algorithm is based on a crank-angle based engine combustion that predicts the in-cylinder pressure from the definition of a burn rate function. The model is then adapted to model-based estimation by applying an extended Kalman filter in conjunction with a recursive least squares estimation scheme. The estimator is tested at certain operating points on a high-fidelity Diesel engine simulator, as well as on experimental data obtained at various operating conditions. The results obtained show the effectiveness of the estimator in reconstructing the cylinder pressure on a crank-angle basis and in rejecting ii

4 measurement noise and modeling errors. Furthermore, a comparative study with a conventional signal processing method shows the advantage of using the derived estimator, especially in the presence of high signal noise (as frequently happens with low-cost sensors). As an extension and further application, this methodology is built upon to cover a wider range of operations as well as transient data. Linear parameter varying techniques using genetic algorithms are utilized to identify the gains of linear spline functions of the LPV-corrector estimator. The LPV-corrector performs well with a relatively small computation burden. The two estimators are examined under both steady state data and transient data, where the comparison criteria include estimation of combustion metrics. Finally, a model-based estimation methodology that facilitates real-time reconstruction of individual in-cylinder pressure utilizing a minimum sensor set is demonstrated. Based on a derived crankshaft speed model incorporated with the pressure model, a sliding mode observer is implemented, wherein chattering is mitigated and the estimation design is validated. Adding disturbances to the model parameter degrades the performance of the SMO, which motivates the development of an adaptive-smo based on the certainty equivalence principle, utilizing the cylinder pressure signal from one cylinder. The estimator was derived analytically and a proof of stability is provided. iii

5 This is dedicated to the UAE iv

6 ACKNOWLEDGMENTS I wish to thank my adviser, Prof. Steve Yurkovich, for his intellectual support, encouragement, and enthusiasm which motivated me to do my best, and for his patience in correcting both stylistic and scientific errors. It is only through his sound advice and guidance that I was able to make the transition from an undergraduate to a graduate level mindset and work ethic. I would like to thank Doctor Marcello Canova for discussing with me various aspects of this thesis and for explaining all thermodynamical and mechanical aspects that I needed for this work. Without him all this would not have been possible. I would also like to thank Prof. Giorgio Rizzoni and Prof. Yann Guezennec who were also instrumental in the help they provided. Indeed their vast experience helped us surpass many hurdles seamlessly. I would also like to thank all the people at CAR that helped me in this project and helped me feel part of the group. Last but not least, I would like to thank my family, my friends, and my scholarship sponsors, for providing the best support and encouragement any person could ever hope for. Finally, I am deeply grateful to my wife and our two little daughters for their constant love and support. v

7 VITA April 17, Born - Al Ain, The United Arab Emirates (UAE) Distinguished Students Scholarship Scholarship Coordination Office (UAE- President Office) B.S. Electrical & Computer Engineering, The Ohio State University Petroleum Institute Study Leaver Program Abu Dhabi National Oil Company (ADNOC) M.S. Electrical & Computer Engineering, The Ohio State University 27-present PhD Student in Electrical & Computer Engineering, The Ohio State University PUBLICATIONS Al-Durra, A., Yurkovich, S., and Guezennec, Y., Gain-Scheduled Control for an Automotive Traction PEM Fuel Cell System. Proceedings of the ASME International Mechanical Engineering Congress and Exposition, 4266, Nov. 27. Al-Durra, A., Canova, M., and Yurkovich, S., Application of Extended Kalman Filter to On-Line Diesel Engine Cylinder Pressure Estimation. Proceedings of the Dynamic System and Control Conference, 2523, Oct. 29. vi

8 FIELDS OF STUDY Major Field: Electrical and Computer Engineering Studies in: Linear Control Nonlinear Control Sliding Mode Control Prof. Steve Yurkovich Prof. Andrea Serrani Prof. Vadim Utkin vii

9 TABLE OF CONTENTS Page Abstract Dedication Acknowledgments Vita List of Tables ii iv v vi xi List of Figures xii Chapters: 1. Introduction and Motivation Introduction CI and SI Background Current Practice in IC Engine Control Issues and Motivations Thesis Overview State of the Art in Cylinder Pressure Modeling and Estimation Introduction Applications of Cylinder Pressure Closed Loop Control Combustion Modes and Flex Fuel Combustion Diagnostics and Misfire Detection Control-Oriented Cylinder Pressure Models Pressure Estimation viii

10 2.4.1 Using the Cylinder Pressure Signal Using Crankshaft Speed Other Sensors Summary Mathematical Tools for Estimation Introduction Extended Kalman Filter Linear Parameter Varying Systems Sliding Mode Observer Adaptive Observer Summary EKF Estimation of Cylinder Pressure Introduction Cylinder Pressure Model for Estimator Design Description of the Model Structure Model Calibration and Validation Design of In-Cylinder Pressure Estimator Implementation Preliminary Validation of the Estimator Extension for Offset Compensation Application to On-Line Cylinder Pressure Estimation Application to Lab-Grade Sensor Data Application to Production-Grade Sensor Data Summary LPV-EKF Estimation for Expanded Operating Regions Introduction Model Structure and Validation Description of the Model Model Calibration and Validation Estimation Schemes Results from Steady State Data Results from Transient Data Summary ix

11 6. Multi-Cylinder Pressure Estimation Introduction Crankshaft Model Structure Estimation Scheme Cylinder Pressure Estimation Design Simulation and Experimental Validation Adaptation Using One Cylinder Pressure Measurement Summary Conclusion Summary Summary of Contributions Future Directions Appendices: A. Nomenclature B. Derivation of In-Cylinder Pressure Model C. Methods for cylinder pressure pegging D. Estimator comparison for all conditions E. Calibration Results for LPV Design F. La Salle / Yoshizawa Theorem Bibliography x

12 LIST OF TABLES Table Page 4.1 Test engine specifications Summary of experimental tests conducted for model calibration Comparison of combustion metrics calculated from the EKF and RLS-EKF estimators, for a representative engine operating point Specifications for the two pressure transducers utilized in the study Test engine specifications D.1 Comparison of the EKF and RLS-EKF estimators, for 11 operating points xi

13 LIST OF FIGURES Figure Page 1.1 General structure for a typical CI engine General prototype for Diesel engine feedback control based on cylinder pressure General scheme for combustion control NOx and soot formation regions on φ-t map [1] Two-envelope pressure model Switching surfaces for multi-dimensional sliding mode EKF estimator scheme Control volume Selection of operating points for model calibration Example of Wiebe function fitting (Mode 6) Cylinder pressure (engine simulator versus estimate) Comparison of model identification results with burn rate curve obtained from the experimental cylinder pressure trace (operating point 8) Validation of cylinder pressure model on experimental data: burn rate profile (test 2) xii

14 4.8 Validation of cylinder pressure model on experimental data: cylinder pressure (test 2) Pressure estimation with augmented EKF (7 bar offset) Pressure estimation with corrected EKF (7 bar offset) (2,1) element in the observability gramian Pressure estimation using EKF with offset Frequency response for Moving Average filter Comparative scheme for on-line pressure estimation Comparison of pressure traces measured in the same cylinder with labgrade sensor and production-type sensor during one engine cycle (Test No. 1, cycle 58) Noise characteristics for lab-grade (left) and production-type (right) pressure sensors Error signature in crank-angle domain: error between estimators outputs and cylinder pressure trace obtained from average of 1 cycles (lab-grade sensor) Comparative results for on-line cylinder pressure estimation from labgrade sensor data (test 3) Error signature in crank-angle domain: error between estimators outputs and cylinder pressure trace obtained from average of 1 cycles (production-grade sensor) Comparative results for on-line cylinder pressure estimation from productiontype sensor data (test 3) GM Ecotec LE Operating points used for model calibration xiii

15 5.3 Metrics comparison for both calibrated models (scheduled-parameters solid, fitted-parameters dotted) The values of the linear spline function at the solid circles completely defines the linear spline function Errors in the estimation of the combustion metrics for steady state data using EKF-RLS Comparison between the actual, input, and EKF-estimated pressures (top) and normalized cumulative burn rate (bottom) for test number LPV corrector gains derived from the steady state experimental data Errors in the estimation of the combustion metrics for steady state data using LPV-corrector Comparison between the actual, input, and LPV-estimated pressures (top) and normalized cumulative burn rate (bottom) for test number The portion of the FTP-cycle on which the two estimators are examined The estimated combustion metrics by EKF during a portion of FTPcycle Comparison between the actual, input, modeled, and EKF-estimated pressures (top) and normalized cumulative burn rate (bottom) for cycle number The estimated combustion metrics by LPV-corrector during a portion of FTP-cycle Comparison between the actual, input, modeled, and LPV-estimated pressures (top) and normalized cumulative burn rate (bottom) for cycle number Schematic of four-cylinder engine showing the sensor configuration utilized in the proposed estimation scheme Engine crank-slider mechanism xiv

16 6.3 Indicated torque for each of the four cylinders Pressure, torque, and speed for a motoring cycle Schematic of the pressure and indicated torque models for a fourcylinder engine A piecewise continuous sliding mode observer gain K(θ) A piecewise continuous observer gain L(θ) Cylinder pressures data using parametric pressure model (solid) and estimated (dotted). Left to right, the peaks represent cylinders 1, 3, 4, and Results using experimental pressure traces with no disturbances. Left to right, the peaks represent cylinders 1, 3, 4, and Errors in the estimation of the cycle-by-cycle combustion metrics with no disturbances (cylinder-1 and test-1) Results using experimental pressure traces with different initial pressures but no disturbances (cylinder-1 and test-1) Results using experimental pressure traces with disturbances. Left to right, the peaks represent cylinders 1, 3, 4, and Errors in the estimation of the cycle-by-cycle combustion metrics with disturbances(cylinder-1 and test-1) Results using experimental pressure traces with disturbances applied to adaptive observer. Left to right, the peaks represent cylinders 1, 3, 4, and Errors in the estimation of the cycle-by-cycle combustion metrics with disturbances and adaptation (cylinder-1 and test-1) Errors in the estimation of the cycle-by-cycle combustion metrics with disturbances and adaptation (cylinder-2 and test-1) xv

17 6.17 Errors in the estimation of the cycle-by-cycle combustion metrics with disturbances and adaptation (cylinder-3 and test-1) Errors in the estimation of the cycle-by-cycle combustion metrics with disturbances and adaptation (cylinder-4 and test-1) Results using experimental pressure traces with disturbances and noisy crankshaft speed signal applied to adaptive observer. Left to right, the peaks represent cylinders 1, 3, 4, and Errors in the estimation of the cycle-by-cycle combustion metrics with disturbances, noise and adaptation (cylinder-1 and test-1) Errors in the estimation of the cycle-by-cycle combustion metrics with disturbances, noise and adaptation (cylinder-2 and test-1) Errors in the estimation of the cycle-by-cycle combustion metrics with disturbances, noise and adaptation (cylinder-3 and test-1) Errors in the estimation of the cycle-by-cycle combustion metrics with disturbances, noise and adaptation (cylinder-4 and test-1) E.1 Calibration for Wiebe parameter a E.2 Calibration for Wiebe parameter Δθ E.3 Calibration for Wiebe parameter m E.4 Calibration for heat transfer scaling factor α E.5 Calibration for polytropic coefficient γ xvi

18 CHAPTER 1 INTRODUCTION AND MOTIVATION 1.1 Introduction The objective of an internal combustion engine is to produce mechanical energy from the chemical energy stored in the fuel. The mixture of fuel and air prior to combustion and the burned products of the combustion are the actual working fluids. The process of igniting this mixture divides the internal combustion engines into two main categories: compression ignition (CI) (mostly with Diesel fuel) and spark ignition (SI) (mostly with gasoline fuel). In the CI engine, the fuel is injected after the air is compressed in the combustion chamber, causing the fuel to self ignite. On the other hand, the mixture in the SI engine is ignited by the spark. Each engine has advantages over the other; mainly, the CI engine tends to have higher efficiency and torque output, whereas the SI engine is usually faster and smoother, with less smoke. For both types of engines, the recent stringent regulations on fuel consumption and exhaust emissions, as well as the recent developments made in materials, injection equipment, and emission treatment systems has motivated researchers to start looking for flexible and accurate electronic control techniques to supervise a complex engine under all conditions. The following sections will provide fundamental information 1

19 about both engines, which will be helpful in understanding the motivation behind the work in this dissertation. 1.2 CI and SI Background CI engines are one of the leading candidates for combustion engines, with high efficiency and low emissions. The common problems that jeopardized their appeal to customers in the 199s (such as poor transient response, low power density, and high smoke emissions) have been significantly mitigated by introducing air systems with VG-turbochargers, controlled EGR, and common-rail fuel injection systems. In fact, the current generation of CI engines can achieve high performance in fuel economy as well as very low CO 2 and smoke levels by introducing advance technologies and more refined control [2]. A modern CI engine for automotive application achieves its performance targets by controlling two systems, namely air handling and fuel injection. The air handling system typically includes a turbocharger, exhaust gas recirculation (EGR), and cooler devices as shown in Figure 1.1. The turbocharger, which is comprised of a variable geometry turbine connected through a shaft to the compressor, utilizes part of the exhaust gas energy to rotate the turbine-compressor, thereby boosting the intake manifold pressure with fresh air, which increases the engine power density. The flow rates and efficiency can be changed by adjusting the angle of the guide vanes of the turbine. The intercooler following the compressor helps to reduce the air temperature, further increasing the power density. The EGR valve allows a specific portion of the exhaust gas to recirculate and gets mixed with the boosted air in the intake manifold. The recirculated exhaust gas 2

20 decreases the combustion temperature, resulting in less NO x emissions [3]. Furthermore, the EGR cooler reduces the temperature of the recirculated exhaust gas and allows for more exhaust gas to be induced to the intake manifold which will help to reduce the combustion temperature, thus producing less NO x. On the other hand, the fuel injection system is a common-rail system, which allows full flexibility in the control of the fuel injection by varying pressure in the rail, injection quantity, number of injections, injection timing, etc. Therefore, it allows for shaping the injected fuel profile that can be used to optimize the combustion mechanism in order to achieve higher torque and lower emissions and mitigate the problem of noise and vibration harshness. Figure 1.1: General structure for a typical CI engine. 3

21 One of the key elements for the success of the CI engine is its ability to control air, EGR, and fuel. The main objective of CI engine control is to satisfy a diverse set of conflicting constraints imposed by providing the required engine torque (steady state and transient) with minimal fuel consumption, while meeting the exhaust emission regulations [4]. This requires an optimal coordination of the EGR, turbocharger, and injection systems. On the other hand, the SI engine is considered a slightly less complicated system (which is why it is typically cheaper than a CI engine). As mentioned earlier, the SI engine has spark plugs to ignite the fuel-air mixture, whereas the CI engine needs to have a complicated fuel injection system of high pressure injection nozzles that need to inject a controlled amount of fuel into the cylinder. Nevertheless, there are similar subsystems in both engines such as the exhaust gas recirculation system. The SI engine is accompanied by an ignition subsystem that must be controlled to ignite the correctly proportioned air/fuel mixture at precisely the right time, resulting in torque production. There are several variables that influence the optimum spark timing, such as engine speed, manifold pressure, ambient pressure, and coolant temperature [5]. The introduction of the three-way catalytic converter (TWC) has greatly improved emission reduction in SI engines. The TWC converts the pollutant compounds CO, HC, and NOx into H 2 O, N 2, and CO 2, which are much less harmful. However, the TWC works efficiently at stoichiometric, 1 which requires a closed loop air-fuel ratio controller. Furthermore, it is this requirement that makes it impossible to obtain an efficient exhaust gas purification using the TWC in a conventional CI engine, because the CI engine works inherently at lean [6]. 1 Air-to-fuel ratio is defined as stoichiometric when the engine is supplied with the right amount of air to achieve perfect combustion. 4

22 1.3 Current Practice in IC Engine Control The presence of various actuators in the engine plant (e.g., the VGT vane guide, the EGR valve, fuel injectors, spark plugs, air throttle, etc.) introduces complex interactions between the subsystems. Therefore, a control strategy is essential to achieve the main objectives with high robustness; meanwhile, it should not utilize a lot of the resources of the electronic control unit (ECU) and hopefully is simple to implement and calibrate [7]. It has been shown in [6] that it is very difficult to achieve all of these objectives, because the tendency of several input-output relations are conflicting (i.e., almost every input has a tradeoff with our objectives). Today s engines are controlled mainly using feedforward maps, with engine speed and torque being the independent variables. These maps are tuned using correction maps to account for the changes in the operating conditions, such as transients, altitude, ambient conditions, etc. The way to obtain these maps is through extensive calibration of the engine for all possible situations. The calibration of the engine control problem is typically divided into various stages. The first step is to calibrate the reference input maps for the control loops to optimize the stationary performance of the engine. This is usually carried out experimentally on an engine or a test bench. A way to calibrate these inputs is described in [6], where the engine map (speed vs. load) is divided into different operating points (n), each with a specific weight (w n ). Then, an objective function (J) is minimized with respect to all reference points, where J can be defined (e.g., for the CI engine) as follows: J = n=n tot n=1 BSF C n (Δt inj, t inj, p cr, φ egr, φ vgt )w n, (1.1) 5

23 where BSF C is the brake specific fuel consumption, Δt inj the injection duration, t inj the injection timing, p cr the rail pressure, φ egr the EGR rate, and φ vgt the VGT position. The minimization is performed with the following emission constraints: n=ntot n=1 (NO x + HC) n w n (NO x + HC) limit, n=ntot n=1 P M n w n P M limit, (1.2) where P M is particulate matter, HC is hydrocarbon, and NOx is nitrogen oxides. After determining the reference values for the control systems, the second stage of the control design problem focuses on building feedback and feedforward controllers for the subsystems, to achieve the desired dynamic response. In the classic control structure, there are mainly three paths: the air path, the EGR path, and the fuel path. The air and EGR paths are usually treated together (often referred to as the air-path control problem ), while the fuel path is treated by some means independently. To this extent, several contributions have been made on the air system control problem (specially for the CI engine), focusing on the optimization of engine torque and emissions [6, 8, 9, 1, 11]. In particular, several authors have focused on the well-known EGR-VGT control problem, proposing several combinations of feedback and feedforward control schemes to operate the VGT and the EGR systems in order to attain the desired boost pressure and intake gas composition [6, 12, 13, 14]. On the other hand, fewer results have appeared on the definition of closed-loop control schemes for the fuel injection system. The modern common rail injection system, currently considered the state of the art, relies solely on open-loop mapping to schedule the injection parameters in relation to the engine operating conditions, while the closed-loop control is typically limited to the rail pressure [15]. Furthermore, the spark advance setting for the SI engine is still controlled based on the measurement 6

24 of several parameters (introduced earlier) and regulated based on extensive engine testing during the calibration phase [16]. 1.4 Issues and Motivations The complexity of the modern engine has evolved considerably in order to adapt to customer demands and emission legislations. This trend is likely to continue in the near future with the introduction of new technologies, such as the following: Advanced turbo-charging devices such as 2-Stage turbochargers and VG compressors; Dual-loop EGR systems and controlled EGR cooling systems; Piezoelectric injection systems (up to 6 injections/stroke); Advanced aftertreatment systems such as SCR, LNT, and DPF; and Spark ignition direct injection engines. As a further stage of development, alternative combustion modes (LTC, HCCI, and PCCI) have been recently introduced as new means to reduce emissions formation. Moreover, flex fuel is another alternative to reduce petroleum mainly gasoline consumption as well as to lower tailpipe emissions. In spite of the complexities, current engines are still working predominantly based on feedforward maps that are scheduled based on two independent variables: engine speed and load torque. In addition, several correction factors are mapped to account for different boundary conditions. All of these maps require a tremendous amount of effort, time, and equipment to be calibrated. 7

25 The control of the fuel injection system is particularly representative of this issue. The absence of direct feedback from the combustion chamber dictates a reliance exclusively on feedforward schemes based on empirical maps, determining the injection parameters to be used in relation to a set of pre-established operating conditions (typically in terms of engine speed and torque demand). This limits the possibility of compensating for errors by adjusting the fuel quantity and injection timing on a cycle-to-cycle basis, as commonly done for SI engines with spark timing control. In summary, the current control schemes are completely blind to the combustion characteristics, which ultimately affect torque, BSFC, emissions, and NVH. Although the spark timing can be controlled via feedback in the SI engine, it is very difficult to measure all of the parameters that affect the optimum spark timing. Furthermore, it is very expensive and time consuming to conduct the engine testing required to map the effects of the parameters on the spark timing [16]. In this sense, a critical variable for combustion analysis and diagnostics is the in-cylinder pressure, which is typically available only on laboratory setups, where very accurate sensors and sophisticated post-processing can be utilized to provide the level of detail required for combustion analysis. However, the recent availability of cost-effective piezoelectric transducers and the progress in on-line computational capability have led to the opportunity of considering the in-cylinder pressure as a feedback signal for engine combustion control and diagnostics [17, 18]. From the cylinder pressure sensor output, signal processing can be applied to calculate the combustion-related metrics for use in closed-loop control. In order to overcome the limitations of open-loop combustion control, a feedback control system can be proposed, based on the in-cylinder pressure. The overall 8

26 scheme allows one to achieve closed-loop combustion control, with the ultimate goal of shaping the combustion to achieve desired combustion characteristics that meet desired torque, BSFC, emissions, and NVH targets. Figure 1.2 shows a diagram of a general control scheme that utilizes the cylinder pressure as a feedback signal for combustion control. The problem of achieving a control system that meets the desired combustion metrics requires a significant effort in the development of methodologies for estimation and control. In the development of the research framework, a concept for closed-loop combustion control for advanced engines that takes advantage of new technologies, mainly in-cylinder pressure sensing, will be explored. The main objective of the study will be to improve estimation of combustion-related variables, which will help improve control of combustion outputs (torque, SFC, emissions, and NVH), and reduce calibration effort by extending feedback control authority. Figure 1.2: General prototype for Diesel engine feedback control based on cylinder pressure. 9

27 1.5 Thesis Overview Illustrating the importance of the in-cylinder pressure variables in characterizing the combustion process in an engine will be the first objective of Chapter 2, which is mainly a literature review. A summary of the in-cylinder pressure applications will be given, focusing more on on-line practices. Furthermore, some of the key works for in-cylinder pressure modeling and estimation will be highlighted. The main objective of this study is to apply estimation theory on estimating the in-cylinder pressure; consequently, several techniques for parameters or state estimation or a combination of the two are applied to either linear or nonlinear models. Therefore, Chapter 3 will introduce four main estimation methods to familiarize the reader: extended Kalman filter, linear parameter varying state estimation, sliding mode observer, and adaptive observer. Chapter 4 presents the first application of estimation theory where a real-time model-based estimation methodology to reconstruct the engine cylinder pressure from the output of a piezoelectric transducer is discussed. Starting with the raw signal from such a sensor, an Extended Kalman Filter (EKF) is designed in a predictor-corrector format to reconstruct the cylinder pressure. In order to do so, a model of the incylinder thermodynamics is built to provide a prediction of the cylinder pressure from the definition of a burn rate function. Then the estimator gains are computed to correct and compensate for measurement and modeling errors. The robustness of the model-based estimator is tested under different input noise conditions and compared with a conventional signal processing method. The EKF-RLSE estimator described in Chapter 4 is designed for specific operating points. In Chapter 5, linear parameter varying (LPV) techniques are used to expand 1

28 the region of estimation to cover the engine operating map, as well as allow for real-time cylinder estimation during transients. The model to be designed is of LPV form, where the scheduled variables are the air-per-cylinder and the engine speed. For comparison to the LPV model, an extended Kalman filter is employed and examined to estimate the engine in-cylinder pressure in transient operating conditions, namely under a Federal Test Procedure (FTP) cycle. Then, an LPV-corrector is designed using a linear spline (LS) structure to be compared with the EKF in terms of accuracy and computational expense. Estimation techniques can be utilized in place of sensors to predict the pressure trace and indicated torque in a multi-cylinder engine. The work conducted in Chapters 4 and 5 assumes the availability of cylinder pressure transducers (lab-grade or production-grade) in each cylinder to obtain an accurate cylinder pressure trace, and consequently combustion metrics. The objective of the work in Chapter 6 is to build on and extend the results of Chapter 4 in estimating the individual in-cylinder pressure traces in a multi-cylinder engine, relying on a very restricted sensor set, namely a crankshaft speed sensor, a single production-grade pressure sensor, and an intake manifold pressure sensor. In doing so, a crankshaft model is developed and a sliding mode observer is employed to estimate the cylinder pressure using only the crankshaft speed fluctuation measurement. Furthermore, as an added enhancement, the cylinder pressure signal from one cylinder is utilized to adapt the friction and heat release parameters for more accurate estimation in all cylinders. 11

29 CHAPTER 2 STATE OF THE ART IN CYLINDER PRESSURE MODELING AND ESTIMATION 2.1 Introduction Although combustion engines have a long history going back to the nineteenth century, they continue to be refined and improved. One of the important variables that characterizes a combustion engine is the in-cylinder pressure, which has been measurable only on laboratory setups, where very accurate sensors and sophisticated post-processing can be utilized to provide the level of detail required for combustion studies. Therefore, most of the work presented in the literature during the 198s and early 199s was focused on using the in-cylinder pressure for combustion analysis and diagnostics. However, the recent availability of cost-effective piezoelectric transducers (whose cost can be up to two orders of magnitude lower than a lab-grade sensor) and the progress in on-line computational capability have led to the exploration of opportunities to use the in-cylinder pressure as a feedback signal for combustion control [17, 18]. Herein, a summary of the in-cylinder pressure applications will be given, focusing more on the on-line practices. Furthermore, some of the key works achieved on in-cylinder pressure modeling and estimation will be highlighted. 12

30 2.2 Applications of Cylinder Pressure In-cylinder pressure provides rich information that can be used for analyzing and controlling the combustion process. In particular, it can be used to obtain the combustion heat release rate (HRR), which is a key consequence of combustion. Consequently, controlling the combustion mechanism requires that combustion metrics for feedback be developed with explicit knowledge of the HRR characteristics. Furthermore, explicit knowledge of the cylinder pressure is of great value for combustion diagnostics and misfire detection Closed Loop Control The absence of direct feedback from the combustion chambers dictates exclusive reliance on feed-forward schemes, based on empirical maps that determine the controlled parameters to be used in relation to a set of pre-established operating conditions (typically in terms of engine speed and torque demand). This limits the possibility of compensating for deviations from the best open-loop calibrations by adjusting the fuel quantity, injection timing, and spark timing on a cycle-to-cycle basis [19]. Furthermore, to run an engine in one of the alternative combustion modes or to use flex-fuel, feedback control is required to maintain a desired condition in the cylinder based on a feedback signal, usually the cylinder pressure. The goal of the combustion control coincides with the main goal of the overall engine controller which typically provides the required engine torque (steady state and transient) with minimal fuel consumption, while meeting the legislated exhaust emission standards. Each factor has a different weight as explained in Section 1.3. Although it is possible to get a feedback signal for the torque, it is almost impossible 13

31 Figure 2.1: General scheme for combustion control. to find a direct feedback signal for the nitrogen oxides (NOx), hydrocarbon (HC), or particulate matter (PM). Therefore, we need to devise metrics that correlate with emission factors. The schematic shown in Figure 2.1 depicts a general prototype used in most of the work done in the literature for feedback combustion control. Several metrics have been used in the literature for cylinder feedback control. These metrics are derived from the pressure signal directly and indirectly through the heat release approximation. Gregory et al. [2] used peak pressure (P max ) and the crank angle location of peak pressure (CA P max ), where the latter is simple to compute, since it does not require a pegging process 2. The most frequently used metric as feedback is the crank angle location for 5% burn fraction (CA 5 ) [21, 22, 23], while the most common choices for 2 Pegging is explained later in Section

32 control inputs are: spark timing, IVC 3 and EVO 4 timing, injection quantity, injection timing, and injection pressure. It is not reasonable to try to use single loop feedback control for these inputs, since each of their effect is coupled. One study used decoupled controllers for the IVC and EVO, each with different time responses (slow and fast, respectively) to mitigate the coupling between the two inputs on the combustion system [2] Combustion Modes and Flex Fuel It is well known that compression ignition (CI) engines produce higher amounts of NOx and PM emissions as a consequence of the high gas temperature during combustion and the rich unburned fuel in some regions in the cylinder, respectively. Figure 2.2 shows the concentration of the NOx and PM regions with respect to cylinder temperature (T ) and equivalence ratio (φ). In order to meet the stringent emission regulations on combustion engines, several alternative combustion mechanisms (LTC, PCCI, and HCCI defined below) have been introduced to reduce NOx and PM. The main idea is to avoid NOx and PM regions by choosing the appropriate combustion temperature and air-to-fuel ratio (AFR) [1]. Low Temperature Combustion (LTC) is intended to control the combustion process to keep the temperature below 16 o K. This is usually done by introducing large amounts of cooled EGR to achieve a lower combustion temperature. Nevertheless, LTC has some drawbacks, such as being limited to low loads, having lower efficiency, and emitting high carbon monoxide (CO) and HC [24]. Another alternative combustion mode is the Premixed Charge Compression Combustion (PCCI) which aims 3 IVC: intake valve closing. 4 EVO: exhaust valve opening. 15

33 Figure 2.2: NOx and soot formation regions on φ-t map [1]. to allow sufficient time for the fuel to premix with the charge by introducing large amounts of EGR. The piston bowl needs to be modified to enable faster mixing and to avoid the adherence of the fuel to the cylinder walls. PCCI allows the high equivalence ratio regions of the mixture and the combustion temperature to be reduced, which in turn reduces the PM and NOx, respectively. Some of the issues with PCCI is misfiring under low load, which causes high HC and CO emissions as well as low fuel efficiency and noise under high load [24]. The last major alternative combustion and probably the most successful mode is the Homogeneous Charge Compression Ignition (HCCI) which can be run in both Diesel or gasoline engines. HCCI is a form of internal combustion in which a premixed charge of fuel, air, and residual is compressed to the point of ignition. The 16

34 main advantage of the HCCI engine is its high fuel efficiency with low NOx due to the lower peak combustion temperature compared to the conventional Diesel engine [25]. The main challenge is that the HCCI engine does not have a combustion initializer. Instead, combustion is achieved by regulating the temperature, pressure, and composition of the mixture which makes it very sensitive to any changes in the cylinder conditions [26]. The main differences between the three combustion modes (LTC, PCCI, and HCCI) are in the composition of mixture and the method used for preparation (conventional, partially premixed, and premixed). The current goal is to achieve multimode combustion in a traditional engine. Mostly, LTC, PCCI, or HCCI combustion operates at idle and low load conditions, while preserving conventional combustion under high load. Controlling the combustion mechanism (decided mode) requires specific combustion metrics for feedback. The computation of the combustion metrics requires explicit knowledge of the HRR characteristics which can be obtained from the in-cylinder pressure. Another important subject of interest is flexible fuel vehicles, which are sometimes called dual-fuel vehicles. These are designed with IC engines that are capable of running on more than one fuel, mostly gasoline mixed with ethanol which can be considered a renewable fuel. However, gasoline and ethanol have different physical and chemical properties which unavoidably affect the vehicle performance. Therefore, engine controllers such as fuel injection and spark timing must be optimized depending on the estimation of the ethanol content which typically ranges from % to 85% [27]. Stefanopoulou et al. demonstrated the possibility of estimating the ethanol content in direct injection engines using latent heat of vaporization (LHV) calculated based 17

35 on in-cylinder pressure measurements [28, 29]. Moreover, the algorithm they derived offered the possibility of calculating the mass air flow (MAF) sensor drift and fuel injection shift as well Combustion Diagnostics and Misfire Detection Knowledge of the in-cylinder pressure online allows for early detection of engine malfunctions as well as misfire detection. Engine diagnostics plays an important role in various fields, especially in large Diesel engines such as the ones in high seas ships which spend months offshore or at remote oil sites in the middle of the desert. A review of different condition-monitoring and fault-diagnosis is given in [3] where Fourier analysis and neural networks methods are studied. Watzenig et al. conducted an interesting study on the possibility of identifying compression losses and blow-by, using nonlinear model-based parameter identification [31]. Misfire detection has been one of the most important goals of On-Board Diagnostics systems (OBD). Moskwa et al. presented detailed studies in [32, 33, 34] on the possibility of detecting a misfire using the crankshaft speed to estimate cylinder pressure and consequently the combustion heat release. Finally, in [35], Schiefer et al. surveyed several other advantages of in-cylinder pressure information such as sensor replacements, higher comfort, acoustic improvements, and improved fuel economy. 2.3 Control-Oriented Cylinder Pressure Models To be able to capture the behavior of the combustion process, crank-angle models are generally employed, allowing for the characterization of the cylinder pressure. Combustion in the cylinder can be represented by a single-zone or multi-zone thermodynamic model. The multi-zone model assumes uniform pressure in the cylinder, but 18

36 the composition, mass, and temperature are calculated separately for the burnt and unburnt gases. Therefore, such a model is too complicated to be applied to modelbased observation (or control) and adds little to the observer design compared to the computational burden. On the other hand, the models that are usually used for the pressure estimator are based on the zero-dimensional model of the in-cylinder pressure from the intake valve closing to the exhaust valve opening (IVC EVO). These models can be obtained by applying the energy conservation law for a closed thermodynamic system to the control volume representing the cylinder, while neglecting leakages and crevice volumes. The mixture of air, fuel, and residuals is treated as an ideal gas with constant specific heats. Although the above assumptions simplify considerably the combustion process (particularly neglecting the fuel evaporation and mixing dynamics), they lead to a simple and computationally efficient model that can be used for real-time estimation and control applications [36, 37], given by dp cyl dθ + γ dv cyl V cyl dθ p cyl = γ 1 ( dqg V cyl dθ dq ) ht dθ, (2.1) where the polytropic coefficient (γ) is usually assumed constant and typically has a value within , and the cylinder volume (V cyl ) and its derivative are calculated using the geometric relations for the crank-slider mechanism [38]. In Equation 2.1, p cyl represents the in-cylinder pressure, θ is the crank angle, and Q g is the apparent gross heat release, which is directly correlated to the fraction of burned fuel during the combustion. The term Q ht, characterizing the heat transfer through the cylinder wall, is typically calculated assuming purely convective heat transfer, by estimating the convection coefficient through the Woschni correlation [39]. The standard convection heat 19

37 transfer is given by Q ht = ha(t cyl T w ). (2.2) Woshni s heat transfer model for estimating the convection coefficient h is [4] h = 3.26(p cyl /1).8 w.8 B.2 T.546 cyl. (2.3) However, the implementation of the above model increases the complexity of the cylinder pressure model considerably. The heat transfer model implies the knowledge of the in-cylinder gas temperature (T cyl ), the cylinder wall temperature (T w ), and the average cylinder gas velocity (w), thus requiring additional models to calculate (or approximate) such variables. In addition, the Woschni correlation expression has several exponents which are computationally expensive. The only parameters that can be obtained up front are the surface area of heat transfer (A), and the cylinder bore (B). Equation 2.1 has another important term that requires identification, which is the apparent gross heat release dq g. The most common way to represent dq g can be written as dq g dθ = m fuelq LHV dx b dθ, (2.4) where m fuel is the fuel mass, Q LHV is the lower heating value, and x b is the mass fraction of fuel burned. For a spark ignition engine, one Wiebe function might suffice, whereas for a compression ignition engine operating a pilot and main injection strategy, the burn rate is modeled through a linear combination of three Wiebe functions [41, 42]; this will be discussed further in Chapters 4 and 5. Another interesting approach to modeling the cylinder pressure of the gasoline engine was exploited in [43, 44]. The model is formulated as an interpolation between 2

38 Figure 2.3: Two-envelope pressure model. two polytropic pressure envelopes, as ( ) γc ( ) γe VIV C Vpeak p(θ) = p IV C f mix(θ) + p peak (1 f mix(θ)), (2.5) V (θ) V (θ) and can be depicted as in Figure 2.3. The function used for the interpolation called mixing function is modeled as a Wiebe function, given later in Equation 4.9. In this expression, p IV C and V IV C are evaluated at IVC while p peak and V peak are evaluated at the peak pressure in the ideal Otto-cycle. Unfortunately, [43, 44] do not give a comparison of this model compared to the first model which is based on combustion physics. 21

39 2.4 Pressure Estimation During the past two decades, there have been several attempts to obtain the incylinder pressure depending on available signals and the application discussed in the following Using the Cylinder Pressure Signal As suggested in [45], processing cylinder pressure data for combustion analysis and control requires several operations to be performed. First, hardware filtering is performed using frequency domain methods to obtain a signal spectrum with the desired frequency content and minimum sampling time to avoid aliasing problems. Next, the data are sampled, typically in the crank angle domain using the 6-2 production encoder mounted on the engine flywheel. Particular attention is required to diagnose crank offset errors, leading to large errors in the computation of heat release and indicated mean effective pressure (IMEP). Typical accuracy requires knowledge of Top Dead Center (TDC) to within 1 degree. Software filtering, based on Bessel or Butterworth filters with zero phase shift, is then applied to remove the noise from the sensor output. Finally, since the piezoelectric transducers measure only relative pressure and the drift varies slowly so that it can be considered constant for each cycle [46], the sampled signal from the sensor must be referenced to a known pressure value by applying an offset to the pressure to make it agree with the pressure under a known condition. To perform this final operation, often known as pegging, two general paths may be followed, namely referencing to the pressure in the intake or exhaust manifold at specific crank angles (i.e., TDC during valve overlap), or referencing through the 22

40 use of model-based algorithms. Using the manifold pressure as a reference signal is a simple and quick method, but may lead to loss of accuracy due to the highly unsteady flow conditions. Therefore, for cylinder pressure reconstruction, model-based algorithms are generally preferred. Most of the methodologies currently adopted for estimating the pressure trace during the compression phase use a polytropic transformation, with either a fixed or variable coefficient [47]. Polytropic Model The voltage signal of the cylinder pressure sensor has a linear dependency on the cylinder pressure (in reality, there is a small hysteresis nonlinearity), given by E(θ) = K s P (θ) + E bias, (2.6) where K s is the sensor gain and E bias is the sensor offset. During the compression stroke but before the combustion starts, the internal combustion engine follows a polytropic process; therefore, the process can be modeled with a polytropic model as: P mot (θ) = [ ] γ V (θref ) P (θ ref) = c(θ)p (θref). (2.7) V (θ) The polytropic model is valid only for the polytropic process which means that the pressure samples must be taken between intake valve closing and start of injection (IVC SOI). The following explains several pegging methods that were introduced in the literature. Methods for Cylinder Pressure Pegging Lee et al. and Klein et al. conducted surveys on several pegging methods in [48] and [46], respectively. The following is a summary of the most frequently used pegging methods in the literature while Appendix C provides more details: 23

41 Two-point referencing: Assuming fixed polytropic coefficient; noise can be reduced by averaging over several points. Three-point referencing: May overcome the drawback of the previous method which relies on the assumption of the fixed polytropic coefficient. This method deals with γ as a variable. Miscalculating the polytropic coefficient will propagate more errors in estimating the sensor pressure offset. Least square method: Several measurements are used through regression calculations to derive the least square projection assuming that the sensor gain K s is known and the polytropic coefficient γ is constant. Least square with a variable polytropic coefficient: The polytropic coefficient is a variable that can be calculated using the polytropic model and a first-order auto-regressive formulation that filters the estimation to achieve a higher degree of robustness. Furthermore, Hedrick et al. applied nonlinear least-square to solve for the offset, assuming a variable polytropic coefficient using Newton methods [49]. Another interesting approach using the extended Kalman filter was applied to the polytropic process by Klein et al. in [5] to estimate the offset. That approach deals with the polytropic coefficient as a changing variable and shows better performance compared to least square methods. Most results presented in the open literature for cylinder pressure signal processing are oriented to heat release analysis or combustion diagnostics, which are typically done off-line by averaging the cylinder pressure measurements over 1 or more cycles. 24

42 2.4.2 Using Crankshaft Speed Crankshaft speed fluctuations have been used to study combustion events in the engine. Moskwa et al. introduced an estimation method for cylinder pressure in multi-cylinder SI engines based on the measurement of the engine speed [33, 51]. This approach uses a sliding mode observer to overcome the problem of nonlinearities in the engine model. A problem of system observability was explored with means to reduce the estimation errors. Furthermore, the possibility of using the estimated cylinder pressure for engine misfire detection was examined. Ponti et al. employed a different approach by extracting combustion pressure from the estimated cylinder pressure and they found a linear relationship between the low harmonics content of the combustion pressure and the difference between synthetic speed that is strictly related to the indicated torque and the synthetic speed in case of a misfire [52]. Guezennec et al. used stochastic models to estimate the cylinder pressure from the acceleration, the velocity, and the crankshaft position [53]. Furthermore, Guzzella et al. presented a frequency response based on an adaptive technique that uses the phase at engine firing frequency as a feedback to correct some parameters in the pressure model [44] Other Sensors Other sensing technologies were researched as alternatives or supplements to cylinder pressure sensors, such as torque sensors and ion current sensors. Larsson et al. showed the possibility of using torque sensors with the cylinder pressure model introduced in Equation 2.5 and showed good results for low speeds only because of torsional resonances which increase with higher engine speeds [43]. On the other 25

43 hand, ion current sensors showed some disadvantages such as: high sensitivity to the AFR and very lean operations, weak signal at high speeds and low load operations, and deposition of carbon between the electrodes which produces a DC offset [54]. 2.5 Summary In-cylinder pressure is one of the most important engine variables for combustion analysis and diagnostics. The recent availability of cost-effective piezoelectric transducers and the progress in on-line computational capability have led to the exploration of using the in-cylinder pressure as a feedback signal for combustion control. However, most results presented in the open literature for cylinder pressure signal processing are oriented to heat release analysis or combustion diagnostics, which are typically done off-line by averaging the cylinder pressure measurements over 1 or more cycles. Therefore, the techniques presented cannot be utilized for closed-loop combustion control on a cycle-to-cycle basis, wherein knowledge of multiple pressure cycles is required. In conclusion, it is evident that a real-time estimation method to reconstruct the pressure trace is a major requirement for feedback combustion control. 26

44 CHAPTER 3 MATHEMATICAL TOOLS FOR ESTIMATION 3.1 Introduction Estimation is one stage of four steps in modeling a system. The other three are representation, measurement, and validation [55]. Estimation is also a fundamental component in designing a full-state feedback control-law, where designing an observer plays a vital rule especially if the full state is not available [56]. Throughout this thesis, several techniques for parameter or state estimation or a combination of the two are applied to either linear or nonlinear models. In this chapter, four main estimation methods are introduced so that the reader becomes more familiar when they are applied. The four major estimation techniques are: extended Kalman filter, linear parameter varying state estimation, sliding mode observers, and adaptive observers. 3.2 Extended Kalman Filter The extended Kalman filter (EKF) will be employed to estimate the cylinder pressure in Chapter 4. Before describing the extended Kalman filter, a quick introduction to Kalman filter theory will be given. The Kalman filter is a recursive minimum mean-squared state filter 5 which was developed by Rudolf Kalman in The 5 Minimizes E{(x ˆx) (x ˆx) Z} which is equivalent to ˆx = E{x Z} based on the fundamental theorem of estimation theory, where is the condition sign 27

45 algorithm is based on a recursive estimation of the conditioned expectation of the state of a linear Guass-Markov process, x(k) given noisy measurements up to time k, represented as discrete linear time-invariant (LTI) system: x(k) = Φx(k 1) + Ψu(k 1) + Γw(k 1), (3.1a) z(k) = Hx(k) + v(k), k =, 1,..., (3.1b) where x R n is the state vector, u R m is the input vector, z R p is the noisy output vector, w R l is the system noise vector, v R p is the measurement noise vector, Φ R n n is the system matrix, Ψ R n m is the input matrix, Γ R n l is the system error matrix, and H R m n is the output matrix. The filter has a predictor-corrector format as ˆx(k k) = ˆx(k k 1) + K z(k k 1), (3.2) z(k k 1) = z(k) H ˆx(k k 1), (3.3) where ˆx(k k 1) is the state predicted using only the model, while z is usually called the innovations process. The steady state Kalman gain matrix K is obtained by solving the algebraic Riccati equation P = ΦP [I H T (HP H T + R) 1 HP ]Φ T + ΓQΓ T, (3.4) so that K = P H T [HP H T + R] 1, (3.5) where P is the state covariance matrix. Assuming w(k) and v(k) are mutually uncorrelated and jointly Gaussian white noise sequences, Q and R can be obtained from 28

46 the second order statistics E[w(i)w(j) T ] = Q(i)δ ij, (3.6) E[v(i)v(j) T ] = R(i)δ ij, (3.7) where δ ij = 1 only if i = j (otherwise it is zero). The steady-state Kalman filter theorem guarantees stability of the estimator given that the system in Equation 3.1 is asymptotically stable (i.e., all the eigenvalues of Φ lie inside the unite circle) and observable (a requirement for designing any observer) and the noise is stationary with known mean and variance [55]. The steady-state Kalman filter is limited to discrete, linear time-invariant systems. It is well known that in practice such systems rarely exist. The Extended Kalman Filter (EKF) is a form of the Kalman filter extended to continuous and nonlinear systems of the form: x(t) = f[x(t), u(t), t] + G(t)w(t), (3.8a) z(t) = h[x(t), u(t), t] + v(t), (3.8b) where f : R n R m R R n is, in general, a smooth nonlinear mapping, G : R R n is a continuous mapping, and h : R n R m R R p is the system output nonlinear mapping. The EKF is implemented in a predictor-estimator as in the regular KF; however, the prediction is obtained by integrating the nonlinear model equation starting from a nominal point ˆx(θ k θ k ) resulting from the k-step estimation to obtain the predicted state ˆx(θ k+1 θ k ): ˆx(θ k+1 θ k ) = ˆx(θ k θ k ) + θk+1 f[ˆx(θ θ k ), u(θ), θ]dθ. (3.9) 29

47 The key behind the EKF is that the nonlinear model is utilized to obtain the predicted state. Then, a linearized model calculated around that point is used to obtain the estimator gain. The system matrices are recalculated at the predicted state value, according to d dθ Φ(θ, θ k) = f(θ)φ(θ, θ k ), (3.1a) Φ(θ k, θ k ) = I n, θ k. (3.1b) To obtain the transition matrix Φ(θ k+1, θ k ), the differential equation above must be solved. For time-varying systems, this task is generally difficult. However if f(θ) is diagonal (as we will see is the case with the cylinder pressure model in Chapter 4), the transition matrix can be calculated according to [57] ( θk+1 ) Φ(θ k+1, θ k ) = exp f(θ)dθ θ k. (3.11) In the following derivation of the EKF, it is assumed that u(θ) is a piecewise continuous function of crank angle for θ [θ k, θ k+1 ]. Furthermore, because the sampling is done at each crank angle, k will be used as a shortened notation for θ k. The transition matrix Φ and the output matrix H are then used to calculate the filter predicted covariance P (k + 1 k), estimated covariance P (k + 1 k + 1), and gain K(k + 1) using P (k + 1 k) = Φ(k + 1, k)p (k k)φ T (k + 1, k) + Q(k + 1, k), K(k + 1) = P (k + 1 k)h T (k + 1)[H(k + 1)P (k + 1 k)h T (k + 1) + R(k + 1)] 1, P (k + 1 k + 1) = [I K(k + 1)H(k + 1)]P (k + 1 k), (3.12) where Q is the model noise covariance and R is the sensor noise covariance. Finally, the estimation equation is obtained using the discretized model with the correction 3

48 factor, according to ˆx(k + 1 k + 1) = ˆx(k + 1 k) + K(k + 1)[z(k + 1) H(k + 1)ˆx(k + 1 k)]. (3.13) It should be mentioned that there is no guarantee of performance for the EKF designed this way, since there is no convergence assurance especially with relatively low sampling rates [58]. Therefore, it is sometimes viewed as an ad hoc filter [55]. Based on the introduced structure, the EKF is derived in this manner for the purpose of estimating the cylinder pressure on-line. The results in Chapter 4 show that with some modifications, the EKF can effectively alleviate the sensor noise, mitigate the effect of model uncertainties, and predict the sensor offset. 3.3 Linear Parameter Varying Systems Linear parameter varying (LPV) systems are a special class of linear time varying systems (LTV). Control design for LPV systems is very similar to the traditional gain scheduling approach used for nonlinear system control. However, with the LPV structure, stability proofs for the controller are possible, which is the main downfall of traditional gain scheduling control. The scheduling parameters vary with the operating conditions, and their values and rate of change must be bounded. A general continuous-time LPV system has the following state space form: x = A(ρ)x + B(ρ)u, (3.14a) y = C(ρ)x + D(ρ)u, (3.14b) where x R n, A R n n, u R m, B R n m, y R p, C R p n, D R p m, and ρ is the scheduling parameter vector. 31

49 Lyapunov functions can be applied to analyze the stability of linear parameter varying systems. The Lyapunov equation for a continuous LPV system is given by A T (ρ)p (ρ) + P (ρ)a(ρ) + P (ρ) <, (3.15) where P (ρ) R n n is the symmetric positive definite Lyapunov matrix. The general approach to solve equation 3.15 is to use the linear matrix inequality (LMI) with the use of numerical methods for the optimization problem [59]. The system dependency on the parameter varying signal can be linear or nonlinear. A special case of LPV systems is to have the parameter varying coefficients as linear spline functions (LS), which has shown some applications in system identification and control of automotive systems [6, 61]. In Chapter 5, the LPV structure will be used to design a stable Kalman filter to estimate the cylinder pressure allowing for wider range of estimation. 3.4 Sliding Mode Observer The sliding mode observer will be utilized in Chapter 6 to estimate the cylinder pressure using the crankshaft speed fluctuation. Through the past two decades, sliding mode control (variable structure systems) has shown great performance in nonlinear-systems control, since it has excellent robustness properties. Sliding mode theory is based on the concept of forcing the system motion into desired manifolds in the system state space. It is an excellent tool for controlling complex and high-order systems with uncertainties such as complex turbochargers (two-stage), where actuators may introduce switching behavior. For more details on sliding mode design and application, one is referred to [62, 63]. 32

50 Figure 3.1: Switching surfaces for multi-dimensional sliding mode. In this method, the controller switches the feedback gains based on the location of the current state in the phase plane. It is desired that the states move to the switching line (or surface for multi-dimension), then slide along the switching line towards the origin as shown in Figure 3.1. Without lose of generality, we will consider a second-order model to simplify the illustration. The switching line is described by σ = Cx 1 + x 2 =, (3.16) where C is some constant which determines the slope of the switching line in the phase plane and therefore the motion dynamics after the switching line is reached. To design the controller and find the feedback gains, a Lyapunov function candidate is chosen as V = σ 2. To achieve the desired trajectory, it is required that V = 2σ σ <. Therefore, a variable structure control law should be designed such that σ σ <, which is called the enforcement condition. 33

51 The sliding mode observer (SMO) is based on the same idea behind the sliding mode control (see [64] for an excellent tutorial). To illustrate the design of a sliding mode observer, consider the LTI system x = Ax + Bu, x R n, (3.17a) y = c 1 x 1 + c 2 x 2, x 1 R l, det(c 2 ) =, (3.17b) where y R p is the output vector, A R n n is the system matrix, B R n m is the input matrix, and C = [c 1 c 2 ] R m n is the output matrix. The system is first transformed into a block-observable form where the output appears as one of the states (similar to reduced-order observer), in the manner ( x1 y ) ( ) Il = c 1 c 2 }{{} T ( x1 x 2 ). (3.18) The system in Equation 3.17 can be transformed into the new co-ordinates as ( x1 y ) = T AT 1 ( x1 y ) + T ( B1 where T is a transformation matrix of appropriate dimension. B 2 ) u, (3.19) Note that the previous step could be eliminated if the output was indeed one of the system states. Having the system in the block-observable form, the SMO can be designed as: ˆx 1 = A 11ˆx 1 + A 12ˆy + B 1 u + LV, (3.2a) ˆy = A 21ˆx 1 + A 22ˆy + B 2 u + V. (3.2b) 34

52 For the system to be observable, the pair (A 21,A 11 ) must be observable [57]. The error ( x = x ˆx) dynamics are x 1 = A 11 x 1 + A 12 y + LV, (3.21a) y = A 21 x 1 + A 22 y + V. (3.21b) where V = Msign( y) and the the variable sliding mode gain (M) is chosen such that y has opposite sign to y to enforce the sliding mode. Once the sliding mode is enforced ( y = ), the control equivalence principle [63] is used V eq = A 21 x 1, (3.22) to obtain the motion equation x 1 = (A 11 LA 21 ) x 1. (3.23) The final stage is to design L such that the motion equation is asymptotically stable ((A 11 LA 21 ) < ). Although the SMO has proven to very helpful in many applications, it suffers from undesirable oscillations called chattering. The main reasons behind the chattering is the limitation in the switching frequency and the unmodeled dynamics of the system. Several solutions have been proposed to mitigate the chattering such as using a boundary layer in [65] and [66]. This method tries to avoid the discontinuous fluctuation in a small neighborhood about the switching surface (s = ). The method does mitigate the chattering; however, the tracking (convergence) becomes sluggish as this boundary become wider. Another approach used in [67] is to design the SMO gain (M) to be a function of the system state (instead of a constant). The method is further illustrated in [68] proving the possibility to reduce the chattering. The SMO 35

53 will be used in Chapter 6 to estimate the cylinder pressure using the crankshaft speed fluctuation and many of the aspects introduced here will be utilized. 3.5 Adaptive Observer Adaptive controllers and observers have been studied extensively through the last 2 years; however, it seems that this field is not mature due to many issues regarding the stability of the designs. Nevertheless, a specific class of systems was proved to be amendable for adaptive control or observer design with stability and robustness criterion as shown for example in [69] and [7]. In this dissertation, a direct 6 adaptive observer is employed to correct the friction and the heat transfer models adaptively. The design will be based on the principle of certainty equivalence which assumes that the uncertain parameters are know when designing the observer. Then, the designer comes back to design dynamics for the uncertain parameters aiming to make their estimates converge to the actual values. The system is assumed to have the form x = Ax + Ψθ, (3.24a) y = Cx, (3.24b) which is similar to the system in Equation 3.17 with uncertainties (θ) and known regression (Ψ) entering the system linearly. At this point, the principle of certainty equivalence will be used to design an observer (Luenberger observer) as ˆx = Aˆx + Ψθ + K(y C ˆx), (3.25) 6 Update law directly modifies the observer parameters. 36

54 and the error dynamics become ˆe = (A KC)e, (3.26) where the observer gain (K) is designed such that A KC is Hurwitz; therefore, the error converges to zero. The next step is to consider the uncertainty as unknown to get ˆx = Aˆx + Ψˆθ + K(y C ˆx). (3.27) In this case the error dynamics contain the uncertainty error ( θ = θ ˆθ) ˆe = (A KC)e + Ψ θ, (3.28) where the uncertainty error dynamics ( θ = ˆθ) must be designed so that the overall error system is asymptotically stable. To study the stability of the error system, Lyapunov analysis is applied with Lyapunov candidate function 7 V = e P e + 1 2ν θ θ, (3.29a) [ V e e + θ Ψ P e ˆθ ]. (3.29b) ν To make V so that the error system is stable, the adaptation dynamics has the form ˆθ = νψ P e. (3.3) There are two issues with this result. First, the adaptation dynamics are a function of the error, which might be unavailable. This can often be avoided by transforming the system into an adaptive observer form. Second, to prove convergence of the 7 Because A KC is Hurwitz by design, there exists P = P T > such that P (A KC) + (A KC) T P I [71] 37

55 observer, V must be negative-definite, which is not always the case. To partially overcome this problem, the La Salle/Yoshizawa theorem (see Appendix F) can be used to prove the convergence of the state, but not the uncertainty parameters. The application of the Adaptive observer will be utilized with the SMO in Chapter 6 to estimate the engine cylinder pressures using crankshaft speed fluctuation and the cylinder pressure measurement from a single cylinder. 3.6 Summary In this chapter, four estimation methods have been presented to familiarize the reader with observers designs. Each one of those techniques will be employed at different points of this thesis. The extended Kalman filter will be applied in Chapter 4 to remove the affect of the cylinder pressure noise on the measurement as well as to estimate the sensor offset. In Chapter 5, linear parameter varying state estimation will be utilized to extend the region of estimation. Sliding mode observers and adaptive observer will be employed in Chapter 6 to overcome the uncertainties in the heat transfer model as well as the friction model. 38

56 CHAPTER 4 EKF ESTIMATION OF CYLINDER PRESSURE 4.1 Introduction As seen in Section 2.4.1, most applications related to the measurement of incylinder pressure are typically done off-line and rely on averaging the cylinder pressure measurements over 1 or more cycles. Herein, a real-time, model-based estimation methodology that reconstructs the engine cylinder pressure from the output of a piezoelectric transducer is discussed. The work to be presented in this chapter is intended as an enabler for the closed-loop control of combustion engines, providing feedback for the CA 5, the engine IMEP, and other metrics that can be used to control the engine output torque, fuel efficiency, or emissions. A typical production cylinder pressure sensor is subject to noise and offset issues. Starting with the raw signal from such a sensor, an Extended Kalman Filter (EKF) is designed in a predictor-corrector format to reconstruct the cylinder pressure. In order to do so, a model of the in-cylinder thermodynamics is built to provide a prediction of the cylinder pressure from the definition of a burn rate function. Then the estimator gains are computed to correct and compensate for measurement and modeling errors. An initial design of the model-based estimator is tested in simulation based on a high 39

57 fidelity engine model [72]. The estimator design is then refined and the study extends to include a thorough validation based on experimental data from a laboratory test bench [73]. Furthermore, some discussion on issues related to on-line cylinder pressure estimation is given, where the closed-loop combustion control requires the pressure trace to be processed on a cycle-to-cycle basis. In this case, the robustness of the model-based estimator is tested under different input noise conditions (first using a very accurate lab-grade pressure sensor, then a low-cost production-type sensor), and compared with a conventional signal processing method. It should be mentioned that the analysis and results shown in this chapter are applied to Diesel engines; however, similar analysis can be utilized for gasoline engines as we will see in the next chapter. 4.2 Cylinder Pressure Model for Estimator Design As mentioned above, the proposed estimation scheme relies on a simplified model of the in-cylinder processes during the compression, combustion and expansion phases. The predicted in-cylinder pressure is then utilized by the estimator to correct the output of the pressure transducer as depicted in Figure 4.1. The predictor-corrector form chosen for the estimator will compensate for the modeling errors generated by the approximations introduced Description of the Model Structure A simplified single-zone model that is based on thermodynamic analysis of the engine in-cylinder processes occurring from intake valve closing to exhaust valve opening (IVC EVO) is considered. Although this assumption simplifies considerably the combustion process (particularly neglecting the fuel evaporation and mixing dynamics 4

58 Figure 4.1: EKF estimator scheme. for direct injection engines), it leads to a simple and efficient model that can be used for real-time, model-based estimation and control. This approach is very well known in the literature [38, 74, 75], and has been applied for control-oriented models for SI and CI engine, including alternative combustion modes such as HCCI [76, 36, 37]. Figure 4.2: Control volume. 41

59 The combustion model is based on the conservation principle applied to a combustion chamber as the control volume shown in Figure 4.2 as follows: Conservation of Mass: dm cyl dt = i m i e m e, (4.1) Conservation of Energy: d(m cyl u cyl ) dt = Q cyl W cyl + i m i h i e m e h e, (4.2) Ideal Gas Law: p cyl V = m cyl RT cyl. (4.3) Under the assumption that no leakage occurs from the cylinder or crevice volumes, the conservation of mass is satisfied ( m cyl ). By combining the energy equation with the gas law and converting to the crank angle domain ( d dt = d dθ dθ = d ω), several dt dθ steps of manipulations can be applied (see Appendix B) to obtain the following model form: dp cyl dθ + γ V cyl dv cyl dθ p cyl = γ 1 V cyl dq n dθ = γ 1 ( dqg V cyl dθ dq ) ht dθ, (4.4) where the cylinder volume and its derivative are calculated using the geometric relations for the crank-slider mechanism [38] in the manner [ V = V c 1 +.5(Cr 1)(R + 1 cos(θ) (R 2 sin 2 (θ)).5 ) ], (4.5a) [ dv (θ) =.5V c (C r 1) sin(θ) + sin(θ)cos(θ) ]. (4.5b) dθ (R 2 sin 2 (θ)).5 The specific heats ratio γ of the gas mixture, typically varying due to heat transfer and the variation of mixture composition, is here assumed equal to 1.37 for simplicity [72]. 42

60 The heat loss term Q ht is conventionally approximated through the Woschni correlation for convective heat transfer as shown in Equations 2.2 and 2.3. However, this approach would increase the complexity of the model considerably, since it requires determination of the in-cylinder gas temperature and cylinder wall temperature. Since the objective of this model is to retain a very simple structure, the cumulative heat transfer is approximated by introducing a scaling factor α, hence assuming the term Q ht is a fraction of the apparent gross heat release Q g (dq ht = αdq g ). Note that this approximation, inadequate for the estimation of the instantaneous heat losses, leads to only marginal errors in the determination of certain combustionrelated variables, such as the 5% combustion timing, the peak pressure and the engine IMEP. This is generally due to the heat transfer becoming relevant only during the final portion of the expansion stroke, after most of the indicated work has been produced. When incorporated into Equation 4.4, the above assumption results in the following expression for the prediction of in-cylinder pressure: dp cyl dθ + γ dv cyl V cyl dθ p cyl = γ 1 [ (1 α) dq ] g V cyl dθ. (4.6) From a control standpoint, the above equation is representative of a Linear Time Varying (LTV) model, whose form will be applied to the estimator design. The apparent gross heat release rate is given in Equation 2.4, repeated here for convenience: dq g dθ = m fuelq LHV dx b dθ, (4.7) where Q LHV is the lower heating value and x b is the mass fraction of burned fuel. For a SI engine, one Wiebe function might suffice as we will see in the next chapter, whereas for a CI engine operating a pilot and main injection strategy, the burn rate 43

61 is typically modeled through a linear combination of three Wiebe functions [41]: dx b dθ = β dφ pilot dφ mainp 1 + β 2 dθ dθ + (1 β 1 β 2 ) dφ main d dθ, (4.8) where φ pilot characterizes the fuel energy released during the pilot injection, and the functions φ mainp and φ maind account for the premixed and diffusive burning phases associated with the main injection. Each function φ i takes the following form [42]: [ ( ) m+1 ] θ θ φ i = 1 exp a Δθ, (4.9) where the parameters, a, m, Δθ, together with β 1 and β 2 of Equation 4.8, are identified using available engine data [72]. The start of combustion θ is assumed equal to the start of injection, hence neglecting the fuel evaporation and ignition delay Model Calibration and Validation In light of the assumptions introduced, the model developed requires calibration on engine data to ensure a reasonable approximation of the in-cylinder pressure. Since we have two different data sets (simulation data and experimental data) for different engines, the calibration and validation were performed for each set separately. Simulation Data In light of the assumptions and approximations introduced, the cylinder pressure model developed contains a number of parameters that require calibration on engine data. For this study, a high-fidelity GT-Power model was adopted as a virtual engine, in lieu of experimental data. The detailed simulator was calibrated and validated on a heavy-duty six cylinder Diesel engine intended for an over-the-road tractor trailer. 44

62 The detailed model relies on a commercial engine simulation package in which the air and exhaust paths are characterized through a 1-D wave-action approach based on a high-order system of discretized partial differential equations [77]. A physically-based combustion model describes the in-cylinder processes, accounting for local mixture conditions in the premixed and diffusive combustion that characterizes direct injection Diesel engines [78]. This model is fully predictive, describing the physical and thermodynamic foundations of the combustion process with very limited calibration requirements. However, its structure is very complex and not suitable for on-line applications. For this reason, the simulator was adopted as a virtual engine to generate incylinder pressure data, allowing for the identification of the parameters of the controloriented model. The calibration was done on 11 operating conditions that approximate the 13-mode tests (steady-state test intended to represent common points in the operation of a heavy-duty engine). The points considered are specified in Figure 4.3, among the complete data set available from the simulations [79]. The model parameters requiring calibration are the heat transfer scaling factor α and the burn rate parameters (Equations ). To this end, an inverse thermodynamic model was applied to the cylinder pressure traces to obtain the net heat release rate, converting Equation 4.4 to the following form [8, 38]: dq n dθ = V cyl dp cyl γ 1 dθ + γ dv cyl γ 1 dθ p cyl. (4.1) From the net heat release rate, the burn rate is obtained from dx b dθ = 1 Q n (1 α) M fuel Q LHV dθ. (4.11) 45

63 Figure 4.3: Selection of operating points for model calibration. First, α is identified by forcing dx b = 1. The calculated burn rate is then used to fit the combustion model based on the three Wiebe functions, determining the related parameters through a least-square curve fitting process [36, 37]. After identifying the model on the 11 operating conditions shown in Figure 4.3, the parameters are scheduled with respect to engine speed, mass of fuel injected and the direct injection parameters (fuel quantity of pilot and main, and injection timing). Figure 4.4 shows the results of the fitting procedure for Mode 6, providing an acceptable level of accuracy for the objectives of this study. 46

64 Figure 4.4: Example of Wiebe function fitting (Mode 6). A validation of the model was constructed by calculating the cylinder pressure for all of the investigated cases and comparing to the values obtained from the engine simulator. Figure 4.5 shows the validation results for Mode 6 and Mode 11, respectively. The estimator model predicts the cylinder pressure well in all the tested cases, with errors limited only to the the combustion and expansion phase where approximations introduced for modeling the burn rate and heat transfer result in small pressure errors. Experimental Data Herein, the calibration and validation are performed on experimental data, collected on a light-duty Diesel engine, with characteristics given in Table 4.1. Eight engine operating points, summarized in Table 4.2, were acquired in steadystate conditions, varying engine speed and torque. 47

65 12 Actual Pressure Model 1 8 Pressure [bar] Crank Angle [deg] (a) Mode 6 12 Actual Pressure Model 1 8 Pressure [bar] Crank Angle [deg] (b) Mode 11 Figure 4.5: Cylinder pressure (engine simulator versus estimate). Each engine cylinder was equipped with a piezoelectric transducer, installed in the glow plug bore and synchronized with an encoder mounted on the engine flywheel. Pressure transducers were also inserted in the intake and exhaust manifolds. For each of the operating conditions, pressure data were acquired for 1 consecutive engine 48

66 Engine Type DI Diesel, Turbocharged, Intercooler Configuration Inline, four-cylinder Displacement 2499 cm 3 Bore and Stroke 92, 94 mm Compression Ratio 17.5:1 Connecting Rod Length 163 mm Valvetrain 4 per cylinder, DOHC IV O,IV C,EV O,EV C 76 o, 246 o, 473 o, 31 o Fuel Injection System Bosch Common-Rail CP3 Max. Power 15kW at 4r/min Max. Torque 32N m at 2r/min Table 4.1: Test engine specifications. Test No. Engine Speed [r/min] Engine Torque [Nm] , 73, 11, , 73, 11, 146 Table 4.2: Summary of experimental tests conducted for model calibration. cycles, with a resolution of 1 crank-angle degree. Although such resolution would be inadequate for an accurate thermodynamic analysis of the in-cylinder pressure, it is however chosen to represent a typical condition commonly found on many vehicles. Limitations in hardware costs often dictate design, requiring compensation for errors originating from the acquisition system. The signal from the pressure transducers was processed in order to perform a basic analysis of the in-cylinder pressure data. First, the raw signal was passed through a low-pass, zero-phase forward and reverse digital filter to remove the high frequency components resulting from the sensor noise. The filtered signal was then pegged using 49

67 the polytropic method with a nonlinear least square solver as follows [47]: P real (θ)v (θ) n = C, (4.12a) P real (θ) = P meas (θ) P offset, (4.12b) P meas (θ) P offset = C, (4.12c) V (θ) n f = P meas (θ) P offset C, (4.12d) V (θ) n min (f 2 ). (4.12e) C,n,P offset Finally, the signal was cycle-averaged to obtain the cylinder pressure trace. A simple inverse thermodynamic calculation was then used to estimate the heat release and burn rate profile that were used for the calibration of the heat transfer scaling factor α and the burn rate parameters in Equations 4.8 and 4.9; from the net heat release rate, the burn rate is obtained using Equation The model parameters may then be evaluated exactly as we did for the simulation data. The model parameters are scheduled with respect to engine speed, mass of fuel injected and the direct injection parameters (fuel quantity of pilot and main, and injection timing). Figure 4.6 shows the results of the fitting procedure adopted for the calibration of the burn rate, comparing the model with the results obtained from the cylinder pressure data at operating condition 8 (see Table 4.2). The validation was performed by comparing the model prediction for all of the tested conditions, showing reasonable approximation of the experimental cylinder pressure trace. Figures 4.7 and 4.8 show the validation results for operating condition 2 (see Table 4.2). As expected, errors are present in the predicted pressure, particularly during the combustion and expansion phase where the approximations introduced for modeling 5

68 Burn Rate [ ] TDC Cum. Burn Rate [ ] Calculated Fit Crank Angle [deg] Figure 4.6: Comparison of model identification results with burn rate curve obtained from the experimental cylinder pressure trace (operating point 8). the burn rate and heat transfer lead to a loss of accuracy. However, the estimation scheme developed is expected to compensate for modeling errors. This will be one of the criteria leading to the development of the Extended Kalman Filter. 4.3 Design of In-Cylinder Pressure Estimator As introduced above, the in-cylinder pressure model described by Equations (4.6) and (4.7) can be formulated as a linear, time-varying system: P (θ) = f(θ)p (θ) + b(θ)u(θ), (4.13) 51

69 .8 Burn Rate[ ] Cum. Burn Rate[ ] Crank Angle [deg] 1.5 Calculated Fit Crank Angle [deg] Figure 4.7: Validation of cylinder pressure model on experimental data: burn rate profile (test 2). where f(θ), b(θ), u(θ) are defined as: f(θ) = γ V cyl dv cyl dθ, b(θ) = γ 1 V cyl Q LHV (1 α), u(θ) = M f dx b dθ. (4.14) The above model form is the initial step towards the design of an estimator for the in-cylinder pressure that corrects the output of the piezoelectric transducer. In this sense, another important variable to consider in the study is the offset affecting the sensor output, which is typically unknown and varying for each pressure cycle. In this study, the offset is considered as an additional state variable P, hence augmenting 52

70 6 5 Actual Pressure Modeled Cylinder Pressure [bar] Crank Angle [deg] Figure 4.8: Validation of cylinder pressure model on experimental data: cylinder pressure (test 2). the model as follows: [ P x = P ] [ ] f(θ) x = x + }{{} F (θ), [ b(θ) ] u(θ), (4.15) z = Hx = [ 1 1 ] x Implementation In order to correct the pressure transducer output, rejecting sensor noise and modeling errors, an Extended Kalman Filter (EKF) is implemented in a predictorcorrector format [55]. The procedure for implementing the extended Kalman filter was introduced in Section 3.2; therefore, discretization is required. As a linear time 53

71 varying system, Equation 4.15 can be represented in the manner θ x(θ) = Φ(θ, θ )x(θ ) + Φ(θ, τ) [ B(τ)u(τ) + w(τ) ] dτ θ, (4.16a) Φ(θ, τ) = f(θ)φ(θ, τ), (4.16b) Φ(θ, θ) = I. (4.16c) To obtain the transition matrix Φ(θ k+1, θ k ), the differential Equation 4.16b must be solved with the initial condition in Equation 4.16c. For time-varying systems, this task is generally difficult. However, because F (θ) is diagonal, the transition matrix can be calculated as in [57], according to Φ(θ, τ) = exp [ θ F (σ)dσ ] ; (4.17) assuming u(θ) is piecewise constant function of crank angle for θ [θ k, θ k+1 ], the τ discretized system will have the following form: [ θk+1 x[k + 1] =Φ(θ k+1, θ k )x(θ k ) + + θk+1 θ k θ k Φ(θ k+1, τ)w(τ)dτ. ] Φ(θ k+1, τ)b(τ)dτ u(θ k ) (4.18) Furthermore, because sampling is executed at each crank angle, k will be used as a shortened notation for θ k, so that we can succinctly write x(k + 1) = Φ(k + 1, k)x(k) + Ψ(k + 1, k)u(k) + w d (k). (4.19) 54

72 The system matrices can be calculated as follows: [ θk+1 Φ(k + 1, k) 1,1 = exp γ θ k V (θ) ] dv (θ) dθ dθ ( ) γ V (θk ), V (θ k+1 ) = exp[γ(lnv (θ k ) lnv (θ k+1 )] = θk+1 ( ) γ V (θk ) γ 1 Ψ(k + 1, k) 1,1 = V (θ k+1 ) V (θ) Q LHV (1 α)dθ θ k = Q LHV (1 α) V (θ k+1 ) γ θk+1 θ k V (θ) γ 1 (γ 1)dθ Q LHV (1 α)(γ 1) V (θ k+1 ) γ V (θ k ) γ 1 Δθ. (4.2) Using a one crank angle resolution (Δθ = 1) and inserting the system matrices from Equation 4.2 into Equation 4.18 renders the model in the following form: x(k + 1) = ( V (k) V (k+1)) γ x + 1 }{{} Φ(k+1,k) z(k + 1) = Hx(k) = [ 1 1 ] x(k). [ QLHV (1 α)(γ 1) V (k) γ 1 V (k+1) γ ] u(k), (4.21) The procedure outlined here allows one to construct the EKF using the cylinder pressure model developed above, based on a recursive scheme. To initialize the estimator, the pressure in the intake manifold at IVC is used as an approximation for the cylinder pressure at the same crank angle Preliminary Validation of the Estimator As a preliminary step, the EKF is applied for on-line estimation of cylinder pressure using the simulation model built in GT-Power. In order to imitate the highfrequency noise signature characteristic of piezoelectric transducers, a random white noise with standard deviation of 1bar was added to the simulated pressure trace. 55

73 Figure 4.9 shows the results for two operating conditions representing high and low signal to noise ratio, with 7 bar as the pressure offset value. This particularly high value was chosen intentionally to exhibit how the estimator is able to overcome the offset only in part, despite the state augmentation. The reason is due to the covariance matrix (Q) used in the EKF calculation, which in this case uses zero variance (deterministic signal) for the offest. This formulation forces the EKF to place little weight on the measurement effect on the offset, with more weight given to the model. In order for the EKF to account for the measurement offset, a non-deterministic formulation is used, thus adding a variance to the Q matrix for the second state. This allows one to account for the slight changes in the offset within each engine cycle. The result of this estimation scheme is shown in Figure 4.1, where the EKF is compared with the input pressure signal (with applied noise and offset) and the pressure trace from the engine simulator. We see that only marginal improvement was achieved. The estimated pressure converges to the actual value around the TDC position, as observed from the errors calculated on the estimator states (pressure and offset) and shown in Figures 4.1(b) and 4.1(d). This behavior can be explained by analyzing the observability gramian, defined as [57] [ W o (k, k + 1) = H HΦ(k + 1, k) ] [ = 1 1 ( V (k) V (k+1)) γ 1 ]. (4.22) By calculating the value of the (lower left) (2,1) element in the matrix (W o21 ) from IVC to EVO, the matrix W o is generally ill-conditioned, except for a small region around the TDC position as shown in Figure That is, the conditionality of the observability matrix improves when the value of the lower left element deviates from 56

74 15 Input Pressure Estimated Pressure Actual Pressure 1 Pressure [bar] Crank Angle [deg] (a) Mode Input Pressure Estimated Pressure Actual Pressure 7 6 Pressure [bar] Crank Angle [deg] (b) Mode 8 Figure 4.9: Pressure estimation with augmented EKF (7 bar offset). 57

75 15 Input Pressure Estimated Pressure Actual Pressure 1 Pressure [bar] Crank Angle [deg] (a) Mode 1 (Comparison of pressure profiles) 5 Pressure Error [%] 5 Error in Estimated Pressure Error in Estimated Offset Crank Angle [deg] (b) Mode 1 (Errors on estimated states) 9 8 Input Pressure Estimated Pressure Actual Pressure 7 6 Pressure [bar] Crank Angle [deg] (c) Mode 8 (Comparison of pressure profiles) 5 Pressure Error [%] 5 Error in Estimated Pressure Error in Estimated Offset Crank Angle [deg] (d) Mode 8 (Errors on estimated states) Figure 4.1: Pressure estimation with corrected EKF (7 bar offset). 58

76 1. This implies that the system will be unobservable during the early stages of the compression stroke, preventing the estimator from converging to the correct states until operation is close to TDC Crank Angle [deg] Figure 4.11: (2,1) element in the observability gramian Extension for Offset Compensation In order to estimate the offset during the compression phase, hence anticipating the region up to the point where the system becomes observable, the pressure estimator derived to this point is augmented with a recursive least square estimation (RLS) scheme. The following is a summary for the formulation of the RLS scheme: The pressure sensor signal can be represented as: P meas (θ) = P real (θ) + P bias. (4.23) During compression and before fuel injection (or spark timing) the process can be modeled with a polytropic model as shown in Equation 2.7, repeated here 59

77 for convenience P real (θ) = ( ) γ V (θref ) P (θ ref). (4.24) V (θ) Substituting Equation 4.24 into Equation 4.23 gives P meas (θ) = ( ) γ V (θref ) P (θ ref) + P bias. (4.25) V (θ) Define the observation vector and the parameter vector to be used in the RLS as [ ( ) γ V (θ h(k) = ref ) 1 V (k) ] [ P (θref ) and ˆx(k) = P bias ]. (4.26) The covariance form [55] of the RLS algorithm is summarized as follows: ˆx(k + 1) = ˆx(k) + K(k + 1) [ z(k + 1) h(k + 1)ˆx(k) ], (4.27) [ ] 1 K(k + 1) = P (k)h (k + 1) h(k + 1)P (k)h 1 (k + 1) +, (4.28) w(k + 1) P (k + 1) = [ I K(k + 1)h(k + 1) ] P (k). (4.29) where K(k) is the RLS estimator gain vector and P (k) is the covariance matrix. Because the signal to noise ratio is low at the beginning of the compression stroke, a lower weight w(k) is assigned initially and then increased as the cylinder pressure increases. The RLS algorithm is utilized until the pilot injection occurs, after which the EKF is invoked. The initial condition for the EKF is simply set to be the last value reached by the RLS. The results of the RLS-EKF scheme are shown in Figure 4.12, for the same engine operating condition considered previously. Compared to the previous results shown in Figure 4.1, the RLS algorithm improves the performance of the estimator, where convergence occurs within 5-6 deg from IVC. Although some fluctuations due to 6

78 the singularity of the observability gramian are still present, the errors observed on the estimated states (pressure trace and offset) after convergence are less than 5%. Furthermore, having the offset estimation converging faster allows the EKF to achieve better estimation around TDC (especially for peak pressure). As a final comparison of the estimators developed, a set of combustion metrics were calculated and summarized in Table 4.3, for one specific engine operating point. The complete results for the all of the tested operating conditions are reported in Appendix D. The signals chosen, namely 5% burn rate location (CA 5 ), IMEP, Peak Pressure (PP) and related location (P P loc ) are considered important feedback signals for closed-loop combustion control [21, 22, 2]. The CA 5 is defined as the crank angle where the burn rate reaches 5% value, and is calculated using an inverse thermodynamic model based on Equation 4.1. In this study, the IM EP is calculated with reference to the sole compression and expansion phases, according to IMEP = W c,i V d = 1 V d EV O IV C pdv. (4.3) Based on the results shown in the table, it is evident that the RLS-EKF scheme provides a more accurate estimation, especially regarding the IMEP and peak pressure. 4.4 Application to On-Line Cylinder Pressure Estimation The estimator developed to this point has been applied to reconstruct the incylinder pressure trace from the output of the pressure transducer using a high-fidelity simulation. In this section, real-time implementation is considered. 61

79 15 Input Pressure Estimated Pressure Actual Pressure 1 Pressure [bar] Crank Angle [deg] (a) Mode 1 (Comparison of pressure profiles) 1 Pressure Error [%] Error in Estimated Pressure Error in Estimated Offset Crank Angle [deg] (b) Mode 1 (Errors on estimated states) 9 8 Input Pressure Estimated Pressure Actual Pressure 7 6 Pressure [bar] Crank Angle [deg] (c) Mode 8 (Comparison of pressure profiles) 1 Pressure Error [%] Error in Estimated Pressure Error in Estimated Offset Crank Angle [deg] (d) Mode 8 (Errors on estimated states) Figure 4.12: Pressure estimation using EKF with offset. 62

80 Combustion Metric Result with EKF Result with RLS-EKF CA 5 Error [CAD] 1 IM EP Error [%] P P Error [%] P P loc Error [CAD] 1 1 Table 4.3: Comparison of combustion metrics calculated from the EKF and RLS-EKF estimators, for a representative engine operating point. In order to obtain a pressure estimation with the desired accuracy for combustion control using a conventional signal processing approach, a synchronous average must be applied to the sensor output signal for several engine cycles, so as to remove the noise related to engine cyclical fluctuations as well as the high frequency noise induced by the piezoelectric transducer. This methodology will be developed here for comparison purposes because it has been often applied to engine combustion control. The idea is to apply a moving average (MA) filter to the pressure signal with a buffer of at least 1 cycles [18]. In mathematical terms, the pressure P i MA at the i-th cycle is calculated as P i MA (θ) = 1 H i+h 1 k=i [ P i cyl (θ) + P i ], (4.31) where P i is the referencing term and the buffer size H was chosen to be 1 engine cycles. Figure 4.13 shows the frequency response of such a filter. It is obvious that the moving average filter is a low-pass filter by nature. Although the use of low-pass filtering in the processing of the cylinder pressure is sufficiently accurate in steady-state engine operating conditions, it presents limitations during transient operations, where the pressure may vary significantly in the 63

81 1.8 Magnitude Freq [Hz] Figure 4.13: Frequency response for Moving Average filter. span of a few engine cycles. The lag and filtering effects caused by the signal processing prevent the controller from responding promptly in adapting the injection parameters to the new operating conditions, with a consequent degradation of the engine performance and emissions. For this reason, a model-based estimator that provides the in-cylinder pressure trace in real-time could have significant advantages, for instance enabling closed-loop combustion control on a cycle-to-cycle basis. With this in mind, the estimator derived in the previous section is applied here to an on-line pressure estimation problem and compared with results from the conventional signal processing approach (moving average filter) described above. Figure 4.14 illustrates a conceptual scheme for the comparative study. The comparative study is done initially with simulation data, then extended to include experimental data from the setup described in Section 4.2.2, with pressure 64

82 Figure 4.14: Comparative scheme for on-line pressure estimation. sensors of different accuracy. The pressure data were collected at the same engine cycle and in the same cylinder, using a standard non-intrusive installation in the glow plug bore. Data acquisition for each sensor was done at the same engine operating conditions, in order to ensure repeatability of the results. In the following, distinction will be made between a lab-grade sensor and a production-type sensor, the former referring to high accuracy pressure transducers typically used in the laboratory environment for precise measurements. The main specifications of the two sensors used in the study are listed in Table 4.4. Specification Lab-grade sensor Production-type sensor Measuring Range [bar] Sensitivity [pc/bar] Natural Frequency [khz] 9 13 Linearity [% FSO] ±.3 ±.8 Thermal Sensitivity Shift o C [%] ±2. ±2. Cyclic Temperature Drift [bar] ±.2 ±.6 Table 4.4: Specifications for the two pressure transducers utilized in the study 65

83 6 5 Actual Pressure Trace Lab Grade Sensor Prod. Grade Sensor 4 Pressure [bar] Crank Angle [deg] Figure 4.15: Comparison of pressure traces measured in the same cylinder with labgrade sensor and production-type sensor during one engine cycle (Test No. 1, cycle 58). Figure 4.15 shows an example where the outputs of the two sensors are compared for one engine operating condition (test 1 in Table 4.2). The signals are represented against the pressure trace averaged over 1 cycles. The higher noise of the production-type sensor is evident, particularly during the onset and development of the combustion. In order to visualize the difference between the noise characteristics and frequency content in both sensors, Figure 4.16 displays comparative traces in the crank angle domain and in the frequency domain. In particular, the lower plots represent the frequency responses of the two sensors, normalized with respect to the maximum frequency content. Although the cost of a production-type sensor is much lower than a lab-grade sensor (up to two orders of magnitude), the noise signature is more 66

84 Noise [bar] Norm. Freq. Content [x1 2 ] Lab Grade Crank Angle [deg] Frequency [Hz] Noise [bar] Norm. Freq. Content [x1 2 ] Production Grade Crank Angle [deg] Frequency [Hz] Figure 4.16: Noise characteristics for lab-grade (left) and production-type (right) pressure sensors. significant, hence requiring more effort in order to correctly estimate the cylinder pressure and related combustion metrics. One of the most important factors in comparing the RLS-EKF estimator with a conventional signal processing method (MA filter) is the throughput (computation and memory burden). In order to have a more accurate comparison, the RLS-EKF estimator algorithm was converted into C code. In terms of computations, the estimator designed using Equation (3.12) directly takes up to five times the elapsed time needed for the MA filter. However, note that it is possible to further manipulate the 67

85 K(k + 1) expression as follows: HP (k + 1 k)h T + R =P (k k) 11 Φ [P (k k) 12 + P (k k) 21 ]Φ 11 Φ 22 + P (k k) 22 Φ Q 11 + Q 22 + R. (4.32) This allows one to save one step (P (k + 1 k)), reducing the elapsed time to about four times that of the MA filter. On the other hand, the MA filter with a buffer of 1 engine cycles uses 2 MB of memory more than the RLS-EKF estimator, due to the buffering requirements Application to Lab-Grade Sensor Data As a starting point for this comparative study, the model-based estimator and the moving average processing method are applied to the experimental data obtained from the engine setup described in Table 4.1, using high-accuracy pressure transducers. In this case, since only minor noise is induced by the lab-grade transducer, the RLS-EKF estimator requires only minor tuning, which results in a reduction of the relative sizes of the elements in the R and Q matrices. The referencing term used by the MA filter in Equation 4.31 was assumed to be equal to the intake manifold pressure averaged over the valve overlap period. It is worth mentioning that this operation requires the intake manifold pressure to be sampled at a high frequency, with the same crank angle resolution as the cylinder pressure. Although such measurement is common in a laboratory setup, it is typically unavailable in production engines, where the standard Manifold Absolute Pressure (MAP) sensor provides only a low-frequency (cycle-averaged) signal. In this case, since the MA filter is unable to compensate for offset errors, an increase in the estimation error must be expected. 68

86 6 4 EKF MA Pressure Error [%] Crank Angle [deg] Figure 4.17: Error signature in crank-angle domain: error between estimators outputs and cylinder pressure trace obtained from average of 1 cycles (lab-grade sensor). In order to find a common baseline, namely an accurate pressure trace to compare the model-based estimator to the MA filter, the signal from the sensor was processed and cycle-averaged over 1 cycles, as discussed in Section The comparison between the estimator prediction and the MA filter output was extended to include the combustion metrics mostly relevant for closed-loop control, namely CA 5, IMEP, peak pressure and related location (P P loc ). Figure 4.17 summarizes the comparison of the two estimation methods in the crank-angle domain, in terms of error calculated on the pressure trace resulting from averaging the 1 cycles. The RLS-EKF estimator achieves superior results, especially during the compression and expansion phases where the estimation of the offset is critical for accuracy. The results for both estimators, in terms of combustion and torque metrics, are shown in Figure The two estimation methods appear equally effective in reconstructing the cylinder pressure and combustion metrics on a cycle-by-cycle basis. 69

87 In particular, the RLS-EKF estimator results in better prediction of CA 5 and peak pressure, since the errors with the MA filter can reach 4 degrees and 7%, respectively. With regard to the IMEP, the error is very similar for both cases, below 4%. It is worth observing that, in this case, the difference between the model-based estimator and the MA filter may not appear very significant. This is because the highly accurate lab-grade sensors used in this study provide a signal with very high signal-to-noise ratio. In this case, the estimator is more effective in estimating the pressure offset rather than reducing the measurement noise Application to Production-Grade Sensor Data Since the goal of our estimator design is that it can be used as a tool for engine combustion feedback control on vehicle applications, it is important to test its effectiveness when a production-grade pressure transducer is used. In this section, the model-based estimator and MA filter are compared when applied to pressure signals obtained from a sensor with much lower signal-to-noise ratio. A simple characterization of the sensor output was shown in Figure 4.15, and compared with the outputs of the lab-grade sensor. The results from the comparative analysis are shown in Figures The errors obtained with the two estimators on the in-cylinder pressure trace are shown in Figure 4.19, in comparison with the pressure trace after averaging the 1 cycles. Similar to Figure 4.17, the model-based estimator achieves higher accuracy, even with higher noise from the output of the sensor. 7

88 5 CA 5 error Max Pressure Error [%] IMEP error [%] MA EKF cycle number (a) Errors in the estimation of the cycle-by-cycle combustion metrics (MA dotted, RLS-EKF solid) 1 1 Occurances 5 Occurances CA 5 Error [CAD] 5 5 CA 5 Error [CAD] 1 1 Occurances 5 Occurances Max Pressure Error [%] 5 5 Max Pressure Error [%] 1 1 Occurances 5 Occurances IMEP Error [%] 5 5 IMEP Error [%] (b) Distribution of the errors on combustion metrics (MA on the left and RLS-EKF on the right) Figure 4.18: Comparative results for on-line cylinder pressure estimation from labgrade sensor data (test 3). 71

89 6 4 EKF MA Pressure Error [%] Crank Angle [deg] Figure 4.19: Error signature in crank-angle domain: error between estimators outputs and cylinder pressure trace obtained from average of 1 cycles (production-grade sensor). Figure 4.2 shows a complete analysis of the combustion metrics calculated over the 1 engine cycles. Here, the accuracy achieved with the model-based estimator is particularly evident. In particular, the prediction of CA 5 occurs within ±1 degree, which is the maximum crank-angle resolution available. The peak pressure is predicted within 2% error, while the IMEP shows a slightly higher error if compared to the results with the lab-grade sensor, reaching 1% for a few cycles. On the other hand, the MA filter appears quite unreliable in estimating the combustion metrics and particularly the IMEP, whose error exceeds 4%. In addition, observe that the MA filter yields slightly worse results during the first 1 cycles (before the moving average can be applied in full), showing that this methodology is not suited for on-line combustion control during transient conditions. 72

90 5 CA 5 error Max Pressure Error [%] MA EKF IMEP error [%] cycle number (a) Errors in the estimation of the cycle-by-cycle combustion metrics (MA dotted, RLS-EKF solid) 1 1 Occurances Occurances CA 5 Error [CAD] 5 5 CA 5 Error [CAD] 1 1 Occurances Occurances Max Pressure Error [%] 5 5 Max Pressure Error [%] Occurances Occurances data IMEP Error [%] IMEP Error [%] (b) Distribution of the errors on combustion metrics (MA on the left and RLS-EKF on the right) Figure 4.2: Comparative results for on-line cylinder pressure estimation from production-type sensor data (test 3). 73

91 4.5 Summary In light of pressure-based closed-loop combustion control applications, this chapter presents an estimation scheme for the real-time reconstruction of the in-cylinder pressure trace. The focus is on the methodology adopted in order to provide such information. The estimator relies on a model-based approach to process the output signal of the cylinder pressure sensor (unfiltered and not referenced) into the actual cylinder pressure, allowing for the calculation of combustion and torque-related metrics (such as CA 5, P max and IMEP ) on a cycle-by-cycle basis. A model of the in-cylinder thermodynamics was defined in a linear time-varying form to approximate the cylinder pressure trace based on the definition of a burn rate function. The model, validated on experimental data from a light-duty Diesel engine, was applied to the design of an estimator to reconstruct the in-cylinder pressure from the output of a piezoelectric transducer. The estimator, combining an extended Kalman filter with a recursive least squares algorithm, was applied to simulation data and proved to be effective in reconstructing the actual pressure trace, determining the correct value of the offset, and rejecting sensor noise and modeling errors. Finally, the model-based estimator was applied to the on-line pressure estimation problem and compared with results from a conventional signal processing approach based on a moving average filter. The analysis was based on experimental data obtained by testing sensors of different accuracy installed in the same engine cylinder. In particular, data from a lab-grade sensor and a production-grade sensor were used to evaluate the estimator performance under different input noise conditions. The results obtained demonstrate the accuracy and real-time capability of the estimator, showing the ability to effectively reconstruct the cylinder pressure trace from the transducer 74

92 signal and to reject the sensor noise, offset and modeling errors. Nevertheless, to be more applicable for real-life situations, the estimator must be designed to operate under transient conditions, which will be the topic of the next chapter. 75

93 CHAPTER 5 LPV-EKF ESTIMATION FOR EXPANDED OPERATING REGIONS 5.1 Introduction The EKF-RLSE estimator described in Chapter 4 was designed for specific operating points. Linear parameter varying (LPV) techniques will be used in this chapter to expand the region of estimation to cover the engine operating map as well as allowing for real-time cylinder estimation during transients. In this sense, a significant challenge is represented by defining a model for the cylinder pressure that covers the entire engine operating map, but which is simple enough to be used for real time estimation methods. The model will be designed to be of linear parameter varying (LPV) form as shown in equation (3.14) where the scheduled variables are, the airper-cylinder (ρ 1 ) and the engine speed (ρ 2 ). For comparison to the LPV model, an extended Kalman filter will be employed using the same design introduced in Chapter 4 and examined to estimate the engine in-cylinder pressure in transient operating conditions, namely under a Federal Test Procedure (FTP) cycle. Then, an LPVcorrector will be designed using linear spline (LS) structure to be compared with the EKF in terms of accuracy and computational expense. The study is applied to a 76

94 4-cylinder Spark Ignition (SI) engine with a Variable Valve Timing (VVT) system. Comparisons with both simulation data from a commercial 1D gas dynamic simulator (GT-Power) and experimental data are provided to facilitate this study. 5.2 Model Structure and Validation As mentioned above, the estimation scheme to be developed relies on a simplified linear parameter varying model of the in-cylinder pressure during compression, combustion, and expansion phases. Therefore, the first stage on this scheme will be to develop such a model that will be utilized by the estimator to correct the output of the pressure transducers Description of the Model The model that we will consider here is similar to the model described in Section and the cylinder pressure differential equation is repeated here for convenience: dp cyl dθ + γ dv cyl V cyl dθ p cyl = γ 1 [ (1 α) dq ] g, (5.1a) V cyl dθ dq g dθ = m dx b fuelq LHV. (5.1b) dθ Since the study will be applied to a spark ignition engine, one Wiebe function will suffice to represent the mass fraction of fuel burned. In fact, a study conducted at the Center for Automotive Research (CAR) at The Ohio State University shows that one Wiebe function will have a lower prediction error compared to two Wiebe functions when scheduling entire engine map. The Wiebe function in Equation 5.2 is described by four parameters: a and m are shaping parameters, Δθ is the combustion duration, and θ is the start of combustion. Since the ignition happens instantaneously in a 77

95 spark ignition engine, it is reasonable to assume θ equal to the spark timing, or [ ( ) m+1 ] θ θ φ(θ) = 1 exp a Δθ. (5.2) The remaining Wiebe function parameters will be modeled as a nonlinear function of the engine operating variables, air-per-cylinder, and engine speed, explained in the next section Model Calibration and Validation As was the case for the model used in Chapter 4, the model requires calibration on engine data to ensure reasonable approximation of the in-cylinder pressure to be useful for cylinder pressure estimation schemes related to this engine. Herein, the calibration and validation are performed on experimental data, collected on the GM Ecotec LE5 engine (Figure 5.1), a four cylinder spark ignition engine, with specifications defined in Table 5.1. Model 2.4l PFI SI Cylinders Inline, four-cylinder Displacement 24 cm 3 Bore and Stroke 88, 98 mm Compression Ratio 1.3:1 Connecting Rod Length mm Valve timing Dual independent cam phasors Max. Power 13kW at 62r/min Max. Torque 222N m at 48r/min Table 5.1: Test engine specifications. A total of 12 engine operating points, depicted in Figure 5.2, were acquired in steady-state conditions, varying engine speed and torque. The data was processed 78

96 Figure 5.1: GM Ecotec LE5. in the same manner as we did in Section to get the most accurate cylinder pressure trace for each operating point. A simple inverse thermodynamic calculation was then used to estimate the heat release and burn rate profile that were used for the calibration of the heat transfer scaling factor α and the burn rate parameters in Equation 5.2, where the burn rate is obtained using the inverse of Equation 5.1. The model parameters are identified using a least squares fit for each burn rate curve obtained from the cylinder pressure trace. The model obtained by applying this procedure will be called the scheduled-parameters model. The detailed results are shown in Appendix 5, which lead us the following simplifications: m can be assumed constant (around 3.4). 79