Torque estimation from in-cylinder pressure sensor for closed loop torque control Systems, Control and Mechatronics PER-SEBASTIAN PETTERSSON ANTON KJELLIN Department of Signals and Systems CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2017 Master’s thesis EX051/2017 Torque estimation from in-cylinder pressure sensor for closed loop torque control PER-SEBASTIAN PETTERSSON ANTON KJELLIN Department of Signals and Systems Division of Systems and Control Automatic control Chalmers University of Technology Gothenburg, Sweden 2017 Torque estimation from in-cylinder pressure sensor for closed loop torque control PER-SEBASTIAN PETTERSSON ANTON KJELLIN © PER-SEBASTIAN PETTERSSON & ANTON KJELLIN, 2017. Supervisor: Johan Engbom, Volvo Group Truck Technology Examiner: Torsten Wik, Department of Signals and Systems Master’s Thesis EX051/2017 Department of Signals and Systems Division of Systems and Control Automatic Control Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: Volvo Trucks D13 engine. Source: Volvo Group Trucks Technology Typeset in LATEX Gothenburg, Sweden 2017 iv Torque estimation from in-cylinder pressure sensor for closed loop torque control PER-SEBASTIAN PETTERSSON & ANTON KJELLIN Department of Signals and Systems Chalmers University of Technology Abstract In-cylinder pressure sensors are gradually moving out of the laboratory and into production type engines, thus making more direct information of the combustion process available for the engine control unit. For several reasons it is important to accurately control the torque produced by the engine. In light of this objective, this thesis propose and studies two methods for estimating the average indicated torque of a heavy duty diesel engine with mea- surements from a single in-cylinder pressure sensor. It is further studied how this estimate can be used as feedback in a torque control loop. For this case study, it is demonstrated that the angular precision when sampling the cylinder pressure is very important for the accuracy of the estimation but that the sampling interval can be moderate and still produce a compelling estimation. Furthermore, it is demonstrated that the sensor drift, characteristic for piezo-electric in-cylinder pressure sensors, could be neglected when estimating the torque. In an engine cell test it is illustrated how the influence from faulty injectors on the output torque could be corrected, by using the estimation of the average indicated torque as feedback. Keywords: In-cylinder pressure sensor, combustion control, torque control, ECU, EMS, ECM, torque estimation. v Acknowledgements When performing this work, we took help and guidance from some excellent people, who deserve our greatest gratitude. First we want to say thank you to Johan Engbom at Volvo for his enthusiasm and commitment in our project. We would also like to thank our supervisor professor Torsten Wik at Chalmers for his guidance and assistance. Deepest gratitude is also due to Ph.D. Marcus Hedegärd at Chalmers, without whose knowledge and assistance this study would not have been successful. Conny Nicander at Volvo deserves our greatest appreciation for supporting our project. Thanks also to Kedar Jaltare and Viktor Andersson for giving us good technical guidelines throughout numerous consultations. Many others have made valuable and inspirational suggestions which have im- proved our work greatly, Igor Lumpus, Mikael Svensson, Bengt Lassesson, Arash Idelchi and others. Anton Kjellin and Per-Sebastian Pettersson, Gothenburg, June 2017 vii Contents 1 Introduction 1 1.1 Thesis purpose and scope . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 System overview 5 2.1 Angular measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 In-cylinder pressure sensor . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Torque estimation methods 7 3.1 Estimation method 1: Riemann sum . . . . . . . . . . . . . . . . . . 9 3.2 Estimation method 2: Tailor made basis . . . . . . . . . . . . . . . . 9 4 Estimation method analysis 11 4.1 Data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.1.1 Reference indicated torque . . . . . . . . . . . . . . . . . . . . 12 4.2 Expected error introduced when using only one ICPS . . . . . . . . . 13 4.3 Design parameters Estimation method 1 . . . . . . . . . . . . . . . . 14 4.4 Design parameters Estimation method 2 . . . . . . . . . . . . . . . . 17 4.5 Pressure measurement offset impact analysis . . . . . . . . . . . . . . 20 4.6 Angle measurement offset impact analysis . . . . . . . . . . . . . . . 21 4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Controller 25 6 Implementation and lab set-up 29 6.1 Engine control unit implementation . . . . . . . . . . . . . . . . . . . 29 6.2 Engine test cell set-up . . . . . . . . . . . . . . . . . . . . . . . . . . 30 7 Results 31 7.1 Torque estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.2 Closed loop torque controller 1 . . . . . . . . . . . . . . . . . . . . . . 33 7.3 Closed loop torque controller 2 . . . . . . . . . . . . . . . . . . . . . . 35 8 Discussion 37 8.1 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8.2 Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 9 Conclusion 39 10 Future Work 41 10.1 Gather reference data and analyze error . . . . . . . . . . . . . . . . 41 10.2 Adaptive reference feed forward controller . . . . . . . . . . . . . . . 41 ix Contents 10.3 Individually weighting cylinder contributions . . . . . . . . . . . . . . 41 10.4 Investigating impact of crankshaft torsion on estimation accuracy . . 42 Bibliography 43 A Appendix 1 I x Notation Abbreviations ECM Engine Control Module GTT (Volvo) Group Trucks Technology ICPS In-Cylinder Pressure Sensor TDC Top Dead Center Capital Letters C Constant value E Torque error F Controller, Transfer function FLP Moving average filter, Transfer function G Plant, Transfer function L Crank lever function scaled by cylinder area L Vector with pre-calculated values from crank lever function Nc Number of cylinders Nf Order of filter T Torque [Nm] T0 Operating point (Nm, RPM) Tf Friction torque [Nm] Tg Indicated/gas torque [Nm] T ∗g Reference gas torque [Nm] Tm Mass torque [Nm] xi Contents Small Letters b Piston diameter [m] e Error er Control error fθ Sample frequency [samples/degree] k Optimized vector of constants l Length of connecting rod [m] p̃ Vector of pressure measurements [Pa] p0 Crank case pressure, approximated to ambient pressure [Pa] pd Pressure drift [Pa] pEx Exhaust pressure [Pa] pg Effective pressure, difference between cylinder and crankcase[Pa] pi Cylinder pressure for cylinder i [Pa] pI Intake manifold pressure [Pa] pmax Expected maximum pressure [Pa] r Distance from crankshaft to connecting rod s Vertical piston position [m] ỹ Vector of measured weights for Estimation method 2 z Frequency domain argument Greek Letters α Crank angle offset [degree] θ Crank shaft angle relative to TDC [degree] θi Crank angle where samples are taken [degree] θI Vector of crank angles where samples are taken [degree] Ωi Function relating flywheel angle to cylinder individual angle for cylinder i Θ 4-stroke engine cycle angular interval [0°, 720°). |Θ| Length of the 4-stroke engine cycle, 720°. ψj Scalar valued basis function Λj,l,m Matrix with spline parameters nφ Number of parameters in Estimation method 2 γ Value that represent the contribution to the mean indicated torque from a sub-interval ω Engine speed [rad/s] ω Mean engine speed [rad/s2] ρ Pearson linear correlation coefficient Diacritical marks ˜ Measurement̂ Approximation or estimate Average value xii 1 | Introduction New alternative bio fuels have started to enter the market and engine manufacturers are seeing a future where their engines will need to be able to run effectively on several different fuels. The properties between these fuels may differ substantially. For example, a difference in energy content between diesel and biodiesel of 7% is common[11]. Together with a continued need to improve engine efficiency and reduce vehicle emissions, for environmental reasons and stricter government regulations, better and more refined combustion control is of great importance [13]. The conventional control of heavy duty diesel engines is based on feed-forward of sensor information of what enters and leaves the combustion chamber. The various control signals are determined using pre-calibrated look-up tables, basically making the control an open-loop structure. The time and resources required for calibration are significant and grows exponentially with the increase of control parameters[16]. However, in case the fuel properties are not what the engine has been calibrated to, or the engine components have changed, the behavior of the engine will differ from what is intended, and from what is optimal. Engine manufacturers are therefore mowing towards including more closed loop control structures where the quantity that is to be controlled is measured, either directly or indirectly. An advantage of closed loop control is also that it inherently requires less calibration. In-cylinder pressure With a few exceptions, for example knock control, the closed loop control structures that has been suggested for combustion control relies heavily on information about the in-cylinder pressure[8][10][12][17]. The in-cylinder pressure is a fundamental variable in both thermodynamics and classical mechanics, and it is possible to ex- tract a lot of other quantities from it, such as heat release, torque, peak-pressure position and exhaust composition[1][3]. Until now few of these methods have been implemented into production type engines. The main reason for this is that the in-cylinder pressure sensors (ICPS) have, until recently, been considered too expensive and fragile and therefore limited to research use only. This has, parenthetically, triggered a research field where methods to estimate the cylinder pressure from other sensors have been studied e.g. by combining crankshaft acceleration and torque measurements [2][6][13]. Recent advancement in sensor technology, now makes it sensible for engine manufactures to include ICPS in their production types engines. Accurate measurements of the pressure from each combustion cycle provides many new opportunities for advanced feedback control and it has been shown that a reduction in both emissions and improved fuel efficiency is possible[17]. 1 1. Introduction Torque control One of the variables that could be extracted from the in-cylinder pressure is the indicated torque (gas torque), Tg, which is the torque that is created by the pressure difference between the cylinder and the crankcase acting on the piston. The gas torque is the biggest contributor to the output torque, T , from the crankshaft T = Tg − Tf − Tm. (1.1) The friction torque, Tf , and the mass torque, Tm, imposed by the piston assembly are the other contributing factors. The mathematical basis for computing the indicated torque from the cylinder pressure is a geometric relationship that can be derived from the geometry of the mechanical linkage between the piston and the crankshaft (Chapter 3). If an al- gorithm could be derived for production type engine control modules it would be possible to give a live in-vehicle estimate of the indicated torque. This estimate could be used in a feedback control structure, making it possible to keep engine behavior even tough fuel or engine properties changes. 1.1 Thesis purpose and scope The purpose of this thesis is to evaluate if a torque controller that uses a live estimate of the average indicated torque, based on measurement from only one ICPS, as feedback is feasible to introduce into production type engines from Volvo Trucks. The available computational power and memory of the engine control unit is scarce. To introduce a new function or sensor it is therefore necessary that the computational complexity of the function and the required sample frequency of the sensor is reasonable. It is also necessary that the function gives reliable output, i.e. the function needs to be robust against possible disturbances. 1.2 Thesis outline In this thesis two methods for estimating the average indicated torque, using an ICPS from one cylinder, are formulated, analyzed and compared. Two feedback torque controllers are also formulated and analyzed. A live test is then presented where the methods have been implemented on a production type engine control module and verified in an engine test cell at Volvo Group Trucks Technology. Chapter 2 contains an overview of the system with its sensors, actuators and other components. In Chapter 3 the two methods for estimating the mean indicated torque are formulated and described. Based on a large data-set containing previous measured cylinder pressure curves and other engine parameters the methods are analyzed with regards to sample frequency, sensor drift, crank angle resolution and robustness to measurement noise. This is described in Chapter 4. The two controllers are presented and derived in Chapter 5. The first controller is a simple integrating controller with a feed-forward of the reference. The second 2 1. Introduction is an extension of the first where a dynamic look-up table that is populated by the integrated error from different operating points is introduced. The implementation of the estimation methods and controllers on the engine control unit later used in the live test is briefly described in Chapter 6. This chapter also contains information about how the measurements were taken during the live test. Chapter 7 contains the result from the live test done on a 13 litre 6 cylinder heavy duty diesel engine from Volvo. The last chapters contains a discussion, presents the conclusions drawn from the work described and presents proposed future work. 3 2 | System overview Diesel engines are compression ignited engines. This means that the fuel injected into the combustion chamber is ignited only by high compression. The torque output is controlled by controlling the air and fuel entering the combustion chamber. This is in contrast to gasoline engines where the fuel is ignited by a spark plug and where also the ignition needs to be controlled . The control of the engine is handled by one central unit called the engine control module (ECM). The ECM is connected to several sensors fitted to the engine and its surroundings, for diagnose and control. The sensors used in this thesis, in addition to the ICPS, are the exhaust pressure sensor, the intake manifold pressure sensor, the ambient pressure sensor and the angle measurement sensors. These are standard sensors used in production type engines. 2.1 Angular measurement For several reasons it is important for the ECM to know the current orientation of the crankshaft. The crank shaft angle, θ, is detected at discrete points on the flywheel and the ECM estimates the angle between these points by extrapolation based on the engine speed. Another detection point on the cam shaft together with missing measurement points on the flywheel, referred to as missing teeth, are used to find the absolute angle. On the engine considered in this thesis there is a detection point each 6° and three gaps separated by 120°. Figure 2.1 illustrates the detection points on the flywheel. Figure 2.1: Illustration of the detection points on the flywheel. The gaps (missing teeth) of detection points that are used to establish absolute position are also illus- trated. The engine considered in this project have three of these gaps, separated by 18 teeth. 5 2. System overview 2.2 In-cylinder pressure sensor The cylinder pressure, pi(θ), is measured by an in-cylinder pressure sensor. An typ- ical ICPS is illustrated in Figure 2.2. The sensor is placed in the cylinder head with the tip directly exposed to the combustion gas pressure. Some different technologies are used, such as piezo-electric, piezo-resistive and optical pressure sensing. Figure 2.2: Combustion chamber pressure sensor. Is fitted in the cylinder with the tip directly exposed to the combustion gas pressure. In this thesis a flush mounted piezo-electric sensor from the manufacturer Kistler is used. The central component in a Piezo-electric pressure sensor is the quartz crystal that generates a charge when exposed to pressure. If the pressure is kept constant the charge will eventually leak to zero. Piezo-electric pressure sensors can therefore only measure dynamic pressures, i.e changing pressures, as opposed to static pressures. Another property of piezo-electric sensors is that the measurement signal drifts over time. 6 3 | Torque estimation methods Figure 3.1 depicts the mechanical linkage between the piston and the crankshaft. Pressure in the cylinder produces a force on the piston which is transmitted, via the connecting rod, to the crankshaft and results in a torque. θ r + l Φ r l s Ac b Figure 3.1: Piston-crankshaft mechanical linkage. The relationship between pressure and the indicated torque for cylinder i is Tg,i(θ) = L(θ)pg,i(θ) (3.1) where pg,i(θ) is the pressure difference between the combustion chamber and the crankcase, i.e. pg,i(θ) = pi(θ) − p0. The pressure in the crankcase, p0, can be approximated to atmospheric pressure [4]. L(θ) states the relation L(θ) = Ac ds dθ , (3.2) where Ac = b2 4 π is the piston area and ds dθ is the crank lever function [4] ds dθ = r sin(θ) 1 + r l cos(θ)√ 1− r2 l2 sin2(θ)  , (3.3) which relates force on the piston to torque on the crankshaft. The indicated torque at the output side of the crankshaft, Tg(θ), is a sum of contributions from the Nc individual cylinders, i.e. Tg(θ) = Nc∑ i=1 Tg,i(θ) = Nc∑ i=1 L(Ωi(θ))pg,i(θ), (3.4) 7 3. Torque estimation methods where Ωi(θ) is cylinder individual angle for cylinder i at flywheel angle θ. The cycle average indicated torque at the output side of the crankshaft is T g = 1 |Θ| ∫ Θ Tg(θ)dθ = Nc∑ i=1 1 |Θ| ∫ Θ L(Ωi(θ))pg,i(Ωi(θ))dθ (3.5) where Θ = [0°, 720°) is the 4-stroke engine cycle and |Θ| = 720° is the length of the cycle. Assuming periodicity in θ of all pressure signals at steady state operation, i.e. pg,i(θ) = pg,i(θ + |Θ|), then T g = Nc∑ i=1 1 |Θ| ∫ Θ L(Ωi(θ))pg,i(Ωi(θ))dθi := Nc∑ i=1 T g,i. (3.6) Using weights, wi, relating the torque contribution from cylinder i to that of one cylinder, x, the total average indicated torque can be expressed in terms of one measured pressure, pg,x(θ) as T g = Nc∑ i=1 wiT g,x (3.7) Algorithms using momentaneous engine speed measurements to calculate the weights have been developed with promising results [5]. In this thesis however, it has been assumed that the angle-pressure relationship in all cylinders are equal, i.e. wi = 1 ∀i, and therefore T g = NcT g,x = Nc |Θ| ∫ Θ L(θ)pg,x(θ)dθ. (3.8) Further, when approximating the pressure inside the crankcase, p0, to atmo- spheric pressure it can be considered constant on the time scale of one engine cycle. Then the contribution from p0 cancels, i.e. T g = Nc |Θ| ∫ Θ L(θ)(px(θ)− p0)dθ = Nc |Θ|  ∫ Θ L(θ)px(θ)dθ − p0 ∫ Θ L(θ)dθ︸ ︷︷ ︸ =0  . (3.9) That is, assuming equal angle-pressure relationship in all cylinders and constant pressure inside the crankcase during the engine cycle, the average indicated torque can be expressed as T g = Nc |Θ| ∫ Θ L(θ)px(θ)dθ, (3.10) using pressure measurements from only on cylinder, x ∈ [1, Nc]. 8 3. Torque estimation methods 3.1 Estimation method 1: Riemann sum The first method considered to approximate the integral in Equation (3.10) is a Riemann middle sum. The continuous pressure signal, p(θ), will be sampled in θ with frequency fs = n |Θ| [samples/degree]. The system will thus have access to the pressure signal at n angle-equidistant points, θi, over the cycle and T g will be estimated using this data according to T g ≈ T̂ g = Nc 1 n n−1∑ i=0 L(θi)p̃(θi), (3.11) with θi = (2i+ 1) |Θ|2n . Once the number of samples, n, and θI := {θi : i = 1, . . . , n} have been decided Nc n L(θi) can be pre-calculated and collected in a column vector, L := Nc n L(θI). The average indicated torque can then be computed from the scalar product T̂ g = LT p̃, (3.12) where p̃ := p̃(θI) is a column vector of measurements. 3.2 Estimation method 2: Tailor made basis In [2] it was assumed that cylinder pressure on a given operating region can be modelled as p̂(θ) = [ψ0(θ), ..., ψnφ ]︸ ︷︷ ︸ ψ(θ) ỹ (3.13) where ψj are scalar valued basis functions in angle θ and ỹ = [1, ỹ1, ..., ỹnφ ] is a vector of corresponding weights that are individual for each pressure curve. For identification of ψj a large set of measured pressured curves were used, and ψj was parameterized using cubic splines, i.e. ψj(θ) = 3∑ m=0 θmΛj,l,m, (3.14) where Λj,l,m are the spline parameters. It was also shown that the weights ỹj, j = 0, ..., nφ can be chosen as cylinder pressures at distinct angles. Using this, a convex optimization problem for the spline parameters, Λj,l,m, in the model could be for- mulated. It was further suggested to choose some of the weights in ỹ to be other measured quantities. Evaluation for the same engine as in the current study showed that as few as nφ = 6 measured parameters gave good accuracy over a selected operating region. The above suggests that the average gas torque could be estimated to a satisfac- tory accuracy by a linear mapping from only a few measured parameters. Explicitly, we have T g ≈ T̂ g = 1 |Θ| ∫ Θ L(θ)p̂(θ)dθ = 1 |Θ| ∫ Θ L(θ)ψ(θ)dθỹ := kT ỹ (3.15) 9 3. Torque estimation methods that is T̂ g = kT ỹ, (3.16) where kT = 1 |Θ| ∫ Θ L(θ)ψ(θ)dθ (3.17) is a constant vector. This method of estimation may require less data to be sampled and processes than Estimation method 1. If this estimate is also accurate and robust then this method may be preferred. For determination of k a set of Np pressure curves measured under different conditions within a specified operating region are used. An optimization problem is then formulated as a least squares problem, i.e min k eTe, (3.18) where e is the model error e = Tg − T̂g =  T g,1 ... T g,Np −  ỹT1 ... ỹTNp k := Tg − Ak, (3.19) where Tg,i is taken as the best possible approximation of the true torque calculated using high frequency sampled data and where row i in A collects measurements from the i:th pressure curve. The unconstrained solution to (3.18) is found as [7] ∇keTe = 0⇒ k = (ATA)−1AT T̃g (3.20) 10 4 | Estimation method analysis Both methods for estimating the average indicated torque presented above have design parameters that affect the precision and computational complexity of the estimation algorithms. A provided data-set taken from the intended engine type is used in this chapter to analyze different choices of parameters. Further, the expected effect of different possible error sources on the estimation algorithms are also studied. 4.1 Data set Analysis and testing of the estimation methods have been based on a large data-set collected from a 13 liter engine in a test cell at Volvo Trucks. The data collected includes pressure and angle measurements sampled in time with high frequency at different operating points T0 (load and speed). Also included are measurements of the atmospheric pressure, pressure in the intake manifold and crank shaft angle. Table A.1 in Appendix A lists information about the data set. The data set includes pressure measurements from all 6 cylinders. Typical indi- vidual pressure signals and calculated torque signals over one engine cycle are shown in Figure 4.1. Figure 4.2 shows selected pressure measurements and indicated torque from one cylinder at different operating points. 3, [deg] 100 200 300 400 500 600 700 P re ss u re , [P a ] #106 0 2 4 6 8 10 12 14 16 18 Pressure signals over one cycle pI pc CYL1 CYL2 CYL3 CYL4 CYL5 CYL6 (a) Pressure measurements. 3, [deg] 100 200 300 400 500 600 700 T o rq u e , [N m ] -4000 -2000 0 2000 4000 6000 8000 Calculated torques over one cycle P Ti 7T CYL1 CYL2 CYL3 CYL4 CYL5 CYL6 (b) Corresponding calculated torque Figure 4.1: Individual signals from all cylinders over one engine cycle. 11 4. Estimation method analysis 3, [deg] -360 -300 -240 -180 -120 -60 0 60 120 180 240 300 360 P re ss u re , [P a ] #107 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Typical pressure signals at diffrent operating points T0 T 0 = [800, 1800] T 0 = [1300, 2100] T 0 = [1800, 2650] (a) Pressure measurements. 3, [deg] -360 -300 -240 -180 -120 -60 0 60 120 180 240 300 360 T o rq u e , [N m ] -2000 0 2000 4000 6000 8000 Calculated torque curves at diffrent operating points T0 T 0 = [800, 1800] T 0 = [1300, 2100] T 0 = [1800, 2650] (b) Corresponding calculated torque signals. Figure 4.2: Signals from one ICPS over one cycle at different operating points. 4.1.1 Reference indicated torque To evaluate the precision of the two estimation methods a reference mean indicated torque is needed. This reference should, as far as possible, represent the true physical indicated torque. The high frequency sampled pressure data in the data set is distorted by high frequency noise. The signal is sampled with at least 3.4 ∗ 103 samples per cycle. All information in the continuous pressure signal up to a frequency of 1.7 ∗ 103 repeti- tions per cycle will therefore be represented in the sampled signal. The continuous pressure signal seems to be practically band limited to 750 repetitions per cycle and all frequency content above this limit is considered as high frequency noise, either from the measurement or the process. A low pass filtered pressure signal with cut-off frequency of 750 repetitions per cycle is considered the best possible representation of the physical continuous pressure signal available and is used to calculate the reference mean indicated torque. Figure 4.3 shows pressure measurement data and the low pass filtered signal at three different operating points. The magnitude of the noise is fairly constant over the cycle except for angles around TDC. This may be actual high frequency pressure variations in the cylinder but considered as process noise and attenuated by filtering. 12 4. Estimation method analysis 3, [deg] -240 -239 -238 -237 -236 -235 P , [P a ] #105 1.55 1.56 1.57 1.58 1.59 1.6 1.61 1.62 1.63 1.64 1.65 Typical pressure signal at operating point: T 0 = [800, 1800] Measured signal Estimated true signal 3, [deg] -240 -239 -238 -237 -236 -235 P , [P a ] -2000 -1500 -1000 -500 0 500 1000 1500 Noise at operating point: T 0 = [800, 1800] 3, [deg] -240 -239 -238 -237 -236 -235 P , [P a ] #105 2.69 2.7 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 Typical pressure signal at operating point: T 0 = [1300, 2100] Measured signal Estimated true signal 3, [deg] -240 -239 -238 -237 -236 -235 P , [P a ] -1500 -1000 -500 0 500 1000 1500 Noise at operating point: T 0 = [1300, 2100] 3, [deg] -240 -239 -238 -237 -236 -235 P , [P a ] #105 2.64 2.66 2.68 2.7 2.72 2.74 2.76 2.78 Typical pressure signal at operating point: T 0 = [1800, 2650] Measured signal Estimated true signal 3, [deg] -240 -239 -238 -237 -236 -235 P , [P a ] -3000 -2000 -1000 0 1000 2000 3000 Noise at operating point: T 0 = [1800, 2650] Figure 4.3: Pressure samples and low pass filtered signals at the three operating points considered. Zoomed on flat intake stroke Reference torque is calculated from the low pass filtered high frequency sampled data as a Riemann middle sum, the error of the reference using all data is then limited by [15] Emax = Nc|Θ|2 24n2 ( 1 n n∑ i=1 |T ′′g |max,i ) (4.1) Approximating the second derivative of a typical torque signal calculated from low pass filtered pressure measurements, |T ′′g |, an estimate of Emax is calculated. From this calculation it is shown that the error in the calculated reference at an operating point with the lowest angular sampling frequency in the data set (fθ ≥ 4.74) is smaller than 0.2 Nm. 4.2 Expected error introduced when using only one ICPS In order to use only one ICPS it has been assumed that the contribution to the average torque from all individual cylinders are equal. The validity of this assump- tion is examined on the data-set. The relative error for the average indicated torque introduced by this assumption, ei,h = 6T g,i − ∑6 j=1 T g,j∑6 j=1 T g,j (4.2) when using measurements from cylinder i ∈ [1, 6] at operating point h is presented in Figure 4.4. Here the average value over the complete test cycle has been used. 13 4. Estimation method analysis The magnitude of the largest relative error found is for cylinder 1 at the operating point T0 = [1800, 1800] where the contribution is on average around 5% off. CYL 1 CYL 2 CYL 3 CYL 4 CYL 5 CYL 6 e -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Relative error, all operating points and all cylinders 7ei;h Figure 4.4: Relative offset of individual cylinder contributions to the mean torque. 4.3 Design parameters Estimation method 1 The parameter to be chosen for Estimation method 1 is θI in Equation (3.12). Equidistant sampling symmetric around TDC is assumed, thus the choice of θI is fully determined by choice of sampling frequency, fθ = n |Θ| , where n is the number of samples per engine cycle. The computational demand is proportional to fθ and the expected error in the approximation is also affected by the choice of fθ. This section investigates how the error stemming from approximating the integral by a sum and error stemming from pressure measurement noise is impacted by choice of fθ. Numerical integration error Figure 4.5 displays the resulting error stemming from approximating the integral by a sum, as a function of sampling frequency for three operating points selected to cover the speed and torque span of the data set. This suggests that the sample interval can be moderate and still produce an accurate estimate. For sampling frequencies higher than 1 18 [samples/degree] the error is well below 1%. 14 4. Estimation method analysis f3, [samples/deg] 1/30 1/24 1/18 1/12 1/6 0 0.02 0.04 0.06 0.08 Magnitude of relative error at operating point: T 0 = [800, 1800] jejmax jejmean f3, [samples/deg] 1/30 1/24 1/18 1/12 1/6 0 0.02 0.04 0.06 0.08 Magnitude of relative error at operating point: T 0 = [1300, 2100] jejmax jejmean f3, [samples/deg] 1/30 1/24 1/18 1/12 1/6 0 0.02 0.04 0.06 0.08 Magnitude of relative error at operating point: T 0 = [1800, 2650] jejmax jejmean Figure 4.5: Magnitude of relative error from data over fθ. Pressure noise error The measured pressure signal, p̃(θ), carries information of the actual pressure, p(θ), distorted by high frequency measurement and process noise, e. p̃(θ) = p(θ) + e (4.3) The impact of the noise on the mean torque estimate using Estimation method 1 can be analyzed T̂ g = Nc 1 n n∑ i=1 L(θi)p̃(θi) = Nc 1 n n∑ i=1 L(θi)p(θi) +Nc 1 n n∑ i=1 L(θi)e = T g + E, (4.4) where E is the resulting error, E = Nc n ∑n i=1 L(θi)e. If e is gaussian, e ∼ N (µ(p̃), σ2(p̃)), then E ∼ N (µe, σ2 E) where µE = Nc n n∑ i=1 L(θi)fµ(p̃(θi)), and σ2 E = N2 c n2 n∑ i=1 L2(θi)f 2 σ(p̃(θi)). (4.5) If e has zero mean, fµ(p̃) = 0, and constant variance, f 2 σ(p̃) = σ2 e , then µE = 0, and σ2 E = N2 c σ 2 e n2 n∑ i=1 L2(θi) (4.6) 15 4. Estimation method analysis and σ2 E is proportional to n−1. Figure 4.6 shows the mean, the standard deviation and the maximum of the error the pressure measurement noise in the data set contributes with, E. The maximal error increases with decreasing sampling frequency but even at sampling frequencies as low as 1 30 [samples/degree] the maximal error found is well below 1%. f3, [samples/deg] 1/30 1/24 1/18 1/12 1/6 #10-4 -2 0 2 4 6 8 Relative of error at operating point: T 0 = [800, 1800] jEjmax 7E