ABSTRACTThe information provided by the in-cylinder pressure signal isof great importance for modern engine management systems.The obtained information is implemented to improve thecontrol and diagnostics of the combustion process in order tomeet the stringent emission regulations and to improvevehicle reliability and drivability. The work presented in thispaper covers the experimental study and proposes acomprehensive and practical solution for the estimation of thein-cylinder pressure from the crankshaft speed fluctuation.Also, the paper emphasizes the feasibility and practicalityaspects of the estimation techniques, for the real-time onlineapplication. In this study an engine dynamics model basedestimation method is proposed. A discrete-time transformedform of a rigid-body crankshaft dynamics model isconstructed based on the kinetic energy theorem, as the basisexpression for total torque estimation. The major difficulties,including load torque estimation and separation of pressureprofile from adjacent-firing cylinders, are addressed in thiswork and solutions to each problem are given respectively.The experimental results conducted on a multi-cylinder dieselengine have shown that the proposed method successfullyestimate a more accurate cylinder pressure over a wider rangeof crankshaft angles.

INTRODUCTIONIn-cylinder pressure data provides extremely valuableinformation for engine control as it is an instantaneous and

direct measure of engine performance and combustionprocess. This information allows several engine monitoringand control capabilities[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. The reliability andcost of cylinder pressure sensors is the driver for usingestimation techniques to indirectly obtain the cylinderpressure value. They have the potential to make availablecylinder pressure data with reasonable accuracy and no extrahardware cost.

The goal of this work is to develop a comprehensive andpractical method, based on the crank shaft dynamic model, toestimate in-cylinder pressure, emphasizing the feasibility andpracticability for the real-time online implementation. Thispaper firstly reviews the major investigations on the in-cylinder pressure estimation research in recent years. Thendifferent dynamics model structures are examined and theircomplexity and applicability are compared. A discrete-timetransformed form of the rigid-body crankshaft dynamicsmodel is constructed based on the kinetic energy theorem, asthe basis expression for total crankshaft torque estimation.Individual torque component analysis to facilitate theextraction of gas torque from total torque is then presented.Due to the limited capability of the rigid-body dynamicmodel in handling the complicated crankshaft speedfluctuation, a process to pre-filter the measured speed signalis found effective to alleviate the limitation. A practicalmethod to estimate the friction and load torque is alsoproposed based on the characteristics of the instantaneoustorque profile during the engine cycle. To further improve the

An Experimental Study on Engine Dynamics ModelBased In-Cylinder Pressure Estimation

2012-01-0896Published

04/16/2012

Feilong LiuDelphi Corporation

Gehan A. J. Amaratunga and Nick CollingsUniversity of Cambridge

Ahmed SolimanUniversity of North Carolina Charlotte

Copyright © 2012 SAE International

doi:10.4271/2012-01-0896

estimation result, an isentropic-process-based correctiontechnique and a combustion-progress based pressurereconstruction over TDC region are also proposed. Thismethod has been experimentally applied on a multi-cylinderdiesel engine test. The test result and quantitative errorevaluation is detailed in this paper, and then followed by theconclusion remark.

BACKGROUNDThe information provided by the in-cylinder pressure signal isof great importance for modern engine management systems.The obtained information is implemented to improve thecontrol and diagnostics of the combustion process in order tomeet the stringent emission regulations and to improvevehicle reliability and drivability. An adequately accuratesignal of cylinder pressure is the prerequisite condition formany promising engine closed-loop control systems[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16], such as:

• Torque-on-demand engine control algorithm

• Onboard real-time combustion diagnostics

• Individual cylinder spark timing control

• Air-fuel balancing between cylinders

Currently the hardware cost of adding such in-cylinderpressure sensors limits its availability for productionapplications [27]. Therefore, the software-based pressureestimation techniques, as an alternative way to measure in-cylinder pressure, are attractive as they have the potential tomake cylinder pressure data available with reasonableaccuracy and no extra hardware cost.

In-cylinder pressure and the associated indicated torqueestimation has been a subject of active research since the1980s and promising progress has been reported. One of theearliest efforts made on developing the engine dynamicmodel to allow crankshaft speed based torque estimation wascarried out by Rizzoni, who introduces a two-step procedureto estimate indicated torque [28]. First, the net torque isobtained by using an engine dynamic and measuredinstantaneous crankshaft speed. Then the indicated torque isderived through the correction of mass torque caused byreciprocating components and friction loss. Another approachutilizing the engine dynamic models to estimate IndicatedMean Effective Pressure (IMEP) was introduced by Lida andAkishino [23]. It is based on using a simplified torsionalvibration model to extract the instantaneous torque from theflywheel speed fluctuation measurement and convert to IMEPby considering each of the action torque components.

In later research, frequency-domain analysis was recognizedas a useful tool to extract the periodic components in the in-cylinder pressure and indicated torque based on the cyclicnature of engine operation. Rizzoni and Connolly [22]

proposed a mapping relationship between speed and torque ateach engine firing harmonic order in the frequency domain,assuming the crankshaft dynamics as a constant linearsystem. They further obtained an empirical correlationbetween mean component and first order component of theindicated torque. A simplified lumped-mass model andfrequency-domain analysis was used by Taraza to obtain thestatistical correlation between amplitude and phase of thesame harmonic components of gas pressure, torque and speed[29, 30].

In the works of Schagerberg et al [31,32,33], a morecomplicated multi degree-of-freedom engine dynamic modelwas used to estimate in-cylinder pressure by utilizing acrankshaft integrated torque sensor. The rigid-body and multidegree-of-freedom models were also compared in the contextof predicting in-cylinder pressure.

Apart from using engine dynamics to correlate the pressureand speed, Cavina et al. validated the existence of coherencebetween the engine block vibration and pressure due toperiodic engine firing and that only 2nd and 4th orderpressure variables are sufficient to reconstruct the meanindicated torque [34]. This is similar to the observation madeby Wagner and Bohme [26], who proposed a physical modelfor the transfer behaviour from pressure to structure-bornesound and an identification method for the unknown systemparameters. The later investigation by Lee et al. revealed abetter coherence can be obtained through speed fluctuationand cylinder pressure or indicated torque [17]. Other researchin the area includes the works of Li et al [35, 36]. Inspired byconstructing a statistical correlation function to avoid thecomplexities of the actual physical system, Guezennecproposed an in-cylinder pressure stochastic estimator basedon polynomial form fitting, which is well suited for real-timeimplementation due to its low computational cost [37]. Theuse of a non-linear observer, particularly the sliding modeobserver, has also been experimented in the crankshaft speedbased torque estimation [21, 38].

In this study the engine dynamic model based estimation ischosen as the basis for the investigation on in-cylinderpressure estimation, due to its advantage of directing theestimation process by a physical deterministic model whilerequiring minimal level calibration effort. This methodphysically correlates the torque excitation and resultant speedvariation through engine mechanical dynamic, hencetheoretically it's feasible to derive an analytic expression toconnect the time-domain crankshaft motion and externalexcitements including in-cylinder pressure and other torquessources. Once the indicated torque, one of the main externalexcitements on the crankshaft dynamics, is solved for throughthis model by using the available crankshaft speedmeasurement, the gas pressure can be readily obtained byutilizing the engine geometry relationship. The mechanicaldynamics model is also a self-contained and naturally

adaptive for different engine operation conditions, since mostof its parameters are the physical structural constant such asinertia of moment, mass, length etc that remain essentiallyconstant under most conditions. This can lead to significantreduction of calibration effort and hence is an obviousadvantage over other estimation techniques.

ENGINE CRANKSHAFT DYNAMICMODELINGThe crankshaft, like other mechanical components, in natureis a structural component with the distributed mass andelasticity. The accurate modelling of its dynamics behaviourrequires a continuous distributed system model that couldhave an infinite number of degrees of freedom, which isimpractical to employ in fast online processing application.The common alternative to simplify this complexity is todiscretize the continuous system into a finite set of rigidbodies that are interconnected with springs and dampers,known as the lumped-mass model. In past research suchmodels have been used to investigate and predict thebehaviour of the engine mechanical system, with the cylinderpressure or individual cylinder torque as the input to themodel and crankshaft motion or other engine dynamic statesas the model output.

The rigid-body model (single degree-of-freedom) and multi-body model (multiple degrees-of-freedom) are most oftenused among the lumped mass models, where the former is thefurther simplified result from the latter by lumping all themasses together to preserve only one degree-of-freedom forcrankshaft speed.

An example of 7 degrees-of-freedom 4-cylinder crankshaftdynamics is depicted in Figure 1. The moment of inertia ofthe crankshaft has been lumped and separated into 7 piecesdenoted by J0 to J6 and each of which has its own angular

position and speed variables, expressed by θ0…6, …6,allowing the deformation analysis on the crankshaft. J0 istotal inertia of the vibration damper and other auxiliaries ofthe crankshaft on its free end. J1 to J4 is the inertia of eachcrank throw corresponding to one cylinder plus the rotationpart (big end) of connecting rod, J5 is the flywheel inertia andJ6 is drivetrain inertia. K is rotational stiffness between twoadjacent crankshaft inertias. Crel is relative friction betweentwo adjacent inertias and Cabs is the absolute friction betweenthe inertia and the non-rotating reference, such as enginebody. T1 to T4 are each cylinder's excitation torque acting onthe crankshaft and Tload is the load torque opposing thecylinder torques.

Figure 1. The 7-DOF lumped-mass crankshaft dynamicmodel of a 4 cylinder in-line engine.

The excitation torque of each cylinder mainly consists of gaspressure torque Tgas, friction torque Tfriction and mass torqueTmass. The mass torque is due to the reciprocating movingparts of the dynamics system, such as piston assembly andreciprocating part (small end) of connecting rod. It isintrinsically dependent on crank angle, its first and secondderivatives (speed and acceleration), which will be furtherdiscussed in later section.

By Newton's 2nd law, the dynamic equation of each lumpedmass can be readily established. The general dynamicequation of the multi-body model can be expressed in thematrix state-space form by combining all lumped massequations as:

Eq.1

Where all the variables are in their matrix forms, and θ, , are the vectors of all inertias' angular position, speed andacceleration respectively. J, K and C are symmetric matricesand referred to as inertia, stiffness and damping matricesrespectively. The torque items, Tgas, Tfriction, Tmass, Tloadare also in their matrix forms representing vectors of allcylinder individual torques.

If all the inertias in Figure 1 are further lumped together, thecorresponding single degree-of-freedom rigid-body modelcould be derived as:

Eq.2

Where J is the total inertia of crankshaft and all the variablesare in the scalar form. The torque items are the sums of allcylinder action torques.

The multi-body model forms a Multi-In-Single-Out (MISO)system. As it allows relative motion between pieces oflumped-mass on the crankshaft, it is able to model the

torsional vibration, which is naturally present in crankshaftspeed signal due to the crankshaft elasticity. However, whenattempt to invert such a model to estimate the individual in-cylinder pressure from the single signal of crankshaft speed,it can be difficult to separate each cylinder's contribution.

The advantage of the rigid body model is that, in this case themulti-cylinder engine is in fact simplified into a virtualsingle-cylinder engine that has multi-firing within one enginecycle. This represents a Single-In-Single-Out (SISO) system,which is much easier to inversely solve the dynamics toestimate indicated torque and in-cylinder pressure from thecrankshaft speed signal. As will be described later, with theaid of signal processing technique and the appropriate methodto separate individual cylinder torque contribution, certainshortcomings of such rigid body model can be overcome andthus make it feasible to estimate in-cylinder pressure formulti-cylinder engines.

For the online application, since all the signals are measuredat discrete-time sampling steps, either in time domain or incrankshaft angle position domain, it is necessary to discretizethe continuous-time model presented in Eq.2 into a discrete-time form. Also, it is desired to substitute the accelerationterm in Eq.2 because the acceleration value is not generallyimmediately available from direct measurement and canintroduce large numerical error when attempt to derive itusing the data of limited sampling rate. Integrating both sidesof Eq.2 over Δθ crank angle can eliminate this accelerationterm and also convert the model into discrete-type form.Equation 3 shows the left side of the integration, whichrepresents the kinetic energy change of the crankshaft during

this Δθ interval, where denotes the instantaneouscrankshaft speed at crankshaft angle θ.

Eq.3

Equation 4 shows the right side of the integration, whichdenotes the work done by the torque during the same crankangle interval. The instantaneous torque at this specific crankangle is assumed nearly constant when the Δθ interval issufficiently small. In this study, 1 crank angle degree (CAD)is chosen as the Δθ interval.

Eq.4

Combining Eq.3 and Eq.4 the complete discrete-time modelform is obtained in Eq.5.

Eq.5

This derived model shows the direct connection betweeninstantaneous crankshaft speed change and total torque actingon the crankshaft. It serves as the basis for the torque andpressure estimation.

ESTIMATION OF THE TORQUECOMPONENTSFigure 2 shows the flowchart of the proposed estimationalgorithm. This method will first estimate the total torqueexerted on the crankshaft. Then the gas torque will beextracted, by estimating and subtracting other torquecomponents from the total torque. Finally the gas torque canbe converted into in-cylinder gas pressure by thedeterministic engine geometry relationship.

Figure 2. The flowchart of the estimation algorithm

The experimental data presented in this work is collectedfrom a production type 2.0-liter 4-cylinder 16-valve commonrail diesel engine (max power rated 97 kW at 3800 rpm, maxtorque rated 325 Nm at 1800 rpm). The instantaneouscrankshaft speed is derived from the timestamp of the

crankshaft angular position measurement at the rate of 1CAD per sample. The intrusive-type non-water-cooledpiezoelectric pressure sensors are installed on the cylinders tomeasure the actual combustion pressure. The amount ofelectric charge output by the pressure sensor is proportionalto the pressure opposed on its sensing diaphragm (−16 pC/barfor the sensor used in this study). A charge amplifier convertsthe charge amount into a voltage signal that will be measuredby the data acquisition system. A driveshaft-embedded torquesensor is used to measure the instantaneous load torque.

TOTAL TORQUE ESTIMATIONRearranging Eq.5 in following form gives the total torqueestimator:

Eq. 6

The inertia value for the tested engine, J, is identifiedexperimentally by the approach in [39]. The value of theinstantaneous crankshaft speed square change,

, is required to solve above equation.However, the direct calculation of this term using themeasured raw crankshaft speed generally presents a highlevel of high frequency noise in the result. One example ofthe measured instantaneous crankshaft speed signal and suchdirectly-calculated term, under the steady state operationcondition, is shown in Figure 3. As shown, it is difficult toobserve the cyclic nature of crankshaft torque profile fromsuch plot.

As can be noticed in the upper part of Figure 3, there issignificant portion of higher frequency components thatoverlap on the basis of the speed fluctuation trend line (thelower frequency component). The higher frequencycomponents indicate the torsional vibration, which is causedby the elasticity and damping properties of the actualcrankshaft. Its frequency is distinctive from that of the basisspeed fluctuation trend line resulted from the engine cyclicgas torque excitement.

This can be further investigated by examining the powerspectrum density (PSD) of the speed square change. As seenin Figure 4, the torsional vibration, which is at a higherengine firing harmonic orders and takes a considerableportion of the spectrum energy, is separate from the low endof the frequency spectral. As handling such higher frequencytorsional vibration is beyond the capability of the rigid-bodymodel, therefore an appropriate signal processing technique isneeded to pre-process the measured raw speed signal first.

Figure 3. Measured instantaneous crankshaft speed andderived speed square change in one engine cycle. Test

condition: 1500rpm, 40%load.

Figure 4. Power Spectrum Densities of measuredinstantaneous crankshaft speed and derived speed square

change. Test condition: 1500rpm, 40%load.

A digital low-pass filter is found effective to separate thehigher order torsional vibration components from the lowerfrequency ones that are of interest. A Finite ImpulseResponse (FIR) filter with 18-harmonic order cut-offfrequency was implemented in this work. The effect offiltering is shown in Figure 5, where the cyclic nature of thespeed square change and thus the total crankshaft torque canbe readily observed.

Figure 5. Filtered instantaneous crankshaft speed andderived speed square change. Test condition: 1500rpm,

40%load

The examples of total torque estimation under several engineconditions, mainly including low to medium engine speedrange, are shown in Figures 6, where the estimated totaltorque is compared with the measured one. Note that there isnot a practical way to directly sense the torque that wasexerted on crankshaft section that is internally inside engine,as no strain gauge or any other means of instrumentation canbe placed inside the high speed rotation crankshaft sectionsinside engine. In this study, to obtain such measured totaltorque, the individual torque components of the measuredtotal torque are either directly sensor-measured (driveshaftload brake torque), derived through other measurement (in-cylinder pressure for gas torque calculation) or analyticallycalculated (mass torque, to be described later), then summedtogether. As shown, for such speed and low range, theestimated total torque can approximate to the measured totaltorque during most crank angle positions.

For the higher engine speed range, the estimated total torquebased on the rigid-body model starts to exhibit larger error,due to the increased level of torsional vibration in the crankshaft speed signal. An example is shown in Figure 7, wherethe experimental data shows the total crankshaft torque startsto exhibit greater discrepancy from the actual measured

crankshaft speed acceleration. To obtain a satisfactory resultfor higher engine speed, a dynamic model that is capable toquantify the torsional vibration can be essential.

Figure 7. Relationship between measured crankshaftacceleration and total torque (3000 RPM)

MASS TORQUEMass torque is a result of the varying speed translationalmotion of the reciprocating masses, including the piston and

Figure 6. Examples of the estimated total torque tested under various conditions.

small end of connection rod. The method to calculate masstorque can be derived from the energy point of view, byconsidering the change in kinetic energy E during Δθ crankangle change, as result of the acting torque Tmass, asexpressed in Eq.7.

Eq.7

The piston motion is generally dynamically modelled as asingle point mass moving along the cylinder axis. Describingthe motion of a connecting rod is however more complex,since it undertakes both translational and rotational motion.Common practice to dynamically model the connecting rod isto construct a statically equivalent model. In such a model,two points of mass connected by a mass-less rod denoted bymA and mB respectively. In later derivations, mA is placed atthe piston pin performing translational motion together withthe piston, and mB is placed at the crank pin rotating togetherwith crankshaft. Such a model is not fully dynamicallyequivalent to the actual connecting rod due to the fact that itsmoment of inertia is not equal to that of original body. But itis chosen in this study due to online implementationrequirement, because the fully equivalent model will lead toonly one mass being placed at either joint, while the motionof the other one will not be a pure translational or rotationalmotion, which makes it more difficult to be applied inkinetics analysis [33].

Therefore the kinetic energy of piston and connecting rodassembly can be expressed via the definition of translationaland rotational kinetic energy:

Eq. 8

Where, mpiston is the piston assembly mass including piston

head, pin and rings, is the piston instantaneous speed and

is the crankshaft instantaneous speed and ImB is themoment of inertia of the rotational part of the connecting rodmodel with respect to the crankshaft rotation axis.

Differentiating both sides of Eq. 8 with respect to crankshaftangle θ and substitute the time-domain items with angle-domain ones gives the expression of the mass torque [39]:

Eq.9

Where the engine geometry relations and canbe readily analytically calculated based on real-time crankangle position [39]. Due to the crank configuration of theinline 4-cylinder engine used in the study, there will be a πphase difference between the engine geometry relations ofany two cylinders that fire successively.

Note that the inertia of the connecting rod mass at crank joint(big end of the connecting rod), ImB = mBr2, can beconsidered as part of the total inertia of moment ofcrankshaft/flywheel assembly J, due to its pure rotationalmotion. Therefore the actual mass torque calculation in thisin-cylinder pressure estimation algorithm will not include thisterm. The final mass torque expression is as shown in Eq.10.

Eq.10

The speed square term in the mass torque expression Eq.13,, increases rapidly when engine speed increases. Hence at

higher engine speeds range it will dominate the magnitude ofmass torque and make it much more influential indetermining the total crankshaft torque. One example of themass torque for a single cylinder is depicted in Figure 8. Thetotal mass torque for the engine, including all cylinders, canbe derived by summing up all the mass torques from allcylinders.

Figure 8. Mass torque profile of individual cylinder. Testcondition: 1500rpm, 40%load.

LOAD AND FRICTION TORQUEESTIMATIONTo date, an accurate predictive expression for instantaneousengine friction has proven difficult to obtain and be used byonline applications. Such expression involves the adequatelymodelling of friction in the piston ring and skirt, valve train,auxiliary and the crankshaft bearing [55,56,57]. Where it'sdesired to obtain the directly measured instantaneous friction

value, either a structurally modified dedicated researchengine or a thorough calibration is necessary for the proposedmethods [58,59]. Meanwhile to infer the mean frictiontorque, modeling it as a polynomial function or lookup mapof engine speed or piston speed, temperature and load etc, isthe common practice [13, 31]. In this study a simplified loadand friction torque (referred as the total loss torque in thisstudy) estimator is proposed based on the observedcharacteristics of crank angle domain torque profiles.

The instantaneous total loss torque can be estimated based onrearrangement of the torque components equation as shownin Eq.11.

Eq.11

In this expression, total torque Ttotal and mass torque Tmasscan be derived by the total torque estimator Eq.6 and themass torque analytic expression Eq.13 respectively. Thoughthe gas torque Tgas is one of the ultimate estimation targets inthis algorithm, however its value at some specific crankshaftangular positions during compression stroke can bereasonably inferred by the adiabatic process model and someinitial in-cylinder conditions.

Though the total gas torque is the combination of individualgas torques from all four cylinders, due to the sequential

firing nature of a multi-cylinder engine, there is typically aperiod within an engine cycle, during which only onecylinder's gas torque dominates the total gas torque. Thetorque produced by the other three cylinders during the sametime can be relatively small and thus neglected whencalculate the total gas torque. For example, as marked in leftpart of Figure 9 and zoomed in the right part, between about50 BTDC to 30 ATDC, only cylinder no.4 is undergoing thecompression stroke and expansion stroke while othercylinders are in either the exhaust or intake strokes. Since thegas torque contribution produced by intake and exhauststroke is much lower than that of compression and expansionstroke, they have no significant effect on the total gas torqueprofile, which means the cylinder no.4 gas torque by itselfcan effectively approximate the total gas torque during thistime period.

To predict individual cylinder gas pressure during thecompression stroke, a number of well established thermaldynamic models can be applied, provided that initialconditions of the trapped in-cylinder charge at the beginningof the compression stroke are known. Manifold AbsolutePressure (MAP) and Intake Air Temperature (IAT), both ofwhich are available for typical engine control system, areused here to infer the initial gas conditions. The gas torqueduring expansion stroke is more difficult to predict due to theunknown gas composition and initial gas condition at end ofthe combustion process. Therefore the expansion period

Figure 9. Illustration on the suitable crank angle position to predict total gas torque.

(mostly the crank angle after TDC) was avoided in the finalimplementation of this method.

The adiabatic process model Eq.12 is used to predict the gaspressure during the compression stroke.

Eq.12

Where P comp (θIVC) is the in-cylinder gas pressure at thecrankshaft position of intake valve closing (IVC) indicatingstart of the gas compression process. This can be measuredby the Manifold Absolute Pressure sensor (MAP) andcorrected by the calibrated volumetric efficiency. V chamber isthe calculated crank-angle-based combustion chambervolume [40]. The value of adiabatic constant, k, is taken to bearound 1.3 in this study for the compression stroke charge[31].

Once the gas pressure is inferred, the gas torque can bereadily obtained via the gas pressure-torque relationexpression Eq.13 [40].

Eq.13

As shown in Figure 10, the predicted pressure profileapproximates the measured one very well except for thegreater discrepancy near TDC. This discrepancy at TDC,however, does not pose a significant difficulty to convert gaspressure into the gas torque, because the gas torque isapproaching zero near TDC due to the fact that the

characteristics of the crank lever approaches 0 at TDC.

With total torque Ttotal and mass torque Tmass being solved,the load and loss torque during this specific crank angleposition can be calculated through Eq.14. It should berealized that the actual total load and loss torque can exhibitfluctuations within one engine cycle even under steady stateengine conditions. One example of the instantaneous loadtorque measurement is shown in Figure 11. For the operationconditions studied in this work, the experimental data showedthat the estimated load torque value, which is estimated forthe specific crank angle position, has typically less than 4%of difference compared to the actual measured engine-cycle-averaged load torque. Therefore such estimated load torque isused as the engine-cycle-averaged load torque.

Figure 11. Measured total loss profile during one enginecycle. Test condition: 1500rpm, 40%load, steady

condition

THE TOTAL GAS TORQUEESTIMATIONBy rearranging the torque balancing equation Eq.2 into thefollowing form, the instantaneous total gas torque profile

Figure 10. Examples of the predicted gas pressure (left) and derived torque (right) during compression stroke, Condition: 1500rpm, 40% load

(summation of all cylinders) can be readily obtained using thetorque components derived previously.

Eq.14

Examples of the total gas torque profile estimation resultsunder different engine conditions are shown in Figure 12. Theerror of the estimated total gas torque profile is evaluated byR.M.S. error over 10 engine cycles with respect to the actualtorque derived from measured cylinder pressure.

INVERSION OF GAS TORQUE INTOIN-CYLINDER PRESSUREA re-arranged crank-slider mechanism equation can be usedto convert the individual cylinder gas torque into in-cylinderpressure:

Eq.15

Where Pi(θ) is in-cylinder pressure of cylinder No.i, Tgas_i isgas torque contributed by that cylinder, Ap is cylinder head

area, is the engine geometry function of pistondisplacement x and crankshaft angle θ.

SEPARATION OF INDIVIDUALCYLINDER GAS TORQUEThe total gas torque derived in previous section is the

summation of all cylinder gas torques, Thus the individual cylinder contribution needs to beextracted before perform the torque-to-pressure inversion.

Figure 12. The example of the estimated total gas torque under various conditions.

The method to extract the gas torque profile of a specificcylinder is similar to the load torque estimation. Theappropriate crank angle span will be identified, during whichthat cylinder is the concurrent firing cylinder and hence itsgas torque dominates the total gas torque profile. As depictedin Figure 14, in order to extract the gas torque of cyld no.4,its dominant crank angle span can be observed as aboutbetween 30 BTDC to 50 ATDC in this shown case. Outsidethis angular range, the total gas torque starts to besubstantially affected by cyld no.3 and cyld no.2.

Figure 13. The dominant CAD span of cyld no.4 on thetotal gas torque profile.

During the aforementioned crank angle span, next firingcylinder generally will undergo the compression stroke andthus its gas compression torque can also affect the total gasduring this time window (in this case it's cyld no.2, afterabout 50 ATDC in Figure 13). By estimating its gascompression torque (via the adiabatic process model andinitial gas condition) and subtracting it from total gas torque,the gas torque of current firing cylinder can be extracted moreaccurately.

Then the in-cylinder pressure profile of the firing cylindercan be reconstructed from the extracted torque profile, basedon the engine geometry relationship according to Eq.18. Theadiabatic model can be applied to expand the obtainedpressure profile into a wider crank angle span covering mostof the compression and expansion stroke.

IN-CYLINDER PRESSUREPROCESSING AROUND TDCAn example of such reconstructed pressure profile is shownin Figure 14. As shown, the reconstructed pressure shows adiscontinuity error around TDC position. This issue is due tothe fact that the characteristics of the engine geometry

function , as the denominator in Eq.18, approaches zero atTDC position.

Figure 14. The reconstructed in-cylinder pressureprofile. Test condition: 1500 rpm, 40% load.

An interpolation method over the TDC position is proposedto overcome the above issue around TDC position. Theinterpolation function, as expressed in Eq.16, transits thepressure profile from the ending part of the compressionstroke into the beginning part of expansion stroke,.

Eq.16

Where f(θ) is the interpolation function and its value evolvesfrom 0 to 100% gradually during the interpolated crank anglespan. Such interpolation in fact models the gas pressureevolution process, during which the in-cylinder chargemixture undergoes the combustion event. One commonlyreferred function serving such purpose is the well knownWiebe function [40], as shown in Eq.17, which wasdeveloped to model the mass fraction burned (MFB) profileduring a combustion event.

Eq.17

Where a and m are adjustable parameters to control the shapeof the Wiebe function to best fit the actual MFB. Theparameters θ, θ0 and Δθ denote the current crankshaft angle,the specified angular position of start of combustion (SOC)and the specified angular duration of the combustionrespectively. By taking the actual MFB measurement, theseangular positions are empirically set to the fixed values of 5BTDC and 30 ATDC respectively for the tested conditions inthis study.

One example of such interpolated pressure profile by usingthe Wiebe function is shown in Figure 15. As shown in theplot, though the peak error over the TDC region is reduced,there are still noticeable discrepancies compared with the

measured pressure profile. This discrepancy indicates theshape of the Wiebe function, which denotes a single-stagedcontinuous heat release process, doesn't exactly match theactual MFB profile encountered in the experiment. Thedouble combustion peak pressures, as can be observed in thefigure and is commonly seen for modern diesel enginenowadays, is due to the employed dual injection strategy,which results in the dual-staged heat release process. Thisimplies the ideal interpolation function should account for theactual instantaneous heat release rate in order to approximatesuch staged heat release process.

Figure 15. Interpolated pressure profile by Wiebefunction. Test condition: 1500 rpm, 40% load

DISCUSSION ON THEINTERPOLATION FUNCTIONHeat release rate is an indicator of the rate of chemical energyrelease from the fuel as a result of combustion. Completeanalysis of heat release rate involves a thoroughthermodynamics analysis of an open boundary system,combustion efficiency, non-quasi-static effect and creviceeffect etc [31]. To give a satisfactory indication ofcombustion progress for online application, a simplifiedexpression of heat release rate can be adequate.

The typical method to calculate the simplified heat releaserate is expressed in Eq.18, from the measured in-cylinderpressure data.

Eq.18Where dQ is the heat release rate and Q is total heat releaseup to θ crank angle from start of combustion (SOC). p and vare instantaneous in-cylinder pressure and combustionchamber volume respectively. k is specific heat ratio. Thenormalized value of accumulated heat release Q, rangingfrom 0 to 100%, can be used as the interpolation function.

One example of the measured heat release rate on the testengine is shown in Figure 16. The heat release rate dQ showstwo peaks around the TDC position, indicating the stagedcombustion process.

Figure 16. The heat release rate obtained from measuredpressure profile. Engine condition: 1500 rpm, 40% load

One example of calculated pressure profile using above heatrelease rate as the interpolation function is shown in Figure17. Compared with the previously Wiebe-functioninterpolated pressure profile in Figure 16, this heat-release-rate interpolated one adequately follows the measuredpressure profile over the TDC area.

Figure 17. Interpolated pressure profile by heat releaserate function. Engine condition: 1500 rpm, 40% load

It should be noted that the Eq. 21 relies on the measured in-cylinder pressure data, which is not available for this onlinepressure estimation application. When the pressure data is notavailable, it is still possible to infer the expected heat releaserate by other existing engine operational parameters, such asthe ECU fuel injection pattern and/or ignition timing (spark-ignition engine only). The fuel injection schedule, includingfuelling quantity of each injection, start of injection timingand duration of injection, is readily available during engineruntime. By taking into account the air/fuel mixture ignitiondelay and combustion time [31], an approximation of the heatrelease rate could be feasible.

ERROR EVALUATIONThe estimation accuracy on the reconstructed in-cylinderpressure is evaluated by calculating the following errors:

• Absolute Pressure (%)

• IMEP (%)

• Peak Pressure Value (%) and its position (CAD)

The error of absolute pressure is defined in Eq.19, whichgives the average error of the pressure value along the entireestimated pressure curve.

Eq.19

The estimated peak pressure position CADpeak are determinedbased on spline interpolation of 0.25 CAD resolutions aroundTDC position based on the estimated pressure profile data, inorder to more accurately locate the peak pressure.

The evaluation result is shown in Table 1 and the detailedcycle-by-cycle error of each group is given in Figure A1 andA2 in Appendix.

As shown in Table 1, the accuracy of the peak pressurelocation is in general less than 5 CAD, which makes thismethod a possible choice for peak pressure location relatedcontrol or diagnostics applications including MBT sparkignition timing control, diesel fuel injection timing control,misfire detection and NOx reduction control etc.

The greater error in IMEP estimation can be due to the errorin load torque estimation, as the load torque estimator relieson other measured or derived torque components at a specificangular window, during which most of them have the lowervalue compared to their peak amplitude. In experiments itwas found that, though often time the relative error is small,the absolute error in those torque components can sometimesbe comparable to their lower value and the level of loadtorque absolute value itself, which in turn translates intohigher relative error of load torque estimation and thus IMEPestimation.

As previously indicated, this method is more suitable for lowto medium engine speed range due to the increasing level of

torsional vibration at higher engine speed. A single set ofdynamic model parameters was used throughout all the tests,which should be realized as a substantial reduction ofcalibration effort over other pressure estimation methods. Thefact that the individual cylinder dominant CAD span is usedthroughout this proposed algorithm also implies that it'ssuitable for an engine with less cylinder numbers, as whencylinder number increases, the period between two successivefiring cylinders becomes shorter therefore make it moredifficult to separate the individual cylinder torquecontribution.

CONCLUSION REMARKSIn the previous work reported in the literature, enginedynamics model was mostly used as a forward approach toinvestigate how the in-cylinder pressure affects the modeloutcome (crankshaft speed in most cases). The invertedapproach, having known crankshaft motion to estimate in-cylinder pressure, has been experimentally investigated inthis study.

In this work a complete algorithm to reconstruct in-cylinderpressure for individual cylinder is proposed based on enginedynamics model. A discrete-time rigid-body crankshaftdynamics model is constructed in this study based on thekinetic energy theorem, as the basis expression for totaltorque estimation. The major difficulties, including the loadtorque estimation and separation of pressure profile frommultiple cylinders, are addressed and solutions to eachproblem given.

To correct the inherent pressure estimation error around TDCresulting from inverting engine geometry characteristics, acombustion-progress based approach is proposed tointerpolate around TDC position of the estimated pressureprofile. Different interpolation functions are compared andthe heat release rate is proven most effective, which alsotheoretically agrees with the physical meaning of suchinterpolation function and therefore indicates a potentialdirection for future work.

Experimental evaluation proved the applicability of suchrigid-body crankshaft dynamics model in the context ofpressure estimation and also indicated its limitation. Thesimple rigid-body model is proven to be effective from low tomiddle engine speed range. To enhance the capability ofrigid-body to handle the complicated real crankshaft speed

Table 1. Error Evaluation of Dynamic Model Based Pressure Estimation Method

fluctuation, a pre-filtering on the raw speed signal isimplemented to cut off the higher frequency torsionalvibration components presented in the crankshaft speedmeasurement. The experiments also show that, as the enginespeed increases, the crankshaft motion become morecomplicated so that a multiple DOF crankshaft model can beneeded to further accurately model the crankshaft dynamicsbehaviour.

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CONTACT INFORMATIONFeilong Liu (corresponding author):

Delphi Corporation3000 University Drive, Mail Code: 483-300-220, AuburnHills, MI 48326, United StatesTel: +1-248-836-0558feilong.liu@delphi.com

ACKNOWLEDGMENTSThe authors are grateful to Denso Corporation for the fundingsupport. We also thank Mr. Toyoji Yagi of Denso for hisinsightful discussion during the study.

DEFINITIONS/ABBREVIATIONS

Notation

Cxy coherence function between x and y

H transfer function

k specific heat ratio

P Pa pressure

Px, Pxy auto power spectral density of x, crosspower spectral density of x and y

Q KJ heat release

T N.m torque

Vc m3 cylinder clearance volume

Vchamber m3 cylinder combustion chamber volume

ω rad/s crankshaft angular frequency

rad, crankshaft angular position,

rad/s, speed and acceleration

rad/s2

rad/s2 crankshaft speed fluctuation

AcronymsCAD

Crankshaft Angular Degree

DFTDiscrete Fourier Transform

DT-FTDiscrete-Time Fourier Transform

ECUElectronic Control Unit

EOCEnd of Combustion

FRFFrequency Response Function

IDFTInvert Discrete Fourier Transform

IMEPIndicated Mean Effective Pressure

IVCIntake Valve Close

MISOMulti-In-Single-Out system

SISOSingle-In-Single-Out system

SOCStart of Combustion

TDCTop Dead Center

TWCThree Way Catalytic converter

WOTWide Open Throttle

APPENDIX

Figure A1. Detailed cycle-by-cycle experiment result of group 1 data.

Figure A2. Detailed cycle-by-cycle experiment result of group 2 data

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