Model Based Engine Map AdaptationUsing EKF

Erik Hockerdal∗,∗∗, Erik Frisk

∗, and Lars Eriksson∗

∗ Department of Electrical Engineering, Linkopings universitet,Sweden, {hockerdal,frisk,larer}@isy.liu.se

∗∗ Scania CV AB, Sodertalje, Sweden, erik.hockerdal@scania.com

Abstract:

A method for on-line map adaptation is developed. The method utilizes the EKF as aparameter estimator and handles parameter aging, operating point dependent model andmeasurement quality. Map adaptation, by construction, gives marginally stable models withlocally unobservable modes, that are handled. The method is also suitable for off-line calibrationof maps where the only requirement of the data is that the entire operating region of the systemis covered. The method is applied to a truck engine where an air mass-flow sensor adaptationmap is estimated based on data from a diesel engine during an ETC. It is shown that anadaptation map can be found in a measurement sequence not specially designed for adaptation.

Keywords: bias compensation, EKF, non-linear, observer, engine map, adaptation

1. INTRODUCTION

Look-up tables and maps are frequently used to describerelations in modern control and diagnosis systems wherephysical models are unavailable, e.g. low level sensor andactuator characteristics.

In engine control systems, maps of different types are com-monly used for example to compensate for changed am-bient conditions and aging engine subsystems like coolerefficiency, injector characteristics, aftertreatment systemsetc. These are typical examples of maps that need continu-ous on-line adaptation to avoid undesired system behavior,like e.g. biased air mass-flow estimates, causing increasedemissions. A related topic of major concern in enginecontrol system development is the calibration process ofthe complex control system with its variety of maps andparameters.

An attractive idea of how to handle these problems isto incorporate system models to aid in the developmentof sufficiently robust and fast adaptation algorithms thatcan be used both on-line to handle system aging and off-line to automatize the calibration process of engine controlsystems (Guzzella and Amstutz, 1998).

Basically map adaptation can be viewed as a desire toreduce and store the compensation of an operating pointdependent estimation error, and a way to reduce station-ary estimation errors in model based observers was devel-oped in Hockerdal et al. (2008b). It utilizes an observabledefault state space model

xt+1 = f(xt, ut) (1a)

yt = h(xt), (1b)

and measurements, y, u, from the system and estimatesa low order bias augmentation Aqq that, when used inobserver design, reduces the stationary estimation errorsof the resulting observer. The result of this method is anobservable augmented model

xt+1 = f(xt − Aqqt, ut) (2a)

qt+1 = qt (2b)

yt = h(xt), (2c)

that can be used with any suitable observer design toconstruct an observer with reduced stationary estimationerrors compared to using the default model directly.

However, since the observer designed in Hockerdal et al.(2008b) treats the bias as a random walk it is not ableto keep track of the changes in bias between operatingpoints over time. That is, as soon as the system changesoperating point, all information about the bias in theprevious operating point is discarded.

An observer that stores information about the bias couldbe useful in applications where the system is such thatinformation with a certain quality only is present in someoperating points. Then information collected at theseoperating points can be used to improve the quality ofthe estimated variables throughout the entire operatingregion of the system, e.g. air mass-flow adaptation in dieselengines with exhaust gas recirculation (EGR) and variablegeometry turbine (VGT) (Hockerdal et al., 2008a). Inengine map adaptation schemes it is also important thatthe algorithm is robust against outliers since occasionalspurious measurements are inevitable.

Hence, the goal is to design a robust observer with memorythat is able to handle old parameters, operating pointdependent models, and varying measurement quality, andoccasional spurious measurements.

2. METHOD OUTLINE

Based on the method developed in Hockerdal et al.(2008b), an information preserving observer, i.e. withmemory, can be obtained by exchanging the assumptionof a bias modeled as a random walk

qt+1 = qt + vt, vt ∼ N (0, Q), (3)

driven by white noise, for a parameterized function, ormap, describing the bias

qt = qfcn(xt, ut, θt)

θt+1 = θt + vt, vt ∼ N (0, Qfcn).

Where qfcn(xt, ut, θt) is a parameterized function or mapwith unknown parameters θ that describes the bias depen-dence on the system states x, and inputs u. The parame-ters are modeled as random walks in the same way as thebias state in (3).

If this is done for the general state space model (2), thefollowing system is obtained,

xt+1 = f(xt − Aqqt, ut) (4a)

θt+1 = θt (4b)

qt = qfcn(xt, ut, θt) (4c)

yt = h(xt), (4d)

where Aqq adjusts the stationary points of the system.Here and henceforth system, measurement, and randomwalk noise terms are left out to increase readability. Notethat this is similar to the formulation used in Hockerdalet al. (2008b), the difference is that the bias states havebeen exchanged for a parameterized function and thefunction parameters are introduced as new states. If (4c) isinserted into (4a), a standard state space form is obtainedwhich means that any suitable observer design can beapplied. For example, one way of estimating states whileat the same time handling unknown parameters is toapply a joint parameter and state estimating extendedKalman filter (EKF) (Kopp and Orford, 1963). There theparameters are introduced as new states with constanttime derivatives and augmented to the original states justas in the system described by (4).

To develop a model like (4) and use it for estimation andidentification entails that some new issues have to be ad-dressed compared to Hockerdal et al. (2008b). Of these themain concern in this paper is how to update the functionparameters, θt, in a controlled manner. Another relatedissue is how to find a suitable parameterization (4c), withits structure and regressors. This is not treated here andthe interested reader is referred to e.g. Lind and Ljung(2008).

Even though any suitable observer design can be appliedto this system, the choice here is to use a stochastic filter.An advantage of stochastic filters, like for example EKF,compared to deterministic observers is that, not only thestate estimate, but also an estimate of the estimation errordistribution is computed. The estimation error statistics isused in the computation of the filter feedback gain, whichgives the stochastic filters natural tuning parameters thatallow filter tailoring to handle unknown state initialization,time dependent model and measurement quality, outlierrejection etc.

For these reasons and because of simplicity the jointparameter and state estimating EKF is used throughoutthis paper.

3. OBSERVABILITY

In all estimation problems observability or, at least, de-tectability of the system at hand is central. Since thesystem considered here can be viewed as an extension

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5Observability of linear interpolated engine map

Measured air mass-flow [kg/s]

Air

mas

s-flow

corr

ection

[%]

(W1, θ1)

(W2, θ2)

(W3, θ3)

(W4, θ4)(W5, θ5)

(W6, θ6)

(W7, θ7)

(W8, θ8)

(W9, θ9)

(W10, θ10)

(W11, θ11)

(W12, θ12)

Fig. 1. Air mass-flow correction map with the grid pointsdenoted with the pair (Wi, θi) corresponding to acorrection factor of θi at a mass-flow of Wi.

of the augmented system addressed in Hockerdal et al.(2008b) it is natural to assume that the system

xt+1 = f(xt − Aqqt, ut) (5a)

qt+1 = qt (5b)

yt = h(xt) (5c)

is observable and analyze how the replacement of (5b)by a parameterized function, (4b – 4c), affects the ob-servability. The observability is therefore directly depen-dent on the properties of the parameterized function (4c)that describes the operating point dependence of the biasqt = qfcn(xt, ut, θt).

If for example (4c) is an engine map, implemented as alook-up table with the grid points as parameters, and aninterpolation algorithm computing the output. Then thesystem will have locally unobservable states (Hermannand Krener, 1977), i.e. parameters that are not usedin the interpolation in the current operating point arenot observable. In Figure 1 the local observability ofthe parameter states in linear interpolation is illustratedusing an air mass-flow correction map from an inline sixcylinder Scania diesel with EGR and VGT (Hockerdalet al., 2008a). If the operating point, in this case defined bythe air mass-flow measurement, lies in the shaded regionof the figure then only the two grid points constituting theborder of the region are observable.

3.1 Growing Estimation Error Covariance

At any given time there are generally, for systems like (4),some parameters θi that are locally unobservable. A prop-erty of systems with locally unobservable modes is thatthe estimation error covariance matrix in an EKF frame-work grows linearly for the unobservable modes if systemnoise for these modes is present. That is, in regions wherethe system seldom operates the estimation error covariancematrix coefficients, corresponding to locally unobservablemodes or parameter states, will grow linearly withoutbound. This linear growth in covariance matrix elementshave two sides. 1) It offers a way to achieve fast update ofold parameters while protecting often updated parametersfrom spurious measurements. 2) It may cause numericalproblems affecting the system stability when consideringthe life-time of the system, which has to be handled.

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Time [s]

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(a) Parameter variance.

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Time [s]

Air

mas

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-[k

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w5

w6

w7

w8

w9

w10

(b) Air mass-flow trajectory.

Fig. 2. The figure shows the evolution of the estimationerror variance for three parameter states.

This effect is illustrated in Figure 2, where the varianceof three parameter states, θ5, θ8, and θ10 from Figure 1are plotted versus time. In Figure 2(a) θ5 corresponds toa parameter that is not observed at all for the studiedtrajectory, while the parameter θ8 is observable during thefirst half of the trajectory and unobservable for the secondhalf. For the parameter θ10 the case is reversed, that is theparameter is first unobservable and then observable.

Experiences from adaptive maps in engine applications,not using the EKF and joint parameter and state es-timation, indicate problems concerning parameter agingand occasional spurious measurements. For example anengine that, during normal operation, does not cover theentire parameter space and only occasionally enters someareas, may suffer from undesired system behavior causedby old parameters. Many of todays adaptation schemesapply the same adaptation algorithm in each update stepand do not adjust the update procedure with respect towhen the parameters were last updated (Wu, 2006; PeytonJones and Muske, 2007). In these cases, a linearly growinguncertainty for seldom updated parameters enables a fastparameter update rate of old parameters without riskinglarge errors in the state estimates. This can in some sensebe thought of as a dynamic forgetting factor similar torecursive least square (RLS) techniques and is a highly de-sirable property in engine adaptation algorithms not only

to handle aging parameters but also to protect updatedparameters from occasional spurious measurements, thatare fairly common in engine applications.

A direct and intuitive way of handling the linear growthof estimation covariance of locally unobservable parameterstates, i.e. 2), is to introduce an upper limit for the cor-responding estimation error covariance matrix elements.A possible upper limit is the initializing error covariancematrix, P0. Since it is desirable to limit the estimationerror covariance of only the locally unobservable param-eters it is appropriate to perform the limitation elementwise, i.e. compare Pi,i to P0 i,i, and limiting Pi,i by settingPi,i = P0 i,i when Pi,i ≥ P0 i,i. It is straightforward toshow (Jaynes, 1996) that the off-diagonal elements in Pdo not affect the estimation error covariance for a singleparameter, and by using P0 as an upper limit, the intro-duction of yet another tuning parameter is avoided.

4. FILTER TUNING – Q VS. QFCN

Even though the method developed here is quite similarto the one developed in Hockerdal et al. (2008b) there aresome important differences. One difference is the rate atwhich the bias and parameter states are updated.

For the observer designed in Hockerdal et al. (2008b) itis necessary for the bias state to change approximately asfast as the system dynamics, otherwise it will not be ableto track a change in system operating point. However, arapidly changing bias state captures also high frequencydisturbances, and is thereby sensitive to outliers, whichmakes the bias state in this method unsuitable for enginemap adaptation.

In an observer utilizing a parameterized function to de-scribe the bias, the parameter states operate with anupdate rate determined by system aging, which is substan-tially slower than for a bias state that has to track changesin system operating point. This makes the observer basedon a model containing a parameterized function or mapless sensitive to temporary disturbances, compared to anobserver using only one state to describe the bias. However,both methods can be used to find an adaptation map, thefirst estimates it directly and the latter after some postprocessing like mean value computations, which makes itless suitable for on-line applications.

Another issue that is, to some extent, straightforwardlyhandled by stochastic filters is initialization of the un-known bias or function parameters. By proper tuning ofthe corresponding elements in the estimation error covari-ance matrix, P0, a temporary faster update of unknownbias or function parameter states is achieved. That is, dueto an initially faster update rate of unknown parameters,in the same way as old parameters are allowed a fasterupdate rate, a rapid convergence of the otherwise quiteslow parameter states is achieved.

5. METHOD EVALUATION

To evaluate the method two studies, a simulation studyand a study utilizing experimental data, are performedwhere the aim is to adapt the air mass-flow sensor indiesel engine, characterized by a 1-D adaptation map. The

simulation study shows the convergence of the methodand includes a minor analysis of model error, and noisesensitivity while the experimental part shows the result ofthe method applied to experimental data.

In both studies a non-linear model of a heavy duty truckengine developed in Wahlstrom and Eriksson (2006) isused together with measurements from an engine in anengine test cell. The model has three states, intake andexhaust manifold pressures, and turbine speed which allare present in the model output together with the air mass-flow through the compressor. The data used are collectedduring a European transient cycle (ETC).

5.1 Observers

Three observers are designed and evaluated both in sim-ulation and on experimental data. The observer designsare: An EKF based on the default model developedin Wahlstrom and Eriksson (2006) directly,

xt+1 = f(xt, ut)

yt = hWcmp(xt),

referred to as Def. An EKF with an extra bias stateintroduced in the measurement equation to reduce theestimation error from the method developed in Hockerdalet al. (2008b),

xt+1 = f(xt, ut)

qt+1 = qt

yt = hWcmp(xt) + qt,

referred to as Aug. A joint state and parameter estimatingEKF based on the default model and a parameterized bias,

xt+1 = f(xt, ut)

θt+1 = θt

yt = (1 + qfcn(hWcmp(xt), θt))hWcmp

(xt),

referred to as Map. where qfcn is presented in Figure 1.

In the simulation study all observers use only feedbackfrom the air mass-flow sensor, hWcmp

(xt), whilst in theexperimental evaluation, feedback from all sensors exceptthe exhaust manifold pressure sensor, x2, is used. Eventhough the model is observable from any of the outputsthe model errors are such that an augmented feedbackis needed in the experimental evaluation. The estimationperformance evaluation is with respect to all states andoutputs, i.e. intake and exhaust manifold pressures, tur-bine speed, and air mass-flow through the compressor.

For the study utilizing experimental data, all observersare augmented with an additional bias state, in additionto those introduced to handle the measurement error,with the purpose of reducing estimation errors due tomodel errors in the compressor model causing incorrectprediction of the compressor mass-flow, i.e. for Def.

xt+1 = f(xt, ∆Wcmp,t, ut)

∆Wcmp,t+1= ∆Wcmp,t

yt =(

x1,t x3,t hWcmp(xt, ∆Wcmp,t

))T

.

5.2 Simulation data

The simulation study serves two purposes, 1) to comparemodeling errors as stationary biases (Hockerdal et al.,

modelETC

yu

yerror

SensorSimulation

Fig. 3. Simulation set-up with sensor error from Hockerdalet al. (2008a).

2008b), or as parameterized functions, and 2) to analyzethe effect of incorrect noise description and model errorsensitivity when using the EKF as a parameter estimator.

Both these utilize the simulation set-up presented inFigure 3 for creating the data. The data is created bysimulating the model with input data from an ETCsegment. The segment is chosen to contain a wide range ofair mass-flows such that a trajectory for which the systemis observable is created. To simulate incorrect air mass-flow measurement a 1-D sensor error map is used in thesimulation that adjusts the air mass-flow according to

Wmeas = (1 + q(Wtrue))Wtrue, (6)

where q(Wtrue) is the engine map presented in Figure 1.The distorted air mass-flow, Wmeas, is then used forfeedback in the observers.

Convergence One property of the estimation bias cor-rection method developed in Hockerdal et al. (2008b) isthat since no information about the bias in each operatingpoint is saved the observer convergence speed depends onthe speed of the bias states. While the extension presentedhere use a parameterized function and the convergencespeed of the filter, when the parameters are adapted, willtherefore not be dependent on the speed of the parameterstates.

To analyze and compare the convergence speed of filtersutilizing parameterized bias to filters utilizing bias statesis a difficult task. A reason for this is that the convergencespeed is highly dependent on the filter tuning. Sincethe maximum errors occur in transients, see for examplethe transient at the time 5 s in Figure 4, they give anindication on how well the estimator is able to tracktransient behavior. Figure 4 presents the estimation errorsfrom Aug., where the bias state has been tuned to be tooslow and does not manage to track the change in bias. Asreferences the estimation errors from Map. and Def. arealso presented. Since a bias state has to be approximatelyas fast as the system dynamics and the parameters as fastas the system aging, filters tuned with these aspects inmind will have about the same performance with respectto estimation quality during normal operation, which isconfirmed in Table 1. Nevertheless, since the bias state isallowed to change much faster than the parameter states,a filter utilizing that method will be more sensitive todisturbances, i.e. outliers as mentioned in Section 4. Afilter with a parameterized function with slow parametersdoes not allow disturbances to affect the estimation ofneither model nor parameter states to the same extentas a filter with a bias state, i.e. have a stronger smoothingeffect.

Figure 5 shows the true and estimated map from Map.,and the correction made by the slowly varying bias fromAug. computed according to

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Mass

-flow

[kg/s]

Time [s]

Def.Aug.Map.

Fig. 4. Estimation error of Aug., where the bias is tooslow which gives large estimation errors in transients,together with estimation errors of Map. and Def.

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Air mass-flow [kg/s]

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Fig. 5. Mass-flow correction estimated by Aug. and Map.

r =qt

yt − qt

.

From this figure it is obvious that Map. manages toestimate a correction map out of a cycle, not speciallydesigned for map adaptation, without any post processing.Also, the correction made by Aug. captures the true mapbut some post processing, like for example mean valuecomputations, is needed to get a map that can be usedfor interpolation etc.

Evolution of adaptation map In an application where themethod is used for engine map adaptation it is importantthat the method converges. However, since the modeldescription probably never will be entirely correct, it isimpossible to converge to something that can be calledthe true map. However, for simulated data this can beachieved. Figure 6 shows the evolution of the adaptationmap over time. The parameters are all initiated to zero,

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Table input values [kg/s]

Adaptation map evolution

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Fig. 6. Adaptation map evolution showing fast conver-gence.

Table 1. Def., Aug. and Map. estimation error.

Meas.Max abs. error Mean error

Def. Aug. Map. Def. Aug. Map.pim[Pa] 13901 1156 1075 2521 47 46pem[Pa] 13482 1071 999 2383 43 42

ntrb[rpm] 3118 446 432 769 23 22Wair[kg/s] 0.02 0.003 0.003 0.003 0.00007 0.00006

indicated by the straight line at time t = 0, and convergesto the true map as the operating region is covered.

In Table 1 it is seen that all measures, both maximumand mean, are approximately the same for Aug. andMap. for all system states and the system output, whileDef. has significantly larger errors. From this it can beconcluded that the estimation performance with respectto the default states and outputs are similar for the twoobservers Aug. and Map. Though, Map. also automaticallyestimates a map that describes the air mass-flow sensorerror, that can be used by other algorithms or functionsin the engine control unit (ECU).

Robustness The robustness of the proposed algorithm isonly briefly analyzed by i) introducing a variety of modelerrors in minor sub models known to have inaccuracies,e.g. the EGR system, ii) tampering with model andmeasurement noise structures and intensity.

The first to study the properties of filter divergence dueto modeling errors and the second to analyze the effectdue to incorrect noise properties. None of the experimentsshowed any tendencies of divergence or lack of convergenceconsistency.

5.3 Experimental data

To see how the method can be expected to work in areal application an evaluation using measurements froman engine in an engine test cell is conducted. Since in thiscase there are other model errors present besides the airmass-flow sensor error an extra bias state is introducedthat compensates for incorrect mass-flow through thecompressor, see Section 5.1. With the introduction of an

Table 2. Mean estimation error using experi-mental data for Def., Aug and Map.

Meas.With ∆Wcmp,t

Without ∆Wcmp,t

Def. Aug. Map. Def. Aug. Map.pim[Pa] 4208 169 -210 13875 13176 13019pem[Pa] -10987 -13746 -14420 -2339 -3454 -3450

ntrb[rpm] 69 16 29 -57 -19 5Wair[kg/s] 0.008 -0.020 -0.020 0.031 0.035 0.035

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Air mass-flow – [kg/s]

Cor

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ion

fact

or–

[%]

q

y−q

q

y−q– stat

Final Map

Fig. 7. By Aug. and Map estimated mass-flow correction.

extra state compensating for the compressor mass-flow,known to be biased, the estimate of the intake manifoldpressure becomes significantly better at the expense of theexhaust manifold pressure estimate, while the estimates ofturbine speed, and air mass-flow is almost unaffected, seeTable 2.

In Section 4 the tuning of Aug. and Map. are discussed,especially the different philosophies of the bias describingstates – the relatively fast bias state in Aug. and the slowmap states in Map. In Figure 7, and Table 2 the similarityin estimation performance between Aug. and Map. is strik-ing, which is an expected result. The mean estimation er-rors for pim and ntrb are reduced while the mean estimationerrors for pem and Wcmp are slightly increased. That is, inabsence of outliers Aug. and Map. are comparable withrespect to output estimation performance. The benefitwith Map. is that it also estimates an adaptation map.

From Figure 7 it is seen that, even though there areunknown model errors present, besides the compressormass-flow, the method manages to estimate a map thatdescribes the difference between modeled and measuredair mass-flow through the compressor well. Figure 7 showsthe correction factor between modeled and measured airmass-flow, similar to what was observed in Figure 5.

Finally, all this shows that an adaptation map can beestimated even though the data used is from the highlytransient ETC, not specially designed for air mass-flowsensor adaptation.

6. CONCLUSIONS

A method for storing bias information from differentoperating points is developed. With this method it ispossible to achieve simultaneous estimation of original

model states and parameters, like for example adaptationof engine maps.

Stochastic filters together with a parameterized bias thathas locally unobservable states is in fact an asset thathandles seldom updated parameters and gives robustnessagainst occasional spurious measurements in ordinary mapadaptation algorithms. The linear growth of estimationerror covariance, that comes as a result of local unobserv-ability of the parameters, also form a potential numericalproblem for the filter and a way to limit this growthwithout extra filter parameters is provided.

The method shows promising results in a simulation study,where it manages to estimate the engine states while atthe same time estimating a parameterized air mass-flowadaptation map. In an evaluation with experimental datait is shown that while maintaining the same estimationquality with respect to mean and maximum absolute error,as the method developed in Hockerdal et al. (2008b), anengine adaptation map can be estimated as well. Thatis, simultaneous state estimation and map adaption isachieved without, for adaptation, specially designed cycles.

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