an extended kalman observer for the in-cylinder air mass flow estimation

8/8/2019 An Extended Kalman Observer for the in-Cylinder Air Mass Flow Estimation

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An Extended Kalman Observer for

the In-Cylinder Air Mass Flow Estimation

Osvaldo Barbarisi1, Alessandro di Gaeta2, Luigi Glielmo2, and StefaniaSantini1

1 Dipartimento di Ingegneria, Universita degli Studi del Sannio in Benevento,{barbarisi, glielmo}@unina.it

2 Dipartimento di Informatica e Sistemistica, Universita degli Studi di NapoliFederico II, {digaeta,santini}@unina.it.

Abstract. Nowadays, every new gasoline car sold in US, Canada and Europe isequipped with a three-way catalytic converter (TWC) in combination with a fuelcontrol system in order to reduce the pollutant emissions. For the maximum TWCconversion efficiency, the engine control has to furnish a stoichiometric air fuelmixture to the cylinders. For this purpose an estimation of the air incoming intocylinders becomes necessary in every driving condition, not only in steady-stateoperations, but especially during rapid throttle transients. A conventional speed-density equation, that evaluates the air incoming engine, depends on the enginespeed and intake manifold pressure. The pressure signal is influenced by sensordynamic and by pumping fluctuations due to periodic moving masses as enginevalves. In this work, an Extended Kalman Filter is proposed as observer of thein-cylinder air mass flow rate.

1 Introduction

Automotive emission regulations are becoming stricter and stricter in USAand Europe. Figure 1 shows the evolution of legal limits on vehicle pollutantemission imposed by European Commitment since 1994 to 2005. Vehicle pol-lutant emissions are tested on special drive cycles composed by urban andextraurban parts.

In order to meet emissions statements, gasoline engines are equipped bya Three Way Catalyst (see figure 2). Since the TWC conversion efficiency ismaximal when a stoichiometric Air-Fuel Ratio (AFR) is fed to cylinders (seefigure 3), a tight control of the injected fuel feds to the cylinders is required.

Conventional AFR control strategies use a feedback control based on thesignal of an oxygen-sensor, placed in the exhaust pipe, to ensure that theAFR will remain in the neighborhood of the stoichiometry value (see figure4). In all AFR controller a in-cylinder air mass flow rate has to be perfectlyestimated in order to calculate the exact fuel amount to be injected.

In current commercial the air mass flow rate calculation is based on themeasure of a intake manifold pressure sensor and engine speed. Unfortunatelyfor the pressure, the low-pass characteristic of automotive commercial sensormakes the pressure signal affected by a delay which introduces, during fast



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Fig. 1. Emission regulation of pollutant emission of vehicle

Fig. 2. Engine and Three Way Catalyst

transients, errors in the air estimation and, consequently, in the fuel quantityto be injected to obtain the stoichiometry ratio of the mixture.If using a fast pressure sensor, another problem arises. This sensor captures

pressure oscillation caused by the periodic motion of valves and pistons. Theamplitude of the fluctuations is about 10% of the mean pressure value andis not negligible. This pumping fluctuations have to be filtered in order toreconstruct the mean pressure necessary for the proper working of the AFR



Extended Kalman Observer for Air Estimation 3

Fig. 3. Conversion efficiency of TWC depends on Air-Fuel Ratio (forcommercial gasoline ΛStoich = 14.56)

Fig. 4. A simple block scheme of the conventional AFR control

control algorithms. This operation again introduces an undesirable delay inthe information. A physically reliable filter, that can be on-line implemented,always would introduce a delay.

Taking into account the above considerations, the goal of this work is todesign an on-line observer, based only on the measure provided by a com-monly used automotive “slow” pressure sensor. The observer will be able toreconstruct the mean value of the real signal without introducing additionaldelay. The non linearity of the manifold dynamics suggested the use of an

Extended Kalman Filter to estimate manifold pressure. The in-cylinder airmass flow rate is computed by the correct pressure information by the wellknown and commonly used speed-density equation [3]. The overall strategyhas been designed and validated on experimental data along an european



4 Barbarisi et al.

drive cycle and fast transient operations in order to test the effectiveness of the approach.

2 Model Plant

In this section, a mathematical model will be provided in order to describedynamics of the inlet manifold. In figure 5, plant representation is shown.The model will be taken in account for the Extended Kalman Filter imple-mentation. Figure 6 shows a diagram block describing manifold dynamics.Any block will be described below.

Fig. 5. Inlet manifold

2.1 Air mass flow through throttle

Under the hypothesis of isentropic flow, the air mass flow rate through thethrottle mat is described by the follow equation (see figure 5) [3]

mat =pa√RT a

C d(α)Aθ(α)β

pm pa

, (1)

where pa and T a are respectively the ambient pressure and temperature; pmis the inlet manifold mean pressure; α is the angle position of the throttleplate; Aθ is the effective section area of the throttle corrected by the dischargecoefficient C d(α), experimentally identified; the term β models the correctiondue to sonic ( pm

pa xc) and subsonic ( pm

pa> xc) regime of the air flow as

β (x) =

γ 12

2

γ+1

12γ+1γ−1

if x xc, 2γγ−1

x2γ − x

γ+1γ

if x > xc,

(2)




Fig. 6. Plant model

with xc =

2γ+1

γγ−1

(γ = 1.46 is the adiabatic coefficient of the air; xc =

0.518).

2.2 Speed Density

In spite of the complexity of the fluid dynamic phenomena occurring duringa transient (due to fast opening or closing of the throttle), the conventionalvolumetric efficiency η (function of the engine working point), identified dur-ing steady-state conditions, is used to describe carefully the inlet air massflow rate. So the speed-density gives an accurate description of the air mass

flow rate through inlet valves

map = η( pm, N )V dN

120

paRT m

, (3)

where V d is engine displacement and N is the engine speed; R is the universalgas constant; T m and pm are the mean manifold temperature and pressure;the volumetric efficient η is highly nonlinear function of the engine velocity(N ) and manifold pressure ( pm). It can be only calculated via experimental.Figure 7 shows the volumetric efficient of a commercial gasoline engine.

2.3 Manifold Model

Considering figure 5, the mass balance inside the inlet manifold is describedby

dmm

dt= mat − map. (4)



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Fig. 7. Volumetric efficiency η( p,m , N )

where mm is the air mass inside the inlet manifold. By using the gas law

pmV m = mmRT m (5)

(where V m is the manifold volume) and neglecting temperature variation wehave

V m ˙ pm = RT mmm. (6)

By considering (4) finally we have the classical filling-empty model [4]

˙ pm =RT mV m

( mat − map) , (7)

2.4 Pumping fluctuations

Pumping fluctuations are caused by any disturbance initiated at the bound-ary of inlet manifold such as moving piston, moving valve and moving throttleplate. These disturbances travel along the pipe experiencing many reflections.When the engine is operated in the steady-state, they finally settle down intoa standing wave.

The source of pumping noise is periodic and so the pumping fluctuationsare frequency locked to the engine event frequency. This is illustrated infigure 8 where a steady pressure signal is shown in angle domain and incorresponding frequency domain.




Fig. 8. Measured pressure signal. First frame: crankshaft angle domain;second frame: magnitude of density spectrum.

A second order model can describe dynamic of the fundamental compo-nent in the the time domain (180[deg] is the engine phase duration).

q + 2ζ 2πN

30 q +2πN

302

q = ν q (8)

Notice that, for time based equation, the fundamental harmonic is varyingwith engine speed as 30

2πN . The model (8) is a oscillatory filter with a poorly

damping (ζ = 0.01) where ν q(t) is the white noise excites it.

2.5 Sensor

Automotive intake manifold air pressure (MAP) sensor can be modeled by afirst order system

˙ pms = − 1τ MAP

pms + 1τ MAP

pm, (9)

where pms is the measured pressure and τ MAP is the time constant.



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3 Intake Air Observer

3.1 Choice of operating domain

Some phenomena in the engine show particular properties if described incrank-shaft angle θ domain expressed in degree unit. Thus, time-based modelsintroduced above will be described in angle-domain.

Generally, given a generic time invariant non linear system

x (t) = f (x(t), u(t))

let

dt =1

ω(θ)dθ

with ω > 0, the relation between time and angle domain is given by

t(θ) =

θ

0

dζ w(ζ )

Let

xθ(θ) = x(t)|t=t(θ) , uθ(θ) = u(t)|t=t(θ) ,

we have

dxθ (θ)

dθ=

1

ω(θ)f (xθ (θ) , uθ (θ)) . (10)

3.2 Discrete model plant in the angle domain

In order to derive the discrete EKF, a discrete angle based plant model hasto be furnished by letting ω = 6N where N is the engine speed expressed inround per minute unit (RPM), whether ω is expressed in deg/s unit.

Let we discretize the state equations (7) by Euler method, (9) by usingbackword method and (8) by using matching zeros/poles method. Thus, bychoosing a sample angle θs = 45deg, we obtain the corresponding plant model

pm(k + 1) = pm(k) +θs

6N

RT mV m

mat

pm(k), α(k)

+

−

map

pm(k), N

+ δm

+ w pm(k)

(11a)

pms(k + 1) =aMAP pms(k) + (1 − aMAP)

pm(k) + q1(k)

+ w ps(k),

(11b)δm(k + 1) =δm(k) + wδm(k) (11c)q1(k+1)q2(k+1)

=Aq

q1(k)q2(k)

+ Gqwq(k), (11d)




with output equations

z(k) = pms + ν p(k), (12)

where aMAP =

1 + θs6N

1τ MAP

−1.

In (11) and (12), the process noise w(k) = ( wpm (k) wpms(k) wδm (k) wq1(k) wq2(k )

and measure noise ν (k) = ν pms(k) have been introduced. Notice that, in or-

der to compensate the unavoidable inaccuracy of the speed density (3) aδm air mismatch parameter has been considered. Since pumping fluctuationsare so explicitly inclose in the model, noises are modelled as white gaussianprocesses with the follow statistical properties

E [wk] = 0, E [wkwk] = Qk, (13)

E [ν k] = 0, E [ν kν k] = rk, (14)

with Qk 0 and rk > 0 respectively positive semi-definite and definite

matrixes.

3.3 Extended Kalman Filter

The model of inlet manifold is described by equations (11) and (12). Fromnow on, we will indicate the state vector as xk = ( pm(k) pms(k) δm q1(k) q2(k) )

and the manipulable input as uk = α(k). So the plant model equations cannow rewritten as

xk+1 = f k(xk, uk) + Gwk (15a)

zk = H xk + vk (15b)

Let the conditioned means be xk|k = E [xk|z0, . . . , zk], xk+1|k = E [xk+1|z0, . . . , zk],and Σ k

|k, Σ k

+1|k the corresponding error covariance matrix; and impose that

the initial state is a gaussian aleatoric variable so x0 ∈ N (x0, P 0). The recur-sive equations of the extended Kalman filter [2] are listed below

xk|k = xk|k−1 + Lk(zk − H xk|k−1), (16a)

xk+1|k = f k(xk|k, uk), (16b)

where

Lk = Σ k|k−1H

H Σ k|k−1H + rk−1

, (16c)

Σ k|k = Σ k|k−1 − Σ k|k−1H

H Σ k|k−1H + rk−1

H Σ k|k−1, (16d)

Σ k+1|k = F kΣ k|kF k + GQkG, (16e)

with

Σ 0|−1 = P 0, (16f)

x0|−1 = x0, (16g)



10 Barbarisi et al.

where F k = ∂f k(x,uk)∂x

x=xk|k

. In figure 9 is shown the block diagram of the

plant model (equations (15)) and the relative EKF block scheme (equations

(16)).

Fig. 9. Block sceme representation of plant model and extend Kalmanfilter

3.4 Numerical results

The EKF has been tested on an experimental data set obtained by a com-mercial vehicle driven along an european extra-urban drive cycle. The vehiclewas equipped with a four cylinders engine 1282cc. The ambient temperatureand pressure, during tests, were respectively 30 [K] and 1.011 · 105 [Pa].

Figure 10 shows the measured throttle angle α and engine speed N , whilein figure 11 is represented the measured pressure and the estimated manifoldpressure by observer. Figure 12 shows the estimated throttle air mass flowrate mat and the in-cylinder air mass flow rate map adjusted by the aircorrection parameter δm.

Notice that EKF estimates the mean manifold pressure upstream sensor;this is more relevant during the transients, as shown in figure 13. It canestimate correctly the in-cylinder air mass flow rate map by the speed-density

(3) (see figure 14). Figure 15 shows a comparison between the estimatedair mass flow rate by the EKF and by speed-density computed using themeasured pressure signal. Last method introduces a percentage error till 40%during transient conditions.




Fig. 10. First frame: measured throttle angle position; Second frame:measured engine speed

Fig. 11. Measured and estimated manifold pressure

4 Conclusions

Conventional AFR control strategies, to ensure the stoichiometry of the airfuel mixture, need a measure of the in-cylinder air mass flow rate. An eventu-



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Fig. 12. First frame: estimated throttle air mass flow rate mat and esti-mated in-cylinder air mass flow rate map; Second frame: in-cylinder aircorrection parameter δm

Fig. 13. Measured and estimated manifold pressure.




Fig. 14. Estimated air mass flow throttle mat and estimated inlet air massmap corrected by parameter of correction δm.

Fig. 15. First frame: in-cylinder air mass flow rate map estimated viaspeed density based on measured pressure and via EKF; Second frame:

percentage error between two methods.



14 Barbarisi et al.

ally error strikes again on an increasing of emission pollutant during transient.In this paper has been designed an algorithm, based on extended Kalman fil-ter, to estimate the air incoming into cylinders. The EKF observes the intakemanifold mean pressure, upstream commercial sensor, and then evaluates thespeed-density equation to furnish an estimate of the air mass flow rate. Thealgorithm has been tested on experimental data set of a commercial vehicleand good air estimate has been reached during steady and transient states.

In future work, is our intention to test the performances of commercialAFR controller in which the air estimation will be substitute by EKF pre-sented. We will wait a reduction of those AFR excursions, depending onuncorrect air estimation, and then a reduction of the emission pollutant.

References

1. Kienke U. and L. Nielsen (2000) Automotive Control System, Springer.

2. Brian D, O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall.3. Heywood J. B. (1988) Internal Combustion Engine Fundamentals. McGraw-

Hill, New York.4. Hendricks H.(1995) Engine Modelling for Control Applications: a Critical Sur-

vey. Control and Diagnostics in Automotive Applications.

an extended kalman observer for the in-cylinder air mass flow estimation

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