Individual Cylinder Characteristic Estimation for a Spark

Individual Cylinder Characteristic Estimation for a

Spark Injection Engine∗

L. Benvenuti§, M. D. Di Benedetto?,S. Di Gennaro? and A. Sangiovanni-Vincentelli †‡

§Dipartimento di Informatica e Sistemistica “Antonio Ruberti”, Universita degli Studi di Roma “La Sapienza”Via Eudossiana 18, 00184, Roma, Italy.

[email protected]

†P.A.R.A.D.E.S., Via San Pantaleo 66, 00186, Rome, Italy.lucab, [email protected]

?Dipartimento di Ingegneria Elettrica, Universita di L’AquilaPoggio di Roio, 67040, L’Aquila, Italy.

[email protected], [email protected]

‡ EECS Dept., University of California at Berkeley, CA. 94720, [email protected]

September 25, 2002

Abstract

Engine control policies are mostly based on the assumption that all injectors have the samebehavior independent of location and aging. In reality, injectors do vary and age. To containvariations around a nominal value, tight tolerances are imposed on the manufacturing process. Evenif the manufacturing process is tightly controlled, the air-to-fuel ratio needed to satisfy emissionconstraints is difficult to achieve due to aging and even slight mismatch among different injectors.To devise control policies that take into account behavior differences among injectors, we need toestimate injector characteristics from measurements that are taken on the engine during its lifetime. In this paper, we present an estimation technique for injector characteristics based on a setof measurements that can be carried out using the sensors present in the car, i.e., intake manifoldpressure, crank-shaft speed, throttle-valve plate angle, injection timings and exhaust air-to-fuelratio, which is measured by a single UEGO sensor placed at the exhaust pipe output.

1 Introduction

Today a great deal of emphasis is placed by automotive engineers on the development of closed-loopengine control systems that meet exhaust emission standards while minimizing fuel consumption andmaximizing driving performance. Meeting emission constraints imply that the air-to-fuel (A/F) ratioof the mix provided to the combustion process by the injection system must be as close as possible tostoichiometric, which corresponds to the amount of air theoretically required to oxidize all the injectedfuel.

∗The work has been conducted with partial support of PARADES, a Cadence, Magneti-Marelli and ST-microelectronics E.E.I.G, by the European Community Projects IST-2001-33520 CC (Control and Computation) andIST-2001-32460 HYBRIDGE, and by CNR PFMADESSII SP3.1.2.

1

Since the air flow is regulated by the throttle valve, controlled by the driver with the acceleratorpedal, the control variable becomes the fuel flow provided by the injectors to the intake manifold.Regulating the fuel flow requires accurate estimation of the characteristics of the injectors.

It is in fact known that injector variability may cause cylinder-to-cylinder differences in the mixturecomposition of the order of 5% [12]. Cylinder-to-cylinder air-fuel maldistribution results in emissionconcentration imbalance between cylinders. These individual emission differences will not necessarilyaverage out to produce an overall result equivalent to that obtained with all cylinders operating atthe same air-to-fuel ratio. This is due to the nonlinear variation of hydrocarbons, nitrogen oxides andcarbon monoxide concentrations with air to fuel ratio [4, 12]. Moreover, even in the case of perfec-tly matched injectors, cylinder-to-cylinder air-fuel maldistribution can arise from different individualcylinder behavior in breathing due to intake manifold structure and valve characteristics.

To reduce the air-fuel maldistribution due to injector characteristics imbalance, injectors areusually required to have close tolerances, up to 1%, resulting in high cost per injector. Injectors arein fact machined to tolerances of the order of 6% and then their tolerance is reduced to 2% (in theoperation nominal range, i.e. when the on-time is greater than 500 µs) by a quite expensive off-linetuning process. Better tolerance can be achieved by an even more expensive process that consists oftesting and appropriately grouping the injectors in the so-called matched injector sets.

To achieve accuracy in controlling the emission concentrations, control policies that can discernthe contribution of each injector and thus overcome the cylinder-to-cylinder air-to-fuel imbalance,have been recently proposed [4, 10, 16]. These policies allow the accommodation of greater injectortolerances and consequently the reduction of the cost per injector. These are inherently closed-loopstrategies, since they allow the tuning of the individual cylinder A/F using the signal of the exhaust-gas-oxygen (UEGO) sensor. In [10], a single switching UEGO sensor is used for this purpose, whilein [16] the estimation of the individual cylinder A/F is obtained from a model inversion of an eight-cylinder engine. A slight different approach is presented in [16], where the individual cylinder A/F isestimated at low engine speed for a six-cylinder engine, using two UEGO sensors. The drawback ofthese control strategies is that they are effective only under steady-state operating conditions. There-fore, in situations where the UEGO sensor signal is not reliable (transients and cold start conditions),they can not be applied and different open-loop strategies have to be designed.

The open-loop strategies are based on the estimation/measurement of the air charged in eachcylinder. Clearly, in this situation, the individual cylinder A/F can be successfully controlled only ifthe individual injector characteristics are known. This paper addresses this problem, providing theestimation of the individual injector characteristics, along with the individual air flow estimation. Morein detail, we propose a method for the estimation of each injector’s characteristics via measurementsof the intake-manifold pressure, the crank-shaft speed, the throttle-valve plate angle, the injectionson-time and the UEGO sensor reading from the exhaust. The estimation processes is carried outduring steady-state conditions, and the resulting parameters can be stored and used in the open-loopstrategies during the transients and the UEGO sensor warm up.

The estimation algorithm is based on a fairly accurate modelling of the cylinder air-filling process,of the exhaust manifold dynamics, and of the UEGO sensor. For the cylinder air-filling process, astandard lumped model is used [11]. To obtain an accurate estimation algorithm, we studied the de-pendence of the air charge dynamics on some characterizing variables (pressure sensor error, throttleoffset area, off-line estimation volumetric efficiency error). For the exhaust manifold dynamics, thetransport delays and the mixing between the air-fuel charges associated with the cylinders were consi-dered. Finally, we considered a linear UEGO sensor modelled as a first order lag plus a delay, a common

assumption in the literature (e.g., [6], [10], [15], [17])1. The idea behind the proposed approach is toestimate first the sensor error and the throttle offset area on the basis of steady pressure measurements,and then, to calculate the average air flow entering each cylinder using these estimations. Using

1The use of higher-order models for the A/F sensor does not change the core of our approach, since the effect of amore complex model is felt only in the computation of the inverse of the model.

2

information from a single linear UEGO sensor, the air-to-fuel ratio for each cylinder is estimated andthe injector characteristics are obtained. In this last step, the UEGO sensor signal is appropriatelysampled in order to invert the exhaust manifold and sensor dynamics to reconstruct the individualcylinder A/F’s.

The paper is organized as follows. In Section 2, we formulate the estimation problem. In Section 3,we show how to estimate the average air-mass flow entering the cylinders [3]. In Section 4, estimation ofthe fuel-to-air ratios is covered. In Section 5, we estimate the injector characteristics and in Section 6,we offer realistic simulations and concluding remarks.

2 Estimation Strategy

In this section, we state our solution approach for the individual injector characteristic estimationproblem for a Spark Ignition (SI) engine.

We are interested in obtaining the characteristics of the individual injector because the variablesavailable for the engine control are the injection profile, i.e., approximately the time in which theinjector is open, and the timing of the spark. The injection profile is used to control the fuel-massflow into the cylinders that determines the behavior of the engine with respect to pollution, fuelconsumption and driveability. The fuel mass per injection pulse can be assumed to be an affinefunction of the fuel injector on-time

mf,i =n

30gi (Ti − Ti,off ) , i = 1, · · · , nc

where Ti is the time the injector is open [s], gi the gain [kg/s] and Ti,off the offset [s] of each injector.Hence, the problem is to estimate gi and Ti,off . In Section 5, we show that these two parameters canbe estimated once the fuel-mass flow mf,i injected in each cylinder is estimated. The key measurementfor determining mf,i is the fuel-to-air ratio F/A given by a single UEGO sensor. The UEGO sensor iscapable of measuring the fuel-to-air ratio γ(t). If indeed there were no mixing of the exhaust gases fromthe various cylinders in the exhaust manifold, the problem would be fairly simple. The complicationsarise from the partial overlap of the exhaust gases from each cylinder.

Let

γi(t) =mf,i(t)ma,i(t)

, i = 1, · · · , nc (1)

denote the fuel-to-air ratio released by the ith cylinder during the exhaust phase, where ma,i(t) is theair-mass flow entering in each cylinder during the intake process, and nc the number of cylinders. Notethat the contribution to cylinder-to-cylinder air-fuel maldistribution of the different cylinder breathingcharacteristics is negligible with respect to the injectors’ variability. Then we assume that the averageair-mass flow ma,i entering each cylinder is the same. Therefore, if we could estimate the ratios (1)and the air-mass flow ma,i, we could obtain the fuel flows mf,i(t). The estimation of γi(t) from theUEGO sensor measurement is the subject of Section 4 and is based on the inversion of

γ(t) = ϕ(γ1(t), · · · , γnc(t)),

a function that will be derived considering that γ(t) depends on the fuel-to-air ratio γc(t) at the runnerconfluence, which, in turns, is a known function f(γ1(t), · · · , γnc(t)) of the fuel-to-air ratios.

The estimation process for the air-mass flow ma,i is described in Section 3. ma,i is a dynamicalfunction of pman, the mean value manifold pressure, and the volumetric efficiency, ηv. If we assumethat

(H) the mean value intake manifold pressure pman and the crankshaft speed n remain constant2 �2Hypothesis (H) is not restrictive since steady-state situations are usually encountered during engine operation, and

reasonably simple tests are available on the on-board computer to detect steady-state operation. Moreover, as shown

3

then, pman = 0 and the average air mass flow ma = ncma,i, entering the nc cylinders, is equal to theaverage air-mass flow ma,th passing through the throttle. The average air-mass flow is a function of thethrottle-plate angle α, which is directly measurable, and of the throttle-offset area A0

th, an unknownvalue slowly varying because of aging. We can measure the manifold pressure with an unknown errorεp, and the volumetric efficiency with an unknown constant error εηv . Hence, the estimation algorithmfor ma,i is based on the estimation of εp, εηv , and A0

th.

3 Air-Mass Flow Estimation

In this section, we show how to estimate the average air-mass flow ma entering the cylinders [3]. Wehave already observed (see the Appendix for a summary of the derivation of these relationships) thatwe need first to estimate the mean value intake manifold pressure pman, from measurements (21), andthe volumetric efficiency ηv, given by (24). Since we can measure pman with an unknown error εp, andwe can estimate ηv with an unknown constant error εηv , the problem is to determine estimations ofthese errors.

We suppose that the steady-state hypothesis (H) holds true, so that pman = 0 and the averageair flow ma, given by (23), is equal to the average air flow ma,th passing through the throttle, givenby (20). Since this last quantity is a function of the throttle offset area A0

th, an unknown value slowlyvarying because of aging, we need to estimate this parameter as well.

The proposed algorithm allows for estimating the errors εp, εηv and the parameter A0th on the

basis of three different steady state measurements. We assume that the variables and parameters ofthe system remain unchanged during the three measurements. We assume to measure the manifoldpressure with an unknown error εp ∈ [−3%, 3%], to estimate the volumetric efficiency ηv with anunknown constant error εηv ∈ [−10%, 10%] and to have an unknown offset area A0

th ∈ [0, 5] mm2 forthe throttle. Finally, we assume to have a F/A feedback control system which regulates the injectionpulse duration Ti in such a way that the air to fuel ratio is close to stoichiometry.

The following subsection 3.1 is devoted to the determination of estimates for εp, εηv and A0th,

while in subsection 3.2 we will determine the average air flow ma.

3.1 Sensor-error and throttle-offset area estimations

Two cases are considered: the case in which the ambient pressure is known and the one when it isunknown.

3.1.1 The case of known ambient pressure

From hypothesis (H), pman = 0 so that ma,th = ma. Then, using (20) and (23), we obtain

Ae,th(α) +A0th =

Vd n pman

120 patm

√RTatm

ηv(n, pman)

β

(pman

patm

)where the expression (22) for Ath(α) has been used. Substituting pman obtained from (21), and using(24) for ηv, we have(

Ae,th(α) +A0th

)(1 + εp) = (1 + εηv)η

estv (n, pm, εp)f(pm, εp, patm, Tatm) n pm (2)

in [13], the manifold pressure in steady-states distributes uniformly around the mean values with quite small standarddeviations. Note also that even if the engine model is intrinsically hybrid [2], hypothesis (H) allows us to neglect thehybrid nature of the SI engine since we will use this model in the case of constant crank-shaft speed and constant pressure.

4

wheref(pm, εp, patm, Tatm) =

Vd

120 patm

√RTatm

1

β

(pm

(1 + εp)patm

) .

If we assume to know patm and Tatm, equation (2) gives all the possible values (εp, εηv , A0th),

which enable the model to return the given measured crank-shaft speed n, throttle plate angle αand measured manifold pressure pm. Assuming to reach three different steady-state situations, say(n1, α1, p1

m), (n2, α2, p2m), (n3, α3, p3

m), then it is possible to find the values (εp, εηv , A0th) that satisfy

equation (2) for the three different cases. To do this, define

Aie,th: = Ae,th(αi), Eηv : = (1 + εηv)

andhi (εp) := ηest

v (ni, pim, εp)f(pi

m, εp, patm, Tatm) ni pim

so that equation (2), for the three steady state situations, gives(Ai

e,th +A0th

)(1 + εp) = hi(εp)Eηv , i = 1, 2, 3. (3)

We can rewrite (3) as

3∑i=1

(A

(i+1)mod 3e,th −A

(i+2)mod 3e,th

)hi(εp) = 0

Eηv =Ai

e,th −Aje,th

hi (εp)− hj (εp)(1 + εp) i 6= j

A0th =

hj (εp)Aie,th − hi (εp)A

je,th

hi (εp)− hj (εp)i 6= j

or ϕ(εp) = 0

Eηv = φij(εp) i 6= j

A0th = ψij(εp) i 6= j

with i, j = 1, 2, 3.

Then, an estimation εp of the sensor error εp can be computed using an iterative process, forexample Newton’s method, to find the zeros of the function ϕ(εp). Hence, we can compute theestimated values for εηv , and the estimate A0

th as follows

εηv =13

3∑i,j=1i>j

φij(εp)− 1, A0th =

13

3∑i,j=1i>j

ψij(εp).

3.1.2 The case of unknown ambient pressure

In this case we assume to measure the ambient pressure with the manifold sensor at key-on. Then wehave patm,m = (1 + εp)patm and equation (2) reduces to

Ae,th(α) +A0th = (1 + εηv)η

estv (n, pm, εp)f(pm, patm,m, Tatm) n pm (4)

wheref(pm, patm,m, Tatm) =

Vd

120 patm,m

√RTatm

1

β

(pm

patm,m

) .

5

Assuming, as before, to reach three different steady-state situations, it is possible to find the values(εp, εηv , A

0th) that satisfy equation (4) for the three different cases. Consider, as before, the functions

Aie,th, Eηv and define

hi (εp) := ηestv (ni, pi

m, εp)f(pim, patm,m, Tatm)nipi

m

so that equation (4), for the three steady-state situations, gives

Aie,th +A0

th = hi(εp)Eηv , i = 1, 2, 3. (5)

We rewrite (5) as

3∑i=1

(A

(i+1)mod 3e,th −A

(i+2)mod 3e,th

)hi(εp) = 0

Eηv =Ai

e,th −Aje,th


A0th =


je,th


or ϕ(εp) = 0

Eηv = φij(εp) i 6= j

A0th = ψij(εp) i 6= j

for i, j = 1, 2, 3.

Hence, as in the previous case, an estimation εp of the sensor error εp can be easily computed.Finally, we have

εηv =13

3∑i,j=1i>j

φij(εp)− 1, A0th =

13

3∑i,j=1i>j

ψij(εp).

It is worth noting that this method relies on the fact that patm does not varies significantly betweenkey-on and the three different steady-state measurements. This may be not the case when drivingin mountainous areas, so that in this case an estimation error would appear due to the variabilityof patm.

3.2 Air-Flow Estimation

Once the estimations εp, εηv and A0th are obtained, an estimation of the air mass flow entering the

cylinders can be computed using (23) and (20) as

ma,i =˜ma(εp, εηv) + ˜ma,th(εp, A0

th)2 nc

, i = 1, · · · , nc (6)

where ˜ma(εp, εηv) =Vd

120RTatmn

pm

(1 + εp)(1 + εηv) η

estv (n, pm, εp)

and

˜ma,th(εp, A0th) =

patm

(Ae,th(α) + A0

th

)√RTatm

β

(pm

(1 + εp) patm

)in the case of known ambient pressure, or


patm,m

(Ae,th(α) + A0

th

)(1 + εp)

√RTatm

β

(pm

patm,m

)

6

when the ambient pressure is measured. It is worth noting that, from the steady–state hypothesis(H), one should have ˜ma = ˜ma,th

3.

4 The Estimation of the Fuel-to-Air Ratios γi

The problem solved in this section is the estimation of the fuel-to-air ratios γi(t), i = 1, · · · , nc, givenby (1), from the output signal γ(t), obtained from a single UEGO sensor. In the following subsection,we derive first a model of the exhaust gases during their motion from the cylinders to the oxygensensor. Then, an estimation algorithm for mf,i is proposed.

4.1 Mathematical Model of the Exhaust Manifold

In this section, we develop a model for an nc-cylinder engine describing the contribution in each cycleof each cylinder to the measured F/A ratio. We take into account the mixing among the air-fuelcharges associated with each cylinder and the time that a single air-fuel charge takes to travel thelength of its exhaust manifold runner. The UEGO sensor dynamics are also considered.

During engine operation, four events take place, i.e. intake, compression, combustion and exhaust.As previously explained, under hypothesis (H) the hybrid nature of the engine can be neglected.Moreover, since we are studying the exhaust gas dynamics, the exhaust phase will be considered thestarting event of the engine dynamics. Referring to the ith cylinder, at the beginning of the exhaustprocess the exhaust valve opens, and the burnt gas goes from the high pressure environment in thecylinder to the lower pressure in the exhaust manifold. During this process, the exhaust gas expandsinto the exhaust manifold volume and travels towards the UEGO sensor. Note that in the exhauststroke, most of the burnt gas in the cylinder is pushed out into the exhaust manifold by the piston.

When the gas moves in the exhaust manifold the main processes that must be taken into accountare

1) the transport of the burnt gases in the exhaust manifold runner;

2) the gas-mixing in the manifold junctions.

The transport delays associated with each of the nc exhaust runners are not equal, since each runnerhas a different length. Moreover, during the gas transport, the gas molecule velocities spread about themean value. While it is not difficult, although mathematically involved, to consider these aspects, forthe sake of simplicity, we will assume that the delays are all equal to a quantity δr, and we will neglectthe velocity dispersion. The effect of the delay differences will be analyzed by a set of simulations.The mixing process can be seen as a merging process at the runner confluence, where the mixtureflows sum up. Hence, the merging process during the travelling towards the sensor will be modelledby a first order system. In summary, the main processes which take place in the exhaust manifold canbe taken into account by the following transfer function

PMIX (s) =e−δr s

1 + τMIXs(7)

with δr, τMIX the transport delay and the time constants of the mixing process. Under the assumptionof plug flow in the exhaust manifold, the average velocity of the exhaust gases is proportional to theengine speed [6], [15]. Thus, the transport delay δr can be assumed inversely proportional to thecrank-shaft speed [7], [10], [17]

δr =k

n.

3In fact, (εp, εηv , A0th) are defined as the solutions of the equations ma(εp, εηv ) = ma,th(εp, A0

th). In (6) we considered

the mean between the values of ˜ma and ˜ma,th in order to reduce influences of possible estimation errors on εp, εηv , A0th.

A study on parameter sensibility may help in fixing a more appropriated weighted average to compute ma,i.

7

However, when the sensor is located close to the exhaust valve, the blast of exhaust gases during theblow down process dominates the transport delay. As a consequence, the exhaust transport delay canbe assumed constant (as shown in table 1 in Ref. [15]).

The UEGO sensor has desirable properties, since it is linear, accurate and gives fast responses.As far as its modelling is concerned, the diffusion process of the oxygen that occurs in the UEGOsensor can be modelled as a first order system with a diffusion delay [6], [9]. Hence, the UEGO sensordynamics are represented by the following transfer function

PUEGO(s) =e−δ

UEGOs

1 + τUEGOs(8)

with δUEGO , τUEGO the diffusion delay and time constant of the diffusion process. As shown in [15],the sensor’s time constant τUEGO varies with the throttle angle but not with the engine speed.

The fuel-to-air ratio at the runner confluence is a known function

γc(t) = f(γ1(t), · · · , γnc(t))

of the ratios γi(t). In fact, since the air mass flow ma,i entering each cylinder is the same, γc(t) dependsonly on the ratios γi(t), on the number nc of cylinders and on the timing of the exhaust process ofeach cylinder. When the only charge present at the UEGO sensor at time t is the ith one, f is simplygiven by the value γi(t), while it is given by the arithmetic mean between γi(t), γj(t) when the chargespresent at the UEGO sensor at time t are the ith and jth ones. More precisely, let T be the timerequired for the crank-shaft to advance 720◦. Then, each phase takes T/4 time and the cycles of twosubsequent cylinders are shifted by T/nc. Therefore, when nc > 4 the contributions to γc(t) due totwo subsequent cylinders i, i+ 1 overlap for a time

∆ =T

l.c.m.(4, nc)

and, during this overlap, we have

γc(t) =γi(t) + γi+1(t)

2.

The function γc(t) is clearly a piece-wise periodic function of period ∆. In the case of four and fivecylinders one has the pictures shown in Figure 1.

On the basis of the previous discussion, the measured normalized fuel-to-air mixture ratio γ(t)can be obtained from

γ(s) = P (s)γc(s) (9)

where

P (s) =e−δs

(1 + τMIXs)(1 + τUEGOs), δ =

k

n+ δUEGO

is the transfer function describing the dynamics of the exhaust manifold, obtained using (7), (8).

4.2 The Estimation Algorithm: the Ideal Case

The dependence of the piece-wise constant signal γc(t) on the time interval ∆ suggests to look for asolution of the problem in the discrete-time context, by considering a sample rate equal to ∆ and adiscrete-time signal γc(q) followed by a zero-order holder. The discrete-time signal γc(q) is shown inFigure 2 in the case of four and five cylinders.

From (9) and considering the inverse Laplace transform, we obtain

γ(t+ δ) = L−1

[1

(1 + τMIXs)(1 + τUEGOs)γc(s)

].

8

Since γc can be seen as a discrete-time signal followed by a zero-order holder, we consider the samplingγs(q) of the signal γ(t+ δ), with period ∆. Hence, we have

γs(q) = Z−1[Q(z)γc(z)

]where Q(z) is the transfer function of the sampled system, consisting of the zero-order holder, themixing and sensor dynamics, and the sampler

Q(z) = Z[L−1

[1

(1 + τMIXs)(1 + τUEGOs)1s

]∣∣∣∣t=k∆

]z − 1z

= 1− τMIX

τMIX − τUEGO

z − 1z − p1

+τUEGO

τMIX − τUEGO

z − 1z − p2

(10)

withp1 = e−∆/τ

MIX , p2 = e−∆/τUEGO .

Note that we consider γ(t+δ) and not γ(t) since the total delay δ, depending on the crank shaft speedand the UEGO sensor characteristics, is a known quantity. Therefore, it is possible to compensate forthe presence of this delay in the estimation algorithm and to work directly on the signal γ(t+ δ).

In what follows, in order to illustrate the procedure, we consider two interesting cases, the4-cylinders and 5-cylinders engines.

4.2.1 A simple case: a four cylinders engine

In the case of a four cylinders engine ∆ = T/4 and the contributions γi(t) do not overlap, see Figure 1.Referring to Figure 2, the Z-transformation of the signal γc(q) is given by

γc(z) = Z[γc(q)

]= γ1

z4

z4 − 1+ γ2

z3

z4 − 1+ γ3

z2

z4 − 1+ γ4

z

z4 − 1. (11)

Considering the transfer function (10) and the input (11) it is possible to compute the output

γs(z) = Z[γs(q)

]= Q(z)γc(z). (12)

This output has the form

γs(z) = γst(z) + γss(z) =R1z

2 +R2z +R3

(z − p1)(z − p2)+

z

z4 − 1

4∑j=1

γss, j z4−j (13)

where γss(z), γst(z) are the Z-transformations of the steady-state and transient components γst(q),γss(q) of the output γs(q), and R1, R2, R3 are appropriate constants. Comparing (12) and (13) onegets

A4

γ1

γ2

γ3

γ4

= B4

γss, 1

γss, 2

γss, 3

γss, 4

(14)

where

A4 =

a1 a2 0 00 a1 a2 00 0 a1 a2

a2 0 0 a1

, B4 =

b1 b2 1 00 b1 b2 11 0 b1 b2b2 1 0 b1

with

a1 =τUEGO

τMIX − τUEGO

(1− p2)p1 −τMIX

τMIX − τUEGO

(1− p1)p2

9

a2 = − τUEGO

τMIX − τUEGO

(1− p2) +τMIX

τMIX − τUEGO

(1− p1), b1 = p1p2, b2 = −(p1 + p2).

Equation (14) represents the relationship existing between the measured signal γss,j , j = 1, · · · , 4, atthe steady-state and the fuel-to-air ratios γj released by each cylinder during the exhaust phase. Thisequation can be solved for the γj ’s; for, we note that A−1

4 exists when

(1 + 2a+ p1)p2 + (1 + 2b)p1 + b+ a 6= 0, a = − τMIX

τMIX − τUEGO

, b =τUEGO

τMIX − τUEGO

namely almost always.

The structure of the matrices in (14) reflects the physics of the exhaust dynamics. The inputcontributions to γss, j is distributed among the four cylinders, and the ai’s, bi’s determine the corre-sponding percentage of contribution. In fact, the runners are supposed to have the same geometry, sothat the contribution of the ith cylinder to γss, j is equal to the contribution of the (i+ 1)th cylinderto γss, j+1.

4.2.2 The five cylinders case

In the case of a five cylinders engine ∆ = T/20 and the contributions γi(t) do overlap, as shown inFigure 1. Considering Figure 2, we have

γc(z) =γ1 + γ5

2z20

z20 − 1+ γ1

z19

z20 − 1+ γ1

z18

z20 − 1+ γ1

z17

z20 − 1+γ2 + γ1

2z16

z20 − 1+ γ2

z15

z20 − 1

+γ2z14

z20 − 1+ γ2

z13

z20 − 1+γ3 + γ2

2z12

z20 − 1+ γ3

z11

z20 − 1+ γ3

z10

z20 − 1+ γ3

z9

z20 − 1

+γ4 + γ3

2z8

z20 − 1+ γ4

z7

z20 − 1+ γ4

z6

z20 − 1+ γ4

z5

z20 − 1+γ4 + γ5

2z4

z20 − 1+ γ5

z3

z20 − 1

+γ5z2

z20 − 1+ γ5

z

z20 − 1

which has to be put in (12). Reasoning as in the four cylinders case, we can write


2 +R2z +R3

(z − p1)(z − p2)+

z

z20 − 1

20∑j=1

γss, jz20−j .

Comparing the two expressions for γs(z) and using the same notation of the previous subsection, wefinally determine

A5

γ1

...γ5

= B5

γss, 1...

γss, 20

(15)

10

where

A5 =

c4 0 0 0 c5c1 0 0 0 0c1 0 0 0 0c2 c3 0 0 0c5 c4 0 0 00 c1 0 0 00 c1 0 0 00 c2 c3 0 00 c5 c4 0 00 0 c1 0 00 0 c1 0 00 0 c2 c3 00 0 c5 c4 00 0 0 c1 00 0 0 c1 00 0 0 c2 c30 0 0 c5 c40 0 0 0 c10 0 0 0 c1c3 0 0 0 c2

, B5 =

b1 b2 1 0 · · · 0 0

0 b1 b2 1 · · · 0 0

......

. . . . . . . . ....

0 · · · b1 b2 1

1 0 · · · 0 b1 b2

b2 1 0 · · · 0 0 b1

with

c1 = (1− p1)(1− p2)

c2 = −(p1 − 1)(p2 −

12

)a−

(p1 −

12

)(p2 − 1)b

c3 =12(p1 − 1)a+

12(p2 − 1)b

c4 = −(p1 − 1)(

12p2 − 1

)a−

(12p1 − 1

)(p2 − 1)b

c5 = −12(p1 − 1)p2a−

12p1(p2 − 1)b.

As in the four cylinders case, the regular structure in this relation reflects the physics of theexhaust dynamics, with the input contributions to γss, j distributed among the five cylinders.

Note that (15) can be solved with respect to the estimates γi because the matrix A5 has a leftpseudo inverse almost always.

4.3 The Estimation Algorithm: Practical Implementation

In order to implement the estimation algorithm, we have to consider the constraints due to the samplingperiod of the Central Control Unit (CCU). Let us assume that the UEGO sensor measurements aresampled with a sampling period ∆s. In general ∆s 6= ∆, so that the desired output measurementsγss, i can be obtained by interpolating the sampled output measurements; note that the number of thenecessary measurements is given by

ns = l.c.m.(4, nc)

so that the CCU has to sample the output for a period ns ∆s. Obviously, this period can be greaterthan the period T in which a cycle takes place. Since the steady–state output is a periodic signal overT , we can get the desired measurements from subsequent cycles. Since we have to ensure independentmeasurements, when ns∆s > T the following must hold

h∆s 6= kT, h = 1, · · · , ns, k = 1, · · · , k

11

where k is smallest integer such that ns∆s < kT . In fact, this condition ensures that two samplingsdo not occur at the same instant tmod T . Since ns, ∆s are given, there exist particular values for thecrank-shaft speed that do not allow to have independent measurements. For example, in a 5-cylindersengine, for which ns = 20, and for ∆s = 4 ms when n ≤ 1500 rpm the samplings are independent;for higher speeds these samplings are not all independent, and in Table 1 are reported the maximumnumber of independent samplings obtainable for the critical speeds.

Critical speed n (rpm) Independent Samplings

1579 191667 181765 171875 162000 152143 142308 132500 122727 113000 103158 193333 93529 173750 84000 154286 74615 134737 195000 65294 175455 115625 166000 5

Table 1: Maximum number of independent samplings obtainable for critical speeds n

Note also that the hypothesis of steady-state measurements is not restrictive since steady-statesituations are usually encountered in the engine behavior, and reasonably simple tests are available onthe on-board CCU to detect steady-state situations. Moreover, in order to minimize the estimationerror, when the steady-state conditions are maintained for a sufficiently long period of time we canuse m ns samplings and compute the mean value for each sampling over m different measurements.

Finally, in the ideal case, equations (14), (15) can be solved with respect to γi in an analytic way.In the real case, in order to minimize the error due to parameter uncertainties and noise, a least squarealgorithm is used. Therefore, one obtains

γ1

γ2

γ3

γ4

= (AT4A4)−1AT

4B4

γss, 1

γss, 2

γss, 3

γss, 4

(16)

12

in the case of 4-cylinders, and

γ1

γ2

γ3

γ4

γ5

= (AT

5A5)−1AT5B5

γss, 1

...

γss, 20

(17)

in the case of 5-cylinders.

5 Injector Characteristics Estimation

In this section we obtain an estimate of the injector characteristics gi, Ti,off . Once the air flow massma,i and the fuel-to-air ratios γi, i = 1, · · · , nc, have been estimated with (6) and (16) or (17), from (1)we obtain the estimates of the injected fuel massmf,i = γi(t) ma,i, i = 1, · · · , nc. (18)

From this relation it is possible to determine the injector characteristics gi and Ti,off , solution ofequation (25) with mf,i replaced by (18) for each steady-state situation. One can easily estimate theinjector characteristics

gi = 103∑

k=1

nk mjf,i − nj mk

f,i

nknj(T j

i − T ki

) , i = 1, · · · , nc

Ti,off =13

3∑k=1

T ki n

k mjf,i − T j

i nj mk

f,i

nk mjf,i − nj mk

f,i

where j = (k + 1) mod 3 and the superscript k = 1, 2, 3 corresponds to a steady-state situation.

6 Experimental Results

Ideally, the quality of the approach should be assessed on a real engine. However, confidentialityabout engine parameters and injector characteristics requested by our industrial partners togetherwith the difficulty of setting up the appropriate instrumentation, made validation on a real engineunfeasible. For this reason, we limit our discussion of experimentation on simulation results. We madesure that the simulation parameters and the results obtained were consistent with reality by havingthe approach scrutinized by Magneti Marelli experts who directed the selection of the critical partsof the experiments. It is conforting that this estimation approach will be adopted in future industrialengine management systems.

The estimation algorithms for the fuel-to-air ratios and for the injector characteristics were appliedseparately to the data obtained from a simulation, to test their efficiency with respect to the precisionof the available measurements.

As far as the fuel-to-air ratios are concerned, the estimation algorithm has been applied to thedata supplied by a model of the exhaust manifold for a 5-cylinders engine with runners of differentlength. This was modelled by choosing a different gain k for each runner delay δr = k/n. The enginespeed is equal to 5000 rpm. It is assumed that there is a complete air-fuel maldistributions, that is(

γ1 γ2 γ3 γ4 γ5

)=(0.90 0.93 1.10 1.08 1.00

).

and the model parameters are the following:

δr,1 = 5 ms, δr,2 = 5.125 ms, δr,3 = 5.25 ms, δr,4 = 5.375 ms, δr,5 = 5.5 ms

13

τMIX = 10 ms, τUEGO = 100 ms, δUEGO = 10 ms.

The mean value of the runners delays has been used in the estimation algorithm, that is δr = 5.25,so that δ = δUEGO + δr = 15.3 ms and the coefficient of matrices A5 and B5 in (15) result to beb1 = 0.8763, b2 = −1.875 and

c1 = 13.488×10−4, c2 = 10.042×10−4, c3 = 3.446×10−4, c4 = 10.191×10−4, c5 = 3.2979×10−4.

In order to render the simulations more realistic we supposed that the signal measured by the UEGOsensor is affected by a noise, mainly due to the chaotic diffusion process. Moreover, we also considereda noise on the measured signal. The amplitudes of the white noises have been appropriately set soto obtain a simulated A/F UEGO output comparable with a signal measured on a real engine in thesame operational situation. Figure 3 shows the typical signal used in the simulation.

As explained in Section 4.3, better results can be obtained by considering mean values for each sam-pling over different measurements, so minimizing the noise effects on UEGO measurements. Clearly,a tradeoff has to be done between the quality (and cost) of the UEGO sensor and the computationaleffort requested by the algorithm, which affects the cost of the central unit.

The result of the the fuel-to-air ratios estimation algorithm (17) is summarized in Table 2.

real value estimated value error %γ1 0.90 0.9007 0.08γ2 0.93 0.9327 0.29γ3 1.10 1.1007 0.07γ4 1.08 1.0707 0.86γ5 1.00 1.0109 1.09

Table 2 : Estimates γi, i = 1, . . . , 5.

As far as the injector characteristics are concerned, the estimation algorithm has been tested onthe single ith injector for which it is known the air-to-fuel ratio A/F . The data are obtained from asimulation and the values of the parameters of the model are

Vd = 1242 cm3, patm = 1013 mbar, Tatm = 300 K, R = 287 KJ/kg K, µc = cp/cv = 1.4

gi = 1.93 10−3 kg/s, Ti,off = 0.75 ms

ηestv = 0.597071 + 5.15605 10−5 n+ 9.30417 10−7 pman.

Moreover, we considered to have

εp = −1.237%, εηv = 3.48%, A0th = 2.95 mm2.

We consider the three steady-state conditions reported in Table 3, and the application of the estimationalgorithm presented in Section 3 (for εp, εηv , A0

th and for ma,i) and in Section 5 (for gi and Ti,off )gives the results summarized in Table 4.

α Ae,th n pm Ti A/F[deg] [mm2] [rpm] [mbar] [ms]67.50 431 2563 985 11.58 14.6426.64 85 1464 904 9.90 14.6620.70 48 2068 601 6.86 14.65

Table 3 : Simulation Data

14

real value estimated value error %gi [g/s] 1.93 1.93 0.1

Ti,off [ms] 0.75 0.75 0.6εp % −1.237 −1.239 0.1εηv % 3.48 3.36 3.5A0

th [mm2] 2.95 2.88 2.2

Table 4 : Estimates

The estimation errors are due to the finite number of data digits used for the measurements. Con-sequently, these errors cannot be avoided unless the accuracy of the sensor is dramatically increased.

Finally, to take into account the effects of cascaded estimations on the overall estimation process,we considered for the same case of Table 3 the higher estimation error obtained in the γi’s estimation,namely an error of 1.10%. The values of the A/F in this new estimation are those in Table 3 multipliedby 1.011. In this case we obtained an error of 1.10% on gi and 0.6% on Ti,off .

7 Conclusions

We presented an estimation technique for injector characteristics based on a set of measurements thatcan be carried out by the sensors present in the car, i.e. intake manifold pressure, crank-shaft speed,throttle-valve plate angle, injections timing and exhaust air-to-fuel ratio, which is measured by a singleUEGO sensor placed at the exhaust pipe output.

The estimation strategy is based on a sequence of estimations of various quantities needed toestimate the characteristics of each injector. In particular, we have determined an estimation chainwhich yields good results when applied to simulated systems. We are in the process of transferringthis approach to our industrial sponsors who are interested in adopting the estimation algorithm onindustrial strength applications.

8 Acknowledgments

This research has been partially sponsored by Magneti Marelli and PARADES, a Cadence, MagnetiMarelli and ST Microelectronics GEIE, and by C.N.R. under Project MADESS II. The authors aregrateful to G. Gaviani, C. Rossi, R. Flora, G. Serra, C. Poggio and their teams at Magneti Marellifor having introduced the problem and for the many discussions. The authors also thank V. Nicolettifor his comments on the practical implementation of the estimation algorithm. We wish to thank theanonymous reviewers who contributed many useful comments to improve the quality of the paper.

Appendix – Mathematical Model of the Intake Manifold for a SparkIgnition Engine

In this Appendix we present the mathematical model of the intake manifold of a SI engine. We makeuse of the so-called Mean Value Model of a spark ignition engine [1, 5, 8, 13, 14], which describes theair dynamics with good accuracy without considering cycle variations. For the sake of simplicity, thetemperature and transient heating effects are not explicitly taken into account.

The intake manifold dynamics describes the mean values of the relevant engine variables, i.e. theintake manifold pressure, the fuel flow-rate inside the cylinder and the crank shaft speed with respectto variations of the engine inputs, namely the throttle plate angle and the injection pulse duration(the spark advance angle is assumed to be constant).

15

Throttle Air Mass Flow Rate Equations. The average air mass flow rate through the throttlema,th [kg/s] can be calculated (see [18]) as the air mass flow m of a compressible fluid through achannel between two areas at different pressures:

m = Cdp1√RT1

A2√1−A2/A1

β

(p2

p1

)(19)

where Cd is a discharge coefficient, R is the ideal gas constant for air [J K−1 kg−1], Ai, pi, Ti are thearea, the pressure, the temperature of section i and

β

(p2

p1

)=

√√√√ 2µc

µc − 1

[(p2

p1

) 2µ c −

(p2

p1

)µc+1µ c

]ifp2

p1≥ rcr

√µc

(2

µc + 1

) µc+12(µc−1)

ifp2

p1< rcr

with rcr =(

2µc+1

) µcµc−1 and µc the specific heat ratio [J/Kg/K].

If we assume the downstream section A2 to be the throttle section and the upstream section A1

a section just before the throttle, then A1 � A2 and equation (19) reduces to

ma,th = Cdpatm√RTatm

Ath(α) β(pman

patm

)(20)

where patm and Tatm are the ambient pressure [N m−2] and the air temperature [K], pman is the meanvalue intake manifold pressure [N m−2], Ath is the throttle area [m2], α the throttle plate angle [rad]and Cd the throttle discharge coefficient.

The mean value manifold pressure pman can be determined from the average of a set of measure-ments, giving

pm = (1 + εp) pman (21)

of the pressure, where |εp| ≤ 3% is the estimate sensor error.

In the sequel we assumeCdAth(α) = Ae,th(α) +A0

th (22)

where Ae,th(α) is the so-called equivalent throttle area and A0th is the offset area, i.e. the area of the

throttle when α = 0, Ae,th(0) = 0. It is assumed that A0th ∈ [0, 5] mm2 is an unknown value slowly

varying because of aging.

Engine Air Pumping Equations. Theoretically, the average air mass flow ma [kg/s] entering the nc

cylinders of the engine is given by

ma = ρairVd

2n

60where ρair is the air density, Vd the displacement volume [m3] and n the crank shaft speed [rpm]. Byusing the ideal gas equation for the air in the manifold

pman

ρair= RTair = RTatm

and taking into account the real supply to the cylinders, we obtain ma as a function of the crankshaftspeed n and of the mean value intake manifold pressure pman

ma =Vd

120RTatmn pman ηv(n, pman) (23)

where ηv is the so-called volumetric efficiency and it is assumed to be a function of n and pman. Inthe sequel we will assume to know the volumetric efficiency up to a constant error |εηv | ≤ 10%, i.e.

ηv (n, pman) = (1 + εηv) ηestv (n, pman) (24)

16

where ηestv is the estimated volumetric efficiency.

Fuel Flow Equations. The fuel mass per injection pulse can be assumed to be a linear function of thefuel injector on-time

mf,i = gi (Ti − Ti,off ) , i = 1, · · · , nc

where Ti is the time the injector is open [s], gi the gain [kg/s] and Ti,off the offset [s] of each injector.Therefore, the mean-value injected fuel mass flow is given by

mf,i =n

30gi (Ti − Ti,off ) , i = 1, · · · , nc. (25)

In this relation the fuel flow dynamics, due to the wall wetting, can be neglected assuming the steadystate measurement condition. Moreover, the small dynamic coupling between the fuel subsystem andthe air subsystem is ignored.

References

[1] C. F. Aquino, Transient A/F Control Characteristics of the 5 Liter Central Fuel Ignition Engine,Technical Report No. 810494, SAE, 1981.

[2] A. Balluchi, L. Benvenuti, M. D. Di Benedetto, C. Pinello, and A. L. Sangiovanni–Vincentelli,Automotive Engine Control and Hybrid Systems: Challenges and Opportunities, Proceedings ofIEEE, Vol. 88, No. 7, pp. 888-912, 2000.

[3] L. Benvenuti, M. D. Di Benedetto, C. Rossi, and A. Sangiovanni-Vincentelli, Injector Characteri-stics Estimation for Spark Ignition Engines, Proceedings of the 37th IEEE Conference on Decisionand Control, Tampa, FL, pp. 1546-1551, 1998.

[4] K. J. Bush, N. J. Adams, S. Dua and C. R. Markyvech, Automatic Control of Cylinder by CylinderAir-Fuel Mixture using a Proportional Exhaust Gas Sensor, Technical Report No. 940149, SAE,1994.

[5] V. Chaumerliac, P. Bidan and S. Boverie, Control-oriented Spark Ignition Engine Model, ControlEngineering Practice, Vol. 2, No. 3, pp. 381-387, 1994.

[6] C.-F. Chang, N. P. Fekete, A. Amstutz and J. D. Powell, Air-Fuel Ratio Control in Spark-IgnitionEngines Using Estimation Theory, IEEE Transactions on Control Systems Technology, Vol. 3,No. 1, pp. 22–31, 1995.

[7] S. B. Choi and J. K. Hedrick, An Observer–Based Controller Design Method for ImprovingAit/Fuel Characteristics of Spark Ignition Engines, IEEE Transactions on Control Systems Tech-nology, Vol. 6, No. 3, pp. 325–334, 1998.

[8] D. J. Dobner, A Mathematical Engine Model for Development of Dynamic Engine Control, Tech-nical Report No. 800054, SAE, 1980.

[9] N. P. Fekete, U. Nester, I, Gruden, J. D. Powell, Model-based Air-Fuel Ratio Control of a LeanMulti-Cylinder Engine, Technical Report No. 950846, SAE, 1995.

[10] J. W. Grizzle, K. L. Dobbins and J. A. Cook, Individual Cylinder Air-Fuel Ratio Control witha Single EGO Sensor, IEEE Transactions on Vehicular Technology, Vol. 40, No. 1, pp. 280-286,1991.

[11] J. W. Grizzle, J. A. Cook and W. P. Milam, Improved Cylinder Air Charge Estimation forTransient Air Fuel Ratio Control, Proceedings of the American Control Conference 1994, Vol. 2,pp. 1568-1573, 1994.

17

[12] J. B. Heywood, Internal Combustion Engine Fundamentals, McGraw-Hill Book Co., Inc., NY,1989.

[13] E. Hendricks and S. C. Sorenson, Mean Value Modelling of Spark Ignition Engines, TechnicalReport No. 900616, SAE, 1990.

[14] E. Hendricks and T. Vesterholm, The Analysis of Mean Value SI Engine Models, Technical ReportNo. 920682, SAE, 1992.

[15] V. K. Jones, B. A. Ault, G. F. Franklin and J. D. Powell, Identification and Air-Fuel RatioControl of a Spark-Ignition Engine, IEEE Transactions on Control Systems Technology, Vol. 3,No. 1, pp. 14–21, 1995.

[16] P. E. Moraal, J. A. Cook and J. W. Grizzle, Single Sensor Individual Cylinder Air-Fuel RatioControl of an Eight Cylinder Engine with Exhaust Gas Mixing, Proceedings of the AmericanControl Conference, San Francisco, CA, 1993.

[17] B. K. Powell, H. Wu and C. F. Aquino, Stoichiometric Air-Fuel Ratio Control Analysis, TechnicalReport No. 810274, SAE, 1981.

[18] C. F. Taylor and E. S. Taylor, The Internal Combustion Engine, Second Edition, Int. TextbookCo., Scranton, Pa., 1970.

18

Since the air flow is regulated by the throttle valve, controlled by the driver with the acceleratorpedal, the control variable becomes the fuel flow provided by the injectors to the intake manifold.Regulating the fuel flow requires accurate estimation of the characteristics of the injectors.

It is in fact known that injector variability may cause cylinder-to-cylinder differences in the mixturecomposition of the order of 5% [12]. Cylinder-to-cylinder air-fuel maldistribution results in emissionconcentration imbalance between cylinders. These individual emission differences will not necessar-ily average out to produce an overall result equivalent to that obtained with all cylinders operatingat the same air-to-fuel ratio. This is due to the nonlinear variation of hydrocarbons, nitrogen ox-ides and carbon monoxide concentrations with air to fuel ratio [4, 12]. Moreover, even in the caseof perfectly matched injectors, cylinder-to-cylinder air-fuel maldistribution can arise from differentindividual cylinder behavior in breathing due to intake manifold structure and valve characteristics.

To reduce the air-fuel maldistribution due to injector characteristics imbalance, injectors areusually required to have close tolerances, up to 1%, resulting in high cost per injector. Injectors arein fact machined to tolerances of the order of 6% and then their tolerance is reduced to 2% (in theoperation nominal range, i.e. when the on-time is greater than 500 µs) by a quite expensive off-linetuning process. Better tolerance can be achieved by an even more expensive process that consists oftesting and appropriately grouping the injectors in the so-called matched injector sets.

To achieve accuracy in controlling the emission concentrations, control policies that can discernthe contribution of each injector and thus overcome the cylinder-to-cylinder air-to-fuel imbalance,have been recently proposed [4, 10, 16]. These policies allow the accommodation of greater injectortolerances and consequently the reduction of the cost per injector. These are inherently closed-loopstrategies, since they allow the tuning of the individual cylinder A/F using the signal of the exhaust-gas-oxygen (UEGO) sensor. In [10], a single switching UEGO sensor is used for this purpose, whilein [16] the estimation of the individual cylinder A/F is obtained from a model inversion of an eight-cylinder engine. A slight different approach is presented in [16], where the individual cylinder A/F isestimated at low engine speed for a six-cylinder engine, using two UEGO sensors. The drawback ofthese control strategies is that they are effective only under steady-state operating conditions. There-fore, in situations where the UEGO sensor signal is not reliable (transients and cold start conditions),they can not be applied and different open-loop strategies have to be designed.

The open-loop strategies are based on the estimation/measurement of the air charged in eachcylinder. Clearly, in this situation, the individual cylinder A/F can be successfully controlled only ifthe individual injector characteristics are known. This paper addresses this problem, providing theestimation of the individual injector characteristics, along with the individual air flow estimation. Morein detail, we propose a method for the estimation of each injector’s characteristics via measurementsof the intake-manifold pressure, the crank-shaft speed, the throttle-valve plate angle, the injectionson-time and the UEGO sensor reading from the exhaust. The estimation processes is carried outduring steady-state conditions, and the resulting parameters can be stored and used in the open-loopstrategies during the transients and the UEGO sensor warm up.

The estimation algorithm is based on a fairly accurate modelling of the cylinder air-filling process,of the exhaust manifold dynamics, and of the UEGO sensor. For the cylinder air-filling process, astandard lumped model is used [11]. To obtain an accurate estimation algorithm, we studied the de-pendence of the air charge dynamics on some characterizing variables (pressure sensor error, throttleoffset area, off-line estimation volumetric efficiency error). For the exhaust manifold dynamics, thetransport delays and the mixing between the air-fuel charges associated with the cylinders were consid-ered. Finally, we considered a linear UEGO sensor modelled as a first order lag plus a delay, a common

2

assumption in the literature (e.g., [6], [10], [15], [17])1. The idea behind the proposed approach is toestimate first the sensor error and the throttle offset area on the basis of steady pressure measurements,and then, to calculate the average air flow entering each cylinder using these estimations. Usinginformation from a single linear UEGO sensor, the air-to-fuel ratio for each cylinder is estimated andthe injector characteristics are obtained. In this last step, the UEGO sensor signal is appropriatelysampled in order to invert the exhaust manifold and sensor dynamics to reconstruct the individualcylinder A/F’s.

The paper is organized as follows. In Section 2, we formulate the estimation problem. In Section 3,we show how to estimate the average air-mass flow entering the cylinders [3]. In Section 4, estimation ofthe fuel-to-air ratios is covered. In Section 5, we estimate the injector characteristics and in Section 6,we offer realistic simulations and concluding remarks.

2 Estimation Strategy

In this section, we state our solution approach for the individual injector characteristic estimationproblem for a Spark Ignition (SI) engine.

We are interested in obtaining the characteristics of the individual injector because the variablesavailable for the engine control are the injection profile, i.e., approximately the time in which theinjector is open, and the timing of the spark. The injection profile is used to control the fuel-massflow into the cylinders that determines the behavior of the engine with respect to pollution, fuelconsumption and driveability. The fuel mass per injection pulse can be assumed to be an affinefunction of the fuel injector on-time

mf,i =n

30gi (Ti − Ti,off ) , i = 1, · · · , nc

where Ti is the time the injector is open [s], gi the gain [kg/s] and Ti,off the offset [s] of each injector.Hence, the problem is to estimate gi and Ti,off . In Section 5, we show that these two parameters canbe estimated once the fuel-mass flow mf,i injected in each cylinder is estimated. The key measurementfor determining mf,i is the fuel-to-air ratio F/A given by a single UEGO sensor. The UEGO sensor iscapable of measuring the fuel-to-air ratio γ(t). If indeed there were no mixing of the exhaust gases fromthe various cylinders in the exhaust manifold, the problem would be fairly simple. The complicationsarise from the partial overlap of the exhaust gases from each cylinder.

Let

γi(t) =mf,i(t)ma,i(t)

, i = 1, · · · , nc (1)

denote the fuel-to-air ratio released by the ith cylinder during the exhaust phase, where ma,i(t) is theair-mass flow entering in each cylinder during the intake process, and nc the number of cylinders. Notethat the contribution to cylinder-to-cylinder air-fuel maldistribution of the different cylinder breathingcharacteristics is negligible with respect to the injectors’ variability. Then we assume that the averageair-mass flow ma,i entering each cylinder is the same. Therefore, if we could estimate the ratios (1)and the air-mass flow ma,i, we could obtain the fuel flows mf,i(t). The estimation of γi(t) from theUEGO sensor measurement is the subject of Section 4 and is based on the inversion of

γ(t) = ϕ(γ1(t), · · · , γnc(t)),1The use of higher-order models for the A/F sensor does not change the core of our approach, since the effect of a

more complex model is felt only in the computation of the inverse of the model.

3

a function that will be derived considering that γ(t) depends on the fuel-to-air ratio γc(t) at the runnerconfluence, which, in turns, is a known function f(γ1(t), · · · , γnc(t)) of the fuel-to-air ratios.

The estimation process for the air-mass flow ma,i is described in Section 3. ma,i is a dynamicalfunction of pman, the mean value manifold pressure, and the volumetric efficiency, ηv. If we assumethat

(H) the mean value intake manifold pressure pman and the crankshaft speed n remain constant2 �

then, pman = 0 and the average air mass flow ma = ncma,i, entering the nc cylinders, is equal to theaverage air-mass flow ma,th passing through the throttle. The average air-mass flow is a function of thethrottle-plate angle α, which is directly measurable, and of the throttle-offset area A0

th, an unknownvalue slowly varying because of aging. We can measure the manifold pressure with an unknown errorεp, and the volumetric efficiency with an unknown constant error εηv . Hence, the estimation algorithmfor ma,i is based on the estimation of εp, εηv , and A0

th.

3 Air-Mass Flow Estimation

In this section, we show how to estimate the average air-mass flow ma entering the cylinders [3]. Wehave already observed (see the Appendix for a summary of the derivation of these relationships) thatwe need first to estimate the mean value intake manifold pressure pman, from measurements (21), andthe volumetric efficiency ηv, given by (24). Since we can measure pman with an unknown error εp, andwe can estimate ηv with an unknown constant error εηv , the problem is to determine estimations ofthese errors.

We suppose that the steady-state hypothesis (H) holds true, so that pman = 0 and the averageair flow ma, given by (23), is equal to the average air flow ma,th passing through the throttle, givenby (20). Since this last quantity is a function of the throttle offset area A0

th, an unknown value slowlyvarying because of aging, we need to estimate this parameter as well.

The proposed algorithm allows for estimating the errors εp, εηv and the parameter A0th on the

basis of three different steady state measurements. We assume that the variables and parameters ofthe system remain unchanged during the three measurements. We assume to measure the manifoldpressure with an unknown error εp ∈ [−3%, 3%], to estimate the volumetric efficiency ηv with anunknown constant error εηv ∈ [−10%, 10%] and to have an unknown offset area A0

th ∈ [0, 5] mm2 forthe throttle. Finally, we assume to have a F/A feedback control system which regulates the injectionpulse duration Ti in such a way that the air to fuel ratio is close to stoichiometry.

The following subsection 3.1 is devoted to the determination of estimates for εp, εηv and A0th,

while in subsection 3.2 we will determine the average air flow ma.

3.1 Sensor-error and throttle-offset area estimations

Two cases are considered: the case in which the ambient pressure is known and the one when it isunknown.

2Hypothesis (H) is not restrictive since steady-state situations are usually encountered during engine operation, andreasonably simple tests are available on the on-board computer to detect steady-state operation. Moreover, as shownin [13], the manifold pressure in steady-states distributes uniformly around the mean values with quite small standarddeviations. Note also that even if the engine model is intrinsically hybrid [2], hypothesis (H) allows us to neglect thehybrid nature of the SI engine since we will use this model in the case of constant crank-shaft speed and constant pressure.

4

3.1.1 The case of known ambient pressure

From hypothesis (H), pman = 0 so that ma,th = ma. Then, using (20) and (23), we obtain

Ae,th(α) + A0th =

Vd n pman

120 patm

√RTatm

ηv(n, pman)

β

(pman

patm

)where the expression (22) for Ath(α) has been used. Substituting pman obtained from (21), and using(24) for ηv, we have(

Ae,th(α) + A0th

)(1 + εp) = (1 + εηv)η

estv (n, pm, εp)f(pm, εp, patm, Tatm) n pm (2)

wheref(pm, εp, patm, Tatm) =

Vd

120 patm

√RTatm

1

β

(pm

(1 + εp)patm

) .

If we assume to know patm and Tatm, equation (2) gives all the possible values (εp, εηv , A0th),

which enable the model to return the given measured crank-shaft speed n, throttle plate angle α

and measured manifold pressure pm. Assuming to reach three different steady-state situations, say(n1, α1, p1

m), (n2, α2, p2m), (n3, α3, p3

m), then it is possible to find the values (εp, εηv , A0th) that satisfy

equation (2) for the three different cases. To do this, define

Aie,th: = Ae,th(αi), Eηv : = (1 + εηv)

andh

i (εp) : = ηestv (ni

, pim, εp)f(pi

m, εp, patm, Tatm) nip

im

so that equation (2), for the three steady state situations, gives(A

ie,th + A

0th

)(1 + εp) = h

i(εp)Eηv , i = 1, 2, 3. (3)

We can rewrite (3) as

3∑i=1

(A

(i+1)mod 3e,th − A

(i+2)mod 3e,th

)hi(εp) = 0

Eηv =Ai

e,th − Aje,th

hi (εp) − hj (εp)(1 + εp) i �= j

A0th =


je,th

hi (εp) − hj (εp)i �= j

or ϕ(εp) = 0

Eηv = φij(εp) i �= j

A0th = ψij(εp) i �= j

with i, j = 1, 2, 3.

Then, an estimation εp of the sensor error εp can be computed using an iterative process, forexample Newton’s method, to find the zeros of the function ϕ(εp). Hence, we can compute theestimated values for εηv , and the estimate A0

th as follows

εηv =13

3∑i,j=1i>j

φij(εp) − 1, A0th =

13

3∑i,j=1i>j

ψij(εp).

5

3.1.2 The case of unknown ambient pressure

In this case we assume to measure the ambient pressure with the manifold sensor at key-on. Then wehave patm,m = (1 + εp)patm and equation (2) reduces to

Ae,th(α) + A0th = (1 + εηv)η

estv (n, pm, εp)f(pm, patm,m, Tatm) n pm (4)

wheref(pm, patm,m, Tatm) =

Vd

120 patm,m

√RTatm

1

β

(pm

patm,m

) .

Assuming, as before, to reach three different steady-state situations, it is possible to find the values(εp, εηv , A

0th) that satisfy equation (4) for the three different cases. Consider, as before, the functions

Aie,th, Eηv and define

hi (εp) : = η

estv (ni

, pim, εp)f(pi

m, patm,m, Tatm)nip

im

so that equation (4), for the three steady-state situations, gives

Aie,th + A

0th = h

i(εp)Eηv , i = 1, 2, 3. (5)

We rewrite (5) as

3∑i=1

(A

(i+1)mod 3e,th − A

(i+2)mod 3e,th

)hi(εp) = 0

Eηv =Ai

e,th − Aje,th


A0th =


je,th


or ϕ(εp) = 0

Eηv = φij(εp) i �= j

A0th = ψij(εp) i �= j

for i, j = 1, 2, 3.

Hence, as in the previous case, an estimation εp of the sensor error εp can be easily computed.Finally, we have

εηv =13

3∑i,j=1i>j

φij(εp) − 1, A0th =

13

3∑i,j=1i>j

ψij(εp).

It is worth noting that this method relies on the fact that patm does not varies significantly betweenkey-on and the three different steady-state measurements. This may be not the case when drivingin mountainous areas, so that in this case an estimation error would appear due to the variabilityof patm.

6

3.2 Air-Flow Estimation

Once the estimations εp, εηv and A0th are obtained, an estimation of the air mass flow entering the

cylinders can be computed using (23) and (20) as

ma,i =˜ma(εp, εηv) + ˜ma,th(εp, A

0th)

2 nc, i = 1, · · · , nc (6)

where ˜ma(εp, εηv) =Vd

120 RTatmn

pm

(1 + εp)(1 + εηv) η

estv (n, pm, εp)

and


patm

(Ae,th(α) + A0

th

)√

R Tatmβ

(pm

(1 + εp) patm

)in the case of known ambient pressure, or


patm,m

(Ae,th(α) + A0

th

)(1 + εp)

√R Tatm

β

(pm

patm,m

)

when the ambient pressure is measured. It is worth noting that, from the steady–state hypothesis(H), one should have ˜ma = ˜ma,th

3.

4 The Estimation of the Fuel-to-Air Ratios γi

The problem solved in this section is the estimation of the fuel-to-air ratios γi(t), i = 1, · · · , nc, givenby (1), from the output signal γ(t), obtained from a single UEGO sensor. In the following subsection,we derive first a model of the exhaust gases during their motion from the cylinders to the oxygensensor. Then, an estimation algorithm for mf,i is proposed.

4.1 Mathematical Model of the Exhaust Manifold

In this section, we develop a model for an nc-cylinder engine describing the contribution in each cycleof each cylinder to the measured F/A ratio. We take into account the mixing among the air-fuelcharges associated with each cylinder and the time that a single air-fuel charge takes to travel thelength of its exhaust manifold runner. The UEGO sensor dynamics are also considered.

During engine operation, four events take place, i.e. intake, compression, combustion and exhaust.As previously explained, under hypothesis (H) the hybrid nature of the engine can be neglected.Moreover, since we are studying the exhaust gas dynamics, the exhaust phase will be considered thestarting event of the engine dynamics. Referring to the ith cylinder, at the beginning of the exhaustprocess the exhaust valve opens, and the burnt gas goes from the high pressure environment in thecylinder to the lower pressure in the exhaust manifold. During this process, the exhaust gas expandsinto the exhaust manifold volume and travels towards the UEGO sensor. Note that in the exhauststroke, most of the burnt gas in the cylinder is pushed out into the exhaust manifold by the piston.

When the gas moves in the exhaust manifold the main processes that must be taken into accountare

3In fact, (εp, εηv , A0th) are defined as the solutions of the equations ma(εp, εηv ) = ma,th(εp, A0

th). In (6) we considered

the mean between the values of ˜ma and ˜ma,th in order to reduce influences of possible estimation errors on εp, εηv , A0th.

A study on parameter sensibility may help in fixing a more appropriated weighted average to compute ma,i.

7

1) the transport of the burnt gases in the exhaust manifold runner;

2) the gas-mixing in the manifold junctions.

The transport delays associated with each of the nc exhaust runners are not equal, since each runnerhas a different length. Moreover, during the gas transport, the gas molecule velocities spread about themean value. While it is not difficult, although mathematically involved, to consider these aspects, forthe sake of simplicity, we will assume that the delays are all equal to a quantity δr, and we will neglectthe velocity dispersion. The effect of the delay differences will be analyzed by a set of simulations.The mixing process can be seen as a merging process at the runner confluence, where the mixtureflows sum up. Hence, the merging process during the travelling towards the sensor will be modelledby a first order system. In summary, the main processes which take place in the exhaust manifold canbe taken into account by the following transfer function

PMIX (s) =e−δr s

1 + τMIX s(7)

with δr, τMIX the transport delay and the time constants of the mixing process. Under the assumptionof plug flow in the exhaust manifold, the average velocity of the exhaust gases is proportional to theengine speed [6], [15]. Thus, the transport delay δr can be assumed inversely proportional to thecrank-shaft speed [7], [10], [17]

δr =k

n.

However, when the sensor is located close to the exhaust valve, the blast of exhaust gases during theblow down process dominates the transport delay. As a consequence, the exhaust transport delay canbe assumed constant (as shown in table 1 in Ref. [15]).

The UEGO sensor has desirable properties, since it is linear, accurate and gives fast responses.As far as its modelling is concerned, the diffusion process of the oxygen that occurs in the UEGOsensor can be modelled as a first order system with a diffusion delay [6], [9]. Hence, the UEGO sensordynamics are represented by the following transfer function

PUEGO(s) =e−δ

UEGOs

1 + τUEGOs(8)

with δUEGO , τUEGO the diffusion delay and time constant of the diffusion process. As shown in [15],the sensor’s time constant τUEGO varies with the throttle angle but not with the engine speed.

The fuel-to-air ratio at the runner confluence is a known function

γc(t) = f(γ1(t), · · · , γnc(t))

of the ratios γi(t). In fact, since the air mass flow ma,i entering each cylinder is the same, γc(t) dependsonly on the ratios γi(t), on the number nc of cylinders and on the timing of the exhaust process ofeach cylinder. When the only charge present at the UEGO sensor at time t is the ith one, f is simplygiven by the value γi(t), while it is given by the arithmetic mean between γi(t), γj(t) when the chargespresent at the UEGO sensor at time t are the ith and jth ones. More precisely, let T be the timerequired for the crank-shaft to advance 720◦. Then, each phase takes T/4 time and the cycles of twosubsequent cylinders are shifted by T/nc. Therefore, when nc > 4 the contributions to γc(t) due totwo subsequent cylinders i, i + 1 overlap for a time

∆ =T

l.c.m.(4, nc)

and, during this overlap, we have

γc(t) =γi(t) + γi+1(t)

2.

8

The function γc(t) is clearly a piece-wise periodic function of period ∆. In the case of four and fivecylinders one has the pictures shown in Figure 1.

On the basis of the previous discussion, the measured normalized fuel-to-air mixture ratio γ(t)can be obtained from

γ(s) = P (s)γc(s) (9)

where

P (s) =e−δs

(1 + τMIX s)(1 + τUEGOs), δ =

k

n+ δUEGO

is the transfer function describing the dynamics of the exhaust manifold, obtained using (7), (8).

4.2 The Estimation Algorithm: the Ideal Case

The dependence of the piece-wise constant signal γc(t) on the time interval ∆ suggests to look for asolution of the problem in the discrete-time context, by considering a sample rate equal to ∆ and adiscrete-time signal γc(q) followed by a zero-order holder. The discrete-time signal γc(q) is shown inFigure 2 in the case of four and five cylinders.

From (9) and considering the inverse Laplace transform, we obtain

γ(t + δ) = L−1

[1

(1 + τMIX s)(1 + τUEGOs)γc(s)

].

Since γc can be seen as a discrete-time signal followed by a zero-order holder, we consider the samplingγs(q) of the signal γ(t + δ), with period ∆. Hence, we have

γs(q) = Z−1[Q(z)γc(z)

]where Q(z) is the transfer function of the sampled system, consisting of the zero-order holder, themixing and sensor dynamics, and the sampler

Q(z) = Z[L−1

[1

(1 + τMIX s)(1 + τUEGOs)1s

]∣∣∣∣t=k∆

]z − 1

z

= 1 − τMIX

τMIX − τUEGO

z − 1z − p1

+τUEGO

τMIX − τUEGO

z − 1z − p2

(10)

withp1 = e

−∆/τMIX , p2 = e

−∆/τUEGO .

Note that we consider γ(t+δ) and not γ(t) since the total delay δ, depending on the crank shaft speedand the UEGO sensor characteristics, is a known quantity. Therefore, it is possible to compensate forthe presence of this delay in the estimation algorithm and to work directly on the signal γ(t + δ).

In what follows, in order to illustrate the procedure, we consider two interesting cases, the4-cylinders and 5-cylinders engines.

4.2.1 A simple case: a four cylinders engine

In the case of a four cylinders engine ∆ = T/4 and the contributions γi(t) do not overlap, see Figure 1.Referring to Figure 2, the Z-transformation of the signal γc(q) is given by

γc(z) = Z[γc(q)

]= γ1

z4

z4 − 1+ γ2

z3

z4 − 1+ γ3

z2

z4 − 1+ γ4

z

z4 − 1. (11)

9

Considering the transfer function (10) and the input (11) it is possible to compute the output

γs(z) = Z[γs(q)

]= Q(z)γc(z). (12)

This output has the form


2 + R2z + R3

(z − p1)(z − p2)+

z

z4 − 1

4∑j=1

γss, j z4−j (13)

where γss(z), γst(z) are the Z-transformations of the steady-state and transient components γst(q),γss(q) of the output γs(q), and R1, R2, R3 are appropriate constants. Comparing (12) and (13) onegets

A4

γ1

γ2

γ3

γ4

= B4

γss, 1

γss, 2

γss, 3

γss, 4

(14)

where

A4 =

a1 a2 0 00 a1 a2 00 0 a1 a2

a2 0 0 a1

, B4 =

b1 b2 1 00 b1 b2 11 0 b1 b2

b2 1 0 b1

with

a1 =τUEGO

τMIX − τUEGO

(1 − p2)p1 −τMIX

τMIX − τUEGO

(1 − p1)p2

a2 = − τUEGO

τMIX − τUEGO

(1 − p2) +τMIX

τMIX − τUEGO

(1 − p1), b1 = p1p2, b2 = −(p1 + p2).

Equation (14) represents the relationship existing between the measured signal γss,j , j = 1, · · · , 4, atthe steady-state and the fuel-to-air ratios γj released by each cylinder during the exhaust phase. Thisequation can be solved for the γj ’s; for, we note that A

−14 exists when

(1 + 2a + p1)p2 + (1 + 2b)p1 + b + a �= 0, a = − τMIX

τMIX − τUEGO

, b =τUEGO

τMIX − τUEGO

namely almost always.

The structure of the matrices in (14) reflects the physics of the exhaust dynamics. The inputcontributions to γss, j is distributed among the four cylinders, and the ai’s, bi’s determine the corre-sponding percentage of contribution. In fact, the runners are supposed to have the same geometry, sothat the contribution of the ith cylinder to γss, j is equal to the contribution of the (i + 1)th cylinderto γss, j+1.

4.2.2 The five cylinders case

In the case of a five cylinders engine ∆ = T/20 and the contributions γi(t) do overlap, as shown inFigure 1. Considering Figure 2, we have

γc(z) =γ1 + γ5

2z20

z20 − 1+ γ1

z19

z20 − 1+ γ1

z18

z20 − 1+ γ1

z17

z20 − 1+

γ2 + γ1

2z16

z20 − 1+ γ2

z15

z20 − 1

+γ2z14

z20 − 1+ γ2

z13

z20 − 1+

γ3 + γ2

2z12

z20 − 1+ γ3

z11

z20 − 1+ γ3

z10

z20 − 1+ γ3

z9

z20 − 1

+γ4 + γ3

2z8

z20 − 1+ γ4

z7

z20 − 1+ γ4

z6

z20 − 1+ γ4

z5

z20 − 1+

γ4 + γ5

2z4

z20 − 1+ γ5

z3

z20 − 1

+γ5z2

z20 − 1+ γ5

z

z20 − 1

10

which has to be put in (12). Reasoning as in the four cylinders case, we can write


2 + R2z + R3

(z − p1)(z − p2)+

z

z20 − 1

20∑j=1

γss, jz20−j

.

Comparing the two expressions for γs(z) and using the same notation of the previous subsection, wefinally determine

A5

γ1

...γ5

= B5

γss, 1...

γss, 20

(15)

where

A5 =

c4 0 0 0 c5

c1 0 0 0 0c1 0 0 0 0c2 c3 0 0 0c5 c4 0 0 00 c1 0 0 00 c1 0 0 00 c2 c3 0 00 c5 c4 0 00 0 c1 0 00 0 c1 0 00 0 c2 c3 00 0 c5 c4 00 0 0 c1 00 0 0 c1 00 0 0 c2 c3

0 0 0 c5 c4

0 0 0 0 c1

0 0 0 0 c1

c3 0 0 0 c2

, B5 =

b1 b2 1 0 · · · 0 0

0 b1 b2 1 · · · 0 0

......

. . . . . . . . ....

0 · · · b1 b2 1

1 0 · · · 0 b1 b2

b2 1 0 · · · 0 0 b1

with

c1 = (1 − p1)(1 − p2)

c2 = −(p1 − 1)(

p2 −12

)a −

(p1 −

12

)(p2 − 1)b

c3 =12(p1 − 1)a +

12(p2 − 1)b

c4 = −(p1 − 1)(

12p2 − 1

)a −

(12p1 − 1

)(p2 − 1)b

c5 = −12(p1 − 1)p2a − 1

2p1(p2 − 1)b.

As in the four cylinders case, the regular structure in this relation reflects the physics of theexhaust dynamics, with the input contributions to γss, j distributed among the five cylinders.

Note that (15) can be solved with respect to the estimates γi because the matrix A5 has a leftpseudo inverse almost always.

11

4.3 The Estimation Algorithm: Practical Implementation

In order to implement the estimation algorithm, we have to consider the constraints due to the samplingperiod of the Central Control Unit (CCU). Let us assume that the UEGO sensor measurements aresampled with a sampling period ∆s. In general ∆s �= ∆, so that the desired output measurementsγss, i can be obtained by interpolating the sampled output measurements; note that the number of thenecessary measurements is given by

ns = l.c.m.(4, nc)

so that the CCU has to sample the output for a period ns ∆s. Obviously, this period can be greaterthan the period T in which a cycle takes place. Since the steady–state output is a periodic signal overT , we can get the desired measurements from subsequent cycles. Since we have to ensure independentmeasurements, when ns∆s > T the following must hold

h∆s �= kT, h = 1, · · · , ns, k = 1, · · · , k

where k is smallest integer such that ns∆s < kT . In fact, this condition ensures that two samplingsdo not occur at the same instant t mod T . Since ns, ∆s are given, there exist particular values for thecrank-shaft speed that do not allow to have independent measurements. For example, in a 5-cylindersengine, for which ns = 20, and for ∆s = 4 ms when n ≤ 1500 rpm the samplings are independent;for higher speeds these samplings are not all independent, and in Table 1 are reported the maximumnumber of independent samplings obtainable for the critical speeds.

Critical speed n (rpm) Independent Samplings

1579 191667 181765 171875 162000 152143 142308 132500 122727 113000 103158 193333 93529 173750 84000 154286 74615 134737 195000 65294 175455 115625 166000 5

Table 1: Maximum number of independent samplings obtainable for critical speeds n

12

Note also that the hypothesis of steady-state measurements is not restrictive since steady-statesituations are usually encountered in the engine behavior, and reasonably simple tests are available onthe on-board CCU to detect steady-state situations. Moreover, in order to minimize the estimationerror, when the steady-state conditions are maintained for a sufficiently long period of time we canuse m ns samplings and compute the mean value for each sampling over m different measurements.

Finally, in the ideal case, equations (14), (15) can be solved with respect to γi in an analytic way.In the real case, in order to minimize the error due to parameter uncertainties and noise, a least squarealgorithm is used. Therefore, one obtains

γ1

γ2

γ3

γ4

= (AT4 A4)−1

AT4 B4

γss, 1

γss, 2

γss, 3

γss, 4

(16)

in the case of 4-cylinders, and

γ1

γ2

γ3

γ4

γ5

= (AT

5 A5)−1A

T5 B5

γss, 1

...

γss, 20

(17)

in the case of 5-cylinders.

5 Injector Characteristics Estimation

In this section we obtain an estimate of the injector characteristics gi, Ti,off . Once the air flow massma,i and the fuel-to-air ratios γi, i = 1, · · · , nc, have been estimated with (6) and (16) or (17), from (1)we obtain the estimates of the injected fuel mass

mf,i = γi(t) ma,i, i = 1, · · · , nc. (18)

From this relation it is possible to determine the injector characteristics gi and Ti,off , solution ofequation (25) with mf,i replaced by (18) for each steady-state situation. One can easily estimate theinjector characteristics

gi = 103∑

k=1

nk mjf,i − nj mk

f,i

nknj(T

ji − T k

i

) , i = 1, · · · , nc

Ti,off =13

3∑k=1

T ki nk mj

f,i − Tji nj mk

f,i

nk mjf,i − nj mk

f,i

where j = (k + 1) mod 3 and the superscript k = 1, 2, 3 corresponds to a steady-state situation.

6 Experimental Results

Ideally, the quality of the approach should be assessed on a real engine. However, confidentialityabout engine parameters and injector characteristics requested by our industrial partners together

13

with the difficulty of setting up the appropriate instrumentation, made validation on a real engineunfeasible. For this reason, we limit our discussion of experimentation on simulation results. We madesure that the simulation parameters and the results obtained were consistent with reality by havingthe approach scrutinized by Magneti Marelli experts who directed the selection of the critical partsof the experiments. It is conforting that this estimation approach will be adopted in future industrialengine management systems.

The estimation algorithms for the fuel-to-air ratios and for the injector characteristics were appliedseparately to the data obtained from a simulation, to test their efficiency with respect to the precisionof the available measurements.

As far as the fuel-to-air ratios are concerned, the estimation algorithm has been applied to thedata supplied by a model of the exhaust manifold for a 5-cylinders engine with runners of differentlength. This was modelled by choosing a different gain k for each runner delay δr = k/n. The enginespeed is equal to 5000 rpm. It is assumed that there is a complete air-fuel maldistributions, that is(

γ1 γ2 γ3 γ4 γ5

)=

(0.90 0.93 1.10 1.08 1.00

).

and the model parameters are the following:

δr,1 = 5 ms, δr,2 = 5.125 ms, δr,3 = 5.25 ms, δr,4 = 5.375 ms, δr,5 = 5.5 ms

τMIX = 10 ms, τUEGO = 100 ms, δUEGO = 10 ms.

The mean value of the runners delays has been used in the estimation algorithm, that is δr = 5.25,so that δ = δUEGO + δr = 15.3 ms and the coefficient of matrices A5 and B5 in (15) result to beb1 = 0.8763, b2 = −1.875 and

c1 = 13.488×10−4, c2 = 10.042×10−4

, c3 = 3.446×10−4, c4 = 10.191×10−4

, c5 = 3.2979×10−4.

In order to render the simulations more realistic we supposed that the signal measured by the UEGOsensor is affected by a noise, mainly due to the chaotic diffusion process. Moreover, we also considereda noise on the measured signal. The amplitudes of the white noises have been appropriately set soto obtain a simulated A/F UEGO output comparable with a signal measured on a real engine in thesame operational situation. Figure 3 shows the typical signal used in the simulation.

As explained in Section 4.3, better results can be obtained by considering mean values for each sam-pling over different measurements, so minimizing the noise effects on UEGO measurements. Clearly,a tradeoff has to be done between the quality (and cost) of the UEGO sensor and the computationaleffort requested by the algorithm, which affects the cost of the central unit.

The result of the the fuel-to-air ratios estimation algorithm (17) is summarized in Table 2.

real value estimated value error %γ1 0.90 0.9007 0.08γ2 0.93 0.9327 0.29γ3 1.10 1.1007 0.07γ4 1.08 1.0707 0.86γ5 1.00 1.0109 1.09

Table 2 : Estimates γi, i = 1, . . . , 5.

As far as the injector characteristics are concerned, the estimation algorithm has been tested onthe single ith injector for which it is known the air-to-fuel ratio A/F . The data are obtained from a

14

simulation and the values of the parameters of the model are

Vd = 1242 cm3, patm = 1013 mbar, Tatm = 300 K, R = 287 KJ/kg K, µc = cp/cv = 1.4

gi = 1.93 10−3 kg/s, Ti,off = 0.75 ms

ηestv = 0.597071 + 5.15605 10−5

n + 9.30417 10−7pman.

Moreover, we considered to have

εp = −1.237%, εηv = 3.48%, A0th = 2.95 mm2

.

We consider the three steady-state conditions reported in Table 3, and the application of the estimationalgorithm presented in Section 3 (for εp, εηv , A0

th and for ma,i) and in Section 5 (for gi and Ti,off )gives the results summarized in Table 4.

α Ae,th n pm Ti A/F

[deg] [mm2] [rpm] [mbar] [ms]67.50 431 2563 985 11.58 14.6426.64 85 1464 904 9.90 14.6620.70 48 2068 601 6.86 14.65

Table 3 : Simulation Data

real value estimated value error %gi [g/s] 1.93 1.93 0.1

Ti,off [ms] 0.75 0.75 0.6εp % −1.237 −1.239 0.1εηv % 3.48 3.36 3.5A0

th [mm2] 2.95 2.88 2.2

Table 4 : Estimates

The estimation errors are due to the finite number of data digits used for the measurements. Con-sequently, these errors cannot be avoided unless the accuracy of the sensor is dramatically increased.

Finally, to take into account the effects of cascaded estimations on the overall estimation process,we considered for the same case of Table 3 the higher estimation error obtained in the γi’s estimation,namely an error of 1.10%. The values of the A/F in this new estimation are those in Table 3 multipliedby 1.011. In this case we obtained an error of 1.10% on gi and 0.6% on Ti,off .

7 Conclusions

We presented an estimation technique for injector characteristics based on a set of measurements thatcan be carried out by the sensors present in the car, i.e. intake manifold pressure, crank-shaft speed,throttle-valve plate angle, injections timing and exhaust air-to-fuel ratio, which is measured by a singleUEGO sensor placed at the exhaust pipe output.

The estimation strategy is based on a sequence of estimations of various quantities needed toestimate the characteristics of each injector. In particular, we have determined an estimation chainwhich yields good results when applied to simulated systems. We are in the process of transferringthis approach to our industrial sponsors who are interested in adopting the estimation algorithm onindustrial strength applications.

15

8 Acknowledgments

This research has been partially sponsored by Magneti Marelli and PARADES, a Cadence, MagnetiMarelli and ST Microelectronics GEIE, and by C.N.R. under Project MADESS II. The authors aregrateful to G. Gaviani, C. Rossi, R. Flora, G. Serra, C. Poggio and their teams at Magneti Marellifor having introduced the problem and for the many discussions. The authors also thank V. Nicolettifor his comments on the practical implementation of the estimation algorithm. We wish to thank theanonymous reviewers who contributed many useful comments to improve the quality of the paper.

Appendix – Mathematical Model of the Intake Manifold for a SparkIgnition Engine

In this Appendix we present the mathematical model of the intake manifold of a SI engine. We makeuse of the so-called Mean Value Model of a spark ignition engine [1, 5, 8, 13, 14], which describes theair dynamics with good accuracy without considering cycle variations. For the sake of simplicity, thetemperature and transient heating effects are not explicitly taken into account.

The intake manifold dynamics describes the mean values of the relevant engine variables, i.e. theintake manifold pressure, the fuel flow-rate inside the cylinder and the crank shaft speed with respectto variations of the engine inputs, namely the throttle plate angle and the injection pulse duration(the spark advance angle is assumed to be constant).

Throttle Air Mass Flow Rate Equations. The average air mass flow rate through the throttlema,th [kg/s] can be calculated (see [18]) as the air mass flow m of a compressible fluid through achannel between two areas at different pressures:

m = Cdp1√R T1

A2√1 − A2/A1

β

(p2

p1

)(19)

where Cd is a discharge coefficient, R is the ideal gas constant for air [J K−1 kg−1], Ai, pi, Ti are thearea, the pressure, the temperature of section i and

β

(p2

p1

)=

√√√√ 2µc

µc − 1

[(p2

p1

) 2µ c −

(p2

p1

)µc+1µ c

]if

p2

p1≥ rcr

√µc

(2

µc + 1

) µc+12(µc−1)

ifp2

p1< rcr

with rcr =(

2µc+1

) µcµc−1 and µc the specific heat ratio [J/Kg/K].

If we assume the downstream section A2 to be the throttle section and the upstream section A1

a section just before the throttle, then A1 � A2 and equation (19) reduces to

ma,th = Cdpatm√R Tatm

Ath(α) β

(pman

patm

)(20)

where patm and Tatm are the ambient pressure [N m−2] and the air temperature [K], pman is the meanvalue intake manifold pressure [N m−2], Ath is the throttle area [m2], α the throttle plate angle [rad]and Cd the throttle discharge coefficient.

16

The mean value manifold pressure pman can be determined from the average of a set of measure-ments, giving

pm = (1 + εp) pman (21)

of the pressure, where |εp| ≤ 3% is the estimate sensor error.

In the sequel we assumeCdAth(α) = Ae,th(α) + A

0th (22)

where Ae,th(α) is the so-called equivalent throttle area and A0th is the offset area, i.e. the area of the

throttle when α = 0, Ae,th(0) = 0. It is assumed that A0th ∈ [0, 5] mm2 is an unknown value slowly

varying because of aging.

Engine Air Pumping Equations. Theoretically, the average air mass flow ma [kg/s] entering the nc

cylinders of the engine is given by

ma = ρairVd

2n

60where ρair is the air density, Vd the displacement volume [m3] and n the crank shaft speed [rpm]. Byusing the ideal gas equation for the air in the manifold

pman

ρair= R Tair = R Tatm

and taking into account the real supply to the cylinders, we obtain ma as a function of the crankshaftspeed n and of the mean value intake manifold pressure pman

ma =Vd

120 RTatmn pman ηv(n, pman) (23)

where ηv is the so-called volumetric efficiency and it is assumed to be a function of n and pman. Inthe sequel we will assume to know the volumetric efficiency up to a constant error |εηv | ≤ 10%, i.e.

ηv (n, pman) = (1 + εηv) ηestv (n, pman) (24)

where ηestv is the estimated volumetric efficiency.

Fuel Flow Equations. The fuel mass per injection pulse can be assumed to be a linear function of thefuel injector on-time

mf,i = gi (Ti − Ti,off ) , i = 1, · · · , nc

where Ti is the time the injector is open [s], gi the gain [kg/s] and Ti,off the offset [s] of each injector.Therefore, the mean-value injected fuel mass flow is given by

mf,i =n

30gi (Ti − Ti,off ) , i = 1, · · · , nc. (25)

In this relation the fuel flow dynamics, due to the wall wetting, can be neglected assuming the steadystate measurement condition. Moreover, the small dynamic coupling between the fuel subsystem andthe air subsystem is ignored.

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References

[1] C. F. Aquino, Transient A/F Control Characteristics of the 5 Liter Central Fuel Ignition Engine,Technical Report No. 810494, SAE, 1981.

[2] A. Balluchi, L. Benvenuti, M. D. Di Benedetto, C. Pinello, and A. L. Sangiovanni–Vincentelli,Automotive Engine Control and Hybrid Systems: Challenges and Opportunities, Proceedings ofIEEE, Vol. 88, No. 7, pp. 888-912, 2000.

[3] L. Benvenuti, M. D. Di Benedetto, C. Rossi, and A. Sangiovanni-Vincentelli, Injector Characteris-tics Estimation for Spark Ignition Engines, Proceedings of the 37th IEEE Conference on Decisionand Control, Tampa, FL, pp. 1546-1551, 1998.

[4] K. J. Bush, N. J. Adams, S. Dua and C. R. Markyvech, Automatic Control of Cylinder by CylinderAir-Fuel Mixture using a Proportional Exhaust Gas Sensor, Technical Report No. 940149, SAE,1994.

[5] V. Chaumerliac, P. Bidan and S. Boverie, Control-oriented Spark Ignition Engine Model, ControlEngineering Practice, Vol. 2, No. 3, pp. 381-387, 1994.

[6] C.-F. Chang, N. P. Fekete, A. Amstutz and J. D. Powell, Air-Fuel Ratio Control in Spark-IgnitionEngines Using Estimation Theory, IEEE Transactions on Control Systems Technology, Vol. 3,No. 1, pp. 22–31, 1995.

[7] S. B. Choi and J. K. Hedrick, An Observer–Based Controller Design Method for ImprovingAit/Fuel Characteristics of Spark Ignition Engines, IEEE Transactions on Control Systems Tech-nology, Vol. 6, No. 3, pp. 325–334, 1998.

[8] D. J. Dobner, A Mathematical Engine Model for Development of Dynamic Engine Control, Tech-nical Report No. 800054, SAE, 1980.

[9] N. P. Fekete, U. Nester, I, Gruden, J. D. Powell, Model-based Air-Fuel Ratio Control of a LeanMulti-Cylinder Engine, Technical Report No. 950846, SAE, 1995.

[10] J. W. Grizzle, K. L. Dobbins and J. A. Cook, Individual Cylinder Air-Fuel Ratio Control witha Single EGO Sensor, IEEE Transactions on Vehicular Technology, Vol. 40, No. 1, pp. 280-286,1991.

[11] J. W. Grizzle, J. A. Cook and W. P. Milam, Improved Cylinder Air Charge Estimation forTransient Air Fuel Ratio Control, Proceedings of the American Control Conference 1994, Vol. 2,pp. 1568-1573, 1994.

[12] J. B. Heywood, Internal Combustion Engine Fundamentals, McGraw-Hill Book Co., Inc., NY,1989.

[13] E. Hendricks and S. C. Sorenson, Mean Value Modelling of Spark Ignition Engines, TechnicalReport No. 900616, SAE, 1990.

[14] E. Hendricks and T. Vesterholm, The Analysis of Mean Value SI Engine Models, Technical ReportNo. 920682, SAE, 1992.

[15] V. K. Jones, B. A. Ault, G. F. Franklin and J. D. Powell, Identification and Air-Fuel RatioControl of a Spark-Ignition Engine, IEEE Transactions on Control Systems Technology, Vol. 3,No. 1, pp. 14–21, 1995.

18

[16] P. E. Moraal, J. A. Cook and J. W. Grizzle, Single Sensor Individual Cylinder Air-Fuel RatioControl of an Eight Cylinder Engine with Exhaust Gas Mixing, Proceedings of the AmericanControl Conference, San Francisco, CA, 1993.

[17] B. K. Powell, H. Wu and C. F. Aquino, Stoichiometric Air-Fuel Ratio Control Analysis, TechnicalReport No. 810274, SAE, 1981.

[18] C. F. Taylor and E. S. Taylor, The Internal Combustion Engine, Second Edition, Int. TextbookCo., Scranton, Pa., 1970.

19

t

t

γc

γ1 γ2 γ3 γ4

γ5

γc

γ4

γ5

γ1 γ2 γ3 γ4

Figure 1: Normalized fuel-to-air ratio γc(t) at the runner confluence (four and five cylinders engines)

20

γ4

γ4γ5

q

∆ =T

4•

••••

q

∆ =T

20

• • • • • • • • • • ••• • • • • • • • • • •

γ5 + γ1

2

γ1 + γ2

2

γ2 + γ3

2

γ3 + γ4

2

γ4 + γ5

2

γ1γ2

γ3

γ5

γ1γ2

γ3

γ4

γc(q)

γc(q)

Figure 2: γc(q) in the case of four and five cylinders

21

5.6 5.65 5.7 5.75 5.8 5.85 5.91

1.001

1.002

1.003

1.004

1.005

1.006

1.007

Figure 3: UEGO sensor output used in simulation

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Documents

Individual Cylinder Characteristic Estimation for a Spark