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F2000A177
Seoul 2000 FISITA World Automotive Congress
June 12-15, 2000, Seoul, Korea
A Nonlinear Dynamic Model of SI Engines for Designing Controller
Paljoo Yoon1) Seungbum Park2) Myoungho Sunwoo3)*
1)2)
3)
Grdauate Student, Hanyang University, Seoul, Korea
Associate Professor, Hanyang University, Seoul, Korea
In this paper, a nonlinear dynamic engine model is introduced, which is developed to represent a SI engine over a wide
range of operating conditions. The model includes intake manifold dynamics, fuel film dynamics, and engine rotational
dynamics with transport delays inherent in the four stroke engine cycles, and can be used for designing engine controllers.
The model is validated with engine-dynamometer experimental data. The accuracy of the model is evaluated by the
comparison of the simulated and the measured data obtained from a 2.0L inline four cylinder engine over wide operating
ranges. The test data are obtained from 42 operating conditions of the engine. The speed range is from 1500[rpm] to
4000[rpm], and the load range is from 0.4[bar] to WOT. The results show that the simulation data from the model and the
measured data during the engine test are in good agreement. Furthermore, this nonlinear engine model is mathematically
compact enough to run in real time, and can be used as an embedded model within a control algorithm or an observer when
a powertrain controller is designed and developed.
Keywords: Engine Control System, Intake Manifold Pressure Dynamics, Fuel Film Dynamics, Engine Rotational
Dynamics
previously were developed under the assumption that the
engine is operated near the stoichiometric AFR only, and
they did not consider the change of MBT with respect to
AFR.
INTRODUCTION
Due to more stringent fuel economy and emission
legislations, improvements in transient engine control
performance have arisen as important issues. To reduce
tail-pipe emissions using a three-way catalytic converter,
the air-to-fuel ratio (AFR) should be controlled very
precisely in both the steady state and transient engine
operations. (Mean AFR variation : ±0.2%) (2). In order
to regulate the AFR precisely, most of the conventional
engine control units currently used adopt the feedforward
control schemes parallel to oxygen sensor feedback.
However, most of the current engine control algorithms
are designed on the basis of steady state engine maps, and
open-loop correction terms are added to compensate for
transient engine dynamics.
Therefore, this type of
controller cannot compensate control gain from
manufacturing tolerance or engine aging, and it does not
provide robustness to external disturbances. Many studies
have conducted to alleviate these problems regarding the
development of the control-oriented dynamic engine
model (5)(7)(10)(12) and model-based control since the
1980s. Moskwa and Hedrick (1992) developed the mean
torque predictive engine model based on the research
results of Dobner (1983). Both presented very nice model
descriptions in the intake manifold and fuel delivery
model. However, the torque function they used to estimate
the maximum possible torque comes from general
simulation results, and which may cause errors for the
application of specific engines. Additionally, the model
validation results they presented showed relatively large
errors. Hendricks and Sorenson (1990) developed a very
accurate mean value engine model based on engine power
instead of engine torque. However, engine torque is more
adequate in describing the dynamic behavior of an engine
than engine power. Most of the controllers dedicated to
vehicle motion control require not engine power, but
engine torque information for the proper engine control
action. Furthermore, most engine models published
This study introduces a nonlinear dynamic engine
model that is applicable to a wide operating range of SI
engines. The dynamic model uses the test data of both
steady and transient operations and nonlinear estimation
techniques. As shown in Figure 1, the engine model
consists of three input variables (throttle angle, fuel flow
rate, and spark timing), one disturbance (load torque), and
three state variables (intake manifold pressure, engine
speed, and fuel mass in the fuel film). The time constants
of mass airflow through the throttle body and torque
production model are relatively smaller than three state
variables, so they are expressed as algebraic relations.
Figure 1. Block Diagram of 3-State Engine Model
EXPERIMENTAL SETUP
For the development of the engine model, 2.0L, an
inline 4 cylinder DOHC engine and an eddy current type
dynamometer are used. The ignition timing, fuel injection
timing and duration are controlled by a PC-based Engine
1
* Corresponding author. e-mail: [email protected]
The discharge (or flow) coefficient, C D is a function of
the throttle angle (i.e., throttle geometry) and the pressure
ratio across the throttle body. It is obtained from the ratio
of the ideal flow rate to the actual flow rate from the
steady state flow test data from various operating
conditions.
Control System (ECS) developed at Hanyang University
(17). The ignition (dwell time and spark timing) and fuel
injection (timing and duration) of each cylinder can be
precisely controlled by the ECS with the use of a
crankshaft encoder and a cam sensor. An eight channel
analog-to-digital converter, which the ECS employs, is
used for the purposes of data acquisition and closed-loop
control. The analog signals from the engine-dynamometer
system, such as intake manifold pressure, mass airflow
rate, AFR, throttle angle, and brake torque are measured at
each engine event. The AFR is measured by a wideband
oxygen sensor. A hot film type airflow sensor is used for
the measurement of mass airflow through the throttle body.
Furthermore, IMEP is calculated with a PC-based
combustion analysis system with a cylinder pressure
sensor.
INTAKE MANIFOLD PRESSURE DYNAMICS
For the derivation of the manifold pressure state
equation, the conservation of air mass in the intake
manifold and the differential form of the ideal gas law are
used.
With simple mathematical manipulations, the
manifold pressure state equation, Eq (5) is obtained.
RTman 

Pman =
 m − m ap 

Vman  at
ENGINE MODELING
The mass airflow rate out of the intake manifold, m ap is
represented by a speed-density algorithm shown in Eq (6).
MASS AIRFLOW RATE THROUGH THROTTLE
BODY
m ap =
(1)
man = ηvol Pman = f1(n) + f 2 (n) Pman
MA is the maximum possible flow rate through the specific
throttle, and TC is a normalized flow as a function of the
cross-sectional area, and PRI is a normalized flow as a
function of pressure ratio. These are as follows:
(
P
A α
MA = amb th max
RTamb
),
TC =
PRI =
γ +1
γ −1   2

 2  2(1−γ )  2 



1 − PR γ  

 γ +1


γ − 1 


 


f1(n) = f10 + f11n + f12n 2 + f13n3 + f14n 4 ,
Ath (α )
Ath (α max )
PRI = 1.0
where
PR =
Pman
,
Pamb
if PR ≥ PCR
(6)
(7)
where
f 2 (n) = f 20 + f 21n + f 22 n 2
Figure 2 shows the normalized air charge with various
engine operating conditions. It can be easily understood
from the figure that the man is approximately proportional
to the manifold pressure at a constant engine speed, and
shows very small variations with the engine speed.
1
1
PRγ
VD
η P
N
120 RTman vol man
Apart from constants, the speed-density relationship is
represented by (ηvol Pman )× N , and the value in the parenthesis,
η vol Pman , is proportional to the actual air charge per stroke
(9). This quantity, η vol Pman , can be called the normalized
air charge, m an . The normalized air charge is obtained by
the steady state engine test, and is approximated with a
polynomial equation.
The flow of air through a throttle valve can be
considered as a special case of the isentropic flow of
compressible fluid (7). The mass air flow rate through the
throttle, m at , including the discharge coefficient, C D can
be rewritten by Eq (1).
m at = C D ⋅ MA ⋅ TC ⋅ PRI
(5)
(2)
otherwise
γ
 γ −1
 2

PCR = 

 γ +1
If the shape of the throttle bore is assumed to be circular
and the projection area of the throttle plate about the
vertical plane is ellipsoidal, the cross-sectional area of
throttle opening, Ath (α ) is represented Eq (3).

cosα  D 2
π
d  d
sin − 1  −
+
D2 − d 2
Ath (α ) = D 2 1 −

4
cosαo 
2
D 2


 d cosα o 
D 2 cosα
+ d
sin − 1
+
D 2 cos 2 α − d 2 cos 2 α o
 D cosα  2 cosα
2 cosα o


(3)
Figure 2 Normalized air charge with Pman and N
Since the SI engine operation is based on the engine
events, an air charge per stroke has more important
meanings than the normalized air charge during the
process of the development of the engine model. As
shown in Eq (8.1), the air charge per stroke is obtained
from the integration of the mass air flow into the intake
port between intake events, but the value obtained during
At the small throttle opening, there is an error between the
ideal cross-sectional area and the actual area due to
machining variation (6). A correction for this error is:
α o = 0.91α o* − 2.59
(4)
2
the steady state test is widely used because of its
mathematically simple form.
m = ∫ t IVC m (t )dt
ac t IVO ap
−
m ff (t ) = e
(8.1)
 kg 
m ap  
30m
ap  VD
 s 
=
=
mac =
 4 RT
N
 rev   min   2 stroke 
man

N





 min   60 s   rev 
where

m
 an

(8.2)
(
120V
man ,
V Dη vol N
120 RT
man m
Pman, ss =
V Dη vol N at





t

−

τ
τ f X m fi (0) + τ f X m fi (t )1 − e f


(12.1)
= constant at step test, and
at steady state
Therefore, fuel flow rate into the cylinder is
m fc (t ) = (1 − X )m fi (t ) +
)
where
(9.1)
1
τf
−
m ff (t ) = m fi (t ) − X ∆m fi (t ) e
t
τ
f
(12.2)
∆m fi (t ) = m fi (t ) − m fi (0)
If the throttle angle is assumed to be constant and only step
fuel perturbation is applied, the equivalence ratio in the
cylinder is represented by Eq (12.3).
where τ man corresponds to the time constant of the intake
manifold pressure state equation represented by the first
order differential equation, and Pman,ss is the manifold
pressure determined from the throttle angle and engine
speed at a steady state.
τ man =
f
m ff (0) = τ f X m fi (0)
An intake manifold pressure state equation, Eq (9.1),
can be obtained by substituting Eq (6) into Eq (5).
dPman
V η N
RTman
= − D vol Pman +
m at α , Pman
120V man
dt
Vman
1 
=−
− Pman, ss 
P

τ man  man
m fi (t )
t
τ
φ c (t ) =
(9.2)
t

−
(A/ F) 
s m (t ) − X ∆m (t ) e τ f

fi
m ac (t )  fi






(12.3)
Figure 3 and Figure 4 show the AFR response to the
perturbations of fuel supply rate according to Eq (12.3).
FUEL FILM DYNAMICS
The characteristics of the fuel injection process is very
complex and is largely influenced by various factors, such
as injection schemes, spray patterns, injection timing, port
geometry, intake manifold wall temperature, fuel rail
pressure, and so on. However it is actually not easy to
implement such complicated fuel delivery models which
consider all aspects mentioned above to the real time
engine control applications. Thus, the first order lumped
parameter model expressed in Eq (10) is widely used to
represent the fuel delivery process (1)(7).
m ff = −
1
Figure 3 Sensitivity of A/F response to X produced by
fuel perturbations with τ f =0.6
m ff + Xm fi
τf
m fv = (1 − X ) m fi
m fc = m fv +
(10)
1
m
τ f ff
In Eq (10), m fi is the injected fuel flow rate from the
injector, X is a fraction of fuel that forms a fuel puddle on
the walls, m ff is mass of fuel in the fuel puddle, and the
actual fuel flow rate, m fc , which enters the cylinder, is the
sum of m fv and the evaporated fuel from the film with the
time constant τ . Therefore, the actual AFR in the
cylinder is given by m ac / m fc , and it can also be represented
by an equivalence ratio ( φ c = 1 / λ ).
f
φ c (t ) = ( A / F ) s
m fc (t )
m ac (t )
Figure 4 Sensitivity of A/F response to τ f produced by
fuel perturbations with X = 0.6
(11)
Figure 3 shows how the AFR response is influenced by the
value of X , Figure 4 shows the influence of the value of
τ f . The AFR at the initial transient period goes up rapidly
due to the influence of fast fuel m fv , and is changed
exponentially according to the evaporation behavior of the
fuel film. As shown in these figures, steady state fuel film
dynamics is independent of fuel delivery model
parameters, X and τ f , and these parameters affect the
In order to understand the dynamic characteristics of
AFR with the variation of important fuel delivery model
parameters ( X and τ f ), the transient response of AFR
with step fuel perturbation is predicted. If Eq (10) is
regarded as a kind of an initial value problem, the solution
is given by
3
Figure 5 and Figure 6 represent the (MBT )λs and
test engine, respectively.
transient AFR only. Furthermore, the changes of AFR at
the initial stage of the transient period are determined by
X , and the AFR response of the successive transient
period is affected by the time constant, τ f . Actually, the
fuel film model parameters, X and τ f , depend on various
engine operating conditions (engine speed, load, AFR and
coolant temperature, and so on), and the engine speed and
coolant temperature give the dominant effects on model
parameters (14).
Eq. (13) represents the typical
relationship between engine operating conditions and the
parameters of fuel film model.
X = a 0 + b0 ( N ) + [a1 + b1 ( N )]T EC
(13.1)
2 + c T 3 ]/ N
τ f = [c 0 + c1TEC + c 2TEC
3 EC
(13.2)
In this study, the AFR under various engine operating
conditions are measured from the step fuel perturbation
test. The wideband oxygen sensor is used to measure the
AFR at the exhaust manifold of the test engine. The
Levenberg-Marquardt Method is employed to estimate the
nonlinear fuel film model parameters (15).
∆MBT (λ )
of
Figure 5 MBT at stoichiometry
TORQUE PRODUCTION MODEL
The combustion process in the cylinder produces
engine torque, and the quantity of the torque produced is
influenced by the AFR of the mixture in the cylinder, spark
timing, and combustion efficiency. The transport delay
with respect to each engine event gives dynamic
characteristics to the torque production model. In this
study, it is assumed that the dynamic torque of the engine
is a function of engine speed, air charge per stroke, spark
timing, and the AFR.
Under this assumption, the
predictive torque production model is derived from the
steady state engine experiments. One of the most
important things in the development of the torque
production model is the identification of the optimal spark
timing (MBT) at each of the operating conditions. Most
engine models published previously(5)(7)(10) were
developed under the assumption that the engine is operated
near the stoichiometric AFR only, and they did not
consider the change of the MBT with respect to the change
of the AFR. The recent advanced engine technologies,
such as lean burn, and gasoline direct injection engine,
motivate the development of the engine model, which is
applicable to a wide range of AFR. The MBT at the
stoichiometric AFR is identified under various engine
speed and load conditions. After that, the identification of
the MBT at various AFR conditions is proceeded. The
difference between the MBT at the stoichiometric AFR
and the MBT at an arbitrary AFR at various engine
operating conditions is denoted by ∆MBT (λ ) . Based upon
this approach, the MBT at an arbitrary AFR can be
represented by Eq (14).
(MBT )λ
s
(
(MBT )λ = (MBT )λ
where
)
(
)
= f m ac , n = m0 + m1n + m 2 n 2 + m3 + m 4 n m ac
s
+ ∆MBT (λ )
Figure 6 MBT variation with respect to
λ
Under the assumption of negligible cylinder-bycylinder engine torque variations, the indicated torque is
obtained from the measurement of IMEP of cylinder #1
using a cylinder pressure sensor (3). If AFR and spark
timing are fixed, the indicated torque of an engine is a
function of engine speed and air charge. Therefore the
indicated torque at the MBT and the stoichiometric AFR
can be represented by Eq (15). Figure 7 shows the test
data and model data generated from Eq (15).
(Tind )MBT , λ
s
(
)
= t 0 + t1n + t 2 n 2 + t 3 + t 4 n m ac
(15)
(14.1)
(14.2)
Figure 7 Indicated torque at MBT and
∆MBT (λ ) = d 0 + d1λ + d 2 λ 2
λ =1
The indicated torque at an arbitrary engine condition is
obtained from Eq (15) multiplied by the efficiencies of the
spark timing and the AFR which reflect the changed
operating conditions.
(MBT )λ : MBT at Stoichiometric AFR
s
(
(MBT )λ : MBT at arbitrary AFR
) ( )
⋅ η ∆SA ⋅ η λ
Tind n, m ac , η ∆SA , η λ = Tind
MBT , λ s
4
(16)
where
η ∆SA = s 0 + s1 (∆SA) + s 2 (∆SA)2
production model, three different transport delays are
introduced. This approach is reasonable for real time
control applications because the combustion process of the
engine occurs much faster than the transport dynamics of
air and fuel. However, these transport delays are subjected
to provide stability problems for feedback systems, which
must be considered in the control synthesis (10).
∆SA = (MBT )λ − SA
η λ = l 0 + l1λ + l 2 λ 2
The η∆SA , shown in Figure 8, denotes the spark timing
efficiency, which is a function of the difference of spark
timing between the MBT at an arbitrary AFR condition
and spark timing at the current operation. The value of
η ∆SA at the MBT is 1. The η λ , depicted in Figure 9,
represents the influence of the AFR to the indicated torque,
and is normalized by 1 at a stoichiometric AFR.
Figure 10 Schematic diagram of torque production model
Figure 8 Spark timing efficiency
TRANSPORT DELAY BETWEEN ENGINE
EVENTS
In the instant of closing the intake valve, the amount
and AFR of the mixture in the cylinder that are available
for torque production are determined. Therefore, it seems
natural to synchronize the sample times with the intake
events. If each event of the engine is concentrated at one
instant of time (5), the useful transport delays used in this
engine model are:
Intake to exhaust delay,
Figure 9 A/F efficiency
Intake to torque delay,
Actually, the indicated torque generated by the
combustion of an in-cylinder mixture is decreased by both
engine friction and pumping loss. The difference between
the indicated torque and the friction and pumping loss
becomes the brake torque. The friction and pumping
torque, T , is estimated from the difference between the
indicated torque measured from the steady state cylinder
pressure and the brake torque obtained from an engine
dynamometer. It is well known that the friction and
pumping loss is a function of the engine speed and intake
manifold pressure (16). As a result of that, the friction and
pumping torque can be approximated by a polynomial of
engine speed and air charge.
(
)
∆ϕ IT1
Injection to torque delay,
Spark to torque delay,
= 489°
= 279°
∆ϕ IT 2
∆ϕ ST
(18)
= 585°
= 75°
ROTATIONAL DYNAMICS OF ENGINE
f /p
T f / p = p 0 + p1n + p 2 n 2 + p 3 + p 4 n m ac
∆ϕ IE
The rotational dynamics of an engine is modeled under
the assumption of a lumped parameter system with
constant inertia. Using Newton's Second Law, the state
equation for engine speed is given by:
J eff
(17)
where
Summarizing the torque production model proposed in
this study, this model has four inputs (engine speed, air
charge, fuel flow rate, and spark timing), six submodels
approximated by polynomials, and brake torque as an
output. The overall structure of the torque production
model developed in this study is shown in Figure 10. In
order to provide the dynamic characteristics of the torque
with
dN (t ) 30 
(t ) − T / (t ) − T (t ) 
=
T
f p
L 
π  ind
dt
( )
(t − ∆t IT1 ) ⋅ η ∆SA (t − ∆t SA ) ⋅ η λ (t − ∆t IT 2 )
Tind (t ) = Tind
MBT , λ s
∆t IT1 =
∆ϕ IT1
6 N (t )
,
∆t IT 2 =
∆ϕ IT 2
6 N (t )
,
∆t ST =
ENGINE MODEL VALIDATION
5
(19)
∆ϕ ST
6 N (t )
To validate the developed engine model, steady and
transient tests in a wide operating range must be carried
out. The algebraic equation derived in this study is an
approximation of experimental data using the LevenbergMarquardt Method (18), which is a kind of nonlinear least
square estimation technique.
The test data and
approximation results are depicted in each figure in a
comparative manner. The sampling interval for the
simulation is selected to be synchronized with the intake
event because of the event-based operation characteristics
of the engine.
ts =
120
N cyl N
Case I : Constant Engine Speed
To prove the effectiveness of the torque production
model proposed in this study, the dynamometer is operated
at constant speed mode. Figure 11 shows the comparison
between the model and the measurement when step throttle
input is applied. The model outputs of the manifold
pressure and the brake torque show in good agreement
with experimental data. However, for the brake torque, a
slight time delay is observed during the transient period
between the predicted and the measured data. The time
delay may be caused by the digital-to-analog conversion
and the signal output process in the dynamometer
controller.
(20)
In order to evaluate the accuracy of the model, test data
from various operating conditions are collected, and a
comparative study is done with the predicted values of
various variables of the model using the same input
profiles (throttle angle, fuel flow rate, and spark timing).
MODEL VALIDATION AT STEADY STATE
In order to confirm the accuracy of the model, the mean
values of the errors in the predictions of the model and the
test data are calculated from 42 operating conditions of the
engine. The speed range is from 1500[rpm] to 4000[rpm],
and the load range is from 0.4[bar] to WOT. The results
given in Table 1 show that the relatively large error (about
10%) is observed at low load conditions but the slight
error is observed in the remaining operating regions. The
larger error observed in the low load region is due to the
characteristics of the eddy current dynamometer, because
it is well known that the eddy current type dynamometer
causes poor torque control performance at low load
conditions. Furthermore, the shaft that connects the engine
and dynamometer is not rigid enough, and thus the
influence of the shaft compliance on the test data is much
larger at low load conditions, further studies are needed to
accommodate the effects of the dynamometer and shaft
compliance on the model performance.
Table 1. Mean errors in the steady state predictions of the
engine model
Variable
% error
N
Pman
m at
-5.01
Figure 11 Validation of engine model at constant engine
speed (N=2000rpm)
5.68
Case II : Constant Load Torque
-3.41
In this case, the dynamometer is set at a constant torque
operation, and the dynamic behaviors of three state
variables are examined. Figure 12 shows the comparison
between the model and the measurement when step
throttle transient is applied. The predicted manifold
pressure, engine speed, and the AFR at the exhaust pipe
show in good agreement with the experimental data.
MODEL VALIDATION AT TRANSIENT STATE
For validation of the engine model during transients, the
dynamometer is operated at a constant speed and a
constant torque mode, respectively. A set of time
responses was recorded using throttle transients. Among
three state variables, the engine speed and the manifold
pressure are measured directly, and the AFR at the exhaust
pipe is measured instead of fuel mass in the film because
the fuel film dynamics is not easy to measure. The torque
production model is verified with the brake torque from
the output of the dynamometer controller.
6
(4) The engine model is validated to an accuracy of
±5% by the comparison of the predicted and the
measured data obtained from wide operating ranges.
(5) This model is mathematically compact enough to run
in real time, and can be used as an embedded model within
a control algorithm or an observer.
(6) For the improvement of the reliability of this model in
low load regions, the effect of the eddy current
dynamometer's characteristics and shaft compliance to the
model accuracy should be considered.
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Fuel Transport Characteristics of SI engines for Transient
Fuel Compensation. SAE Paper 950067
[15] Sunwoo, M., Yoon, P. and Ju, J. 1998. A Study on the
Design of Fuel Film Compensator and A/F Controller in SI
Figure 12 Validation of engine model at constant load
torque
CONCLUSION
A nonlinear dynamic engine model which is applicable
to a wide operating range of SI engines is developed by
using the steady and transient test data and a nonlinear
estimation technique. The following conclusions are
derived in this study.
(1) The engine model is comprised of three input
variables (throttle angle, fuel flow rate, and spark timing)
including one disturbance (load torque). The model is
described with three state variables (intake manifold
pressure, engine speed, and fuel mass in the fuel film), and
major subsystems (mass airflow through throttle, torque
production model, etc.) of the engine expressed in
algebraic equations are included.
(2) The applicable operating range of the engine model is
expanded by the inclusion of the changes of the MBT with
respect to AFRs, and it also can be extended to the lean
burn operations.
(3) The influence of fuel delivery model parameters to
transient AFR characteristics is examined through the
analysis of the sensitivity of AFR response to model
parameters. The AFR change at the initial stage during the
transient period is determined by X , and the response of
the successive transient period is mainly affected by time
constant, τ f .
7
Engines. Autumn Conference Proceeding of KSAE, pp.
154～159 (in Korean)
[16] Taylor, C.F. 1985. The Internal Combustion Engine
in Theory and Practice. Vol. 1, 2nd ed. MIT Press
[17] Yoon, P., Kim, M. and Sunwoo, M. 1998. A Study on
Design and Development of an Engine Control System
based on Crank Angle. Transactions of KSAE, Vol. 6,
No .4 (in Korean)
[18] Zielinski, T.J. and Allendoerfer, R.D. (1997) Least
Squares Fitting of Nonlinear Data in the Undergraduate
Laboratory. J. of Chemical Education, Vol. 74, No. 8
8