Proceedings of the American ControlConference

Chicago Illinois • June 2000

Modeling and Control of a Variable Valve Timing Engine

Lawrence Mianzo Huei Peng

Graduate Student Associate Professor
Visteon University of Michigan
Advanced Powertrain Department of Mechanical Engineering
Control Systems Group and Applied Mechanics
17000 Rotunda Drive, Dearborn, MI 48121 Ann Arbor, MI 48109-2125
lmianzo~visteon.com hpeng~umich.edu

Abstract could be improved. More recently, in [6] it was shown

that using the flexibility of an electromechanical val-
A cylinder-by-cylinder model of an experimental vari- vetrain to run in 2, 3, and 4 valve modes, as well as
able valve timing 4-cylinder engine has been developed. by using cylinder deactivation, fuel economy consump-
The model includes the cylinder and manifold mass, tion could be improved over a wide range of operating
temperature, burned gas residual, and pressure dynam- conditions.
ics, including combustion effects, as well as the valve
actuator dynamics. The cylinder-by-cylinder model More recent attention has focused on the control-
is used to obtain a cycle-averaged mapping between oriented modeling and the control of variable valve
torque at a given engine speed and intake valve timing, timing engines. [7] modeled the cylinder-by-cylinder
which is suitable for future control design implementa- breathing dynamics of a careless engine system, includ-
tions. ing higher frequency runner dynamics using the forced
oscillator model. The model was then used to obtain a
cycle-averaged model between valve duration and lift
and mean-valued cylinder charge. In the companion
1 Introduction paper, [7], the cycle-averaged model was used with the
cylinder-by-cylinder model to develop and demonstrate
Recently, variable valve timing engines have attracted a a control approach to estimate cylinder charge online
lot of attention because of their ability to control valve and use that information to update a feedforward map
events independent of crank shaft rotation, allowing for between charge demand and valve lift and duration.
reduced pumping loss (work required to draw air into However, the model used in [7] and [8] did not include
the cylinder under part-load operation), and increased modeling of the exhaust process and the effects of com-
torque performance over a wider range than conven- bustion on cylinder pressure, and therefore the effects
tional spark-ignition engine. Variable valve timing also of combustion on the breathing process and torque pro-
allows control of internal exhaust gas recirculation (by duced. This paper attempts to to demonstrate a more
control of the valve overlap), allowing for control of complete cylinder-by-cylinder model that includes the
NOx emissions produced during combustion. exhaust valves, manifold, and breathing process, as
well as the in-cylinder combustion.
Several detailed studies have been performed to show
the benefits of variable valve timing engines. [1] de-
scribes a mechanical variable valve timing system, and
points to the possibility of controlling the air flow into 2 In-Cylinder Dyanmics
the engine via the valve openings, thus eliminating the
throttle and reducing pumping losses. In [2] and [3], The in-cylinder dynamics consist of the 4 states of
varying the intake valve timing was shown to reduce cylinder pressure, temperature, mass, and burned gas
pumping mean effective pressure while improving fuel residual fraction, as described in [9]. The cylinder pres-
economy and NOx emissions. [4] describes an electro- sure is obtained from the perfect gas law
hydraulic valvetrain based engine and provides a good
Pc~Vcyt =rncytRTcyl (1)
summary of the benefits of camless engines. [5] showed
that by optimizing valve events at part-load conditions, Where Pcyt is the cylinder pressure, Vcul is the cylinder
volumetric efficiency, fuel economy, and NOx emissions volume, mcut is the cylinder mass, R is universal gas

0-7803-5519-9100 $10.00 © 2000 AACC 554

constant, and T~ut is the cylinder temperature. Here, QLHV is the lower heating value of the fuel,
which is a measure of the energy of the fuel, and can
Differentiating (1) with respect to time, we obtain be found in fuel property tables, mb is the mass of the
burned fuel, which is given as the product of the mass
dR
P~ytV~ut+ PcutVc~t= ¢ncu,RT~ut+ m~yt--~T~ut fraction burned, Xb and the injected fuel, rail

(2) mb = Xbmif (11)

And therefore
drab
d R dF~ut,~ . + m~ytRT"cut (3) dt - &bmil + xbrhii (12)
+mcu~ dF~ut - ~ -~c~,
One common method of obtaining the mass fraction
burned, Xb, is to use an empirically fit function of
mass fraction burned versus crank angle, 0, such as
q-mcyl R1 - R2 pcytT~u t + mcutRLu, (4) the Wiebe function described in [9], given as

Where F~ul is the fraction of burned gas in the cylinder

Xb = 1 -- e x p [ - - a ( ~ s ° ) m + l ] (13)
and
R = F~utR1+ (1 - Fcut)R2 (5)
Where 00 is the crank angle at the start of combustion,
Dividing the LHS of Equation (4) by Pc~tVcytand the A0 is the total combustion duration, and a and m are
RHS by m~uzRT~ut, we have correlation parameters.

t:)cyl : [ZhcYl ~'cyl + ~R1 Fcyl - ~]Pcul (6) Noting that the total energy is equal to d(m~ulu)/dt,
and the internal energy, u, is a function of t e m p e r a t u r e
and burned gas fraction, then
The cylinder mass rate of change from conservation of
mass, assuming the convention that flow rate is positive _ d(rncutu) du
into the control volume, is dt - rhc~tu + m c y t - ~ = rhcylU
Ou OF~yh Ou OT~yz )
r h ~ = ~h~,~ + m ~ (7) (14)
0-----7 +
Where rhi~ and r h ~ are the mass flow rates through
Noting the following identities for internal energy
the intake and exhaust valves, respectively. The flow
through the valves can be modeled as flow through an
u = F ~ u l + (1 - F~yl)u2 (15)
orifice as follows:
And the specific heat, Cv, which is defined as
rh = A~Hd(P, , P2) (S)
du
where A e l ! is the effective flow areas of the orifice and cv = ~ (16)
P~ and P2 are the upstream and downstream pressures
and d is the differential pressure constant. Then, Equation (14) becomes

The cylinder temperature, from the 1st Law of Ther- J~ ---- mcyl u -}- mcylCv~~cyl "~ m c y l ( ~ i -- u2)PCyll (17)
modynamics as described in [9], is given by
Setting Equation (9) equal to (17) and solving for 2bout
= Qw - ~V @ ~2inhin q- IJ2exhex -]- Och = Ow
-P~,V~t + rhi~hi. + r h ~ h ~ + Qch (9) m~ylcvTcyl = Qw - Pc~If/c~,+ dni,~hin + rhe~hex
Where /~ is the total energy in the system (~w is the +Q h - ,hc tu - m c y l ( u l - u2)k l, (18)
rate of heat transfer through the cylinder wall, rdr is
The dynamics of the burned gas fraction in the cylinder
the rate of work done on the piston, hi,~ and he~ are
are described as
the enthalpy of the flow through the intake and ex-
haust flows, and Q~h is the combustion heat release
dmcytFcyl _ rhcytFcyl + mcylFcyt = rhi~Fi~c~t
rate, given by dt
drab f~
+ he Fcyzo (19)
Qch = - - ~ - ~ L H V (10) + m i n ( (mcyt(1 - Fcyt) ) s o c , (mif A F R ) s o c ) 2 b

555
Where the function rain(.) evaluates to ( m i I A F R ) at W h e r e Yt2throttle is the mass flow rate through the throt-
the start of combustion (SOC) if the mixture is lean tle, and i is the cylinder index. For the manifold tem-
or (mcul(1 - Fcyt)) at SOC, if the mixture is stoichio- perature state, the manifold volume is constant and
metric or rich. This is done since only the part of the heat transfer through the manifold walls are neglected
mixture that is stochiometric will burn completely, i.e,
excess fuel is not burned and if the fuel is lean, only m i c v T i ----rithrottlehthrottle "4- l:ninhin
the portion of unburned gases in the cylinder stoichio- - r i i u - mi(ul - u2)/~i (26)
metrically proportional to the fuel will burn. Also,
riinFie,~t and rie~F~yt+~e are mass flow rate of the The burned gas fraction rate of change in the intake
burned gases across the intake and exhaust valves, re- manifold is given by
spectively. If the flow of burned gases to the cylinder dmi Fi
changes direction, the sign of the mass flow is auto- -- r i i F i + m i F i = rithrottleFthrottte~i
dt
matically taken care of but the burned fraction of the
+riinF~ocut (27)
flow should be that of the manifold if flow is into the
cylinder, or that of the cylinder if the flow is out of the Where
cylinder, this is accomplished by letting the fraction,
r i i ~ F i ~ y t , be defined as I F throttle if 7hthrottle > 0
(28)
Fthrottle++i : Fi if Tt2throttle < 0
F~ if rim > 0
Fi~cyt = F~ut if riin ~ 0 (20) We will assume Fth~ottte = 0, since the intake air con-
tains no burned component. Solving Equation (27) for
And
/re if rie~ > 0
Fc~t~e = Fcut if rie~ < 0 (21)
rniFi = rith~ottleFthTottte~i + riinFiocyt - riiFi (29)
Where Fi and Fe are the intake and exhaust manifold The exhaust manifold is analogous to the intake man-
fraction, respectively. ifold, with analogous states, Pe, me, Ve, Te, and Fe,
which are the exhaust manifold pressure, mass, vol-
Solving Equation (19) for F~yl, the rate of change of
ume, temperature, and fraction, except that flow is
burned fraction in the cylinder can be obtained from
through the exhaust pipe instead of the throttle.
m~ul[Zcuh = riinFiocyl -t- rie~F~,++e- ri~utFcut
The cylinder mass rate of change from conservation of
+min((m~yt(1 - F~ut))soc, (mifAFR)soc)gcb (22) mass is
rite = r i ~ + ri~pipe (30)
The burned gas fraction rate of change in the exhaust
3 Manifold Dynamics
manifold is given by
The manifold dynamics also consist of 4 states: mani- m e r e = riexFcyt~e + riepipeFe~epipe - rieFe, (31)
fold pressure, temperature, mass, and burned gas resid-
ual fraction, analogous to the cylinder dynamics. The Wh~re
manifold pressure is obtained from
= ~ Fevipe if riepipe > 0
Fe++evWe [ Fe if ri~pipe _< 0
(32)
R1 R2 ~.-

PiVi = riiRTi + mi------ff-~ri + miRT"i (23)

Where Pi, mi, Vi, T/, and Fi, are the intake manifold
pressure, mass, volume, temperature, and fraction. 4 Rotational Dynamics

Dividing the LHS of Equation (23) by PiVi and the The cylinder pressure acting against the piston creates
RHS by miRT/ a force. The force is composed of an inertial component
and a component due to pressure acting on the piston
t5/= [ + ~ + (24) area and is given by

Fpiston = Fpress + Finertial (33)

The intake manifold mass rate of change from conser-
vation of mass is Or, more explicitly,

rii : rithrottle q" E riim (25) p, - 7rB 2

Fpiston = (Pcvl - atrn)T (meH)A (34)
i - -

556
Where Pat,n is atmospheric pressure, B is the cylinder
bore, m e / f is the effective mass, and A is the piston
acceleration, which can be obtained from
L
A = Rw2[cosO + (1 -- )~2sin20)l/2 (35)

AcosO
+ RgzsinO[1 + (1 - )~2sin20)l/2J1 V
where R is the crank radius, and A is the ratio of crank
radius to connecting road length, and w and 5: are the
angular velocity and acceleration, respectively.
F i g u r e 1: Variable Valve Timing Engine Model
The net indicated torque for each piston, Ti is equal to 1500 RPM. 4.0 Bar BMEP
the product of the tangential component of the force
and the crank radius, so
AsinOcosO
Ti = Fpi,ton[sinO + (1 - )~2sin20)l/i]R (36) 25

The angular acceleration is then given by

1
= I-~H(ETi-Tm/-T, --T, oad) (37)

Where Tmf is the mechanical friction loss torque, Ta lO FI ~

is the essential accessory loss torque, Ttoad is the load

torque, and IelI is the effective inertia. 5 // "x

o
-400 -300 -200 -100 0 100 200 300 400
Crank Arigle (Degrees)
5 Simulation Results
F i g u r e 2: Peak Pressure Comparison at 650 RPM and
A 24 state model was implemented in Simulink. A 1.5 BMEP
block diagram of the model is shown in Figure 1. The
control inputs to the model are change in valve com-
mand, spark, and fuel injection duration from a nom-
inal value. For example, intake valve closing (IVC)
crank angle, the control can only affect the plant on
nominal was chosen as bottom dead center before com- a per-cycle basis. Therefore, for controller design, a
bustion (crank angle = -180 degrees, using the conven- relationship is required between valve timing and net
tion that combustion top dead center is 0 degrees). So indicated torque on a cycle average basis. The cylinder-
a change in IVC of 20 degrees would be a delay in in- by-cylinder model is used to obtain this relationship.
take valve closing to -160 degrees. The map is then used in combination with the valve ac-
t u a t o r dynamics, rotational dynamics, and torque loss
Experimental data was only available at low speeds
models for design and evaluation of the controller.
and load conditions, because of the durability of the
experimental engine. Peak pressure was chosen as the Because the throttle position and air inlet to the cylin-
model correlation parameter because of its direct im-
der is no longer controlled by driver input, a nonlin-
pact on net indicated engine torque. Comparison of
ear map between pedal position and desired torque
experimental and model peak pressures are shown in
demand is typically used. This demand map requires
Figure 2 and 3.
the work of experienced vehicle calibrators, and for the
purpose of this study is assumed to already exist. The
net indicated torque demand is then used as the de-
6 Control Architecture sired signal to be tracked. Because the control is only
available on a per-cyle basis, the average net indicated
A proposed control architecture is shown in Figure 4. torque per cycle, rather than the instantaneous torque
Although the cylinder-by-cylinder model predicts in- is required. The cycle averaged torque is obtained as
stantaneous torque and cylinder pressure at a given by integrating the net integrated torque over the engine

557
 2500 RPM, 7.0 Bar BMEP
50 r n r 1 T Torque VS. IVO and IVC at 2500 RPM

2OO.
35
E180.

'1oo -
~2s
140-

,5 ~
100-
1~°~
1o i! x

DiRa lye [ D ~ r e ~ } -100 -40

o Delta IVO (DegiNs)
Crank Angle (Degrees)

F i g u r e 5: Torque vs. IVO and IVC at 2500 R P M

F i g u r e 3: Peak Pressure Comparison at 2500 RPM and
7.0 BMEP

Acknowledgment

The authors would like to thank Ibrahim Haskara for

his help with the solver for the thermodynamic prop-
erties and help debugging, and Brett Collins for the
contribution of the friction/inertia model.

References
F i g u r e 4: Control Approach [1] Lenz, H. P., Wichart, K. and Gruden, D., "Variable
Valve Timing - A Possibility to control Engine Load with-
out Throttle" SAE Paper 880288, 1988
cycle. [2] Tuttle, J. H., "Controlling Engine Load by Means of
Late Intake Valve Closing" SAE Paper 80079~, 1980
T~.~- 1 ~ (~-~Ti)dO (38) [3] Tuttle, J. H., "Controlling Engine Load by Means of
~cycle ycle i
Late Intake Valve Closing" SAE Paper 820408, 1982
Where i is the cylinder index. The relationship be- [4] Schecter, M. M. and Levin, M. B., "Camless Engine"
tween valve timing and torque and air charge was ob- SAE Paper 960581, 1996
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state, cycle-averaged map. The torque output at 2500 Timing Strategies for Optimum Engine Performance and
RPM is shown in Figure 5. Because of it's dominant Fuel Economy" Proceedings of the Energy Sources Technol-
effect on volumetric efficiency and load control, only ogy Conferene and Exhibition, January, 1994
intake valve timing will be varied. [6] Pischinger, M., "A New Opening" Engine Technology
International, 2000 Annual Review
[7] Ashhab, M. S., Stefanopoulou, A. G., Cook, J. A.,
and Levin, M. B., "Control-Oriented Model for Careless
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DSCD, pp. 179-186, 1999; and to appear ASME Journal of
Dynamic Systems, Measurement, and Control.
A cylinder-by-cylinder model of 4-cylinder vararible
[8] Ashhab, M. S., Stefanopoulou, A. G., Cook, J. A.,
valve timing engine has been developed and correlated
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for peak pressure at several engine speed and load con- II)," in Proceedings of 1999 IMECE, DSCD, pp. 187-194,
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558