VALIDATION OF THE LINEAR COMBUSTION ENGINE MODEL

S. Plšek*, P. Deutsch **, O.Vysoky***

Department of control engineering, FEE, CTU, Karlovo namesti 13, Prague 2
* e-mail: stanislav.plsek@email.cz
**e-mail: Pavel.Deutsch@ricardo.com
***e-mail: vysokyo@feld.cvut.cz

Abstract: This paper describes validation of the linear combustion engine model and al-
gorithm design of air/fuel ratio control for this type of engine. Linear combustion en-
gine transforms chemical energy of hydrocarbon fuel to electric energy. Main parts of
the engine are linear motor-generator and two opposite cylinders. All these parts are
situated in one line. There is no rotating but only linear motion. A big benefit of this
engines’s concept is its mechanic simplicity which leads to high lifetime. The possible
application of the linear combustion engine is in hybrid vehicles as a power unit. Model
is firstly validated with a prototype of the linear combustion engine and then is used for
developing and testing of algorithms for air fuel ratio control.

Keywords: Thermodynamic model, linear engine, air fuel ratio control.

During LCE operation, the pistons are accelerated by
a combustion mixture and move from one side to the
1 INTRODUCTION opposite side. The released energy is partially used to
compress the fuel mixture in the opposite cylinder.
The linear combustion engine (LCE) presented here This action is repeated periodically. The difference
is a two-stroke, two cylinder combustion engine. The between the energy released by the combustion mix-
LCE prototype employs two 50ccm cylinders with ture and the energy consumed by mixture compres-
the direct fuel injectors. These cylinders are from sion and mechanical losses is drained from the sys-
scooter Aprillia SR 50 Ditech. The linear motor- tem as electric energy by a linear motor-generator
generator is a product of VUES Company and is which is also used as a starter during the start of
driven through the 3-phase power bridge with IGBT LCE. The engine's output is directly in a form of
transistors. A functional schematic of the LCE is electric power. Finally, the motor-generator also al-
shown in Figure 1. lows the prevention of the LCE from stopping when
a misfire occurs.

Fig. 1. Schematic of the linear combustion engine.

 of the model are covered by block Mechanics. Block
e-motorgen simulates function of linear motor-
generator. Block AFR simulates function of linear
lambda sensor.

Name Description Units

x_ign Position of ignition [mm]
x_inj Position of injection [mm]
deltaT_f Duration of fuel preparing [ms]
alpha Throttle opening degree [°]

Tab.1. List of the LCE model inputs

Fig. 2. Prototype of the linear combustion engine
Inputs of the overall model are identical with adjust-
able parameters of the prototype. The list of input
2 MODEL OF LINEAR COMBUSTION ENGINE parameters is shown in table 1.

The model is proposed as an open thermodynamic Name Description Units

system and describes the thermodynamic, fluid-flow, x Piston position [m]
heat-transfer and combustion phenomena that govern N_inst Instantaneous engine [m/s]
performance aspects of the combustion engine. The speed
model assumes the gas inside the open system AFR Air/Fuel ratio [-]
boundary is uniform in composition and state at each p_man Intake manifold pressure [Pa]
point in time and state and composition change with
T_man Intake manifold [K]
time due to heat transfer, work transfer and mass temperature
flow across the boundary. The basic equations of the
stroke/s Number of strokes per [1/s]
LCE model are mass and energy conservation, with second
time as the independent variable.

Tab. 2. List of the LCE model outputs

Although the model simulates much more parameters

(i.e. pressure and temperature in the cylinders), only
measurable parameters on the linear combustion
engine prototype are taken out of the model. The list
of output variables is shown in table 2. This model
was accepted from Deutsch, P. - Vysoký, O. In-cycle
thermodynamic model of linear combustion engine.

Fig. 3. Model of the linear combustion engine 3 VALIDATION

Without appropriate measurement equipments, it was

The model is composed of seven particular blocks not possible to make identification of LCE parame-
(Figure 3). These blocks of model are consistent with ters. At the beginning of validation were identified
individual parts of prototype. The first block is In- proportions of particular segments of LCE. Some
take block. This block simulates pressure and tem- parameters are based on values from manufactures
perature of the intake manifold. It is followed by (e.g. parameters of linear motor-generator). Then
Cylinder-left and Cylinder-right and outlet block is were measured time behaviours of piston position
Exhaust. The cylinder block simulates development together with pressures in the intake manifold, in the
of mixture, pressure and temperature in the cylinder right scavenging chamber and in the right cylinder.
during overall combustion cycle as well as exhaust These data were then used to identify rest model pa-
gases mass flow rate into the exhaust manifold and rameters.
fresh air mass flow rate from the intake manifold.
The exhaust block simulates pressure and tempera-
ture of the exhaust manifold. The mechanic features
3.1. Maximum volume of scavenging chamber 3.2. Parameters of Wiebe function
This parameter was not measured directly. Maximum The mass fraction burned profile is often represented
volume of scavenging chamber Vsc_MAX was calcu- by the Wiebe function:
lated from time behaviours of piston position x to-
gether with pressure in the scavenging chamber p. ⎡ ⎛ t − t ign ⎞
m +1
⎤
With the assumption of ideal gas and of constant xb = 1 − exp ⎢− a⎜⎜ ⎟⎟ ⎥. (3)
temperature is process in scavenging chamber iso- ⎢⎣ ⎝ Δt ⎠ ⎥⎦
thermal:
This profile is similar to the pressure ratio profile.
p ⋅ V = konst . (1) The pressure ratio is defined using the ratio between
pressure from a firing cycle pf and the pressure from
Immediate volume V is function of piston position: a motored cycle pm:
⎛S ⎞ p f ( x)
Vsc = Vsc _ MAX − A p ⎜ − x ⎟ . (2)
PR( x) = −1. (4)
⎝2 ⎠ p m ( x)
S is a cylinder stroke and Ap is an area of the piston.
Parameters Δt, m and a have been fitted according to
measured pressure ratio. Time behaviors of pressure
from the firing cycle and the pressure from the mo-
120 0,02 tored cycle are painted in figure 6.
115 0,015
2,5
110 0,01

105 0,005 2
p
p [kPa]

x [m]

100 0 x 1,5
p [MPa]

pf(t)
95 -0,005
pm(t)
1
90 -0,01
čas zápalu
85 -0,015 0,5

80 -0,02
0
0 0,01 0,02 0,03 0,04 0,05
0 0,005 0,01 0,015 0,02 0,025 0,03
t [s] t [s]

Fig. 4. Time behaviors of piston position x together

Fig.6. Pressure in the cylinder for firing and mo-
with pressure in the scavenging chamber p
tored cycle
Calculation was made for segment of the time behav-
Mass Fraction burned profile for model and for pro-
iors when passive valve between intake manifold and
totype is shown in the figure 7.
scavenging chamber and the inlet scavenging port
between scavenging chamber and cylinder are 100

closed. This segment is shown in figure 5. 90

80
70
108 0,01
60
106 0,008 prototype
fb [%]

50
0,006 model
104 40
0,004 30
102
0,002 20
p [kPa]

100 p
x [m]

0 10
98 x
-0,002 0
96 0 0,002 0,004 0,006 0,008 0,01
-0,004
t [s]
94 -0,006
92 -0,008
90 -0,01
Fig.7. Mass Fraction burned profile for model and
0 0,002 0,004 0,006 for prototype
t [s]

3.3. Comparison between model and prototype

Fig. 5. Segment of the time behaviors for calcula-
tion volume The comparisons between the measured and simu-
lated pressure in the cylinder and pressure in the
scavenging chamber for corresponding operating
points can you see in figures 8 and 9.
 3,5 m& a
3 m& f
lambda = . (6)
2,5
⎛ m& a ⎞
⎜ ⎟
p [MPa]

2 prototype
⎜ m& ⎟
1,5 model
⎝ f ⎠S
1

0,5
4.2. Lambda sensors
0
0 0,01 0,02 0,03 0,04 There are primarily two types of lambda sensors.
t [s] They both work by measuring the oxygen concentra-
tion in the exhaust gases. The exhaust gas oxygen
Fig. 8. Simulated and measured time behavior of sensor is a discrete sensor with switching point
cylinder pressure during one cycle around the stoichiometric value. The other type of
sensor is the universal exhaust gas oxygen sensor,
120 sometimes called a linear oxygen sensor. The behav-
iour of this sensor is continuous and we can measure
115
an absolute lambda value over the engines whole
110 operating range. Control algorithm is developing for
p [KPa]

105
prototype air fuel ratio control with universal exhaust gas oxy-
model gen sensor. But lambda sensors don’t instantaneously
100 adjust to a change in air fuel mixture. That because
95 of two reasons: first the sensor has a time constant of
about 0.1 s and then one has to consider the transport
90
0 0,01 0,02 0,03 0,04
delay when exhaust gases travel from the cylinder to
the location of the sensor.
t [s]

Fig. 9. Simulated and measured time behavior of 4.3. H-∞ loop-shaping design
scavenging chamber pressure during one cycle This procedure is essentially a two–stage process.
First, the open loop plant is augmented by pre- and
post- compensators to give a desired shape to the
4 DESIGN OF AIR FUEL RATIO CONTROL singular values of the open loop frequency response.
This could be based on an initial controller design.
Algorithm for air fuel ratio control was developing Then the resulting shaped plant (initial loop shape) is
by method H∞ loop-shaping design. Actuating vari- robustly stabilized (“robustified”) with respect to the
able for air fuel control is time duration of prepared quite general class of factor uncertainly using H-∞
fuel. optimization.
Initial loop shape was made by method Root Locus
4.1. Definition of lambda Design. This designed compensator (W) was used as
pre-compensator for robustly stabilization (Ks). G is
The air/fuel ratio is defined as the ratio between air nominal linear model from model LCE.
mass flow and fuel mass flow:

m& a
AFR = . (5)
m& f
Stoichiometry is the optimal mixture of air and fuel
in which, when ignited, all of the carbon and hydro-
gen would completely burn, yielding only carbon and Fig. 10. A practical implementation of the loop
water. The stoichiometric value depends on the qual- shaping design
ity of the gasoline, but is normally between 14.57
and 14.70. The parameter lambda is the AFR normal-
ized with the stoichiometric value which yields: 4.4. Simulations results
Demonstration of lambda control on reference value
and lambda control with various position of ignition
can you see on the figures 11 and 12.
 Skogestad, S. – Postlethwaite, I.. Multivariable feed-
back control: analysis and design. John Wiley
and Sons, 2nd edition, 2005.
Brandstetter, M. Robust Air-Fuel Ratio Control For
Combustion Engines - PhD thesis, University of
Cambridge, 1996.
Němeček, P. Lineární spalovací motor – PhD mini-
mum. Praha, 2004
Fig. 11. Lambda control on reference value Jeřábek, J. Model a řídicí algoritmy pro lineární spa-
lovací motor - diploma thesis. Praha, 2005.

Fig. 12. Lambda control with various position of

ignition

5 CONCLUSION

A nonlinear time-based in-cycle thermodynamic

model of a linear combustion engine was introduced
in this paper. This model has been developed as a
compact system in Matlab/Simulink and can be used
as one component of a more complex system. The
main benefit of the in-cycle thermodynamic model
introduced in this paper is its ability to be an ade-
quate tool for development of modern in-cycle con-
trollers of the LCE. This model and this control will
use for tuning of running LCE.

ACKNOWLEDGEMENTS

This research has been supported within the Re-

search Programme No.MSM6840770038: Decision
making and Control for Manufacturing III by the
Ministry of Education of the Czech Republic. This
research has been supported by the MSMT project
No. 1M0568 Josef Bozek’s Research Center of En-
gine and Automotive Technology II.

6 REFERENCES

Deutsch, P. - Vysoký, O. In-cycle thermodynamic

model of linear combustion engine. In Proceed-
ings of the 2006 IEEE, International Conference
on Control Application [CD-ROM]. Piscataway:
IEEE, 2006, s. 2430-2435. ISBN 0-7803-9796-7.