m’C
P
Fig. 3. Closed loop control
In the open loop control, the classic control of a diesel engine (Hafner, 2000) is done
according to the diagram in fig. 2, the optimal values of the actuators are updated by
memorized static maps. Then a predictive corrector (Hafner, 2001) is generally used in order
to compensate the engine dynamic effects.
In the closed loop control (fig. 3), the engine is controlled by error signals which are the
difference between the predicted or measured air mass flow rate and the intake pressure,
and their reference values. The controller uses memorized maps as reference, based on
engine steady state optimization (Hafner, 2001; Bai, 2002). The influence of the dynamic
behavior is integrated by several types of controller (PI, robust control with variable
parameters, …) (Jung, 2003).
Our work proposes practical solutions to overcome and outperform the control insufficiency
using static maps. The advantage of this approach is to be able to propose dynamic maps
capable of predicting, “on line”, the in-air cylinders filling. Therefore the optimal static maps
in fig. 1 and 2 can be replaced by optimal dynamic ones.
We suggest a mathematical optimization process based on the mean value engine model to
minimize the total pollutants production and emissions over dynamic courses without
deteriorating the engine performance. We used the opacity as a pollution criterion, this choice
was strictly limited due to the available data, but the process is universal and it can be applied
individually to each pollutant which has physical model or to the all assembled together.
This optimization’s procedure is difficult to be applied directly in “on line” engines’
applications, due to the computation difficulties which are time consuming. Consequently, it
will be used to build up a large database in order to train a neural model which will be used
instead. Neural networks are very efficient in learning the nonlinear relations in complex
multi-variables systems; they are accurate, easy to train and suitable for real time applications.
All the simulations results and figures presented in this section were computed using
Matlab development environment and toolboxes. The following section is divided to four
subsections as follows: I Engine dynamic modeling, II Simulation and validation of the
engine’s model, III Optimization over dynamic trajectories, IV Creation of Neural network
for “on line” controller.
4.1 Engine dynamic modeling
Diesel engines can be modeled in two different ways: The models of knowledge quasi-static
(Winterbonne, 1984), draining-replenishment (Kao, 1995), semi mixed (Ouenou-Gamo, 2001;
Younes, 1993), bond graph (Hassenfolder, 1993), and the models of representation by
transfer functions (Younes, 1993), neural networks (Ouladssine, 2004).
Optimized Method for Real Time Nonlinear Control
173
Seen our optimization objective, the model of knowledge will be adopted in this work. The
semi-mixed model is the simplest analytic approach to be used in an optimization process.
The Diesel engine described here is equipped with a variable geometry turbocharger and
water cooled heat exchanger to cool the hot air exiting the compressor, but it doesn’t have
an exhaust gas recirculation system that is mainly used to reduce the NOx emissions.
Consequently the engine is divided to three main blocks: A. the intake air manifold, B. the
engine block, C. the opacity (Omran, 2008a).
4.1.1 Intake air manifold
Considering air as an ideal gas, the state equation and the mass conservation principle gives [4]:
dP
a
a
V
r. a
T . c
m
ma0 (31)
dt
c
m is the compressor air mass flow rate, a
m 0 is the air mass flow rate entering the engine,
Pa, Va and Ta are respectively the pressure, the volume and the temperature of the air in the
intake manifold and r is the mass constant of the air. m a 0 is given by:
m
a 0
V ma 0, th (32)
m a 0, th is the theoretical air mass flow rate capable of filling the entire cylinders’ volume at
the intake conditions of pressure and temperature:
Vcyl.ω.P
a
m
a0,th
(33)
4 π r a
T
Vcyl is the displacement, ω is the crankshaft angular speed, and ηv is the in-air filling
efficiency given by:
2
(34)
v
η
0
α
1
α ω α2ω
Where αi are constants identified from experimental data. The intake temperature Ta is
expressed by:
a
T
1 eηch
c
T
e
η ch w
T ater (35)
Tc is the temperature of the air at the compressor’s exit. Twater is the temperature of the
cooling water supposed constant. ech is the efficiency of the heat exchanger supposed
constant. The temperature Tc is expressed by:
γ 1
γ
a
P
1
(36)
c
T
0
T 1
1
0
P
c
η
4.1.2 Engine block
The principle of the conservation of energy applied to the crankshaft gives:
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Applications of Nonlinear Control
d 1
J θ 2
ω
e
r (37)
dt 2
J(θ) is the moment of inertia of the engine, it is a periodic function of the crankshaft angle due
to the repeated motion of its pistons and connecting rods, but for simplicity, in this paper, the
inertia is considered constant. Pe is the effective power produced by the combustion process:
η .m .P
e
e
f
ci (38)
m f is the fuel flow rate, Pci is the lower calorific power of fuel and ηe is the effective
efficiency of the engine modeled by [5]:
2
c c λ c λ c λ w
1
2
3
4
e
η
λ
(39)
2
2
2
2
c
5 λ w c6 λ w
7
c λ w
ci are constants, and λ is the coefficient of air excess:
m
a0
λ
(40)
m f
Pr is the resistant power:
r
Crω (41)
Cr is the resistant torque. Fig. 4 represents a comparison between the effective efficiency
model and the experimental data measured on a test bench. The model results are in good
agreement with experimental data.
0.5
Model at 800 RPM
0.45
Model at 1200 RPM
Model at 1600 RPM
0.4
Model at 2000 RPM
Exp. Data at 800 RPM
0.35
Exp. Data at 1200 RPM
Exp. Data at 1600 RPM
0.3
Exp. Data at 2000 RPM
ciency
ffe 0.25
ctive E
0.2
ffeE
0.15
0.1
0.05
00
10
20
30
40
50
60
70
80
90
100
Air Excess
Fig. 4. Comparison between the effective efficiency model results and the experimental data
at different crankshaft angular speed.
Optimized Method for Real Time Nonlinear Control
175
4.1.3 Diesel emissions model
The pollutants that characterize the Diesel engines are mainly the oxides of nitrogen and the
particulate matters. In our work, we are especially interested in the emitted quality of
smokes which is expressed by the measure of opacity (Fig. 5) (Ouenou-Gamo, 2001):
2
m
3
m w
4
m
m5 w
6
m
Opacity m w m
m
1
a
f
(42)
mi are constants identified from the experimental data measured over a test bench.
100
80
60
Opacity
[%] 40
20
2500
0
15 20
2000
25 30 35
1500 Crankshaft Angular
40 45
Speed [rpm]
Air/Fuel Ratio
50 55
1000
60
750
Fig. 5. Graphical representation of the opacity computed using (32) and a constant fuel flow
rate equal to 6 g/s.
4.1.4 System complete model
Reassembling the different blocks’ equations leads to a complete model describing the
functioning and performance of a variable geometry turbocharged Diesel engine. The model
is characterized by two state’s variables ( Pa, w), two inputs ( m f , Cr) and the following two
differential equations representing the dynamic processes:
a
dP
a
V
r a
T c
m
mao
dt
(43)
d 1
2
Jω
e m f
c
P i Cr w
dt 2
4.2 Model validation
The test bench, conceived and used for the experimental study, involves: a 6 cylinders
turbocharged Diesel engine and a brake controlled by the current of Foucault. Engine’s
characteristics are reported in table 2.
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Applications of Nonlinear Control
Stroke [mm]
145
Displacement [cm3]
9839.5
Volumetric ratio
17/1
Bore [mm]
120
Maximum Power [KW]
260
at crankshaft angular speed [rpm]
2400
Maximum torque [daN.m]
158
at crankshaft angular speed [rpm]
1200
Relative pressure of overfeeding [bar]
2
Table 2. Engine Characteristics
Different systems are used to collect and analyze the experimental data in transient phase
and in real time functioning: - Devices for calculating means and instantaneous measures, -
a HC analyzer by flame ionization, - a Bosch smoke detector and - an acquisition device for
signal sampling. The use of these devices improves significantly the quality of the static
measures by integration over a high number of points.
Fig. 6 and 7 show a comparison between two simulations results of the engine complete
model and the experimental data. The inputs of the model are the fuel mass flow rate
and the resistant torque profiles. The output variables are: the pressure of the intake
manifold Pa the crankshaft angular speed ω and the opacity characterizing the engine
pollution. The differential equations described in section 4.1.4 are computed simultaneously
using the Runge-Kutta method. The simulations are in good agreement with the
experimental data.
Inputs data
10
1500
]/s
orque
]1000
t
5
.m
uel [g
[N 500
F
0
esistant
0
50
100
R
00
50
100
t [s]
Simulation results
t [s]
Experimental data
2000
2.5
30
]
re
]
] 2
% 20
ine
rpm [1500
ressu bar
ng
p [
ity [
E
1.5
p a
10
pac
speed
O
1000
Intake
1
0
0
50
100
0
50
100
0
50
100
t [s]
t [s]
t [s]
Fig. 6. Simulation 1: Comparison between the complete engine model and the experimental
data measured on the test bench.
Optimized Method for Real Time Nonlinear Control
177
Inputs data
15
1500
]/s 10
orque
]1000
.m
ant t
uel [g 5
[N 500
F
st
esi
0
R
0
0
100
200
300
0
100
200
300
t [s]
t [s]
Simulation results
Experimental data
2500
2.5
12
]
]
]
ne
rpm
2
% 10
2000
ressure
ngi
bar [
E
ke p p1.a5
acity [ 8
speed [
ta
Op
1500
In
6
0
100
200
300
10
100
200
300
0
100
200
300
t [s]
t [s]
t [s]
Fig. 7. Simulation 2: Comparison between the complete engine model and the experimental
data measured on the test bench.
4.3 Optimization process
4.3.1 Problem description
When conceiving an engine, engines developers have always to confront and solve the
contradictory tasks of producing maximum power (or minimum fuel consumption) while
respecting several pollution’s constraints (European emissions standard). We are only
interesting in reducing the pollutants emissions at the engine level, by applying the optimal
“in-air cylinders filling”. Consequently, the problem can now be defined; it consists in the
following objective multi-criteria function:
Maximize "Power"
(44)
Minimize "Pollutants"
This multi-objective optimization problem can be replaced by a single, non dimensional,
mathematical function regrouping the two previous criteria:
Poll
i
f
dt
dt
(45)
max
i Po
i
ll ,max
P is the engine effective power, Polli is a type of pollutant, and the indication max
characterizes the maximum value that a variable can reach. The integral represents the heap
of the pollutants and power over a given dynamic trajectory. This trajectory can be, as an
example, a part of the New European Driving Cycle (NEDC).
In this chapter we will only use the opacity as an indication of pollution seen the simplicity
of the model and the priority given to the presentation of the method, but we should note
that the optimization process is universal and it can involve as many pollution’s criteria as
we want. The function "objective" becomes:
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Applications of Nonlinear Control
Op
f
dt
dt
(46)
max
ma
Op x
4.3.2 Formulation of the problem
The problem consists therefore in minimizing the following function "objective" over a
definite working interval [0, t]:
ci
P
e mf dt
P
max
f
(47)
1
m
2
m
3
m w m 4
5
m w
6
m
w
m
a
mf
dt