A Generalized Compressor Power Model for Turbocharged ... Articles/A... · W fdC wt w( outout inin)...
Transcript of A Generalized Compressor Power Model for Turbocharged ... Articles/A... · W fdC wt w( outout inin)...
Tao Zeng Mechanical Engineering
Michigan State University East Lansing, Michigan 48824
Email: [email protected]
Devesh Upadhyay and Harold Sun Ford Motor Company
Dearborn, Michigan 48124 Email: [email protected]
Guoming G. Zhu Mechanical Engineering
Michigan State University East Lansing, Michigan 48824
Email: [email protected]
ABSTRACT Control-oriented models of automotive turbocharger
compressors typically describe the compressor power assumming
an isentropic thermodynamic process with a fixed isentropic
efficiency and a fixed mechanical efficiency for power
transmission between the turbine and compressor. Although these
simplifications make the control model tractable, they also
introduce additional errors due to unmodeled dynamics,
especially when the turbocharger is operated outside its normal
operational region. This is also true for map-based approaches,
since these supplier-provided maps tend to be sparse or
incomplete at the boundary operational regions and often ad-hoc
extrapolation is required, leading to large modeling error.
Furthermore, these compressor maps are obtained from the
steady flow bench tests, which introduce additional errors under
pulsating flow conditions in the context of internal combustion
engines. In this paper, a physics-based model of compressor
power is developed using Euler equations for turbomachinery,
where the mass flow rate and compressor rotational speed are
used as model inputs. Two new coefficients, speed and power
coefficients are defined. This makes it possible to directly estimate
the compressor power over the entire compressor operating range
based on a single analytic relationship. The proposed modeling
approach is validated against test data from standard
turbocharger flow bench, steady state engine dynamometer as
well as transient simulation tests. The validation results show that
the proposed model has adequate accuracy for model-based
control design and also reduces the dimension of the parameter
space typically needed to model the compressor dynamics.
NOMENCLATURE P : Pressure [N/m
2]
T : Temperature [K]
W : Work [J]
m : Mass flow rate [kg/sec]
: Torque [Nm]
: Compressor angular velocity [rad/sec]
: Gas density[kg/m3]
: Universal gas constant[J kg−1
K−1
]
A: Geometric area [m2]
: Isentropic index
Subscripts:
1, in: represent the compressor upstream or inlet condition.
2, out: represent the compressor downstream or outlet
condition.
Symbols not in this list are defined locally in the text.
INTRODUCTION It is common for plant models, used in the air path control of
turbocharged (TC) diesel engines, to assume that the power
consumed by the compressor is an isentropic thermodynamic
process. The actual power is then derived from either a map-based
compressor isentropic efficiency [1]-[3] or an empirically fitted
isentropic efficiency [6],[8],[9]. A common alternative approach
is to define the compressor power as a first order dynamic with an
ad-hoc time constant with the turbine power as the source term [4],
[5]. Map-based approaches of compressor power, on the other
hand, rely on the overall TC system efficiency available as a series
(for various vane positions) of supplier provided maps [7] applied
to the calculated turbine power. Alternately empirically fitted
compressor efficiencies are typically regressions as a 2nd
or 3rd
order polynomial in the Blade Speed Ratio (BSR). The
polynomial coefficients are often dependent on the shaft speed
[8]-[11] and are identified against the populated regions of the
manufacturer supplied maps. These polynomial models are,
A GENERALIZED COMPRESSOR POWER MODEL FOR TURBOCHARGED INTERNAL COMBUSTION ENGINES WITH REDUCED COMPLEXITY
Proceedings of the ASME 2016 Dynamic Systems and Control Conference DSCC2016
October 12-14, 2016, Minneapolis, Minnesota, USA
DSCC2016-9792
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therefore, also subject to extrapolation for operating conditions
beyond the manufacturer supplied test points. Note that the typical
manufacturer provided compressor performance map is based on
hot gas flow bench test data. These tests are performed under
steady flow conditions, hence, practical applications could deviate
from these maps under pulsating flow when coupled to an Internal
Combustion (IC) engine [1], [2], [13]. Another issue, often
encountered when operating at light load conditions, is the
sparsity of manufacturer provided compressor performance maps.
This is as indicated in Figure 1, for a sample turbocharged engine.
As a result, operating the compressor outside the available map
requires extrapolation under the assumption of a convex hull
extension of the supplied map. Several investigations into
extrapolation methods, based on these assumptions are reported in
open literature as in [14], [15].
The traditional compressor power is computed from the
standard isentropic efficiency equation in (1) based on the inputs
as indicated in Figure 2.
Figure 1. Operating range deficit between mapped and desired engine operational
ranges
11
1
air
air
in
outairpinis
C
CP
PcTmW
(1)
Figure 2. Traditional approach for deriving compressor power
In (1), the compressor isentropic efficiency must be defined.
In a review of existing literature, compressor isentropic efficiency
is commonly modeled using empirical fits based on manufacturer
provided compressor maps. In [9], the efficiency is expressed as a
polynomial function of the non-dimensional mass flow rate. The
polynomial coefficients are three individual functions of Mach
number. In [26] the efficiency is modeled using an elliptical fit
based on mass flow rate and pressure ratio. This model depends on
the maximum efficiency and the location in terms of corrected
mass flow rate and compressor pressure ratio. In [23], the
efficiency model form [26] is further modified for pressure ratio
variation. In [25] the corrected compressor work is defined as a
polynomial function of corrected mass flow rate and the model is
further fitted with experimental data. TABLE 1 summarizes these
approaches and our assessment of these models for performance
in the interior as well as the exterior of the supplier maps.
TABLE 1. Comparison among different modelling approaches
Efficiency prediction
method
Performance
within the
mapped area
Performance
outside the
mapped area
Mapping Maps provided by
manufacturer Good No data
Compressor
Models
based on
empirical
fitting
Jensen et al. [9] Good
Subject to
extrapolation
methods and
base map
granularity.
General
performance
varied form
average to poor.
Guzzella-Amstutz [27] Good
Kolmanovsky [10] Good
Andersson [23] Good
Canova et al. [24],[25] Good
Eriksson et al. [26] Good
Sieros et al.: Simple
linear [28] Good
Sieros et al.: Generalized
linear 1 [28] Good
Sieros et al.: Generalized
linear 2 [28] Good
In this paper, the compressor power model is developed based
on the Euler equations for turbomachinery. The model uses the
mass flow rate and compressor speed as inputs. It is found that
with the proposed approach a quadratic analytic function of two
parameters, the speed and power coefficients, is adequate to
characetrize the compressor performance over the entire operating
range. The model is validated using hot gas flow bench test data,
steady-state engine dynamometer test data, and transient
simulation data. Results indicate that the proposed reduced order
model is suitable for both steady state, pulsation flow conditions
as well as transient operation.
The paper is organized as follows. Section II discusses the
generalized compressor model based on the Euler equations with
different slip factors and Section III provides validation results
from turbocharger flow bench, turbocharged engine dynamometer
and transient simulation data. The last section adds some
conclusions.
GENERALIZED COMPRESSOR POWER MODELING
Compressor power model based on turbomachinery Euler equation
A typical centrifugal compressor geometry layout is shown in
Figure 3, and the velocity triangles of the gas flow at the
compressor impeller inlet and outlet are shown in Figure 4, and
are adapted from [1], [17]. Using the Euler equations the
compressor power can be expressed as:
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1122 CUCUmrvrvmW ininoutoutC (2)
For the centrifugal impeller, it is assumed that the air enters
the impeller eye in the axial direction so that the initial angular
momentum of the air at the inlet of the compressor can be assumed
to be zero ( 011 CU ) [1], [17]. The ideal compressor power
equation can then be reduced to:
22_ CUmW idealC (3)
or equivalently:
22_ CRmW idealC (4)
Ideally, the flow exits the blade with a back sweep angle 2
with some slip. The theoretical absolute flow velocity can be
expressed as
222,2 tan rCUC (5)
where, 2rC is impeller outlet flow radial velocity. Neglecting slip
loss for the radial direction, the ratio between 2C and 2C in a
impeller is defined as slip factor as shown in (6); The slip factor
depends on a number of factors such as the number of impeller
blades, the passage geometry, the impeller eye tip to impeller exit
diameter ratio, the mass flow rate as well as compressor speed
[17].
2
2
C
C
(6)
With the slip factor the compressor power equation (4) becomes:
222222 tan_
rCURmCRmWidealC
(7)
For simplicity, 12 rr CC is assumed, where 1rC is impeller inlet
radial velocity. This leads to (constant inlet density assumption)
112212
A
m
A
mCC rr
(8)
Substituting (8) into (7), the compressor power can be formulated
as
2
11
22222
tan_
mA
RmRW
idealC
(9)
As a result of flow and ‘windage’ losses, the actual required
input work is greater than the theoretical value necessary to
achieve the target flow rate [33]. To account for this, power loss
factor and friction loss term fk are introduced to (10):
fC WWWidealC
_
(10)
In this paper, is assumed to be constant. Based on the study
from [17] and [34], the friction loss for impeller and diffuser are:
33 mkkmkW fdfiff (11)
where fik and fdk are friction constants for impeller and diffuser,
respectively. Including the friction losses, the actual desired
compressor power can now be written as:
32
11
22222
tanmkm
A
RmRW fC
(12)
Now, define two new parameters: the power coefficient
PowerC and speed coefficient SpeedC as below:
3m
WC C
Power
(13)
m
CSpeed
(14)
From (12), (13) and (14), the new compressor power can be
expressed in terms of these two coefficients as a quadratic
function:
fSpeedSpeedPower kCA
RCRC
11
22222
tan
(15)
Figure 3. Centrifugal compressor layout
Figure 4. Velocity triangles of a centrifugal compressor at the rotor inlet and outlet
Figure 5. Expected relationship between the speed and power coefficients from the
equation (14), n is different compressor operating point
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Expected variation of Cpower with Cspeed based on the
relationship in (15) is shown in Figure 5. The range of this
variation will be influenced by variation in one or more of the
operating/design parameters as indicated in Figure 5. Different
compressor designs will result in different impeller wheel
diameters, power loss factor, slip factor, and exit area. For a given
compressor design and inlet condition, the inlet gas density 1 as
well as the parameters and fk may be assumed to be constant.
The power coefficient, as in (15), can then be expressed compactly
as a function of the speed coefficient and the slip factor for a given
operating condition, i.
, n,, i CfC iiSpeediPower 21 , ,__ (16)
Equation (16) indicates that the power coefficient is a function of
speed coefficient and the slip factor for a given compressor
design. One of the goals of this work was to investigate the
prediction capabilities of the power and speed coefficients in
conjunction with an appropriate slip factor model such that the
dimension of the parameter space in (16) may be reduced from
two to one is indicated in (17).
SpeedPower CfC (17)
Equation (18) shows the compressor power models in terms
of the expanded forms of Cpower and Cspeed. It is clear that in order
to achieve a form as in (17) the slip factor must be considered
either as a parameter fixed by design or defined in terms of the
speed and/or power coefficient to maintain the homogeneity of the
compressor power model with respect to Cpower and Cspeed. The
effectiveness of such approximations will therefore need to be
evaluated.
fC k
mA
R
mR
m
W
11
222
223
tan (18)
Slip factor investigation
Several slip factors models are readily available in open
literature [17, 22, 29-32]. The slip factor typically depends on the
compressor design parameters as well as the operating conditions,
such as: compressor rotational speed and mass flow rate. This led
to the development of several empirical slip factor models as in
[17, 22, 29-32]. In order to maintain the impact of flow conditions
on slip, non-constant slip factor candidate models, as proposed in
[22], [29], and [32] are investigated in this work:
Slip 1: 10 e-1*0.5-tan
12cos
-2
211
2
Z
RA
m (Reffstrup) (19)
Slip 2:
10
tan
cos/1
211
2
22
A
mR
RZ
(Stodola)
(20)
Slip 3: 10132
R
ma
(Stahler) (21)
where a is a design parameter in (21) related to the flow exit
angle and is a constant for a given impeller design. Z is the number
of impeller blades. Integrating these three slip models into the
proposed compressor power model (15), a generalized compressor
power model can be established as follows:
322
1 SpeedSpeedPower CCC (22)
where coefficients 1 , 2 , and 3 depend on the selected slip
factor model and are defined in TABLE 2. The model (22) is
referred to as the generalized compressor power model in the rest
of this paper. Note that the three coefficients in (22) take constant
values and are fixed by compressor design under the assumption
of constant or slowly varying inlet conditions. Using the
definitions of speed and power coefficients, the proposed
compressor power model (22) becomes:
32
2
13
mmm
WC
(23)
Equations (22) and (23) are the proposed analytical models
for the compressor power. The proposed model shows that the
compressor operation can be represented as an analytic quadratic
function rather than a two dimensional map. The proposed model
has only two inputs: mass flow rate and TC rotational speed. For
practical applications, these two parameters are available as
measurements. Alternately the speed can also be obtained by
solving turbocharger rotor dynamic differential equations as in
[3]-[5], and the compressor mass flow rate could be modeled with
shaft speed and pressure ratio as inputs. In this paper, the
compressor power model is validated under the former
assumption of the two parameters are available as measurements.
TABLE 2. Model coefficients for various slip models
Sli
p m
odel
1
1
2cos-2
22 0.5e.50
ZR
2
5.00.5etan 2cos
-2
11
22
Z
A
R
3
2
11
2tan
Ak f
Sli
p m
odel
2
1 )cos/1( 222 ZR
2 11
22 tan
A
R
3 fk
Sli
p m
odel
3 1 2
2R
2
11
22
2
tan
A
R
R
a
3 fkRA
a
211
2tan
MODEL VALIDATION
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Model validation using standard flow bench test data
Data from standard hot gas flow bench tests for three different
production compressors was used to verify the proposed model
and the relationship between Cpower and Cspeed The test setup for
generating these data sets can be found in [12]. The inlet
conditions of compressors are considered as fixed [21], which
agrees with the constant inlet condition assumption used here. The
test matrix for the three different compressors is shown in Figure
6. The minimum test speed for Compressor-1 and Compressor-2
are 30,000 RPM, while the lowest speed for Compressor-3 is
46,000 RPM.
Power and speed coefficients are calculated based on (13) and
(14) for the test data points as in Figure 6. The values are shown in
Fig 7 (labeled as “compressor-i test data”) for the three different
compressor designs considered in this work. The Cpower vs Cspeed
characteristic curves for each compressor can be characterized as
a unique quadratic function. This representation is an important
and significant order reduction when considering the large
two-dimensional efficiency maps typically used for defining
compressor power [8],[11],[23],[25]-[27]. In order to identify the
model (22), a fitting cost function is defined as follows:
n
i
iWiWJ
1
2
measmodel
(24)
where n is the number of steady state compressor operating points
from hot gas flow bench test results; modelW is the calculated
compressor power using (23); and measW is the standard
compressor power calculated using (25) with measured inputs:
inoutpmeas TTcmW (25)
Figure 6. Turbo flow bench test ranges for 3 different compressors
The design parameters for Compressor-1 were available from
the supplier. The first step was to identify the power loss
coefficient and friction loss power for each slip model candidate
for Compressor-1. A Least-Squares optimization was used to
identify , fk and a such that the cost function (24) is
minimized.
The identified parameters are shown in TABLE 3. The
identified power loss parameters ( , fk and a ) are in a
reasonable range, indicating that compressor input power
necessary to achieve the desired flow rate must be larger than the
ideal power calculated in (4). The identified parameter a in Slip
Model-3 agrees with the result from [29]. The maximum friction
loss power identified was around 2.38kW over the entire
compressor operating range. Based on the fitting results in
TABLE 3. , the Slip Model-3 offers the best fit (lest error). The
identified model for Compressor-1 using the Slip-Model 3 is
shown in Figure 7. Fitting results confirm that the model structure
proposed in (22) can be generalized for modeling centrifugal
compressor power.
TABLE 3. Identified model coefficient for Compressor-1with different slip factor
models
fk a R2
Slip model 1 1.295 19040 - 0.968
Slip model 2 1.323 6374 - 0.968
Slip model 3 1.351 11390 1.334 0.989
Since compressor design parameters were not available for
the other two compressors (Compressor-2 and Compressor-3), we
used the generalized model structure (22) to identify the 3
parameters 1 , 2 and 3 directly and the identified values are
shown in TABLE 4. It is interesting to note from Figure 7 that the
three different compressor designs lead to three distinct
characteristic curves. This is expected and reflects the design
differences between the different compressors. In fact this
representation also allows a quick and easy method to compare
different compressor designs. As an example, it is clear that for a
given mass flow rate and TC shaft speed the compressor power
requirement follows the trend, Power1 > Power2 > Power3,
indicating that Design-1 may perhaps have a larger loss relative to
the other designs. Also note that it appears that Compressor-3 may
offer the widest operating range of the three designs considered
here.
TABLE 4. Identified model coefficient for compressor-2 and compressor-3 with
model structure (22)
1 2 3 R2
Compressor 2 0.001762 10.15 15600 0.995
Compressor 3 0.001179 0.01179 13900 0.996
Logarithmic scale plots offer new insights. The Log scale
plots for the three compressor designs are shown in Figure 7, and
are represented by three corresponding straight lines. This follows
directly from the Log scale representation of the generalized
model as (26)
qCpC SpeedPower logloglog (26)
In (26), p is the slope of each characteristic line and q is the y-
intercept for each design. Equation (26) can also be formulated as:
pSpeed
qPower CC 10 (27)
or
2 4 6 8 10 12 14 16
x 104
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
TC Speed [RPM]
Com
pre
ssor
ma
ss f
low
ra
te[k
g/s
]
Compressor-1
Compressor-2
Compressor-3
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pqC
mm
W
10
3 (28)
Using the same fitting method as described in the previous
section, identified p and q and the associated results are shown in
TABLE 5. One important advantage of the straight line model is
that (27) can, ideally, be obtained from only two compressor
operating conditions (one in high load, one in low load). This
method significantly reduces the experimental burden during the
early stages of compressor development.
Figure 7. Model validation for three different compressors
TABLE 5. Identified model coefficient for three compressor with (27)
p q R2
Compressor 1 2.51 5.795*10^(-6) 0.994
Compressor 2 2.50 4.769*10^(-6) 0.989
Compressor 3 2.36 1.375*10^(-5) 0.996
In summary, model validation results confirm the
effectiveness of generalized compressor power structure proposed
in this work. These results also demonstrate that for a centrifugal
compressor the operating characteristics can be reduced to a single
analytic quadratic function in linear scale and a straight line in
logarithmic scale.
Model validation based on steady state engine dynamometer test data
In this section, Compressor-1, identified earlier, was tested on
the engine dynamometer with a heavy duty turbocharged diesel
engine. Detailed engine and test setup information can be found in
[19] and [20]. The steady state test covers the whole engine
operating range (185 testing points). The engine speed changes
from 500 to 3,500 RPM. The engine brake torque varies from 10
to 1200 Nm. The compressor operating range (corresponding to
each engine operating point) results in a mass flow rate between
0.01 and 0.45 kg/s and a compressor speed between 5.9k and 109k
RPM. The compressor operating range (above 30,000 RPM) from
the engine dynamometer tests is a subset of the standard
turbocharger flow bench test data for the same operating range as
shown in Figure 8.
Figure 8. Test ranges of TC flow bench and engine dyno tests
Figure 9. Comparison of the standard TC flow bench and engine steady state
dynamometer test data for Compressor-1
The test bench data set was limited to 30kRPM TC speed at
the low end. Validation results in Figure 9 show the Cpower vs Cspeed
curves for the engine data, the flow bench test data and the fitted
model (based on the test bench data). Since the engine data set
extended to lower load conditions (TC RPM < 30k), the
0 0.5 1 1.5 2 2.5
x 106
0
1
2
3
4
5
6
7x 10
4
Speed coeff
Pow
er
co
eff
Compressor-1 test data
Compressor-1 model
Compressor-2 test data
Compressor-2 model
Compressor-3 test data
Compressor-3 model
106
102
103
104
105
Speed coeff
Pow
er
co
eff
Compressor-1 test data
Compressor-1 model
Compressor-2 test data
Compressor-2 model
Compressor-3 test data
Compressor-3 model
0 2 4 6 8 10 12 14
x 104
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
TC speed [RPM]
Com
pre
ssor
ma
ss f
low
ra
te [
kg/s
]
TC flow bench test
Engine steady state dyno test
0 2 4 6 8 10 12
x 105
0
1
2
3
4
5
6
7x 10
4
Speed coeff
Pow
er
co
eff
TC Flow bench test
Engine steady state dyno(above 30000 RPM)
Engine steady state dyno(below 30000 RPM)
Model
106
102
103
104
105
Speed coeff
Pow
er
co
eff
TC Flow bench test
Engine steady state dyno(above 30000 RPM)
Engine steady state dyno(below 30000 RPM)
Model
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characteristic curves for the engine data are plotted in two sets to
cover both operating regimes around the 30k RPM TC speed. It is
clear from Figure 9 that while the model adequately reproduces
the engine performance for the operating condition with TC speed
> 30K RPM, there is significant error for operating conditions
with TC speed < 30kRPM. These operating conditions are typified
by low mass flow rates and investigations reveal that the
measurements available from the engine at these low load
conditions suffer from severe variability and are therefore quite
suspect. An additional source of error could be related to the heat
transfer effect from the hot end (turbine) adding a positive bias to
the compressor outlet temperature. This effect of heat transfer on
compressor efficiency under light load operating conditions was
verified by experimentally and reported in [35], [36]. The higher
compressor outlet temperature leads to a higher calculated
compressor power (based on measurements), which makes the
measured power coefficient (below 30,000RPM) to be higher than
the model predicted value, as is observed in Figure 9. This is also
confirmed when considering transient test data from a GT-Power
based engine simulation as discussed below.
. Figure 10. Transient simulation validation
Transient Model Validation through GT-Power simulations
In order to verify the behavior of the proposed reduced order
model, transient engine simulation data is used in this section.
The main reason for using data from engine simulation was to
avoid measurement errors and biases, as discussed earlier.
GT-power engine simulation results used in this section are based
on the same engine with Compressor-1 connected to a variable
geometry turbocharger (VGT). The GT-power simulation setup
and model validation can be found in [20]. In this transient
simulation, the engine follows the target torque based on the pedal
position. The VGT vane feedback control is used to track the
target boost pressure generated from calibrated maps. A US06
driving cycle was studied for this case as shown in Figure 10.
Compressor speed varies from 14,000 to 105,000 RPM for this
transient simulation. The relationship between the power and
speed coefficients shows reasonable agreement between data and
model prediction for Compressor-1 with Slip Model 3.
CONCLUSION A compressor power model, based on the Euler
turbomachinery equations and realistic assumptions was
developed. Two new performance coefficients, the power and
speed coefficients were proposed as an alternative to multiple
performance maps. The proposed correlation between Cpower and
Cspeed is especially useful in defining the compressor power
necessary for regulating a desired compressor mass flow rate.
This can then be translated into a VGT vane position or the assist
demand in assisted boosting systems. This relationship can also be
easily used to compare compressor design variants and
compressor power requirements. The model is validated against
data sets from standard turbocharger flow bench tests, steady-state
engine dynamometer tests as well as transient engine simulations.
Validation results indicate that the proposed model provides
accurate compressor power prediction over a broad range of
compressor operating conditions and provides for an easy and
reliable extrapolation for operating conditions outside the
standard mapping domain. Further, the proposed model reduces
the dimensionality of the parameter space typically necessary for
such applications. The reduced order, reduced complexity model
is especially useful for the control applications. Future work will
focus on improving prediction accuracy in the face of
measurement noise as previously discussed.
REFERENCES [1] Zinner K., Supercharging the Internal Combustion Engine,
Berlin, Springer, 1978.
[2] Watson N., and Banisoleiman K., “A variable-geometry
turbocharger control system for high output diesel engines,”
SAE Technical Paper 880118, 1988.
[3] Kolmanovsky I., et al. “Issues in modelling and control of
intake flow in variable geometry turbocharged
engines,” Chapman and Hall CRC research notes in
mathematics, 1999, pp. 436-4.
[4] Upadhyay D., Utkin V. I., and Rizzoni G., “Multivariable
control design for intake flow regulation of a diesel engine
using sliding mode,” Proceedings of IFAC 15th Triennial
World Congress, 2013.
[5] Jankovic M. and Kolmanovsky I., “Robust nonlinear
controller for turbocharged diesel engines,” The proceedings
of 1998 American Control Conference, 1998.
[6] Wahlström J., Eriksson L., and Nielsen L., “EGR-VGT
control and tuning for pumping work minimization and
emission control,” IEEE Transactions on Control System
Technology, 18(4) 2010, pp. 993-1003.
0 100 200 300 400 500 6000
2
4
6
8
10
12x 10
4
Time [s]
Turb
och
arg
er
Sp
eed
[R
PM
]
Turbocharger speed
2 3 4 5 6 7 8 9
x 104
0
5000
10000
15000
Speed coeff
Pow
er
co
eff
GT simulation
Model
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