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International Journal of Automotive Technology, Vol. ?, No. ?, pp. ?−?(year) Copyright © 2000 KSAE
Serial#Given by KSAE
GAUSSIAN PROCESS REGRESSION FEEDFORWARD CONTROLLER
FOR DIESEL ENGINE AIRPATH
Volkan ARAN1,3, Mustafa UNEL2,3*
1) FORD OTOSAN Sancaktepe Engineering Center, Akpınar Mah. Sancaktepe Istanbul Turkey
2) Integrated Manufacturing Technologies Research and Application Center,
Sabanci University, 34956, Istanbul, Turkey.
3) Faculty of Engineering and Natural Sciences, Sabancı University, Istanbul Turkey
(Received date ; Revised date ; Accepted date ) * Please leave blank
ABSTRACT−Gassian Process Regression (GPR) gives emerging modeling opportunities for diesel engine control.
Recent serial production hardwares increases online calculation capabilities of the engine control units. This paper
presents a GPR modeling for feedforward part of the diesel engine airpath controller. A variable geotmetry turbine
(VGT) and an exhaust gas recirculation (EGR) valve outer loop controllers are developed. The GPR feedforward
models are trained with a series of mapping data with physically related inputs instead of speed and torque as in
conventional case. A physical model free and calibratable controller structure is proposed for hardware flexibility.
Furthermore, a discrete time sliding mode controller is utilized as a feedback controller. Feedforward modeling and
the subsequent airpath controller are implemented on the physical diesel engine model.
KEY WORDS:Gaussian process regression, Feedforward control, Discrete time sliding mode control, Airpath
control
NOMENCLATURE
i
P
: Intake manifold pressure
x
P
: Exhaust manifold pressure
c
P
: Compressor power
a
P
: Ambient pressure
R: Ideal gas constant
i
T
: Intake manifold temperature
x
T
: Exhaust manifold temperature
a
T
: Ambient temperature
i
V
: Intake manifold volume
: Turbocharger time constant
ci
W
: Compressor mass airflow
xi
W
:Exhaust gas recirculation mass flow
ie
W
: Engine inlet gas mass flow
xt
W
: Turbine inlet gas mass flow
f
W
: Fuel mass flow
c
: Isentropic compressor efficiency
T
: Turbine total efficiency
c: specific heat of air
ff
u
: feedforward control term
fb
u
: Feedback control component
EGR
Ar
: Exhaust gas recirculation valve area
xt
h
: Exhaust gas enthalpy
1
u
: Controlled input 1, Area of EGR
2
u
: Controlled input 2, Area of VGT
: isentropic ratio
VGT
r
: VGT vane position
EGR
r
:EGR valve position
* Corresponding author. munel@sabanciuniv.edu
Author
1. INTRODUCTION
Emissions control is probably the most challenging part
of the current diesel engine development process.
Tailpipe emissions is a result of aftertreatment and
engine out (feedgas) emissions. Both tailpipe and engine
out emissions are closely related to the engine airpath
control performance. One of the most harmful kind of
exhaust emission gases is nitrogen oxides (NOx).
Exhaust gas recirculation (EGR) system is a major
engine out NOx reduction element in diesel engines
(Heywood 2000). Higher combustion temperature
favours NOx formation. Combustion temperature
reduction requires lower oxygen concentration and
increased gas heat capacity which are mainly achieved
by utilization of the EGR. However, lower temperature
and reduced oxygen concentration boosts formation of
another diesel engine emission type called particulate
matter (PM) that threatens the human health. The
described trade-off emphasizes the importance of
precise airpath control.
Fresh air is pumped to the engine via turbocharger.
Modern diesel engines utilize geometry turbochargers
(VGT) for higher boost build up performance and
optimized pumping loss. Turbocharger harvests the
waste heat after exhaust stroke and uses the energy for
pumping air into the engine. VGT actuator governs the
energy that is being harvested through the turbine and
changes the exhaust manifold and intake manifold
pressures. EGR line gas flow is driven by the pressure
difference between intake and exhaust manifolds and
shares the total exhaust flow with the turbine. As a
result, VGT and EGR systems are closely coupled. Also
the airpath has non-minimum phase behavior which
creates challenge in obtaining inverse models
(Kolmanovsky 1997).
Although selection of output is another line of
research examined by several authors (e.g. Nieuwstadt
et. al. 2000, Wahlstrom and Eriksson 2013), mass air
flow (MAF) through compressor and manifold air
pressure (MAP) are the common selection of controlled
outputs of diesel engine airpath. In the practical
applications, desired values of MAF and MAP signals
are interpolated from predefined (calibrated) tables
whose axes are speed and injected fuel quantity or
desired inner torque. When one neglects low pressure
EGR or multi turbocharger configurations, diesel engine
airpath control problem can be defined as tracking MAF
and MAP desired values via manipulation of EGR and
VGT actuators despite disturbances of other engine
dynamics.
Due to its complex nature, diesel engine airpath has
been an interesting plant for control research for
decades. However, PID control with extensive gain
scheduling structure is the most common in the
industrial application softwares. Since sensors have
inevitable delayed nature and fast tracking is crucial for
engine performance and emissions, feedforward term
plays an important role in the airpath control problem. A
recent airpath feedforward control study is the dynamic
feedforward control with predetermined optimum tables
(Mancini et. al. 2014). This study utilizes speed and fuel
quantity based static feedforward maps and applies an
optimized dynamical correction on them. Suggested
implementation is explicit. Changes in the boundary
conditions such as backpressure and inlet depression is
not taken into the account and main problems of the
static mapping are unresolved while its transients are
improved.
Gaussian process regression (GPR) models are being
used for online inverse modeling of the robotic systems
(Schreiter et. al. 2016). In an automotive application,
inner loop dynamics of the throttle valve is represented
by nonlinear autoregression with exogenous inputs
(NARX) model whose nonlinear part is a GPR
(Bischoff et al. 2014). Diesel engine fuel systems
dynamics are modelled with local gaussian process
regression in (Tietze 2015) for offline model based
calibration. Current generation of an ECU supplier has
an advanced modeling unit in its ECU and online
simulation of GPR models become practical for the
automotive industry. This is a new capability for the
powertrain control development and its application
areas is expected to be broadening.
A calibratable and physical model free control
approach is sought in our work. Singularity free and
accurate inverse model for the airpath is known to be a
hard problem; therefore a data driven inherently smooth
modeling approach is favorable. On the other hand,
mapping feedforward terms with respect to the physical
states rather than operation points makes calibration
procedure robust to the boundary condition variations
such as backpressure. GPR can be seen as a gray-box
modeling procedure since it is physically interpretable
and contains prior information itself instead of a total
abstraction. This nature of the model distinguishes from
other modeling approaches from calibratability point of
view. Authors initiated feasibility study for GPR EGR
inverse model recently (Aran and Unel 2016). However,
it was only a modeling study and control aspects were
not discussed. Current study includes VGT as well, and
develops both a GPR based feedforward controller and a
discrete time sliding mode feedback controller
(Sabanovic et. al. 2003). The controller is preferred
since it does not require computation of equivalent
Author
control. All the modeling and control studies are
realized on a modeling environment called Virtual
Drive (VD). The Virtual Drive was developed and
enhanced based on (Unver et. al. 2016) and became an
inhouse vehicle and powertrain modeling software of
Ford Otosan Powertrain Controls team.
The organization of the text is as follows: Diesel
engine airptha control problem is stated in Section 2,
and the Gaussian process regression feedforward
controller ise developed in Section 3. Discerete time
sliding mode feedback control is provided in Section 4.
Both modeling and control simulations are givenh in
Section 5. Finally, the paper is concluded with several
remarks and possible future directions are indicated.
2. DIESEL ENGINE AIRPATH CONTROL
PROBLEM
A basic diesel engine airpath model is based on ideal
gas law, isentropic compressor work, conservation of
mass and throttle equation for the layout given in Fig. 1.
An engine simulation model requires 12 states to
capture dynamics of the whole engine system (Unver et.
al. 2016). However, airpath models for control can be
constructed with three states (Jankovic and
Kolmanovsky 1998, Jung et. al. 2005) or one can
include a fourth state if the throttle is included.
Figure 1 Airpath schematic of Diesel Engine (Jung et. al.
2005).
Equations (1)-(3) represent a widely used state
equations for the intake manifold pressure
i
P
, exhaust
manifold pressure
x
P
and compressor power
c
P
. MAF,
MAP, EGR position, VGT position and charge air
cooler out gas temperature sensors are generally
available in the modern serial production diesel engines.
i
e
i
iexici
i
i
iP
T
T
WWW
V
RT
P
.
.)( +−+=
(1)
i
x
x
xtxifie
x
x
xP
T
T
WWWW
V
RT
P
.
.)( ++−+=
(2)
)(
1
.
ctc PPP −=
(3)
Assuming constant temperatures (i.e.,
.
i
T
,
.
x
T
are zero)
and following the steps in the literature (Jung et. al.
2005), one can reach the control affine representation of
the form
111
.),(),( uPPbPPfP xicii +=
(4)
23122
.)(),(),( uPbuPPbPPfP xxixix ++=
(5)
243
.)(),,( uPbPPPfP xcxic +=
(6)
where
1
u
and
2
u
are control inputs which are EGR and
VGT valve areas, respectively. If MAF (Wci ) is
selected as one of the controlled outputs, then the output
equation for the MAF can be written as
)( a
i
P
P
airc
c
ci c
P
W=
(7)
In light of (4) and (6), one can obtain the following
state- space form:
uxbxfx )()( +=
(8)
3. FEEDFORWARD CONTROLLER FOR THE
AIRPATH
Engine development process gives the opportunity of
operation region mapping. That means one can obtain
nearly complete prior information of possible operation
points and related inputs. These mappings are done for
steady state operation points and also emission
modeling design of experiments includes almost all
feasible operation zone. If steady mappings, i.e. states
for which
0=x
dt
d
, are available with complete state
and controlled values, then the control effort required to
conserve the measured states are known. Therefore, in
light of (8), the feedforward control can be determined
by setting
0=x
; i.e.
Author
)(
)(
)()(0 xb
xf
ffff uuxbxf −=+=
(9)
Conventionally speed and inner torque based maps are
used in the industry for the estimation of feedforward
term. This study proposes a Gaussian Process
Regression model based on physically related inputs
such as
x
P
,
i
P
and
xi
W
.
Inverse actuator model for EGR (Wahlstrom and
Eriksson 2011) based on normal operation conditions is
given by (10). In this equation
EGR
Ar
represents area of
the EGR valve which is directly related to the EGR
valve position
)( EGR
r
, which is the output of the
inverse actuator model. Obtaining desired accuracy for
the EGR flow requires introduction of further
parameters and their tuning in the aforementioned study.
−
−
−
=2
1
1
1
opt
x
i
i
xxi
EGR
P
P
P
RTW
Ar
(10)
Energy flow from turbine to compressor can be used for
VGT inverse model. Total efficiency for VGT based on
vane position can be defined as (11) using steady state
turbine compressor energy balance.
)(
)(
xxtxt
c
VGTT ThW P
r=
(11)
State equation (3) can be rewritten in terms of efficiency
as in (12).
))()()((
1
.
a
i
P
P
airciVGTTxxtxtccWrThWP −=
(12)
As a result of presented physical modeling, input
channels for the inverse EGR model are selected as
xi PP /
,
i
P
and
xi
W
. VGT inverse model inputs are
ci
W
,
i
P
,
x
T
, respectively, and its output is the VGT
vane position
)( VGT
r
.
3.1. Gaussian Process Regression
It is assumed that the inverse actuator system is a zero
mean Gaussian process model. A Gaussian process is a
collection of random variables, any finite number of
which has a joint Gaussian distribution (Rasmussen and
Williams 2006)). The noise is assumed to be additive
independent and identically distributed, and the ouput y
is feedforward control value. The formulation detailed
in (Rasmussen and Williams 2006) will be followed in
this section.
EGR and VGT channels are seperately modelled in
multi-input single-output (MISO) fashion. Let the
relationship between inputs (
x
:
xi PP /
,
i
P
,
xi
W
;
ci
W
,
i
P
,
x
T
) and the output (y:
VGTEGR rr ,
) be given as:
+= )(xfy
,
),0( 2
N
(13)
Prior covariance on the noisy output observations
i
y
and
j
y
is defined as
ijjiji xxkyy
2
),(),cov( +=
(14)
Covariance function
),( ji xxk
is defined over input
samples
i
x
and
j
x
, and
ij
is the Kronecker delta
function. Definition of
),( ji xxk
for the squared
exponential covariance term is given as
rr
dji
T
exxk 5.0
),( −
=
(15)
where horizontal scale parameter
d
is a scalar and
r
is a scaled input sample given by
T
n
ji
jiji l
xx
l
xx
l
xx
rnn
−
−−
=...
21
2211
(16)
where length scale parameters
j
l
determine the weights
between input channels.
For an experiment of
m
samples, one can construct the
following covariance matrix that will be used in
subsequent analysis:
=
),(.........
...),(......
),(...),(),(
),(
12111
mm
ji
m
xxk
xxk
xxkxxkxxk
XXK
(17)
Author
Length scale “
l
” and horizontal scale “
d
” are the
main parameters of the model and they are called
hyperparameters. These parameters are found by
maximum likelihood estimation. Training values are
used for finding hyperparameters and they are also
embedded into the model through K matrix. The test
values are the simulation inputs, current states in our
case, whose outputs are calculated. Test inputs are
denoted by
*
x
. The covariance vector between
simulation point and the training points is represented
as:
T
m
xxkxxkxxkk ),(...),(),( *2*1** =
(18)
Predicted output
*
y
(uffegr or uffvgt ) is calculated with
(19).
yIKky T12
** )( −
+=
(19)
For efficient simulation equation, (19) can be rewritten
as
T
ky ** =
(20)
where
is a vector of size
m
, and can be calculated as
)\( yLLT
=
(21)
where L is retrieved through cholesky decomposition,
)( 2IKcholeskyL
+=
(22)
Parameter optimization procedure utilizes the following
maximum likelihood cost function
)2log(2/))(log(5.0)|(log
nLtraceyXyp T−−−=
(23)
3.2. Modeling Details
Gaussian process regression requires a space filling
design of experiment (DoE) for the inputs. Test data is
collected with engine mapping simulations by setting
speed and desired torque to grid points and waiting for
10 seconds settling, then averaging values of the last 30
seconds. Test grid of 417 points from the engine
operation region is shown in Figure 2.
Boost delay is the characteristic of the turbocharger
system, therefore mapping tests are repeated with 90%
and 80% of the base calibration MAP values as shown
in Figure 3.
Figure 2. Engine mapping operation points
Figure 3. Three mapping boost values
Training points are selected with a bin logic. Input data
is divided into bins of equal intervals and 3 values (i.e.
minimum, maximum and median of the bin) from each
bin is taken as training samples. A sample bin for EGR
model is shown in Figure 4.
Figure 4. A sample training data selection bin
Author
Training points are selected with the described logic and
rest of the data are left for the validation. Total number
of 179 training samples are selected for VGT and 1252
points are left for validation. EGR modeling required
more training data (i.e. 312 samples for training and
1164 samples for validation) yet resulted in lower
accuracy than the VGT inverse model. Model training is
done with “fitrgp” function in MATLAB. Exact method
is used with squared exponential kernel utilizing auto
relevance determination in the form of (15) and (16).
Hyperparameters
,, d
l
and
are extracted from
“fitrgp” function and simulations are executed using
equation (20).
3. DISCRETE TIME SLIDING MODE
CONTROL
One of the aims of this paper is finding a flexible
architecture in terms of related hardware’s physical
details. Although modelling of airpath is described in
terms of simplified physical equations, this information
is used only for input selection. Similarly extensive use
of physical modelling is avoided in the feedback control
as well. A discrete-time sliding mode controller
developed by (Sabanovic et.al. 2003) is employed in
this work. This controller does not require computations
of equivalent control, and therefore detailed physical
modelling is not necessary. For a control affine system
as given in (4-6), a sliding surface can be defined as
(24). The discrete-time sliding mode control law is
given in (25). Controller sensitivity matrix B is the only
plant related information.
)()( xxCxx refref −+−=
(24)
))1()1(()()1()( 1−+−−−= −tDtGBtutu
(25)
where
D
and
C
are design parameters and
x
G
=
.
The whole control effort consists of GPR feedforward
and sliding mode type feedback controller as depicted in
Fig.7.
Figure 5 Overall control diagram
Simulation results are presented in the next section for
MAF and MAP outputs.
4. RESULTS
Simulation model is a 13L heavy duty diesel engine
model. WHTC (world harmonized test cycle) is the
certification test cycle for dynamometer homologation
of heavy duty diesel engines in Europe (UN ECE 2013).
Thus, WHTC is selected for controller performance
analysis. Normalized speed (n_norm) and torque cycle
(M_n) is presented in Fig. 8.
Figure 6 WHTC in speed (n_norm) and torque
(M_norm) (UN ECE 2013)
Validation results for VGT and EGR valve position
estimations are depicted in Fig. 7 and Fig. 8 respectively.
Although less training samples are used for VGT
inverse model fitting (feedforward for MAP control), its
validation accuracy is higher than EGR inverse model
(feedforward for MAF control).
Figure 7. Validation Fit Results for VGT
Author
Figure 8. Validation Fit Results for EGR
Since there are cross-talks between model based
feedforward term and feedback control, individual
performance of the sliding mode feedback controller is
analyzed first. Base performance of the sliding mode
feedback controller is checked on WHTC cycle as
depicted in Fig.9-10 where a 60 sec section
corresponding to high torque gradients is illustrated.
Figure 9 MAF tracking results for WHTC
It is clear that feecback controller provides satisfactory
performance for MAF channel while MAP control
could be improved by VGT feedforward term. Thus,
feedforward plus sliding mode feedback control for both
MAF and MAP outputs are applied in WHTC
simulation. Results for the same 60 sec section are
depicted in Fig. 11-12. Comparing these results with Fig.
9-10, it is clear that MAP channel performance is
significantly improved; but MAF performance is getting
not better due to small oscillations caused by EGR
feedforward.
Figure 10 MAP tracking results for WHTC
Figure 11 MAF tracking results for WHTC
Figure 12 MAP tracking results for WHTC
In the last simulation, feedforward is used only for MAP
channel through VGT and results are depicted in Figs.
13-14. It can be seen that MAP performance is
improved as before, and MAF tracking is still as good
as Fig. 9.
Author
Figure 13 MAF tracking results for WHTC
Figure 14 MAP tracking results for WHTC
Control efforts for the last simulation are presented in
Fig. 15.
Figure 15 Control efforts (VGT and EGR)
Overall tracking performances of the implemented
controllers with various metrics for the whole WHTC
test cycle which ends in 1800 seconds are tabulated in
Table 1.
Table 1 Results Summary
MAF
MAP
Method
bestfit
rmse
nrmse
R2
bestfit
rmse
nrmse
R2
FB (SMC)
86,86
39,67
0,03
0,98
58,72
236,96
0,10
0,83
FB+FF
(SMC+GPR)
85,63
43,58
0,03
0,98
66,55
187,18
0,08
0,89
FB+FF*
(SMC+GPR)
87,32
38,06
0,03
0,98
66,23
188,98
0,08
0,89
FB (SMC): Sliding mode feedback controller
FF (GPR): Gaussian process regression feedforward
FF*: Feedfoward is used only for VGT.
5. CONCLUSIONS
Gaussian process regression (GPR) models are being
available in real time applications of automotive serial
production hardware. Proposed models require less data
points than previous table based applications.
Feedforward has an important role in delayed systems
such as diesel engine boost (MAP) build up and its
calculation via GPR modeling improves control
performance significantly as can be seen from presented
simulation results. However, MAF control through EGR
does not show a similar characteristic, and therefore a
sliding mode feedback controller is applied without an
EGR feedforward term due to its superior performance
on WHTC.
As a future study, improved GPR models for EGR will
be investigated. GPR models for feedback control will
also be studied.
ACKNOWLEDGEMENT− The authors would like to thank
Ford Otosan Powertrain Control Software team for their
support on the plant model.
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4. Reports and User Guide
UN ECE (2013) E/ECE/324/Rev.1/Add.48/Rev.6-
E/ECE/TRANS/505/Rev.1/Add.48/Rev.6
