Control Engineering Practice 21 (2013) 1007–1019
Contents lists available at SciVerse ScienceDirect
Control Engineering Practice
journal homepage: www.elsevier.com/locate/conengprac
Spark ignition engine control strategies for minimising cold start fuel
consumption under cumulative tailpipe emissions constraints
D.I. Andrianov n, C. Manzie, M.J. Brear
Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia
art ic l e i nf o
a b s t r a c t
Article history:
Received 6 March 2012
Accepted 14 March 2013
Available online 11 May 2013
This paper proposes a methodology for minimising the fuel consumption of a gasoline fuelled vehicle
during cold starting. It first takes a validated dynamic model of an engine and its aftertreatment reported
in a previous study (Andrianov, Brear, & Manzie, 2012) to identify optimised engine control strategies
using iterative dynamic programming. This is demonstrated on a family of optimisation problems, in
which fuel consumption is minimised subject to different tailpipe emissions constraints and exhaust
system designs. Potential benefits of using multi-parameter optimisation, involving spark timing, air–fuel
ratio and cam timing, are quantified. Single switching control policies are then proposed that perform
close to the optimised strategies obtained from the dynamic programming but which require far less
computational effort.
& 2013 Elsevier Ltd. All rights reserved.
Keywords:
Engine control
Multiple-criterion optimisation
Multivariable control
Optimal trajectory
Dynamic programming
Validation
1. Introduction
Road vehicles with internal combustion engines are a significant source of air pollution (Seinfeld, 2004). The pollutants found
in the exhaust include carbon monoxide (CO), nitrogen oxides (NO
and NO2, also referred to as NOX), unburned hydrocarbons (HC)
and particulates. These substances present significant environmental and health risks, and are therefore regulated. To improve
the air quality and account for the increasing number of road
vehicles, the allowable emissions limits are continually tightened.
In most vehicles with gasoline fuelled engines, these emissions
limits are achieved with the use of three-way catalysts in the
exhaust system, designed to convert engine-out CO, NOX and HC
emissions to CO2, H2 O and N2. However, the chemical processes
involved depend strongly on the catalyst temperature. Whilst the
conversion efficiency of a hot catalyst can be high, a cool catalyst
performs poorly. Consequently, cold start emissions play a critical
role in meeting emissions standards.
Engine control is a cost effective approach to limit cold start
emissions, whilst avoiding the need for additional or upgraded
hardware. A common strategy is to retard the spark timing. This
enables more heat to be rejected into the exhaust, which heats the
catalyst more quickly. Another approach is to raise the engine's
idle speed to produce an increased number of combustion events,
and thus higher enthalpy input to the catalyst. Both of these
approaches, however, result in increased fuel consumption. More
n
Corresponding author. Tel.: + 79111608224.
E-mail address: d-andrianov@ya.ru (D.I. Andrianov).
0967-0661/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.conengprac.2013.03.011
generally, maximising vehicle fuel economy whilst meeting emissions standards is a key and ongoing problem faced by all car
manufacturers.
Manufacturers have traditionally relied heavily on experimentation to identify engine control set-points during cold starting
(the work of Dohner, 1978 is an early example). However, since
every cold start test must be followed by a cooling period, these
approaches are both time consuming and costly. To reduce both
the amount and duration of the testing required, a variety of
model-based methods have been proposed. Some of these do not
directly consider tailpipe emissions (Benz, Hehn, Onder, &
Guzzella, 2011; Keynejad & Manzie, 2011b; Sanketi, Zavala, &
Hedrick, 2006; Shaw & Hedrick, 2003; Sun & Sivashankar, 1997),
whilst other methodologies make use of black-box (e.g. Cohen,
Randall, Tether, VanVoorhies, & Tennant, 1984) or phenomenological (e.g. Kang, Kolmanovsky, & Grizzle, 2001; Kolmanovsky,
Siverguina, & Lygoe, 2002; Kum, Peng, & Bucknor, 2011) models.
However, indirect consideration of tailpipe emissions in the
optimisation can yield inaccurate or misleading results. Furthermore, use of black-box and phenomenological models can be
impractical, as a significant amount of engine testing can be
required for their calibration.
This paper takes a different approach. The minimisation of fuel
consumption under cumulative tailpipe emissions constraints is
viewed as a dynamic optimisation problem, involving a computationally practical and validated cold start model of a spark ignition
engine, an exhaust system and a three-way catalyst (Andrianov
et al., 2012). In contrast to other numerical optimisation
approaches, which use black-box or phenomenological models of
similar functionality (e.g. Bérard, Cotta, Stokes, Thring, & Wheals,
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D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
ϑint
ϑovlp
Nomenclature
Variables
eX
_ cyl
m
_ fuel
m
_ X;out
m
N
ncat
p
T
tf
tsw
u
uc
x
z
α
λ
τbrake
θ
θadv
ϑexh
normalised emissions (mol X/kg fuel)
exhaust mass flow rate (kg/s)
fuel mass flow rate (kg/s)
mass flow rate of tailpipe emissions X (kg/s)
engine speed (rad/s)
number of nodes in the catalyst model
pressure (Pa)
temperature (K)
final time constant (s)
switching time (s)
integrated model input vector
engine control vector
integrated model state vector
integrated model algebraic vector
throttle angle (deg)
normalised air–fuel ratio
brake engine torque (N m)
spark timing (CAD BTDC)
spark advance from MBT spark timing (CAD)
exhaust valve closing angle (CAD)
2000; Cohen et al., 1984; Fiengo, Glielmo, Santini, & Serra, 2002;
Fussey, Goodfellow, Oversby, Porter, & Wheals, 2001; Sanketi et al.,
2006), this work takes advantage of physics-based modelling to
significantly reduce the amount of engine testing required. In the
following sections the methodology for obtaining optimised
engine control strategies is demonstrated on several examples,
where spark timing, air–fuel ratio and cam timing are subject to
optimisation. The control policies found are then compared and
validated experimentally when possible.
intake valve closing angle (CAD)
valve overlap (CAD)
Subscripts
cp
cyl
em
eng
g
idp
im
in
max
MBT
min
out
connecting pipe
exhaust port gas conditions
exhaust manifold
lumped engine conditions
gaseous phase
result of iterative dynamic programming
intake manifold
gas conditions at the inlet
maximum value
maximum brake torque
minimum value
gas conditions at the outlet
Superscripts
ref
⋆
reference
optimal, optimised
The validated model of Andrianov et al. (2012) includes the
appropriate control setpoint to fuel consumption and tailpipe
emissions functionality. The accuracy in simulating cumulative
fuel consumption and tailpipe emissions under transient driving
conditions is of order 2% and 10% respectively with respect to
experimental results. In this section this model is first briefly
described, and then the optimal engine control problem is
formulated.
2.1. The integrated model (Andrianov et al., 2012)
2. Problem formulation
To enable the dynamic optimisation, a computationally practical model capable of simulating fuel consumption and legislated
tailpipe emissions as a function of the engine control setpoints is
required. Throughout this work it is assumed that perfect setpoint
controllers are in place to deliver the developed trajectories.
The structure of the model used is shown in Fig. 1. The engine is
represented by a second order mean value model similar to that of
Keynejad and Manzie (2011a) and considers fluid, thermal and
mechanical domains. It calculates the intake manifold pressure
pim, exhaust port gas temperature Tcyl and fuel mass flow rate
_ fuel as a function of throttle angle α, engine speed N, normalised
m
Fig. 1. Structure of the combined engine, emissions, exhaust and aftertreatment system model.
D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
air–fuel ratio setpoint λ, spark timing θ, and cam timing ϑint and
ϑovlp . To simulate driving conditions using representative engine
speed and torque trajectories, the dynamometer control system
model is used. This model is implemented as a PI controller,
adjusting the engine's throttle angle α to track the reference torque
τref
for some prescribed engine speed Nref. Engine-out CO, NO
brake
and HC emissions are approximated by static functions of the
engine state and control setpoints. Nitrogen dioxide (NO2) emissions from spark ignition engines are negligible (Heywood, 1988)
and are therefore not considered. Other emissions (O2 and H2)
required by the catalyst model are estimated based on chemical
equilibrium calculations, whilst ensuring consistency between the
exhaust gas composition modelled and the λ setpoint.
The exhaust manifold and the pre-catalyst connecting pipe are
both modelled using a lumped parameter approach. Warm-up
behaviour and heat transfer between the exhaust gas and inner
surfaces, and between outer surfaces and the ambient air are
considered. These models estimate the gas temperature drop from
the exhaust port to the catalyst inlet. The three-way catalyst is
modelled using a variation of the one-dimensional physics-based
approach of Pontikakis and Stamatelos (2004). The model considers substrate warm-up dynamics, heat and mass transfer
between the exhaust gas and the washcoat, thermal conduction
in the substrate, heat release from exothermic reactions, oxygen
storage and a reduced order chemical kinetic scheme with 10
reactions.
The overall model is represented by a system of differential
algebraic DAEs
_ ¼ F integ ðxðtÞ; zðtÞ; uðtÞÞ;
xðtÞ
ð1aÞ
0 ¼ Ginteg ðxðtÞ; zðtÞ; uðtÞÞ;
ð1bÞ
where functions Finteg and Ginteg enclose the model equations,
xðtÞ∈R35 , zðtÞ∈R3 and u∈R6 are state, algebraic and input vectors
respectively. Whilst this model encloses a significant number of
equations, it has been shown (Andrianov et al., 2012), nonetheless,
that such formulation provides a reasonable compromise between
the model's accuracy and complexity. An alternative integrated
model, for a closely coupled catalyst, is formulated such that the
outlet boundary conditions of the exhaust manifold specify the
inlet conditions for the catalyst. This model is represented by
_ ¼ F integ;ccc ðxðtÞ; zðtÞ; uðtÞÞ;
xðtÞ
ð2aÞ
0 ¼ Ginteg;ccc ðxðtÞ; zðtÞ; uðtÞÞ;
ð2bÞ
where functions F integ;ccc and Ginteg;ccc differ from Finteg and Ginteg
with the exclusion of the connecting pipe dynamics. Consequently,
the state and algebraic vectors become xðtÞ∈R34 and zðtÞ∈R2 . In
both model formulations the input vector u is specified by
uðtÞ ¼ ½τref
; N ref ; λ; θ; ϑint ; ϑovlp T ;
brake
ð3Þ
and the outputs are given by
_ CO;out ; m
_ NO;out ; m
_ HC;out T ¼ H integ ðxðtÞ; uðtÞÞ;
_ fuel ; m
½m
ð4Þ
_ X;out is the mass flow rate of tailpipe emissions X.
where m
Importantly, only fully warm engine maps and the results from a
single transient test are required for complete calibration of the
integrated model. For further details on the modelling, model
calibration and validation see Andrianov (2011) and Andrianov
et al. (2012).
2.2. Optimal control problem formulation
If tcyc is the duration of the drive cycle, specified by engine
brake torque τref
and engine speed Nref inputs, and vector u⋆
c ðtÞ
brake
is the optimal trajectory of the engine control setpoints, with
1009
variables in uc constrained within physically reasonable ranges,
then the dynamic optimisation problem is formulated as
Z tcyc
_ fuel ðuc ; tÞ dt;
Jðuc Þ ¼
ð5aÞ
m
0
u⋆
c ðtÞ ¼ arg min Jðuc ðtÞÞ;
ð5bÞ
uc ðtÞ
such that
uc ¼ ½λ; θ; ϑint ; ϑovlp T ;
ð5cÞ
; Nref ; uc T ;
u ¼ ½τref
brake
ð5dÞ
λmin ≤λðtÞ ≤λmax ;
ð5eÞ
θmin ≤θðtÞ ≤θmax ;
ð5fÞ
θMBT þ θadv;min ≤θðtÞ ≤θMBT þ θadv;max ;
ð5gÞ
ϑint;min ≤ϑint ðtÞ ≤ϑint;max ;
ð5hÞ
ϑovlp;min ≤ϑovlp ðtÞ ≤ϑovlp;max ;
Z
0
t cyc
ð5iÞ
_ tp ðuc ðtÞ; tÞ dt o mtp;max :
m
ð5jÞ
Eq. (5f) constrains the ignition timing within a range supported by
an ECU, whilst (5g) sets acceptable drivability characteristics and
limits engine knock. Maximum brake torque spark timing θMBT ðtÞ
is specified as a static function of the engine state and control
setpoints. Cumulative tailpipe emissions are limited by mtp;max ¼
½mCO;out;max ; mNO;out;max ; mHC;out;max T . Depending on whether an
under-floor or a closely coupled catalyst configuration is required,
either (1) and (4) or (2) and (4) form the equality constraints
for (5).
2.3. Modified control problem formulation
The development of solutions to (5) over the full duration
of the NEDC cycle (t cyc ¼ 1180 s) is computationally very
intensive. Thus, an approximation for this optimisation problem
is considered.
To reduce the computational requirements and improve the
time resolution of the optimised control policies, the cycle is
limited to the first 400 s, which includes catalyst light-off and
covers engine warm-up from cold to fully warm operating conditions. Furthermore, the number of control input variables in uc
considered for optimisation is reduced. To distinguish the variables
to be optimised, uc is partitioned into vectors uco and ucu , containing engine control setpoints that are optimised and unchanged
(preset to a production ECU strategy) respectively,
uc ¼ ½uco ; ucu T :
ð6Þ
If (5j) are implemented using barrier functions
bi ðuco Þ ¼
8
>
<0
if
>
:m
1
i;out;max
R tf
0
_ i;out ðuco ; tÞ dt
m
if
R tf
0
_ i;out ðuco ; tÞ dt o mi;out;max ;
m
0
_ i;out ðuco ; tÞ dt≥mi;out;max ;
m
R tf
ð7aÞ
which are 0, if the constraints (5j) are satisfied, and are greater
than or equal to 1, when they are violated, the optimal control
problem (5) is approximated as
Z tf
_ fuel ðuco ; tÞ dt þ wcon
J s ðuco Þ ¼
∑
bi ðuco Þ;
ð7bÞ
m
i ¼ CO;NO;HC
0
u⋆
co ðtÞ ¼ arg min Jðuco ðtÞÞ;
uco ðtÞ
ð7cÞ
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D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
such that
u¼
½τref
; N ref ; ucu ; uco T ;
brake
ð7dÞ
λmin ≤λðtÞ ≤λmax ;
ð7eÞ
θmin ≤θðtÞ ≤θmax ;
ð7f Þ
θMBT þ θadv;min ≤θðtÞ ≤θMBT þ θadv;max ;
ð7gÞ
ϑint;min ≤ϑint ðtÞ ≤ϑint;max ;
ð7hÞ
ϑovlp;min ≤ϑovlp ðtÞ ≤ϑovlp;max :
ð7iÞ
In this formulation tf ¼ 400 s. In order to simulate hard constraints,
the value for wcon selected is several orders of magnitude larger
than the overall fuel consumption.
As previously, the equality constraints are specified either by
(1) and (4) or (2) and (4), depending on whether an under-floor or
a close-coupled catalyst configuration is required. The brake
torque τref
and engine speed Nref prescribe the driving condibrake
tions, and the unoptimised control variables ucu are assigned
trajectories previously obtained from an NEDC test with the
production control strategy. This paper considers several compositions of uco , two exhaust system configurations and tailpipe
emissions limits based on Euro-3 and Euro-4. These cases are
presented in Table 1.
Note that the valve overlap ϑovlp in Case 6 is part of ucu , which is
prescribed. The intake valve closing angle ϑint can therefore be
limited by the physical constraints in the cam timing mechanism,
in addition to (7h). To consider a wider range of possible ϑint in the
optimisation, the valve overlap is specified in terms of ϑint , when
the exhaust cam-shaft is in the fully advanced state. Consequently,
the following applies:
8 ref
< ϑovlp
if ϑint −ID þ ϑref
4 ϑexh;min ;
ovlp
ϑovlp ¼
ð8Þ
: ϑexh;min −ðϑint −IDÞ if ϑint −ID þ ϑref ≤ϑexh;min ;
ovlp
where ID is the intake duration and ϑexh;min is the most advanced
exhaust valve closing angle. For the engine used in this work
ϑexh;min ¼ −4:51 ATDC and ID ¼ 2561.
3. Proposed solution approach
The integrated models (1) and (2) are of relatively high order.
Consequently, indirect methods (involving calculus of variations or
Pontryagin's minimum principle), direct methods (where a
dynamic optimisation problem is converted to a static problem)
and dynamic programming (Bellman, 1954) are impractical for
obtaining solutions to (7) in reasonable time frames. In this paper
a simplified variation of dynamic programming called iterative
dynamic programming (Luus, 2000) is used. However, whilst this
procedure possesses good speed, flexibility and convergence
properties, it does not guarantee convergence to a global optima,
but is nonetheless needed to make the optimisation computationally feasible.
The state-space is initially allocated by solving the model
equations subject to some nominal engine control policy uco ðtÞ,
then by trajectories perturbed from this policy. For the optimisation cases considered in this work, 3–5 trajectory perturbations
were typically used at each iteration. The inputs tested at each of
the discretised values of the state vector x are chosen to lie in the
Table 1
Optimisation cases considered.
Case
Optimisation vector
Control constraints
Emission constraints
Model equations
1
uco ¼ ½θðtÞ
θmin ¼ −101
θmax ¼ 501
θadv;min ¼ −301
θadv;max ¼ 51
mCO;out;max ¼ 22:8 g
mNO;out;max ¼ 1:29 g
mHC;out;max ¼ 1:58 g
(based on Euro-3)
(1) and (4)
under-floor catalyst
2
uco ¼ ½θðtÞ
θmin ¼ −101
θmax ¼ 501
θadv;min ¼ −351
θadv;max ¼ 101
mCO;out;max ¼ 22:8 g
mNO;out;max ¼ 1:29 g
mHC;out;max ¼ 1:58 g
(based on Euro-3)
(2) and (4)
close-coupled catalyst
3
uco ¼ ½θðtÞ; λðtÞ
θmin ¼ −101
θmax ¼ 501
θadv;min ¼ −351
θadv;max ¼ 101
λmin ¼ 0:9
λmax ¼ 1:1
mCO;out;max ¼ 22:8 g
mNO;out;max ¼ 1:29 g
mHC;out;max ¼ 1:58 g
(based on Euro-3)
(1) and (4)
under-floor catalyst
4
uco ¼ ½θðtÞ; λðtÞ
θmin ¼ −101
θmax ¼ 501
θadv;min ¼ −351
θadv;max ¼ 101
λmin ¼ 0:9
λmax ¼ 1:1
mCO;out;max ¼ 22:8 g
mNO;out;max ¼ 1:29 g
mHC;out;max ¼ 1:58 g
(based on Euro-3)
(2) and (4)
close-coupled catalyst
5
uco ¼ ½θðtÞ; λðtÞ
θmin ¼ −101
θmax ¼ 501
θadv;min ¼ −351
θadv;max ¼ 101
λmin ¼ 0:9
λmax ¼ 1:1
mCO;out;max ¼ 9:9 g
mNO;out;max ¼ 0:69 g
mHC;out;max ¼ 0:79 g
(based on Euro-4)
(2) and (4)
close-coupled catalyst
6
uco ¼ ½θðtÞ; ϑint ðtÞ
θmin ¼ −101
θmax ¼ 501
θadv;min ¼ −351
θadv;max ¼ 101
ϑint;min ¼ 46:51ABDC
ϑint;max ¼ 98:51ABDC
mCO;out;max ¼ 22:8 g
mNO;out;max ¼ 1:29 g
mHC;out;max ¼ 1:58 g
(based on Euro-3)
(2) and (4)
close-coupled catalyst
D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
proximity of the recorded best control policy from the previous
iteration. The range of input values tested is reduced with each
iteration, making the discretisation of the inputs more refined. A
typical input range contraction factor used was 0.85.
Whilst these modifications to dynamic programming can significantly reduce its computational requirements, use of finely
resolved time grids can require a large number of simulations,
which can be time demanding. Use of rough grids, however, limits
the frequency at which the optimised control inputs can be varied,
and as a result, some higher frequency characteristics of optimal
trajectories, such as rapid spark timing switching or air–fuel ratio
pulsation, might be missed. In this paper temporal resolutions of
Δt ¼ 20 s and Δt ¼ 10 s are used across the 400 s drive cycle
duration considered. Whilst these resolutions limit the ability of
the solution to capture higher frequency information, they also
enable more general trends in the engine control setpoints to be
established.
1011
4. Experimental methods
All model calibration and validation work, as well as implementation of control schemes were carried out on a 4 l Ford BF
engine and a Horiba-Schenck Titan 460 kW transient dynamometer. The exhaust system comprised of a cast iron exhaust
manifold, a connecting pipe and an aged catalyst (75 h Ford 4mode schedule, equivalent to roughly 80,000 km of road driving).
A photo of the test rig is presented in Fig. 2.
Normalised air–fuel ratio measurements were taken using
Bosch LSU 4.9 wide-band sensor. If the engine was required to
operate at a non-stoichiometric air–fuel ratio, the sensor was
utilised in the feedback loop of a PI controller, adjusting the
injection duration. Fuel flow measurement was performed using
AVL KMA 4000. Exhaust gas composition results presented were
obtained from a Horiba vehicle certification grade emissions
bench.
5. Simulation and experimental results
Solutions to (7) are developed for each of the cases listed in
Table 1. The results are explicitly validated using experimental
data whenever possible, whilst trends in the optimised trajectories
are identified and examined.
5.1. Case 1: spark timing solution for an under-floor catalyst
5.1.1. Δt ¼ 20 s grid
The dynamic optimisation problem (7) was solved using an
evenly spaced temporal grid of Δt ¼ 20 s resolution, requiring
roughly a day of computation on a typical desktop PC. The solution
was specified in terms of spark timing offset relative to a production control strategy previously recorded over the same driving
conditions, which enabled to consider this production strategy as a
reasonable initial guess of the control solution in the first iteration
of iterative dynamic programming. The resulting optimised spark
timing is presented in Fig. 3 along with the constraints (7g).
Fig. 2. Test rig.
10
Spark adv. from MBT (CAD)
5
0
-5
-10
-15
-20
-25
Speed (km/h)
-30
limits
-35
60
30
0
0
50
100
150
200
250
300
350
400
Time (s)
Fig. 3. Spark timing strategy optimised using Δt ¼ 20 s grid under Euro-3 constraints for the under-floor catalyst.
1012
D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
300
Fuel consumed (g)
250
modelled
measured
200
150
100
50
0
0
50
100
150
200
250
Time (s)
300
350
400
35
30
feedgas measured
CO (g)
25
Euro-3 constraint
feedgas modelled
20
15
tailpipe measured
10
tailpipe modelled
5
0
0
50
100
150
200
250
Time (s)
300
350
400
350
400
2.5
2
NO (g)
feedgas modelled
1.5
Euro-3 constraint
1
feedgas measured
tailpipe modelled
0.5
observed differences are not only the result of the modelling
assumptions used, but also the consequence of imprecise control
policy implementation, caused primarily by phasing errors
between the spark timing strategy and the prescribed engine
torque and speed trajectories.
The spark timing policy is characterised by a period of initially
significant retard relative to MBT timing, where it is in close
proximity of the retard constraint. This is followed by a transition
to near MBT timing. The maximum indicated thermal efficiency is
observed at the MBT spark advance. Thus, retarded ignition leads
to a reduction in the efficiency. Under such conditions, more heat
is rejected with the exhaust, which causes the exhaust enthalpy to
increase and enables the catalyst warm-up time to be reduced.
This strategy also tends to reduce NO and HC emissions from the
engine, which provides additional benefits.
During warm idle events the ignition is again retarded with the
intention to operate within the calibrated region of the engine
model, whilst minimising fuel use. It is therefore not clear at this
stage whether use of MBT ignition is beneficial during these
periods. However, as warm idle events are marked by low fuel
usage, optimisation results involving an exhaustively calibrated
engine model are expected to be comparable.
Simulated and measured fuel consumption, as well as precatalyst (feedgas) and post-catalyst (tailpipe) emissions are presented in Fig. 4. While catalyst light-off is observed roughly 60 s
after engine start, Fig. 3 shows that the optimised spark timing
remains retarded from MBT until approximately 130 s. The additional influx of heat appears to be required to raise the catalyst
temperature further, in order to achieve higher pollutant conversion efficiencies during the acceleration event at roughly 120 s.
Near MBT operation later in the cycle facilitates better fuel
economy at the expense of reduced exhaust enthalpy input into
the catalyst. However, by that stage the catalyst is sufficiently
warm to maintain a working temperature from the exothermic
reactions taking place in the washcoat.
By the end of 400 s, cumulative HC emissions are almost at the
Euro-3 limit, suggesting that further reduction in fuel consumption may be limited by the emissions constraints. This tradeoff
therefore emphasises the opportunity to improve fuel economy if
the emissions are below the limits.
tailpipe measured
0
0
50
100
150
200
250
Time (s)
300
5
5.1.2. Δt ¼ 10 s grid
To test whether the solution developed in Section 5.1.1 is time
grid independent, the spark timing strategy was reproduced on a
grid with intervals Δt fixed at 10 s. This required roughly 3 days
10
5
Spark adv. from MBT (CAD)
3
feedgas measured
2
Euro-3 constraint
1
0
feedgas modelled
tailpipe measured
tailpipe modelled
0
Δ t = 10 sec
Δ t = 20 sec
limits
-5
-10
-15
-20
-25
-30
0
50
100
150
200
250
Time (s)
300
350
400
Fig. 4. Measured and calculated cumulative (a) fuel consumption, and (b) CO,
(c) NO and (d) HC emissions obtained using a spark strategy for the under-floor
catalyst.
This trajectory was implemented on the engine. Simulated and
measured fuel consumption and tailpipe emissions are shown
in Fig. 4, demonstrating reasonable agreement. Note that the
-35
Speed (km/h)
HC (g)
4
60
30
0
50
100
150
200
Time (s)
250
300
350
400
Fig. 5. Spark timing strategy optimised using Δt ¼ 10 s and Δt ¼ 20 s grids under
Euro-3 constraints for the under-floor catalyst.
D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
of computation on a typical desktop PC, approximately 3 times
longer than with Δt set to 20 s. The resulting spark timing policy,
demonstrated in Fig. 5, is characterised by a more significant
310
Fuel consumed (g)
300
290
280
270
260
250
-15
-10
-5
0
5
10
Constant spark timing advance from optimal (CAD)
15
1013
retard in the initial phase of the cycle, and a much sharper and
earlier transition from highly retarded to near MBT spark advance.
The benefit in terms of the overall fuel consumption is roughly
1.4% with respect to the strategy with Δt ¼ 20 s.
5.1.3. Validation of local optimality
To test the spark timing solution, the optimised spark timing
was offset by −101, −51, 51 and 101. The perturbed control
strategies were implemented on an engine and the results are
shown in Fig. 6, where the offset of 01 corresponds to the
optimised trajectory. While the trajectories with retarded spark
timing appear to satisfy the emissions constraints, they result in an
overall fuel consumption increase. Conversely, trajectories with
more advanced spark timing improve the fuel economy, but
violate the hydrocarbon limits. The minimum achievable fuel
consumption, that allows all the emissions constraints to be
satisfied, corresponds to an offset of 01. Therefore, the experiments
confirm the local optimality of the control policy. Of course, this is
not an exhaustive evaluation of optimality, which would require
an impractically large number of experiments.
Cumulative tailpipe CO (g)
25
Euro-3 constraint
20
15
10
5
0
-15
Cumulative tailpipe NO (g)
1.4
-10
-5
0
5
10
Constant spark timing advance from optimal (CAD)
15
0.8
Note that such spark timing automatically satisfies ECU and
drivability constraints (7f) and (7g). Prior to time tsw, the strategy
is specified by the most retarded spark timing possible under
these constraints. It is then switched to approach MBT timing. The
optimisation problem is reformulated as
0.6
uco ¼ ½θsw ðt sw ; tÞ;
ð10Þ
0.4
t⋆
sw ¼ arg min J s ðuco Þ;
ð11Þ
Euro-3 constraint
1.2
1
t sw
0.2
0
-15
and solved, yielding
-10
-5
0
5
10
Constant spark timing advance from optimal (CAD)
15
2.8
Cumulative tailpipe HC (g)
5.1.4. Switching result
As the results of Sections 5.1.1 and 5.1.2 appear to be converging
towards a bang–bang type policy for spark timing, there is interest
in investigating the merit of such a policy directly. The potential
benefits of exclusively searching for a switched policy is that the
search space is significantly reduced, with only the switching
times required. This results in an obvious speed up in developing
the solution.
The maximum retard from MBT is prescribed by drivability
constraints, leading to a policy defined exclusively in terms of the
switching time tsw in the following form:
(
maxð−101; θMBT ðtÞ−301Þ for t ot sw ;
θsw ðt sw ; tÞ ¼
ð9Þ
for t≥t sw ;
minð501; θMBT ðtÞÞ
2.4
2
1.6
Euro-3 constraint
1.2
0.8
0.4
0
-15
-10
-5
0
5
10
Constant spark timing advance from optimal (CAD)
15
Fig. 6. Measured cumulative (a) fuel consumption and (b) CO, (c) NO and (d) HC
tailpipe emissions for perturbed spark timing strategies.
t⋆
sw ¼ 61:3 s:
ð12Þ
Fig. 7 shows the effect of the switching time tsw on the overall fuel
economy and tailpipe emissions as simulated by the model. Whilst
earlier switching tends to improve the fuel economy, it also
appears to violate cumulative tailpipe emissions limits, as less
heat is made available for the catalyst. Later switching causes more
heat to be rejected into the exhaust, which generally reduces
overall tailpipe emissions, but increases fuel consumption. The
switching time t ⋆
sw is therefore a balanced compromise between
cumulative tailpipe emissions and fuel economy.
The performance of the switching strategy appears to approach
that of the policy from Section 5.1.2, with the minimum value of
the cost function Js falling within 0.4% of the result produced using
iterative dynamic programming. However, only minutes of computation were required to solve this static optimisation problem,
as opposed to days in the case of Section 5.1.2. Furthermore, the
approximate solution might lead to real-time implementable
policies, with the switching time specified, for example, by
representative temperatures or species in the exhaust.
1014
D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
25
20
15
10
5
0
2
Tailpipe NO
(g)
Tailpipe CO
(g)
Fuel (g)
310
290
270
250
Euro-3 constraint
1
Tailpipe HC
(g)
0
4
3
2
1
0
Euro-3 constraint
Euro-3 constraint
0
20
40
60
80
100
Time of spark timing switch (s)
120
140
Fig. 7. The effect of spark timing switching time on the overall fuel consumption and tailpipe emissions for the under-floor catalyst.
Speed (km/h)
Spark adv. from MBT (CAD)
10
0
-10
-20
-30
close-coupled catalyst
under-floor catalyst
limits
-40
60
30
0
0
50
100
150
200
Time (s)
250
300
350
400
Fig. 8. Spark timing strategies optimised under Euro-3 constraints for the under-floor and close-coupled catalysts.
5.2. Case 2: spark timing solution for a close-coupled catalyst
In Section 5.1 it was demonstrated that as far as the cost
function Js is concerned, there are small differences between the
results of optimisation using Δt ¼ 20 s and Δt ¼ 10 s. Therefore,
to reduce excessive computational requirements and to ensure
consistency between all of the results, all further studies are
limited to Δt ¼ 20 s grids. For this case roughly 1 day was
required for the development of the solution on a typical
desktop PC.
Fig. 8 demonstrates the spark timing solution for the closely
coupled catalyst. The duration of the initial spark timing retard is
reduced relative to the optimised control policy for the under-floor
catalyst. This is because close-coupling avoids some of the heat
losses that occur for the under-floor catalyst, in the exhaust system
between the engine manifold and the catalyst, and so can heat the
catalyst with less spark retard.
As seen in Fig. 9(a), the spark timing is heavily retarded during
the first idle event. This increases the mass flow rate of the exhaust
and the catalyst inlet gas temperature, which helps to bring the
catalyst to its operating temperature quicker.
Fig. 9(b)–(d) indicates that while the new control strategy
results in lower overall engine-out CO emissions, engine-out NO
and HC emissions are significantly increased. However, as the
catalyst is exposed to higher inlet exhaust enthalpy, its conversion
efficiency is enhanced, which results in very similar cumulative
D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
5.2.1. Switching result
As in Section 5.1.4, approximating the spark timing solution by
a switching strategy and optimising the switching time yielded
300
Fuel consumed (g)
250
t⋆
sw ¼ 32:5 s:
under-floor cat.
200
close-coupled cat.
100
50
5.3. Case 3: spark timing and λ solution for an under-floor catalyst
150
0
50
100
150
200
250
Time (s)
300
350
400
35
feedgas
30
CO (g)
25
Euro-3 constraint
20
15
under-floor cat.
10
close-coupled cat.
under-floor cat.
tailpipe
close-coupled cat.
5
0
0
50
100
150
200
250
Time (s)
300
350
400
3.5
3
feedgas
2.5
NO (g)
ð13Þ
The minimum value of the cost function Js agreed closely with the
iterative dynamic programming result, which is consistent with
earlier findings. As previously, only minutes of computation were
required.
0
close-coupled cat.
2
1.5
Euro-3 constraint
1
under-floor cat.
tailpipe
close-coupled cat.
0.5
under-floor cat.
0
0
50
100
150
200
250
Time (s)
300
350
400
5
4.5
4
3.5
HC (g)
1015
feedgas
3
close-coupled cat.
2.5
under-floor cat.
2
Euro-3 constraint
1.5
under-floor cat.
1
tailpipe
close-coupled cat.
The solution to Case 3 was developed using Δt ¼ 20 s time
discretisation, requiring roughly 3 days of computation on a
typical desktop PC. It is presented in Fig. 10. Simulated emissions
are compared against single spark timing optimisation results of
Section 5.1.1 in Fig. 11.
The λ control policy is initially lean and spark timing is retarded
to help reduce engine-out CO and HC emissions, whilst maintaining similar levels of NO. In later parts of the cycle, however, λ is
characterised by multiple switches, and the engine operates lean
during most of the time. This compromises the NO conversion
efficiency in the hot catalyst in favour of the fuel economy and
lower tailpipe CO and HC emissions, such that both NO and HC
emissions fall onto the Euro-3 limits by the end of the 400 s of
NEDC conditions considered.
The trends in the spark timing strategy closely resemble those
of Section 5.1. It is characterised by an initial period of significant
retard, transition towards MBT, and then near MBT operation.
With this strategy catalyst light-off occurs roughly at 60 s, whilst
the spark timing is retarded significantly from the MBT setpoint
for another 70 s. This provides additional heat for the catalyst,
increasing its conversion efficiency during some of the later higher
power events.
The overall fuel consumption saving relative to the single
parameter spark timing optimisation of Section 5.1.1 is approximately 6.1%, which is achieved at the cost of increased cumulative
NO tailpipe emissions.
5.3.1. Switching result
Fig. 10 suggests that a bang–bang strategy with a single switch
is inappropriate for approximating the λ trajectory. Approximating
λ by a trajectory with multiple switches is computationally more
involved and is not considered in this paper. Consequently, if only
the spark timing solution is approximated by a switching strategy,
whilst the λ trajectory is fixed to the iterative dynamic programming solution, the optimised switching time evaluates to
t⋆
sw ¼ 57:3 s:
ð14Þ
The minimum value for the cost function Js is then within 2% of the
iterative dynamic programming result, which is consistent with
previous observations.
5.4. Cases 4 and 5: spark timing and λ solution for a close-coupled
catalyst
0.5
0
0
50
100
150
200
250
Time (s)
300
350
400
Fig. 9. Modelled cumulative (a) fuel consumption, and (b) CO, (c) NO and (d) HC
emissions using spark timing strategies for under-floor and close-coupled catalysts.
tailpipe emissions to the optimisation case with the under-floor
catalyst. The time to catalyst light-off remains almost fixed at 60 s
for the two exhaust systems.
Solutions to (7) for Cases 4 and 5 are presented in Fig. 12 and
have taken each roughly 3 days to generate on a typical desktop
PC. The spark timing strategies differ by almost a constant crankangle during the first 60 s. Consequently, fuel consumption resulting from the Euro-4 strategy is 3.7% higher than of the Euro-3
strategy, as indicated in Fig. 13(a). The small differences in the λ
trajectories have minor effects on the fuel economy.
Fig. 13(b)–(d) demonstrates modelled cumulative emissions.
As previously, whilst all of the cumulative tailpipe emissions
constraints are fulfilled, both NO and HC emissions lie on the
1016
D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
Spark adv. from
MBT (CAD)
10
0
-10
-20
-30
-40
1.1
λ
1
Euro-3 strategy
limits
0.9
Speed
(km/h)
60
30
0
0
50
100
150
200
250
300
350
400
Time (s)
300
35
250
30
spark
200
150
100
Euro-3 constraint
20
feedgas
15
tailpipe
10
50
0
feedgas
25
spark and lambda
CO (g)
Fuel consumed (g)
Fig. 10. Spark timing and λ control strategies optimised under Euro-3 constraints for the under-floor catalyst.
spark
5
0
50
100
150
200
250
Time (s)
300
350
0
400
tailpipe
spark and lambda
0
50
100
150
200
250
Time (s)
300
350
400
5
4
3.5
4
feedgas
2.5
2
spark and lambda
1.5
feedgas
3
2
Euro-3 constraint
Euro-3 constraint
1
spark
spark and lambda
0.5
0
HC (g)
NO (g)
3
tailpipe
spark
0
50
100
150
200
250
Time (s)
0
300
350
400
tailpipe
1
spark
spark and lambda
0
50
100
150
200
250
Time (s)
300
350
400
Fig. 11. Modelled cumulative (a) fuel consumption, and (b) CO, (c) NO and (d) HC emissions using spark timing, and spark timing and λ strategies for an under-floor catalyst.
respective limits by the end of 400 s. Apart from the reduced
catalyst warm-up time and its enhanced conversion efficiency,
another major contribution towards meeting the Euro-4 constraints comes from the reduction of engine-out emissions, partly
caused by a more significant spark retard early in the cycle.
5.4.1. Switching result
Consideration of a switching spark timing trajectory, with the λ
trajectory set to the solution of iterative dynamic programming,
results in
t⋆
sw ¼ 8:0 s
t⋆
sw ¼ 35:5 s
and
ð15Þ
ð16Þ
with respect to Euro-3 and Euro-4 constraints respectively. In both
of these cases the minimum values of cost functions Js are within
1% of iterative dynamic programming results.
D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
1017
Spark adv. from
MBT (CAD)
10
0
-10
-20
Euro-3
Euro-4
limits
-30
-40
1.1
λ
1
Euro-3
Euro-4
limits
0.9
Speed
(km/h)
60
30
0
0
50
100
150
200
250
300
350
400
Time (s)
Fig. 12. Spark timing and λ optimised under Euro-3 and Euro-4 constraints for the close-coupled catalyst.
300
25
Euro-3 constraint
20
200
Euro-3
150
15
Euro-4 policy
Euro-4 constraint
10
Euro-3 policy
100
50
5
0
0
0
50
100
150
200
250
Time (s)
300
350
400
4
5
3.5
4.5
Euro-4 policy
1.5
1
Euro-3 policy
0.5
Euro-4 policy
0
50
100
150
200
250
Time (s)
Euro-3 constraint
tailpipe
Euro-4 constraint
50
350
200
250
Time (s)
300
350
400
3
2.5
2
1.5
1
400
150
feedgas
0
Euro-3 constraint
tailpipe
Euro-4 constraint
Euro-3 policy
0.5
300
100
3.5
HC (g)
Euro-3 policy
2
0
tailpipe
4
feedgas
2.5
0
Euro-3 policy
Euro-4 policy
3
NO (g)
feedgas
Euro-4
CO (g)
Fuel consumed (g)
250
Euro-4 policy
0
50
100
150
200
250
Time (s)
300
350
400
Fig. 13. Modelled cumulative (a) fuel consumption, and (b) CO, (c) NO and (d) HC emissions using spark timing and λ strategies for the close-coupled catalyst.
5.5. Case 6: spark and cam timing solution for a close-coupled
catalyst
The study examines the benefits of including cam timing in the
dynamic optimisation. The solutions to Cases 2 and 6 are compared in Fig. 14. The solution to Case 6 took roughly 3 days to
generate on a typical desktop PC.
The optimised spark timing strategies from the single and two
parameter optimisation studies agree closely during most of the
drive cycle. Some of the most significant differences in the spark
timing (and cam timing) are observed between 80 and 120 s.
However, Fig. 15 shows that neither fuel economy nor emissions
are greatly affected, as these instances are characterised by a low
fuel flow. Simulated cumulative tailpipe NO emissions increase
only slightly due to the increase in the respective engine-out
emissions, but remain below the specified emissions limits.
The improvement in overall fuel economy of only 1.3% over the
single parameter spark timing optimisation result and reasonable
D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
IVC
Spark adv. from
(CAD ABDC)
MBT (CAD)
1018
10
0
-10
-20
-30
-40
100
80
60
40
Speed
(km/h)
Overlap
(CAD)
50
40
30
20
10
0
60
spark and cam
spark
limits
30
0
0
50
100
150
200
Time (s)
250
300
350
400
300
35
250
30
spark
200
spark and cam
150
100
0
50
100
150
200
250
Time (s)
300
tailpipe
spark
350
spark and cam
0
400
0
50
100
150
200
250
Time (s)
300
350
400
5
4
feedgas
4
3.5
HC (g)
3
NO (g)
15
5
4.5
2.5
spark and cam
2
1.5
Euro-3 constraint
50
100
spark
3
spark and cam
2
Euro-3 constraint
150
200
250
Time (s)
0
300
350
tailpipe
1
tailpipe
spark and cam
spark
0
feedgas
spark
1
0.5
0
Euro-3 constraint
20
10
50
0
feedgas
25
CO (g)
Fuel consumed (g)
Fig. 14. Spark and cam timing optimised under Euro-3 constraints for the close-coupled catalyst.
400
0
50
100
150
200
250
Time (s)
300
350
400
Fig. 15. Modelled cumulative (a) fuel consumption, (b) CO, (c) NO and (d) HC emissions using spark, and spark and cam timing strategies for the close-coupled catalyst.
agreement of the optimised and production cam timing trajectories suggest that the ECU cam control strategy in this engine is
already near the expected optima.
The performance of the control policies developed are compared in Table 2 in terms of the fuel consumption saving relative to
the Case 1 iterative dynamic programming result with Δt ¼ 20 s.
Subscripts sw and idp identify switching trajectories and those
derived using iterative dynamic programming respectively. As a
guide, the 400 s period considered in the optimisation accounts for
approximately 30% of the fuel consumed during the entire NEDC
drive cycle.
6. Conclusion
This paper presented a methodology for minimising the fuel
consumption of a gasoline fuelled vehicle during cold starting and
subject to cumulative emissions constraints. This problem was
D.I. Andrianov et al. / Control Engineering Practice 21 (2013) 1007–1019
Table 2
Fuel consumption saving over 400 s of NEDC for the engine control strategies
developed (with respect to the iterative dynamic programming result of Case 1).
Iter. dyn. programming
θ, λ
θ, ϑ
UFC, Euro-3
CCC, Euro-3
CCC, Euro-4
0% [1]
7.4% [2]
–
6.1% [3]
11.9% [4]
8.6% [5]
8.7% [6]
–
Switching policy
Optimisation variables
θsw
θsw , λidp
0.2% [1]
7.0% [2]
–
4.6% [3]
11.4% [4]
8.1% [5]
[n], optimisation case n; UFC, under-floor catalyst configuration; CCC, close-coupled
catalyst configuration; –, could not meet emissions constraints.
solved numerically using a validated transient model of the engine
and exhaust system reported in a previous study (Andrianov et al.,
2012).
Including the model calibration, this methodology should
generate optimised results significantly more quickly than many
other approaches in the open literature. One and two parameter
optimisations using iterative dynamic programming were achievable on a desktop PC within 1–3 days, which suggests that
parallelisation would enable both higher dimensional optimisation
and finer temporal resolution in practical time frames. Single
switching control policies were also proposed, and these were
found to perform close to the optimised strategies obtained from
the iterative dynamic programming but with far less computational effort.
The methodology also quantified how the physics in this
problem imposed limits on the achievable fuel consumption.
These were as follows, all which are consistent with industrial
practice.
There was a trade-off between the lowest achievable fuel
likely due to the production cam control strategy being already
well optimised.
Optimisation variables
θ
UFC, Euro-3
CCC, Euro-3
CCC, Euro-4
1019
consumption and the emissions standard achievable with a
given engine and exhaust system design. This is an important
trade-off, since catalysts (and in particular their precious metal
loading) are a significant cost that needs to be justified in terms
of total vehicle performance.
Close coupling a given catalyst reduces fuel consumption for a
given emissions standard. This is because the reduced heat
losses between the engine and the catalyst enables the engine
to operate more efficiently for a greater part of the drive cycle,
whilst maintaining high enough enthalpy exhaust for acceptable catalyst performance.
Multiple parameter optimisation yielded significant fuel consumption savings when spark timing and air–fuel ratio were
used. This primarily arose from frequent use of lean mixtures
throughout the drive cycle conditions considered. The inclusion
of cam timing did not yield significant benefits, which was
Acknowledgements
This research was supported by the Advanced Centre for
Automotive Research and Testing (ACART, www.acart.com.au),
the Ford Motor Company of Australia and Australian Research
Council Grants FT0991117 and FT100100538.
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