Input redundant internal combustion engine
with linear quadratic Gaussian control
and dynamic control allocation
J.P.R. Jongeneel
January 2009
DCT doc.no.: 2009.023
Input redundant internal combustion engine
with linear quadratic Gaussian control
and dynamic control allocation
Traineeship report
DCT doc. no.: 2009.023
J.P.R. Jongeneel
ID number: 0610188
September 2008 - January 2009
Participating Institutes:
The University of Melbourne
Eindhoven, University of Technology
Dr. C. Manzie ∗
Prof. dr. D. Nesic
Prof. dr. H. Nijmeijer
∗
†
Dep. of Mechanical Engineering
Dep. of Electrical Engineering
Parkville 3010
Victoria, Australia
www.unimelb.edu.au
†
Dep. of Mechanical Engineering
Postbus 513
5600 MB Eindhoven
The Netherlands
www.tue.nl
Preface
This report is the result of a traineeship, carried out at the University of Melbourne, Australia. This three month traineeship is part of the Master’s degree in Mechanical Engineering
at Eindhoven University of Technology, The Netherlands.
I would like to thank my supervisors, Chris Manzie (Mechanical Engineering department,
UoM) and Dragan Nesic (Electrical Engineering department, UoM) for their supervising
assistance and their encouraging guidance during research. Together with my colleagues,
they were always willing to answer my questions.
Finally, I would also like to give thanks to Henk Nijmeijer (TU/e), who gave me the opportunity for being in Australia. I had a really great time there, I made a lot of friends and
enjoyed the beautiful nature.
Roelof Jongeneel
i
ii
Abstract
Nowadays, many cars have a variable cam phasing device to advance or retard the camshaft
in order to tune the engine on-line. If the intake camshaft is constantly changed between
advanced or retarded position, one can achieve a longer opening time of the air intake valves.
This report elaborates on the idea of using throttle, as well as variable valve opening duration
to regulate the air mass flow through an internal combustion engine. The main scope is on
controlling such a configuration.
First, a mean value nonlinear engine model has been worked out. To include the effect of
an intake valve opening prolongation in the mean value model, it has been parametrised in
the volumetric efficiency. The volumetric efficiency is a measure for the amount of fresh air
which flows into the cylinder at the inlet stroke. A map has been created using simulations
on a detailed discrete event engine model.
Next, the obtained nonlinear model has been linearised. Based on the linear model, a linear
quadratic Gaussian (LQG) controller with integrated error state has been designed in order
to track a prescribed engine speed and to accommodate for load disturbances. To manage
the two actuators in a more suitable way, a dynamic control allocation technique has been
used to redistribute the control signals to the engine model inputs.
Finally, the nonlinear model has been evaluated using the LQG-controller and dynamic
input allocator. The addition of variable valve opening duration results in a significantly
faster transient when applying a step in the engine speed reference or a step in load torque
disturbance. The dynamic input allocator does not improve the transient time, however, it
makes the camshaft oscillation approach zero in steady state.
iii
iv
Contents
Preface
i
Abstract
iii
Contents
v
Nomenclature
vii
1 Introduction
1
2 Literature Review
2.1 Engine Control . . . . . . . . . . . .
2.2 Engine Modelling . . . . . . . . . . .
2.3 Modelling the Volumetric Efficiency
2.4 Redundant Actuator . . . . . . . . .
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3 Modelling the Engine
3.1 Nonlinear Engine Model . . . . . . . . . .
3.1.1 Mass Flow Intake Manifold In . . .
3.1.2 Mass Flow Intake Manifold Out . .
3.1.3 Mass Flow Exhaust Manifold In .
3.1.4 Mass Flow Exhaust Manifold Out
3.1.5 Engine Torque . . . . . . . . . . .
3.1.6 Nonlinear Engine Model . . . . . .
3.2 Linearised Engine Model . . . . . . . . .
3.3 Comparing the Models . . . . . . . . . . .
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4 Controlling the Engine Model
4.1 LQG Controller . . . . . . . . . . . . . . . . . .
4.1.1 Augmented Plant . . . . . . . . . . . . . .
4.1.2 Optimal State Feedback . . . . . . . . .
4.1.3 Kalman Filter . . . . . . . . . . . . . . .
4.1.4 LQG: Combined LQR and LQE . . . . .
4.1.5 Controller Example . . . . . . . . . . . .
4.2 Dynamic Input Allocator . . . . . . . . . . . . .
4.3 Simulating the Closed Loop System . . . . . . .
4.4 Influence of the Variable Valve Opening Duration
4.5 Influence of the Allocator . . . . . . . . . . . . .
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5 Conclusions and Recommendations
5.1 Conclusions . . . . . . . . . . . . .
5.2 Potential Automotive Applications
5.3 Shortcomings . . . . . . . . . . . .
5.4 Recommendations . . . . . . . . .
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31
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A Valve Opening Duration
35
B Friction Model
B.1 Component Mechanical Friction Models
B.1.1 Crankshaft Friction . . . . . . .
B.1.2 Reciprocating Friction . . . . . .
B.1.3 Valvetrain Friction . . . . . . . .
B.1.4 Auxiliary Friction . . . . . . . .
B.2 Total Mechanical Friction Model . . . .
37
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Bibliography
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41
vi
Nomenclature
Symbols
A
A
F s
Cd
Hl
J
Lv
ṁ
ncyl
nr
P
R
t
T
V
Vd
α
αcl
γ
ζ
ηi
ηvol
κ
λ
τe
τl
ϕ
Ψ
ωe
Area
Stoichiometric air/fuel ratio
Discharge coefficient
Lower heating value
Moment of inertia
Valve lift
Mass flow
Number of cylinders
Number of revolutions per cycle
Pressure
Specific gas constant
Time
Temperature
Volume
Displacement volume of a cylinder
Throttle angle
Throttle angle at closed position w.r.t. normal axis
Measure for the valve opening prolongation
Spark angle
Indicated engine efficiency
Volumetric efficiency
Ratio of specific heats
Normalised air/fuel ratio
Torque produced by engine
Load torque
Camshaft angle
Nonlinear function to model sonic flow
Engine speed
vii
[m2 ]
[-]
[-]
[J kg−1 ]
[kg m2 ]
[m]
[kg s]
[-]
[-]
[Pa]
[J kg−1 K−1 ]
[s]
[K]
[m3 ]
[m3 ]
[◦ ]
[◦ ]
[◦ ]
[◦ ]
[-]
[-]
[-]
[-]
[N m]
[N m]
[◦ ]
[-]
[rad s−1 ]
Subscripts
a
amb
CAC
e
exh
eff
em
f
im
o
ST P
th
Air
Ambient
Cylinder air charge
Engine
Exhaust
Effective
Exhaust manifold
Fuel
Intake manifold
At linearising or operating point
At standard temperature and pressure
Throttle
Notation
ˆ
˙
⋆
Estimated
Time derivative
Augmented vector or matrix
Steady state
Abbreviations
afmep
BDC
cfmep
fmep
IC
IVO
LQG
pmep
rfmep
TDC
vfmep
Auxiliary friction mean effective pressure
Bottom dead center
Crankshaft friction mean effective pressure
Friction mean effective pressure
Internal combustion
Intake valve opening
Linear quadratic Gaussian
Pump mean effective pressure
Reciprocating friction mean effective pressure
Top dead center
Valvetrain friction mean effective pressure
viii
Chapter 1
Introduction
At the time Nicolaus August Otto invented his famous 4-stroke engine in 1876, it was not
running very efficiently. During the following years, even till nowadays, many engineers and
researchers have been working on improving engine design, tuning and control to obtain a
higher torque output, less fuel demand, better pollutant emissions and other goals. Observed
data [Fer98] shows a conversion efficiency of 4% for engines built around 1900, whereas 2000’s
engines can run at 32% conversion efficiency for driving energy.
Since the creation of embedded systems, more and more electronics and control techniques
have been implemented in cars to obtain better performance and efficiency increase.
The Mechanical Engineering department of the University of Melbourne is also involved in
engine research. It has test facilities (Figure 1.1) to perform engine tests. Currently, they
are working on designing, modelling and testing a hydrogen fueled engine.
Figure 1.1: The engine test facilities at the University of Melbourne. A Ford
6-cylinder, 4 liter engine is installed.
Within the engine research group, the question raises if one can control an IC engine using
not only the common throttle valve, but also by adjusting valve timing. By using both
1
2
actuators, the engine will be able to achieve a faster response to a prescribed setpoint, and
it also gives more freedom in tuning for optimal engine design criteria.
Adjusting valve timing is possible since modern engines often have a variable cam phaser
device to retard or advance the camshaft. Both actuators have an effect on the airflow
through the engine, which is also known as “the breathing process”. If we consider the
engine as a volumetric pump, the airflow is directly related to the engine speed, and when
running at stoichiometric fuel consumption, the mass airflow sets the fuel demand and thus
the amount of mechanical power produced.
Changing valve timing can be performed in various ways. Possible adjustments are changing
opening time and having that, adapting the closing time to define the time the valves are
open. This applies to both the intake camshaft as well as the exhaust camshaft. The
time between the exhaust valve opens and the intake valve closes, sets the overlap. We
will restrict ourselves by only considering the intake valve opening duration as a varying
parameter because this might mostly influence the airflow.
We consider the case, we want to track a certain engine speed, using both throttle valve
angle and intake valve opening duration as inputs. We neglect the actuator dynamics. Can
we achieve a better tracking of the engine speed by exploiting both actuators?
As a matter of fact, there are now two actuators to control the same airflow. This means,
there is a redundant actuator. In literature, the problem of designing a controller for such a
plant is treated by suitably allocating the controller outputs to the plant inputs in order to
exploit the actuators in a desired way. Control allocation is being used in several applications
in practice. It is ideal for controlling, for example, a dual stage actuator with one long stroke,
but slow actuator and one short stroke, but fast one.
In this report, we will try to apply the dynamic allocation theory to the redundant actuated
engine problem. We expect a change in throttle angle to have a slow effect on the intake
manifold pressure, whereas the valve timing has a fast effect, but due to its small range, it
is not capable to fully control the engine.
The assignment is to investigate the effect of having a variable valve opening time on an
internal combustion engine in a closed loop control environment. Secondly, see if a dynamic
control allocation technique can be useful to obtain a better exploitation of the engine input
actuators and results in a better tracking behaviour.
The research, proceeding from this assignment, is described in this report and organised as
follows: First, we identify the research gap of this project in Chapter 2. Then, an engine
model is drawn up and evaluated in Chapter 3. Next, in Chapter 4, a controller is designed
together with the dynamic control allocation part. Finally, the closed loop control system
is tested and evaluated in simulations. To finalise the project, conclusions are made as well
as recommendations for further research in Chapter 5.
Chapter 2
Literature Review
2.1
Engine Control
Already before the invention of the internal combustion engine, the speed of a steam engine
was controlled by a centrifugal governor (or Watt’s fly-ball), attached to the throttle valve,
to maintain a constant engine speed.
Nowadays, the throttle valve is actuated either mechanically by the driver or electrically
using control. Since the 1980’s, a variable valve timing system is implemented in most
production cars, allowing to advance or retard the camshaft while the engine is operating.
A lot of research has been published on the possibilities and achievements of having camshaft
phasing. See for example [AKS01] and [Ste96].
Another technology and direction of research is in camless valves engines. These engines
use an electro-mechanical or hydraulic device to actuate each valve individually. As infinite
variable valve timing is possible, the intake valves can be used to throttle the engine. This
eliminates the need for a throttle valve actuator. See for example [KHR05] and its overview
on camless engines.
The research described in this report elaborates the idea to use variable intake valve opening
duration together with the standard throttle valve actuator to throttle the engine. The focus
will be on control.
To regulate the air flow through the intake valves using a camshaft, the duration of valve
opening has to be variable. In [LMR00] the idea of a variable cam velocity1 has been
1 By giving the camshaft a varying angular velocity, it is constantly retarding and advancing. If we slow
down the angular velocity if the valve is open, we achieve a longer opening time. This means we have to
speed up if the valves are closed. In this research project, a similar concept is used as described in [LMR00].
It is explained in more detail in Appendix A.
3
4
described, to obtain different valve opening durations. The invention has been validated
with experiments on a diesel engine and shows an increase in torque up to 8.2% using
variable intake valve closing.
2.2
Engine Modelling
Regarding engine modelling, we can distinguish between mean value models and discrete
event models. A discrete event engine model considers the ignition at a discrete point in time
and contains a model for the combustion of fuel and conversion into mechanical power and
heat. Typically, the model requires small computation steps of less than microseconds or a
few crankangle degrees. In a mean value engine model the fast dynamics, for example the incylinder pressure and in-cylinder temperature – which are highly varying due to combustion
in a closed volume – are replaced by their mean value. By doing this, computation can be
much faster. Moreover, as the mean value model is described in continuous time, it can be
applied with conventional control techniques.
Most research projects on engine control start with drawing up a (mean value) model, after
which control is being applied. Usually, the focus is on flows (of air, heat and mechanical
work). A mean value model can be build up by modelling a number of sub-systems separately
such as intake and exhaust manifold, cylinder chamber, turbocharger and others.
For each sub-system, the flow can be modelled by fitting a function to a set of measurement
data, parametrised to the variables of interest. This has been done in [Ste96]. Here, for
example, the air mass flow through the throttle valve ṁα is modelled as
ṁα = f (α) g(Pim ) ,
containing two second-order fitting functions f and g, parametrised to throttle angle α and
intake manifold pressure Pim , respectively.
To obtain a more general description for the flows in each sub-model, all known relationships
between parameters (pressures, temperatures, gas properties) can be included. Automotive
engineering handbooks such as [Hey88] and [GuO04] can be of a great help. Relations which
are difficult to express in parameters can be exchanged by fitting functions (efficiencies,
calibration functions). A control project which follows this approach is for example [And05].
To continue the throttle valve air mass flow example; with the latter approach, the mass
flow is described as
Pamb
im
ṁα = √
Aeff (α) Ψ PPamb
.
R Tamb
The ambient pressure Pamb , ambient temperature Tamb and specific gas constant R are single
parameters. The effective cross-sectional area at the throttle valve Aeff(α) is a nonlinear
im
fitting function, dependent on the throttle valve angle α. Likewise, Ψ PPamb
is a fitting
function to model sonic flow, as a function of the normalised intake manifold pressure Pim .
5
CHAPTER 2. LITERATURE REVIEW
2.3
Modelling the Volumetric Efficiency
A preliminary part of this ‘control’ project is to investigate the effect of having varying valve
opening duration. Because we want to apply control, we choose to use a mean value model
describing the engine by a set of equations. The term in a mean value model where the effect
of having a different valve opening duration is visible, is the volumetric efficiency, denoted
by ηvol .
The volumetric efficiency is a unitless ratio which describes how well a cylinder is filled
with air, with respect to its standard displacement volume. In literature, it is common to
parametrise the volumetric efficiency in a fitting function to the parameters of interest, such
as spark angle, valve overlap or cam angle. See for example [AKS01] and [LCS07] where the
effect of cam phasing (i.e. retard or advance the camshaft) has been mapped experimentally.
Experimentally mapping the volumetric efficiency takes place by measuring the engine speed
ωe and the air mass flow ṁim , pressure Pim and temperature Tim at the intake manifold.
If the displaced volume Vd and the number of cylinders ncyl are known and specific gas
constant R is assumed to be constant, the volumetric efficiency ηvol can be derived by
ηvol = ṁim
R Tim 4π
.
Pim Vd ωe ncyl
A map for ηvol (ωe , Pim ) can be made by exciting with a sequence of different ωe and Pim .
No similar report in mapping the volumetric efficiency to a variable valve opening duration
has been found. So, in order to design a mean value model which includes variable valve
opening duration, an appropriate function for volumetric efficiency has to be determined
either analytically, experimentally or via simulation.
2.4
Redundant Actuator
If a system has a redundant actuator, the controller has to actuate multiple plant inputs
based on the information of less plant outputs. Such a controller (for example a SIMOcontroller) can be designed using classical or modern controller design techniques. The
controller gain for each controller output channel is designed using the same feedback information. This does not always result in the most optimal cooperation between the controller
outputs. An example can be the cooperation of a big actuator with a small fine tuning
device. All inputs of such a system can be exploited in a more suitable way using control
allocation techniques. This means that the control signals are optimised and redistributed,
after which they enter the plant model.
Regarding the control allocation techniques, two alternatives can be followed, namely to
decide the control inputs directly using an adapted optimal control, or either first face the
overall control effort using optimal control and then redistribute the control inputs using
6
control allocation. [HaG05] discusses both ways and states the advantage of having the
modular design, because of configuring and tuning. The modular design is depicted in
Figure 2.1 with the controller, control allocation and plant model in a closed loop feedback
configuration.
reference
output
controller
allocator
plant model
Figure 2.1: Configuration when regulation and allocation are performed separately.
Application fields where input redundancy is at hand and control allocation is applied are
in reconfigurable flight control, ships and underwater vehicles, hard-disk drives and general
dual stage actuators, and recently in the modern Tokamak nuclear fusion reactors. Several
methods for control allocation have been proposed in literature and can be classified in
direct control allocation, daisy chain allocation, redistributed pseudo-inverse, constrained
quadratic or linear programming. See [Bod02] for a survey. All these techniques adopted in
practice can be considered as static. That means the distribution of the control signal over
the inputs is fixed in time.
Some of the mentioned techniques have been extended and improved to work as a dynamic
allocator. This means the allocation strategy is being adjusted on-line based on the operating
conditions of the system model and actuators. A dynamic control allocation has advantages
as it can add integrators or differentiators to the distribution gain, it can have a frequency
division in the control law and it is able to deal with actuator dynamics.
Elaborated ideas on dynamic control allocation are published. For example, the dynamic
allocator proposed in [Joh04] recovers the pseudo-inverse allocation asymptotically. In
[LSY05] a dynamic allocation method was proposed based on sampled data using Model
Predictive Control. A method for dynamic allocation, proposed in [Har02], is based on a
constrained quadratic programming problem and gives a frequency dependent control distribution. [Zac07] gives a contribution in control allocation, by proposing a dynamic allocator
in the form of a linear, first-order filter, which can be added to an existing controller. By
modifying it to a nonlinear filter, it is also able to deal with rate and saturation limits.
Chapter 3
Modelling the Engine
To apply control, a mean value model will be used. First, a nonlinear engine model will be
built and later on, this model will be linearised in order to design a linear controller.
3.1
Nonlinear Engine Model
A 4-stroke, spark ignition, internal combustion engine will be considered, without intercooler
or turbocharger. We consider the cylinders with on both sides either the intake or the exhaust
manifold. Figure 3.1 displays a scheme of this model.
ϕ + ∆ϕ
ϕ
α
Pamb
ɺ imin
m
Pim
ɺ emin
m
ɺ imout
m
Pem
Pamb
ɺ emout
m
ωe
Figure 3.1: The engine model scheme, showing the states and inputs.
To model the air mass flow, it suffices to have two differential equations for Ṗim and Ṗem . It
is mathematically shown by [SNM09] that a 13th -order turbocharged engine model can be
approximated by a 4th -order model for control purposes. Temperatures are slowly changing
compared to pressures and therefore temperatures Tim and Tem can be replaced by their
steady state, mean values Tim,o and Tem,o . In addition, to track the speed, a differential
equation for ω̇e is needed. For each of the air mass flow terms ṁ(·) and the net produced
7
8
engine torque τe in (3.1), appropriate functions have to be found.


Ṗim









Ṗem









 ω̇e
=
κRTim,o
(ṁimin − ṁimout )
Vim
=
κRTem,o
(ṁemin − ṁemout )
Vem
=
1
(τe − τl )
Je
(3.1)
Parameter
h V iis the volume of the specified cavity and R is the specific gas constant, which
J
is 286.9 kgK
for air. The specific heat ratio is denoted by κ, which is 1.40 [−] for air at
engine pressures. Je is the engines inertia and τl is the load torque affecting the engine such
as road slope, wind disturbance and drive train losses. Je is assumed to be known and τl
can be seen as a disturbance.
3.1.1
Mass Flow Intake Manifold In
For the mean value model, the expression for ṁimin , the mass-flow at the inlet of the intake
manifold [GuO04], is
ṁimin
=
Aeff (α)
=
Ψ
Ψ
Pim
Pamb
Pim
Pamb
=
≈
Pamb
im
Aeff (α) · √
· Ψ PPamb
RTamb
cos (α + αcl )
Cdth A(α) = Cdth Ath 1 −
cos (αcl )
 r κ+1
κ−1

2


κ
for

κ+1


s


κ1
κ−1

κ

P
Pim
2κ
im

1
−
for
 Pamb
κ−1
Pamb
s
κ
2
κ+1
κ+1 κ−1
Pim
1 − exp 9
−9
Pamb
(3.2)
(3.3)
Pim
Pamb
1≥
≤
Pim
Pamb
2
κ+1
>
κ
κ−1
2
κ+1
(3.4)
κ
κ−1
(3.5)
The effective throttle area Aeff (α) in (3.2) is a function of the throttle valve opening angle
α. In literature, it often consists of a fitted function over experimental data, but it is here
replaced by (3.3) with Cdth being the discharge coefficient.
The nonlinear function for Ψ is needed because the flow reaches sonic conditions in the
narrowest part at high pressure differences. Pressures are denoted by P . As a piecewise
function is not ideal for control purpose, approximation (3.5) can be used. Compare both
functions (3.4) and (3.5) in Figure 3.2.
9
CHAPTER 3. MODELLING THE ENGINE
0.685
0.6
Ψ
0.4
0.2
Ψpiecewise
Ψ
approximate
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Pressure ratio Pim/Pamb
0.7
0.8
0.9
1
Figure 3.2: Comparison of the piecewise and its approximated function for Ψ
at κ = 1.4.
3.1.2
Pim
Pamb
Mass Flow Intake Manifold Out
The expression for ṁimout , the mass-flow at the outlet of the intake manifold, consists of
the mass flow through the engine cylinder. The engine can be considered as a volumetric
pump with efficiency ηvol .
ṁimout = ηvol (Pim , ωe , γ)
Pim Vd ωe ncyl
RTim,o 2πnr
(3.6)
where Vd is the displaced cylinder volume, ncyl is the number of cylinders, nr the number of
revolutions per cycle (nr = 2 for a 4-stroke engine), ηvol is the volumetric efficiency of the
engine and ωe is the engine speed in radians per second.
The volumetric efficiency ηvol is a dimensionless parameter that describes how well the
cylinder chamber is filled with air. It takes typically values between 0.4 and 0.8 and depends
on many factors, including engine speed, intake manifold pressure, air-fuel ratio and valve
timing. To involve valve timing in the model, a parametrisation of ηvol has to be made.
At first, this problem has been approached analytically. If the intake valve lifts, it can be
considered as a curtain of which air can flow through. If we integrate that air mass flow
function to crankshaft angle ϕ over one full engine cycle; and after multiplying it with half
the engine speed, the mass-flow in time domain is known:
ṁimout
∂mimout ∂ϕ
Pim ωe
= ncyl niv πDv √
=
∂ϕ
∂t
RTim 2
I0 n o
(ϕ)
Ψ PPcim
·Cdiv (Lv )·Lv (ϕ) dϕ (3.7)
2π
Here, Lv (ϕ) is the valve lift per crankshaft angle, Pc the in-cylinder pressure, Dv is the
valve diameter and niv the number of intake valves per cylinder. By equalising equation
(3.6) to (3.7), an expression for ηvol should be found. However, in-cylinder pressure Pc is
very difficult to describe analytically, through which this method does not give the desired
results.
Finally, in order to map the volumetric efficiency, simulations have been done with a discreteevent model, written in Modelica language and implemented in the multi-physical software
package Dymola (Figure 3.3). The model contains sub-systems in the thermodynamic,
10
thermal and mechanical domain, including the fuel system, ignition system, cooling and
lubrication system. It consist of a large number of equations and differential equations
with equal number of unknown variables. The Dymola model is based on a physical Ford
6-cylinder engine, installed in the test lab. The model has been made at the University of
Melbourne during research on engine cold start problem, described in [Key09].
Figure 3.3: The global graphical scheme of the single cylinder Dymola model. The
intake valve duration γ can be set.
To involve the effect of valve opening duration in the model, a parameter γ has been introduced. It stands for the maximum prolongation, measured in relative degrees of the
camshaft. In the discrete-event Dymola model, the valve lift is implemented by a look-up
table for every crankshaft angle. By setting a different value for γ, the camshaft angular
velocity is superpositioned with a back and forth movement which results in a different value
for valve lift being looked up during simulation.
To keep the intake valve opening (IVO) angle at about the same place, the camshaft profile will be shifted with 0.6γ. The camshaft profile causes the valves being lifted. Figure 3.4 shows the results of a simulation. The lift of the valves are plotted for γ ∈
{−20, −10, 0, +10, +20}, as a function of the angle of the crankshaft. For the case γ = 0◦ ,
the corresponding cylinder chamber volume and the mass flows in and out of the cylinder
are plotted.
Valve opening duration parameter γ can vary between − ω2513
≤ γ ≤ ω2513
rad
rad , based on the
e[ s ]
e[ s ]
◦
maximum speed of the camshaft phasing actuator of 200 /s. The derivation and explanation
of the concept of varying camshaft angular velocity can be found in Appendix A.
11
CHAPTER 3. MODELLING THE ENGINE
0.012
intake cam
exhaust cam
cam profile
0.01
0.008
0.006
γ = −20 −10 0
0.004
+10
+20
0.002
0
90
180
1
360
450
540
630
683
720
spark
x 10
0.5
0
0
90
power stroke
mass flow [kg/s]
270
−3
3
chamber volume [m ]
0
0.03
180
BDC
exhaust stroke
270
180
270
360
TDC
450
intake stroke
540
BDC
compression stroke
630
720
TDC
540
630
720
ṁimout
ṁemin
0.02
0.01
0
0
90
360
450
crankshaft angle [dgr]
Figure 3.4: The lift of the valves, and the cylinder chamber mass flows and volume are plotted, as a function of the crankshaft angle. TDC and BDC denote
the crankshaft position when the piston is at the top and bottom of its travel,
respectively.
γ = −20
γ = −10
γ=0
γ = +10
γ = +20
1
0.9
ηvol [−]
0.8
0.7
0.6
0.5
0.4
0.3
1000
8
7
1500
6
5
4
x 10
P
im
[Pa]
4
2000
3
2
2500
ωe [rpm]
Figure 3.5: Simulation results for the volumetric efficiency ηvol (Pim , ωe , γ).
12
In the Dymola model, the volumetric efficiency is calculated with the assumption of ambient
gas constant and ambient temperature in the intake manifold, by
ηvol =
VCAC
VCAC,ST P
=
mCAC RT
PCAC
mCAC,ST P RT
PCAC,ST P
→
mCAC Pamb
mCAC,ST P Pim
with CAC standing for the cylinder air charge and STP for the conditions at standard temperature and pressure. The volumetric efficiency is independent for the number of cylinders
being modeled. The result of the simulations1 can be seen in Figure 3.5.
Note that the valve opening duration is not calibrated to its optimal position. According
to Figure 3.4 it seems that the intake valve is still open if the cylinder is already moving
upwards, and therefore some air flows back. It has to be mentioned that an engine is not
only optimised to maximum torque delivery – associated with the highest airflow rate – but
also to fuel efficiency and pollutant gasses.
Finding a parametrised function for the simulation results as depicted in Figure 3.5, is
not easy. It requires a parametrisation of ηvol to the three depending variables Pim , ωe
and γ. For simplification, there has been chosen to take a certain operating point, namely
ωe = 1500 rpm and Pim ≈ 40 kPa, such that the number of depending variables is reduced
to only one, which is the parameter γ. The reduced number of datapoints with a fitted
line are depicted in Figure 3.6. The relationship of the valve opening prolongation γ to the
volumetric efficiency ηvol can be described by the second-order function
ηvol (γ) = −0.00038571γ 2 − 0.015357γ + 0.67571.
(3.8)
0.9
simulation data
data fit
0.8
0.6
η
vol
[−]
0.7
0.5
0.4
0.3
−15
−10
−5
0
γ [°]
5
10
15
Figure 3.6: Simulation results for ηvol (γ)|ωe =1500rpm;Pim ≈40kPa with a fitting function.
The almost perfect fitting reveals that the underlying equations in the Dymola model can
show a perfect second-order relationship. This is however not easy to determine, because of
the large number of nested equations and the discrete process of looking up the valve lift in
a table.
1A
ζ=
single cylinder, open loop model has been used, with parameters set to α = 0.5◦ , τl = 100/6 Nm and
47◦ .
13
CHAPTER 3. MODELLING THE ENGINE
3.1.3
Mass Flow Exhaust Manifold In
The expression for ṁemin , the mass-flow at the inlet of the exhaust manifold, is the same
as the flow of air out of the intake manifold (3.6), plus a small amount of fuel input:
!
1
ṁemin = ṁimout 1 +
(3.9)
λ FA s
where λ is the normalised air/fuel ratio λ =
ma
mf ( A
F )
ratio, which is 14.64 for gasoline.
3.1.4
and
s
A
F s
is the stoichiometric air/fuel
Mass Flow Exhaust Manifold Out
The expression for ṁemout , the mass-flow at the outlet of the exhaust manifold (exhaust
pipe), is similar to the mass flow through the throttle valve (3.2), namely
Ψ
ṁemout
=
≈
Pamb
Pem
Pem
Cdexh Aexh · p
· Ψ PPamb
em
RTem,o
s
κ+1 κ−1
2
Pamb
κ
1 − exp 9
−9
κ+1
Pem
(3.10)
where Cdexh is the discharge coefficient and Aexh is the exhaust pipe cross-sectional area.
Like in (3.2), the approximation for Ψ is used.
3.1.5
Engine Torque
The net produced engine torque τe from (3.1) is a nonlinear function depending on a lot of
variables.
Consider the engine produced power and power losses:
τe ωe
ṁf
ηi
= ηi ṁf Hl −
=
ncyl Vd ωe
(pmep + fmep)
2πnr
(3.11)
ṁemin
A
λ F
s
im
= ηi (ωe , PPamb
, λ, ζ)
pmep
= Pem − Pim
fmep
= cfmep + rfmep + vfmep + afmep
Here, mf is the mass of the fuel burnt per combustion cycle, with lower heating value Hl ,
which is 44.0 MJ/kg for gasoline. The indicated engine efficiency ηi is a function of many
im
variables. The engine speed ωe , intake manifold pressure ratio PPamb
, normalised air/fuel
14
0.4
0.3
0.2
0.1
i
η [−] (λ = 1, ζ = 37°)
0.5
2500
0
6
2000
5
4
4
x 10
1500
3
P
im
[Pa]
2
1000
ωe [rpm]
Figure 3.7: Calibration function for indicated engine efficiency ηi (Pim , ωe )|λ=1,ζ=37◦
ratio λ and spark angle ζ are fitted in a third-order polynomial based on data from the
considered I6 Ford engine. For λ = 1 and ζ = 37◦ , this polynomial is visualised in Figure
3.7.
pmep stands for pumping mean effective pressure, and is basically the engine pumping
loss, expressed in ‘pressure’. The expression for friction mean effective pressure (fmep)
is a detailed friction model, containing crankshaft friction (cfmep), reciprocating friction
(rfmep), valvetrain friction (vfmep) and auxiliary friction (afmep). The derivation of the
friction model is worked out in Appendix B. The Dymola model uses the same friction
model and this has been designed according to [SaH03]. The resulting expression for fmep
is a function of the engine speed ωe :
fmep = 1.05·105 + 67.0ωe − 0.0637ωe2 +
1
3.07·105
+ 1.81ωe2 + 0.125·10−3ωe3 .
ωe
(3.12)
The expression for the engine torque now becomes:
τe = ηi (·)
3.1.6
ṁemin Hl
ncyl Vd
−
(Pem − Pim + fmep)
A
ω
2πnr
λ F s e
(3.13)
Nonlinear Engine Model
By substituting equations (3.2), (3.6), (3.9), (3.10), (3.13) in (3.1) and moreover using
κ = 1.40 and nr = 2, we get the final set of three first-order nonlinear differential equations
15
CHAPTER 3. MODELLING THE ENGINE


Ṗim =










Ṗem =










 ω̇e =
κRTim,o
Vim
κRTem,o
Vem
1
Je
Pamb
Pim Vd
Pim
cl )
√
Cdth Ath 1− cos(α+α
0.685
1−exp
9
−9
−ηvol (γ) RT
cos(α )
P
im,o
RTamb
cl
Pim Vd
ηvol (γ) RT
im,o
ηi (Pim ,ωe ) λ
1
(A
F )s
ωe ncyl
4π
Hl
ωe
1+ λ
1
( FA )s
Pim Vd
ηvol (γ) RT
im,o
amb
−Cdexh Aexh √ Pem
RTem,o
ωe ncyl
4π
1+ λ
1
(A
F )s
−
ωe ncyl
4π
0.685 1−exp 9 PPamb
−9
em
ncyl Vd
(Pem −Pim +fmep)−τl
4π
,
(3.14)
with ηvol , ηi and fmep being
ηvol
=
−0.00038571γ 2 − 0.015357γ + 0.67571
ηi
=
2
0.178 + 0.220·10−5 Pim + 0.278·10−10 Pim
+ 0.247·10−2 ωe − 0.502·10−7 ωe Pim . . .
−12
2
−5 2
2
−0.117·10
ωe Pim − 0.997·10 ωe + 0.354·10−9 ωe2 Pim − 0.268·10−14 ωe2 Pim
fmep
=
1.05·105 + 67.0ωe − 0.0637ωe2 +
3.2
3.07·105
ωe
1
+ 1.81ωe2 + 0.125·10−3 ωe3 .
Linearised Engine Model
The engine model will be controlled using a linear quadratic Gaussian controller. This
LQG controller can only be derived from a linear model. In the next section, the nonlinear
model (3.14), will be linearised around a certain operating point and presented into the
state-variable form
(
ẋ = Ax + Bu + Bd d
(3.15)
y = Cx + Du + Dd d
T
T
where x = [Pim Pem ωe ] is the plant state, u = [α γ]
disturbance input and y = [ωe ] is the plant output.
is the plant input, d = [τl ] is a
The state values of an equilibrium point can be found by equalising (3.14) to zero and
assuming parameter values as listed in Table 3.1. These parameter values are chosen to be
the same as used in the Dymola model, which are extracted from measurements on a real
engine.



0 =








0 =










0 =
κRTim,o
Vim
κRTem,o
Vem
1
Je
Pamb
Pim Vd
Pim
cl )
√
Cdth Ath 1− cos(α+α
0.685
1−exp
9
−9
−ηvol (γ) RT
cos(α )
P
im,o
RTamb
cl
Pim Vd ωe ncyl
ηvol (γ) RT
4π
im,o
ηi (Pim ,ωe ) λ
1
(A
F )s
Hl
ωe
1+ λ
1
( FA )s
Pim Vd
ηvol (γ) RT
im,o
amb
−Cdexh Aexh √ Pem
ωe ncyl
4π
RTem,o
1+ λ
1
(A
F )s
−
ωe ncyl
4π
Pamb
0.685 1−exp 9 Pem −9
ncyl Vd
(Pem −Pim +fmep)−τl
4π
(3.16)
16
Symbol
A
Value
14.64
0.00385
F s
Aexh
Ath
Cdth
Cdexh
Hl
Je
ncyl
Pamb
R
Tamb
Tim
Tem
Vim
Vd
Vem
αcl
ηi
ηvol
κ
λ
π0.0702
4
0.85
0.7
44.0 · 106
0.15
6
101325
287.327
288.15
288.15
593
0.004
0.0006638
0.004
7
f (Pim , ωe )
f (γ)
1.40
1
Unit
[-]
[m2 ]
[m2 ]
[-]
[-]
[J/kg]
[kg m2 ]
[-]
[Pa]
J
]
[ kgK
[K]
[K]
[K]
[m3 ]
[m3 ]
[m3 ]
[◦ ]
[-]
[-]
[-]
[-]
Table 3.1: Used parameter values to linearise the model around.
Substituting all the values of Table 3.1 into equation (3.16) and choosing γo = 0◦ , αo = 5◦
and τl,o = 50 Nm as operating conditions, lead to the following equilibrium point:
Pim,o
Pem,o
ωe,o
=
=
=
26264 Pa
101633 Pa
1606 rpm
Linearising the equations leads to the following

−12.62
0
"
#

A B Bd
−2330
 27.66
= 

0.02249
−0.002113
C D Dd
0
0
(3.17)
system matrices for equation (3.14):

−1965 83283 7510
0

4319
0
−16512
0

 (3.18)
0.2317
0
−11.99 −6.6667 
1
0
0
0
The linearised system (3.18) is fully controllable as well as observable, as the rank of P and
O are both equal to the number of states.
h
i
h
iT
P =
B AB A2 B
O =
C CA CA2
rank(P) = 3
rank(O) = 3
17
CHAPTER 3. MODELLING THE ENGINE
The eigenvalues λ and corresponding eigenvectors v of system (3.18) are
"
λ1 λ2 λ3
v1 v2 v3
#



= 


−6.2 − 1.7i
−2330
−6.2 + 1.7i

−1
−1
0

.
−0.0058 + 0.0016i −0.0058 − 0.0016i
−1 
0.0033 − 0.0009i
0
0.0033 + 0.0009i
(3.19)
They show, there is a stable oscillating pair λ1,2 and a very stable real eigenvalue λ3 . As
a matter of fact, solving the set of equations is a stiff problem. The eigenvector v3 reveals
that this stiffness is caused by the differential equation for Ṗem . The pressure in the exhaust
manifold Pem,o is close to Pamb due to a relative unrestricted flow through the exhaust pipe.
3.3
Comparing the Models
Figure 3.8 shows the open loop response to various input steps on the nonlinear model and
the linearised model. The disturbance input is set to τl = 50 Nm.
Figure 3.9 shows the steady state output values of the engine models, for different input
values α and γ. Both, the nonlinear engine system and its linearised system are observed
in an uncontrolled open loop setup. The model’s output value in steady state condition, is
plotted for a sequence of different input values for α and γ.
nonlinear plant
linearised plant
160
Engine speed ωe [rad/s]
Engine speed ωe [rad/s]
180
140
120
100
260
240
220
200
180
nonlinear plant
linearised plant
80
0
0.5
1
1.5
Time [s]
2
2.5
160
3
0
0.5
(a) α = 3, γ = 0
2
2.5
3
180
Engine speed ωe [rad/s]
Engine speed ωe [rad/s]
1.5
Time [s]
(b) α = 7, γ = 0
180
170
160
150
nonlinear plant
linearised plant
140
1
0
0.5
1
1.5
Time [s]
2
(c) α = 5, γ = 15
2.5
3
170
160
150
nonlinear plant
linearised plant
140
0
0.5
1
1.5
Time [s]
2
2.5
3
(d) α = 5, γ = −15
Figure 3.8: Open loop response of the engine model after a step is being applied
from α = 5 and γ = 0 to the indicated input values.
The transient of the linearised model is a bit different than that of the nonlinear model at
the boundaries of the operating range, as can be seen in Figure 3.8. From Figure 3.9, it
can be concluded that the steady state conditions of the linear model are quite the same as
18
nonlinear engine model
linearised engine model
linearising point
280
260
engine speed ωe [rad/s]
240
220
200
180
160
140
120
7
100
6
80
5
15
10
5
0
valve prolongation γ [deg]
4
−5
−10
−15
3
throttle angle α [deg]
Figure 3.9: Steady state output of the nonlinear and the linearised engine model
for different inputs.
the nonlinear model because the linearised surface does not differ much from the nonlinear
surface, especially around the linearising point.
Note the difference in slope of the surfaces in both orthogonal directions in Figure 3.9. A
change in throttle angle α has a bigger influence on the system than a change in valve
opening duration γ. This shows that our engine model can not be regulated by variable
valve timing alone as it can not reach the full range of operation in steady state. Within
the limits of γ, the engine speed can only be manipulated in a range of about 20 rad/s,
keeping the throttle valve opening fixed. However, it does not say that variable valve timing
is useless or gives no contribution, because it might affect the transient.
Chapter 4
Controlling the Engine Model
In this chapter, the engine model presented in Chapter 3, will be evaluated in a closed loop
setup as depicted in Figure 4.1. A controller will be designed to actuate the two engine
inputs and having an error feedback. A dynamic input allocation block will redistribute
those input signals for optimal exploitation. Finally, the closed loop setup will be simulated.
d
r
LQG
controller
yc1
yc 2
input
allocator
u1
u2
engine model
y
y
Figure 4.1: Closed loop control diagram.
4.1
LQG Controller
Because the idea of the project is to exploit the two actuator inputs using the dynamic
control allocation technique described in [Zac07], we chose to follow this paper in designing
the controller. [Zac07] elaborates its allocation theory accompanied by examples, where
first a Linear Quadratic Gaussian (LQG) controller is designed. The LQG theorem has also
been applied to the linearised engine model. Note that, according to [Zac07, Footnote 1],
the dynamic input allocation is not limited to one specific linear controller, but can be used
with any locally Lipschitz controller.
The engine controller will be designed, guaranteeing asymptotic tracking of a constant reference engine speed. This will be accomplished by having a negative unit feedback and
19
20
d
r +
e
−
x̂
∫e
∫
Gx  T
− 
 Gi 
 
u
w
v
y
Linearised
engine model
y
B
+
+
+
x̂ɺ
A
∫
x̂
L
C
−
Kalman
filter
+
ŷ
y
Figure 4.2: LQG controller.
by adding an integrated tracking error state. Figure 4.2 shows a diagram of the described
controller.
The LQG problem is also known as the separation theorem because the LQG problem and
its solution can be separated into two distinct parts, namely a state feedback part and a
Kalman filter.
4.1.1
Augmented Plant
The first step in designing our controller is to augment the plant – which is the linearised
engine model – with an integrated error state. The error is defined as e = r − y, where y is
the output of the plant and r is a constant reference engine speed.
"
ẋ
e
#
=
"
Ax + Bu
r − (Cx + Du)
#
=
"
A 0
−C 0
#"
x
R
e
#
+
"
B
−D
#
u+
"
0
I
#
r
(4.1)
= A x + B u + Ir
A and B are the augmented state-space matrices and x are the corresponding new states.
21
CHAPTER 4. CONTROLLING THE ENGINE MODEL
4.1.2
Optimal State Feedback
An optimal state feedback regulator, also known as Linear Quadratic Regulator, is designed
based on the true states. We assume an unconstrained quadratic programming problem
as the pressures are naturally bounded, the engine speed is tracked and both inputs can
be bounded with the dynamic control allocator if needed. We also assume an infinite time
horizon, as there is no strict point in time tf at which the output has to reached a certain
working point, i.e. tf = ∞. This results in the following cost function to be minimised:
JLQR =
Z∞
0
xT Qx + uT Ru dt
(4.2)
subject to ẋ = A x + Bu
where Q and R are weighting matrices to be chosen.
The solution of this equation is of the form u = −Gx, where G = R−1 B T P and P = P T >
0 is the unique positive definite solution of the continuous time matrix algebraic Riccati
equation
AT P + P A − P BR−1 B T P + Q = 0.
Finally, the feedback control gain G can be separated into state and integrated error gains
[Gx Gi ].
Q and R are chosen weighting matrices such that Q = QT ≥ 0 and R = RT > 0; often in
a block diagonal form. The diagonal elements of Q and R can be chosen different from
each other in order to value the trade-off between control error and control effort. Putting
more weight on Q, minimises the quadratic integral for the smallest transient area, whereas
putting more weight on R, optimises for the smallest input needed.
By choosing appropriate diagonal elements, the matrix R can be adapted to give more weight
to a certain input relative to the other. The matrix Q can put weight on the states x, which
R
means we can promote or penalise the integrated error state e above the states x, as well
as the states individually.
These matrices can be seen as design variables, and the cost function (4.2) will be set by
certain given design criteria. Regarding this design example, we take
R =
"
1 0
0 1
#
,



Q = 

1
0
0
0
0
1
0
0
0 0
0 0
1 0
0 100



.

This means that we put more weight on minimising the error area, which makes sense and
is commonly used.
22
4.1.3
Kalman Filter
The optimal regulator problem discussed so far, is deterministic. The feedback gain is
calculated using the fact that all states are supposed to be known. A practical problem
exists in the fact that there are disturbances which are acting upon the process and affecting
the states. This can also be seen as unmodelled behaviour, which has an unpredictable
behaviour on the system. A second fact is that measuring the output with a feedback sensor
also inducing uncertainties and disturbances to the system.
The effect of disturbances will be taken into account by extending the process description
(3.15) by means of the addition of zero-mean, white Gaussian process and measurement
noise, such that we obtain
ẋ = Ax + Bu + Bd d + w
y = Cx + Du + Dd d + v
(4.3)
where w(t) is the process noise and v(t) the measurement noise.
We define W and V as the covariance matrices W = E w(t) wT(t) and V = E v(t) v T(t) ,
where E denotes the expected value. W and V are diagonal matrices for zero-mean Gaussian
white noise signals. Because there is currently no information available about the disturbances affecting the system and the method of measuring the output, W and V are assumed
to be the identity matrix.
In order not to have feedback from the states containing noise, ‘deterministic’ states have
to be estimated. The estimated states can be reconstructed using a Linear Quadratic Estimator, i.e. a Kalman filter of the following form:
x̂˙ = Ax̂ + Bu + L (y − ŷ)
ŷ = C x̂.
(4.4)
The Kalman filter is minimising the steady state error covariance between the true states
and the estimated states
JLQE =
n
o
T
lim E (x(t) − x̂(t)) (x(t) − x̂(t))
t→∞
(4.5)
The Kalman gain L is determined through solving L = SC T V −1 , where S = S T > 0 is found
by solving the algebraic Riccati equation
0 = AS + SAT − SC T V −1 CS + W.
23
CHAPTER 4. CONTROLLING THE ENGINE MODEL
4.1.4
LQG: Combined LQR and LQE
Finally, the overall controller is constructed. The combined observer and control gain is
described by the following equations:
R
u = −Gx x̂ − Gi e
ŷ = C x̂
x̂˙ = Ax̂ + Bu + L (y − ŷ)
R
= Ax̂ − BGx x̂ − BGi e + Ly − LC x̂
This yields the final controller, represented in state space format:
" #
"
#"
# "
#
" #
x̂˙
A − BGx − LC −BGi
x̂
L
0
R
=
+
y+
r
e
0
0
e
−I
I
h
u =
4.1.5
−Gx −Gi
i
"
x̂
R
e
(4.6)
#
Controller Example
When selecting state weight Q, input weight R, process noise weight W and measurement
noise weight V , respectively as



Q = 

1
0
0
0
0
1
0
0
0 0
0 0
1 0
0 100



,

R =
"
1 0
0 1
#
,

1 0 0


W =  0 1 0 ,
0 0 1

V
=
h
1
i
;
the resulting controller is
"
Ac Bc Br
Cc Dc Dr
#





= 





−83615
982.7
1861.6
−8350 −23.381 0

1295
−16603 −37051 164591 0.0767 0 

0.943
−10.37 −30.00 119.52
0.1911 0 
.
0
0
0
0
−1
1 


−0.9969 −0.0661 −0.1803 0.7987
0
0 
−0.0768 0.8644
2.5055 −9.9681
0
0
(4.7)
24
4.2
Dynamic Input Allocator
For designing the dynamic input allocator, consider the linear system description (3.15).
[Zac07] distinguishes strongly and weakly input redundant systems. A plant is strongly
input redundant if the dynamic input allocator does not affect the state-response of the
plant. The plant output responses coincide for all times for both systems with and without
B ]) 6= ∅.
input allocator. This condition hold if it satisfies ker([ D
A plant is weakly input redundant if the dynamic input allocator does not affect the steady
state output response of the plant, however it does affect the transient. This condition holds
if it satisfies
ker(P ⋆ ) 6= ∅
where P ⋆ is the steady state transfer which is defined as
−1
P ⋆ = lim C (sI − A) B + D .
s→0
In case of our engine application only the weakly input redundant condition holds. We
define a matrix B⊥ such that the image of B⊥ is equal to the nullspace of P ⋆ :
im(B⊥ ) = ker(P ⋆ )
#
"
h
i
−0.0094
.
= ker 45.35 0.43
=
1
The dynamic input allocator will be defined as
(
T
ẇ = −KB⊥
W̄ u
u = yc + B⊥ w
(4.8)
(4.9)
(4.10)
with yc being the output of the controller and u the input of the engine model. Rewritten
in state-space format gives
"
#
"
#
T
T
Aa Ba
−KB⊥
W̄ B⊥ −KB⊥
W̄
=
,
(4.11)
Ca Da
B⊥
I
where K and W̄ are suitable matrices to be explained next. If we choose W̄ as a diagonal
matrix, it is possible to promote or penalise certain plant inputs. The higher a diagonal
element has been chosen, the more that input will be penalised.
Selecting a suitable K, can be useful to penalise certain input directions, if there are multiple
redundant inputs. By altering K proportionally, the speed of the dynamic allocator can be
changed, as long as the whole system with input allocation is internally stable and the plant
output response converge to a steady state value independent of any converging external
signals r(·) and d(·).
25
CHAPTER 4. CONTROLLING THE ENGINE MODEL
Parameter K only affect the speed of the allocation and not the steady state conditions.
This can be seen from the expression for the steady state plant input allocation
−1 T ⋆
T
u⋆ = I − B⊥ B⊥
W̄ B⊥
B⊥ W̄ yc
which is independent of K.
When selecting K = [5] and W̄ = [ 10 01 ]; the input allocator

"
#
−5
0.0470
Aa Ba

=  −0.0094
1
Ca Da
1
0
block results in

−5

0 .
1
(4.12)
The structure of the allocator block reveals that it sums up all its inputs and divides that
integrated cumulative to its outputs, according to the directions of B⊥ .
4.3
Simulating the Closed Loop System
Finally the obtained controller, allocator and linearised engine system are implemented in
MATLAB Simulink. The model can be seen in Figure 4.3. The additions of constants are
needed because the model is linearised around {αo , γo , τl,o } = {5, 0, 50} with corresponding
equilibria ωe,o = 168.21 and the closed loop needs to be compensated for that.
[alpha_0 gamma_0]
[alpha_0 gamma_0]
[5 0]
[5 0]
controller output
omega_e_0.
plant input
omega_e_0
-C-
Step to ...
w_e [rad/s]
-C-
x' = Ax+Bu
y = Cx+Du
controller
x' = Ax+Bu
y = Cx+Du
x' = Ax+Bu
y = Cx+Du
50
input allocator
tau_l
50
linear engine model
plant output
= omega_e
tau_l_0
Figure 4.3: Closed loop implementation in Simulink.
Figure 4.4 shows the system in Simulink, where the linearised engine model is replaced by
the nonlinear engine model.
To evaluate the controlled engine setup and to compare the linearised model with the nonlinear engine model, some simulations have been carried out. Figure 4.5 shows the behavior
of both systems under changing reference input or disturbance input. These simulations
are without input allocation, done by setting K = [0] which is equivalent to removing the
26
[alpha_0 gamma_0]
[alpha_0 gamma_0]
[5 0]
[5 0]
controller output
plant input
omega_e_0.
[alpha_0 gamma_0]
-C-
[5 0]
x' = Ax+Bu
y = Cx+Du
Step to ...
w_e [rad/s]
x' = Ax+Bu
y = Cx+Du
controller
input allocator
engineNonlinearModel
S-Function
50
plant output
= omega_e
tau_l
-Comega_e_0
Figure 4.4: Closed loop implementation in Simulink where the linearised engine
model has been replaced by the original nonlinear model.
allocation block in Simulink. The used linear plant and controller state-space matrices are
(3.18) and (4.7):
"
A B
C
"
Bd
D Dd
#
Ac
Bc
Br
Cc
Dc Dr



= 

#
−12.62
0
−1965 83283
7510
0
27.66
−2330
4319
0
−16512
0
0.02249 −0.002113 0.2317
0
−11.99 −6.6667
0

−83615
1295
0.943
0
0
982.7
−16603
−10.37
0
1
1861.6
−37051
−30.00
0
0
0
−8350
164591
119.52
0




= 



 −0.9969 −0.0661 −0.1803 0.7987
−0.0768 0.8644
2.5055 −9.9681
0
−23.381
0.0767
0.1911
−1
0
0
0
0
0
1



,

(3.18)





.



0 
0
(4.7)
Note that using a linear controller to control the nonlinear engine model is not valid anymore,
when applying a step in load torque from 50 [Nm] to more than 130 [Nm]. The solution
does not converge to a steady state value anymore.
27
CHAPTER 4. CONTROLLING THE ENGINE MODEL
260
180
250
170
240
160
230
150
ω ; nonlinear plant
e
ω ; linear plant
Engine Output
Engine Output
e
220
210
200
190
140
130
120
110
180
100
ωe ; nonlinear plant
170
90
ω ; linear plant
e
160
0
1
2
3
4
5
6
7
8
9
80
10
15
0
3
4
5
6
7
8
9
10
10
α
α
5
0
Controller Output
Controller Output
2
15
10
γ
−5
α ; nonlinear plant
γ ; nonlinear plant
α ; linear plant
γ ; linear plant
−10
−15
1
0
1
2
3
4
5
Time
6
7
8
9
5
0
γ
−5
α ; nonlinear plant
γ ; nonlinear plant
α ; linear plant
γ ; linear plant
−10
−15
10
(a) Step in speed from 168 to 250 [rad/s]
0
1
2
3
4
5
Time
6
7
8
9
10
(b) Step in speed from 168 to 100 [rad/s]
180
240
170
230
160
220
150
210
ω ; nonlinear plant
e
ω ; linear plant
Engine Output
Engine Output
e
140
130
120
110
200
190
180
170
100
160
ωe ; nonlinear plant
90
150
ω ; linear plant
e
80
0
1
2
3
4
5
6
7
8
9
140
10
15
0
3
4
5
6
7
8
9
10
10
α
α
5
0
Controller Output
Controller Output
2
15
10
γ
−5
α ; nonlinear plant
γ ; nonlinear plant
α ; linear plant
γ ; linear plant
−10
−15
1
0
1
2
3
4
5
Time
6
7
8
9
(c) Step in load torque from 50 to 100 [Nm]
5
0
γ
−5
α ; nonlinear plant
γ ; nonlinear plant
α ; linear plant
γ ; linear plant
−10
10
−15
0
1
2
3
4
5
Time
6
7
8
9
(d) Step in load torque from 50 to 0 [Nm]
Figure 4.5: Linear controller simulated with the linear and nonlinear engine model,
with the allocator off; during a step in speed or load torque at t = 1s.
10
260
260
250
250
240
240
230
230
Engine Output
Engine Output
28
220
210
200
190
220
210
200
190
180
180
170
170
ω
ω
e
0
2
4
6
8
10
12
14
16
18
e
160
20
8
8
6
6
4
4
2
2
Controller Output
Controller Output
160
0
−2
−4
−6
2
4
6
8
10
12
14
16
18
20
0
−2
−4
−6
−8
−8
α
γ
−10
−12
0
0
2
4
6
8
10
Time
12
14
16
18
α
γ
−10
−12
20
(a) without variable valve opening duration
0
2
4
6
8
10
Time
12
14
16
18
20
(b) with variable valve opening duration
260
260
240
240
Engine Output
Engine Output
Figure 4.6: Linear controller simulated with the nonlinear engine model, without
input allocation, during a step in speed from 168 to 250 [rad/s] at t = 1s.
220
200
180
220
200
180
ω
ω
e
160
0
1
2
3
4
5
6
7
8
9
e
160
10
0
−5
−10
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
Time
6
7
8
9
0
−5
−10
0
1
2
3
4
5
6
7
8
9
10
5
Engine Input
5
Engine Input
1
5
Controller Output
Controller Output
5
0
0
−5
α
γ
−10
0
1
2
3
4
5
Time
6
7
(a) without input allocation
8
9
0
−5
α
γ
−10
10
(b) with input allocation
Figure 4.7: Linear controller simulated with the nonlinear engine model, with and
without input allocation, during a step in speed from 168 to 250 [rad/s] at t = 1s.
10
29
CHAPTER 4. CONTROLLING THE ENGINE MODEL
4.4
Influence of the Variable Valve Opening Duration
Figure 4.6 shows the advantage of using an extra control input, namely the addition of a
variable valve opening duration. After giving a step in the reference speed, it takes 2.1
seconds to settle out, whereas the throttle valve configuration takes about 14.2 seconds to
settle out towards its steady state value, within 1% of the applied step (that is here to
250 ± 0.82 rad/s). The addition of variable valve opening duration results in a more than
six times smaller settling time in this example.
While giving a step in the load torque disturbance, the same result is achieved regarding a
shorter settling time.
The controller of Figure 4.6a has been constructed by removing the second column of matrix
B of (4.1) during the controller design. It means that we change the SIMO-controller to a
SISO-controller. The dynamic allocation is switched off by selecting K = [0].
Both simulations uses the same Q and R matrices to compare the two, having the same
control error and control effort.
4.5
Influence of the Allocator
Next, the dynamic control allocator is activated having K
the used allocator block is

"
#
−5
0.0470
Aa Ba

=  −0.0094
1
Ca Da
1
0
= [5]. The state-space model of

−5

0 .
1
(4.12)
Figure 4.7 shows the effect of having dynamic input allocation. Having dynamic control
allocation, the output of the controller is not equal to the input of the engine model. Two
extra subplots show the actual signals into the engine model, which are the redistributed
signals coming from the allocator.
It is stated in Section 4.2 and it can be seen in Figure 4.7, that the allocator does not have
an effect on the steady state output response, but it does affect the transient.
The practical implementation of the dynamic allocation does not have an effect on the
settling time of the engine speed: both about 2 seconds. It is even slightly worse due to the
larger overshoot.
30
Chapter 5
Conclusions and
Recommendations
5.1
Conclusions
In this report, a nonlinear model of an internal combustion engine has been presented. It
is designed for control purposes, involving the effect of a intake valve opening prolongation.
The latter is accomplished by parametrising the volumetric efficiency to a parameter γ which
is a measure for the amount of valve opening prolongation.
A linear engine model has been derived by substituting all parameters from a real engine
into the set of equations and linearising it. The nonlinear model and the linearised model
are proven to be applicable for the purpose they have been designed for, namely in a closed
loop control environment.
The engine model, actuated by throttle and controlled by a linear quadratic Gaussian controller, is able to track a prescribed engine speed profile and accommodate for varying load
torque disturbances.
The addition of variable valve opening duration results in a substantial faster settling time
of the engine speed while tracking a certain engine speed trajectory or accommodate for
load torque disturbances.
The practical implementation of dynamic control allocation does not have an effect on the
settling time of the engine speed. An advantage is that the amount of sweeping the camshaft
(denoted by γ) becomes almost zero in steady state. This is desired because it takes energy
to constantly phasing the camshaft in order to get a change in valve opening duration.
31
32
5.2
Potential Automotive Applications
The concept of using a variable cam phaser in a way to prolong the valve opening, seems to
give an improvement in engine performance and also gives more freedom to tune the engine
parameters. To implement this system there is no need to make changes to the hardware of
the engine. Only changes in the management system and in the actuation of the variable
cam phaser are required.
Some car manufacturers already see the advantage of having more freedom in timing the
opening and closing of intake and exhaust valves. This is shown by the upcoming interest
and prototypes of unthrottled, camless engines. They use a electro-mechanical actuator for
each valve. The application presented in this report, can be used to ‘approach’ this freedom
in timing, using only standard components. Therefore, it is a cheap solution and easily
implementable in new engine designs, however it does not provide the full functionality of
a camless engine. The valves of a camless valve system can be opened at any time or even
stay closed, whereas the variable valve timing system is set to certain constrains.
Some ideas about parametrisation of the volumetric efficiency and control presented in this
report, can be useful during research on camless engines.
The advantage of implementing the varying valve opening duration system, is a faster response to a requested engine speed. This gives a better sense to the driver while cruising on
a certain speed and it can also be useful while changing gear with an automatic gearbox.
5.3
Shortcomings
In the work, presented in this report, several assumptions and simplifications have been
made. Also, some results are dubious; such as the quantification of the volumetric efficiency,
describing the effect of a varying intake valve opening duration in the engine model. The
quantification has been done by simulation using a detailed engine model, programmed
in the multi-physical software package Dymola. The mapping of the volumetric efficiency
appears to be dependent on the chosen computational step size of the Dymola model, which
should not be possible. Secondly, this Dymola model is only valid for engine speeds between
1000 rpm and 2500 rpm and intake manifold pressures between 20 kPa and 80 kPa, through
which the engine model presented in this report is only trustful while operating in this range.
During controller design, assumptions are made on the nature and intensity of disturbance
signals affecting the system and are not physically validated. Besides this, the actuator
dynamics of the throttle valve and camshaft phasing device are neglected or assumed to be
perfect, that is, having no inertia and no rate or saturation limits.
CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS
5.4
33
Recommendations
In order to have more reliable results on the influence of involving valve opening prolongation,
the shortcomings have to be eliminated. The Dymola model has to be checked, and if
necessary to be improved. It would be better to perform measurements on the real engine. Or
possibly, reconstruct the volumetric efficiency using available measurement data, obtained
during test with different cam shapes.
The disturbances and uncertainties have to be quantified to come to a better controller
design. Besides, the system has to deal with the electro-mechanical limits of the actuators,
so consider the torque and positioning bounds. The maximum torque an actuator can apply
to overcome its inertia, sets the acceleration and this defines the minimum time period an
actuator is able to travel to a new prescribed position. The magnitude and rate saturation
limits of input channel γ, the measure for the amount of sweeping the camshaft, are defined
in Appendix A. The theory on the dynamic allocation technique described in [Zac07], also
treats the implementation of such saturation limits by modifying the allocator structure.
This report focusses on controlling an engine, having a throttle valve and variable valve
timing, and evaluates the ‘mechanical’ outputs speed and torque. It would also be interesting
to evaluate the setup on fuel efficiency, pollutant gasses and others things of importance in
engine design.
Finally, it would be great to perform tests on a real engine. To do this, the actuation of the
variable camshaft phaser has to be modified to produce a sinusoid with feedback. Before
testing, the controller has to work well and being robust.
34
Appendix A
Valve Opening Duration
To be able to manipulate the valve opening angle and lift duration, a device can be used
called a camshaft phaser. A picture of such an actuator can be seen in Figure A.1. This
particular example is actuated by applying oil pressure to the cam phaser, via an oil control
valve, actuated by a dc-motor [Del09].
Figure A.1: Picture of a cam phaser and its implementation.
All of nowadays engines are equipped with some kind of cam phaser device. Most of them
are open loop controlled in two or three positions i.e. advanced or retard position. We will
assume, we have a device that can be positioned in every position between −20◦ and +20◦
and with a maximum speed of 200◦/s.
Consider the case where we change the camshaft to advance position during the opening of
the intake valve and change it back to retard position during the closing of the intake valve.
This results in a longer opening time of the intake valve. This case is illustrated in Figure
A.2. Of course, a shorter opening time can be achieved by actuating the other way around.
Due to the maximum actuator speed, there is a limit in the prolongation which can be
achieved, and intuitively this depends on the engine speed. This limit is next to be defined.
35
36
Lv
valve lift
exhaust
valve
intake
valve
cam angle ϕ
relative cam change
∆ϕ
γ
cam angle ϕ
200 ° s
relative cam velocity
∆ϕɺ
cam angle ϕ
Figure A.2: The middle plot shows the angular shift of the camshaft with respect
to the normal cam angle. The lower plot its derivative. The upper plot shows the
influence on the valve lift.
First start with the equations for the cam angle change and its derivative.
∆ϕ = −γ cos ϕ
d∆ϕ
= γ ϕ̇ sin ϕ
dt
(A.1)
with ϕ̇ = 12 ωe because of the fact that the speed of the camshaft is half the speed of the
crankshaft at a four-stroke engine.
Finally, let us equalise (A.1) to 200◦/s at ϕ = 90◦ as shown in Figure A.2.
1
γ 12 ωe sin 90◦ 360
= 200◦ /s
◦
γ = 200◦ /s
2π
360◦
180◦ · ωe rad
s
=
2513
ωe rad
s
(A.2)
As an example, the maximum amount of sweeping the camshaft is γ = ±15◦ at 1600 rpm
(= 168 rad/s).
Appendix B
Friction Model
To simulate the engine model, it has to contain a friction model which approaches real engine
friction.
First, the engine has pumping losses. It is the result of flow resistances while gasses are
pushed out and pulled into the cylinders during exhaust and intake strokes. When writing
this friction in terms of pressures, it is equal to the difference of the exhaust manifold
pressure Pem and the intake manifold pressure Pim :
pmep = Pem − Pim
(B.1)
Second, when the engine is running, solid surfaces are moving relative to each other, causing
mechanical friction, whether the rubbing surfaces are lubricated or not. Also, auxiliary
components which are running along with the engine, inducing mechanical friction to the
engine. The engine accessories are the oil pump, the water pump, the fuel pump and the
alternator.
B.1
Component Mechanical Friction Models
The mechanical friction losses of the rubbing engine components are divided into three
component groups:
• Crankshaft: Main bearings, front and rear main bearing oil seals.
• Reciprocating: Piston skirts, piston rings, connecting rod bearings.
• Valvetrain: Camshafts, cam followers and valve actuation mechanisms.
For each of the component groups, an expression for the losses, expressed in pressures, are
drawn up. These are corrected by a factor 1000
nc to convert them to pressures in [Pa] and
37
38
the independency of the number of cylinders. The parameters, used in the equations, are
explained in Table B.1. If a parameter has a constant value, corresponding to the real Ford
engine, it is also listed in this table.
Symbol
Definition
afmep
B
Cff
Crf
Coh
Com
cfmep
Db
fmep
Lb
Lv
µs
N
nb
nc
nv
pmep
rfmep
S
Sp
Toil
vfmep
Auxiliary friction mean effective pressure
Bore
Constant “Flat follower”
Constant “Roller follower”
Constant “Oscillating hydrodynamic”
Constant “Oscillating mixed”
Crankshaft friction mean effective pressure
Bearing diameter
Friction mean effective pressure
Bearing length
Maximum valve lift
Scaling term for oil viscosity
Engine speed
Number of bearings
Number of cylinders
Number of valves
Pumping mean effective pressure
Reciprocating friction mean effective pressure
Stroke
Mean piston speed
Oil temperature
Valvetrain friction mean effective pressure
Value
92.25
400
0.0151
0.5
21.4
80.1
32.4
11
340−Toil
148.4
nc + 1
6
4nc
99.31
2SN
60
321
Unit
[Pa]
[mm]
[-]
[-]
[-]
[-]
[Pa]
[mm]
[Pa]
[mm]
[mm]
[-]
[rpm]
[-]
[-]
[-]
[Pa]
[Pa]
[mm]
[m/s]
[K]
[Pa]
Table B.1: List of symbols and their values used in the friction model.
B.1.1
Crankshaft Friction
The expression for the crankshaft friction is:
cfmep =
Db
N Db3 Lb nb
D 2 N 2 nb
1.22·10 2
+ 3.03·10−4µs
+ 1.35·10−10 b
2
B Snc
B Snc
nc
5
!
1000
nc
(B.2)
The first term gives the friction of the main bearing seals. The second term encounters the
main bearing hydrodynamic friction. A viscosity scaling term µs is included to compensate
for oil temperature variations, affecting the oil viscosity. The last term accounts for the
turbulent dissipation, the work required to pump fluids through flow restrictions.
39
APPENDIX B. FRICTION MODEL
B.1.2
Reciprocating Friction
The expression for friction, caused by the reciprocating motion is:
rfmep =
!
Sp
1000 1
N Db3 Lb nb 1000
4
−4
2.94·10 µs
+ 4.06·10 1+
+ 3.03·10 µs
(B.3)
B
N B2
B 2 Snc
nc
2
The first term gives the piston friction assuming hydrodynamic lubrication. The second
term is for the piston ring friction under mixed lubrication. The last term accounts for
friction from the hydrodynamic journal bearing of the connecting rod.
B.1.3
Valvetrain Friction
The expression for the valvetrain friction is:
vfmep =
500 nv
N nv
N nb
+ Cff µs 1 +
+ Crf
+ ...
244µs 2
B Snc
N
Snc
Snc
!
0.5
L1.5
nv
500 Lv nv 1000
v N
Coh µs
+ Com 1+
BSnc
N
Snc
nc
(B.4)
The first term represents the camshaft bearing hydrodynamic friction. The next two terms
predict friction resulting from relative motion between the cam lobe and the cam follower.
The fourth term predicts friction caused by relative motion between valvetrain components
such as the valve lifter in the lifter bore or the valve in the valve guide. The fifth term
represents the oscillating mixed lubrication friction.
B.1.4
Auxiliary Friction
The expression to model the friction of auxiliary components is a calibrated function:
afmep =
8.63·10
−7
3
N − 5.20·10
−3
2
N + 30.5N + 601
!
1
nc
(B.5)
The calibration constants were determined from measurement data of the Ford engine. The
negative coefficient for the squared engine speed term results from the fact that less work is
required from the oil pump while running into the middle engine speed range.
40
B.2
Total Mechanical Friction Model
The total friction mean effective pressure is defined as:
fmep = cfmep + rfmep + vfmep + afmep
(B.6)
Substituting equations (B.2), (B.3), (B.4) and (B.5) into (B.6) and moreover, using the
60
parameter values listed in Table B.1 and the fact that N = 2π
ωe ; the following function for
fmep is obtained:
fmep = 1.05·105 + 67.0ωe − 0.0637ωe2 +
1
3.07·105
+ 1.81ωe2 + 0.125·10−3ωe3
ωe
where ωe is the engine speed in radians per seconds and fmep is measured in Pa.
(B.7)
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