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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 enginewith linear quadratic Gaussian control
and dynamic control allocation
Traineeship reportDCT doc. no.: 2009.023
J.P.R. JongeneelID number: 0610188
September 2008 - January 2009
Participating Institutes:
The University of Melbourne Eindhoven, University of Technology
Dr. C. Manzie ∗ Prof. dr. H. Nijmeijer
Prof. dr. D. Nesic †
∗ Dep. of Mechanical Engineering Dep. of Mechanical Engineering† Dep. of Electrical Engineering Postbus 513
Parkville 3010 5600 MB Eindhoven
Victoria, Australia The Netherlands
www.unimelb.edu.au www.tue.nl
Preface
This report is the result of a traineeship, carried out at the University of Melbourne, Aus-
tralia. 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 oppor-
tunity 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 3
2.1 Engine Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Engine Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Modelling the Volumetric Efficiency . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Redundant Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Modelling the Engine 7
3.1 Nonlinear Engine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.1 Mass Flow Intake Manifold In . . . . . . . . . . . . . . . . . . . . . . . 8
3.1.2 Mass Flow Intake Manifold Out . . . . . . . . . . . . . . . . . . . . . . 9
3.1.3 Mass Flow Exhaust Manifold In . . . . . . . . . . . . . . . . . . . . . 13
3.1.4 Mass Flow Exhaust Manifold Out . . . . . . . . . . . . . . . . . . . . 13
3.1.5 Engine Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.6 Nonlinear Engine Model . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Linearised Engine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Comparing the Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Controlling the Engine Model 19
4.1 LQG Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.1 Augmented Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.1.2 Optimal State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1.3 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.1.4 LQG: Combined LQR and LQE . . . . . . . . . . . . . . . . . . . . . 23
4.1.5 Controller Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Dynamic Input Allocator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Simulating the Closed Loop System . . . . . . . . . . . . . . . . . . . . . . . 25
4.4 Influence of the Variable Valve Opening Duration . . . . . . . . . . . . . . . . 29
4.5 Influence of the Allocator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
v
5 Conclusions and Recommendations 31
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Potential Automotive Applications . . . . . . . . . . . . . . . . . . . . . . . . 32
5.3 Shortcomings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.4 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
A Valve Opening Duration 35
B Friction Model 37
B.1 Component Mechanical Friction Models . . . . . . . . . . . . . . . . . . . . . 37
B.1.1 Crankshaft Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
B.1.2 Reciprocating Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
B.1.3 Valvetrain Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
B.1.4 Auxiliary Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
B.2 Total Mechanical Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . 40
Bibliography 41
vi
Nomenclature
Symbols
A Area [m2](
AF
)
sStoichiometric air/fuel ratio [-]
Cd Discharge coefficient [-]
Hl Lower heating value [J kg−1]
J Moment of inertia [kgm2]
Lv Valve lift [m]
m Mass flow [kg s]
ncyl Number of cylinders [-]
nr Number of revolutions per cycle [-]
P Pressure [Pa]
R Specific gas constant [J kg−1 K−1]
t Time [s]
T Temperature [K]
V Volume [m3]
Vd Displacement volume of a cylinder [m3]
α Throttle angle [◦]
αcl Throttle angle at closed position w.r.t. normal axis [◦]
γ Measure for the valve opening prolongation [◦]
ζ Spark angle [◦]
ηi Indicated engine efficiency [-]
ηvol Volumetric efficiency [-]
κ Ratio of specific heats [-]
λ Normalised air/fuel ratio [-]
τe Torque produced by engine [Nm]
τl Load torque [Nm]
ϕ Camshaft angle [◦]
Ψ Nonlinear function to model sonic flow [-]
ωe Engine speed [rad s−1]
vii
Subscripts
�a Air
�amb Ambient
�CAC Cylinder air charge
�e Engine
�exh Exhaust
�eff Effective
�em Exhaust manifold
�f Fuel
�im Intake manifold
�o At linearising or operating point
�STP At standard temperature and pressure
�th Throttle
Notation
� Estimated
� Time derivative
� Augmented vector or matrix
�⋆ Steady state
Abbreviations
afmep Auxiliary friction mean effective pressure
BDC Bottom dead center
cfmep Crankshaft friction mean effective pressure
fmep Friction mean effective pressure
IC Internal combustion
IVO Intake valve opening
LQG Linear quadratic Gaussian
pmep Pump mean effective pressure
rfmep Reciprocating friction mean effective pressure
TDC Top dead center
vfmep 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
1By 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 in-
cylinder 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 mα is modelled as
mα = 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
mα =Pamb√R Tamb
Aeff (α)Ψ(
Pim
Pamb
)
.
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
fitting function, dependent on the throttle valve angle α. Likewise, Ψ(
Pim
Pamb
)
is a fitting
function to model sonic flow, as a function of the normalised intake manifold pressure Pim.
CHAPTER 2. LITERATURE REVIEW 5
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 mim, 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 = mim
R Tim
PimVd
4π
ωencyl
.
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 SIMO-
controller) can be designed using classical or modern controller design techniques. The
controller gain for each controller output channel is designed using the same feedback infor-
mation. 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.
plant modelcontroller allocator
outputreference
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 distri-
bution. [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.
α
ɺinimm
ɺoutimmimP emP
eω
ɺinemm
ɺoutemm
ambP ambP
∆+ϕ ϕ ϕ
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 Pim and Pem. 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 ωe is needed. For each of the air mass flow terms m(·) and the net produced
7
8
engine torque τe in (3.1), appropriate functions have to be found.
Pim =κRTim,o
Vim
(mimin− mimout
)
Pem =κRTem,o
Vem
(memin− memout
)
ωe =1
Je
(τe − τl)
(3.1)
Parameter V is the volume of the specified cavity and R is the specific gas constant, which
is 286.9[
JkgK
]
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 mimin, the mass-flow at the inlet of the intake
manifold [GuO04], is
mimin= Aeff (α) · Pamb√
RTamb
· Ψ(
Pim
Pamb
)
(3.2)
Aeff (α) = CdthA(α) = Cdth
Ath
(
1 − cos (α + αcl)
cos (αcl)
)
(3.3)
Ψ(
Pim
Pamb
)
=
√
κ(
2κ+1
)κ+1κ−1
for Pim
Pamb≤(
2κ+1
)κ
κ−1
(
Pim
Pamb
)1κ
√
2κκ−1
[
1 −(
Pim
Pamb
)
κ−1κ
]
for 1 ≥ Pim
Pamb>(
2κ+1
)κ
κ−1
(3.4)
Ψ(
Pim
Pamb
)
≈
√
κ
(
2
κ + 1
)
κ+1κ−1
(
1 − exp
(
9Pim
Pamb
− 9
))
(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 Cdthbeing 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.
CHAPTER 3. MODELLING THE ENGINE 9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.685
Pressure ratio Pim
/Pamb
Ψ
Ψpiecewise
Ψapproximate
Figure 3.2: Comparison of the piecewise and its approximated function for Ψ(
Pim
Pamb
)
at κ = 1.4.
3.1.2 Mass Flow Intake Manifold Out
The expression for mimout, 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.
mimout= ηvol(Pim, ωe, γ)
PimVd
RTim,o
ωencyl
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:
mimout=
∂mimout
∂ϕ
∂ϕ
∂t= ncylnivπDv
Pim√RTim
ωe
2
0∮
2π
{
Ψ(
Pc(ϕ)Pim
)
·Cdiv(Lv)·Lv(ϕ)
}
dϕ (3.7)
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 discrete-
event 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 in-
troduced. 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 pro-
file will be shifted with 0.6γ. The camshaft profile causes the valves being lifted. Fig-
ure 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 − 2513ωe[ rad
s]≤ γ ≤ 2513
ωe[ rads
], based on the
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.
CHAPTER 3. MODELLING THE ENGINE 11
0 90 180 270 360 450 540 630 683 7200
0.002
0.004
0.006
0.008
0.01
0.012
γ = −20 −10 0 +10 +20
spark
cam
pro
file
intake camexhaust cam
0 90 180 270 360 450 540 630 7200
0.5
1x 10
−3
BDC TDC BDC TDCpower stroke exhaust stroke intake stroke compression stroke
cham
ber
volu
me
[m3 ]
0 90 180 270 360 450 540 630 720
0
0.01
0.02
0.03
mas
s flo
w [k
g/s]
crankshaft angle [dgr]
mimout
memin
Figure 3.4: The lift of the valves, and the cylinder chamber mass flows and vol-
ume 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.
1000
1500
2000
250023
45
67
8
x 104
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ωe [rpm]P
im [Pa]
η vol [−
]
γ = −20
γ = −10
γ = 0
γ = +10
γ = +20
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,STP
=
mCACRTPCAC
mCAC,ST P RT
PCAC,ST P
→ mCAC
mCAC,STP
Pamb
Pim
with CAC standing for the cylinder air charge and STP for the conditions at standard tem-
perature 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) = −0.00038571γ2 − 0.015357γ + 0.67571. (3.8)
−15 −10 −5 0 5 10 150.3
0.4
0.5
0.6
0.7
0.8
0.9
γ [°]
η vol [−
]
simulation datadata fit
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 single cylinder, open loop model has been used, with parameters set to α = 0.5◦, τl = 100/6 Nm and
ζ = 47◦.
CHAPTER 3. MODELLING THE ENGINE 13
3.1.3 Mass Flow Exhaust Manifold In
The expression for memin, 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:
memin= mimout
(
1 +1
λ(
AF
)
s
)
(3.9)
where λ is the normalised air/fuel ratio λ = ma
mf(AF )
s
and(
AF
)
sis the stoichiometric air/fuel
ratio, which is 14.64 for gasoline.
3.1.4 Mass Flow Exhaust Manifold Out
The expression for memout, 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
memout= Cdexh
Aexh · Pem√
RTem,o
· Ψ(
Pamb
Pem
)
(3.10)
Ψ(
Pamb
Pem
)
≈
√
κ
(
2
κ + 1
)
κ+1κ−1
(
1 − exp
(
9Pamb
Pem
− 9
))
where Cdexhis 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:
τeωe = ηimfHl −ncylVdωe
2πnr
(pmep + fmep) (3.11)
mf =memin
λ(
AF
)
s
ηi = ηi(ωe,Pim
Pamb, 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
variables. The engine speed ωe, intake manifold pressure ratio Pim
Pamb, normalised air/fuel
14
1000
1500
2000
2500
2
3
4
5
6
x 104
0
0.1
0.2
0.3
0.4
0.5
ωe [rpm]P
im [Pa]
η i [−]
(λ =
1, ζ
= 3
7° )
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 − 0.0637ω2e +
3.07·105
ωe
+ 1.81ω12e + 0.125·10−3ω3
e . (3.12)
The expression for the engine torque now becomes:
τe = ηi(·)memin
λ(
AF
)
s
Hl
ωe
− ncylVd
2πnr
(Pem − Pim + fmep) (3.13)
3.1.6 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
CHAPTER 3. MODELLING THE ENGINE 15
Pim =κRTim,o
Vim
[
CdthAth
(
1− cos(α+αcl)cos(αcl)
)
Pamb√RTamb
0.685(
1−exp(
9 Pim
Pamb−9))
−ηvol(γ) PimVd
RTim,o
ωencyl
4π
]
Pem =κRTem,o
Vem
[
ηvol(γ) PimVd
RTim,o
ωencyl
4π
(
1+ 1
λ( AF )
s
)
−CdexhAexh
Pem√RTem,o
0.685(
1−exp(
9Pamb
Pem−9))
]
ωe = 1Je
[
ηi(Pim,ωe)1
λ(AF )
s
Hl
ωe
{
ηvol(γ) PimVd
RTim,o
ωencyl
4π
(
1+ 1
λ(AF )
s
)}
−ncylVd
4π(Pem−Pim+fmep)−τl
]
,
(3.14)
with ηvol, ηi and fmep being
ηvol = −0.00038571γ2 − 0.015357γ + 0.67571
ηi = 0.178 + 0.220·10−5Pim + 0.278·10−10P 2im + 0.247·10−2ωe − 0.502·10−7ωePim . . .
−0.117·10−12ωeP2im − 0.997·10−5ω2
e + 0.354·10−9ω2ePim − 0.268·10−14ω2
eP 2im
fmep = 1.05·105 + 67.0ωe − 0.0637ω2e + 3.07·105
ωe+ 1.81ω
12e + 0.125·10−3ω3
e .
3.2 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
{
x = Ax + Bu + Bdd
y = Cx + Du + Ddd(3.15)
where x = [Pim Pem ωe]T
is the plant state, u = [α γ]T
is the plant input, d = [τl] is a
disturbance input and y = [ωe] is the plant output.
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 =κRTim,o
Vim
[
CdthAth
(
1− cos(α+αcl)cos(αcl)
)
Pamb√RTamb
0.685(
1−exp(
9 Pim
Pamb−9))
−ηvol(γ) PimVd
RTim,o
ωencyl
4π
]
0 =κRTem,o
Vem
[
ηvol(γ) PimVd
RTim,o
ωencyl
4π
(
1+ 1
λ( AF )
s
)
−CdexhAexh
Pem√RTem,o
0.685(
1−exp(
9Pamb
Pem−9))
]
0 = 1Je
[
ηi(Pim,ωe)1
λ(AF )
s
Hl
ωe
{
ηvol(γ) PimVd
RTim,o
ωencyl
4π
(
1+ 1
λ(AF )
s
)}
−ncylVd
4π(Pem−Pim+fmep)−τl
]
(3.16)
16
Symbol Value Unit(
AF
)
s14.64 [-]
Aexh 0.00385 [m2]
Athπ0.0702
4 [m2]
Cdth0.85 [-]
Cdexh0.7 [-]
Hl 44.0 · 106 [J/kg]
Je 0.15 [kg m2]
ncyl 6 [-]
Pamb 101325 [Pa]
R 287.327 [ JkgK
]
Tamb 288.15 [K]
Tim 288.15 [K]
Tem 593 [K]
Vim 0.004 [m3]
Vd 0.0006638 [m3]
Vem 0.004 [m3]
αcl 7 [◦]
ηi f(Pim, ωe) [-]
ηvol f(γ) [-]
κ 1.40 [-]
λ 1 [-]
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 = 50Nm as operating conditions, lead to the following equilibrium point:
Pim,o = 26264 Pa
Pem,o = 101633 Pa
ωe,o = 1606 rpm
(3.17)
Linearising the equations leads to the following system matrices for equation (3.14):
[
A B Bd
C D Dd
]
=
−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 0 1 0 0 0
(3.18)
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.
P =[
B AB A2B]
rank(P) = 3
O =[
C CA CA2]T
rank(O) = 3
CHAPTER 3. MODELLING THE ENGINE 17
The eigenvalues λ and corresponding eigenvectors v of system (3.18) are
[
λ1 λ2 λ3
v1 v2 v3
]
=
−6.2 + 1.7i −6.2 − 1.7i −2330
−1 −1 0
−0.0058 + 0.0016i −0.0058− 0.0016i −1
0.0033 + 0.0009i 0.0033− 0.0009i 0
. (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 Pem. 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 = 50Nm.
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.
0 0.5 1 1.5 2 2.5 3
80
100
120
140
160
180
Eng
ine
spee
d ω
e [ra
d/s]
Time [s]
nonlinear plantlinearised plant
(a) α = 3, γ = 0
0 0.5 1 1.5 2 2.5 3160
180
200
220
240
260
Eng
ine
spee
d ω
e [ra
d/s]
Time [s]
nonlinear plantlinearised plant
(b) α = 7, γ = 0
0 0.5 1 1.5 2 2.5 3140
150
160
170
180
Eng
ine
spee
d ω
e [ra
d/s]
Time [s]
nonlinear plantlinearised plant
(c) α = 5, γ = 15
0 0.5 1 1.5 2 2.5 3140
150
160
170
180
Eng
ine
spee
d ω
e [ra
d/s]
Time [s]
nonlinear plantlinearised plant
(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
3
4
5
6
7
−15−10
−50
510
15
80
100
120
140
160
180
200
220
240
260
280
throttle angle α [deg]valve prolongation γ [deg]
engi
ne s
peed
ωe [
rad/
s]nonlinear engine modellinearised engine modellinearising point
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.
r
y
1uy
engine modelLQG
controllerinputallocator
1cy
d
2u2cy
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 ref-
erence engine speed. This will be accomplished by having a negative unit feedback and
19
20
r e
y
x Tx
i
G
G
−
u y+−
y
Kalmanfilter
Linearisedengine modele∫∫
B
∫
C
L
A
−
+y
x
xɺ
++
+
w vd
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.
[
x
e
]
=
[
Ax + Bu
r − (Cx + Du)
]
=
[
A 0
−C 0
][
x∫
e
]
+
[
B
−D
]
u +
[
0
I
]
r
= Ax + B u + Ir
(4.1)
A and B are the augmented state-space matrices and x are the corresponding new states.
CHAPTER 4. CONTROLLING THE ENGINE MODEL 21
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
=
∞∫
0
(
xT Qx + uT Ru)
dt
subject to x = Ax + Bu
(4.2)
where Q and R are weighting matrices to be chosen.
The solution of this equation is of the form u = −Gx, where G = R−1BT P and P = PT >
0 is the unique positive definite solution of the continuous time matrix algebraic Riccati
equation
AT P + PA − PBR−1BT 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
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 1 0
0 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
x = Ax + Bu + Bdd + w
y = Cx + Du + Ddd + 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) vT(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 distur-
bances 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 Esti-
mator, i.e. a Kalman filter of the following form:
˙x = Ax + Bu + L (y − y)
y = Cx.(4.4)
The Kalman filter is minimising the steady state error covariance between the true states
and the estimated states
JLQE
= limt→∞
E{
(x(t) − x(t)) (x(t) − x(t))T}
(4.5)
The Kalman gain L is determined through solving L = SCT V −1, where S = ST > 0 is found
by solving the algebraic Riccati equation
0 = AS + SAT − SCT V −1CS + W.
CHAPTER 4. CONTROLLING THE ENGINE MODEL 23
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:
u = −Gxx − Gi
∫
e
y = Cx˙x = Ax + Bu + L (y − y)
= Ax − BGxx − BGi
∫
e + Ly − LCx
This yields the final controller, represented in state space format:
[
˙x
e
]
=
[
A − BGx − LC −BGi
0 0
] [
x∫
e
]
+
[
L
−I
]
y +
[
0
I
]
r
u =[
−Gx −Gi
]
[
x∫
e
]
(4.6)
4.1.5 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 1 0
0 0 0 100
, R =
[
1 0
0 1
]
, W =
1 0 0
0 1 0
0 0 1
, V =
[
1]
;
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
input allocator. This condition hold if it satisfies ker([ BD ]) 6= ∅.
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
P ⋆ = lims→0
(
C (sI − A)−1
B + D)
.
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 ⋆) (4.8)
= ker([
45.35 0.43])
=
[
−0.0094
1
]
. (4.9)
The dynamic input allocator will be defined as
{
w = −KBT⊥Wu
u = yc + B⊥w(4.10)
with yc being the output of the controller and u the input of the engine model. Rewritten
in state-space format gives
[
Aa Ba
Ca Da
]
=
[
−KBT⊥WB⊥ −KBT
⊥W
B⊥ I
]
, (4.11)
where K and W are suitable matrices to be explained next. If we choose W 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(·).
CHAPTER 4. CONTROLLING THE ENGINE MODEL 25
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
u⋆ =(
I − B⊥
(
BT⊥WB⊥
)−1BT
⊥W)
y⋆c
which is independent of K.
When selecting K = [5] and W = [ 1 00 1 ]; the input allocator block results in
[
Aa Ba
Ca Da
]
=
−5 0.0470 −5
−0.0094 1 0
1 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.
50
tau_l_0
50
tau_l
plant output = omega_e
plant input
-C-
omega_e_0.
-C-
omega_e_0
x' = Ax+Bu y = Cx+Du
linear engine model
x' = Ax+Bu y = Cx+Du
input allocator
controller output
x' = Ax+Bu y = Cx+Du
controller
[5 0]
[alpha_0 gamma_0]
[5 0]
[alpha_0 gamma_0]
Step to ...w_e [rad/s]
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 non-
linear 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
50
tau_l
plant output = omega_e
plant input
-C-
omega_e_0.
-C-
omega_e_0
x' = Ax+Bu y = Cx+Du
input allocator
controller output
x' = Ax+Bu y = Cx+Du
controller
[5 0]
[alpha_0 gamma_0]
[5 0]
[alpha_0 gamma_0]
Step to ...w_e [rad/s]
engineNonlinearModel
S-Function
[5 0]
[alpha_0 gamma_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):
(3.18)
[
A B Bd
C D Dd
]
=
−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 0 1 0 0 0
,
(4.7)
[
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
.
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.
CHAPTER 4. CONTROLLING THE ENGINE MODEL 27
0 1 2 3 4 5 6 7 8 9 10160
170
180
190
200
210
220
230
240
250
260
Eng
ine
Out
put
ωe ; nonlinear plant
ωe ; linear plant
0 1 2 3 4 5 6 7 8 9 10−15
−10
−5
0
5
10
15
α
γ
Con
trol
ler
Out
put
Time
α ; nonlinear plantγ ; nonlinear plantα ; linear plantγ ; linear plant
(a) Step in speed from 168 to 250 [rad/s]
0 1 2 3 4 5 6 7 8 9 1080
90
100
110
120
130
140
150
160
170
180
Eng
ine
Out
put
ω
e ; nonlinear plant
ωe ; linear plant
0 1 2 3 4 5 6 7 8 9 10−15
−10
−5
0
5
10
15
α
γC
ontr
olle
r O
utpu
t
Time
α ; nonlinear plantγ ; nonlinear plantα ; linear plantγ ; linear plant
(b) Step in speed from 168 to 100 [rad/s]
0 1 2 3 4 5 6 7 8 9 1080
90
100
110
120
130
140
150
160
170
180
Eng
ine
Out
put
ωe ; nonlinear plant
ωe ; linear plant
0 1 2 3 4 5 6 7 8 9 10−15
−10
−5
0
5
10
15
α
γ
Con
trol
ler
Out
put
Time
α ; nonlinear plantγ ; nonlinear plantα ; linear plantγ ; linear plant
(c) Step in load torque from 50 to 100 [Nm]
0 1 2 3 4 5 6 7 8 9 10140
150
160
170
180
190
200
210
220
230
240
Eng
ine
Out
put
ω
e ; nonlinear plant
ωe ; linear plant
0 1 2 3 4 5 6 7 8 9 10−15
−10
−5
0
5
10
15
α
γ
Con
trol
ler
Out
put
Time
α ; nonlinear plantγ ; nonlinear plantα ; linear plantγ ; linear plant
(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.
28
0 2 4 6 8 10 12 14 16 18 20160
170
180
190
200
210
220
230
240
250
260
Eng
ine
Out
put
ωe
0 2 4 6 8 10 12 14 16 18 20−12
−10
−8
−6
−4
−2
0
2
4
6
8
Con
trol
ler
Out
put
Time
αγ
(a) without variable valve opening duration
0 2 4 6 8 10 12 14 16 18 20160
170
180
190
200
210
220
230
240
250
260
Eng
ine
Out
put
ωe
0 2 4 6 8 10 12 14 16 18 20−12
−10
−8
−6
−4
−2
0
2
4
6
8
Con
trol
ler
Out
put
Time
αγ
(b) with variable valve opening duration
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.
0 1 2 3 4 5 6 7 8 9 10160
180
200
220
240
260
Eng
ine
Out
put
ωe
0 1 2 3 4 5 6 7 8 9 10
−10
−5
0
5
Con
trol
ler
Out
put
0 1 2 3 4 5 6 7 8 9 10
−10
−5
0
5
Eng
ine
Inpu
t
Time
αγ
(a) without input allocation
0 1 2 3 4 5 6 7 8 9 10160
180
200
220
240
260
Eng
ine
Out
put
ωe
0 1 2 3 4 5 6 7 8 9 10
−10
−5
0
5
Con
trol
ler
Out
put
0 1 2 3 4 5 6 7 8 9 10
−10
−5
0
5
Eng
ine
Inpu
t
Time
αγ
(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.
CHAPTER 4. CONTROLLING THE ENGINE MODEL 29
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 = [5]. The state-space model of
the used allocator block is
[
Aa Ba
Ca Da
]
=
−5 0.0470 −5
−0.0094 1 0
1 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 con-
troller, 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 re-
sponse 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 33
5.4 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
relative cam change
relative cam velocity
valve lift
vL
exhaustvalve
intakevalve
cam angle ϕ
cam angle ϕ
cam angle ϕ
γ
ϕ∆
ϕ∆ ɺ s200 °
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 cosϕ
d∆ϕ
dt= γϕ sin ϕ (A.1)
with ϕ = 12ωe 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.
γ 12ωe sin 90◦ 1
360◦= 200◦/s
γ = 200◦/s2π
180◦ · ωe
[
rads
]360◦ =2513
ωe
[
rads
] (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 1000nc
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 Value Unit
afmep Auxiliary friction mean effective pressure [Pa]
B Bore 92.25 [mm]
Cff Constant “Flat follower” 400 [-]
Crf Constant “Roller follower” 0.0151 [-]
Coh Constant “Oscillating hydrodynamic” 0.5 [-]
Com Constant “Oscillating mixed” 21.4 [-]
cfmep Crankshaft friction mean effective pressure [Pa]
Db Bearing diameter 80.1 [mm]
fmep Friction mean effective pressure [Pa]
Lb Bearing length 32.4 [mm]
Lv Maximum valve lift 11 [mm]
µs Scaling term for oil viscosity 340−Toil
148.4 [-]
N Engine speed [rpm]
nb Number of bearings nc + 1 [-]
nc Number of cylinders 6 [-]
nv Number of valves 4nc [-]
pmep Pumping mean effective pressure [Pa]
rfmep Reciprocating friction mean effective pressure [Pa]
S Stroke 99.31 [mm]
Sp Mean piston speed 2SN60 [m/s]
Toil Oil temperature 321 [K]
vfmep Valvetrain friction mean effective pressure [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 =
(
1.22·105 Db
B2Snc
+ 3.03·10−4µs
ND3bLbnb
B2Snc
+ 1.35·10−10D2bN2nb
nc
)
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.
APPENDIX B. FRICTION MODEL 39
B.1.2 Reciprocating Friction
The expression for friction, caused by the reciprocating motion is:
rfmep =
(
2.94·102µs
Sp
B+ 4.06·104
(
1+1000
N
)
1
B2+ 3.03·10−4µs
ND3bLbnb
B2Snc
)
1000
nc
(B.3)
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 =
(
244µs
Nnb
B2Snc
+ Cffµs
(
1 +500
N
)
nv
Snc
+ Crf
Nnv
Snc
+ . . .
Cohµs
L1.5v N0.5nv
BSnc
+ Com
(
1+500
N
)
Lvnv
Snc
)
1000
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 =
(
8.63·10−7N3 − 5.20·10−3N2 + 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
parameter values listed in Table B.1 and the fact that N = 602π
ωe ; the following function for
fmep is obtained:
fmep = 1.05·105 + 67.0ωe − 0.0637ω2e +
3.07·105
ωe
+ 1.81ω12e + 0.125·10−3ω3
e (B.7)
where ωe is the engine speed in radians per seconds and fmep is measured in Pa.
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