On optimal control of the wastegate in a
turbocharged SI engine
ROBERTO ARGOLINI - VIVIANA BLOISI
Master’s Degree Project
Stockholm, Sweden June 2007
XR-EE-RT 2007:014
Sommario
The project aims to improve positive torque transient response through more
advanced wastegate controllers than what are used today. All controllers are
developed for a standard General Motors turbocharged engine. In many turbocharged SI engines, a wastegate is used for preventing the turbine to overrun
and to decrease the pumping loss. Today, the wastegate is controlled by a PI
controller, which tries to fulfill a compromise between fuel consumption and
torque response by regulating the wastegate position.
A nonlinear Mean Value Engine Model (MVEM) of this engine, with 13
states and linearized in 45 different working points, is used. The original model, implemented in Matlab/Simulink, has been enriched with new features,
like lambda and spark advance efficiencies and the related exhaust temperature correction.
The project aims to do a theoretical analysis to find the optimal control of
wastegate position, investigating also spark retard and fuel enrichment during
a positive torque transient. First a solution for achieving optimal wastegate
control is designed, based on Linear Quadratic (LQ) approach. Since the optimal control strategy is expected to vary quite much for different working
points, a gain scheduling architecture has been investigated.
An independent lambda controller has been developed, in order to maximize the lambda efficiency and quicken the torque response during transient.
Since the system operates near a constraint boundary, another solution based on Model Predictive Control (MPC) of the wastegate has been investigated.
The MPC design has been extended also to a MIMO formulation, adding the
throttle and the air to fuel ratio as control inputs, and the trade off between fast
torque response and fuel economy is analyzed. A complete realtime MPC implementation, with the capability for automatic code generation in the dSpace
microAutobox environment, requires the model, now with 13 states, to be reduced to a minimum state space order. The extent of model reduction that is
required and the possible performance deterioration have been investigated.
iii
Indice
1 Introduction
1
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.4
Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 Engine Model
5
2.1
Mean Value Engine Model . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Incompressible Flow restrictions . . . . . . . . . . . . . . . . . .
8
2.2.1
Air Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2.2
Intercooler . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2.3
Exhaust System . . . . . . . . . . . . . . . . . . . . . . . .
9
Compressible Flow Restrictions . . . . . . . . . . . . . . . . . . .
10
2.3.1
Throttle flow . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.3.2
Wastegate flow . . . . . . . . . . . . . . . . . . . . . . . .
10
Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.4.1
10
2.3
2.4
2.5
2.6
2.7
2.8
Mass flow into the cylinders . . . . . . . . . . . . . . . . .
2.4.2
Volumetric efficiency . . . . . . . . . . . . . . . . . . . . .
11
2.4.3
Exhaust Manifold Out Temperature . . . . . . . . . . . .
11
Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.5.1
Compressor - Pressure Model . . . . . . . . . . . . . . . .
12
2.5.2
Compressor - Air Mass Flow Model . . . . . . . . . . . .
12
2.5.3
Compressor - Efficiency Model . . . . . . . . . . . . . . .
12
Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.6.1
Turbine - Pressure Model . . . . . . . . . . . . . . . . . .
13
Model Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.7.1
Control volumes . . . . . . . . . . . . . . . . . . . . . . .
14
2.7.2
Turbo Shaft Dynamics . . . . . . . . . . . . . . . . . . . .
15
Torque generation . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
v
vi
I NDICE
2.9
Fuel path and (A/F)-ratio . . . . . . . . . . . . . . . . . . . . . .
19
2.10 Fuel consumption . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3 Simulink Model
21
3.1
Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.2
Car . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.3
Engine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.3.1
Model Simplifications . . . . . . . . . . . . . . . . . . . .
25
3.3.2
Model Inputs . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.3.3
Model States . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.3.4
Model Output . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.3.5
Maps for parameters representation . . . . . . . . . . . .
28
Performance Limiting Factors . . . . . . . . . . . . . . . . . . . .
28
3.4.1
Knock . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.4.2
Surge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.4.3
Limits on air to fuel ratio . . . . . . . . . . . . . . . . . .
29
Engine Management System (ECM) . . . . . . . . . . . . . . . .
30
3.4
3.5
3.6
3.7
3.5.1
Map-Based Control . . . . . . . . . . . . . . . . . . . . . .
31
3.5.2
Throttle control . . . . . . . . . . . . . . . . . . . . . . . .
32
3.5.3
Lambda Control . . . . . . . . . . . . . . . . . . . . . . .
33
3.5.4
Fuel Control . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.5.5
Spark Advance Control . . . . . . . . . . . . . . . . . . .
34
Turbo Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.6.1
Wastegate Control . . . . . . . . . . . . . . . . . . . . . .
36
System Modifications . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.7.1
Bypass Valve . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.7.2
Lambda and SA Efficiency . . . . . . . . . . . . . . . . . .
37
3.7.3
Temperature Correction Map . . . . . . . . . . . . . . . .
39
3.7.4
Volumetric Efficiency . . . . . . . . . . . . . . . . . . . . .
39
3.7.5
WG Pressure valve . . . . . . . . . . . . . . . . . . . . . .
42
3.8
Changes in the ECM . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.9
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4 Validation of the NonLinear Model
45
4.1
Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.2
Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.3
Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
I NDICE
vii
5 Analysis of linearized model
55
5.1
Linearized Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.1.1
Model linearization . . . . . . . . . . . . . . . . . . . . . .
56
5.2
Validation of the linearized model . . . . . . . . . . . . . . . . .
58
5.3
Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
5.3.1
SISO Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
63
5.3.2
Controllability and observability . . . . . . . . . . . . . .
65
Open Loop system behavior . . . . . . . . . . . . . . . . . . . . .
66
5.4.1
Step in the WG . . . . . . . . . . . . . . . . . . . . . . . .
66
5.4.2
Pulse in the WG with constant throttle . . . . . . . . . . .
66
5.4.3
Pulse in the WG with step in throttle . . . . . . . . . . . .
69
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
5.4
5.5
6 Optimal Wastegate Control
73
6.1
Spark-ignited engines . . . . . . . . . . . . . . . . . . . . . . . . .
73
6.2
Turbocharging . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
6.3
Wastegate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
6.3.1
Transient response and turbo lag . . . . . . . . . . . . . .
76
6.3.2
Open Loop System Characteristics . . . . . . . . . . . . .
79
6.3.3
Dealing with constraints . . . . . . . . . . . . . . . . . . .
80
6.3.4
Control Problem formulation . . . . . . . . . . . . . . . .
81
6.4
Linear Quadratic Regulation (LQR) . . . . . . . . . . . . . . . . .
82
6.5
Optimal State Feedback . . . . . . . . . . . . . . . . . . . . . . .
83
6.5.1
Stability and robustness . . . . . . . . . . . . . . . . . . .
84
6.6
LQR for non-zero set points . . . . . . . . . . . . . . . . . . . . .
86
6.7
Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6.8
Integral action in LQ control . . . . . . . . . . . . . . . . . . . . .
88
6.9
Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
6.9.1
LQ on Linearized System . . . . . . . . . . . . . . . . . .
91
6.9.2
LQR on Nonlinear System . . . . . . . . . . . . . . . . . .
94
6.10 Gain Scheduling Strategy for LQ Control . . . . . . . . . . . . .
99
6.10.1 Variable engine speed . . . . . . . . . . . . . . . . . . . . 101
6.10.2 Variable engine speed with 3 controllers . . . . . . . . . . 105
6.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7 Lambda and SA Control
109
7.1
Lambda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.2
Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2.1
Static Lambda Control . . . . . . . . . . . . . . . . . . . . 111
viii
I NDICE
7.2.2
Dynamic Lambda Control . . . . . . . . . . . . . . . . . . 112
7.3
Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.4
Spark Advance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
116
8 MPC
119
8.1
119
MPC Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1
8.2
Parameters Tuning . . . . . . . . . . . . . . . . . . . . . . 122
Implementation in Simulink using Mpc Toolbox . . . . . . . . .
123
8.2.1
Building the MPC object . . . . . . . . . . . . . . . . . . . 124
8.2.2
Prediction Model . . . . . . . . . . . . . . . . . . . . . . . 126
8.2.3
Offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2.4
State Estimation . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2.5
MPC Computation . . . . . . . . . . . . . . . . . . . . . . 128
8.2.6
Setting MPC parameters . . . . . . . . . . . . . . . . . . . 128
8.3
System Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.4
Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.5
8.4.1
MPC on the Linearized Model . . . . . . . . . . . . . . . 133
8.4.2
Pumping Losses . . . . . . . . . . . . . . . . . . . . . . . 140
8.4.3
MPC on the Nonlinear Model . . . . . . . . . . . . . . . . 140
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9 Model Reduction
149
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.2
Model Reduction Process . . . . . . . . . . . . . . . . . . . . . . 150
9.3
Hankel Singular Values . . . . . . . . . . . . . . . . . . . . . . . .
150
9.4
Overview of Model Reduction Techniques . . . . . . . . . . . . .
151
9.5
9.4.1
Truncation and Residualization . . . . . . . . . . . . . . . 152
9.4.2
Balanced truncation and residualization . . . . . . . . . . 154
9.4.3
Optimal Hankel norm approximation . . . . . . . . . . . 154
9.4.4
Balanced stochastic model truncation (BST) . . . . . . . . 155
Model Reduction Procedure . . . . . . . . . . . . . . . . . . . . . 155
9.5.1
Number of states to use in the reduced model . . . . . . 156
9.5.2
Model Reduction Techniques. Comparison: Additive Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.6
Model Reduction Technique: Multiplicative Error Method . . . 160
9.6.1
9.7
Validate the model . . . . . . . . . . . . . . . . . . . . . . 161
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
I NDICE
ix
10 Multivariable System Analysis
10.1 Directions in multivariable systems . . . . . . . . . . . . . . . . .
171
171
10.2 Scaling the system . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
10.3 Singular value decomposition . . . . . . . . . . . . . . . . . . . .
174
10.4 Multivariable Poles and Zeros . . . . . . . . . . . . . . . . . . . .
175
10.5 Input-output controllability . . . . . . . . . . . . . . . . . . . . .
176
10.5.1 MISO case . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.5.2 MIMO case . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.6 SVD Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . .
180
10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
11 MIMO Control
185
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
11.1.1 Best Fuel Consumption: formulation of the problem . . . 186
11.2 Control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
189
11.2.1 Layout 1: Fuel Optimal Control . . . . . . . . . . . . . . . 189
11.2.2 Layout 2: Torque Optimal Control . . . . . . . . . . . . . 189
11.3 Simulations with MIMO MPC . . . . . . . . . . . . . . . . . . . . 189
11.3.1 Simulation on the Linear Model . . . . . . . . . . . . . . 190
11.3.2 Considering Pumping Losses . . . . . . . . . . . . . . . . 192
11.3.3 Simulation on the Nonlinear Model . . . . . . . . . . . . 195
11.3.4 Comparison of the two controller versions . . . . . . . . 195
11.4 MIMO MPC on the Reduced Order Model . . . . . . . . . . . . 198
11.5 MIMO MPC for α, WG and λ . . . . . . . . . . . . . . . . . . . .
200
11.6 Numerical errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
11.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
12 Conclusion
207
Bibliografia
211
List of Notations
λst : λ stoichiometric
Kef f : efficiency coefficient
ṁair : air mass flow
ṁf uel : fuel mass flow
mAir−Req : requested air mass
mF uel−Req : requested fuel mass
MP red : predicted torque
nr : number of crank revolutions for each power stroke per revolution
rpm: rounds per minute
V̇a : volume flow rate of air into the intake system
V̇d : rate at which volume is displaced by the piston,
WG: wastegate
Numerical values:
λstec =14.65
qHV = 44 ∗ 106 J/kg
Vd = 2 ∗ 10−3 m3
R = 287.2J/(kgK)
nr =2
η=0.3469
a0 =857.748579470353/1000
a1 =-5216.7334951752
xi
List of Abbreviations
BMEP: Brake Mean Effective Pressure
IMEP: Indicated Mean Effective Pressure
FMEP: Friction Mean Effective Pressure
PMEP: Pumping Mean Effective Pressure
RHP: Right Half Plane
ECM: Electronic Control Manager
ECU: Electronic Control Unit
MIMO:Multiple Input Multiple Output
MISO:Multiple Input Single Output
SIMO:Single Input Multiple Output
SISO:Singol Input Single Output
SA: Spark Advance
SI: Sparked Ignited
SVD: Singular Value Decomposition
xiii
Capitolo 1
Introduction
This Master’s Degree Project is performed at Automatic Control Group at KTH,
Stockholm, in cooperation with General Motors Power Train, Södertälje (Sweden).
The project concerns optimal control of a turbocharged SI engine, to improve
positive torque response during the transient.
1.1 Background
The ever increasing governmental regulation of emissions and driver demands
for fuel economy and drivability emphasize the need for advanced engine technology and control. Among many other measures to improve the fuel consumption, one method is to apply a turbocharger. Investigations have shown a
considerable potential for improvement in this area. In turbocharged engines
energy from exhaust gases is used to compress the intake air and produce a
higher intake manifold pressure. This is done by a turbine connected to the exhaust. The increased intake pressure results in an increased amount of air into
the combustion chamber and thus increases the power output that is available.
A turbo also increases the inlet air temperature which can be cooled by an intercooler. The turbocharger speed and the resulting pressure after the compressor
is controlled using a wastegate.
A wastegate is a valve that diverts exhaust gases away from the turbine
wheel. Diversion of exhaust gases causes the turbine to lose speed, which in
turn reduces the rotating speed of the compressor. The pressure above atmospheric pressure that the turbocharger is creating in the engine’s inlet manifold
1
2
C APITOLO 1. I NTRODUCTION
is called boost pressure and the primary function of the wastegate is to stabilize
this boost pressure, to protect the engine and the turbocharger.
Today, the wastegate is mechanically actuated, by means of a pressure valve
which simply opens up the wastegate when the pressure over it is too high,
and can be controlled by a simple PI controller. In all the other cases it is
closed. This thesis investigates the possibility for a more advanced control of
the wastegate, to improve the responsiveness of turbocharged SI engines.
The wastegate in state-of-the-art turbocharged engines is always closed in
order to achieve the highest possible pressure before the turbine and thus the
highest possible turbocharger speed. The higher the turbocharger speed, the
smaller is the turbo lag, as the torque build-up time is highly sensitive to the
turbocharger inertia. The wastegate is only opened to keep the boost pressure
below a certain limit, i.e. for boost pressure control.
One possible way to reduce time lag is to control the turbo so that the turbo always keeps a relatively high speed and then use the throttle to reduce
the flow into the intake manifold and the manifold pressure. In this way the
driveability is improved by reducing the turbo lag but the fuel consumption is
increased, since the intake manifold pressure is increased.
The turbo increases the intake manifold pressure and also the intake manifold temperature and with an intercooler the charger density is increased,
which results in higher power output from the engine. An intercooler also
gives better engine efficiency, since it can give the same mass flow through the
engine at a lower pressure which in its turn requires less turbine pressure and
in this way reduces the pumping losses.
1.2 Prior Work
This project continues the investigation started by Ida Kristoffersson in her
Master Thesis[1]. For more details on engine modeling, [2] and [3] are recommended. The problem of controlling the wastegate is handled in [4], with a
pressure based approach, and in [5]. Multivariable design and nonlinear control are extensively treated in [6],[7] and [8]. MPC is implemented in Matlab,
using the MPCToolbox, based on [9].
1.3 Problem Formulation
In this work the control strategy is based on the idea of using turbocharging and wastegate to reduce pumping losses during the transient and so im-
1.3. P ROBLEM F ORMULATION
3
prove the torque response. Further investigations for improving the system
responsiveness concerned also the air-to-fuel ratio and the spark advance.
The first part of the project focused on analyzing the main characteristics
of a turbocharged engine, from a control point of view, in order to be able to
understand the problems and so design the requested controllers.
A nonlinear Mean Value Engine Model (MVEM), with 13 states and linearized in 45 different working points, has been first studied and then used
for simulations and control design. The original model, implemented in Matlab/Simulink, has been enriched with new features, like spark advance and
lambda efficiencies and exhaust temperature correction depending on spark
advance. A validation of the modified nonlinear model has been performed,
recording data from the car in a driving test.
To check that the linearized model correctly captured the main dynamics
of the system, it was compared and validated with respect to the nonlinear
one. A mathematical analysis of the linearized model in all the 45 operating
points was carried through, in order to point out the main characteristics and
fundamental limitations of the system.
The proposed solution for achieving optimal wastegate control is based
on Linear Quadratic (LQ) approach. For such controller a torque based design is applied. Reference tracking is achieved adding a feedforward action
in the LQ controller. To guarantee disturbance rejection, it was also enforced
with integral action and with an anti wind-up solution, related to hard input
constraints.
Since the optimal control strategy is expected to vary quite much for different working points, a gain scheduling strategy based on engine working
conditions, has been designed.
In order to improve the torque transient response, also the air-to-fuel ratio influence has been investigated and an independent lambda controller has
been developed, in order to maximize the lambda efficiency during transient.
Since the system operates near a constraint boundary, another solution based on Model Predictive Control (MPC) has been investigated. The first step
has been considering only the wastegate as manipulated variable. Later on
a multivariable approach has been considered, adding the throttle as control
input and the fuel consumption as a second output. A mathematical analysis
of the resulting MIMO system has been performed, based on singular value
decomposition.
The MPC was then extended to the MIMO formulation and some constraints on the states were also included, in order to consider physical limi-
4
C APITOLO 1. I NTRODUCTION
tations of the system. Different MPC configurations have been designed, to
manage the trade-off between fast torque response and low fuel consumption.
As last step, the air-to-fuel ratio was added to the manipulated variables,
to verify if it is possible to get some benefit improving the transient torque
response.
The complete realtime MPC implementation, with the capability for automatic code generation in the dSpace microAutobox environment, requires
the model, now with 13 states, to be reduced to a minimum state space order.
The extent of model reduction that is required and the possible performance
deterioration caused by the reduction have been investigated.
1.4 Thesis outline
Chapter 2 covers the modeling of a turbocharged SI engine and Chapter 3 describes the Simulink model including the car, the driver and the ECM. In Chapter 4 the validation of the nonlinear model is shown, by comparing simulation
results and data from a test on the car. Chapter 5 deals with the analysis of the
linearized model, which is then used as a basis for the investigation of transient
torque response and the possible approaches to its improvement. In Chapter 6
different ways to improve the engine response are investigated and a strategy
based on LQ control is shown. Chapter 7 deals with control of Lambda and
Spark Advance, and presents a strategy to improve torque response by maximizing the Lambda efficiency. A strategy based on model predictive control
(MPC) is investigated thoroughly in Chapter 8 with a model-based approach.
In Chapter 9 the applicability of the strategy is discussed and a reduced order
model is derived, in order to make the MPC computationally lighter. A different formulation for the problem is given in Chapters 10 and 11, where multivariable control theory is used and MPC is designed to control also throttle
and Lambda together with the wastegate.
Capitolo 2
Engine Model
Turbocharged SI engines are a major possibility in the current trend of downsized engines with preserved drivability performance. Downsizing refers to
reducing the size of the engine without a loss in the amount of torque produced, by adding a turbocharger, which connects a turbine placed before the
exhaust system to a compressor placed after the air filter. This way the energy in the exhaust gases is utilized to compress the incoming air more and the
engine can produce more torque.
Considering control and supervision it is favorable to have a mean value
model to be used. Such models of turbo engines are similar to those of naturally aspirated engines, but there are some special characteristics, e.g. the interconnected gas flows, the intercooler, the difference in relative sizes between
the gas volumes (compared to naturally aspirated engines), the turbo, and the
wastegate.
2.1 Mean Value Engine Model
In order to perform a model-based investigation and optimization of the dynamic behavior of a turbocharged SI engine, a good model is necessary. A
highly suitable model class is then mean value engine models (MVEMs) which
describe the average behavior of the engine over one to several thousands of
engine cycles[3] [2].
Fig. 2.1 shows a sketch of the turbocharged SI engine under consideration.
The components that have to be modeled are: air filter, compressor, intercooler,
throttle, engine, turbine, wastegate, and a lumped model for the catalyst and
5
6
C APITOLO 2. E NGINE M ODEL
exhaust. One benefit of this modelling strategy is that every component can
be separately identified and then the engine model is built using the separate
components. Each component is described in terms of equations, constants,
parameters, states, inputs and outputs.
For mean value models, that describe variations slower than an engine cycle, the frequency range is within 0.1-50 Hz. Parameters in mean value models
are usually physically interpretable. This is advantageous in simulations of
principle concern.
General guidelines for MVEMs are:
• Changes that take in the order of 10-1000 cycles to reach their final state
are expressed by differential equations.
• Changes that are faster are expressed by static relations.
The basic principle used here is that each engine component shown in Fig.2.1 is
modeled as a flow restriction (or pump) that affects the thermodynamic state of
the fluid (gaseous). The standard calculation causality of almost all restrictions
is that they take pressure difference as input and gives the flow as a result. The
volumes between the restrictions are modeled as reservoirs where normally
the pressure in each reservoir represent a state in the model.
Whereas for the control volume there is only one type of model, for the
restrictions there are two main components, namely compressible and incompressible fluids.
Abbreviated Nomenclature
It is natural to accompany the component based modeling with a system to
name the variables. The main variables in the table to the left are specified by
a subscript from the table to the right referring to its own component or the
component before. As an example, Tc omp refers to the temperature after the
compressor.
Variables
ṁ
Mass flow
P
Power
p
Pressure
T
Temperature
Tq
Torque
Π
Pressure ratio
2.1. M EAN VALUE E NGINE M ODEL
7
Subscripts
amb
Ambient
af
Air filter
em
Exhaust mannifold
es
Exhaust system
Tq
Torque
ic
Intercooler
im
Intake manifold
t
Turbine
th
Throttle
The following sections quickly describe the modelling of some important
subsystems present in the MVEM, based on the model developed by Per Anderson in [3] and the components related to turbocharging are given in the
turbo section (see [3] for more details).
Figura 2.1: Sketch of a turbocharged SI-engine, that illustrates how the engine
is divided into subsystems to enable physical modeling.
8
C APITOLO 2. E NGINE M ODEL
2.2 Incompressible Flow restrictions
In many engine components the fluid flow can be regarded as incompressible
and in-viscid. Under such conditions the pressure head losses caused by bends,
valves, and sudden changes in area approximately follow the same relation
Δp = pbef ore − paf ter = ξ(Re)
RTbef ore ṁ2
pbef ore
(2.1)
where the ξ only has a weak dependence on the Reynolds number (Massey,
1998). In most cases ξ(Re) · R can be regarded as constant, here denoted H, and
the equation can be used as a general model for incompressible flow through
a restriction. Models for the air filter, intercooler, and exhaust system are all
based on this relation.
2.2.1 Air Filter
The air filter is important to model since it produces a pressure drop that can
be as high as 5%, which propagates trough the intake system and influences
the whole engine. A sketch of the air filter is shown in Fig.2.2. It consists of
three sudden changes in area, the filter itself, and the pipes in and out of it. As
described above, Eq. 2.1 can be used to model pressure loss for several kinds
of flow restrictions, which gives the following pressure loss over the air filter:
Δpaf = pamb − paf = Haf
Tamb ṁ2air
pamb
(2.2)
p_amb
m_air
T_amb
p_af
T_af
m_air
Figura 2.2: Sketch of the air filter showing the changes in flow area.
2.2.2 Intercooler
A heat exchanger, called intercooler, is used after the compressor to cool the
compressed air back to near ambient temperature. It increases air density
2.2. I NCOMPRESSIBLE F LOW RESTRICTIONS
9
which is advantageous, but in SI engines it is mainly used for knock reduction. To achieve efficient cooling of the charge air, the tubes in the intercooler
need to be rather thin so that the air is exposed as much as possible to the cooling medium. The result is that some of the gain in intake air density is lost.
Therefore, beside the obvious need to model the outlet air temperature, also a
model for the pressure head loss is needed.
Intercooler - Pressure Head Loss
Intercooler pressure head loss is modeled by [10]
Δpic = Hic
Tcomp ṁ2air,ic
pcomp
(2.3)
where ṁair,ic is the air-mass flow through the intercooler.
Intercooler - Temperature Head Loss
For all practical purposes of intercoolers the flow rate of the cooling fluid ṁcool
is greater than the air mass flow, ṁair , through it. This suggests, (Holman,
1997), that the following equation can be used to measure the efficiency of the
intercooler
=
Tcomp − Tic
Tcomp − Tcool
(2.4)
Solving this equation for Tic yields the desired expression for the intercooler
outlet temperature, expressed in terms of the temperatures and the intercooler
efficiency
Tic = Tc − (Tcomp − Tcool )
(2.5)
Tcool is the temperature of the cooling medium, in this case ambient air, Tcool =
Tamb . In order to predict the intercooler outlet temperature, a model for the
intercooler efficiency, , is needed. We are not interested in details here, the
description will not go deeper, but more can be found in [2].
2.2.3 Exhaust System
The exhaust system consists of changes in flow area and flow direction and the
head losses can therefore also be modeled in the same way as the air filter, i.e.
using Eq. 2.1:
Δpt = pt − pamb = Hes
Tes,in ṁ2air
pt
(2.6)
where Tes,in is the temperature of the gases flowing into the exhaust system.
10
C APITOLO 2. E NGINE M ODEL
2.3 Compressible Flow Restrictions
Isentropically compressible mass flows (Taylor, 1994) are used to describe the
flow through the throttle and the wastegate.
2.3.1 Throttle flow
The throttle is modeled with a butterfly type model,see Figure 2.3, where the
angle of the throttle plate controls the flow area. In engine under consideration
the throttle is controlled by the ECU through an electrical servo, thus providing high flexibility in determining the behavior of the vehicle by designing
different profiles for the throttle behavior.
a
a0
Figura 2.3: Left: the throttle body. The angle a is the actual angle related to the
throttle housing, while the angle a0 defines when the throttle is closed. Right:
the flow areas are representd by the two moon crest shaped areas.
2.3.2 Wastegate flow
Wastegate flow is modeled using the standard model for a compressible restriction, with variable area.
2.4 Engine
The engine produces a port air.mass flow, exhaust mass flow and exhaust temperature.
2.4.1 Mass flow into the cylinders
The air mass flow out of the intake manifold into the cylinders depends on
many parameters, but mostly on engine speed N, intake manifold pressure
pim , and temperature Tim :
ṁac (N, pim , Tim ) = ηvol (N, pim , . . .)
Vd N pim
nr RTim
(2.7)
2.5. C OMPRESSOR
11
where ηvol is the volumetric efficiency, a parameter used to describe the effectiveness of the engine’s ability to induct new air into the cylinders; Vd is the
displaced volume for the whole engine; nr is the number of engine revolutions
per cycle and R is the gas constant.
2.4.2 Volumetric efficiency
The intake system consisting of air filter, intercooler and throttle plate, intake
port and intake valve restricts the amount of air that the engine can induct. The
measure for the effectiveness of an engine’s induction process is the volumetric
efficiency, ηvol . The volumetric efficiency is defined as the volume flow rate of
air into the intake system, V̇a , divided by the rate at which volume is displaced
by the piston, V̇d ,
ηvol =
V̇a
ṁa nr
=
ρai Vd ηcyl N
V̇d
(2.8)
where ρai is the air density in the inlet, ṁa is the air mass flow, and nr is the
number of crank revolutions for each power stroke per revolution (nr = 1 for
a two stroke engine and nr = 2 for a four stroke engine). The engine speed n
is given in revolution per seconds. An equivalent definition of the volumetric
efficiency is to use the mass of air in one cylinder directly
ma
ρai V̇d
(2.9)
The definition are equivalent but the mass of air is not easily determined.
The inlet density can be either be taken as the intake system density. Then
ηvol measures the efficiency for the pumping over the inlet valves and ports.
Or ρ can be taken as atmospheric density, and then it measures the efficiency
for the whole intake.
2.4.3 Exhaust Manifold Out Temperature
The temperature out of the exhaust manifold before entering the turbine can be
modeled with sufficient accuracy as a function of air mass flow Tem = f (ṁaf )
when the engine operates at stoichiometric conditions, (Eriksson, 2002).
2.5 Compressor
Compressor performance is determined in a flow bench where the compressor
is run together with the turbine in a number of points and the state changes of
12
C APITOLO 2. E NGINE M ODEL
the fluid are measured. From this data a performance map of the compressor is
constructed. The performance map is usually implemented in a look-up table
which serves as the actual compressor model.
2.5.1 Compressor - Pressure Model
f an adiabatic process is assumed, compression from a state with pressure p01
and To1 to another state, with pressure p02 requires an amount of specific energy, wi , that obeys the following relation between the pressure ratio and the
specific energy. In this case p0 1 = paf and p0 2 = pcomp , so
Πc =
γ
pcomp
win γ−1
= (1 +
)
paf
cp T01
(2.10)
An expression for the work can be derived, see e.g. (Mtiller et al., 1998), by
considering the Euler equations for compressor blades and the losses due to
friction and incidence, resulting in
win = U22 (s1 (
ṁair 2
ṁair
) + s2 (
) + s3 )
U2
U2
(2.11)
wehre U2 is the rotor tip speed and si are constants that are tuned to measured
data.
2.5.2 Compressor - Air Mass Flow Model
n mean value models it is desirable to represent the flow through the compressor as a function of pressure ratio and turbine shaft speed, rather than as in Eqs
2.10 and 2.11. To do this, the equations can be inverted to
ṁair = a1 U2 ∓
a2 U22
γ−1
+ a3 Taf (Πc γ − 1)
(2.12)
where ai are tuning constants.
2.5.3 Compressor - Efficiency Model
he efficiency is defined by the ratio of the isentropic and the actual specific
input work
γ−1
( ppc ) γ − 1
win
ηc =
= afTc
win,act
Taf − 1
where win is modeled by Eq.2.11.
(2.13)
2.6. T URBINE
13
2.6 Turbine
Dimensionless quantities are used when describing the turbine performance
and the same definitions, as for the compressor, are applied. Here it is worth to
stress that the dimensionless expressions are of higher importance for the turbine since it is exposed to more extreme variations in inlet conditions compared
to the compressor; Temperatures vary from 500 K up to 1300 K and pressures
at turbine inlet from atmospheric conditions up to 300 kPa.
For turbines the corrected mass flow and efficiency is normally plotted versus pressure ratio
Pem
pt
= l/Πt . Turbine efficiency is determined from the flow
conditions and calculated using
ηt =
1−
Tt
Tem
t
1 − ( ppem
)
(2.14)
γ−1
γ
In addition, the mechanical efficiency of the turbocharger might also be included in the turbine efficiency. Look-up tables derived from the map may serve as
turbine models but it is also interesting to study parameterizations.
2.6.1 Turbine - Pressure Model
eglecting the corrected quantities and studying the mass flow alone shows that
there exists a relationship between mass flow and pressure ratio for the turbine
that is practically independent of turbine shaft speed. The connection between
the flow and pressure ratio is almost independent of the compressor speed,
therefore the speed dependence is neglected and the corrected air mass flow is
modeled as a function of only the pressure ratio Πt =
pt
pem .
The parallel flow
in the wastegate also has a large effect and makes the validation of the turbine
model difficult, when using engine data.
2.7 Model Dynamics
Two kind of dynamics are included in the model:
- Control volumes. All receivers are modeled as control volumes with
filing and emptying dynamics using pressure and temperature as states.
- Turbo shaft dynamics. The axle connecting compressor and turbine has
its own dynamics, driven by the turbine torque and loaded by the compressor and friction.
14
C APITOLO 2. E NGINE M ODEL
2.7.1 Control volumes
Control volumes are placed in between all flow restricting components and
they determine some of the engines dynamic characteristics.
A control volume is a thermodynamical volume that stores mass and energy. It has fixed volume V and as a storage it has filling and emptying dynamics.
The increase or decrease o0f the mass is determined by the inlet and outlet air
mass flow ṁin and ṁout . The difference between the inflow and outflow of
mass (ṁin − ṁout ) directly gives the rate of change in mass in the system
dm
= ṁin − ṁout
dt
(2.15)
Energy is also conserved and stored in the system. Considering the energy
there might be heat transfer Q̇ but there is no mechanical work transfer to the
control volume. Energy is also transferred to and from the system through
the in-and outflows. For the open system that we are considering the first law
(energy conservation) gives the following rate of change of the internal energy
dU
= Ḣin − Ḣout − Q̇
dt
(2.16)
Since is difficult to measure mass and energy, these equations are manipulated and rewritten in terms of temperature and pressure. Assuming that
• the gas is ideal (pV = mRT )
• cp andcv are constant
the temperature can be determined from the internal energy and the mass
through
U = mu(T ) = [cv − constant] = mcv T
(2.17)
and the pressure can be determined from the ideal gas law
pV = mRT
(2.18)
Furthermore the enthalpy flows are given by
Ḣin = ṁin cp Tin andḢout = ṁout cp Tout
(2.19)
where the temperature of the outflowing gas is the same as that in the control
volume, i.e. Tout − Tin . Equations 2.15 to 2.19 give a system of equations that
can be solved to give all information on the system and differential equations
2.7. M ODEL D YNAMICS
15
can be formulated so that mass, pressure and temperature are the states in the
model (see [2] for the detailed equations).
Control Volume
H
H
p
T
m_in
T_in
U
m
V
m_out
T_out
Q
Figura 2.4: The control volume is marked with dashed line and in it the pressure, temperature, and mass states are shown. There are two flows across
boundaries. There is no volume change (V is constant), except those from the
flow, so there is no mechanical work exchange.
2.7.2 Turbo Shaft Dynamics
revious sections described models for compressor and turbine separately and
to connect the two submodels Newton’s second law for a rotating system is
used:
Tqt − Tqcomp = I ω̇
(2.20)
where Tqt is the driving torque and Tqcomp denotes braking torque acting on the
rotating parts of the turbo charger spinning with the angular velocity ω. The
power and torque are connected through P = Tq ω so equation 2.20 becomes:
Pt − Pcomp = I ω̇ω
(2.21)
If the turbine and the compressor are treated as two separate thermodynamic
systems, Pt and Pcomp can be calculated from the first law of thermodynamics. The heat transfer with the surroundings and change in potential energy is
neglected.
−P = ṁ(hout − hin ) = ṁcp (Tout − Tin )
(2.22)
For the compressor, the net amount of produced power is negative, i.e. the
compressor consumes energy, while for the turbine, the net amount of produced energy is positive.
16
C APITOLO 2. E NGINE M ODEL
2.8 Torque generation
How large the torque is, depends on the work produced and consumed in the
engine [2]:
Me =
Wig − Wf r − Wpump
nr 2π
(2.23)
where Wig is the indicated gross work produced, i.e. the mechanical work produced, Wf r is the friction work consumed by the engine components as well as
auxiliary devices, and Wpump is the pumping work due to difference in intake
and exhaust manifold pressures, while nr is the number of engine revolutions
per cycle. There exist different expressions for Wig , Wf r and Wpump . Often
they are modeled as follows:
Wig = Vd · mf˙uel qHV
2 min (λ, 1)
ηe ,
Ns
Vd
(2.24)
2 N
N
Wf r = Vd · 0.97 + 0.15
,
+ 0.05
1000
1000
Wpump = Vd · (pem − pim ).
ηe is the engine raw efficiency, qHV the heating value (the amount of energy
that the combustion of one unit of fuel can release when the fuel is in gaseous
phase), Vd the displacement volume, N the engine speed in revolutions per
minute and Ns the engine speed in revolutions per second. The indicated gross
work is coupled to the energy in the fuel and to operating conditions, which
gives the model in the first equation above. In the equation (2.24) the pumping
work is modeled as proportional to the difference in exhaust manifold pressure and intake manifold pressure. The friction model comes from [11]. It
accounts for three different kinds of friction: boundary friction (independent
of engine speed), hydrodynamic friction (proportional to speed) and turbulent
dissipation (proportional to speed squared).
Torque is a valuable measure of a particular engine’s ability to do work
but it depends on engine size. Another useful parameter is the mean effective
pressure (MEP) which is normalized with engine size and has the same unit as
pressure [force per unit area][12]
M EP =
work produced per cycle
volume displaced per cycle
(2.25)
Gross indicated work (IM EPg ) is defined as the work delivered during
the compression and expansion strokes while Net indicated work (IM EPn )
2.8. T ORQUE GENERATION
17
implies the work generated during the entire four-stroke cycle.
The difference between net and gross IMEP is called Pumping Mean Effective Pressure, PMEP:
P M EP = IM EPg − IM EPn
(2.26)
From eq. 2.26 it is evident that a positive PMEP means negative work on the
Figura 2.5: Cylinder pressure versus V/Vmax for a turbocharged SI-Engine.
crankshaft. PMEP and IMEP can also be expressed by the areas formed in the
pressure versus volume diagram, see Figure 2.5:
• PMEP = Area B + Area C
• IMEP= Area A + Area C
The usable power is usually referred to as brake power, so inserting the
brake torque into the definition of MEP, the brake mean effective pressure
(BMEP) is obtained:
BM EP =
Pb nr
2πMb nr
=
Vd ccyl N
Vd ccyl
(2.27)
BMEP is estimated by subtracting pumping (PMEP) and friction work (FMEP)
18
C APITOLO 2. E NGINE M ODEL
from the supplied work from the fuel (IMEP):
BM EP = IM EP − P M EP − F M EP
(2.28)
In turbocharged engines the turbine power is generated by the energy in
the exhaust flow. Figure 2.6 gives a schematic view of the theories behind
turbocharged engines. Loop 1 to 4, in Figure 2.6 is the engine power stroke.
Figura 2.6: Schematic cylinder pressure versus volume diagram, showing the
available exhaust energy.
The driving power of the turbine is the energy available at point 4 when the
piston is at BDC. Area AT represents the available energy, which could be produced by expanding the exhaust gases to atmospheric pressure after the exhaust valves have opened. This energy is used to drive the compressor so that
the intake pressure can be raised above the atmospheric pressure to the charging pressure, which requires work, area Ap . For the ideal case area AT equals
area Ap . In other words, the power generated by the turbine will be used in the
compressor. The power is related to the mass flow through the turbine, ṁ and
enthalpy, h , according to [10]:
P = ṁ(h4 − h5 ) = ṁcp (T4 − T5 )
(2.29)
2.9. F UEL PATH AND (A/F)- RATIO
19
where T is the temperature and cp the specific heat.
The choice of turbine for a specific engine configuration is a compromise
between low and high load performance of the engine. For commercial automotive purposes low load performance is most often preferred. In order to
ensure sufficient pressure at the turbine inlet at low load, the inlet area to the
turbine needs to be limited. However at high load a small inlet area will generate too high rotational speed of the turbine. Hence, it is necessary to limit the
turbine rotational speed during high load due to mechanical constraints. This
can be done by the wastegate, which leads part of the flow past the turbine,
thereby decreasing the pressure at the turbine inlet.
2.9 Fuel path and (A/F)-ratio
The fuel flow into the cylinder is measured and controlled by a fuel injector,
that is an electrically controlled valve. Only a fraction of the injected fuel is
inducted into the cylinder directly, due to the wall wetting phenomenon, i.e.,
some amount of fuel is deposited on the intake walls either as a film or as
puddles and does not mix with the air.
The result is that the following (A/F) mixture enters the cylinder
λ=
ṁac 1
A
ṁf c F
s
(2.30)
where ṁac and ṁf c are the air and fuel flow into the cylinder respectively.
2.10 Fuel consumption
The fuel mass flow to the cylinder is proportional to the air mass flow ([3]):
ṁair
ṁf uel = A .
λ F s
where
A
F s
is the stoichiometric air/fuel ratio.
An expression for the air mass flow is:
ṁair = (a0 pim + a1 )
V d Ns
RTim nr
(2.31)
where Vd is the displacement volume, Ns the engine speed in revolutions per
second, R is the gas constant and nr the number of engine revolutions per cycle.
The constants a0 and a1 are identified in [3] with the Matlabfunction lscurvefit
20
C APITOLO 2. E NGINE M ODEL
which uses a nonlinear least-squares technique. If the state Tim and the input
signal Ns is assumed constant, ṁair can be expressed as a linear function of the
state pim :
ṁair = k1 pim + k2
Vd Ns
k1 = a0 RT
n
im r
(2.32)
Vd Ns
k2 = a1 RT
im nr
Assuming λ = 1, also the fuel mass flow is a linear function of pim :
ṁair
ṁf = A = k1 pim + k2
(2.33)
F s
2.11 Conclusions
A mean value engine model (MVEM) is compiled in this chapter. It is a parametrized component based model of the engine’s intake system and exhaust
system, including the turbocharger and wastegate. Each component of the engine is described in terms of equations, constants, parameters, states, inputs
and outputs. This class of models describe the average behavior of the engine
over one to several thousands of engine cycles, so is very suitable for control
design.
Capitolo 3
Simulink Model
This thesis is based on the model of a turbocharged SAAB 9-3 L850 2.0T, developed at GM Power Train. The model is implemented in Simulink and has
three fundamental components:
• the driver, who is basically modeled by the accelerator pedal signal;
• the car, which contains the Mean Value Engine Model (MVEM) and the
block describing Transmission and Chassy. The MVEM was developed
by Per Andersson [3];
• the ECM, which contains all the controllers.
The top level diagram is shown in figure 3.1.
3.1 Driver
The behavior of the driver is modeled through the acceleration pedal, the gear
box and the clutch. Setting some parameters inside the driver block it is possible to simulate different driving scenarios: acceleration, constant speed or
braking. The focus of the thesis is on improving the torque response during
the transient, so the acceleration phase is considered and a step in the pedal is
given as input to the engine model. The clutch and the gears will be disregarded, and only the signal representing the acceleration pedal will be considered
as an input coming from the driver to the ECM.
3.2 Car
The car is represented by two main blocks:
21
22
C APITOLO 3. S IMULINK M ODEL
P lantD ataB us
Memory2
ECM_1_11_7_8
ECM
BUS
Interrupt G enera tor
ECM
Driv er
D riv er
Simple driver
S AAB 9-3 L 850 2.0T
Ma nua l 5 G ea rs
Figura 3.1: Top level diagram of the plant model implemented in Simulink.
Acc_Ped
t
Clutch
Up
Down
t
Gear
3rd
2nd
1st
t
Figura 3.2: Driver: acceleration phase, from first to third gear. 0=clutch down,
1=clutch up.
3.3. E NGINE S TRUCTURE
23
• the Engine Model, that will be treated in detail in the next section;
• the Transmission and Chassi model: in this block the clutch and the
shafts are assumed to be stiff and the transmission and the final drive
are assumed to multiply the torque by the conversion ratio.
Making some change inside this block we had the possibility to fix the
engine speed as constant.
3.3 Engine Structure
The components included in the model are the intercooler, engine, throttle,
wastegate, turbine, compressor, exhaust system and various pipes or manifold
connecting these components. These pipes or manifolds can be considered as
control volumes where the pressure and temperature of the gas inside depends
on the mass-flows into and out of the volume. Mass-flow are determined by
restrictions which are components that, given the pressure and the temperature
before and after the restriction, determine the mass-flow and temperature of
the flow. The restrictions and control volumes are listed next:
Restrictions
Air-Filter
Compressor
Intercooler
Throttle
Engine
Turbine/wastegate
Exhaust system
Control volumes
Pipe between air-filter and compressor
Pipe between Compressor and Intercooler
Pipe between Intercooler and Throttle
Intake manifold connecting the the throttle and cylinders
Exhaust manifold connecting the cylinders and the turbine/WG
Pipe between turbine/WG and the exhaust system
To complete the model there are only two components left. First there is the turbocharger shaft, which is modeled as a rotating inertia, and second the engine
torque model. Below the components are divided into groups:
<V eE V P R _T_AmbientAir>
<V eAAP R _p_AmbientAir>
T up
p down
1
T flow
T flow
Amb
p up
T up
p down
p up
W _es Out
p_af
p_es
T_es
[W_esO ut]
[p_t]
[T _t]
p_af
[p_a f]
Q_dot
p [P a]
p_es
p [P a]
W _out [kg/s ]
T _in [K ]
T_es
2
dT
[T _a f]
dT
W _in [kg/s ]
W _in [kg/s ]
T [K ]
T [K ]
Q_dot
W _out [kg/s ]
dp
dp
T _in [K ]
<u_W G [0-100]>
[dp_t]
[dT _t]
T _c
p_af [P a]
W _c
T _af [K ]
[w_tc]
<A_ef f [0-2c m2]>
T_af
w_tc
p_bef_c mp
W _t
T _in [K ]
p_em
[W_esIn]
W _t
T_em
[T _em]
[dw_tc]
Tq_driv ing
dw_tc
T_c
Tq_braking
p [P a]
w_tc
[p_c]
p_c
W _in [kg/s ]
T [K ]
m flow
m flow
eta_c
p_c [P a]
w_tc
W _out [kg/s ]
[dp_c]
T_em p_em
T _up
[T _c]
p_ic
2
[T _im]
[p_im]
[p_ic]
[T _ic]
p_ic
[dp_em]
W _c y l
W _in [kg/s ]
T_ic
<V<V
eP eP
HS
<V
HRSeTR
_phi_IntC
R _phi_E
QR<V
<w_e>
<lambda>
_m_F
eS
xhC
amP
<phi_Of
PK
amP
uelR
osCAllC
_phi_F
os
eq>
f sAllC
yetN
lsinalAdv
>ormal>
y ls > anc >
p_emWG _open
<V eAAP R _p_AmbientAir>
<V eE V P R _T_AmbientAir>
T_c
T _fwd_flow [K ]
Q_dot
dT
p_es
T_ic
p_down
W _ic
W _ic
[dT _em]
[p_em]
q_driving_me
q_bra king_me
1
[dT _c]
dp
T q_c
w_tc [rad/s ]
u_wg
T _trb
[dT _a f]
p_amb
[dp_a f]
T [K ]
W _out [kg/s ]
T_af
p [P a]
Q_dot
p_af
dT
T _in [K ]
W _c omp
w_e
W _c y l
[W_cyl]
dp
torque_WG open
[T q_cs]
Tq_c s
p_imWG _open
p down
T [K ]
dT
m f low
T_in [K ]
W _th
W _c y l
W _out [kg/s ]
Q_dot
W _in [kg/s ]
ef f ec tiv e area
T f low
p up
T up
p [P a]
[dp_im]
T_im
p_im
[dT _im]
p_im
<A_th>
T_ic
dT _i dp_ic
Q_dot
dT
W _c omp
p em
[W_comp]
L_req
W _af
T _em
T q_t
T _amb
mF low e
T_af
T ti
p_up
[W_ic]
IMOP
W _ic
F uel mas s flow [kg/s ]
T _c ool [K ]
dp
W _in [kg/s ]
T _in [K ]
p [P a]
phi_IgnOffs et
W _af
p_after_c mp
m_es
E MOP
T q_e
W _out [kg/s ]
dp
T_mix
phi_F inalAdvanc e
[W_a f]
T im
T [K ]
p im
air F low
[W_th]
24
C APITOLO 3. S IMULINK M ODEL
Figura 3.3: Simulink implementation of MVEM. The model is implemented
with control volumes connected to restrictions. In the center the turbocharger
shaft dynamics block connects compressor and turbine/wastegate blocks. The
shadowed backdrop indicated that dynamics (states) are present in the blocks.
p_em
3.3. E NGINE S TRUCTURE
25
1. Restrictions. These determine a mass-flow through the restriction with a
temperature, given pressure and temperature before and after the restriction.
• Incompressible flow restrictions. Air-filter, intercooler and exhaust
system.
• Compressible flow restriction. Throttle and wastegate.
• Custom restrictions. These restrictions are specific applications:
Engine Determines the port air-mass flow, exhaust mass-flow and
temperature of the gases to the exhaust manifold. This block
also models the engine torque.
Compressor The compressor is a part of the turbocharger and it is
considered as a restriction as it produces a mass-flow through
the compressor and models the mass-flow temperature. It also
produces a loading torque to the turbine.
Turbine The turbine part of the turbocharger produces a mass-flow
through the turbine, models its temperature, and describes the
torque produced by the turbine.
2. Adiabatic mixer. This component is used to adiabatically mix flows from
two different restrictions, such as when the flows through the turbine and
wastegate meet. The outputs are mass-flow and flow-temperature.
3. Model dynamics. Components which have one or more states.
• Control volume: A control volume is used to connect two restrictions. It has two states, pressure and temperature. Control volumes
are located between the restrictions: air-filter, compressor, intercooler, throttle, turbine, and exhaust system. Inputs: mass-flows in and
out of the control volume together with the flow temperatures.
• Inertia with friction: The axle in the turbocharger is modeled as a
rotating inertia, where the inertia is the total inertia of the axle, compressor and turbine. The angular velocity of the shaft is represented
by a state. The inertia is powered by the torque from the turbine and
loaded by the compressor and axle friction. The inputs are: driving
torque and loading torque. Output is speed.
3.3.1 Model Simplifications
Some general simplifications and assumptions are made:
26
C APITOLO 3. S IMULINK M ODEL
- Flows run only in forward direction.
- No heat transfer to/from the gas inside of the control volumes.
- No compressor bypass valve.
- All gases are considered to be ideal.
3.3.2 Model Inputs
Inputs to the model are:
Name
Description
Unit
α
Throttle angle
mm2
N
Engine speed
RP M
Wastegate opening. Range 0 − 1
–
λ
Normalized air/fuel ratio
–
pa
Ambient pressure
Pa
Ta
Ambient temperature
K
uwg
3.3.3 Model States
The model has states for pressures and temperatures in each control volume
and one state for the turbocharger speed:
State
Description
paf
Pressure after air-filter
Taf
Temperature after air-filter
pcomp
Pressure after compressor
Tcomp
Temperature after compressor
pic
Pressure after intercooler
Tic
Temperature after intercooler
pim
Intake manifold pressure
Tim
Intake manifold temperature
pem
Exhaust manifold pressure
Tem
Exhaust manifold temperature
pt
Pressure after turbine
Tt
Temperature after turbine
ωTC
Turbocharger speed
Figure 3.4 shows a sketch of a turbocharged SI-engine, that illustrates how
the engine is divided into subsystems to enable physical modeling and indicates all the states.
3.3. E NGINE S TRUCTURE
27
p_af
T_af
Intercooler
p_ic,
T_ic
p_comp, T_comp
Air filter
Compressor
m_c
Throttle
w_t
Exhaust
manifold
Intake
manifold
T_im
p_im
m_em
m_im
Turbine
T_em
p_em
u_wg
p_t
T_t
Catalyst &
Exhaust system
Engine
Figura 3.4: Sketch showing the components that are modeled in a turbocharged
SI-engine.
3.3.4 Model Output
If the state Tim and the input signal N are assumed constant, the engine torque
can be expressed as a linear function of the states pim and pem [1]:
Tq = k3 pim + k4 pem + k5
qHV min (λ,1)
Vd
2 k3 = nr 2π k1 Ns A
ηe + 1
Vd
λ
k4 =
d
− nVr 2π
k5 =
Vd
nr 2π
F
s
min (λ,1)
ηe
Vd
k2 N2s qHV
A
λ
F
2 N
N
+ 0.05 1000
− 0.97 + 0.15 1000
s
Furthermore it is assumed that the input signal λ does not change. The assumptions are only made for a certain operating point, i.e. the values of Tim ,
N and λ can change between operating points.
400
T intake M
350
300
250
200
150
100
50
0
0
5
10
15
20
25
Figura 3.5: Tim can be considered constant.
30
28
C APITOLO 3. S IMULINK M ODEL
3.3.5 Maps for parameters representation
Some of the components depend in a non-linear way on their inputs and might
be difficult to derive analytically a function describing their behavior. If measurements data is available for these it is convenient to use maps or look up
tables instead of a functional representation.
3.4 Performance Limiting Factors
A number of factors limit the performance of an engine; both the compressor
ratio and air to fuel ratio are very important for the efficiency but there are
some factors limiting their potential.
3.4.1 Knock
Knock is a fundamental problem in spark ignited engines. It can easily destroy an engine if the knocking is allowed to continue. Knock is a resonance
phenomenon in the cylinder that originates from an abnormal combustion
where some of the gas auto ignites before the flame front has arrived and burns
uncontrollably.
Normal combustion
Under normal conditions the combustion is ignited by a spark plug that gives
a small flame kernel. The flame kernel increases in size and develops into a
turbulent flame that propagates through the full chamber and is extinguished
at the cylinder walls. The complete combustion process takes around 60° to
complete, which corresponds to a flame propagation that is much slower than
the speed of sound. Therefore the cylinder pressure can be considered to be
constant in the cylinder. The part of the air-fuel mixture that is in front of the
flame front and has not yet burned is called end-gas.
Knocking combustion
Knock is the name given to the noise which is transmitted through the engine structure due to essentially spontaneous ignition of a portion of the endgas. When this abnormal combustion process takes place there is an extremely
rapid release of much of the chemical energy in the end-gas, causing a very
high local pressure and the propagation of pressure waves across the combustion chamber. Knock is a disturbing source of noise and if it is let to continue,
can significantly damage the engine.
3.4. P ERFORMANCE L IMITING FACTORS
29
Two theories explain the origin of knock phenomenon: the autoignition and
the detonation theory. The first states that when the fuel-air mixture in the end
gas region is compressed to sufficiently high pressures and temperatures, the
fuel starts to oxidize spontaneously in part or all of the end-gas region. The
second says that under knocking conditions the flame front accelerates to sonic
velocity and burns the end gas much faster than under normal conditions.
Knock can be controlled using the spark advance.
3.4.2 Surge
The compressor operation is limited by the so called surge line. It shows in the
compressor map the border of a region with unstable operation of the compressor flow. A centrifugal compressor can only generate a pressure increase when
there is a flow through it. An increase in pressure ratio over the compressor
opposes the flow and gives a decreased mass flow. When the pressure ratio becomes too high the compressor can not maintain the flow necessary to generate
the pressure increase and the flow through the compressor breaks down. When
the flow breaks down there can be an uneven distribution of forces around the
compressor wheel. The instabilities induce vibrations in the compressor wheel
and turbo shaft which can cause the blades to hit the casings or destroy the
bearings.
Surge can easily occur when the throttle is suddenly closed. The remedy is
to add a surge valve which reduces the pressure after the compressor by either
a return pipe which leads the flow back in front of the compressor or a pipe that
releases it directly to the ambient atmosphere. The surge valve is controlled by
the engine control system which detects if there is a risk for surge and then
opens the valve.
3.4.3 Limits on air to fuel ratio
The fundamental limitation for SI engines operating with homogeneous charge
is that there is a combustion stability limit around λ = 1.3 where the fuel can
no longer be ignited and combusted. Other limits are also imposed by the legislators, based on the levels of pollutant emissions from the engine.The current
emission after treatment systems can not effectively handle pollutant emissions
other than λ = 1.
30
C APITOLO 3. S IMULINK M ODEL
3.5 Engine Management System (ECM)
An engine management system must meet and compromise between several
goals, such as: produce good driveability, maximize the engine performance,
minimize the fuel consumption, and give low emissions. Some of the steps
towards this are: metering the amount of air and fuel according to quantity
and ratio, forming a combustible mixture, distributing the air-fuel mixture to
the cylinders, igniting the mixture at the optimum position, controlling exhaust
emissions and controlling the non-exhaust emissions.
The control system comprises the control of the A/F ratio, the preparation
of the air to fuel mixture, ignition timing, and ignition energy, that all influence the engine performance regarding the combustion process and emission
formation. Several sensors are positioned on the engine to measure the engine
state. The signals are measured by the electrical control unit (ECM) that calculates and executes the control actions.
The ECM is a complex control unit, that contains many controllers; among
them the most important are:
• Throttle Control
• WG Control
• Lambda Control
• Fuel Control
• Spark Advance Control
As illustrated in Fig. 3.6, the ECM is formed by two main blocks: a Time Based
Unit an an Angular Based Unit. They both receive inputs from the driver (Acceleration pedal position) and from the engine (mainly the engine speed N and
the inlet air mass flow).
The Time Based unit generates the requested Lambda, that goes into the
Angular Based and is used to calculate the requested fuel mass, the requested throttle position (see 3.5.2) and the signal to control the wastegate. The
Angular based unit works with signals that refer to fuel and spark advance.
The requested fuel mass is calculated through lambda request (coming from
the Time based unit), given the engine speed and the inlet air mass flow. The
3.5. E NGINE M ANAGEMENT S YSTEM (ECM)
31
section related to SA, takes as input from the Time Based unit the signal corresponding to the best spark advance timing (phi-MBT) and gives as outputs
two signals (phi-offset and phi-ignition) that will be used to calculate the SA
efficiency.
The aim of the thesis is to improve transient response through more advanced controllers than what are used today, investigating if and how new
different control of throttle, lambda and wastegate may quicken the torque response. So the Fuel Control and Spark Advance Control, implemented now
on the car, will not be changed and a brief description will be given in the
following sections.
Phi Offset
KNOCK
Pct Throttle
THROTTLE CONTROL
Pct WG
WG CONTROL
L Req
LAMBDA CONTROL
Phi Ignition
AIR
Acc Pedal
v Vehicle
N
m Air Per Cyl
TORQUE
FUEL
M Pred
L Req
SPARK
Phi MAX Offset
Phi min Offset
Phi MBT
TIME
BASED
FUEL
N
m Air Per Cyl
PHIIGNITION
t Injection
FUEL
m Fuel Req CONTROL
SA
Phi Offset
Phi Ignition CONTROL
Phi Offset Normal
ANGULAR
BASED
Figura 3.6: ECM
3.5.1 Map-Based Control
In current practice, lookup tables or maps are used extensively in the control
system for parameter representation and directly for control (c.f. Section 2.1),
even though there is a trend towards more model based techniques. When
maps are used for control the basic idea is to use measured quantities, like engine speed and intake manifold pressure (i.e. load), as inputs and deliver the
32
C APITOLO 3. S IMULINK M ODEL
control action directly as output. These maps usually implement open loop
controllers and are obtained through calibration in an engine test bench. The
procedure is simple but time consuming. The engine is controlled to the desired grid point in speed and load and the control outputs are tuned to achieve
desired goals. Control outputs are then stored directly in the map.
The advantage with maps is that they are computationally efficient, easy
to implement and easy to obtain if an engine is available. A major disadvantage is that they do not offer any extrapolation; when an engine is re-designed
all maps might have to be re-calibrated which is a time consuming process.
Another control paradigm is model based control where models are used to
describe the functions and the complex interactions between inputs and outputs. Models are interesting since they offer the benefit of extrapolation and in
general require less measurements for calibration. Models are also interesting
since they directly show the structure of the model and point out the important
effects.
3.5.2 Throttle control
The Throttle control determines the throttle area request to satisfy the demand
coming from the acceleration pedal. Figure 3.7 shows the path from the pedal
(which represents the request of the driver) to the requested air mass flow.
M_Pred
Acc_Pedal
Pedal
to
Torque
m_Fuel_Req
x
Torque
to
Fuel
K_eff
L_st
x
x
m_Air_Req
L_Req
Figura 3.7: Signals path from the pedal to the air mass request.
The pedal position (AccP edal ) is converted into predicted torque (MP red ),
that takes into account the efficiency coefficient (Kef f ). This coefficient considers the influence of Lambda and Spark Advance on the torque, in fact depending on the values of these two variables, the effective torque coming out
from the engine is modified. Torque is then converted into requested fuel mass
flow (mF uel−Req ) and finally to requested air mass flow (mAir−Req ) through
the definition of lambda (see.2.30):
λReq =
ṁair
ṁf uel λst
(3.1)
Then the mAir−Req goes into the throttle controller block, which basically is
3.5. E NGINE M ANAGEMENT S YSTEM (ECM)
m_Air_Req
m_Air
+ e_Air
-
PID
+
+
THROTTLE
AREA
REQUEST
33
Th_Pst_Req
Figura 3.8: Throttle controller.
a PID acting on the error between the air mass reference mAir−Req and actual
value mAir , and finally gives the throttle request.
3.5.3 Lambda Control
In an engine management system for SI engines there are several controllers
and two of the most important are A/F control and ignition control. The main
purpose of the A/F control loop is emission reduction. To achieve good emission reduction is necessary to use both feedback and feedforward. Feedback is
necessary for maintaining λ in the narrow band around λ = 1 where the catalyst operates best. Only an open loop will not be sufficiently accurate due to
model uncertainties. Feedforward is necessary for fast transient response and
it determines the basic amount of fuel to inject.
The main purpose of the ignition control loop is to get good fuel economy
while avoiding knock. At low loads the engine runs with the most fuel efficient ignition while at high loads engine knock can pose a limit on the ignition
timing and it has to be set later.
3.5.4 Fuel Control
The amount of fuel is controlled by the fuel injection system, which is responsible for mixture formation, mixture transport and mixture distribution. By
mixture formation is meant the process of preparing the air and fuel mixture:
some of the fuel drops that enter the inlet manifold evaporates (which is desirable), while some other forms a film on the manifold walls (which is not desirable). Mixture transport is the transport of air-fuel mixture to the cylinders
and mixture distribution is the distribution of air and fuel to each cylinder.
When the driver accelerates, the throttle position changes and so the inlet
air flow changes. The fuel injection controls the amount of fuel injected so that
at all times the air to fuel ratio is kept in a narrow band around λ = 1. This
value of λ is chosen since gives good driveability and low emissions. The fuel
control is based on the speed density principle [2], which determines the air mass
flow into the cylinders m˙ac through the volumetric efficiency, given the engine
34
C APITOLO 3. S IMULINK M ODEL
displacement volume, intake density and engine speed
ṁac = ηvol ρi Vd
N
nr
where the intake manifold density ρi is determined from the pressure and temperature in the intake using the ideal gas law. Then how much fuel to inject is
determined based on the air flow into the cylinders and the requested λ.
ṁf i =
ṁac
A
λReq
F s
Once the fuel mass flow is obtained, it is possible to calculate the desired
injection opening time
tinj =
ṁf i
+ t0 (ubatt )
N c1
where ubatt is the battery voltage, t0 is the opening and closing time of the valve
and c1 is a constant including pressure influence (which is constant), injector
constant, number of cylinders and the number of strokes per cycle.
3.5.5 Spark Advance Control
Spark advance control deals with determination of the engine position where
the spark plug shall ignite the air-fuel mixture and start the combustion [13][14].
It is thus used to position the combustion and pressure trace relative to the
crank shaft motion: work is lost to heat transfer and to compression if it is
placed too early and expansion work is lost if it is placed too late. Spark timing is one of the most important factors affecting the engine performance. The
instant moment when the spark happens in the cycle is really crucial in determining the produced power and torque. The combustion event must be properly located relative to top-center (TC) in order to obtain maximum torque. The
combustion process starts before the end of the compression stroke, continues
through the early part of the expansion stroke and ends after the point in the
cycle at which the peak cylinder pressure occurs. The combined duration of
the flame development and propagation process is typically between 30 and
90 crank angle degree.
Thus, optimal spark advance positions the pressure trace in a way that
compromise between the effects mentioned above. To define the position of
the in-cylinder pressure relative to TDC, the peak pressure position (PPP) is
used, Figure 3.9. The PPP is the position in crank angle where the in-cylinder
pressure takes its maximum value.
3.5. E NGINE M ANAGEMENT S YSTEM (ECM)
35
Figura 3.9: The PPP (Peak Pressure Position).
If the start of the combustion process is progressively advanced before TC,
the compression stroke work transfer (from the piston to the cylinder gases)
increases. Also, if the end of the combustion process is progressively delayed
by retarding the spark timing, the peak cylinder pressure occurs later in expansion stroke and the peak is smaller. So the work transfer from the cylinder gases
to the piston is reduced. When the magnitudes of these two opposing trends
just offset each other, the maximum brake torque (MBT) is produced. This is also
the point where the optimum timing occurs. Also at this point, the maximum
brake power and minimum brake specific fuel consumption is obtained. Any
advanced or retarded timing from this optimum point will produce a lower
torque.
Figura 3.10: Three different pressure traces resulting from three different spark
advances. The optimal spark advance is close to SA2
36
C APITOLO 3. S IMULINK M ODEL
However the optimum spark timing depends on the rate of the flame de-
velopment and propagation and the length of the flame travel path across the
combustion chamber. In addition, MBT timing depends on speed where the
spark must be advanced whenever speed increases in order to maintain optimum timing due to the increasing combustion process duration in crank angle
degrees. Also, optimum spark timing depends on load; as the load and intake
manifold pressure are decreased, the spark timing must be further advanced
to maintain optimum engine performance.
3.6 Turbo Control
Two additional important control loops are used in turbocharged engines, the
first is the surge control where the goal is to avoid surge and the second is the
charge control that is achieved using a wastegate. Surge control will not be
investigated here, but this thesis will focus on the wastegate control.
3.6.1 Wastegate Control
The wastegate is primarily used to control the boost pressure, i.e. the pressure
after the compressor. It is tuned to give the desired torque curve. As previously
mentioned the wastegate controller is situated in the Time Based section of
the ECM. It presents an activation block, which let the wastegate controller be
active only if a certain boost pressure is reached, otherwise the wastegate is
kept closed. When this threshold is overcome, then the controler is activated.
A PI plus a feedforward (FF) action is implemented. The FF action is basically a
p_im_Req
p_bef_Th
PI
+
N
m_air
u_wg
+
FF
Figura 3.11: The WG controller is realized with a PI plus a feedforward (FF)
action.
look up table based on engine speed and air mass flow, which gives the steady
state values for the wastegate. The PI acts so that the difference between the
pressure before throttle and the requested intake manifold pressure is equal to
a desired constant value.
3.7. S YSTEM M ODIFICATIONS
37
The basic operating principle is to open the wastegate when the pressure
after the compressor goes over a certain limit. This basic functionality can be
achieved using mechanical designs. However the performance can be better
matched to the engine and fuel if the wastegate could be opened whenever is
needed. A problem with a conventional mechanical design of the wastegate
controller is that the wastegate opening is only determined by the wastegate
spring stiffness and pressure differences. Whit an electrically controlled valve
the current fuel and engine conditions can be better matched and higher efficiency can be reached without risking to destroy the engine. The boost pressure
can be increased until the ECM senses knock. The wastegate is then opened so
that the pressure in the system decreases and knock is reduced.
3.7 System Modifications
In Chapter 2 the mean value engine model developed by Per Andersson was
presented, based on [3]. The engine under consideration includes some new
features, so the model needs to be slightly changed. The most significative
changes concern the compressor by-pass valve (not included in the previous
model), the influence of lambda and spark advance efficiencies on torque and
the effect of spark advance on temperature and the volumetric efficiency. The
new model was developed by Andreas Jerhammar at GM PowerTrain (Sweden).
3.7.1 Bypass Valve
A compressor bypass valve (CBV), also known as a compressor relief valve,
is a vacuum-actuated valve designed to release pressure in the intake system
of a turbocharged car when the throttle is lifted or closed. This air pressure
is re-circulated back into the non-pressurized end of the intake (before the
turbo). Basically the bypass valve has the same function as the wastegate, it
avoids surge for the compressor while the wastegate prevents the turbine from
overrunning.
3.7.2 Lambda and SA Efficiency
In the new model the efficiencies of λ and SA (ηSA and ηλ ) have been added.
They have an impact on the torque, so when calculating the BMEP we need to
38
C APITOLO 3. S IMULINK M ODEL
take into account the efficiencies. So equation (2.28) is modified as follows:
BM EP = IM EP · ηSA · ηλ − P M EP − F M EP
(3.2)
Lambda Efficiency
1.4
1.2
Efficiency
1
0.8
0.6
0.4
0.2
0
0.8
1
1.2
Lambda
1.4
1.6
1.8
2
Figura 3.12: Lambda Efficiency
As can be seen from the Plot 3.12, ηλ has its maximum value at (λ ≈ 0.9),
so maximum torque and power is achieved when the mixture is slightly rich.
This effect can be utilized to increase the torque at full load; when the air flow
has reached its limit the torque can be increased slightly by adding more fuel.
SA Efficiency
The spark advance (SA) is used to position the combustion in relation to the
crank shaft rotation and has a direct influence on the engine efficiency and the
generated torque. In order to get the maximum power the spark should occur
at the MBT point, so that SA efficiency gives one; for all other values ηSA is
lower than one (see fig. 3.13).
An early SA results in an early pressure build-up and also a lower pressure
during the later part of the expansion stroke. A too early build-up that occurs
before TDC counteracts the piston motion and lowers the output power. Furthermore it has a lower pressure during the expansion phase which results in a
lower output power. The lower pressure values for the cycles with the earlier
3.7. S YSTEM M ODIFICATIONS
39
Spark Advance Efficiency
¬
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
−40
−35
−30
−25
−20
−15
−10
−5
0
[deg]
Figura 3.13: Spark Advance Efficiency
ignition timing are due to the conversion to work of the internal energy coming
from the combustion.
Late SA results in a later pressure build-up and a higher pressure during
the expansion stroke. If the SA comes too late than the gain for higher pressure
during the later part of the expansion stroke can not compensate for the losses
in the early part of expansion stroke.
Figures 3.14-3.16 show the comparison between the model including or not
the efficiencies for Lambda and SA. The thin line refers to the model with efficiencies, while the thick line refers to the model without efficiencies. It can be
seen that, as explained above, ηλ and ηSA influence the engine torque.
3.7.3 Temperature Correction Map
The outgoing temperature from the engine will increase the efficiency and
torque from the turbine, so a temperature correction table as been added, depending on spark advance and air mass (see fig.3.17).
3.7.4 Volumetric Efficiency
The new volumetric efficiency has been calibrated experimentally and is calculated by means of a look up table (see Sig.3.18), which takes as inputs pim and
40
C APITOLO 3. S IMULINK M ODEL
400
30
friction torque
grass torque
300
200
100
0
0
5
10
10
24
22
5
10
5
10
200
5
0
−5
26
20
0
with efficiency
without efficienc y
300
net torque
pumping torque
15
28
100
0
0
5
−100
10
0
Figura 3.14: Thin line: model with efficiencies, thick line: model without
efficiencies.
5
T exhaust M
1500
2
1.5
1
1.3
P exhaustsyste m
x 10
0
x 10
5
5
1.2
1.1
1
0
5
1000
500
0
10
0
with efficiency
without efficienc y
1500
T exhaustsyste m
P exhaust M
2.5
10
5
10
5
10
1000
500
0
0
Figura 3.15: Thin line: model with efficiencies, thick line: model without
efficiencies.
41
1000
0
0
5
10
0.95
M fuel kg/comb
0.9
4
0
5
x 10
10
0
5
5
10
5
10
5
10
20
0
0
x 10
−4
1
0
10
0
40
2
0
with efficiency
without efficiency
0.5
−4
2
0
1
Area WG
Lambda
1
PHi final advance
Throttle
2000
Phi offset normal
3.7. S YSTEM M ODIFICATIONS
0
Figura 3.16: Thin line: model with efficiencies, thick line: model without
efficiencies.
¬ 30
Tem 600
¬ 20
¬ 15
¬ 10
¬ 6
¬ 4
¬ 2
¬ 0
¬ −4
¬ −8
400
200
0
−200
30
20
SA
10
0
−10
0
200
400
600
800
mair
Figura 3.17: Correction map for the exhaust temperature.
1000
42
C APITOLO 3. S IMULINK M ODEL
N.
volumetric
efficiency
¬ 5900
¬ 4900
¬ 3100
¬ 5400
¬ 3900
¬¬
2500
2100
¬
¬1700
1500
¬¬1000
800
N
pim
Figura 3.18: Volumetric efficiency. Look up table.
3.7.5 WG Pressure valve
Today the WG is actuated by a simple pressure valve (see Sec. 6.3.1) which
can be open only if a certain pressure over it is reached and is represented by a
block in the Simulink model. At the beginning one parameter inside the block
(Dynamic-Force) was set to 200, but the torque behavior was very strange, so
it was changed to 0. The comparison of the wastegate signal using DynamicForce equal to 200 or 0 is shown in Fig.3.20, where also the response of the
system without valve model is plotted. It can be seen that only when the WG
valve model is removed, the signal before and after the valve is the same. Instead, when considering the valve model, the signal that the controller thinks
to manipulate is not the same that goes into the system, but is modified by the
valve model (a schematics is shown in Fig.3.19). Consequently, also net and
pumping torque are different when considering or not the valve and when using different values for the parameter Dynamic-Force(Fig.3.21).
Now we assume to substitute this valve with an electronic one, which can
Aeff WG
Valve
Model
Areal WG
Figura 3.19: The signal entering and going out from the valve model are
different.
3.7. S YSTEM M ODIFICATIONS
A eff WG
x 10
43
4
Dyn force 0
Dyn forse 200
without valve model
1
0
4x 10
−4
4.5
5
5.5
6
6.5
7
7.5
8
4.5
5
5.5
6
6.5
7
7.5
8
A real WG
2
1
0
−1
4
Figura 3.20: Top: wastegate position before the valve model.
wastegate position after the valve model.
Bottom:
400
net torque
300
200
100
dyn force 0
dyn force 200
without valve model
0
−100
4
4.5
5
5.5
6
6.5
7
7.5
8
4
4.5
5
5.5
6
6.5
7
7.5
8
pumping torque
15
10
5
0
Figura 3.21: Top: net torque. Bottom: pumping torque. The figure shows
how the responses are different when changing the value of the parameter
Dynamic-Force.
44
C APITOLO 3. S IMULINK M ODEL
be open and closed whenever is needed. So the simulations will be performed
removing the block representing the pressure valve in the Simulink model.
3.8 Changes in the ECM
Since the system has been modified, also the ECM must be modified in order
to avoid mismatches between the two models. So Lambda and SA efficiencies
have been added into the ECM when calculating the predicted torque, which
will be used to calculate the requested fuel and requested air, as described in
Section 3.5.
3.9 Conclusions
The Simulink implementation of the model is discussed in this Chapter. The
model describes a turbocharged SAAB 9-3 L850 2.0T, and was developed at
GM Power Train (Sweden). Some new features were added to the model, so
their meaning and their implementation is presented in the Chapter.
Capitolo 4
Validation of the NonLinear
Model
This chapter discusses the validation of the nonlinear model, which has been
realized by comparing the data recorded during a driving test on the car, and
the results of simulations in Simulink. The driving test was performed on
a SAAB 9-3 L850 2.0T, equipped with a turbocharger, at GM PowerTrain in
Södertälje (Sweden).
4.1 Experiment setup
To validate the nonlinear engine model a test on the car has been performed.
The idea is to drive the car in the street equipped with many sensors and
than collect some data. We have to collect both inputs, like driver pedal position, and outputs, like for example Tim , pim and pic . Then we give the same
inputs to our model and we check if the outputs are more and less the same.
We have been following the steps below:
1. Decide which signals to record
2. Set the car for a driving test
3. Use Control desk to set all the signals that we want to record
4. Run the car and save the data
5. Analyze off -line the collected data
6. Compare them with the model
45
46
C APITOLO 4. VALIDATION OF THE N ON L INEAR M ODEL
1. Recorded signals
To be able to validate the model, in addition to the signals coming from the
sensors, all the inputs and outputs to and from the engine need to be recorded
so that it will be possible to simulate the same testing conditions in Simulink
ambient.
So the following sets of signals have been recorded:
• Inputs for the engine:
– Throttle Area
– N
– Lambda
– Spark Advance
– uwg
– Bypass valve
• Inputs from the driver:
– Pedal Position (obs. the gear is not required since we have N)
• Inputs from the ECU:
– Torque Request
– Air/Mass Request [mg/comb]
– Fuel Mass Flow [Kg/s]
• Outputs:
– Predicted Torque
– Vehicle speed
2. Car setting
The test was performed on a turbocharged SAAB 9-3 2.0T. All the states have
been measured, except for the turbine speed, for which there was no sensor
available. So there were 6 pressure sensors and 6 temperature sensors, and
of course all the inputs (i.e. throttle position, Lambda, Fuel, Spark advance,
wastegate position, engine speed). Especially the temperature sensors that we
use don’t have a really high accuracy, about 5 degrees , and they are quite slow
so we expect to see some difference in their value in compare with the model.
4.1. E XPERIMENT SETUP
47
Software & Hardware.
To record all the required signals from the sensors, two very powerful and flexible units, called M icroAutoBox and RapidP roBox, replace the ECU usually
used in the car. M icroAutoBox is a real-time system for performing fast function prototyping from scratch. It operates without user intervention, just like
an ECU. The RapidP ro hardware works as an extension to dSPACE prototyping systems . It has a compact and robust mechanical design, it is ideally suited
for in-vehicle use, and can also be used on test benches and in laboratories. Application programs are stored in nonvolatile memory, allowing M icroAutoBox
to start up autonomously after power-up. A PC or notebook can be connected
temporarily to the ECU for program download and data analysis. The positive
Figura 4.1: Schematic picture of measurement setup. A PC or notebook can be
temporarily connected to the ECU for program download and data analysis.
aspect to use this configuration is that we can design all our controllers in a
normal Matlab/ Simulink environment and then directly implement them into the car. Real-Time Workshop and Stateflow Coder automatically generate
C code from Simulink block diagrams and Stateflow systems. Together with
dSPACE’s Real-Time Interface, these tools provide a seamless transition from
your block diagram to dSPACE’s real-time hardware.
Once that all the sensors have been connected, we have been using a software to visualize and analyze all the data coming from the car during the
test.
3. Use Control desk to set all the signals that we want to record
The CalDesk software is the heart of the dSPACE calibration system, with a
graphical user interface for handling measurement, calibration, and diagnostic
tasks (see fig. 4.2). Apart from ECU calibration, CalDesk is also designed for
rapid control prototyping, data acquisition, and data analysis tasks. Interfaces
to dSPACE prototyping systems, measurement modules, and vehicle control
48
C APITOLO 4. VALIDATION OF THE N ON L INEAR M ODEL
system are provided. Several devices and interfaces can be accessed in parallel.
The software can directly be installed in a normal Laptop that can be connected
Figura 4.2: CalDesk user interface. While recording the data, this interface
gives the possibility to visualize the desired signals and so check their behavior
in real time.
to the Dspace hardware installed in the car. The frontpage of the software can
be set so that meanwhile the car runs, you can easily have a look of all the
variables that you are interested in and then check if everything is going well
and in the same time record all the data in the computer.
4. Running the car and saving data
Since we have to use the data to validate the engine model, we tried to reproduce, when collecting data, the same scenarios that we simulate with the
Simulink model. The same assumption adopted for the simulations in Simulink are then made for the test on the car. So the gear is fixed and the engine
speed is constant. According with this, the following simulations have been
performed on the car:
• Fixed engine speed at 1500rmp, fixed gear, acceleration
• Fixed engine speed at 3000rmp, fixed gear, acceleration
• Fixed first gear, acceleration
4.2. D ATA PROCESSING
49
• Fixed second gear, acceleration
• Fixed third gear, acceleration
• General driving
4.2 Data processing
Control desk gives the possibility to record directly the data in a Matlab format
and so transfer them directly into Matlab space to be analyzed. The problem
is that the structure coming from Control space make is not exactly easy to
use. So a Script in Matlab (by R. Backman) has been used to convert the data
in a normal Matlab format. To use them as input for the Simulik model, one
more step is needed: to create a Structure with time for all the inputs and states
so that has been possible to use the block From workspace, as inputs for the
model.
Moreover, we could not use directly the signals recorded from the car, but
some units conversion were needed to adapt them to the inputs required by
the model.
4.3 Validation
All the data collected in the driving tests have been used to validate the model
that we are using for control design. A Simulink model has been created trying
to isolate just the engine from all the big and complex model that describes also
the control unit and the body dynamics of the car. In this way is possible to give
as inputs to the engine the same inputs that the real engine received during the
test and then compare the results.
Fixed first gear, acceleration.
The acceleration phase is considered, as can be seen from Fig.4.3, showing the
acceleration pedal from the driver (up) and the car speed [Km/h] (down).
Figures 4.4 - 4.5 show the inputs for the engine: air fuel mass, lambda, throttle,
spark advance, wastegate area and signals calculated in the ECM: requested air
mass, engine losses, predicted torque.
Figure 4.6 shows the comparison between the recorded data (thin line) and
the signals from the simulation of the model (thick line). From the left bottom corner: pim , pcomp , pic , Tim . It can be seen that the sensors are not able to
50
C APITOLO 4. VALIDATION OF THE N ON L INEAR M ODEL
Pedal from ca r
150
100
50
0
0
5
10
15
20
0
5
10
15
20
speed from ca r
120
100
80
60
40
20
500
0
0
5
10
15
20
1
0.5
M fuel kg/s
1
0
x 10
5
10
15
20
0
0
−1
0
5
10
15
20
0
5
10
15
20
0
5
10
15
20
40
20
0
100
0.5
0
1
−4
Area WG
Lambda
1.5
PHi final advance
Throttle
1000
Phi offset normal
Figura 4.3: Fixed first gear scenario. Acceleration pedal from the driver (up)
and car speed [Km/h] (down).
5
10
15
20
50
0
Figura 4.4: Inputs for the engine.
51
VeETQC M EngLoss ECM car
4.3. VALIDATION
1000
500
VeMAFR m AirPerCylAirMeter
0
0
5
10
15
20
1500
60
50
40
30
20
10
0
5
10
15
20
0
5
10
15
20
300
VeETQC M Pred ECM ca
r
m AirReq ECM car
1500
1000
500
0
0
5
10
15
20
200
100
0
−100
−200
Figura 4.5: Signals calculated in the ECM
5
4
P before thro t
3
2
1
0
3
P intake M
x 10
0
x 10
5
10
15
2
1
0
5
10
15
20
0
5
10
15
20
400
1
0
5
5
2
0
x 10
3
0
20
T intake M
P after compresso r
4
5
10
15
20
350
300
250
Figura 4.6: Fixed first gear, acceleration.Thin line: recorded data. Thick line:
signals from the model.
52
C APITOLO 4. VALIDATION OF THE N ON L INEAR M ODEL
follow the system dynamics when it is fast, but generally the the results are satisfactory. Analogous considerations hold for the plots related to the following
simulations.
Fixed engine speed at 1500rpm, fixed gear, acceleration.
Fig.4.7 show the results for the simulation with fixed engine speed at 1500rpm
and fixed gear. Again the recorded data (thin line) are compared with the signals from the simulation of the model (thick line). Left bottom corner: pressure in the intake manifold, pressure after compressor, pressure before throttle,
Temperature in the intake manifold.
5
1.1
P before thro t
x 10
1.05
1
0.95
P intake M
15
0
x 10
5
10
15
5
1.06
1.04
1.02
0
5
10
15
20
0
5
10
15
20
4
300
290
10
5
0
x 10
1.08
1
20
T intake M
P after compresso r
1.1
0
5
10
15
20
280
270
260
Figura 4.7: Fixed engine speed at 1500rpm and fixed gear, acceleration. Thin
line: recorded data. Thick line: signals from the model.
Fixed engine speed at 3000rpm, fixed gear, acceleration.
Figure 4.8 refers to simulation with fixed engine speed at 3000 rpm and fixed
gear. Thin line represents the recorded data and thick line the signals from the
model. From the left bottom corner: pressure in the intake manifold, pressure
after compressor, pressure before throttle, Temperature in the intake manifold.
4.4. C ONCLUSIONS
5
1.6
P before thro t
1.4
1.2
1
0.8
15
P intake M
x 10
0
x 10
5
10
15
5
model
car
1.2
0
5
10
15
20
0
5
10
15
20
4
300
290
10
5
0
x 10
1.4
1
20
T intake M
P after compresso r
1.6
53
0
5
10
15
20
280
270
260
Figura 4.8: Fixed engine speed at 3000rpm, fixed gear, acceleration. Thin line:
recorded data. Thick line: signals from the model.
4.4 Conclusions
A test to validate the nonlinear model is described in this Chapter. Plots showed that the nonlinear model describes the true nonlinear system with a good
level of accuracy. So it will be linearized in the next chapter and the resulting
linearized model will be used for control design.
Capitolo 5
Analysis of linearized model
Since the model is highly nonlinear, it has been linearized around 45 operating
points. Control design requires linear models with good accuracy, so this chapter focuses on the linearization process and analysis of the resulting linearized
model, which has been validated through a comparison with the nonlinear one.
5.1 Linearized Model
An important observation is that several state equations have dependance on
both intake and exhaust conditions. A consequence of the couplings is that it is
not sufficient to only model the intake side on TC SI-engines, hence the model
includes the exhaust side and turbocharger speed dynamics.
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
p˙af
T˙af
pcomp
˙
˙
Tcomp
p˙ic
T˙ic
pim
˙
˙
Tim
pem
˙
˙
Tem
p˙t
T˙t
ωTC
˙
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
fpaf (paf , Taf , pcomp , ωTC , pamb , Tamb )
⎤
⎥
⎥
⎥
⎥
fpcomp (paf , Taf , pcomp , Tcomp , pic , ωTC )
⎥
⎥
⎥
fTcomp (paf , Taf , pcomp , Tcomp , pic , ωTC )
⎥
⎥
fpic (pcomp , Tcomp , pic , Tic , pim , α)
⎥
⎥
⎥
fTic (pcomp , Tcomp , pic , Tic , pim , α)
⎥
⎥
⎥
fpim (pic , Tic , pim , Tim , pem , N, α, λ)
⎥
⎥
fTim (pic , Tic , pim , Tim , pem , N, α, λ)
⎥
⎥
fpem (pim , Tim , pem , Tem , pem , N, λ, uwg , Tamb ) ⎥
⎥
fTem (pim , Tim , pem , Tem , pem , N, λ, uwg , Tamb ) ⎥
⎥
⎥
⎥
fpt (pem , Tem , pt , Tt , ωTC , uwg , pamb )
⎥
⎥
fTt (pem , Tem , pt , Tt , ωTC , uwg , pamb )
⎦
fTaf (paf , Taf , pcomp , ωTC , pamb , Tamb )
fωTC (paf , Taf , pcomp , Tcomp , pem , Tem , pt , ωTC )
55
(5.1)
56
C APITOLO 5. A NALYSIS OF LINEARIZED MODEL
Starting from the non-linear process model ẋ = f (x, u), given by eq.(5.1), a
linearized model of the system has been derived, around 45 operating points.
5.1.1 Model linearization
The linearized model has been developed at General Motors Power Train, (Södertälje), based on [3]. The operating points have been chosen in such a way
that they reflect the non-linear dynamics of the model and describe the input
u0 of the model necessary to determine the stationary point x0 . As the model
consists of control volumes, whose dynamics depend on the air-mass flows; it
is natural to choose the linearization points across the range of air-mass flow
through the engine. The air-mass flow is basically determined by multiplying
the engine speed N by the intake manifold pressure pim . An excellent candidate to describe the operating points is therefore an engine map as it is commonly measured for different engine speeds and different manifold pressures
(see Fig.5.1).
16
x 10
4
14
12
p_im
10
8
6
4
2
0
1000
1500
2000
2500
3000
N
3500
4000
4500
5000
Figura 5.1: Engine map. For a fixed engine speed N, the operating points only
depend on the intake manifold pressire pim .
In the hypothesis of constant Tim , for every stationary point (x0 , u0 ), the
linearized model is:
ẋ = Ax(t) + Bu(t)
y(t) = Cx(t)
(5.2)
5.1. L INEARIZED M ODEL
57
where the states and the inputs are given respectively by x and u vectors:
⎡
⎤
paf
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
x=⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Taf
pcomp
Tcomp
pic
Tic
pim
pem
Tem
pt
Tt
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
u=⎢
⎢
⎢
⎢
⎣
α
N
wg
λ
pamb
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(5.3)
Tamb
ωTC
and the matrices are given by
A = fx (x0 , u0 )
B = fu (x0 , u0 )
C = gx (x0 , u0 )
D = gu (x0 , u0 )
(5.4)
Above the notation fx (x0 , u0 ) refers to the Jacobian of f (x, u) with respect to
the states x evaluated in (x0 , u0 ). So at the end there are 45 systems like (5.2),
with 12 states.
Actually the ambient pressure and temperature are uncontrollable inputs
and can be seen as disturbances. In fact pamb can be considered as constant and
Tamb varies very slowly compared to the other variables, so can be treated as
constant as well. Then the B matrix can be split into two parts and the system
can be rewritten as
ẋ = Ax(t) + Bu(t) + Bd ud (t)
(5.5)
y(t) = Cx(t)
where
⎡
α
⎤
⎥
⎢
⎢ N ⎥
⎥
u=⎢
⎢ wg ⎥
⎣
⎦
λ
ud =
pamb
Tamb
From now on disturbances term will not be considered and the term Bd will be
neglected.
58
C APITOLO 5. A NALYSIS OF LINEARIZED MODEL
The aim of this project is to improve the engine torque response, so a torque-
based control strategy will be used, based on the error between the actual
torque coming from the engine and a given reference torque signal. So a natural choice for the system output is the torque itself. But another good variable
to judge the system performances is the fuel consumption, since the emissions
have to be monitored and cannot exceed the limits imposed by the legislation.
With the hypothesis of constant temperature in the intake manifold, both
the torque and the fuel mass flow can be expressed as linearly dependent on
states, according to Sections 2.10 and 2.8:
Tim = const ⇒
k1 x7 (t) + k2
yf uel = yT q = k3 x7 (t) + k4 x8 (t) + k5
(5.6)
and the C matrix can be set:
C=
000000
k1 0 0 0 0 0
0 0 0 0 0 0 k3 k4 0 0 0 0
(5.7)
The D matrix is a (2 × 12) zero matrix.
Depending on control strategy fuel consumption can be neglected and the
torque is then the only output of the system. So in the following sections and
in the next chapters two different cases will be considered:
1. M ISO system ⇒
ẋ = Ax(t) + Bu(t)
yT q = k3 x7 (t) + k4 x8 (t) + k5
2. M IM O system ⇒
⎧
⎪
⎪
⎨ ẋ = Ax(t) + Bu(t)
yf uel = k1 x7 (t) + k2
⎪
⎪
⎩ y = k x (t) + k x (t) + k
Tq
3 7
4 8
5
5.2 Validation of the linearized model
The linearized model was developed by GM Power Train, based on [3]. Now
the nonlinear model was a bit changed, as explained in Chapter 4, so here a validation is carried out, to check if the linearized model still represents the nonlinear behavior. First, the linearized model is validate for stationary conditions
5.2. VALIDATION OF THE LINEARIZED MODEL
59
and then step response experiments are used for the dynamics. A comparison
between the responses of the nonlinear system and the linearized model shows
that the behavior of the linearized model is similar to the nonlinear one.
Consider as example the operating point number 15.
1. Static simulation. Figures 5.2 - 5.3 show a static simulation with all the
inputs constant and set to their steady state values. After the initial transient, during which the nonlinear system has to reach the steady state,
2
128.5
1.5
Lambda
129
128
127.5
1
0.5
127
0
126.5
−0.5
0
5
10
1
2001
0.5
2000.5
rpm
Area WG
Throttle
the linearized model well represents the nonlinear system.
0
−0.5
−1
0
5
10
0
5
10
2000
1999.5
0
5
10
1999
Figura 5.2: Static simulation. All the inputs are constant and equal to the steady
state value. Operating point 15 is considered.
2. Step in Throttle and in wastegate. First a positive step (acceleration)
is given to the throttle (at t=2.5sec) with the wastegate closed, then (at
t=8sec) a step is given to the wastegate too and it is opened. All the
other inputs are constant and equal to their steady state values (Fig.5.4).
Operating point 15 is considered. Solid line: linearized model. Dashed
line: nonlinear system. The pressure after compressor increases when
opening the throttle, i.e. accelerating, and decreases when opening the
WG. The same is valid for the temperature after compressor (Fig.5.5).
C APITOLO 5. A NALYSIS OF LINEARIZED MODEL
T before thro t
P before thro t
60
x 10
1.4
5
1.2
1
0
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
320
300
280
0
x 10
P intake M
1.5
5
1
0.5
0
250
2
200
1
Lambda
Throttle
Figura 5.3: Linearized (solid line). Non Linear model (dashed line). The initial
discrepancy is due to different initial condition for linear and nonlinear models.
After the transient the responses are very similar.
150
100
x 10
5
−1
10
5
10
0
5
10
2001
2000.5
4
2
0
0
−5
rpm
Area WG
6
0
0
2000
1999.5
0
5
10
1999
Figura 5.4: Step in Throttle and in wastegate. Inputs. At t=2.5sec a positive
step is given to the throttle, and at t=8sec the WG is opened.
5.2. VALIDATION OF THE LINEARIZED MODEL
5
320
T before compressor
x 10
1.005
1
0.995
P after compresso r
1.8
0
x 10
5
300
0
5
10
0
5
10
5
400
1.6
1.4
1.2
1
310
290
10
T after compresso r
P before compressor
1.01
61
0
5
10
380
360
340
320
300
T before thro t
P before thro t
Figura 5.5: Step in Throttle and in wastegate. States. Linearized (solid line).
Non Linear model (dashed line). The initial discrepancy is due to different
initial condition for the two models.
2
x 10
5
1.5
1
0
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
350
300
250
P intake M
2
0
x 10
5
1
0
0
Figura 5.6: Step in Throttle and in wastegate. States. Linearized (solid line).
Non Linear model (dashed line). The initial discrepancy is due to different
initial condition for the two models.
62
C APITOLO 5. A NALYSIS OF LINEARIZED MODEL
x 10
1200
1.4
1.2
1
0
5
x 10
800
600
0
5
10
0
5
10
5
1500
1.06
1.04
1.02
1
1000
400
10
T exhaustsyste m
1.08
P exhaustsyste m
5
T exhaust M
P exhaust M
1.6
0
5
1000
500
0
10
Figura 5.7: Step in Throttle and in wastegate. States. Linearized (solid line).
Non Linear model (dashed line). The initial discrepancy is due to different
initial condition for the two models.
5.3 Linear Analysis
Depending on the number of outputs chosen for the system, the size of the
transfer matrix will change consequently. Since the goal is to improve the
torque response, it is natural to pick the torque as output of the system, then
the model has four inputs and one output (MISO case) and the transfer matrix
is a one row vector with four elements:
GM ISO = [Gα GN Gwg Gλ ]
(5.8)
Since it is not possible to exceed the limits imposed by the legislation for the
gas emissions, an alternative could be considering also fuel consumption. In
this case (MIMO case) the system becomes:
GM IM O =
Gf uel
GTq
=
Gα−f uel
GN −f uel
Gwg−f uel
Gλ−f uel
Gα−Tq
GN −Tq
Gwg−Tq
Gλ−Tq
Since the system is linearized around 45 operating points, there are 45 different transfer matrix G(s), so depending on the operating point also poles
and zeros are different. Looking at the engine map in Fig.5.1 it is shown that
5.3. L INEAR A NALYSIS
63
for a constant engine speed N, the operating points vary with the only pim .
So, for each fixed value of N, the number of corresponding operating points is
reduced, as summarized in the table below:
N [rpm]
OP
1200
1÷4
1500
5 ÷ 11
2000
12 ÷ 18
2600
19 ÷ 25
3000
26 ÷ 32
3500
33 ÷ 39
4700
40 ÷ 45
To simplify the analysis only some engine speeds will be considered for the
simulations, and consequently only the corresponding operating points (op):
• N=1500 rpm : op 5 to 11
• N=2000 rpm : op 12 to 18
• N=3000 rpm : op 26 to 32
5.3.1 SISO Analysis
We focus now on the transfer function between WG and torque, Gwg . All the
transfer functions present stable poles and one zero in the right half plane. As
shown in Fig.5.8, fixing the engine speed (N=constant) and moving from one
operating point to the next increasing the pim , the non-minimum phase zero
moves to higher frequencies.
Take as example operating point 12. The transfer function from WG to
torque is given by:
Gwg =
201.0408(s + 1413)(s + 719.2)(s + 330.3)(s + 102.5)(s + 14.89)
···
(s + 1413)(s + 719.2)(s + 505.6)(s + 234)(s + 103.4)(s + 27.49)
(s + 9.213)(s − 5.541)(s + 1.454)(s + 1.042)(s + 0.8799)(s + 0.7336)
(s + 9.184)(s + 1.453)(s + 0.9235)(s + 0.8737)(s + 0.7334)(s + 0.6406)
All the poles are stables, while there is one zero in the right half plane, so the
system is non-minimum phase, as can be seen from Fig.5.9, showing the inverse response of the system when a step is applied to the wastegate. The
transfer function Gwg has 12 stable poles and one zero in the RHP. From control theory is known that a positive zero imposes limitations on the closed loop
64
C APITOLO 5. A NALYSIS OF LINEARIZED MODEL
root locus WG to Tq
0.4
0.2
13
12
14
15
16
17
18
0.1
0
−0.1
−0.2
0
5
10
15
20
Real Axis
25
30
35
Figura 5.8: Positive zeros. Operating points 12 to 18.
Step Response
0.5
G_wg_Tq_1
op12
0
−0.5
Amplitude
Imaginary Axis
0.3
−1
−1.5
−2
−2.5
−3
−1
0
1
2
3
4
Time (sec)
5
6
Figura 5.9: Inverse response.
7
8
9
5.3. L INEAR A NALYSIS
65
system bandwidth, which cannot be higher than half the frequency of the RHP
zero [6]. It is not possible to group poles and zeros based on N or pim , but varying the operating points poles and zeros move along all the real axis. Anyway
the RHP zero stays approximately in the range 5 − 30rad/s. So the closed loop
bandwidth should be lower than 2.5 − 15rad/s, i.e. approximately 0.5 − 5Hz.
A desirable bandwidth would be around 15Hz, so a solution to overcome the
RHP zero limitation is needed.
5.3.2 Controllability and observability
The concepts of controllability and observability describe how state vectors in
the state space are influenced by inputs and how they show up in the output. They are also important for the control synthesis with observe and state
feedback.
Controllability
The state x∗ is said to be controllable if there is an input that in finite time
gives the state x∗ from the initial state x(0)=0 [15]. The system is said to be
controllable if all states are controllable.
Criterion: the system 5.2 is controllable if and only if the controllability
matrix
C(A, B) = [B AB A2 B . . . An−1 B]
(5.9)
has full rank (n is the order of the system).
To check the controllability the Matlab function ctrbf has been used. The
system is controllable for all the 45 operating points.
Observability
The state x∗ = 0 is said to be unobservable if, when u(t)=0, t ≥ 0 and x(0) = x∗ ,
the output is y(t) ≡ 0, t ≥ 0 [15]. The system is said to be observable if it lacks
unobservable states.
Criterion: the system 5.2 is observable if and only if the observability matrix
O(A, C) = [B AB A2 B . . . An−1 B]T
(5.10)
has full rank.
The function obsvf was used to check the observability in Matlab: the system is observable for all the operating points.
66
C APITOLO 5. A NALYSIS OF LINEARIZED MODEL
5.4 Open Loop system behavior
The idea is to investigate the open loop system behavior, in order to understand
what we should expect from the control action.
5.4.1 Step in the WG
A positive step is given to the WG, and the torque response of the open loop
system is compared for different operating points. Fig.5.10 shows the torque
response for the linearized system, for the operating points from 26 to 30. The
same simulation is performed also on the nonlinear system (see Fig. 5.11). It
can be seen that, increasing the operating point, the effect of the WG on the
torque increases, for both linear and nonlinear system.
20
0
−20
−40
−60
Increasing
operating
point
−80
−100
−120
4
4.5
5
5.5
6
6.5
7
Figura 5.10: Positive step in WG. Linearized system. Operating point 26-30.
Increasing the operating point the WG influence on the torque increases.
5.4.2 Pulse in the WG with constant throttle
Let now investigate what happen to states and output when giving a pulse to
the WG. The experiment is carried on both the linearized and the nonlinear
system. We fully open the WG and then we close it when the benefit the we
get in the torque response is finished. Figures 5.12 - 5.13 show the comparison
between the linearized system response when the WG is fully closed (thin line)
and when a pulse is given to the WG (thick line). Operating point 13 is considered. Figures 5.14-5.15 refer to the nonlinear system. It can be seen that there
are two different dynamics for the torque response when opening and closing
the WG. When we open the WG we are in the operating point, so the derivatives of all the states are zero, while when we close the WG, we have the same
5.4. O PEN L OOP SYSTEM BEHAVIOR
67
20
0
−20
−40
Increasing
operating
point
−60
−80
−100
−120
4
4.5
5
5.5
6
6.5
7
Figura 5.11: Positive step in WG. Nonlinear system. Operating point 26-30.
Increasing the operating point the WG influence on the torque increases.
72
Torque
71
70
69
68
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
5
5.5
6
6.5
7
7.5
8
8.5
9
1
WG
0.8
0.6
0.4
0.2
0
4.5
Figura 5.12: Torque response to a pulse in the WG. Linearized model.
C APITOLO 5. A NALYSIS OF LINEARIZED MODEL
5.9
x 10
4
5.8
5.7
4.5
1.1
x 10
5
5.5
6
6.5
7
7.5
8
8.5
9
5
WG pulse
WG closed
1
0.9
4.5
pumping losse s
P exaustmanifold
P intake manifold
68
5
5.5
6
6.5
7
7.5
8
8.5
9
5
5.5
6
6.5
7
7.5
8
8.5
9
8
7
6
4.5
Figura 5.13: Pressures behavior for the linearized model. The input is a pulse
in the WG.
74
Torque
73.5
73
72.5
72
9.5
WG
2
x 10
10
10.5
11
11.5
12
12.5
13
10
10.5
11
11.5
12
12.5
13
−4
1
0
9.5
Figura 5.14: Torque response to a pulse in the WG. Nonlinear model.
6
x 10
69
4
5.95
5.9
9.5
1.08
x 10
10
10.5
11
11.5
12
12.5
13
5
WG closed
WG pulse
1.06
1.04
9.5
pumping losse s
P exaustmanifold
P intake manifold
5.4. O PEN L OOP SYSTEM BEHAVIOR
10
10.5
11
11.5
12
12.5
13
10
10.5
11
11.5
12
12.5
13
8
7.5
7
9.5
Figura 5.15: Pressures behavior for the nonlinear model. The input is a pulse
in the WG.
output condition (the torque is back to the same value then when we opened
the WG), but the state derivatives are not zero anymore. Even if we are in the
linear system, we do not get a symmetric behavior for the torque when giving
a positive or negative step to the WG. Physically what happens is that, when
we open the WG, pumping losses are reduced and then the torque increases,
but at the same time also the pim decreases a lot, and than it takes a lot of time
before reaching the same steady state level. Since the dynamics of the torque
depends on pim , this can be the explanation for the slow torque response to the
negative WG step.
5.4.3 Pulse in the WG with step in throttle
Now we want to check experimentally how is the system behavior when moving at the same time throttle and WG. A step is given to the throttle, while a
pulse is given to the WG, as in the previous scenario. The idea is that the system
should behave in different ways depending on the conditions of the engine.
Simulation on both linear and nonlinear system are performed, considering
low and high load scenarios.
70
C APITOLO 5. A NALYSIS OF LINEARIZED MODEL
Low Load
A step of 50mm2 in the throttle position is simulated, starting from a torque
of 45Nm, at 2000rpm. Results are compared to the case with the WG totally
closed. Figures 5.16 and 5.17 refer to the linear and nonlinear system response
respectively.
In both cases the behavior is quite similar: there is a small imPlant Outputs
70
65
torque
60
55
50
45
0.9
0.95
1
Time1.05
(sec)
1.1
1.15
Figura 5.16: Linearized system. Torque response. Blue line: WG totally closed.
Green line: pulse in the WG.
provement at the beginning of the transient, but as soon as the WG is closed,
the torque decreases and becomes lower than the case with WG totally closed.
High Load
The same simulation is performed now on the nonlinear system, with the same
input features, but starting from 175Nm. Fig.5.18 shows the torque response.
In this case, compared with the low load, the drop in the torque is bigger, and
so the torque takes too much time before recovering the correct value.
5.5 Conclusions
The linearized model, derived by Per Andersson [3], is presented in this Chapter. Since new features were added to the nonlinear model, a test to validate
5.5. C ONCLUSIONS
71
70
65
Torque
60
55
50
pulse WG
closed
45
9.9
10
10.1
10.2
10.3
10.4
10.5
10.6
10.7
Figura 5.17: Nonlinear system. Torque response.
220
215
210
Torque
205
200
195
WG closed
pulse WG
190
185
180
175
9.5
10
10.5
11
11.5
12
Figura 5.18: High load. Nonlinear system. Torque response.
72
C APITOLO 5. A NALYSIS OF LINEARIZED MODEL
the linearized system was performed, showing that the linearized model has a
good level of accuracy.
The analysis of the linearized system is then presented in the second part
of the Chapter, with a discussion on the open loop system characteristics.
Capitolo 6
Optimal Wastegate Control
Today, the wastegate is actuated by a pressure valve, which can be opened only
in certain pressure conditions. As explained in Chapter 3, it is now controlled
by a PI plus a feedforward action, which is activated when the right pressure
level over it is reached. In all the other cases is closed. The aim of this thesis
is to investigate if and how a more advanced control of the wastegate may
quicken the torque response. This chapter presents a solution based on LQR
design, enforced with integral action to guarantee disturbance rejection and
zero steady state error. Moreover a solution to avoid saturation is included.
6.1 Spark-ignited engines
The load in SI engines is controlled by throttling the air mass flow to the engine,
which influences the pressure in the intake manifold. This method is simple
and entails excellent dynamic behavior, but induces pumping losses. Especially for low loads the efficiency of the engine becomes very poor. At low load
the gas exchange consumes a lot of the invested energy, whereas close to full
load the pumping losses are reduced. A high potential for improving the fuel
consumption comes from either reducing part-load operation or increasing the
efficiency while operating at low load by reducing pumping losses.
The mechanical power which an engine can produce is determined by its
efficiency, the fuel burnt, and the lower heating value of the fuel.
Pe = ηe · ṁf · qHV
(6.1)
In order to use a three-way catalytic converter for the exhaust gas aftertreatment of spark-ignited (SI) engines, the fuel has to be burned at a stoichiometric
73
74
C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
air-to-fuel ratio (λ = 1). A certain power demand thus leads directly to a
corresponding demand for air mass flow.
ṁair = ṁf ·
A
·λ
F stec
(6.2)
As a first approach the air mass flow can be modeled as the product of the
volume flow, the density of the air in the intake manifold and a volumetric
efficiency.
ṁair = V̇f · pim · ηvol =
pim
ωe
·V ·
· ηvol
4π
RTim
(6.3)
Combining equations 6.1-6.3 yields:
Pe =
pim
ωe
1
·V ·
· ηvol · ηe · A
· qHV
4π
RTim
( F )stec · λ
(6.4)
This equation reveals the dominant parameters of the power production.
The rotational speed (ωe ) is limited by dynamical forces resulting from the
moving pistons and valves.
The displaced volume (Vd ) directly influences the dimensions of the engine,
which again is coupled to the friction losses.
Pressure in the intake manifold (pim ) for naturally aspirated engines, this
pressure is limited by the ambient pressure.
Equation 6.4 shows that the product of Vd , pim and ωe characterizes the
power output of the engine. All three parameters can be chosen within certain
limits. As the displaced volume directly correlates to the friction losses, it has
to be reduced for better efficiency. This demands higher pressures in the intake manifold or higher rotational speeds of the engine. As higher rotational
speeds are not accepted by the consumer, the only practicable way is to increase the pressure before the engine. This approach is known as downsizing
and supercharging (DSC).
6.2 Turbocharging
As mentioned above downsizing and supercharging can influence the partload efficiency positively. A smaller displaced volume reduces the dimensions
of the engine which results in lower friction losses. As an additional benefit,
the operating temperature is reached faster due to the smaller thermal mass of
the engine block. Turbocharging is one type of supercharging and includes a
6.3. WASTEGATE
75
turbine in the exhaust path, used to drive a radial compressor, which pumps
air into the intake manifold. Figure 6.1 shows a simple system overview.
E
C
T
Figura 6.1: Turbocharging.
The use of a turbocharger results in the following advantages and disadvantages [16]:
⊕ improved part-load efficiency, as exhaust enthalpy is used to drive the
compressor
⊕ easily applicable
⊕ easily controllable (one additional actuator, but no additional sensor is
necessary)
⊕ cost-efficient and robust production techniques available
⊕ no connection to the crankshaft, but a placing close to the exhaust openings of the engine block is advantageous
increased back pressure decreases full-load efficiency
low boost pressure (boost is quantifiable as the pressure above atmospheric pressure that the turbocharger is creating in the engine’s inlet
manifold) for low engine speeds
delayed boost pressure rise in transient operating condition
delayed light-off of the catalytic converter due to the heat extraction from
the exhaust gas through the turbine
6.3 Wastegate
As mentioned in the introduction, a turbocharger can be applied to achieve the
necessary boost pressure for a desired torque at the crank shaft. The concept
of turbocharging provides the advantage of better fuel efficiency, but entails
76
C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
load [Nm]
(a)
Original naturally
aspired engine
typical operation range
speed[rpm]
load [Nm]
(b)
“Downsized”
--> Reduced
engine displacement
speed[rpm]
load [Nm]
(c)
“Downsized and
Supercharged”
--> fuel recovery
speed[rpm]
Figura 6.2: Positive effect of Downsizing and Supercharging on load.
some new challenges as well. As the turbocharger-engine system is laid out
for medium to high power demands, the boost pressure for low air mass flows
is poor, i. e., at low engine speeds the engine only develops a low torque.
A more serious problem arises from the low-pass characteristic, which is
brought into the system by the inertia of the turbocharger. In order to build
up boost pressure, the turbocharger has to speed up after a tip-in by the driver. First the boost pressure reaches ambient pressure very fast, but then the
turbocharger has to speed up before the desired torque can be reached.
6.3.1 Transient response and turbo lag
When the driver hits the accelerator, it takes a time lag before the driver feels
the engine torque increasing: this time is called turbo lag and it is larger for
a turbo than for a naturally aspirated engine. Looking at the intake manifold
pressure, it is possible to recognize two possible dynamics (see figures 6.3 and
6.4).
In the first phase the dynamics is dominated by the throttle movements dynamics, filling of the intake manifold, and emptying of the volumes around
the intercooler. The second dynamics is dominated by the turbo charger speed
buildup, since the mass flow through the compressor depends strongly on the
turbocharger speed. Therefore the turbo lag is mainly due to the slower speed
6.3. WASTEGATE
P intake Manifold
2
x 10
77
5
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
300
net torque
200
100
0
−100
Figura 6.3: Turbo lag. Up: intake manifold pressure. Down: net torque.
P intake Manifold
2
x 10
5
1.5
1
0.5
0
4.5
5
5.5
6
6.5
7
5
5.5
6
6.5
7
300
net torque
200
100
0
−100
4.5
Figura 6.4: Turbo lag. Detail. Up: intake manifold pressure. Down: net torque.
78
C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
dynamics of the turbocharger.
The wastegate in state-of-the-art turbocharged engines is always closed in
order to achieve the highest possible pressure before the turbine and thus the
highest possible turbocharger speed. The higher the turbocharger speed, the
smaller is the turbo lag, as the torque build-up time is highly sensitive to the
turbocharger inertia. The wastegate is only opened to keep the boost pressure
below a certain limit, i. e., for boost pressure control.
One possible way to reduce time lag is is to control the turbo so that the
turbo always keeps a relatively high speed and then use the throttle to reduce
the flow into the intake manifold and the manifold pressure. In this way the
driveability is improved by reducing the turbo lag but the fuel consumption is
increased, since the intake manifold pressure is increased.
The turbo increases the intake manifold pressure and also the intake manifold temperature and with an intercooler the charger density is increased which
results in higher power output from the engine. An intercooler also gives better engine efficiency, since it can give the same mass flow through the engine
at a lower pressure which in its turn requires less turbine pressure and in this
way reduces the pumping losses.
In a spark-ignited (SI) engine an increased charge temperature results in a
higher knock tendency which can be detrimental. Therefore different measures
are taken to reduce the knock tendency, such as cooling the intake air, spark advance control and wastegate control.
The simplest control for a wastegate is to supply boost pressure directly
from the charge air side to the wastegate actuator. A small hose can connect
from the turbocharger compressor outlet to the nipple on the wastegate actuator. The wastegate will open further as the boost pressure pushes against the
force of the spring in the wastegate actuator until equilibrium is obtained. A
boost-valve takes as input the pressure before and after compressor and, based
on a PWM signal, gives the desired pressure to the WG actuator. Then the WG
actuator converts the pressure in a linear movement which actuates the WG
valve to short cut the turbine.
The purpose of this thesis is to investigate the possibility to add a more
advanced control by integrating an electronic boost controller, i.e. substitute
the pressure valve used today with an electronic valve which can be opened
and closed whenever is needed.
6.3. WASTEGATE
79
T
C
Boost
Valve
PWM
Pamb
Pboost-valve
WG Actuator
Figura 6.5: WG actuator. A boost-valve takes as input the pressure before
and after compressor and, based on a PWM signal, gives the desired pressure
to the WG actuator. Then the WG actuator converts the pressure in a linear
movement which actuates the WG valve to short cut the turbine.
6.3.2 Open Loop System Characteristics
Insight into system behavior is critical to developing a suitable controller simple enough for feasible implementation in an automotive system.
First consider the response to throttle, whose behavior is very similar to the
behavior of typical naturally aspired gasoline engines. Torque and pressures
exhibit a positive steady state gain to throttle opening. This applies to boost
pressure, pic , as well as intake pressure, pim , and exhaust pressure, pem , due to
the natural feedback present in the system via the turbocharger.
Now focus on the wastegate [4]. The DC gain from wastegate to torque is
negative and, fixing engine speed, its absolute value increases with pim (see
Fig.6.6). When considering transient behavior, the torque response to a change
in wastegate valve lift is non-minimum phase. Wastegate movement causes a
rapid change in exhaust pressure, which affects the cylinder flow via two phenomena. The first is through the formation of an adverse pressure gradient
across the engine. The second is due to the effect on pcomp via boosting and is
governed by the dynamics of the turbocharger. The two phenomena have opposing effects and since the slower phenomenon related to the turbocharger is
dominant in steady state, cylinder flow, hence torque, exhibit a non-minimum
phase behavior in response to wastegate.
The transfer function Gwg has a negative dc gain, so it means that opening
the WG (positive step) the torque decreases, while closing it (negative step)
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C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
0
−100
−200
−300
−400
−500
−600
−700
N=1200
−800
−900
0
N=1500
5
N=2000
10
15
N=2500
20
N=3000
25
N=3500
30
35
N=4700
40
45
Figura 6.6: DC gain from wastegate lift to torque, for all the 45 operating points.
For a fixed engine speed, the DC gain is bigger (in absolute value) for higher
intake manifold pressure.
the torque increases. However, the inverse response due to the positive zero
says that at the beginning of the step, when the WG is open the torque increases, and when the WG is closed the torque decreases. So the idea could be to
open the WG for a short while during the transient, to get a higher torque, and
then close it again to track torque reference value. Giving a pulse in WG, the
torque increases at the beginning as expected, but then it goes down, due to
the inverse response, reaching a lower steady state value and this is of course
undesirable.
The character of the system dynamics shows little variation with operating
condition. The magnitude of the response, however, does vary significantly
with engine speed and load.
6.3.3 Dealing with constraints
In the nonlinear model the wastegate can vary between 0 (WG closed) and 100
(WG open), while in the linearized model the wastegate position is normalized
to be between 0 and 1, where 0 corresponds to totally closed and 1 corresponds
to fully open:
umin = 0 ≤ uwg ≤ 1 = umax .
(6.5)
6.3. WASTEGATE
81
There are two ways of dealing with it:
1. Reduce the performance demands so that a linear controller never violates the limits.
2. Modify the design to account for the limits.
Option (1) implies that either the actuator was oversized in the first place or one
is unnecessarily compromising the performance. So option (2) is considered
here for the LQR design (section 6.4), modifying a given linear design adding
anti wind-up and saturation limiting circuite.
6.3.4 Control Problem formulation
The mean value model of the turbocharged system developed in Chapter 2 has
twelve dynamical states, namely air filter pressure, paf , and temperature, Taf ,
boost pressure, Pcomp , and temperature, Tcomp , intercooler pressure, pic , and
temperature, Tic , intake manifold pressure, pim , exhaust manifold pressure,
pem , and Temperature Tem , exhaust system pressure, pt , and temperature, Tt ,
and turbocharger speed, ωtc . The inputs to the system are throttle angle, α,
engine speed N, wastegate valve lift, uwg , air to fuel ratio λ, ambient pressure pamb and temperature Tamb . The primary objective of this system is to
deliver desired brake torque, Tq , determined form the driver pushing the acceleration pedal, at best fuel economy, subject to emissions constraints. Since
emissions are subject to strong legislative limitations, fuel consumption will be
considered as an additional output for the system.
The ambient conditions (Tamb , pamb ) cannot be controlled and are therefore
classified as disturbances. Although all inputs are potentially useful for increasing the agility of a vehicle, not all of them can be used. The air-to-fuel
ratio cannot be chosen arbitrarily, because a catalytic converter is used for the
exhaust gas after treatment, which requires a stoichiometric combustion, i. e.,
λ = 1 in steady state. Moreover, a constant engine speed is used, to isolate the
results from gearbox and vehicle influences. So finally the control signals are
throttle and wastegate. The system inputs and outputs are modeled as follows:
DISTURBANCES
INPUT
OUTPUT
Tamb
Non Controllable:
Tq
pamb
N
Controllable:
λ
α
uwg
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C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
Transient performance is judged with respect to the response time, measured
as the time delay from tip-in to the instant when the torque reaches a value of
90% of the maximum that is achievable at that speed.
The control problem can then be faced with different strategies:
1. Considering only the wastegate as input ⇒ SISO problem
2. Considering both the control inputs (α and uwg ) ⇒ MISO control problem
3. Considering one control input (uwg ), and two outputs (Tq and ṁf uel ) ⇒
SIMO control problem
4. Considering two control inputs (α and uwg ), and two outputs (Tq and
ṁf uel ) ⇒ MIMO control problem
e
C_wg
+
ECM
u_wg
N
Lambda
alpha
Tq
PLANT
T_amb,
p_amb
Figura 6.7: SISO control problem
The rest of this chapter will deal with the SISO formulation.
6.4 Linear Quadratic Regulation (LQR)
The question is how to open and close the wastegate in a way to maximize the
torque response during the transient; an optimal control problem has thus to
be solved.
In optimal control one attempts to find a controller that provides the best
possible performance with respect to some given measure of performance, e.g.,
the controller that uses the least amount of control-signal energy to take the
output to the reference value.
6.5. O PTIMAL S TATE F EEDBACK
83
In general, optimality with respect to some criterion is not the only desirable property for a controller. One would also like stability of the closed-loop
system, good gain and phase margins, robustness with respect to unmodeled
dynamics, etc. In the following sections we study controllers that are optimal
with respect to energy-like criteria. These are particularly interesting because
the minimization procedure automatically produces controllers that are stable
and somewhat robust.
First a full state feedback architecture based on LQR design is considered
[17]. Then, the system dynamics are augmented with integral of the error between the desired and measured torque to satisfy the requirement of steadystate tracking and finally an anti-saturation solution is adopted to prevent the
control signal from hitting the constraints.
6.5 Optimal State Feedback
For the control design a Linear Quadratic criterion as been chosen, to minimize
a trade-off between the size of the control error e = y − yd and the size of the
control input u [15].
ẋ(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
(6.6)
Let the system from control input u to controlled output y be given by (6.6)
where
• x(t) ∈ Rn is the state vector;
• u(t) ∈ Rp is the input;
• y(t) ∈ Rm is the output.
By assumption the whole state x can be measured and therefore it is available
for control. The criterion to be minimized is
2
2
T
J = y Q + uR =
y Qy + uT Ru dt
(6.7)
for a positive definite matrix R and a positive semidefinite matrix Q. The optimal regulator then can be written in terms of the simple state feedback law
u(t) = −K OP T x(t)
(6.8)
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C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
where K OP T is a m × n matrix given by
K OP T = R−1 B T S
(6.9)
and S is the unique, positive semidefinite, symmetric solution to the continuous time Algebraic Riccati Equation (ARE)
AT S + SA + M T QM − SBR−1 B T S = 0
(6.10)
6.5.1 Stability and robustness
The state-feedback control law (6.8), results in a closed-loop system of the form
ẋ = (A − BK OP T )x
(6.11)
A crucial property of LQR controller design is that this closed-loop is asymptotically stable[6], i.e. all the eigenvalues of (A − BK OP T ) have negative real
part, as long as the following two conditions hold:
1. The system 6.6 is controllable.
2. The system 6.6 is observable.
For an LQR-controlled system it is well known (Kalman, 1964; Safonov and
Athans, 1977) that, if the weight R is chosen to be diagonal, the sensitivity
function (measured at K OP T x) S = (I + K OP T (sI − A)−1 B)−1 satisfies
σ̄(S(jω)) ≤ 1, ∀ω
(6.12)
as can be seen in Fig.6.8. From this it can be shown that the system will have
a gain margin equal to infinity, and a (minimum) phase margin of 60 degrees
in each plant input control channel. This means that in the LQR-controlled
system u = −K OP T x, a complex perturbation diag {ki ejθi } can be introduced
at the plant inputs without causing instability.
Since the controlled system, from WG to torque is a single-input plant,
the above shows that the Nyquist diagram of the open-loop regulator transfer function L = K OP T ((sI − A)−1 )B will always lie outside the unit circle
with center -1 [Kalman ,1964],see Fig.6.9.
6.5. O PTIMAL S TATE F EEDBACK
85
S
0
Magnitude (dB)
−10
−20
−30
Increasing
op
−40
−50
Phase (deg)
−60
90
60
30
0
10
−2
10
−1
10
0
1
10
10
Frequency (rad/sec)
2
10
3
10
4
10
5
Figura 6.8: Sensitivity Function S = (I + K OP T (sI − A)−1 B)−1 .
Nyquist (L)
800
600
increasing
op
Imaginary Axis
400
200
0
−200
−400
−600
−800
0
100
200
300
400
Real Axis
500
600
700
800
Figura 6.9: Nyquist diagram of the loop gain L = K OP T ((sI − A)−1 )B.
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C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
6.6 LQR for non-zero set points
The LQR itself does not guarantee reference tracking so, since the aim is to follow the torque reference signal, a modification is required.The formally correct
way to treat the servo problem as an optimization problem, is to include a model for the frequency contents of the reference signal r in the state space model
[15]. This is done by describing r as the output of a linear system driven by
white noise and the state vector will have to be extended to be able to describe
also the reference signal. If the controlled output is y = C1 x and the reference
signal is r = C2 x, the quantity to penalize is e = r − y = [−C1 C2 ]x = Cx.
Since r is known, it is to be included in the measured outputs
ȳ =
y
r
This is quite a laborious process and is worthwhile only if the reference signal
has some very specific properties (like a typical resonance frequency). In the
case r is a step or a piecewise constant, then the result is much simpler, as stated
below.
Assume that ud (r) is a well-defined constant control signal that in stationarity and absence of disturbances would give y(t) ≡ r. We can define the error
variables
ũ(t) = u(t) − ud ,
which satisfy
x̃(t) = x(t) − xd ,
ỹ(t) = y(t) − r,
˙
x̃(t)
= Ax̃(t) + B ũ(t)
ỹ(t) = C x̃(t)
(6.13)
and solve the LQR problem for the error variables:
2
2
J = ẽQ + ũR =
T
ẽ Qẽ + ũT Rũ dt
(6.14)
Thus the optimal solution becomes [15]:
uOP T (t) = −K OP T x(t) + KrOP T r(t)
with
† †
KrOP T = Hcl (0)−1 = C(BK OP T − A)−1 B = C(Acl )−1 B
(6.15)
6.7. T UNING
87
Here Hcl (s) is the closed-loop transfer function matrix:
Hcl (s) = C(sI − Acl )−1 B.
and † denotes pseudoinverse. This choice of KrOP T means that the static gain
of the closed loop system is the identity matrix, so in steady state y = r.
Kr
Figura 6.10: LQ control configuration for reference tracking.
6.7 Tuning
The matrix Q weights the size of the output error. If the entries or Q are large,
then small deviation of the error from zero will cause the cost to increase significantly - thus the control will be tight (high gain). The matrix R weights the
control. If the entries of this matrix are large, then the control action will be
small - opposite to what happens with Q. If we want to increase the size of the
control action we can reduce the entries of this matrix.
The state vector includes pressures, temperatures and turbine speed, the
units used for the different components of x make the values for these variables
numerically very different from each other, so the weights Q are chosen to be
diagonal with [18]
Qii =
1
, i ∈ {1, 2, . . . , n}
maximum acceptable value of xi
(6.16)
Since the wastegate can only vary between 0 and 1, the input weigth was set
to R=1. In this way that all the variables in x and u have the same importance
when minimizing the J criterion.
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C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
6.8 Integral action in LQ control
LQ control does not guarantee integral action per sé, and therefore offsets and
constant disturbances may not be rejected in steady state. So integral action
needs to be enforced in the controller design, including integrators at the output and considering the regulator problem for the augmented system (Fig.
6.11):
q̇ = y,
x̂ =
x
(6.17)
q
The augmented system becomes:
d
x
y
C
x=Ax+Bu
-Kp
1
s
-Ki
Figura 6.11: Integral action in LQ control.
x̂˙ = Âx̂(t) + B̂ û(t)
(6.18)
y(t) = Ĉ x̂(t)
where
x̂ =
x
q,
 =
A
0
C
0
,
B̂ =
B
0
Ĉ =
,
C
0
.
Note that if the integrators are included at the control error instead of the
output, the A matrix is augmented with -C instead of C.
The controller then becomes
u(t) = −K̂
OP T
x̂(t) =
Kp
Ki
x
q
(6.19)
Once the integral action is added, non-zero set point tracking can be achieved
(Fig.6.12).
The LQ controller works on the linearized model, so when connected to
6.8. I NTEGRAL ACTION IN LQ CONTROL
89
d
x
x=Ax+Bu
C
y
-Kp
-Ki
1
s
-yd
-Kr
Figura 6.12: Tracking with integral action.
the nonlinear system, it is necessary to scale the states so that the controller
gets as input the variation of the states from the steady state value. The same
consideration holds for the reference which must be scaled with respect to its
steady state value, i.e. δTq = k3 δdx7 + k4 δx8 .
Considering that the control input has a saturation, when adding the integral term the wind-up phenomenon has to be considered. Two schemes have
been considered:
1. The difference between the control input before and after the saturation
block is multiplied with a constant Ksat and then added to the term coming form the error. In this way the signal entering the integrator is smaller
or bigger depending if the input signal saturates. Figure 6.13 shows the
Simulink implementation of the strategy.
Figura 6.13: Simulink implementation of strategy 1.
2. The signal after the saturation is sent to a comparison block, together with
the error between actual and reference torque (Fig.6.14) [19]. The two
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C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
signals are then compared with the saturation limits and the resulting
signal is used to reset the integrator thus avoiding wind up. The logic is
shown in Fig.6.15.
Figura 6.14: Simulink implementation of strategy 2.
Figura 6.15: Logic operations for the anti wind-up.
6.9 Simulations
Simulations are performed first on the linearized system and then on the nonlinear model. All the controllers are designed as described in the previous
sections, including integral action and anti wind up setting.
In the simulated scenarios only the engine is considered, so the engine block
is extracted from the bigger Simulink model, described in Chapter 3. The inputs to the engine are not coming from the ECM but need to be given as external inputs. The LQ controller provides the input uwg . The throttle is given
either as a step or as a signal shaped like it was coming from the ECM during
an acceleration phase. The other inputs are considered constant. In this way
6.9. S IMULATIONS
91
only the signals of interest in each simulation are varied, and any interference
from the ECM is avoided.
Some modifications are made in the Simulink model, to be able to set the
desired conditions for the simulations and few assumptions are formulated, as
pointed out below:
- The Simulink model is modified in the Transmission and Chassi block, to
be able to set constant engine speed.
- The block representing the WG actuator is removed since, by assumption,
the mechanical valve is substituted by an electronic valve, which can be
open and close regardless of the pressure level over it.
- The combustion is considered to occur at stoichiometric conditions, i.e.
λ = 1.
- The ambient pressure and temperature are considered constant.
- The fuel mass flow is not given as an external input to the engine, but it
is calculated through the air mass flow going out from the cylinders and
the air to fuel ratio:
ṁf uel =
ṁair
λ · λstec
6.9.1 LQ on Linearized System
Results of the simulations for the linearized system are discussed below. The
torque reference is given as a step. The throttle signal is given as a step as well,
and its amplitude is calculated so that it would give a torque higher than the
reference value. In this way it can be seen if the WG acts, both in transient and
steady state, in order to reach the correct set point.
1. Reference tracking. Two different operating points are considered: 15
(see Fig.6.16) and 12 (see Fig.6.17).
Operating point 15 represents a quite high load condition for the engine,
and the WG is kept closed during all the transient. Operating point 12,
instead, represents an engine working condition with low pressures and
low torque, and the LQ controller opens the WG for a short while during
the transient.
Many simulations have been performed on the linearized system, and
what we can conclude is that, in most of the cases, the LQ controller keeps
the WG closed during transient. Only for low load, i.e. operating points
92
C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
with very low pressures, the controller opens the WG for a short while
during transient. A possible explanation could be that, in such a case,
the pumping losses are high, so opening the WG it is possible to reduce
them significantly. In addition, since the pressures are low, when the
WG is opened they decrease not so much, and so when the WG is closed
again, they rise quite fast and so not so much torque is lost.
Torque
100
50
0
4
4.5
5
5.5
6
4
4.5
5
5.5
6
4
4.5
5
5.5
6
Throttle
150
100
50
0
WG
0.4
0.2
0
Figura 6.16: Operating point 15. Reference tracking. From the top: torque,
throttle, WG. It can be seen that the LQ controller keeps the WG closed during
transient.
2. Disturbance rejection. A scenario in which the LQ is tested for disturbance rejection is shown in Fig.6.18. To completely reject disturbances, it
was necessary to add the integral action in the LQ problem formulation.
A step is given to the throttle, keeping the torque reference constant. The
controller reacts opening the wastegate and so bringing back the torque
to the reference value.
3. Anti wind up. A simulation to check the anti wind-up configuration is
performed. From t=2sec to t=7sec we force the control input to saturate to
0. Since the torque reference is set to be higher than what the system can
actually reach, the LQ controller makes the WG to go below 0. Suddenly,
at t=7sec, the torque reference is lowered, under the actual torque signal.
So the LQ controller should immediately opens the WG to follow the
6.9. S IMULATIONS
93
Torque
150
100
50
0
4
4.5
5
5.5
6
6.5
7
7.5
8
4
4.5
5
5.5
6
6.5
7
7.5
8
4
4.5
5
5.5
6
6.5
7
7.5
8
Throttle
100
50
0
WG
1
0.5
0
Figura 6.17: Operating point 12. Reference tracking. From the top: torque,
throttle, WG. The LQ controller opens the WG for a short while during transient. A possible explanation is that for low load operating points, the pressures are very low and so when the WG is opened they decrease not so much,
and so when the WG is closed again, they rise quite fast and so not so much
torque is lost. Moreover the pumping losses are high, so opening the WG can
help to reduce them significantly.
94
C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
4
Torque
2
0
throttle disturbance
−2
1
1.5
2
2.5
3
1
1.5
2
2.5
3
1
1.5
2
t
2.5
3
6
4
2
0
Wg
1
0.5
0
Figura 6.18: Operating point 15. A throttle disturbance is added at t=2sec and
the controller correctly opens the WG to bring back the torque to the set point.
reference, since the throttle is not changed. Fig.6.19 shows that the anti
wind-up configuration works correctly: the controller opens the WG and
the torque is brought back to its reference value.
6.9.2 LQR on Nonlinear System
The controller is now connected to the nonlinear system, obtained by extracting
the engine block from the big nonlinear Simulink model. We want to check the
influence of the WG on the system response, so the other inputs to the engine
are set manually, while uwg is determined by the controller.
1. Torque reference from 150 to 200Nm. An acceleration is simulated on
the nonlinear system. The reference signal is a step from 150 to 200Nm.
Inputs given to the engine are shown in Fig.6.20. The operating point
selected for the simulation is number 15, which means 2000rpm. A disturbance in torque is added to the output at t=7sec, after the transient is
finished. Figure 6.21 shows that the reference value is reached in less than
1 second and the controller correctly opens the WG when the disturbance
comes, to bring back the torque to the set point.
6.9. S IMULATIONS
95
Torque
30
20
10
0
1
2
3
4
5
6
7
8
9
10
Throttle
20
without antiwindup
with antiwindup
10
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
WG
1
0.5
0
Figura 6.19: Anti wind-up: at t=7sec the torque reference is suddenly lowered
and the controller correctly opens the WG to follow the reference.
102
Pamb
Throttle
300
200
100
2
4
6
8
100
98
10
0
−2
2
4
6
8
10
19
2
Area WG
N
6
8
10
4
6
8
10
4
6
8
10
20
2001
2000
1999
4
21
Tamb
Lambda
2
2
2
4
6
8
10
2
x 10
−4
1
0
2
Figura 6.20: Torque reference from 150 to 200Nm. Inputs. At t=5sec a step
in throttle is given. At t=7sec a disturbance is added to the output and the
controller correctly opens the WG at that time, to bring back the torque to the
set point.
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C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
210
200
190
Torque
180
170
160
150
140
2
3
4
5
6
7
8
9
10
Figura 6.21: Torque reference from 150 to 200Nm. Torque. At t=7sec a disturbance is added to the output. The controller brings back the torque to the set
point.
2. Torque reference from 20 to 220Nm Now a different kind of simulation
is performed. A bigger step in torque is given, from 20 to 220Nm. The
big difference with the previous scenario is that now the throttle signal
given to the engine is similar to the one that the ECM would produce for
the same reference step. Still a unique LQ controller is used, designed on
the operating point 15. Fig.6.22 shows the input signals. Fig.6.23 shows
that the torque gets the set point in less than 1 second. At t=7sec the WG
is opened to compensate the torque disturbance, added to the output,
as in the previous scenario. Finally some plots of the state are shown in
Fig.6.24 - 6.25, to check that constraints on the exhaust manifold temperature and turbine speed are respected: Tem < 1300K and ωt < 18000rad/s.
3. Torque reference from 0 to 300Nm. Now a full dynamics simulation is
performed, so the engine accelerates from 0 to 300Nm, and the aim is to
check if only one LQ controller can be enough to cover the whole nonlinear dynamics. Again the throttle signal is shaped like the signal coming from the driver during an acceleration phase (Fig.6.26). The output
reaches the reference but, as expected, it has a quite bad response, with
6.9. S IMULATIONS
97
102
Pamb
Throttle
2000
1000
0
2
4
6
8
100
98
10
0
−2
2
4
6
8
19
10
2
Area WG
N
6
8
10
4
6
8
10
4
6
8
10
20
2001
2000
1999
4
21
Tamb
Lambda
2
2
2
4
6
8
x 10
−4
1
0
10
2
2
Figura 6.22: Acceleration phase. Inputs. The throttle signal is similar to the
signal that would be produced from the ECM to bring the torque from 20Nm
to 220Nm. At t=7sec a disturbance is added to the output and the controller
opens the WG.
250
200
Torque
150
100
50
0
2
3
4
5
6
7
8
9
10
Figura 6.23: Acceleration phase. Torque. Reference step from 20Nm to 220Nm.
The torque gets the set point in less than 1 second. At t=7sec a disturbance is
added to the output.
98
C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
5
T before thro t
320
1.2
1
0.8
15
P intake M
x 10
2
4
x 10
6
8
290
2
4
6
8
10
2
4
6
8
10
15000
5
2
300
4
10
0
310
280
10
turbine speed
P before thro t
1.4
4
6
8
10000
5000
0
10
Figura 6.24: Acceleration phase. States. Turbine speed is constrained. It must
be ωt < 18000rad/s.
5
T exhaust M
1100
1.4
1.2
1
1.08
P exhaustsyste m
x 10
2
x 10
4
6
8
2
4
6
8
10
2
4
6
8
10
1100
1.04
1.02
2
900
5
1.06
1
1000
800
10
T exhaustsyste m
P exhaust M
1.6
4
6
8
10
1000
900
800
700
Figura 6.25: Acceleration phase. States. Tem is constrained. It must be
Tem < 1300K
6.10. G AIN S CHEDULING S TRATEGY FOR LQ C ONTROL
102
Pamb
Throttle
2000
1000
0
2
4
6
8
100
98
10
0
2
4
6
8
10
19
2
Area WG
N
4
6
8
10
4
6
8
10
4
6
8
10
20
3001
3000
2999
2
21
Tamb
Lambda
2
−2
99
2
4
6
8
10
2
x 10
−4
1
0
2
Figura 6.26: Torque reference from 0 to 300Nm. One LQ. All inputs are constant
and the throttle signal is shaped like the signal coming from the driver during
an acceleration phase.
an overshoot of 20 Nm with respect to the set point (Fig.6.27).
4. Disturbance rejection. So far ambient pressure and temperature have
been considered constant, but in normal driving situations the environmental conditions can vary. It is reasonable to set the ambient pressure
as constant, but it is realistic to simulate a change in the ambient temperature and check how the controller reacts to constant disturbances. So a
step is given to Tamb , while all the other inputs are not varied (Fig. 6.28).
The controller correctly opens the wastegate to reject the disturbance, and
so the torque is brought back to the steady state value (Fig.6.29).
6.10 Gain Scheduling Strategy for LQ Control
Since the system is strongly nonlinear, is natural to think about several controllers to cover the whole dynamics of the system. The idea, depicted in
Fig.6.30, is then to design a linear quadratic controller for each operating point,
and then implement a kind of scheduler that, depending on the engine working condition, decides every time which controller to use. In this way the selected LQ controller will always work not so far from the operating point it has
100
C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
320
310
Torque
300
290
280
270
260
4.5
5
5.5
6
6.5
7
Figura 6.27: Torque reference from 0 to 300Nm. One LQ. Torque. The output
reaches the reference but it shows an overshoot of 20 Nm.
102
Pamb
Throttle
91.7805
91.7805
2
4
6
8
100
98
10
0
−2
2
4
6
8
10
20
4
Area WG
N
6
8
10
4
6
8
10
4
6
8
10
30
2001
2000
1999
4
40
Tamb
Lambda
2
2
2
4
6
8
10
2
x 10
−5
2
0
2
Figura 6.28: Disturbance rejection. Inputs. A step is given to Tamb at
t=5sec while the other inputs are constant. The controller correctly opens the
wastegate to reject the disturbance.
6.10. G AIN S CHEDULING S TRATEGY FOR LQ C ONTROL
101
93
92.8
92.6
Torque
92.4
92.2
92
91.8
91.6
91.4
2
3
4
5
6
7
8
9
10
Figura 6.29: Disturbance rejection. Torque. At t=5sec a disturbance in Tamb is
added. The controller brings back the torque to the steady state value.
been designed for. For example, assuming to have N=2000rmp, there are 7 operating points (12-18) and this mean that 7 LQ controllers will be designed. The
scheduler is implemented as a look up table which selects the current operating point based on pim . All the states and references are scaled in a different
way for each controller, so that each LQ receives the correct states and inputs
variation in relation with its operating point.
An example of Simulink implementation is shown in Fig.6.31. In the picture 7 blocks, corresponding to the 7 LQ controllers, can be seen. The three
input signals passed to the system (states, references and non manipulated inputs) are scaled as explained previously. The switch is controlled by the signal
coming from the scheduler.
A first set of simulations is performed with constant engine speed, so the
current operating point is selected based on the only pim , as can be seen from
Fig.6.32. Giving the same inputs than the case with only one controller (Fig.6.33),
the oscillation in the torque is now reduced, as can be seen from Fig.6.34.
6.10.1 Variable engine speed
So far a fixed engine speed has been used, to simplify the analysis. Now a more
realistic simulation is performed, varying N from 2000 rpm to 3000 rmp. The
102
C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
States
P intake manifold
Engine speed
Closest
Operating
point
Control action
Figura 6.30: The scheduler, depending on the engine working condition (basically pim and N, decides every time which controller to use. In this way the
selected LQ controller works always not so far from the operating point is has
been designed for.
Figura 6.31: Simulink: implementation of the switching strategy. Only one LQ
controller at a time is activated, corresponding to the selected operating point.
The 7 blocks to the left, correspond to the 7 LQ controllers. The three input
signals passed to the system (states, references and non manipulated inputs)
are scaled in a different way for each controller, so that each LQ receives the
correct states and inputs variation in relation with its operating point.
6.10. G AIN S CHEDULING S TRATEGY FOR LQ C ONTROL
103
N
3001
3000
P intake Manifold
2999
2
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
x 10
5
1
0
op
35
30
25
Figura 6.32: Switching Strategy. A look up table takes as input the intake
manifold pressure and selects the corresponding operating point. Different
operating points are selected, and consequently different LQ controllers are
activated.
102
Pamb
Throttle
2000
1000
0
2
4
6
8
100
98
10
0
−2
2
4
6
8
10
19
2
Area WG
N
6
8
10
4
6
8
10
4
6
8
10
20
3001
3000
2999
4
21
Tamb
Lambda
2
2
2
4
6
8
10
2
x 10
−4
1
0
2
Figura 6.33: Switching Strategy. Inputs are the same than the case with only
one LQ.
104
C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
350
300
250
Torque
200
150
100
50
0
−50
2
3
4
5
6
7
8
9
10
Figura 6.34: Switching Strategy. The torque follows the reference and the
overshoot is removed.
switching strategy is then extended with another lookup table which considers
also the engine speed, together with the intake manifold pressure.
Two possibilities of implementing a control architecture with so many controllers are investigated:
- all the controllers are enabled during all the simulation and the scheduler
has to select which wastegate signal to activate time by time (Fig.6.35).
- through the use of Simulink ’case’ blocks only one controller is enabled at
a time, while the other are ”frozen”. Then the scheduler has to activate
both the wastegate signal and the LQ controller.
The simulations performed with the two different strategies are not so different. Of course it is computationally lighter to use the ’case’ blocks, but since the
LQ calculation are not so demanding, in our case it does not make difference.
If the LQ controllers would be changed with heavier control algorithms, then
it might be worth to consider only the switching strategy with ”case” blocks.
When changing the engine speed from 2000 rpm to 3000 rpm, the system
goes through many operating points. As explained in Chapter 5, for each engine speed there is a bunch of operating points, depending on different pim . So
at N=2000 rpm, 7 LQ controllers are calculated, from operating point 12 to 18,
and at N=3000 rpm other 7 LQ controllers are calculated, from operating point
6.11. C ONCLUSIONS
105
Figura 6.35: Switching strategy for the LQ controller. All the LQ controllers receive the correct states variations, obtained subtracting the steady state value
to the states coming from the nonlinear system. All the controllers are calculated, but only one is activated at a time, depending on the selected operating
point.
26 to 32. Figures 6.36-6.38 point out the operating point switching strategy, and
show the inputs and the torque response.
6.10.2 Variable engine speed with 3 controllers
To simplify the control architecture, the same simulation is now performed
with only 3 LQ controllers instead of 7. The torque response is shown in Fig.
6.39. It is interesting to note, from the detail in Fig.6.40, that the torque responses are quite similar when using 7 or 3 LQ controllers, so it is not necessary to
use one LQ controller for each operating point, but the control structure can be
simplified.
6.11 Conclusions
In this chapter a linear quadratic (LQ) controller has been designed for the
WG valve, to investigate if a more advanced control can improve the transient
torque response.
The LQ controller has been designed adding an integral action for reference
tracking and an anti wind-up to avoid overflow problems due to input hard
constraints. First, a single LQ controller has been designed, and it has been
tested both on the linear and nonlinear system. The simulations showed that,
on the linear system, in the 90% of the operating points the controller keeps the
WG closed during the transient. With the nonlinear system, one controller is
not enough to cover all the dynamics, so a gain scheduling strategy has been
106
C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
102
Pamb
Throttle
2000
1000
0
2
4
6
8
100
98
10
0
−2
2
4
6
8
19
10
2
Area WG
N
6
8
10
4
6
8
10
4
6
8
10
20
3000
2500
2000
4
21
Tamb
Lambda
2
2
2
4
6
8
x 10
−4
1
0
10
2
2
Figura 6.36: Variable engine speed. Inputs. A ramp is given to the engine speed
(N), from 2000 to 3000 rpm.
N
3000
2500
P intake Manifold
2000
2
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
x 10
5
1
0
op
40
20
0
Figura 6.37: Variable engine speed. For each engine speed there is a bunch of
operating points, depending on different pim .
6.11. C ONCLUSIONS
107
350
300
250
Torque
200
150
100
50
0
2
3
4
5
6
7
8
9
10
Figura 6.38: Variable engine speed. Torque Response. 7 LQ controllers are
calculated for the operating points corresponding to N=2000rpm (op 12 to 18),
and other 7 LQ controllers are calculated for N=3000 rpm (op 26 to 32).
350
300
250
Torque
200
150
100
50
0
4.5
5
5.5
6
6.5
7
Figura 6.39: Variable engine speed with 3 controllers. Torque Response.
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C APITOLO 6. O PTIMAL WASTEGATE C ONTROL
308.5
308
307.5
Torque
307
306.5
306
305.5
305
304.5
304
5.92
5.93
5.94
5.95
5.96
5.97
5.98
5.99
6
Figura 6.40: Comparison: torque with 7 and 3 LQ controllers. Torque responses
are quite similar when using 7 or 3 LQ controllers, so the control structure can
be simplified using only 3 LQ controllers.
implemented, with different LQ controllers designed for different operating
points.
In terms of torque response there are not big advantages using a more complex controller, but there are other positive aspects. It avoids the need for a
feedforward action in the WG controller, which would have to be tuned on the
specific engine. Moreover the WG valve, now independent from pressuresactivation constraints, can be used for disturbance rejection and help the throttle controller to follow the desired torque level.
Capitolo 7
Lambda and SA Control
In Chapter 3 some modification to the original model were described, in particular was mentioned that efficiencies for lambda and SA have been added,
both in the engine and in the ECM. As explained there, these two parameters
influence the torque response, so here we want to investigate if and how we
can improve the torque transient during the acceleration, controlling lambda
and SA so that the efficiency is maximized.
7.1 Lambda
Lambda is the air/fuel equivalence ratio, and describes the relation between
the mass of air and the mass of fuel injected in the engine for the combustion,
given normalized with the stoichiometric mixture:
λ=
(A/F )
(A/F )s
(7.1)
The stoichiometric ratio varies between 13.27 and 17.23; here λs = 14.64 is
used. Due to limits imposed by legislators on the levels of pollutant emissions
from the engine, in steady state should be λ = 1, which means that we have
the same amount of fuel and air compared with the stoichiometric reaction.
Looking at the lambda efficiency map (Fig.7.1) can be seen that the maximum
is achieved when the mixture is slightly rich λ ≈ 0.9. So this effect can be
utilized during acceleration and the torque can be increased by adding more
fuel.
In the ECM there is a controller that is supposed to regulate lambda, so
in the next sections a new control strategy will be discussed. As described in
109
110
C APITOLO 7. L AMBDA AND SA C ONTROL
1.4
1.2
Efficiency
1
0.8
0.6
0.4
0.2
0
0.8
1
1.2
Lambda
1.4
1.6
1.8
2
Figura 7.1: Lambda efficiency map: the maximum is achieved when the
mixture is slightly rich λ ≈ 0.9.
Chapter 3, lambda has influence on both fuel consumption and torque efficiency. In fact the total torque of the engine can be calculated as
BM EP = IM EP · ηSA · ηλ − P M EP − F M EP
(7.2)
So it is clear that the closer lambda efficiency is to the maximum of the curve,
the higher torque we can get, keeping all the other inputs at the same value. For
example, if we request BrakeTorque=100Nm, which corresponds to air mass
of 400mg/comb, the throttle control should ask for 400mg/comb. If we then
increase the efficiency by 1.05 with fuel enrichment, the driver still requests
100Nm, but now the throttle should provide only 400/1.05 = 380 mg/comb.
So it will take less time to reach the torque reference, since the air would reach
380mg/comb faster than 400mg/comb, specially in boosted region.
On the other hand, keeping the lambda efficiency at its maximum means to
have λ ≈ 0.9, i.e. a rich mixture and consequently pollutant emissions. In fact
under rich conditions the amount of air is insufficient for complete combustion
of the fuel, and gives rise to unburned hydrocarbons HC and CO [3]. What we
want to do is then to maximize the lambda efficiency only during the transient
and then try to keep it as close as possible to 1.
7.2. C ONTROL D ESIGN
111
7.2 Control Design
7.2.1 Static Lambda Control
The lambda controller implemented now in the ECM is basically a static map,
which receives as inputs the engine speed and the measured air mass flow, and
gives as output the maximum lambda λmax that can be used for each operating
point. This represents a physical limit that can not be ignored. The lower
lambda constraint is fixed on λmin = 0.6, but of course it corresponds to a very
bad working condition, where the efficiency is really low, as we can be seen
from Fig.7.1.
First λmax is compared to 1; then the minimum of these two values is compared with λmin , to be sure that either λmax and λmin are respected. Using this
static strategy to control lambda, a transient like the one shown in Fig.7.2 is
obtained.
Lambda Reques t
1
0.98
0.96
0.94
0.92
0.9
transient
0.88
0
1
2
3
4
5
6
7
8
9
Figura 7.2: Normal behavior for lambda Request when using a static control
strategy. Considering the lambda efficiency, this is not the best, since there is
a region during the transient where λ = 1 and so we are not using the best
efficiency.
From efficiency point of view this is not the best, since we have a region
during the transient where we are not actually using the best efficiency. Since
the transient is a very short time, 1-2 second maximum, we can enrich the fuel
112
C APITOLO 7. L AMBDA AND SA C ONTROL
Figura 7.3: Static control strategy. A static map receives as inputs N and the
measured air mass flow, and gives as output λmax . λmax is then compared to 1,
and the minimum of these two values is compared with λmin = 0.6, to be sure
that either λmax and λmin are respected.
for a while, because this will result in an increase of the torque and the relative
increase of the fuel will be very small and for few seconds.
The idea is then to use the best lambda efficiency during the transient which
corresponds, according with the map, at λ = 0.9 or, more exactly, to the minimum between 0.9 and λmax , that anyway has to respected (Fig.7.4).
/DPEGD
G\QDPLF
'\QDPLFFRQWUROOHU
PLQ
PLQ
6WDWLFK FRQWUROOHU
6WDWLF
Figura 7.4: Lambda control schematics. During transient, λ is equal to the
minimum between the value going out from the dynamic controller and the
value coming from the static map, due to engine’s cooling constraints.
7.2.2 Dynamic Lambda Control
It is a sort of Logical control based on the lambda efficiency map that has to
work only during the transient. The only problem in doing this is how to detect
that we are in a transient and not in a steady state. The big problem is to find
two signals, one of these as reference and the other one as measurement of the
real engine condition, to detect the transient. Since this model is quite realistic,
it is difficult to find two signals in ECM that at steady state have the same
value.
Let’s check the possibilities:
• ṁair : this is the real feedback signal that the throttle control uses. In
the ECM there is also mairpredicted that together with ṁair establishes the
7.3. S IMULATIONS
113
error that the Throttle control has to deal with. These approach has two
problems:
- the error even in steady state is not zero
- mairpredicted is calculated with λrequest that is actually what we have
to set.
• pim : this could be a good signal because we have both pim from the engine and pimpredicted . The problem is that pimpredicted is calculated with
mairpredicted and then back to λrequest .
• Tq : is a measurement of the brake torque in the engine, and in ECM we
find Tqpredicted . The problem is that in steady state the error is not equal
to 0, so this mean that the controller will try to keep lambda equal to 0.9
even when we are in steady state. From simulations was seen that loosing
a little of effectiveness it is possible to bypass this problem considering
to use the controller until the error is bigger than 5 for example. This of
course is an empiric approach to solve the problem but seems the only
solution since at steady state will never be Tqpredicted = Tq .
Another possible solution is to try to be independent from the reference
Tqpredicted and try to detect the transient based only on Tq , that is the signal
coming from the engine. To do this we use a discrete operator z −n and we
create two signals: the actual Tq and the Tq n samples before, where n is a
control parameter. In this way, when in n samples Tq changes of a certain
amount of units, for example 5, means that there is a transient and then lambda
is set to 0.9. Once steady state is reached, then Tq will be not so different from
the Tq of n samples before and than we will switch off the controller. The
strategy is depicted in Fig.7.5
7.3 Simulations
An acceleration phase is simulated, using a throttle signal similar to the one
that would be produced by the ECM during acceleration. The behavior of
static and dynamic controller is compared, in terms of requested lambda and
torque response. Fig.7.6 shows that, as soon as the transient starts, the dynamic controller (dashed line) brings lambda to 0.9, while it would be higher for
the static version (full line). In this way the efficiency is maximized and consequently the torque response during the transient is slightly higher, as can be
seen from Fig.7.7 and 7.8.
114
C APITOLO 7. L AMBDA AND SA C ONTROL
Figura 7.5: Simulink implementation of the Dynamic control strategy. The
torque signal is used to detect the transient and so to activate the controller.
Throttle
1500
Static controlle r
Static+dynamic controlle r
1000
500
0
4.5
5
5.5
6
6.5
7
5
5.5
6
6.5
7
Lambda
1
0.95
0.9
0.85
4.5
Figura 7.6: Static (full line) versus Dynamic (dashed line) controller: Lambda
request. During transient the Dynamic controller requires λ = 0.9.
7.3. S IMULATIONS
115
350
300
250
Torque
200
150
100
static controlle r
static+dynamic controlle r
50
0
−50
4.5
5
5.5
6
6.5
7
Figura 7.7: Static versus Dynamic controller: Torque.
200
static controlle r
static+dynamic controlle r
190
180
Torque
170
160
150
140
130
120
5
5.05
5.1
5.15
5.2
5.25
Figura 7.8: Detail. Static versus Dynamic controller: torque. Using the
Dynamic controller, it is possible to gain 1-3% in the torque response.
116
C APITOLO 7. L AMBDA AND SA C ONTROL
7.4 Spark Advance
The influence of SA on engine performances was presented in Chapter 3.5.5,
where the importance of the MBT was discussed. In this section a further investigation will be carried out, to find a possible control strategy for SA in order
to improve the torque during the transient.
MBT is the point where the optimum spark timing occurs. It gives maximum brake power and minimum brake specific fuel consumption. Usually
SA = 15 − 30BT DC, but the optimum timing depends on:
- Load: if load and pim decrease, SA timing must be further advanced
- N: if N increases, SA increases as well because there is less time for the
mixture to complete combustion at proper time, since the piston travels
faster.
The problem is that if SA increases, then the end gas temperature increases as
well, thus resulting in a greater knock tendency.
If SA is too large, during high engine load conditions, increases the likelihood of engine detonation and increases combustion temperature and pressure. The result is higher HC and N Ox output and worse performance.
If SA is too small, only partial combustion of air/fuel charge is handled,
resulting in very poor performance and fuel economy, and in higher CO levels.
It is then clear that the optimum SA timing is a trade off and a SA controller
is responsible to regulate it in the ECM. This controller is optimized for steady
state conditions, so the idea is now to check if something better can be achieved
during the transient.
It is not possible to further anticipate the SA, in order to prevent knock, but
it can only be retarded, until -10 degrees, as depicted in Fig.7.9. The idea is to
retard SA to get lower exhaust temperature and so increase the turbine speed
and get higher torque.
Simulations were performed to verify that idea, but the results showed that
the increase in the turbine speed is practically null and is not possible to gain
anything in terms of torque.
7.5 Conclusions
In this Chapter the influence of air to fuel ratio (λ) and spark advance (SA) on
the transient torque response is analyzed. Investigations showed that is not
possible to gain anything in terms of torque acting on the SA, while a better
7.5. C ONCLUSIONS
117
SA
40
30
Knock!
20
10
OK
0
−10
0
1
2
3
4
5
6
7
8
9
Figura 7.9: SA signal. ¡due to knock limitations, it is not possible to go lift the
signal during transient, but it can only be lowered until -10°.
control strategy for λ can give some advantage. The idea is to add a dynamic controller acting only during the transient, and try to maximize the lambda efficiency to get a faster torque response. Simulations showed a possible
improvement of 1-3% in the torque response during transient.
Capitolo 8
MPC
The control strategy presented in this chapter is based on Model Predictive
Control (MPC), and shows the results of simulations performed on both linear
and nonlinear system.
MPC uses a model of the system to predict its future behavior, and then
optimizes a quadratic performance index based on the prediction [9][20]. At
each sampling time, starting at the current state, an open-loop optimal control
problem is solved over a finite time horizon. The first element of the calculated
control signal is applied to the system, which operates with this input until the
next sampling instant. At this time step, new measurements are used to update the problem, and the computation is repeated starting from the new state
and over a shifted horizon, leading to a moving horizon policy. This behavior
leads to a closed-loop approach. By operating this way, MPC plans ahead to
avoid future constraints violations, and compensates for model errors and unknown disturbances at each sampling instant, by revising the plan made at the
previous time step after receiving new information. The idea is visualized in
Fig.8.1.
8.1 MPC Overview
Model Predictive Control (MPC) is widely adopted in industry as an effective means to deal with large multivariable constrained control problems. The
main idea of MPC is to choose the control action by repeatedly solving on line
an optimal control problem. This aims at minimizing a performance criterion
over a future horizon, possibly subject to constraints on the manipulated inputs
and outputs, where the future behavior is computed according to a model of
119
120
C APITOLO 8. MPC
desired
trajectory
y
actual
output
k
current
step
time
N
Prediction
Horizon
max
u
min
past
control
moves
M
Control
Horizon
Figura 8.1: Controller State at the kth sampling instant. At each sampling time,
the first element of the calculated control signal is applied to the system, which
operates with this input until the next sampling instant. At this time step,
new measurements are used to update the problem, and the computation is
repeated starting from the new state and over a shifted horizon, leading to a
moving horizon policy.
8.1. MPC O VERVIEW
121
the plant. Issues arise for guaranteeing closed-loop stability, to handle model
uncertainty, and to reduce on-line computations.
Consider the discrete-time linear time invariant system
x(t + 1) = Ax(t) + Bu(t)
y(t) = Cx(t)
(8.1)
where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control input vector, and
y(t) ∈ Rp is the output vector. A ∈ Rn×n , B ∈ Rn×m and the pair (A,B) is
stabilizable.
The problem of controlling the states to the origin while fulfilling the constraints
ymin ≤ y(t) ≤ ymax ,
umin ≤ u(t) ≤ umax
(8.2)
at all time instants t ≥ 0 can be solved with MPC.
Model predictive control (MPC) solves at each time t the optimization problem[9]
min xt+N |t +
u
N
−1
xt+k|t Qxt+k|t + ut+k Rut+k
(8.3)
k=0
subject to
ymin ≤ yt+k|t ≤ ymax ,
k = 1, . . . , N
umin ≤ ut+k ≤ umax ,
k = 0, . . . , M − 1
ut+k = Kxt+k|t ,
M ≤k ≤N −1
xt|t = x(t)
xt+k+1|t = Axt+k|t + But+k ,
yt+k+1|t = Cxt+k+1|t ,
(8.4)
k≥0
k≥0
assuming that a full measurement of the state x(t) is available at the current
time t. Here xt+k+1|t denotes the predicted state vector at time t+k, obtained
by applying the optimization vector
U = [ut , . . . , ut+M −1 ] ∈ Rs
(8.5)
with s = mM . Here N is the prediction horizon and M is the control horizon
(see Fig.8.1). P 0, R = R and Q = Q 0 are optimization weights and K is
a feedback gain.
Thus the optimizer minimizes the cost function (8.3) over N time steps. For
M steps the control signal is calculated by the optimizer limited by the constraints in (8.4) and for the rest N − M steps the control signal is fixed and
depends on the choice of the feedback gain. Rewriting the optimization prob-
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C APITOLO 8. MPC
lem (8.3)- (8.4), it can be expressed as quadratic programming (QP) problem
[9]
minU
1 2 U HU
+ x (t)F U
subject to GU ≤ W + Ex(t)
(8.6)
where U is the optimization vector defined by (8.5). The matrices H = H 0,
F , G, W and E are obtained from Q, R, P and the model matrices A, B and C.
A simple on-line algorithm for MPC is
1. Measure the current state of the system x(t).
2. Calculate the control signal sequence U by solving (8.6).
3. Apply the first element of the control sequence u(t) to the system (8.1).
4. Time update: t = t + 1.
5. Go to step one and repeat.
8.1.1 Parameters Tuning
Feedback Gains and Terminal Weights
The matrices K and P can be chosen in different ways. One possibility that
ensures stability of the closed loop system is to use P and K that satisfy the
Riccati equation [21]
K = −(R + B P B)−1 B P A,
P = (A + BK) P (A + BK) + K RK + Q.
(8.7)
In this way, after M time steps, the control is switched to an unconstrained LQR
with weights Q and R [9]. So now the parameters Q, R and N need to be tuned.
Horizons
As defined in the previous section, N is the number of future time steps over
which the system behavior is predicted, while M is the number of future moves
in the control signal considered by the optimizer [22]. The number of control
signals determines the dimensions of the optimization problem, so the choice
of the horizons is very important.
8.2. I MPLEMENTATION IN S IMULINK USING M PC T OOLBOX
123
The control horizon is always equal to, or smaller than the prediction horizon. The smaller is N , the more aggressive is the controller. If a large M
is chosen, the controller is closer to the optimal (infinite-horizon) but the increased number of free variables makes the optimization problem computationally more demanding. A smaller M reduces the problem dimensions and
so makes it easier to be solved. Unfortunately there are no rule for choosing
the prediction horizons, but they have to be tuned by trial and error..
Optimization Weights
As well as for the horizons tuning, also to chose the optimization weights there
are no rules, but they are determined by trial and error. The choice depends on
the states of the system and on the whole desired behavior.
8.2 Implementation in Simulink using Mpc Toolbox
Figura 8.2: The MPC has been implemented in Simulink, using the Model
Predictive Control Toolbox in Matlab.
The MPC has been implemented in Simulink, using the Model Predictive
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C APITOLO 8. MPC
Control Toolbox in Matlab [23]. Fig.8.2 shows the graphical user interface
(GUI) of the toolbox. One advantage of an MPC Toolbox design (relative to
classical multi-loop control) is that it generalizes directly to plants having multiple inputs and outputs. Moreover, the plant can be non-square, i.e. having an
unequal number of actuators and outputs. The MPC-toolbox library contains
a block representing the controller, as shown in Fig.8.3.
Input signals to the controller are measured outputs (MO), reference signals
and measured disturbances (MD). Measured disturbances are independent inputs whose values are measured and used for feedforward compensation. Output signals from the controller are the optimal manipulated variables (MV). So
in our case, since the controlled variable is the wastegate, it becomes:
MO = Tq
M D = [α, N, λ, pamb , Tamb ]
M V = uwg
Note that Model Predictive Control Toolbox requires the model to be linear
and time invariant (LTI), whereas the controller gets its inputs from a nonlinear
model.
Figura 8.3: MPC block used in Simulink. The MPC Controller Block receives
the current measured output, reference signal, and measured disturbance signal, and outputs the optimal manipulated variables by solving a quadratic
program.
8.2.1 Building the MPC object
The controller block is based on an MPC object, which stores all the parameters
defining the MPC control law (prediction horizon, weights, constraints, etc.).
An MPC object is built in two steps. The first step happens at construction
of the object when the object constructor ’mpc’ is invoked, or properties are
changed by a set command. At this first stage, only basic consistency checks
are performed, such as dimensions of signals, weights, constraints, etc. The
second step happens at initialization of the object, namely when the object is
8.2. I MPLEMENTATION IN S IMULINK USING M PC T OOLBOX
125
used for the first time by simulating the system. At this second stage, further
checks are performed, such as a test of observability of the overall extended
model.
The construction of the MPC object requires some properties to be set, as explained in the following lines. Given the states space realization of the system
(model), inputs and outputs can be set with the following code:
model = setmpcsignals(model, M D , [1 2 4 5 6 ], M V , [3], OV [1 2])
which creates three structures, whose fields contain the constraints for the
corresponding variables and their rates:
M V (1) = struct( M in , 0, M ax , 1, RateM in , −1, RateM ax , 1)
OV (2) = struct( M in , 0, M ax , 350)
The controller is then built through the command:
controller = mpc(model, T s, N p, N u)
which discretizes the state space continuous model with sampling time Ts . The
prediction horizon Np expresses the number of sampling steps of prediction
and the control horizon Nu is the number of free control moves. Using the
command ’setname’ is possible to set signal names:
setname(controller, input , 1, alpha )
setname(controller, input , 2, N )
setname(controller, input , 3, W G )
setname(controller, input , 4, lambda )
setname(controller, input , 5, pamb )
setname(controller, input , 6, T amb )
setname(controller, output , 1, mf uel )
setname(controller, output , 2, torque )
Weights are stored in a structure and can be defined through the following
instruction:
W eights = struct( Input , 0, M anipulatedV ariablesRate , 1, Output , [0 100])
Note that all weights must be greater than or equal to zero. If all weights on
manipulated variable increments are strictly positive, the resulting QP problem
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C APITOLO 8. MPC
is always strictly convex. If some of those weights are zero, the Hessian matrix
of the QP problem may become only positive semidefinite.
8.2.2 Prediction Model
MV (manipulated variables) u(k)
MD (measured disturbances) v(k)
white
noise
Unmeasured
disturbance
model
UO (unmeasured outputs) yu(k)
Plant
Model
d(k)
UMD
(unmeasured
disturbances)
d(k)
MO (measured outputs) ym(k)
CV
(controlled
variables)
x(k)
Figura 8.4: Model Used for Optimization in the MPC Toolbox.
The linear model used in the MPC Toolbox for prediction and optimization
is depicted in Fig.8.4 The model consists of:
1. A model of the plant to be controlled, whose inputs are the manipulated
variables, the measured disturbances, and the unmeasured disturbances
2. A model generating the unmeasured disturbances
The model of the plant is a linear time-invariant (LTI) system described by the
equations
x(t + 1) = Ax(t) + Bu u(t) + Bv v(t) + Bd d(t)
ym (t) = Cm x(t) + Dvm v(t) + Ddm d(t)
(8.8)
yu (t) = Cu x(t) + Dvu v(t) + Ddu d(t) + Duu u(t)
where x(t) is the state vector of the plant, u(t) is the vector of manipulated
variables (MV), i.e. the command inputs, v(t) is the vector of unmeasured
disturbances (UD) entering the plant, ym (t) is the vector of measured outputs
(MO) and yu (t) is the vector of unmeasured outputs (UO). The overall output
vector y(t) collects ym (t) and yu (t). In the above equation d(t) collects both
state disturbances and output disturbances.
8.2. I MPLEMENTATION IN S IMULINK USING M PC T OOLBOX
127
8.2.3 Offsets
Since the matrices A,B,C,D of the model are obtained by linearizing the non
linear system
ẋ = f (x, u, v, d)
y = h(x, u, v, d)
at some nominal value x = x0 , u = u0 , v = v0 , d = d0 . The linearized model
is affected by the constant terms F = f (x0 , u0 , v0 , d0 ) and H = h(x0 , u0 , v0 , d0 ).
For this reason the MPC algorithm internally adds a measured disturbance
v = 1, so that F and H can be embedded into Bv and Dv , respectively, as additional columns. Note Nonzero offset values d0 for unmeasured disturbances,
while relevant for obtaining the linearized model matrices, are not relevant for
the MPC problem setup. In fact, only d − d0 can be estimated from output
measurements
8.2.4 State Estimation
As the states x(k), xd (k) are not directly measurable, predictions are obtained
from a state estimator. In order to provide more flexibility, the estimator is
based on the model depicted in Fig.8.5. A linear state observer, based on the
UO
(unmeasured outputs)
MV (manipulated variables) u(k)
MD (measured disturbances) v(k)
white
noise
Unmeasured
disturbance
model
xd1(k)
Plant
Model
d(k)
CV
(controlled
variables)
d(k)
UMD
(unmeasured
disturbances)
+
yu(k)
+
y(k)
x(k)
white
noise
Output
disturbance
model
xd2(k)
+
+
Measurement
noise
UMD
white
model
(unmeasured
noise
disturbances)
xd1(k)
UO
(measured outputs)
ym(k)
Figura 8.5: Model Used for State Estimation in the MPC Toolbox.
measured output, then provides estimates of the state of the plant model, the
overall state of the input and output disturbance model and the state of the
measurement noise model.
After the combination of plant and disturbance models has been converted
to discrete-time, delay-free and state-state form the optimization problem can
be solved.
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C APITOLO 8. MPC
8.2.5 MPC Computation
If the problem is unconstrained, then is possible to compute the solution analytically, while if the problem is constrained the solution is computed by solving a quadratic program, using the QP solver coded in the qpsolver.dll function
(see [23] for details)
8.2.6 Setting MPC parameters
All the parameters defining the MPC control law (prediction horizon, weights,
constraints, etc.) are stored in the MPC object, whose properties are listed
below.
Model and Horizons
Control interval - Elapsed time between successive controller moves (in time
units). Often called the sampling period or sampling interval.
Prediction Horizon - Range of each controller prediction, expressed as an integer number of future control intervals.
Control horizon - Number of moves computed for each manipulated variable.
Normally, control horizon prediction horizon.
Constraints
It is possible to specify constraints (bounds) on manipulated variables and outputs. Constraints can be hard (no violation allowed) or soft (some violation
allowed). The degree of softness can be adjusted by the user . The design tool
interprets a blank table cell as (−∞) for a lower bound, or (∞) for an upper
bound.
Constraints on Manipulated Variables (MVs) :
• Minimum - MV’s lower bound
• Maximum - MV’s upper bound
• Max Down Rate - maximum amount the MV can decrease in one
move (≤ 0)
• Max Up Rate - maximum amount the MV can increase in one move
(≥ 0).
8.2. I MPLEMENTATION IN S IMULINK USING M PC T OOLBOX
129
Constraints on Output Variables :
• Minimum - output’s lower bound
• Maximum - output’s upper bound
Weight Tuning
These weights strongly influence controller performance. Each table cell applies to a particular signal. The slider can be used to affect overall performance.
Input Weights Table Weight : penalizes deviations of a manipulated variable
(MV) from its nominal value. Use as follows:
• Zero (or blank) - No penalty. MV can move anywhere between its
upper and lower bound.
• Positive - increase the weight to keep the MV near its nominal value.
Rate Weight : penalizes changes in the MV. Use as follows:
• Zero - No penalty. MV can move anywhere between its upper and
lower bound. Warning: this might allow excessive movement.
• Positive - increase the weight to reduce movement in this MV.
Output Weights
Weight : penalizes deviation of an output from its setpoint (or reference). Use
as follows:
• Zero - No penalty, e.g., when the output does not have a setpoint.
• Positive - increase the weight to keep the output nearer its setpoint.
Example:
Weights=struct(’Input’,1,’ManipulatedVariablesRate’,1,’Output’,10)
set(controller,’Weights’,Weights,’ManipulatedVariables’,MV,
’OutputVariables’,OV)
Fast-Robust Slider Control
This adjusts the weights on all variables simultaneously. Moving the slider
to the left increases MV rate penalties relative to setpoint penalties, which often increases robustness, but makes disturbance rejection and setpoint tracking
more sluggish. The effect is nonlinear so it is needed to run trials to determine
the best setting.
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C APITOLO 8. MPC
Adding Integral Action
The MPC formulation does not guarantee integral action, so it has to be included. This can be done by either tuning the weight ’Integrator’ when setting the
properties of the MPC object:
Weights=struct(’Input’,1,’ManipulatedVariablesRate’,1,
’Output’,10,’Integrator’,1)
or by moving the slider control: the closer is the bar to 1 the stronger is the
resulting integral action.
8.3 System Consistency
MPC works on the linearized model, so it is important for the linearized system
to br consistent with the nonlinear one. The system is strongly nonlinear, and
includes different kinds of nonlinearities:
- as outlined in Chapter 2, the equations describing the system dynamics
are strongly nonlinear,
- constraints on the control input:
| uwg |≤ 100
- nonlinearities throughout the system: hysteresis, switches, saturations,
look up tables,. . .
So to be able to use MPC on the nonlinear model, it is important for the nonlinear model to be consistent with the linearized one, at least locally. A small step
is given so that only the nonlinearity due to the input constraints is excited.
So a first check is performed, as summarized in Figures 8.6 -8.9. A small reference step (0 to -1Nm) is given to the linearized system. Constraints are scaled
so that the input saturates. The same step is given to the nonlinear system and
constraints are scaled in the same way.
Since the system response has the
same behavior in both cases, it means that the nonlinear model is locally consistent with the linearized one. The problem is that is not possible to conclude
anything about the consistency of the two models for larger steps, when also
the other nonlinearities are excited.
8.3. S YSTEM C ONSISTENCY
131
0.12
0.1
0.08
0.06
0.04
0.02
0
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.5
Figura 8.6: uwg . Linear System. Constraints are scaled so that the input
saturates.
Torque response in a linear syste m
137
136.5
136
135.5
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.5
Figura 8.7: Torque. Linear System. A small reference step (0 to -1Nm) is given
to the system.
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C APITOLO 8. MPC
WG response non linear syste m
12
10
8
6
4
2
0
6.5
6.6
6.7
6.8
6.9
7
7.1
7.2
7.3
7.4
7.5
Figura 8.8: uwg . Non Linear System. The same step given to the linearized
system, is given to the nonlinear system and constraints are scaled in the same
way.
Torque response non linear syste m
137
136.8
136.6
136.4
136.2
136
135.8
135.6
135.4
6.5
6.6
6.7
6.8
6.9
7
7.1
7.2
7.3
7.4
7.5
Figura 8.9: Torque. Non Linear System. The system response has the same
behavior for both linearized and nonlinear systems, so the nonlinear model is
locally consistent with the linearized one.
8.4. S IMULATIONS
133
8.4 Simulations
Following the same strategy used in Chapter 6 for the LQ controller, the simulations will be performed first on the linearized model and then on the nonlinear system.
Let recall the hypothesis on which simulations are based:
• Tim is assumed to be constant
• we focus on engine operations with λ=1
• a constant engine speed test case was used to isolate the results from
gearbox and vehicle influences
• a torque based control is designed, so the reference signal is given in
terms of torque
• uwg can only vary between 0 (fully closed) and 1 (fully open)
• the strongest constraints on the states are:
- Tem < 1300K
- ωt < 18000rad/s
8.4.1 MPC on the Linearized Model
One operating point is considered at a time, with fixed engine speed. As expected, simulations show that on the linearized system the MPC gives excellent
results, both for reference tracking and disturbance rejection.
1. Fixed engine speed (1500rpm). Operating point 6 is considered first. At
t=1sec a step of 50Nm is given to the torque reference, starting from the
steady state value. The MPC opens the whole WG at t=1, when the step
starts, for a very short while (about 0.2sec) and then keeps it closed until
t=3sec, when the torque reaches the reference value (Fig.8.10). After that,
in steady state the WG is kept a bit open. The torque response is nice and
smooth, and it reaches the reference in approximately 1 second (Fig.8.11).
Then operating point 9 is considered (see Fig.8.12 and 8.13). The WG
behavior is now different from operating point 6. In fact the MPC keeps
the WG closed at the beginning of the transient and then opens it, in the
same way as the LQ does.
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C APITOLO 8. MPC
WG
alpha
Plant Inputs
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
Figura 8.10: 1500rpm. Operating point 6. Inputs. At t=1sec a step of 50Nm is
given to the torque reference. The MPC opens the whole WG at t=1, when the
step starts, for a very short while (about 0.2sec) and then keeps it closed until
t=3sec, when the torque reaches the reference value. After that, in steady state
the WG is kept a bit open.
Plant Outputs
100
90
torque
80
70
60
50
40
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
Figura 8.11: 1500rpm. Operating point 6. The torque response is nice and
smooth, and it reaches the reference in approximately 1 second.
8.4. S IMULATIONS
135
Plant Inputs
300
alpha
250
200
150
100
0.2
WG
0.15
0.1
0.05
0
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
Figura 8.12: 1500rpm. Operating point 9. Inputs. The MPC keeps the WG
closed at the beginning of the transient and then opens it, in the same way as
the LQ does.
Plant Outputs
0.03
mfuel
0.028
0.026
0.024
0.022
200
190
torque
180
170
160
150
140
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
Figura 8.13: 1500rpm. Operating point 9. Outputs. Up: fuel mass flow. Down:
torque.
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C APITOLO 8. MPC
2. Fixed engine speed (3000rpm). For the first time, in this simulation also the fuel consumption is considered as output and is weighted in the
MPC. The aim is to check if, giving anyway a fixed step in throttle as in
the previous cases, is possible to use different WG control strategies in
order to consider also the fuel consumption and not only the torque behavior. Operating point 30 is selected and the weights are changed using
the slider in the MPC Toolbox, as shown in the table below.
Slider=0.2
Slider=0.4
Slider=0.7
Qwg
0.6
0.9
10
QRAT Ewg
16
11
1
QT q
0.06
0.1
1
Fig.8.14 shows how the WG signal changes for different values of the
slider, keeping the same throttle signal: if the slider is close to 1, the MPC
keeps the WG closed for more time during transient. The output behavior
is shown in Fig.8.15: the closer is the slider to 1, the higher is the weight
on the torque. Consequently, the torque response is faster, but also the
fuel consumption is higher. Viceversa if the slider is set to a value close
to 0: the torque is slower in the transient, but the less fuel is burnt.
3. Rejection to measured disturbances. Operating point 15. A disturbance
in engine speed is simulated and a negative step is given to N, starting
from the steady state value (2000rpm) at t=5sec (Fig.8.16). The controller
fully opens the WG at t=5sec and keeps it to 1 until the torque reaches the
reference value again (Fig.8.17).
4. Rejection to unmeasured disturbances. If no integral action is added,
the MPC does not completely reject the unmeasured disturbances, but
the torque response exhibits a static error, as pointed out in Fig.(8.18). In
fact the MPC keeps the WG a bit open in steady state (Fig.8.19). The same
figures show that when adding the integral action to the MPC, it correctly reacts to disturbance and there is no static error in the torque. As explained in Section 8.2.6 the integral action can be included by adjusting
the Estimate Slider Control in the MPC Toolbox. Increasing the value of
this parameter, the weights on inputs and outputs change, as well as the
weight on the manipulated variable rate. A good tuning of this parameter allows the controller to correctly react to disturbances and bring back
the torque to the reference value.
8.4. S IMULATIONS
137
alpha
Plant Inputs
0.4
WG
0.2
0.7
0.5
1
1.5
2
2.5
3
Time (sec)
Figura 8.14: 3000rpm. Operating point 30. Influence of the Slider Control on
the system. The closer is the slider to 1, the higher is the weight on the torque,
and so the faster is the torque response. Consequently, when slider=0.7 (red
line) the MPC keeps the WG closed at the beginning of the transient for more
time than in the other cases.
Plant Outputs
0.07
mfuel
0.065
0.06
0.055
0.05
0.045
220
210
torque
200
0.4
190
0.2
180
0.7
170
160
0.5
1
1.5
2
2.5
3
Time (sec)
Figura 8.15: 2000rpm. Operating point 30. Fuel and Torque. Influence of the
Slider Control on the system. Red line: slider=0.7; blue line: slider=0.4; green
line: slider=0.2. The closer is the slider to 1, the faster is the torque response,
and so the higher is the fuel consumption.
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C APITOLO 8. MPC
WG
N
alpha
Plant Inputs
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
Figura 8.16: Rejection to a disturbance in engine speed at t=5sec. The controller
fully opens the WG at t=5sec and keeps it to 1 until the torque reaches the
reference value again.
Plant Outputs
320
300
280
torque
260
240
220
200
180
160
140
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
Figura 8.17: Rejection to measured disturbances. Torque. At t=5sec a disturbance in engine speed is added and the MPC correctly rejects it, bringing back
the torque to the reference value.
8.4. S IMULATIONS
139
Plant Outputs
160
torque
155
150
static
error
145
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
Figura 8.18: If no integral action is added, the torque response exhibits a static
error (blue line). Adding the integral action to the MPC, it correctly rejects the
disturbance (green line).
WG
alpha
Plant Inputs
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
Figura 8.19: Throttle (up) and uwg (down). When no integral action is added
(blue line), the MPC keep the WG more open in steady state, thus leading to a
static error in the torque. Instead, adding the integral action to the MPC (green
line), it uses the WG for disturbance rejection.
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C APITOLO 8. MPC
8.4.2 Pumping Losses
Let analyze now the effect of the WG on the pumping losses, considered as an additional output of the system, together with torque and fuel
(Fig.8.20-8.22). Different WG behaviors are compared, considering the
same throttle. If the WG is open for a short while at the beginning of the
transient, then the torque increases fast, due to reduced pumping losses. The drawback is that, as soon as the WG is closed again, then the
torque slows down, due the non minimum phase behavior which relates
WG and torque. Instead, if the WG is kept totally closed, then the torque
increases a bit slower at the beginning, since pumping losses are higher,
but it is not affected by the drawback of the inverse response, and so in
the second part of the transient is higher.
Plant Inputs
46
alpha
44
42
40
38
1.2
1
WG
0.8
0.6
0.4
0.2
0
−0.2
0.8
1
1.2
Time 1.4
(sec)
1.6
1.8
2
Figura 8.20: Comparing different WG behaviors considering the effect on
pumping losses. Red line: the WG is kept totally closed. Blue line: the WG
is open for a short while at the beginning of the transient. The throttle signal is
the same for both cases.
8.4.3 MPC on the Nonlinear Model
Now the MPC is connected to the nonlinear model of the engine, extracted
from the big Simulink model. We want to isolate the controller from vehicle
and ECM influence, so all the other inputs to the engine are fixed, while the
8.4. S IMULATIONS
141
Plant Outputs
0.0115
mfuel
0.011
0.0105
0.01
0.0095
11
10.5
MO3
10
9.5
9
8.5
8
0.95
1
1.05
1.1
1.15
Time (sec)
1.2
1.25
1.3
Figura 8.21: Outputs: fuel consumption (up) and pumping losses (down). If
the WG is open for a short while at the beginning of the transient (blue line),
pumping losses are reduced with respect to the case with the WG totally closed
(red line).
wastegate position is controlled by the MPC. Together with reference tracking,
also disturbance rejection is tested, adding non measured disturbances and
tuning the integrator weight in the MPC in order to guarantee rejection.
1. Acceleration phase with real throttle. The simulations on the linearized
system were performed giving a step in throttle, instead now an acceleration phase is simulated, with a ”real” throttle signal, i.e. the throttle signal
is similar to the signal that would be produced by the ECM in response to
an acceleration from the driver (see Fig. 8.23). All the other inputs to the
engine are kept constant. At t=5sec the controller fully opens the WG for
a short while, and then closes it until approximately 6 seconds, i.e. when
the torque reaches the reference value. Then the WG is kept a bit open in
steady state. The torque response is nice and it shows only a small overshoot; it reaches the reference in approximately 1 second (see Fig.8.24).
Figures 8.25-8.26 show the behavior of some states of the system. It can
be seen that the most important constraints on the states (Tem < 1300K
and ωt < 18000rad/s) are not violated.
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C APITOLO 8. MPC
Plant Outputs
55
torque
50
45
40
Time (sec)
Figura 8.22: If the WG is open for a short while at the beginning of the transient
(blue line), then the torque increases fast, due to reduced pumping losses. The
drawback is that, as soon as the WG is closed again, then the torque slows
down, due the non minimum phase behavior which relates WG and torque.
Instead, if the WG is kept totally closed (red line), then the torque increases a
bit slower at the beginning, since pumping losses are higher, but is higher in
the second part of the transient.
8.4. S IMULATIONS
143
102
Pamb
Throttle
2000
1000
0
2
4
6
8
100
98
10
0
−2
2
4
6
8
19
10
2
Area WG
N
6
8
10
4
6
8
10
4
6
8
10
20
2002
2000
1998
4
21
Tamb
Lambda
2
2
2
4
6
8
x 10
−4
1
0
10
2
2
Figura 8.23: Non Linear model. Step in Torque 20-220Nm. The throttle signal simulates the acceleration from the driver and the other inputs are kept
constant. At t=5sec the controller fully opens the WG for a short while, and
then closes it until approximately 6 seconds, i.e. when the torque reaches the
reference value.
250
200
Torque
150
100
50
0
4
4.5
5
5.5
6
6.5
7
Figura 8.24: Non Linear model. Step in Torque 20-220Nm. The torque response
is nice and it shows only a small overshoot; it reaches the reference in 1 second
approximately.
144
C APITOLO 8. MPC
5
T before thro t
320
1.2
1
0.8
15
P intake M
x 10
2
4
x 10
6
8
290
2
4
6
8
10
2
4
6
8
10
15000
5
2
300
4
10
0
310
280
10
turbine speed
P before thro t
1.4
4
6
8
10000
5000
0
10
Figura 8.25: Non Linear model. Step in Torque 20-220Nm. States. The
constraint on the turbine speed (ωt < 18000rad/s) is not violated.
5
1100
T exhaust M
1.3
1.2
1.1
1
1.08
P exhaustsyste m
x 10
2
x 10
4
6
8
2
4
6
8
10
2
4
6
8
10
1100
1.04
1.02
2
900
5
1.06
1
1000
800
10
T exhaustsyste m
P exhaust M
1.4
4
6
8
10
1000
900
800
700
Figura 8.26: Non Linear model. Step in Torque 20-220Nm. States. The
constraint on the exhaust manifold temperature (Tem < 1300K) is not violated.
8.5. C ONCLUSIONS
145
2. Reference tracking and unmeasured disturbance rejection. Now a step
is given to the throttle, as in the linear case, at t=5sec and the torque
reference is lifted from 150 to 200Nm (Fig.8.27). Then, at t=7sec, a non
measured disturbance is added to the torque to test disturbance rejection.
First the weight on the integral action in the MPC is set to 1 and can be
seen that the torque goes back to the reference but with a static error
(Fig.8.28). Increasing the integrator weight up to 1000, the disturbance is
rejected and the static error is eliminated, as depicted in Fig.8.29.
102
Pamb
Throttle
300
200
100
2
4
6
8
100
98
10
0
−2
2
4
6
8
1
Area WG
N
6
8
10
4
6
8
10
4
6
8
10
20
19
10
2002
2000
1998
4
21
Tamb
Lambda
2
2
2
4
6
8
10
2
x 10
−4
0.5
0
2
Figura 8.27: Non Linear model. At t=7sec a disturbance is added to the torque
and the MPC correctly opens the WG to reject it.
8.5 Conclusions
Since the system has constraints on the input, on the output and also on some
states, a model predictive approach has been investigated to handle these limitations.
In this chapter a SISO-SIMO approach has been studied. First, the WG was
used to improve the torque response and then, considering also the fuel as
output, different ways to control the WG were considered, in order to reduce
the fuel consumption. As LQ analysis showed, the possibilities to improve the
torque response, controlling the WG in a different way than to keep it closed
146
C APITOLO 8. MPC
210
200
Torque
190
180
170
160
150
2
3
4
5
6
7
8
9
10
Figura 8.28: Non Linear model. Disturbance rejection. Without integral action
in the MPC (Integrator weught=1) the torque response shows a static error.
210
200
Torque
190
180
170
160
150
2
3
4
5
6
7
8
9
10
Figura 8.29: Non Linear model. Disturbance rejection. Increasing the integrator weight up to 1000, the disturbance is rejected and the static error is
eliminated.
8.5. C ONCLUSIONS
147
all the time during the transient, are very few and not so realistic. Normally,
opening the WG during the transient, it is lost much more that what is possible
to get in terms of torque, and so it seems not to be worth.
Obviously, MPC handles constraints much better that LQ, but the linear
model represents the nonlinear one only in a neighborhood of the equilibrium
point were the system has been linearized. So when we simulate it on the
nonlinear system the results are not as perfect as on the linear one. What is
interesting instead, is to check which benefit can be achieved using a MIMO
strategy, as shown in the next chapter.
Capitolo 9
Model Reduction
The linearized model of the system has 12 states, so it is not reasonable to think
to a possible real time implementation of the MPC designed in the previous
chapter. So here a model reduction is presented [6][24], showing that it is possible to reduce the model to 4 states, keeping all the properties of the original
one, and under some assumptions it can be reduced to 3 states only.
9.1 Introduction
In the design of robust controllers for complicated systems, model reduction
fits several goals:
1. To simplify the best available model in light of the purpose for which the
model is to be used-namely, to design a control system to meet certain
specifications.
2. To speed up the simulation process in the design validation stage, using a smaller size model with most of the important system dynamics
preserved.
3. Finally, if a modern control method such as LQG or H∞ is used for which
the complexity of the control law is not explicitly constrained, the order of the resultant controller is likely to be considerably greater than is
truly needed. For this purpose , one can either reduce the order of the
plant model prior to control design, as is done in this thesis, or reduce
the controller in the final stage.
The central problem is: given a high order linear time invariant stable model
G, find a low order approximation Gred such that the infinity norm of the dif149
150
C APITOLO 9. M ODEL R EDUCTION
ference, G − Ga ∞ is small. By the model order, the dimension of the state
vector in a minimal realization is meant, that is called McMillan degree[6].
The scope of this section is to find a new model that maintains the main
characteristics of the original one, but with lower order. Since the scope of
the thesis is to use an MPC controller in real time, 12 states dimension is too
much and makes application in real time impossible, so the lower will be the
order of the redacted model the faster will be the response of the controller
when implemented in the car. The best solution will be the lowest model order
whose response guarantees a small difference with the original model.
9.2 Model Reduction Process
For control purposes, it is generally enough to have an accurate model near
the crossover frequency. For simulation, it is enough to capture the essential
dynamics in the frequency range of the excitation signals. This means that it is
often possible to find low order approximations of high order models.
A model reduction task typically involves the following steps:
- Analyze the important characteristics of the model from its time or frequency domain responses obtained from |step| or |bode|, for example.
- Determine an appropriate reduced order by plotting the Hankel singular values of the original model (|hankelsv|) to determine which modes
(states) can be discarded without sacrificing the key characteristics.
- Choose a reduction algorithm. Some reduction methods available in the
toolbox are: |balancmr|,|bstmr|,|dcmatch|and |hankelmr| The methods
employ different measures of ’closeness’ between the original and reduced models and the choice is application-dependent. Their relative
merits will be investigated in the following sections.
- Validation: results are validated by comparing the dynamics of the reduced model to the original. Reduction parameters (choice of model order, algorithm, error bounds etc.) may need to be adjusted if the results
are not satisfactory.
9.3 Hankel Singular Values
In control theory, eigenvalues define a system stability, whereas Hankel singular values define the ”energy” of each state in the system. Keeping larger
9.4. O VERVIEW OF M ODEL R EDUCTION T ECHNIQUES
151
energy states of a system preserves most of its characteristics in terms of stability, frequency, and time responses. Model reduction techniques presented here
are all based on the Hankel singular values of a system. They can achieve a
reduced-order model that preserves the majority of the system characteristics.
Mathematically, given a stable state-space system (A,B,C,D), its Hankel singular values are defined as [6]
σH =
λi (P Q)
(9.1)
where P and Q are controllability and observability grammians satisfying
AP + P AT = −BB T
AT Q + QA = −C T C
9.4 Overview of Model Reduction Techniques
Robust Control Toolbox offers several algorithms for model approximation and
order reduction. These algorithms let control the absolute or relative approximation error, and are all based on the Hankel singular values of the system.
Robust control theory quantifies a system uncertainty as either additive or multiplicative types. These model reduction routines are also categorized into two
groups[23]:
Additive Control Method The reduced order model has a bounded by an error criterion.
Multiplicative Error Method The reduced order model has a multiplicative or
relative error bounded by an error criterion.
In other words, some model reduction routines produce a reduced order model
Gred of the original model G with a bound on the error G − Gred ∞ , the peak
gain across frequency. Others produce a reduced order model with a bound on
the relative error , the peak gain across frequency. Others produce a reduced
order model with a bound on the relative error G−1 (G − Gred )∞ .
These theoretical bounds are based on the ”tails” of the Hankel singular
values of the model[6], i.e.,
Additive Error Bound:
G − Gred ∞ ≤ 2
n
k+1
σi
(9.2)
152
C APITOLO 9. M ODEL R EDUCTION
where σi denotes the ith Hankel singular value of the original system G.
Multiplicative (Relative) Error Bound:
G−1 (G − Gred )∞ ≤
n
(1 + 2σi (
1 + σi2 + σi )) − 1
(9.3)
k1
where σi denotes the ith Hankel singular value of the phase matrix of the model
G.
In the following section different methods will be introduced rapidly and
compared, also considering different orders for the reduced model.
To compare the models between each other singular values plots and step
responses are used, from each inputs of the system to the output.
9.4.1 Truncation and Residualization
Let (A, B, C, D) be a minimal realizationof a stable
system G(s) and partition
x1
where x2 is the vector of n-k
the state vector x, of dimension n, into
x2
states which we wish to remove. With appropriate partitioning of A, B and C,
the state-space equations become
⎧
⎪
⎪
⎨ ẋ1 = A11 x1 + A12 x2 + B1 u
ẋ2 = A21 x1 + A22 x2 + B2 u
⎪
⎪
⎩ y(t) = C x + C x + Du
1 1
(9.4)
2 2
Truncation
s
s
A kth order truncation of the realization G = (A, B, C, D) is given by Ga =
(A11 , B1 , C1 , D) [6]. If A is in Jordan form then it is easy to order the states so
that x2 corresponds to high frequency or fast modes.
Assume that A has been diagonalized so that:
⎡
⎢
⎢
⎢
A=⎢
⎢
⎣
λ1
0
...
0
0
..
.
λ2
..
.
...
..
.
0
..
.
0
0
...
λn
⎤
⎡
bT1
⎤
⎥
⎢ T ⎥
⎥
⎢ b2 ⎥
⎥
⎢
⎥
⎥ B = ⎢ . ⎥ C = c1
⎥
⎢ .. ⎥
⎦
⎣
⎦
bTn
c2
...
cn
(9.5)
Then if the λi s are ordered so that | λ1 |<| λ2 |< · · · the fastest modes are
removed from the model after truncation. The difference between G and Ga
9.4. O VERVIEW OF M ODEL R EDUCTION T ECHNIQUES
153
following a kth order model truncation is given by[6]:
n
G − Ga =
1=k+1
and therefore
n
G − Ga ∞ ≤
i=k+1
ci bTi
s − λi
(9.6)
σ(ci bTi )
| Re(λi ) |
(9.7)
An advantage of modal truncation is that the poles of the truncated model are
a subset of the poles of the original model and therefore retain any physical
interpretation.
Residualization
In truncation all the states and dynamics associated with x2 are disregarded.
Suppose instead of this to simply set ẋ2 = 0, so x2 is residualized in the statespace equations. One can then solve for x2 in terms of x1 and u, and back
substitution of x2 then gives [6]:
−1
ẋ1 = (A11 − A12 A−1
22 A21 )x1 + (B1 − A12 A22 B2 )u
−1
y = (C1 − C2 A−1
22 A21 )x1 + (D − C2 A22 B2 )u
(9.8)
Let assume A22 is invertible and define
Ar = A11 − A12 A−1
22 A21
Br = B1 − A12 A−1
22 B2
Cr = C1 − C2 A−1
22 A21
(9.9)
Dr = D − C2 A−1
22 B2
s
The reduced order model Ga (s) = (Ar , Br , Cr , Dr ) is called a residualization
s
of G(s) = (A11 , B1 , C1 , D) . Usually (A, B, C, D) will have been put into Jordan
form, with the eigenvalues ordered so that x2 contains the fast Modes. An important property of residualization is that it preserves the steady-state gain of
the system, Ga (0) = G(0). This should be no surprise since the residualization
process sets derivatives to zero, which are zero anyway at steady-state. It is
clear from the discussion above that truncation is to be preferred when accuracy is required at high frequencies, whereas residualization is better for low
frequency modelling.
154
C APITOLO 9. M ODEL R EDUCTION
9.4.2 Balanced truncation and residualization
Both methods depend to a large extent on the original realization and the the
Jordan form has been used. A better realization, with many useful properties,
is the balanced realization which will be considered now. A balanced realization
is an asymptotically stable minimal realization in which the controllability and
observability Gramians are equal and diagonal. Let (A, B, C, D) be a minimal
realization of a stable rational transfer function G(s), then (A, B, C, D) is called
balanced if the solutions to the following Lyapunov equations [6]
AP + P AT + BB T = 0
AT Q + QA + C T C = 0
(9.10)
are P = Q = diag(σ1 , σ2 , . . . , σn ) = Σ, where σ1 ≥ σ2 ≥ . . . ≥ σn > 0.
Any minimal realization of a stable transfer function can be balanced by a
simple state similarity transformation and routines for doing this are available
in Matlab. In a balanced realization the value of each σi is associated with a
state xi of the balanced system. The size of σi is a relative measure of the contribution that xi makes to the input-output behavior of the system. Therefore
if σ1 σ2 then the state x1 effects the input-output behavior much more than
x2 or indeed any other state because of the ordering of the σi . After balancing
a system, each state is just as controllable as it is observable, and a measure of
a state’s joint observability and controllability is given by its associated Hankel
singular value. This property is fundamental to the model reduction methods.
So now we can use the previous technics but just removing states having little
effects on the system’s input-output behavior. A precise statement of the error
bound is given in the following theorem[6]:
Theorem 1 Let G(s) be a stable rational transfer function with Henkel singular values
σ1 > σ2 , . . . , > σN where each σi has multiplicity ri and let Gka (s) be obtained by
truncating or residualizing the balanced residualization of G(s) to the first (r1 + r2 +
. . . + rk ) states. Then
G(s) − Gka )∞ ≤ 2(σk+1 + σk+2 + . . . , σN )
(9.11)
9.4.3 Optimal Hankel norm approximation
In this approach to model reduction the problem that is directly addressed is
the following: given a stable model G(s) of order n, find the reduced order
model Gkred (s) of degree k such that the Hankel norm of the approximation
error G − Gkred H is minimized.
9.5. M ODEL R EDUCTION P ROCEDURE
155
The Henkel norm of any stable transfer function E(s) is defined as[6]
1
E(s)H = ρ 2 (P Q)
(9.12)
where P and Q are the controllability and observability Gramians of E(s).
9.4.4 Balanced stochastic model truncation (BST)
This methods is a Multiplicative error method, implemented in Matlab through
the routine bstmr. The error bound is computed based on Hankel singular
values of the phase matrix of G. For a stable system these values indicate the
respective state energy of the system. Hence, reduced order can be directly
determined by examining these values. With only one input argument G, the
function will show a Hankel singular value plot of the phase matrix of G and
prompt for model order number to reduce. This method guarantees an error
bound on the infinity norm of the multiplicative G−1
red (G − Gred )∞ or relative
error G−1 (G − Gred )∞ for well-conditioned model reduction problems[6]:
G−1 (G − Gred )∞ ≤ +
n
(1 + 2σi (
1 + σi2 σi )) − 1
(9.13)
k+1
9.5 Model Reduction Procedure
The problem is that we have 45 different operating points, and so we have 45
different models G = (A, B, C, D) to reduce. It would be desirable to make a
general model reduction that works fine for each point, i.e. a general technique
that can be used for each operating point with good results in average over
the 45 operating points. Of course this will not give the best model with the
minimum number of states for each point, but it will be something that in
average works fine for all the systems (Ai , Bi , Ci , Di ) with i = 1 . . . 45. So
the steps below have been followed:
1. Decide how many states to use in the reduced model.
2. Choose a model reduction technique trying to minimize in average the
infinity norm.
3. Validate the model.
156
C APITOLO 9. M ODEL R EDUCTION
9.5.1 Number of states to use in the reduced model
The system has 6 inputs (α, N, W G, λ, Pamb , Tamb ), one output (Tq ) and 12
states. The first thing has been to analyze the important characteristics of the
model from its time or frequency-domain responses,and has been found that in
the transfer function between WG and torque Gwg (s) there is a right half plane
zero. This require to be really careful in the reduction process and to check if
this zero moves when reducing the model.
The last two inputs (Pamb , Tamb ) are not very important because in many
cases their transfer function to the torque is really small, and they are almost
constant,especially the temperature. So a system with only 4 inputs and 1 output will be considered. During our simulation in the linearize system the engine speed and Lambda are constant, for hypothesis, but in a more general case
they could vary so these two inputs will be kept also in the reduced system.
What we have to do now is to determine an appropriate reduced order by
plotting the Hankel singular values of the original model (—hankelsv—) to
determine which modes (states) can be discarded without sacrificing the key
characteristics. To decide how many states to use we have been initially looking, at some operating point, the Hankel singular value, starting for example
from N = 3000rpm; that was one of the speed that we have been considering,
and plotting some of them. Depending on the operating point, the dominant
singular value could be 2, 3 or 4. This means that the number of states affecting the inputs-outputs dynamics is not the same in all the operating points. To
bypass this problem, a script in Matlab has been created to calculate the Hankel singular value of all the operating points and to find out, in average, the
number of states having an important effect in the input-output dynamics (see
Fig.9.1). It seems that in most of the cases we could reach 3 states, but since it is
an average this mean that there could be some cases requiring 4-5 states, so we
decided to consider 4 states to be sure to get a good model for each operating
point.
9.5.2 Model Reduction Techniques. Comparison: Additive Methods
To choose which technique to use, has been considered which are the most
important frequencies in the dynamic of our engine. For control purposes, it
is generally enough to have an accurate model near the crossover frequency.
For simulation, it is enough to capture the essential dynamics in the frequency
range of the excitation signals. It seems that the main dynamics inside the in-
9.5. M ODEL R EDUCTION P ROCEDURE
157
avarage of HSV over the 45 op
140
120
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
10
11
12
Figura 9.1: Average of Hankel singluar values, over all the 45 operating points
take manifold are around 15-20Hz, so we will consider a good reduced model
that approximates well the original system between 1-1000 rad/sec. All algorithms provide bounds on the approximation error. For additive-error methods
like |balancmr|, the approximation error is measured by the peak (maximum)
gain of the error model |G − Gred | across all the frequencies. This peak gain is
also known as the H-infinity norm of the error model. As first we have been
comparing the additive methods, as balanced DCgain , balanced truncation
and Hankel norm approximation.
Dc gain method gives a non zero D matrix, so it can not be chosen. There is
never a direct term D from input to output in MPC models. This follows from
the fact that, at a given sample k, the output is already given and can hence not
be affected by the present input uk which is to be calculated by the optimizer.
According with this we start to prefer Balanced truncation and Hankel singular
norm decomposition that have D=0. Also this method is really good at low
frequency, since has the property to preserve the DC gain of the system, but
looses most of the dynamics at high frequency. It seems that depending of
the operating point one is better than the other. To overcome this problem, as
for the Hankel singular values, we plot a sort of average over the 45 different
operating points to have an idea of what happens in all the cases, and if there
158
C APITOLO 9. M ODEL R EDUCTION
method
Average Infinity Norm
hankel
balanced truncation
hankel-Throttle
balanced truncation-Throttle
hankel-WG
balanced truncation-WG
0.33009
0.32935
0.002264
0.000983
0.20554
0.28994
Tabella 9.1: Average norm over the 45 operating points. Comparison between
Hankel and balanced truncation method, both for the throttle and the WG.
is a technique that works better than the others in most of the cases (see table
9.1). Even in the frequency range of our interest, it seems that both Hankel
norm approximation and Truncation give a good approximation of the original
model in all the operating points and the resulting static error is not so big.
Op 15 WG 4 states
60
Magnitude (dB)
40
20
0
−20
Phase (deg)
−40
180
sys
bal
han
90
0
−90
−2
10
10
0
10
2
10
4
Frequency (rad/sec)
Figura 9.2: Bode diagram of Gα with 4 states for original system, Truncation
and Hankel method. Operating point 15.
9.5. M ODEL R EDUCTION P ROCEDURE
159
Op 15 throttle 4 states
0
Magnitude (dB)
−10
−20
−30
−40
Phase (deg)
−50
0
sys
bal
han
−45
−90
−2
10
10
0
10
2
10
4
Frequency (rad/sec)
Figura 9.3: Bode diagram of Gwg with 4 states for original system, Truncation
and Hankel method. Operating point 15.
Op 15 WG 4 states
−20
sys−bal
sys−han
−30
Singular Values (dB)
−40
−50
−60
−70
−80
−90
−100 −2
10
10
0
2
10
10
4
Frequency (rad/sec)
Figura 9.4: Singular value plot for the errors G − GT runc and G − GHankel
160
C APITOLO 9. M ODEL R EDUCTION
9.6 Model Reduction Technique: Multiplicative Error Method
We have been also trying to use a multiplicative error method such as |bstmr|.
This algorithm emphasizes relative errors rather than absolute ones. Now, we
look at the singular values for multiplicative (relative) errors (using the ’mult’
option of |hankelsv|), see figure 9.5.
Hankel SV of Phase Matrix
1
0.9
0.8
0.7
abs
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
Order
8
9
10
11
12
Figura 9.5: Hankel singular values of Phase Matrix. Operating point 11.
For multiplicative-error methods such as |bstmr|, the approximation error is measured by the peak gain across frequency of the relative error model
|G \ (G − Gred )|. So we will have the biggest absolute error where G has higher gain. So again just have a look at some plots and see what happen in our
frequencies range of interest. Instead to decide before which number of state
we want , we set the maximum relative error and we let the algorithm find
how many states we need to be inside this value. To achieve at least 5% relative accuracy, what is the lowest order we can get? The function |reduce| can
automatically select the lowest-order model compatible with our desired level
of accuracy. The algorithm has picked a 4-state reduced model |Gred |. This
respects what we were expecting looking the Hankel singular phase matrix
values.
9.6. M ODEL R EDUCTION T ECHNIQUE : M ULTIPLICATIVE E RROR M ETHOD 161
Having a look for example at the Op18 at the two most important transfer
function Gwg and Gα we can see that the relative error is quite small in the
range of frequency that we want to consider.
Op.16 rel WG 4 states bst method
−40
Singular Values (dB)
−45
−50
−55
−60
−65
−70 −2
10
10
0
2
10
10
4
Frequency (rad/sec)
Figura 9.6: Bode diagram of relative error for Gwg . Operating point 16.
Now we can compare this new model with the additive methods and see
what happens. The results are quite similar, the additive method BST keeps the
relative error bounded, so is good when the absolute value becomes smaller
too. On the other hand, when the absolute value is small is also less important
from the control point of view. Transfer functions with absolute value of -30/40dB can be neglected.
So we can say that with 4 states all the analyzed methods are good. The
BST method is chosen for the validation, and in the following sections it will
be compared with the original system.
9.6.1 Validate the model
We want now to validate our results by comparing the dynamics of the reduced model to the original. We may need to adjust our reduction parameters
(choice of model order, algorithm, error bounds etc.) if the results are not satisfactory. To compare the results the step response and the singular values of
162
C APITOLO 9. M ODEL R EDUCTION
Op.16 rel throttle 4 states bst method
−20
−25
Singular Values (dB)
−30
−35
−40
−45
−50
−55 −2
10
10
0
2
10
10
4
Frequency (rad/sec)
Figura 9.7: Bode diagram of relative error for Gα . Operating point 16.
Op 16 Throttle 4 states
−30
Magnitude (dB)
−40
−50
−60
−70
Phase (deg)
−80
0
sys
bal
bst
han
−45
−90
−2
10
10
0
10
2
10
4
Frequency (rad/sec)
Figura 9.8: Bode diagram for Gα . All methods. Operating point 16.
9.6. M ODEL R EDUCTION T ECHNIQUE : M ULTIPLICATIVE E RROR M ETHOD 163
Op 16 Throttle 4 states
0.03
0.025
0.015
sys
bal
bst
han
0.01
0.005
0
0
2
4
6
8
Time (sec)
Figura 9.9: Step response for Gα . All methods. Operating point 16.
Op.16 WG 4 states
60
Magnitude (dB)
40
20
0
−20
−40
180
Phase (deg)
Amplitude
0.02
sys
bal
bst
han
90
0
−90
−2
10
10
0
10
2
10
4
Frequency (rad/sec)
Figura 9.10: Bode diagram for Gwg . All methods. Operating point 16.
164
C APITOLO 9. M ODEL R EDUCTION
Op 16 WG 4 states
50
sys
bal
bst
han
0
Amplitude
−50
−100
−150
−200
−250
0
2
4
6
8
Time (sec)
Figura 9.11: Step response for Gwg . All methods. Operating point 16.
some operating points have been plotted, and the attention has been focused
on the most important transfer functions that could be used for the design of
the control: Gα , Gwg and Gλ . Fig.9.13 shows the step response. The singular
values of the error between the selected method (BST) and the original system
are considered in Fig.9.12, and it can be seen that the result is satisfying. Figures 9.14 and 9.15 show the Bode diagram of Gα and Gwg respectively. It can
be seen that the results are pretty much the same for the reduced model and
for the original system.
As last check also the positive zeros in the transfer functions of the reduced
model have been calculated, to check if their frequency is not so changed from
the original model (see Fig.9.2). This is important since they make a limitation
in the close loop bandwidth of the system.
9.7 Conclusions
The results achieved with 4 states are quite good and there might be the possibility to do something better. We have to consider that in those operating
points in which the module of the transfer function is really low, -30, -40dB,
the effect of the transfer function to the system is really small and then not so
9.7. C ONCLUSIONS
165
Op.14 singular values sys−bst
0
Singular Values (dB)
−20
−40
−60
−80
−100
−120 −2
10
10
0
2
10
10
4
Frequency (rad/sec)
Figura 9.12: Singular value plot of the error between the original system and
the BST reduced model.
Op.14 step response
From: In(1)
From: In(3)
From: In(4)
Amplitude
sys
bst
0
5
0
5
0
Time (sec)
Figura 9.13: Step response for BST method.
5
166
C APITOLO 9. M ODEL R EDUCTION
Op.14 Throttle − Torque
10
Magnitude (dB)
0
−10
−20
−30
−40
−50
0
Phase (deg)
sys
bst
−45
−90
−1
10
10
0
10
1
10
2
10
3
10
4
Frequency (rad/sec)
Figura 9.14: Comparison of Bode diagram of Gα for original system and the
BST reduced model.
Op.14 WG−Torque
Magnitude (dB)
40
20
0
−20
Phase (deg)
−40
180
sys
bst
90
0
−90
−2
10
10
0
10
2
10
4
Frequency (rad/sec)
Figura 9.15: Comparison of Bode diagram of Gwg for original system and the
BST reduced model.
9.7. C ONCLUSIONS
N=1200
(0p 1 to 4)
N=1500
(0p 5 to 11)
N=2000
(0p 12 to 18)
N=2600
(0p 19 to 25)
N=3000
(0p 26 to 32)
N=3500
(0p 33 to 39)
N=4700
(0p 40 to 45)
RHP ZERO (rad/s)
ORIGINAL MODEL HANKEL 4 states
2.2
2.2
6
6.2
9.7
9.9
13.6
13.8
0.07
0.06
3.9
4.04
7.7
7.9
12.2
12.4
17.9
18
23
22.3
23.9
23.4
5.2
5.4
11
11.2
15.8
16
20.6
20.6
23.6
22.9
27.3
26.7
28.5
28.04
7.1
7.3
13.4
13.6
18.3
18.4
20.7
20.7
23.4
23.5
26.1
25.8
26.7
26.4
9
9.2
15.5
15.7
18.6
18.7
21.4
21.3
22
21.9
25.2
25
25.8
25.6
10.8
11
17.1
17.2
19.4
19.6
21.4
21.4
22.3
22.3
23.9
24.3
23.5
24.1
12.6
12.7
18.2
18.3
19.9
20.1
20.2
20.3
19.9
19.4
20.4
20
167
BST 4 states
2.2
6.1
9.8
13.8
0.06
3.9
7.8
12.2
17.9
23
23.9
5.3
11.1
15.9
20.6
13.9
23.6
26.07
26.7
9.1
15.6
18.6
16.1
21.4
22.07
9.1
15.6
18.6
21.4
22
25.1
25.7
10.9
17.2
19.4
21.4
22.4
23.9
23.5
12.6
18.2
19.9
20.3
20.1
20.6
Tabella 9.2: Frequency of the right half plane zero (rad/s) for the 45 operating
points. The original system is compared with Hankel and BST methods (4
states).
168
C APITOLO 9. M ODEL R EDUCTION
important. So we can have a look to understand what happens to the transfer function between WG -Torque and Throttle-Torque, where these are above
-10dB for example. This means that we have to consider operating point 11 or
18 to check the WG behavior, and operating point 12 or 15 to check the throttle
one.
Op.16 WG 3 states
60
Magnitude (dB)
40
20
0
−20
Phase (deg)
−40
180
sys
bal
bst
han
90
0
−90
−2
10
10
0
10
2
10
4
Frequency (rad/sec)
Figura 9.16: Gwg for 3 states. Comparison:Hankel, Truncation and BST methos.
Operating point 16.
As is possible to see from the plots 9.16-9.17, all methods give a very good
result, making the hypothesis that we are interested only in transfer function
with a significant module. Under this assumption, 3 states seem to be a reachable final goal. The BST method works quite well especially where the module
is really high, this because it tries to minimize the relative error instead of the
absolute one, like Hankel or Truncation, and from the control point of view this
is a good property. On the other hand, of course the BST will give a static error
bigger than the other two methods, but for the control strategy point of view
this does not seem to be a big problem.
As shown in the previous sections, we cannot use 3 states for all the operating points. It changes a lot among the operating points, in some cases
even 2 states could be enough. We will consider 4 states as final result, but
9.7. C ONCLUSIONS
169
Op.12 throttle 3 states
Magnitude (dB)
0
−20
−40
Phase (deg)
0
sys
bal
han
bst
−45
−90
−1
10
10
0
10
1
10
2
10
3
10
4
Frequency (rad/sec)
Figura 9.17: Gα for 3 states. Comparison: Hankel, Truncation and BST methos.
Operating point 12.
the possibility of analyzing the best solution for each operating point can also
be considered.
Capitolo 10
Multivariable System
Analysis
This chapter presents the analysis of the system using a different problem formulation, based on multivariable control. Depending on the selected inputs
and outputs, different strategies for the control problem are possible:
• SISO ⇒ one input (uwg ) and one output (torque)
• SIMO ⇒ one input (uwg ) and two outputs (fuel consumption and torque)
• MISO ⇒ two inputs (α and uwg ) and one output (torque)
• MIMO ⇒ two inputs (α and uwg ) and two outputs (fuel consumption and
torque)
The SISO and SIMO formulations were presented through the previous chapters, while now MISO and MIMO problems are analyzed.
10.1 Directions in multivariable systems
For a SISO system y = Gu, the gain at a given frequency is simply
| y(ω) |
| G(jω)u(ω) |
y0
=
=
=| G(jω) |
| u(ω) |
| u(ω) |
u0
(10.1)
The gain depends of the frequency ω, but since the system is linear it is
independent of the input magnitude | u(ω) |. Things are not quite as simple for
MIMO systems where the input and output signals are both vectors, and we
need to sum up the magnitudes of the elements in the vector by use of some
171
172
C APITOLO 10. M ULTIVARIABLE S YSTEM A NALYSIS
norm. If we select the vector 2 − norm, the usual measure of length, then at a
given frequency ω the magnitude of the vector input signal is
u(ω)2 =
| uj (ω) |2 =
u210 + u220 + · · ·
(10.2)
j
and the magnitude of the vector output signal is
y(ω)2 =
| yi (ω) |2 =
2 + y2 + · · ·
y10
20
(10.3)
i
The gain of the system G(s) for a particular input signal u(ω) is then given by
the ratio
2 + ···
y 2 + y20
y(ω)2
G(jω)u(ω)2
=
= 10
u(ω)2
u(ω)2
u210 + u220 + · · ·
(10.4)
Again the gain depends on the frequency ω and again it is independent of the
input magnitude u(ω)2 . However, for a MIMO system there are additional
degrees of freedom and the gain depends also on the direction of the input u.
The largest gain for any input direction is equal to the maximum singular
value
σ̄(G) = max
u=0
Gu2
= max Gu2
u2
u2 =1
(10.5)
and that the smallest gain for any input direction is equal to minimum singular
value
σ(G) = min
u=0
Gu2
= min Gu2
u2
u2 =1
(10.6)
The last identities in (10.5) and (10.6) follow because the gain is independent of
the input magnitude for a linear system.
When plotting the singular values of the system, it can be seen that they
mostly depend on pim . In fact changing operating point, the singular values
change as well: their magnitude bode plot lifts up when switching to operating
points with the same N but higher pim . So for those operating points with the
same pim they are more or less the same. As an example, figure 10.1 shows the
singular value for the operating points 12 ÷ 18. To go on with the analysis it is
better to scale the system, so that conclusions can be drawn about input-output
controllability.
10.2. S CALING THE SYSTEM
173
Singular Values
60
Singular Values (dB)
40
20
p_im
0
−20
−40
−60
−2
10
10
0
2
10
Frequency (rad/sec)
10
4
10
6
Figura 10.1: Singular values. Op from 12 to 18. The magnitude plot lifts up for
increasing values of pim
10.2 Scaling the system
Let the unscaled linear model of the process in deviation variables be
ŷ = Ĝû
(10.7)
where a hat is used to show that the variables are in their unscaled original
units. A useful approach for scaling is to make the variables less than one
in magnitude [6]. This is done by dividing each variable by its maximum
expected or allowed change:
y = ŷ/ymax , u = û/umax
(10.8)
where
• ymax – largest expected change in reference value
• umax – largest allowed input change
Since the expected or allowed variation of the variables about their nominal
value is not symmetric, then the smallest variation should be used for umax
and ymax .
174
C APITOLO 10. M ULTIVARIABLE S YSTEM A NALYSIS
For MIMO systems each variable in the vectors ŷ and û may have a differ-
ent maximum value, in which case Dy and Du become diagonal scaling matrices. This ensures, for example, that all errors (outputs) are of about equal
importance in terms of their magnitude.
The corresponding scaled variables to use for control purposes are then
y = Dy−1 ŷ, u = Du−1 û
(10.9)
On substituting 10.9 into 10.7 we get the scaled transfer functions
G = Dy−1 GDu
(10.10)
then yields the following model in terms of scaled variables
y = Gu
(10.11)
The system has been linearized around 45 operating points, so there are
45 different linear systems. For each of them the scaling matrices Du and Dy
have to be set. The maximum reachable torque is around 300Nm, but it can
not be assumed that it will vary up to the maximum value for each operating
point since the linearized models are valid only in a neighborhood of the nominal values. So the largest expected change in the reference signal can not be
300Nm, but it has to be selected depending on the operating point.
Let take as example operating point 12. Assuming that the WG can vary
in the symmetric interval [-1,1], and the throttle (in mm2 ) between [0,1350] the
scaling procedure gives:
uwgmax = 1
αmax = min(| 0 − ᾱ,12 |, | 1350 − ᾱ,12 |)
¯
¯
yf uel−max = min(f uelmax − | f uel
|, | 0 − f uel
,12
,12
|)
yT q−max = min(T qmax − | T̄q,12 |, | 0 − T̄q,12 |)
where the bar indicates the steady state value, and the subscript means operating point 12.
10.3 Singular value decomposition
The magnitudes of the eigenvalues for MIMO systems do not provide a useful means of generalizing the SISO gain | G(jω) |. The problem is that the
eigenvalues measure the gain for the special case when the inputs and the out-
10.4. M ULTIVARIABLE P OLES AND Z EROS
175
puts are in the same direction, namely in the direction of the eigenvectors. So
analysis based on singular values is needed.
The singular value decomposition (SVD) is defined according to [6].
Consider the transfer matrix in 10.15, with 6 inputs and 1 output, evaluated
at ω = 0, so that G(0) = Ḡ is a constant 1 × 6 matrix. So Ḡ can be decomposed
into its singular value decomposition:
G = U ΣV H
(10.12)
The column vectors of U represent the output directions of the plant, while
the column vectors of V represent the input directions. These input and output
directions are related through the singular values. To see this, note that since V
is unitary we have V H V = I, so (10.12) may be written as [6]
GV = U Σ
Define u1 = ū , v1 = v̄, u2 = u and v2 = v .Then it follows that
Gσ̄ = σ̄ ū Gσ = σu
(10.13)
The vector v̄ corresponds to the input direction with largest amplification, and
ū is the corresponding output direction in which the inputs are most effective.
The direction involving v̄ and ū is sometimes referred to as the ’strongest’, ’high
gain’ or ’most important’ direction.
The largest gain σ̄ is for an input in the direction v̄, while the smallest gain σ
is for an input in the direction v. Since the plant has more inputs than outputs,
the additional input singular vectors v tells us in which directions the input
will have no effect on the output.
For dynamic systems the singular values and their associated directions
vary with frequency and the frequency range corresponding to the closed loop
bandwidth is of main interest. A desirable bandwidth is around 15Hz, that
means approximately 60rad/s, so will be focused on the frequency range from
1rad/s up to 1000rad/s.
10.4 Multivariable Poles and Zeros
The transfer functions from WG to torque in the MISO and MIMO cases are:
GM ISO = [Gα GN Gwg Gλ ]
(10.14)
176
C APITOLO 10. M ULTIVARIABLE S YSTEM A NALYSIS
yfuel
uwg
s
u
s
1
u
0
uthrottle
1
ytorque
Figura 10.2: Outputs (right plot) resulting from use of u 2 = 1 (unit circle
in left plot) for the system G. The maximum (σ̄) and minimum (σ) gains are
obtained for u = v̄ and u = v respectively.
GM IM O =
Gf uel
GTq
=
Gα−f uel
GN −f uel
Gwg−f uel
Gλ−f uel
Gα−Tq
GN −Tq
Gwg−Tq
Gλ−Tq
(10.15)
Poles have negative real part, so the system is always stable, while there are no
transmission zero.
Obviously it would be easier to design a controller on the SISO transfer
function Gwg , but on the other hand, the SISO transfer function gives a RHP zero, so it imposes some limitation on the achievable bandwidth. The multivariable analysis shows that the positive zero disappears when considering more
than one input, so the idea is now to try to get advantage from this information
manipulating the system configuration.
10.5 Input-output controllability
Input-output controllability of a plant is the ability to achieve acceptable control performance, that is, to keep the outputs within specified bounds or displacements from their references, in spite of unknown but bounded variations,
such as disturbances, using available inputs and available measurements[6].
Controllability is independent of the controller and is a property of the plant
alone, so the model G will be analyzed to find out what control performance
can be achieved.
The minimum singular value of the plant σ(G), evaluated as a function of
frequency, is a useful measure for evaluating the feasibility of achieving acceptable control. Inputs and outputs have been scaled according section 10.2,
then we can, with a manipulated input of unit magnitude, achieve an output
of magnitude of at least σ(G) in any output direction. We generally want σ(G)
as large as possible, and at least to avoid input saturation we want σ(G) larger
10.5. I NPUT- OUTPUT CONTROLLABILITY
177
than about 1 at all frequencies where control is required[6]. If σ(G) is below
1, it means that to move the output between -1 and 1, an input larger that 1 is
needed, but input only can be smaller or equal to 1, so the input would saturate
on its constraints.
Plotting the singular value diagram of the scaled transfer function G for
all the operating points, it comes out that in most of the cases it lies below
1. So a different approach is used, based on fixing the desired bandwidth for
the system and then finding the corresponding maximum allowed set point
change so that the minimum singular value lies above 1, for all the frequencies
before the desired bandwidth. The strategy will be explained in the following
sections, first for the MISO case and then for the MIMO problem.
10.5.1 MISO case
For each operating point we proceeded as follows:
1. Scale the system with respect to the input
Gisc = GDu
(10.16)
2. Fix a desired bandwidth
ω ∗ = 60rad/s ≈ 15Hz
3. Plot the singular value of (Gisc ) and find the value X of the singular
value at the frequency ω ∗ ;
4. Set Dy = X, in this way the singular value plot will be lifted by a factor
X and at the frequency ω ∗ it will be 0dB.
In this way we get all the allowed set point changes, given the desired bandwidth.
10.5.2 MIMO case
When two outputs are considered, then the SVD gives two singular values.
From the plot in figure (10.5) it can be seen that is not possible to control both
the outputs independently, since the minimum singular value is very small
and the plot lies below 0dB for all the frequency. Looking at the U vector, the
maximum gain direction of the system refers to the situation in which both the
178
C APITOLO 10. M ULTIVARIABLE S YSTEM A NALYSIS
Singular Values
40
30
Singular Values (dB)
20
Gisc(60rads)=X
10
0
60rad/s
−10
−20
−2
10
10
−1
10
0
1
10
Frequency (rad/sec)
10
2
10
3
10
4
Figura 10.3: (Gisc ): Singular value plot.
Dy
70
60
50
40
30
20
10
0
0
10
20
30
40
50
operating
point
Figura 10.4: Maximum allowed set point changes for each op.
10.5. I NPUT- OUTPUT CONTROLLABILITY
179
Singular Values
20
0
Singular Values (dB)
−20
−40
−60
−80
−100
−120
−140
−2
10
10
−1
10
0
1
10
Frequency (rad/sec)
10
2
10
3
10
4
Figura 10.5: Singular value plot. Op 12.
torque and the fuel increase. That is quite obvious, since it is not physically
possible to increase the torque while decreasing the fuel.
To find the largest allowed set point change, we need now to consider the
minimum singular value σ and require it to be 0dB at the desired bandwidth
ω ∗ . So the procedure is slightly different from the MISO case, since now we
need to consider the output directions as well. The procedure becomes:
1. scale the system with respect to the input
GM IM Oisc = GM IM O Du
(10.17)
2. fix a desired bandwidth (ω ∗ );
3. plot the singular values of (GM IM O )isc and find the value x for which the
minimum singular value at the frequency ω ∗ is 0dB;
4. pick the minimum input vector u corresponding to the minimum gain
direction and multiply u with x.
5. set
Dy =
dT q 0
0 df uel
=
x 0
0 x
u1
u2
=
xu1 0
0 xu2
(10.18)
180
C APITOLO 10. M ULTIVARIABLE S YSTEM A NALYSIS
Singular Values
40
20
60rad/s
Singular Values (dB)
0
−20
x
−40
−60
−80
−100
−120
−2
10
10
−1
10
0
1
10
Frequency (rad/sec)
10
2
10
3
10
4
Figura 10.6: Singular value plot for GM IM Oisc .
In this way we get all the allowed set point changes for both the torque and the
fuel, given the desired bandwidth.
Fuel
U
Torque
u1
U
u2
Figura 10.7: Output directions of the plant.
10.6 SVD Controller
We want to use the information got from the analysis of the multivariable system in order to design a simpler SISO controller instead od a multivariable controller. The idea is to look for a linear fixed combination of inputs and use it like
a fictitious input. In this way the real inputs to the system cannot be moved
independently, but only accordingly the linear combination proportions. A
10.6. SVD C ONTROLLER
1.2
181
−6
x 10 Maximum Torque variation in each Op. without hiting constraints
dTq
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
Figura 10.8: Maximum allowed torque variation for each op.
2.5
x 10
−3
Maximum fuel variation in each Op. without hiting constraints
dfuel
2
1.5
1
0.5
0
0
10
20
30
40
Figura 10.9: Maximum allowed fuel variation for each op.
50
182
C APITOLO 10. M ULTIVARIABLE S YSTEM A NALYSIS
natural choice for this combination is the one that maximizes the output of the
system. Looking back to the SVD, the scaled system at zero frequency can be
rewritten as Gsc = U ΣV . The maximum gain input direction is given by the
column of vector V corresponding to the maximum singular value (V̄ ). So we
can multiply V̄ for the original scaled system Gsc and get a 1 × 1 transfer function, from the fictitious input ū to the torque (see Fig.10.10). The relation needs
now to be rewritten in terms of the real system G, so we get:
S = Dy−1 GDu V̄
(10.19)
Poles and zero of the new transfer function S are all stable and the RHP zero
u
e
C
V
Dy-1 GMISO D
SISO
Figura 10.10: SVD controller
disappears. In this way we can still design a SISO controller but we can get
rid of the RHP zero limitations. The problem of this kind of controller is that
is difficult to handle constraints. The first problem is that, as soon as one input
hits the constraints, then there is no control on that input and so it is like having
only 1 input again and the RHP zero will be back. The other problem is that the
SVD controller uses a fixed combination of inputs, so when one saturates, also
the other one cannot be moved independently. For this reason the controller
design will not be based on this SVD controller, but the MPC will be used also
for the MIMO case, as explained in the next chapter.
10.7 Conclusions
This chapter presented the analysis of the control problem using two different
formulations:
• MISO control problem: two control inputs (throttle and wastegate) and
one output (torque).
• MIMO control problem: two control inputs (throttle and wastegate) and
two outputs (fuel consumption and torque).
At the end of the Chapter a control strategy based on singular value decomposition is described and discussed.
10.7. C ONCLUSIONS
183
The big advantage of a MISO-MIMO approach is that it gives the possibility to handle the RHP zero, present in the transfer function between WG and
torque, and this is a good property for control.
Capitolo 11
MIMO Control
Multi-variable control is investigated in this Chapter, as a potential means to
alleviate the trade off imposed by the SISO control structure. In particular
a multi-variable MPC is designed. The possibility for the controller to act
on both the wastegate and the throttle, allows for a better use of the wastegate, which could be also used to increase the torque. Due to strong emissions governmental limitations, also the fuel consumption is then considered
as output.
11.1 Introduction
Figure11.1 shows the control problem formulation for the MISO case: two inputs (throttle and WG) and one output (torque). The wastegate actuator can
only move between 0 and 1, where 0 means close and 1 means open. At steady
state the wastegate is zero, so it can not be closed any further. That means
that the controller action is limited and it can only help to reduce the torque.
Designing the MPC on the MISO system gives one more degree of freedom:
now the controller can act on both the wastegate and the throttle, and so can
decide for a better use of the wastegate, which could be also used to increase
the torque.
So far the attention was focused only on the torque, since the driver demands for good engine’s performances in terms of available torque. But also
emissions should be considered, since legislation imposes very strong limitations. So also fuel consumption is included as output, and the resulting control
problem, shown in Fig.11.2, is now a MIMO problem: two inputs (throttle and
wastegate) and two outputs (torque and fuel consumption). The following dis185
186
C APITOLO 11. MIMO C ONTROL
cussions are based on [5]. The throttle signal uth affects the intake manifold
u_wg
e
MISO
C
alpha
Tq
PLANT
N
ECM Lambda
T_amb,
p_amb
Figura 11.1: MISO control formulation.
uwg
MIMO
alpha
C
N
Tq
PLANT
Fuel_mass
ECM Lambda
Tamb,
p
amb
Figura 11.2: MIMO control formulation.
pressure pim , while the wastegate signal uwg affects the exhaust back pressure
pem and the boost pressure pcomp . When the system has to reach a certain state,
e.g., when the torque Tq and engine speed N are fixed, one degree of freedom
remains. This degree of freedom can either be used to keep the compressor
speed at its highest possible level, which provides a fast transient response, or
to lower the back pressure, which ensures good fuel economy. As shown in
[5], the control problem can be faced with two different approaches, that will
be described in section 11.2.
11.1.1 Best Fuel Consumption: formulation of the problem
As shown in Chapter 2, the engine torque is modeled on the basis of the work
produced, where net work Wn is determined from the gross indicated work
Wig , produced by the high-pressure part of the engine cycle, minus the sum of
pumping and friction work Wp + Wf during one cycle.
Tq =
Wig − Wp − Wf
Wn
=
4π
4π
(11.1)
11.1. I NTRODUCTION
187
The gross indicated work is
Wig = mf qHV ηig
where qHV is the lower heating value of the fuel and mf is the injected fuel
mass per engine cycle. The gross indicated efficiency ηig depends on the thermodynamic cycle and heat transfer. It is assumed to be independent of pim and
pem . The pumping work is
Wp = Vd (pem − pim )
(11.2)
and the friction work is
Wf = Vd F M EP (N, ṁf )
where the friction mean effective pressure FMEP depends mainly on the speed.
The slight dependence on engine load is captured by the variable ṁf . The
FMEP model used here is based on the ETH model (Inhelder, 1996; Stöckli,
1989). With this parametrization, the friction model as well is independent of
pim and pem .
The brake-specific fuel consumption BSFC is defined as the fuel mass flow
ṁf divided by the generated power P,[5].
BSF C =
ṁf
ṁf
=
P
Tq 2πN
(11.3)
where N is the engine speed in revolutions per second. One problem with the
definition of BSFC is that there is a singularity at zero torque. Therefore it is
advantageous to look at
1
BSF C
=
T q2πN
ṁf
which then has to be maximized for
best fuel efficiency. Optimizing the cruising scenario with constant speed for
the best fuel economy is thus the same as maximizing
Tq
ṁf
. For cruising we now
also consider the maximization under limited resources, that is a desired fuel
flow ṁf,des , which now becomes
max
T q(uth , uwg , ṁf )
subject to ṁf (uth , uwg ) = ṁf,des
(11.4)
A constant fuel flow corresponds to a constant air flow, since we are restricting engine operation to stoichiometric conditions. This leads to the following
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C APITOLO 11. MIMO C ONTROL
formulation of the problem
max
T q(uth , uwg , ṁa )
subject to ṁa (uth , uwg ) = ṁa,des
(11.5)
Studying the torque model under the constraints of constant speed and constant fuel mass flow and with the modeling assumptions, we see that the only term affected by the control inputs is the pumping work. Thus, maximizing 11.1 for constant fuel mass is identical to minimizing pumping work Wp .
Dividing Wp by the constant Vd yields the following function to be minimized:
U (pem , pim ) = pem − pim
In turbocharged engines it is possible to have a pim that is higher than pem ,
which means that the pumping actually produces energy. It is now of great
interest to see if the controller can take advantage of this effect to increase the
engine output torque and fuel economy.
By manipulating the expressions of the air mass flow and the volumetric
efficiency, the following function can be defined
f e(pim ) =
pem
= rc −
pim
ṁ
apim
γ
(11.6)
where a is a positive parameter. This function is always positive, and it is
monotonously increasing with pim , since
fe (pim ) = (rc −
where rc −
ṁ
apim
ṁ γ−1 γ ṁ
)
>0
apim
ap2im
(11.7)
> 0 follows from 11.6 and from the fact that the air mass flow
is positive. This relation says that pim monotonously increases with pem .
The function to be minimized, U (pim , pem ), can now be rewritten as a function with only one variable, V (pim ), using the definition of fe (pim )
V (pim ) = pem − pim = pi m(fe (pim ) − 1)
So the optimal solution is to minimize pim .
11.2. C ONTROL DESIGN
189
11.2 Control design
The fuel optimality statement says that it is always favorable to lower pim and
pem as much as possible. The strategy indicated by this statement is to open the
wastegate as much as possible. The controller to achieve this can be described
as follows.
11.2.1 Layout 1: Fuel Optimal Control
The analytical investigation showed that under the mentioned simplifications,
it is always advantageous to keep the pressures before and after the engine as
low as possible. This leads directly to the fuel-optimal controller that opens
the waste gate as much as possible without reducing the desired air mass flow.
Starting from low load the wastegate is fully open and the throttle is used to
control the load. Only after the throttle is fully open the waste gate starts to
close and controls the intake manifold pressures above ambient pressure.
11.2.2 Layout 2: Torque Optimal Control
The torque-optimal strategy tries to keep the speed of the turbo charger as high
as possible. In current series production cars it is implemented. The goal of this
strategy is to keep the turbocharger on the highest possible speed. This is due
to the fact that the rotational dynamics of the turbocharger are the limiting factor for the time response. This is achieved by closing the wastegate as much
as possible until an appropriate boost pressure after the compressor pcomp is
reached. This level lies above the maximum of pim , since the intercooler and
the open throttle act as flow restrictions. In this strategy the throttle is used for
load control and the wastegate is only used to limit the boost pressure.
11.3 Simulations with MIMO MPC
In Chapter 8, the MPC was designed to control the wastegate, based on the
SISO control problem formulation. Now the MPC is designed for the MIMO
formulation, and so there are two control inputs (α and uwg ) and two outputs
(torque and fuel consumption). Simulations results are shown afterward.
Let recall the hypothesis on which simulations are based:
• Tim is assumed to be constant
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C APITOLO 11. MIMO C ONTROL
• we focus on engine operations with λ=1
• a constant engine speed test case was used to isolate the results from
gearbox and vehicle influences
• a torque based control is designed, so the reference signal is given in
terms of torque
• uwg can only vary between 0 (fully closed) and 1 (fully open)
• the strongest constraints on the states are:
- Tem < 1300K
- ωt < 18000rad/s
11.3.1 Simulation on the Linear Model
We start now to analyze the linear case, in order to better understand the system behavior and interpret the results of the controller. The system is strongly
nonlinear, so is very difficult to design a suitable controller directly, skipping
the analysis of the linearized system.
Comparing Layout 1 and layout 2
The aim is to design one MPC and try to modify it, so that it could switch between the ideal controllers described by layout 1 and layout 2 in the previous
section. In order to do this, one option is to tune the weights on the output
variables (QT q and Qf uel ) so that it is possible to switch between the two layouts, and see if WG and throttle act as expected. For the Fuel Optimal Control
the fuel consumption is weighted much more than torque, and viceversa for
the Torque Optimal Control (Fig.11.3 and 11.4 show a comparison between the
two layouts).
Another possibility is to use in the MPC Toolbox to set a pa-
rameter (Fast-Robust Control Slider) (see Ch.8) which tunes the weights based
on requirements of robustness or fast response. It means that the weights were
initially set to MV=[0 0], MVrate=[1 1] and Output=[0 100], i.e. Torque Optimal Control, since QT q > Qf uel . Moving the slider in the direction of higher
robustness, we expect the weight on fuel to increase and consequently a slower
response in torque, i.e. Fuel Optimal Control. So:
• with QT q < Qf uel , the WG is more open during the transient and in
steady state too, and also the throttle is more open, so the fuel is lower,
both in transient and steady state, but the torque response is slower.
11.3. S IMULATIONS WITH MIMO MPC
191
WG
alpha
Plant Inputs
0
0.2
0.4
0.6
0.8
1
Time (sec)
1.2
1.4
1.6
1.8
2
Figura 11.3: Inputs. Up: throttle. Down: WG. Blue line: Torque Optimal
Control (QT q > Qf uel ). Green line: Fuel optimal Control (QT q < Qf uel ).
torque
mfuel
Plant Outputs
0
0.2
0.4
0.6
0.8
1
Time (sec)
1.2
1.4
1.6
1.8
2
Figura 11.4: Outputs. Up:Fuel consumption. Down: Torque. Blue line: Torque
Optimal Control (QT q > Qf uel ). Green line: Fuel optimal Control (QT q <
Qf uel ).
192
C APITOLO 11. MIMO C ONTROL
• with QT q > Qf uel the WG is kept more closed, thus causing an higher
fuel but a faster torque response.
Simulation showed that results are the same that we expected, but obviously
the results in term of response time and values of torque, throttle and WG, have
to be interpreted considering that the system is linearized.
11.3.2 Considering Pumping Losses
Let see now what happens to the pumping losses: another simulation is performed, similarly to the previous one, adding the pumping losses to the outputs of the system, together with torque and fuel. Operating point 27 is considered and a step of 50Nm is given to the torque reference, starting from the
steady state value. To be able to see the pumping losses in the graphic interface
of the MPC Toolbox, we add them as output and we do not weight them, so
that they will not be considered in the optimization problem. Then we have
now three possible configurations:
1. fast response
2. fuel economy
3. trade off
as can be seen from Fig.11.5- 11.7.
Analytical Explanation
Now we try to understand what came out from the simulations from the analytical point of view, and to explain it looking at the pressures involved in the
system.
We require constant torque at steady state. The net torque can be written
as:
Tnet = Tgrass − Tf ric − Tpump
(11.8)
where Tpump = f (pim ). At the same time, if lambda is fixed to 1, then Tgrass ≈
ṁair ≈ ṁf uel . So if we reduce the pumping losses, then the requested ṁair
decreases thus decreasing the requested grass torque too; consequently the fuel
decreases, but the net torque is the same at steady state (see Fig.11.8). All this
implies that the WG has to be kept open as much as possible, so that pim is
minimized. The requested air mass flow will be satisfied by the throttle, which
will be more opened.
11.3. S IMULATIONS WITH MIMO MPC
193
Plant Inputs
300
250
alpha
200
150
100
fast response
trade off
50
fuel reduction
0
1
WG
0.5
0
−0.5
0.9
1
1.1
1.2
Time (sec)
1.3
1.4
1.5
Figura 11.5: Adding pumping losses. Inputs. Operating point 27 is considered
and a step of 50Nm is given to the torque reference. Blue line: fast response;
green line: trade off; red line: fuel economy.
Plant Outputs
0.045
mfuel
0.04
0.035
0.03
0.025
140
130
torque
120
110
100
90
80
Time (sec)
Figura 11.6: Fuel consumption and torque. Blue line: fast response; green line:
trade off; red line: fuel economy.
194
C APITOLO 11. MIMO C ONTROL
Plant Outputs
9
8
7
pumping
6
5
4
3
2
0.9
1
1.1
Time1.2
(sec)
1.3
1.4
1.5
Figura 11.7: Pumping losses. Blue line: fast response; green line: trade off; red
line: fuel economy.
7SXPS
7RWDOWRUTXH
0BDLUBIORZ
0BIXHOBIORZ
Figura 11.8: If the pumping torque is reduced by opening the WG, then the
requested grass torque is reduced too. Consequently the requested ṁair decreases thus decreasing the requested fuel. The net torque at steady state will
be the same and the requested air mass flow will be satisfied by opening more
the throttle.
11.3. S IMULATIONS WITH MIMO MPC
195
In terms of pressures we can write:
ṁf uel = k1 pim − k2
Tq = k3 pim − k4 pem − k5
(11.9)
Wpump = Vd (pem − pim ) = f (pim )
where k1 , . . . , k5 are positive constants.
11.3.3 Simulation on the Nonlinear Model
We look now at the nonlinear system. It has a lot of non linearities, like saturations, look up tables, switches, . . . , and so we expect that using only one MPC,
based on just one linear system, it will not move so far from the operating point
it is designed for. We give a step to the reference torque, from 100 to 150Nm
and tune the MPC in order to simulate the same two layouts described previously. It is also necessary to tune the Overall Gain Slider Control in the MPC
Toolbox, depending on the operating condition range. There are two possibilities for that, tuning the parameter Set Estimator when defining the MPCobject,
or directly in the user interface.
Also for the nonlinear system happens what we expected, i.e. the WG has
different behaviors depending if the goal is to maximize torque response or fuel economy. Figures 11.9-11.11 show how the system response changes when
changing the weights on the outputs. If QT q Qf uel the torque response is
faster and the MPC keeps the WG closed all the time (Time Optimal control).
The drawback is that the fuel consumption is higher as well as the pumping
losses. Instead, if QT q Qf uel , the WG is fully open almost all the time and
the throttle is used to control the load and bring back the torque to the reference. The torque response is then slower, but the advantage is in term of lower
pumping losses and reduced fuel consumption, both during the transient and
in steady state (Fuel Optimal control).
11.3.4 Comparison of the two controller versions
The engine with the fuel-optimal controller has a longer response time since it
starts with a lower speed of the turbocharger, but it has better fuel economy at
cruising speed. A solution would be to connect the choice of the control strategy to an ”Eco Button”, with which the driver could choose between the good
fuel economy and the fast transient response. As seen from the simulations,
the MPC can handle this trade-off in a good way.
196
C APITOLO 11. MIMO C ONTROL
102
Pamb
Throttle
500
0
2
4
6
8
100
98
10
0
−2
2
4
6
8
20
Area WG
N
6
8
10
4
6
8
10
4
6
8
10
38
36
10
2002
2000
1998
4
40
Tamb
Lambda
2
2
2
4
6
8
10
2
x 10
−5
10
0
2
Figura 11.9: Simulation on Non Linear model.Inputs. Dashed-dotted line: if
QT q Qf uel the torque response is faster and the MPC keeps the WG closed
all the time (Time Optimal control). Full line: if QT q Qf uel the WG is fully
open almost all the time; the torque is slower but fuel consumption is reduced.
160
Torque
140
120
100
80
2
3
4
5
6
7
8
9
10
pumping torque
6
fast respons e
fuel optimizatio n
5
4
3
2
1
2
3
4
5
6
7
8
9
10
Figura 11.10: Simulation on Non Linear model.Torque. Dashed-dotted line:
fast response, full line: fuel economy.
11.3. S IMULATIONS WITH MIMO MPC
197
air mass flow
0.04
0.035
0.03
0.025
0.02
2
x 10
2.5
fuel kg/s
3
4
5
6
7
8
9
10
−3
2
fast respons e
fuel optimizatio n
1.5
2
3
4
5
6
7
8
9
10
Figura 11.11: Simulation on Non Linear model.Fuel. Dashed-dotted line: fast
response, full line: fuel economy.
x 10
5
320
T before thro t
P before thro t
1.3
1.2
1.1
1
11
2
x 10
4
6
8
10
4
9
8
7
2
4
6
8
10
316
314
312
310
2
4
fast respons e
fuel optimizatio n
10000
turbine speed
P intake M
10
318
6
8
10
6
8
10
8000
6000
4000
2
4
Figura 11.12: Simulation on Non Linear model. States. Dashed-dotted line:
fast response, full line: fuel economy. The constraint on the turbine speed
(ωt < 18000rad/s) is not violated.
198
C APITOLO 11. MIMO C ONTROL
5
1060
T exhaust M
1.2
1.15
1.1
1.05
1.04
P exhaustsyste m
x 10
2
x 10
4
6
8
5
1.035
1.03
1.025
1.02
2
4
6
8
1040
1020
1000
980
10
2
4
fast respons e
fuel optimizatio n
1040
T exhaustsyste m
P exhaust M
1.25
10
6
8
10
6
8
10
1020
1000
980
2
4
Figura 11.13: Simulation on Non Linear model. States. Dashed-dotted line:
fast response, full line: fuel economy. The constraint on the exhaust manifold
temperature (Tem < 1300K) is not violated.
11.4 MIMO MPC on the Reduced Order Model
In Chapter 9 a reduced order model was derived, in order to lighten the computational weight of the control problem, especially related to MPC in real
time. So now the simulations are performed using the MPC designed on the
reduced order model (M P Creduced ), and it is shown that results obtained with
the original system are very close to the ones obtained with the reduced order model, thus confirming that a ”smaller” model can thoroughly replace the
original one.
Simulation on the Nonlinear Model
Now the MPC designed on the original model (12 stats) and on the reduced
order model (6 and 4 states) is used. For simplicity, let call them M P Coriginal ,
M P Creduced6 and M P Creduced4 respectively. Operating point 15 is considered.
The torque reference is a step, from 100Nm to 150Nm and step disturbances
are added to the torque to test rejection. It can be seen that the torque correctly
goes back to the reference and that the original MPC gives the same responses
than the reduced version, both for torque and fuel (see Fig.11.14-11.15).
11.4. MIMO MPC ON THE R EDUCED O RDER M ODEL
102
Pamb
Throttle
200
100
0
2
4
6
8
100
98
10
0
2
4
6
8
10
4
6
8
10
6
8
10
6
8
10
20
19 s
4 state
2
4
6 state s −4
x 10
12 state
4 s
Area WG
N
1
0
−1
2
21
Tamb
Lambda
2
−2
199
2
4
6
8
2
0
10
2
4
Figura 11.14: Reduced order model.
Disturbance rejection.
Inputs.
M P Creduced4 and M P Creduced6 react to disturbances in the same way than
M P Coriginal , when a disturbance is added to the throttle signal.
Torque
140
120
80
3
fuel kg/s
4 state s
6 state s
12 state s
100
2
x 10
3
4
5
6
7
8
9
10
3
4
5
6
7
8
9
10
−3
2.5
2
1.5
2
Figura 11.15: Reduced order model. Disturbance rejection. System responses for the MPC designed considering 12, 6 or 4 states are compared. The
torque correctly goes back to the reference and the original MPC gives the same
responses than the reduced version, both for torque and fuel.
200
C APITOLO 11. MIMO C ONTROL
11.5 MIMO MPC for α, WG and λ
In Chapter 7 was showed that is possible to control λ in a more efficient way
and improve the torque response during the transient. So here the MPC design
is augmented, in order to to control also λ, together with α and WG.
The transfer function from λ to torque (Gλ ) has a positive zero and so gives
rise to an inverse response, if used as a single input. However, combining λ
with α and WG, and controlling them independently, the RHP zero disappears.
Due to emissions constraints Lambda can not be smaller than 1 in steady
state, so the idea is to decrease lambda, i.e. maximize its efficiency, only during
the transient. The linearized model does not include the lambda efficiency but,
it still matches the nonlinear system in a good way. Looking at Gλ , can be seen
that if λ increases, the torque increases too. Look now at Lambda efficiency
(see Fig.11.16). If we consider only λ ∈ [0.9, 1], we get the same behavior than
1.4
1.2
Efficiency
1
0.8
0.6
0.4
0.2
0
0.8
1
1.2
Lambda
1.4
1.6
1.8
2
Figura 11.16: Lambda efficiency.
the transfer function Gλ with respect to the torque. So we can require the MPC
to respect the constraint 0.9 ≥ λ ≤ 1.
The last problem to solve is to implement the MPC so that λ is used only
during the transient. To do this we use the parameter target in the MPC Toolbox, and we write:
MV(Lambda).TARGET=1
which means that Lambda will be equal to 1 in steady state.
Figures 11.17 and 11.18 show the inputs and the outputs, respectively, for
the operating point 15. It can be seen that, obviously, the solution which mini-
11.6. N UMERICAL ERRORS
201
mizes the fuel keeps λ = 1 during the transient, since λ < 1 would mean more
fuel.
Plant Inputs
300
alpha
250
200
150
100
1
WG
0.5
0
−0.5
−1
1
lambda
0.95
0.9
0.85
0.8
0.9
0.95
1
1.05
1.1
Time (sec)
1.15
1.2
1.25
1.3
Figura 11.17: MIMO MPC for α, WG and λ. Inputs. Blue line: fast response;
green line: trade off; red line: fuel economy.
11.6 Numerical errors
Many times during simulations we noticed strange behavior of the MPC Toolbox. In 70% of the cases the behavior has been correct, but anyway it could
be interesting to go through a small analysis of the possible causes of these
problems.
Some common examples of strange behavior of the MPC toolbox could be
described as follows:
• The control input produced from MPC changes suddenly during steady
state, even if no other input in the system is changed, and even if it is
controlling the same linear system that has been loaded as plant of the
MPC.
• Opening and closing the MPC Toolbox interface in Simulink, the MPC
Toolbox gives different simulation results.
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C APITOLO 11. MIMO C ONTROL
Plant Outputs
0.04
mfuel
0.038
0.036
0.034
0.032
200
torque
190
180
170
160
0.9
0.95
1
1.05
1.1
Time (sec)
1.15
1.2
1.25
1.3
Figura 11.18: MIMO MPC for α, WG and λ. Inputs. Outputs. Up: fuel mass
flow. Down: torque. Blue line: fast response; green line: trade off; red line: fuel
economy.
The plant is strongly nonlinear (many look up tables, switches, saturations,
hysteresis) so we cannot expect that a controller based on a linear system works
perfectly. But anyway the two examples of strange behavior, mentioned above,
seem not to be related to nonlinearities, but more to some numerical problem.
The condition number of a matrix measures the sensitivity of the solution
of a linear equations system to errors in the data [6]. It gives a clue of the accuracy for matrix inversion and linear equation solution. A condition number
close to 1 indicates a well-conditioned matrix [23]. So the condition number
of the system matrix A has been calculated, in order to check the possibility of
numerical problems during our simulations.
First of all, since the MPC Toolbox uses a discrete system for the plant, the
original system has been discretized with the same sampling time that MPC
Toolbox use, i.e. 0.0125 sec. The resulting condition number is:
Cond1 = 3.6042e8
This number seems to be high and could be a possible reason for the strange
behavior of the Toolbox. So we balanced the system to get a better conditioned
A matrix. To do this we use the function ssbal in Matlab [23].
11.7. C ONCLUSIONS
203
Doing this we get:
Cond2 = 1.5219e6
The number is still high. The idea is then to use the reduced model, derived
in Chapter9 and see if something better can be obtained. The reduced system
with 4 states is discretized first, and then the condition number is calculated:
Cond3 = 17.3016
The number now is much better, so we can try to see what happens using this
model inside the MPC Toolbox (see Fig.11.19-11.20)
1
Pamb
Throttle
1
0
−1
0
5
−1
1012 states model
0
4 states model
1
Tamb
Lambda
1
0
−1
0
5
10
0
5
10
0
5
10
0.4
Area WG
N
5
0
−1
10
1
0
−1
0
0
5
10
0.2
0
−0.2
Figura 11.19: The same inputs are given to the 12 states model (thin line) and
to the 4 state model (thick line).
11.7 Conclusions
A multivariable approach has been investigated to check if any benefit can be
obtained by controlling more than one input of the system at the same time,
i.e. throttle, WG and λ.
An MPC controller has been used and many simulations on both linear and
nonlinear system have been performed. The use of more inputs gives many advantages in the terms of fuel consumption and fast torque response trade off.
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C APITOLO 11. MIMO C ONTROL
torque respons e
137.5
reference
4 state model
12 state model
137
Torque
136.5
136
135.5
135
134.5
0
2
4
6
8
10
Figura 11.20: Torque. Using the 12 states model (thin line), the MPC has
a strange behavior at the beginning of the simulation. These problems are
eliminated if using the 4 states model (thick line).
11.7. C ONCLUSIONS
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In fact using the MPC is possible to simulate different scenarios, getting a faster
torque response but using more fuel, or trying to minimize the fuel consumption and getting a slower torque response. So, setting different parameters, is
possible to have different car behaviors with one controller. Opening or closing the WG in a certain percentage, is possible to reduce the pumping looses
and so get the same net torque with less air mass flow, which means, keeping
lambda equal to 1, less fuel consumption.
A qualitative investigation on the linear system, using also the air to fuel
ratio (λ) as manipulated variable, has been performed. Simulations showed
that the use of λ as third controlled variable can give a benefit in terms of torque
response during the transient. The drawback is that MPC is really sensitive to
the strong nonlinearities present in the system, so is very difficult to tune it.
Capitolo 12
Conclusion
In this thesis the possibility of a new and more advanced strategy for the
wastegate control in a turbocharged engine was studied. The concept of turbocharging provides the advantage of better fuel efficiency, but entails some
new challenges as well.
Today the wastegate is actuated by means of a simple pressure valve which
can be opened only when a certain pressure level over it is reached and is controller by a PI plus a feedforward action tuned on the engine. Here is presented
the possibility of integrating an electronic boost controller, i.e. substituting the
pressure valve used today with an electronic valve, which can be opened and
closed whenever is needed and independent from the particular engine. The
idea is to investigate which kind of benefits can be obtained with this solution.
The main characteristics of a turbocharged engine were investigated and
analyzed from a control point of view, in order to be able to understand the
problem and so design the requested controllers.
A nonlinear Mean Value Engine Model (MVEM) has been first studied and
then used for simulations. The original model has been enriched with new
features. Different volumetric efficiency models have been tested, both in the
control unit and in the engine model. The one tuned directly on the car resulted
to be the most suitable for modeling and control design. Spark advance and
lambda efficiencies were added to the model, as well as exhaust temperature
correction depending on spark advance, in order to result more realistic. In this
way also the torque behavior better fits the real response.
To check that the new modified model still represented the real system, a
validation has been performed, comparing data recorded from the car and data
got from the Simulink model, giving the same inputs. Simulations showed that
the model has a good level of accuracy.
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C APITOLO 12. C ONCLUSION
The nonlinear model is described by 45 different linear systems, each one
with 6 inputs (throttle, engine speed, wastegate, lambda, ambient pressure,
ambient temperature), 2 outputs (torque and fuel mass flow) and 13 states
(pressures, temperatures and turbine speed). The assumption of constant intake manifold was made, in order to express the torque and the fuel consumption as a linear function of the states. Ambient pressure and temperature are
considered as disturbances. Since the controllers will be based on the linearized
model, is very important that it captures the main dynamics of the system. Simulations showed that results change for the chosen operating point. Generally,
due to the strong nonlinearities of the system (look up tables, switches, saturations, . . . ), the linearized model is consistent with the nonlinear one just very
close to the point it was linearized on.
A mathematical analysis of the linearized model in all the 45 operating
points was carried out, in order to point out the main characteristics and fundamental limitations of the system, from the control point of view. All the 45
linearized systems are full controllable and observable, have stable poles, and
are nonminimum phase. The main limitation for the wastegate control strategy is represented by the right half plane zero in the transfer function between
wastegate and torque. Consequently the bandwidth is limited between 1-5Hz,
meanwhile the main system dynamics, represented by the intake manifold system, is around 15Hz. This led to investigate a MIMO approach to overcome
this problem.
Control design was performed on a SAAB 9-3 L850 2.0 turbocharged engine. The question is how to open and close the wastegate in a way to maximize the torque response during the transient: an optimal control problem has
thus to be solved. By designing choice a torque based strategy is used.
First, Linear Quadratic (LQ) techniques, including integral action of the error between the desired and measured torque, anti wind-up and feedforward
gain, have been applied, with full state feedback architecture, since all the state
were available for measurements. Simulations were performed first on the linearized system. In 90% of the cases the LQ controller kept the wastegate closed
during the transient. Only in simulation of low loads (for example operating
point 12) the WG is opened for a short while at the beginning of the transient.
At low load the pumping losses are higher, and can be significantly reduced
opening the WG. At the same time, the pressures in the system are not so high,
especially the intake manifold pressure, and so the drop is not as big as in high
load conditions. The torque, compared with keeping the WG closed all the
time, increases at the beginning, when the WG is opened, but then it drops
209
down when the WG is closed again. It is possible to conclude that, depending
on the difference between the torque gained at the beginning of the transient
and lost at the end, could be worth or not opening the WG.
Reference tracking and disturbance rejection were simulated and the torque
correctly followed the set point, with different rising time depending on the
the selected operating point. The same simulations were performed also on
the nonlinear system, showing that a gain scheduling strategy was necessary
to cover all the system dynamics (0-300Nm). First, fixed engine speed was considered, using 7 LQ controllers, one for each operating point and a scheduler
based on the intake manifold pressure. Then a more realistic scenario, with
variable engine speed, was simulated, using different sets of LQ controllers
for each engine speed. In both cases, the scheduler correctly selected the LQ
controller and simulations showed better results than using just one controller.
Further investigations showed that is possible to use only 3 LQ controllers for
each engine speed, without performance deterioration.
Generally the LQ approach to improve the transient torque response showed
just small benefits at low load conditions in compare with PI control strategy.
Bigger advantages are obtained in terms of independence of the controller from
the specific engine used, and disturbance rejection.
An independent controller for lambda was also designed to maximize the
lambda efficiency during the transient. It showed improvements of 1-3% in the
transient torque response, without big increase of the fuel consumption.
Since the system operates near a constraint boundary, constraints are an
inescapable part of the problem formulation and a more general control design,
including the constraints from the beginning is suitable. So another solution
based on Model Predictive Control (MPC) has been investigated.
Simulation of the SIMO scenario, considering torque and fuel consumption as output, showed that is possible to tune the controller in different ways
in order to get different transient behaviors of the WG and consequently of
the system response. It is possible then to have faster torque or less fuel consumption. The MPC shows fast rejection to both measured and unmeasured
disturbances. For the MIMO case, the MPC managed in a good way the trade
off between fast response and fuel economy. The different is that now, using
throttle and WG as controlled inputs, is possible to reduce the fuel consumption both in transient and steady state. Keeping the WG more open is possible
to generally reduce the pumping losses, and so have less air mass flow into the
engine to reach the same net torque. Less air mass flow requested means, considering lambda equal to 1, less fuel consumption. Simulations showed that is
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C APITOLO 12. C ONCLUSION
very hard to tune the MPC Matlab Toolbox on the nonlinear model, probably
due to the strong nonlinearities present in the system (lookup tables, switches,
saturations, . . . ).
The MPC was extended to consider also lambda as manipulated variable
and to include constraints on the states. The simulations performed on the
linearized system showed that, when the target is fuel economy, lambda is
correctly kept equal to 1, and throttle and WG are more open.
Different model reduction techniques over the linearized system were used,
like Hankel norm approximation, balanced residualization, balanced truncation and BST. Considering to reduce all the 45 linearized systems to the same
number of states, a model with 4 states was derived, without significant performance deterioration, achieving generally a maximum relative error of 5%.
MPC has been tested also on the reduced order model, comparing results
achieved when using the plant with 12, 6 or 4 states. Responses are pretty
much the same.
The MIMO approach showed benefits in compare with using independent
controllers designed on the SISO system, as implemented now in the car control unit. Of course this leads to a more complex design, but the MPC approach
seems to be a suitable solution. The MPC Toolbox results not to be enough reliable for future industrial application, so a solution could be to use a different
implementation for the MPC. The Simulink model used is very accurate but
perhaps too complex for control design. The idea could be to use a ”smaller”
nonlinear system, able to capture the main dynamics of the system, to simplify
then the controller development and afterwards test it on the detailed original
model.
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