houtzager_thesis_v3.
Towards Data-Driven Control
for Modern Wind Turbines
Ivo Houtzager
TOWARDS DATA-DRIVEN CONTROL
FOR MODERN WIND TURBINES
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op
vrijdag 2 december 2011 om 10:00 uur
door
Ivo HOUTZAGER
werktuigkundig ingenieur
geboren te Krimpen aan den IJssel
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. ir. M. Verhaegen
Copromotor:
Dr. ir. J. W. van Wingerden
Samenstelling promotiecommisie:
Rector Magnificus,
Prof. dr. ir. M. Verhaegen,
Dr. ir. J. W. van Wingerden,
Prof. dr. L. Y. Pao,
Prof. dr. ir. M. Steinbuch,
Prof. dr. ir. J. Hellendoorn,
Prof. dr. ir. G. A. M. van Kuik,
Prof. dr. M. Lovera,
voorzitter
Technische Universiteit Delft, promotor
Technische Universiteit Delft, copromotor
University of Colorado
Technische Universiteit Eindhoven
Technische Universiteit Delft
Technische Universiteit Delft
Politecnico di Milano
This dissertation has been completed in partial fulfilment of the requirements of the
Dutch Institute of Systems and Control (DISC) for the graduate studies.
The work presented in this thesis has been partly supported by the Dutch Technology
Foundation (STW) under project number: TMR5636.
The work presented in this thesis has been partly supported by the [email protected] consortium under project number: 2007-014.
ISBN: 978-90-5335-490-2
Copyright © 2011 by I. Houtzager.
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including
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permission from the copyright owner.
Printed by Ridderprint in The Netherlands
Acknowledgements
I
the last four years, my family and friends frequently comment me about what an
extraordinary job I do by myself. They assume that the person writing this thesis
is doing all the work on his own, when in reality various people contributed to the
research in different ways. Doing something as complicated as a Ph.D. research is not
something you do by yourself. Doing something as complicated as a Ph.D. research is
not something you do by yourself, I know.
N
First, I would like to thank all my colleagues within the DCSC department. Michel,
I am grateful you gave me the opportunity of achieving this and improving my
writing and presentation skills. Jan-Willem, I really appreciated your tireless support
and guidance and it was pleasure working with you. Your enthusiasm, constructive
criticism, and willingness to make time for me whenever I needed, made me confident
in doing my research. Gijs, Justin, Jianfei, Xiukun, Rufus, Balasz, Navin, Pieter, and
Patricio, your comments, suggestions, and feedback have been very valuable and I
enjoyed working with you. Other colleagues I would particularly like to mention are
Paolo, Andrea, Jacopo, Pawel, Ali, Amol, Stefan, Aleksandar, Ruxandra, and Federico,
with whom I spent enjoyable times both inside and outside the office.
Furthermore, I would like to thank all my colleagues within the DUWIND organization. Gijs, Wim, Teun, Thanasis, and Carlos, I really appreciate the fruitful
discussions about wind turbines and your support during the experiments. I would
like to thank Stoyan from ECN for providing me with the simulation data for the
identification experiments. I am grateful to the members of my Ph.D. committee for
providing me with constructive remarks, which helped me to improve this thesis. I
would like to thank the members of the STW user committee for their input during
project meetings and for hosting some of these meetings. I would also like to thank
those who did not help me directly with this work, but contributed to make these
years funnier and less boring. Sebastiaan, Barend, Bas, Edwin, Rianne, Erik and
other KrRB lifeguards, with whom I always enjoyed swimming on Thursday evening.
Charles and Alida, thank you very much for providing excellent beers afterwards.
Last, but certainly not least, I would like to thank my parents Bart and Henny
and my brother Leon and his partner Sjoerd for their support, encouragement, and
understanding.
Ivo Houtzager
Rijswijk, November 2011
vii
viii
Contents
Acknowledgements
vii
1 Introduction
1
1.1 Introduction to modern wind turbines . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Phenomenon of the wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.2
The history of wind energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.3
State-of-the-art in wind turbine designs . . . . . . . . . . . . . . . . . . .
4
1.1.4
Trends in wind turbine designs . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2 Introduction to modern wind turbine control . . . . . . . . . . . . . . . . . . . .
7
1.2.1
State-of-the-art control for power maximization . . . . . . . . . . . . .
7
1.2.2
State-of-the-art control for load minimization . . . . . . . . . . . . . .
11
1.2.3
Developments in active load reduction . . . . . . . . . . . . . . . . . . . .
12
1.3 Introduction to data-driven modelling and control . . . . . . . . . . . . . . . .
15
1.3.1
First-principles modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.3.2
Data-driven modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.3.3
Data-driven control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.4 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.5 Main contributions and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2 Predictor-Based Subspace Identification for the Model-Based Control of Wind
Turbine Systems
25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2 System description, assumptions and notations . . . . . . . . . . . . . . . . . .
28
2.2.1
System description and assumptions . . . . . . . . . . . . . . . . . . . . .
28
2.2.2
Innovation form descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.3
Lifted system description and notations . . . . . . . . . . . . . . . . . . .
29
2.3 Batchwise identification with VARX-based PBSIDopt . . . . . . . . . . . . . . .
32
ix
x
Contents
2.3.1
VARX predictor with finite past window . . . . . . . . . . . . . . . . . . .
32
2.3.2
The relation between the VARX predictor and the state . . . . . . . .
33
2.3.3
The VARX-based PBSIDopt algorithm . . . . . . . . . . . . . . . . . . . . .
34
2.4 Batchwise identification with VARMAX-based PBSIDopt . . . . . . . . . . . . .
36
2.4.1
VARMAX predictor with finite past window . . . . . . . . . . . . . . . . .
36
2.4.2
The relation between the VARMAX predictor and the state . . . . .
37
2.4.3
The VARMAX-based PBSIDopt algorithm . . . . . . . . . . . . . . . . . . .
38
2.5 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.5.1
Identification results with the SUSCON wind turbine model . . . .
40
2.5.2
Identification results with the TURBU wind turbine model . . . . .
43
2.5.3
Identification results with the PHATAS wind turbine model . . . . .
48
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3 Recursive Identification with Application to the Real-Time Closed-Loop Tracking of Flutter
53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.2 Recursive identification with RPBSIDpm . . . . . . . . . . . . . . . . . . . . . . . .
55
3.2.1
Step 1: Adaptive filtering of the Markov parameters . . . . . . . . . .
55
3.2.2
Step 2: The selection of the weight and state vector basis . . . . . .
56
3.2.3
Step 3: Adaptive filtering of the system matrices . . . . . . . . . . . . .
58
3.2.4
The RPBSIDpm algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.3 Practical implementation in real time . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.3.1
Regularization of the Markov estimation problem . . . . . . . . . . . .
60
3.3.2
Computational cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.4 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.4.1
Slowly time-varying case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.4.2
Abrupt-change case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.5 Experimental study on 2D-airfoil system . . . . . . . . . . . . . . . . . . . . . . . .
66
3.5.1
Description of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.5.2
Experimental modelling and control . . . . . . . . . . . . . . . . . . . . . .
69
3.5.3
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
xi
Contents
4 Rejection of Periodic Wind Disturbances on an Experimental Smart Rotor
Test Section
79
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.2 System description, assumptions and notations . . . . . . . . . . . . . . . . . .
81
4.2.1
System description and assumptions . . . . . . . . . . . . . . . . . . . . .
81
4.2.2
Lifted system description and assumptions . . . . . . . . . . . . . . . .
82
4.3 Repetitive control with lifted LQG design . . . . . . . . . . . . . . . . . . . . . . .
83
4.3.1
Output-feedback formulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.3.2
Multiple memory loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
4.3.3
The lifted LQG solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.4 Experimental study on “smart” rotor section . . . . . . . . . . . . . . . . . . . . .
89
4.4.1
Description of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . .
89
4.4.2
Experimental modelling and control . . . . . . . . . . . . . . . . . . . . . .
90
4.4.3
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5 Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances 103
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Description of the simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2.1
Specifications of the reference wind turbine
. . . . . . . . . . . . . . . 104
5.2.2
Simulation environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.3
Baseline control for above-rated operation . . . . . . . . . . . . . . . . . 106
5.2.4
Individual pitch control for load reduction . . . . . . . . . . . . . . . . . 108
5.2.5
Wind conditions during the simulations . . . . . . . . . . . . . . . . . . . 112
5.3 Lifted Repetitive Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3.1
System descriptions and assumptions . . . . . . . . . . . . . . . . . . . . 114
5.3.2
Lifted repetitive control by LQG design . . . . . . . . . . . . . . . . . . . . 116
5.3.3
Model linearisation for the lifted RC design . . . . . . . . . . . . . . . . 119
5.3.4
Extensions to RC for uncertain period time . . . . . . . . . . . . . . . . . 120
5.3.5
Guidelines for selection of the parameters . . . . . . . . . . . . . . . . . 122
5.4 Simulation results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.4.1
Performance of individual pitch controller . . . . . . . . . . . . . . . . . 123
5.4.2
Performance of lifted repetitive controller . . . . . . . . . . . . . . . . . . 125
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 Conclusions and Recommendations
133
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
xii
Contents
A On the Consistency of VARX Model Estimation in Closed Loop
139
A.1 Consistency when the true system is the VARX model . . . . . . . . . . . . . . 139
A.2 Consistency when the true system is the innovation state-space model . 140
A.2.1 Controller description and assumptions . . . . . . . . . . . . . . . . . . . 140
A.2.2 The relation between the VARX and Markov parameters . . . . . . . 141
A.2.3 The asymptotic behaviour for a finite VARX predictor . . . . . . . . . 141
B Spectral Analysis for the Verification of Models in Closed Loop
143
B.1 Transfer function description and assumptions . . . . . . . . . . . . . . . . . . . 143
B.2 Consistency with conventional spectral analysis . . . . . . . . . . . . . . . . . . 144
B.3 Spectral analysis for closed-loop estimation . . . . . . . . . . . . . . . . . . . . . 144
C Sequentially Semi-Separable Matrix Descriptions and Computations
145
C.1 Sequentially semi-separable matrix descriptions . . . . . . . . . . . . . . . . . . 145
C.2 Sequentially semi-separable matrix computations . . . . . . . . . . . . . . . . . 147
C.3 SSS matrix construction for efficient repetitive control synthesis . . . . . . 152
Bibliography
153
List of Abbreviations
169
Summary
171
Samenvatting
173
Curriculum Vitae
175
CHAPTER
1
Introduction
To reduce the cost per kWh, the wind turbine manufactures want to increase the energy yield of wind farms as much as possible. One of the
current developments is to increase the rotor diameter to maximize the
energy yield, because of the quadratic relationship between rated power and
rotor diameter, and also because of the increased wind resources at higher
altitudes. Consequently, modern wind turbines have become the largest
rotating machines on earth, with a rotor blade larger than the wingspan of
an Airbus A380. Due to the increase of rotor size, and also due to fewer
restricting planning constraints, wind farms are more often being relocated
offshore where maintenance is difficult and expensive. Therefore, the need of
an active reduction of loads on wind turbines is becoming more important.
This thesis focuses on the development of data-driven modelling and modelbased control algorithms for the load reduction on modern wind turbines.
To provide some additional background information, and to better motivate
the research, this introductory chapter will start with a brief overview of the
basic principles and trends in the field of wind energy. After this overview,
the state-of-the-art control in modern wind turbines and the developments
for load reduction are reviewed. Further, data-driven modelling and control
is introduced. The chapter finishes with the problem formulation, the main
contributions, and a brief outline of the remaining chapters of this thesis.
1.1
B
Introduction to modern wind turbines
addressing the more technical control and modelling aspects of wind
turbine technology, an attempt is made to give a short general introduction to
modern wind turbines. This involves a very brief explanation of the phenomenon
of wind, an historical part explaining the development of wind energy, the state-ofthe-art in wind turbine designs, as well as a part dealing with trends in wind turbine
designs. Since wind energy is still a rapidly developing area, the overview will be
EFORE
1
2
Chapter 1: Introduction
far from exhaustive. It is also by no means the intention to give a full historical
review of wind turbines, but merely to mention some major milestones in their
development and to give examples of the historical exploitation of wind energy. For
more background information the reader is referred to the books Burton et al. (2011);
Manwell et al. (2010).
1.1.1
Phenomenon of the wind
The dictionary describes wind as “an airflow in the atmosphere” (Van Dale, 2009).
Around Earth there is an amount of air located in a shell, which covers the whole
planet: the atmosphere. The layer rests on the surface of the Earth, because it is
subjected to gravity. The air pressure is the weight of an air column, which reaches
from the surface of Earth to the border of the atmosphere. On a specific time there are
areas with a low and high pressure. The sun warms a specific region of the earth, this
warm air has a lower density then cold air, so air will move and pressure differences
arrive. As soon as there is a difference in pressure between two locations, nature tries
to restore the balance. Air flows from high pressure areas to the low pressure areas.
This movement is called wind. The wind speed will be faster if the pressure difference
over a certain distance is larger.
The force from pressure difference applied on an air particle is not the only one;
also the friction between the air current and the surface, the shape of pressure areas,
and the Earth rotation gives forces that influences the displacement of air. All of these
influences in horizontal wind speed can be represented in a Van der Hoven spectrum.
In Figure 1.1, a typical Van der Hoven spectrum for the horizontal wind speed is given.
A high value indicates a significant change in wind speed over the corresponding time
period. Although this figure is obviously height and site-specific, there are distinctive
similarities. An overall phenomenon is the spectral gap between time periods of 10
minutes to 2 hours, that separates the long-term macro-meteorological effects, due
to the gradient between pressure systems and diurnal winds, with short-term micrometeorological effects, due to local wind gusts and turbulence.
In the lower layers of the atmosphere the wind speed is strongly influenced by
the friction with the surface of the Earth. The roughness of the landscape, like
buildings and trees, can reduce the wind speed and increase the turbulence intensity
considerably. The wind shear effect is a roughness-induced variation of the wind
speed at different heights above the surface, and represents for the effect of the wind
getting a lower velocity as we move closer to the surface. Turbulence is very fast
irregular wind flow, like whirls or vortexes, with different frequencies (to a maximum
period of 1 min), amplitudes, directions, and may intersect each other. Water surfaces
are considered very smooth, therefore wind speed at sea is faster and less turbulent
than at the same height on land (DWIO, 2009).
Wind turbines interact with these wind effects to capture its kinetic energy and
converting it into usable energy. For this reason, these wind effects play also an
important role in the control objectives for wind turbines. A clear distinction in
control objectives can made between the long-term wind effects to be tracked for
improving the power output and with the short-term turbulence to be rejected for
3
1.1 Introduction to modern wind turbines
reducing the fatigue loads. Control is successfully used in wind turbines to track the
long-term macro-meteorological wind effects and even the short-term wind gusts for
an improved power output. However, the irregular wind flows of turbulence decreases
the effectiveness of wind turbines, and imposes a lot of wear and tear on the wind
turbine. To reduce these fatigue loads, feedback control can be applied to damp the
natural frequencies of the wind turbine structure that are in the same frequency range
as the turbulence spectrum.
macro-meteorological
5
micro-meteorological
annual
peak
synoptic
peak
4
magnitude (m/s)
power control
bandwidth
gust
peak
3
diurnal
peak
2
turbulence
1
spectral
gap
1year
0
10−8
10−7
4days
10−6
1day
10−5
1min
1hour
10−4
10−3
frequency (Hz)
10−2
10s
10−1
100
101
Figure 1.1: Typical Van der Hoven spectrum for the horizontal wind speed. (Green
Rhino Energy, 2011; Kühn, 2001; Van der Hoven, 1957)
1.1.2
The history of wind energy
Wind energy was the most important source of energy in the Netherlands, before the
invention of the steam engine. During The Golden Age (17th century) of the Netherlands, wind energy played an important part in the economic growth. Windmills were
used to saw log timber, grind grain, and to pump water for irrigation or to prevent
the ocean from flooding the low-lying land. Especially, the introduction of the windpowered sawing of log timber into planks was important for the economic growth,
as it enabled to get thinner planks at a 30 times faster rate than before (Bonke et al.,
2004). This invention eventually led to the end of the wood-sawyers guilds1 , who were
manually sawing the log timber, but resulted in considerable more Dutch sail ships.
These ships were even lighter and faster than any of the competing countries, and this
property eventually led to the nickname: “flying Dutchmen” (Top, 2005).
1 Dutch: “houtzagers gilden”. In that time it was very common in Holland to take your profession as
surname, as probably did my name-related ancestors.
4
Chapter 1: Introduction
In the late nineteenth century the first wind turbines for generating electricity
were developed and constructed by C.F. Brush, which is one of the founders of today’s
General Electric. This wind turbine had a rotor diameter of 17 meters and 144 rotor
blades and had an electric output of only 12kW. It was the Dane P. la Cour, who later
discovered with experiments, that faster moving wind turbines with just a few rotor
blades are more efficient for electricity production than slow moving wind turbines.
Wind turbines for generating electricity need to operate at high speeds, but do not
require much torque or turning force. Between 1940-1950, the first “modern” twobladed and three-bladed wind turbines were developed in Denmark.
Nowadays, the development of wind energy seems to be unstoppable. Wind
energy was rediscovered after the oil crisis in 1973, and currently is a competitive form
of renewable energy. It is clean, economic, practical and has renewable interaction
with the environment and draws attention to politics, business circles and individuals.
It is becoming an aesthetically pleasing symbol of a better future, especially when
compared with the effects of acid rain, global climate change, radioactivity, land and
water contamination associated with conventional energy sources. However, wind
energy is currently still more expensive than the fossil fuel alternatives; like, natural
gas, oil, and coal. With the future improvements in wind power output, the rising cost
of fossil fuels, and that current rates of exploitation are expected to deplete within the
next centuries, it is only a question of when, not if, wind energy can compete against
the conventional energy sources.
1.1.3
State-of-the-art in wind turbine designs
The majority of modern wind turbines have three blades and have the rotor positioned upwind with a horizontal axis. This Horizontal-Axis Wind Turbine (HAWT)
design is also usually called the “Danish” concept. The main components of a HAWT
that are visible from surface are the tower, nacelle, and rotor (with hub and blades),
as shown in Figure 1.2. The rotor blades capture the kinetic energy of the wind and
transform it into torque. The nacelle houses the generator driven by the rotor, with
possibly a gearbox in between. A yaw mechanism keeps the rotor facing the wind.
This is called an upwind design and has the advantage that the wind shade of the
tower, also called the tower shadow, is smaller before the tower than behind. The
disadvantage of upwind turbines is that the rotor blades need to be made stiffer, and
placed at some distance from the tower to prevent the blades hitting the support
structure. They provide excellent methods for adjusting the rotor speed for optimal
power output at different wind speeds and minimizing the loads by regulating the
pitch of the rotor blades and the generator torque.
The advantages of a three-bladed rotor are that the rotor is better balanced and
has a greater aesthetic appeal. The disadvantages are that they cost more, weigh more,
and can be difficult to install. One-bladed and two-bladed wind turbines exist and are
cheaper and lighter, by saving the cost of one or two blades. They can rotate at higher
speeds, which reduces the cost of the gearbox, and they are easier to install. However,
these wind turbines can be noisier and endure heavy loads when the blades passes the
front of the tower. One of the current solutions to avoid these loads is to install a teeter
5
1.1 Introduction to modern wind turbines
mechanism, which is a special hub bearing that allows for the teeter movement of the
entire rotor (similar to the tilt movement when the rotor is in the upright position).
rotation
rotor blades
pitch
nacelle
tilt
hub
yaw
yaw mechanism
tower
Figure 1.2: Schematic representation of a modern offshore wind turbine
1.1.4
Trends in wind turbine designs
The power available in the wind Pw (t ) passing through an area A with a mean speed
V (t ) over the area can be given as:
1
Pw (t ) = ρAV (t )3 ,
2
(1.1)
where ρ is the density of air. A simple way of modelling the ability of a wind turbine
to capture wind energy is by defining a power coefficient as the ratio of extracted
power and wind power as: C p = P/Pw . With this definition, the power output P (t )
of a wind turbine can be given by the well-known expression (Burton et al., 2011; Pao
and Johnson, 2011);
1
(1.2)
P (t ) = ρπR 2C p V (t )3 ,
2
where R is the rotor radius. A doubling of the rotor diameter leads to a four-time
increase in power output. The influence of the wind speed is even greater. With a
double increase of wind speed leads to an eight-fold increase in power. So, the trend
with wind turbines is to increase the rotor diameter as much as possible and place the
wind farms in areas with the highest wind speeds; namely, offshore locations, as the
best onshore locations are already taken. The influence of the power coefficient C p is
6
Chapter 1: Introduction
limited by the maximum achievable value of C p,max ≈ 0.59, called the Betz limit (Betz,
1966). The power coefficient of modern wind turbines reaches values of about 0.45.
In Figure 1.3, the trend is illustrated with an 1.6 MW wind turbine with a rotor
diameter of 60 m realized in 1996 to a 6 MW ENERCON E-126 wind turbine with a
rotor diameter of 127 m realized in 2008. A 7 MW VESTAS V-164 wind turbine with a
rotor diameter of 164 m is announced to be realized in 2012. Thus it is clearly visible
that wind turbine manufactures increase the rotor diameter to maximize the energy
yield. Modern wind turbines have become the largest rotating machines on earth,
with a rotor blade larger than the wingspan of an Airbus A380 (79.8 m).
164m
127m
60m
33m
’86
0.03
’88
’90
’92
’94
’96
’10
0.3
0.5
1.3
1.6
2
’00
’02
’04
4.5
5
’06
’08
6
’10
’12
7MW
Figure 1.3: The rotor diameter and power output at the moment of market introduction (van Kuik, 2001; van Wingerden, 2008).
Offshore locations offer significant advantages. First, the mean wind speeds are
higher and there is less turbulence. Second, the expansion on offshore locations are
less restricted than on land, due to less planning and noise constraints and fewer land
use pressures. For these reasons, the turbine size can be increased more and larger
wind farms can be developed. Early experiences (since 2001) has demonstrated the
technical feasibility of large offshore wind farms, in spite of that wind power at sea
calls for maintenance that is different from those onshore, because the accessibility to
the turbines is more difficult. Therefore, the more demanding offshore environment
requires more reliable wind turbines.
One of the research trends in wind turbine designs is to equip wind turbines only
with two blades and position the rotor downwind instead (Fairley, 2009). As discussed
in Subsection 1.1.3, the two-bladed downwind designs have a number of significant
advantages compared with the current three-bladed upwind design. The use of only
two blades will give a lighter rotor that can operate at higher rotational speeds, is
cheaper to produce (minus one blade), and will be easier to install. Placing the rotor
downwind will eliminate the tower clearance issues, such that more flexible blades
and higher rotational speeds are possible. The success of this design will heavily
1.2 Introduction to modern wind turbine control
7
depend on active control techniques (including the work in this thesis) to compensate
the disadvantages; namely, the increased tower shadow, and the unbalance of the
rotor. A teeter mechanism can ensure that the overall load on the two blades remain
the same, since a difference in load will cause a change in teeter angle. Thus it ensures
that the shaft moment will be zero around the teeter axis. The load reducing effect
has previously shown to be approximately 20% related to the flapwise rotor blade
moment (Rasmussen and Kretz, 1992). However, the physical properties of the teeter
bearing become more complex when it is used for larger rotating systems due the
increasing gyroscopic effects and aerodynamic forces (Larsen et al., 2007).
1.2
Introduction to modern wind turbine control
This section provides a technical insight of the state-of-the-art control in modern
wind turbines and the future developments in active load reduction. The exact
control methods and strategies used in commercial wind turbines are protected by
the manufacturers. However, it is expected that the work of Bossanyi (2000, 2003a);
van der Hooft et al. (2003) forms the basis for the widespread control approaches in
the industry. For this reason, this outlined control procedure will form the baseline
for comparing the control strategies developed in this thesis. The control objectives
considered in wind turbines are twofold: power maximization and load minimization.
With the background information in Subsection 1.1.1, the control objectives could
be formulated as tracking the long-term wind effects for maximising the power output
and rejecting the short-term turbulence induced vibrations for minimizing the fatigue
loads, see Figure 1.1. From a practical point of view the above objectives are not
very convenient, as the wind speed measurements are generally not available and the
description of fatigue damage is highly non-linear and its prediction using rainflow
counting and the Miner’s rule gives a large uncertainty (Veldkamp, 2006). If the wind
speed is measured, for example by an anemometer on the nacelle, this measurement
is generally not useful for control. The measurement will be affected by the wind
turbine wake and turbulence, therefore it is not representative for the mean wind
speed over the rotor swept area. The non-linearity of the fatigue process makes it
difficult to formulate an appropriate cost to be minimized (Hammerum et al., 2007).
For both reasons, the objectives are pursued by other measures and the use of models.
1.2.1
State-of-the-art control for power maximization
Wind turbines with variable-speed operation are the most dominating type of newly
installed wind turbines in the last years. This is due to their very attractive features,
given by the presence of the power converter. When implementing a frequency
converter between the generator and the electrical grid, it is possible to decouple the
rotational speed from the grid frequency. Compared with fixed-speed turbines, the
variable-speed turbines have the advantage that at lower wind speeds, called below
rated, the rotor speed can vary to maintain an optimal power production. At low wind
speeds the reduced rotor speed also reduces the aerodynamically generated acoustic
noise and the stresses on the mechanical structure, since sudden wind gusts can be
absorbed by an increase in rotor speed.
8
Chapter 1: Introduction
The use of variable-speed wind turbines has increased the importance of power
control, also called torque control, depending if the reference input of the generator
is the power P (t ) or the generator torque Tge (t ), see also Figure 1.4. These variables
are related by the fundamental relation as:
P (t ) = Tge (t ) Ωge (t ) ,
(1.3)
where Ωge (t ) is the rotational generator speed. The basis for power control is the
wind turbine model in (1.2). This wind turbine model is simplified by introducing
a static power coefficient C p to represent the efficiency of the wind turbine under
certain operational conditions. For this purpose, the effects of the dynamics to the
efficiency are neglected. The strategy of the power controller is to achieve the optimal
power coefficient C p,opt by moving along the power coefficient surface C p (λ (t ) , θ (t ))
by adjusting the collective pitch angle θ (t ) and the tip speed ratio λ (t ). The tip speed
ratio is the ratio between the mean wind speed and rotor speed and is defined as:
λ (t ) =
Ωge (t ) R
ν V (t )
,
(1.4)
where ν is the gearbox ratio.
β1
ϕ
V1
Tro
θ1
J ro
Ωro
J ge
Tge
Ωge
M y ,1
M tilt
M yaw
x fa
x sw
θ2
θ3
Figure 1.4: Schematic representation of the variables in wind turbine control
In Figure 1.5, the power coefficient surface is given for the UPWIND 5MW (also
referred to as the NREL 5MW) wind turbine described in Jonkman et al. (2008). The
power coefficient surface C p (λ (t ) , θ (t )) can be calculated with the blade element
momentum method used in most wind turbine modelling software. If the pitch
angles are kept constant at fine pitch (in this case at θfine = 0 deg), the optimal
9
1.2 Introduction to modern wind turbine control
power coefficient C p,opt can be found at an optimal value of the tip speed ratio λopt ,
see Figure 1.5. Now, the optimal generator speed Ωge (t ) can be described with the
following equation:
ν λopt V (t )
Ωge (t ) =
.
(1.5)
R
The optimal static relationship between the varying wind speed V (t ) and the maximum power capture from the wind is already given by the model in (1.2). Rewriting (1.5) and substituting into (1.2) gives the optimal reference torque Tge,opt (t ) by
tracking the generator speed Ωge (t ) as:
Tge,opt (t ) = K λ Ωge (t )2 =
ρπR 5C p,opt
2ν 3 λopt 3
Ωge (t )2 .
(1.6)
The fine pitch is normally selected to be around 2–3 deg higher to compensate for
possible blade contamination. When the blades are covered with insects or ice, the
fine pitch can be lowered to obtain some additional “spare” power, such that the gain
K λ can be kept at its nominal value (Spruce, 2004).
power coefficient C p
0.5
C p,opt
0.4
0.3
0.2
0.1
0
20
20
15
15
10
tip speed ratio λ
λopt
10
5
5
0
0
pitch angle θ (deg)
Figure 1.5: Power coefficient surface for the UPWIND 5MW wind turbine.
The quadratic rule (1.6) with some additional filters gives smooth and stable
control in almost all variable-speed wind turbines (Johnson et al., 2006). However,
for heavy rotors the large inertia could prevent it from changing fast enough to follow
the wind optimally, because the dynamics of the rotor are neglected in the model. It
is possible to manipulate the generator torque to cause the rotor to accelerate faster
when required. In Bossanyi (2000), a feedback control method is described where
the required generator speed is estimated using aerodynamic torque reconstruction.
In Holley et al. (1999) it is shown that this would capture maximal 3% more energy
from fast wind gusts, but it demands huge torque variations, making it in most cases
impractical. Still an improvement can be expected by the identification of the power
coefficient surface, due to the sometimes large difference between model and reality.
10
Chapter 1: Introduction
The power control strategy is thus fundamentally based on the static optimal
curve in (1.6), and illustrated in Figure 1.6. This quadratic relationship is not used
in the whole below rated area. For very low wind speeds it is not worthwhile to
produce energy. With the UPWIND 5MW wind turbine, this is the case for wind
speeds lower than 4 m/s. For start-up, the generator speed in trajectory A–B is tracked
using a PI torque controller. Because of design restrictions, like tower clearance
and acoustic noise, an upwind wind turbine reaches the maximum rotor speed at
a relatively low wind speed. If the wind speed increases further, it is desirable to
increase the generator torque without any further speed increase, in order to capture
more energy from the wind. For this purpose, a PI torque controller is implemented
to track the generator speed in the trajectory C–D. Finally, in above-rated condition
D the generator torque is kept constant and the rated generator speed is tracked by
adjusting the pitch, for example when the wind speed is still increasing the pitch
angle is increased to lower the power coefficient to keep the rotor speed at rated.
In Figure 1.5, it is clearly visible that the power coefficient is lower for an increasing
collective pitch angle.
50
45
Tge,max
D
generator torque Tge (kNm)
40
35
C
30
25
20
15
H
F
10
G
B
5
0
E
A
0
50
100
Ωge,max
150
generator speed Ωge (rad/s)
Figure 1.6: Generator torque-speed curve for the UPWIND 5MW wind turbine.
In Bossanyi (2000, 2003a), also an improvement in load reduction is recommended
to the generator torque-speed trajectory A–B–C–D. The improvement is that the logic
describing the different sections of the generator torque-speed curve can also be easily
extended to implement so-called “speed exclusion zones”. These zones can be used to
avoid speeds at which blade passing frequency would excite the tower resonance, by
adding additional generator speed set-points with some logic for switching between
them, see the trajectory E–F–G–H in Figure 1.6.
To catch the maximum wind power, horizontal-axis upwind wind turbines have
to be directed towards the wind. This is accomplished actively with an electrical
1.2 Introduction to modern wind turbine control
11
or hydraulic servo system that rotates the nacelle with rotor on the yaw bearing,
see Figure 1.2. The wind turbine is said to have a yaw error, if the rotor is not
perpendicular to the wind. Almost all manufactures of upwind turbines do not control
their yaw error continuously; they prefer to brake the yaw mechanism whenever
it is unused. The servo is activated only when the mean relative wind direction
exceeds some predefined limits. A misaligned wind turbine would extract less energy
compared with a wind turbine which has no yaw error. More important is that a wind
turbine with yaw error is subjected to larger periodic loads than wind turbines which
are yawed in a perpendicular direction against the wind. Yaw control is separated
from other control, because of the many mechanical limitations. For example, the
limitation in yaw angle due to the entanglement of the power cables in the tower.
1.2.2
State-of-the-art control for load minimization
Until the advent of MW-scale wind turbines in the 90’s, stall regulation predominated
to keep the loads within acceptable limits, as all other control options were considered
too complex. Stall-regulated wind turbines have a rotor with blades that stall the flow
around the blade in high wind speeds. This means that increasing wind speeds automatically induce increasing drag forces that limit the captured energy, therefore these
wind turbines are stable by design. With new control technology, it became possible
to introduce active stall-regulated wind turbines that automatically adjust the pitch
to assist the stall at keeping the rotor speed constant when generating (occasionally
double speed, with a lower rotor speed for lighter winds). With the increasing size of
wind turbines, the stall-induced vibrations (vibrations which occur as the blade enters
stall) were becoming problematic, for example the negative aerodynamic damping in
deep stall causes the tower vibrations to rise to the shutdown level (Spruce and Turner,
2010).
Collective pitch regulation is now the favoured option for the largest modern wind
turbines. This considerable improvement allowed that blade designs are significantly
lighter, due to the lower load spectrum, and have more energy yield, as their design
is not any longer optimized for stall characteristics. This is also one of the reasons
that allowed modern wind turbines to run at variable rotational speed, but created
a dependence on an active collective pitch control system to keep the loads within
acceptable limits, therefore these modern wind turbines are not stable by design.
Currently, the number of control loops in modern wind turbines for load reduction
have increased considerably, see Figure 1.7. Most modern wind turbines have these
control loops decomposed with care into the following, separate, and almost uncoupled Single-Input and Single-Output (SISO) control loops (Bossanyi et al., 2009):
• Generator speed regulation using the generator torque
• Generator speed regulation using the collective pitch
• Torsional shaft damping using the generator torque
• Fore-aft tower damping using the collective pitch
• Sidewise tower damping using the generator torque
12
Chapter 1: Introduction
ẋ fa
fore-aft
tower control
Ωge,ref+
pitch control
+ + θ1,2,3
−
torque control
shaft
torsion control
+
Tge
+
wind turbine
Ωge
+
+
sidewards
tower control
ẋ sw
Figure 1.7: Block diagram of the feedback loops for wind turbine control
In Chapter 5, each controller of these control loops is described in more detail
for the UPWIND 5MW wind turbine. Many of these controllers can be designed
using the classical loop-shaping methods of controller tuning. With some help of
wind turbine modelling software, a linearised model of each of the SISO open-loop
dynamics is extracted. The loop is then closed by a linear control transfer function
with the measured signal as input and the control action as output. The classical
loop-shaping techniques are then used to tune the controller transfer function, such
that the closed-loop behaviour is sufficiently responsive and yet sufficiently stable, for
example by examination of the stability margins and the use of root-locus, Nyquist,
Bode and step response plots (Bossanyi et al., 2009).
1.2.3
Developments in active load reduction
With the increase in rotor size and the relocation offshore, where the environmental
conditions are heavier and maintenance is difficult and expensive, the importance of
active reduction of fatigue loads will increase. The purpose of such active control is
to increase the lifespan of the wind turbine, without making the construction heavier,
so that the wind turbine becomes more cost-effective. The collective pitch control
1.2 Introduction to modern wind turbine control
13
method assumes that wind gusts are active on the whole rotor area. Therefore, it
can only handle slow wind changes that have an effect on the entire rotor. As the
rotor size increases, the wind speed variation across the rotor swept area becomes
larger. For further load reduction it is necessary to react to wind speed variations
in a more detailed way; for example, each blade individually with Individual Pitch
Control (IPC) (Bossanyi, 2003b, 2005; Selvam et al., 2009; van Engelen, 2006), or even
at several radial distances spanwise along each blade with “Smart” Rotor Control
(SRC) (Andersen et al., 2010; Barlas and van Kuik, 2010; Buhl et al., 2005; Lackner and
van Kuik, 2010b).
The goal for both concepts is to increase the lifetime of the wind turbine. When
the lifetime constraint is met, the concept provides the designer with an increased
envelope which can be used to, for instance, mount the rotor downwind (for eliminating tower clearance issues), and allow lighter components or increase the mean
loading on the blades which increases the power conversion. However, the lifetime
of a wind turbine becomes also dependent on the wear and tear of the pitch/smart
actuators. Therefore it is required that the control variations on these actuators are
kept to a minimum. A preliminary study in Tavner et al. (2011) investigated the effects
of the stochastic wind fluctuations, i.e. turbulence, on pitch mechanism failures. It
was observed from statistical analysis that the cross-correlations between failures and
the turbulence intensity coefficients are much larger than to the mean wind speed
deviation coefficients. This could indicate that the current pitch control are tuned in
such a way that the pitch actuators are acting to heavily on the turbulence.
Individual pitch control
The idea for Individual Pitch Control (IPC) seems very attractive and almost readily
implementable, since many of the larger wind turbines already have individual pitch
actuators, although controlled collectively, for the regulation of the rotor speed.
However, the performance of the IPC method is restricted by the limited bandwidth
of the pitch actuator and they only affect the load on the whole blade (Lackner
and van Kuik, 2010a). For this IPC concept, researchers are mainly investigating the
use of feedback control after transforming (and decoupling) the rotating frame to a
fixed non-rotating frame (Bossanyi, 2003b, 2005; van Engelen, 2006). This so-called
Coleman transformation makes it possible to apply Single-Input and Single-Output
(SISO) loop-shaping control techniques to generate smooth near-sinusoidal control
signals to compensate for the 1P and 2P disturbance harmonics (Bossanyi, 2009).
For higher harmonics the use of loop-shaping techniques become more challenging, because more coupling exists between the transformed measurement channels.
Model-based control design techniques should be used to overcome this problem, as
proposed in Selvam et al. (2009). Also at these higher harmonics, a larger amount
of phase lag (delay) from the input-output dynamics have to be compensated. As
the generated control signals for the 1P and 2P harmonics are near sinusoidal, the
phase lag can be compensated during the calculation of the reverse of the Coleman
transformation. However, this becomes considerably more challenging at higher
harmonics. The Coleman transformation with loop-shaping has also successfully
been applied on an experimental two-bladed wind turbine in Bossanyi and Wright
14
Chapter 1: Introduction
(2009); Bossanyi et al. (2010), but the use of model-based control design methods
is challenging as the dependence of the dynamics on the azimuth is not necessarily
lowered.
“Smart” rotor control
The “Smart” Rotor Control (SRC) concept is borrowed from the helicopter industry
(Straub et al., 2009), where multiple actuators with a significantly higher bandwidth,
like trailing-edge flaps or translational/inflatable Micro-Electro-Mechanical (MEM)
tabs, are proposed to reduce the loads by manipulating the airflow locally and
consequently the aerodynamic forces. In Figure 1.8(b) an illustrative example is given
of a wind turbine with “smart” rotor blades. The rejection of disturbances using SRC
is already a research topic for several years. The feasibility of load reduction using
feedback control applied on an experimental “smart” airfoil with trailing-edge flaps
was demonstrated in Bak et al. (2007); van Wingerden et al. (2008).
(a) Translational/inflatable MEM tab
(b) Trailing-edge flaps
Figure 1.8: An illustrative example of a wind turbines with the “smart” rotor concept.
At the tip of the blade a number of additional control devices are placed; for example,
the MEM-tabs and trailing-edge flaps illustrated in (a) and (b), respectively.
With the increase of sensors and actuators on each blade, the loopshaping control
techniques with the Coleman decoupling are not directly applicable. Possible solu-
1.3 Introduction to data-driven modelling and control
15
tions are to use a different representation of the Coleman transformation together
with model-based control as proposed in Thomsen et al. (2010), or to use the
distributed feedback controllers proposed in Engels et al. (2010); Rice and Verhaegen
(2010a). The difficulty for the synthesis of these kinds of controllers is that a special
structured model is required, which is not always directly available. Also feedback
controllers will typically give a lot of control variations to the actuators. Therefore,
the wind turbine designers are reluctant to implement these new concepts directly
in practice. An alternative is to compensate only the periodic disturbances using
feedforward control, as successfully demonstrated on two rotating blades in van
Wingerden et al. (2010, 2011). In this method the amplitudes of 1P, 2P, 3P, and 4P
sinusoidal basis functions are adaptively fitted using the blade-root bending moment
measurements from the previous rotation, and the frequencies are obtained by
measuring the rotational speed. Furthermore, the method can easily be adapted to
be used in combination with feedback control for both IPC and SRC. In Chapter 5,
we will extend the development of this adaptive feedforward control by proposing
the use of repetitive control in the form of a lifted feedback controller, which takes
in addition the periodic time-varying MIMO dynamics and non-sinusoidal periodic
load disturbances into consideration.
1.3
Introduction to data-driven modelling and control
This section provides a brief explanation of the modelling methodologies used for
wind turbine control design and its verification. It is by no means the intention to
give a full detailed overview of all the modelling aspects, but merely to mention the
advantages, disadvantages, and challenges in the different modelling methodologies.
The use of models has a long tradition in the wind energy community. In the industry,
by far the most important purpose of the use of models is to lower the risk of
failures. In Failure Modes and Effects Analysis (FMEA) (IEC, 1985), the risk of failure
is defined as: risk = severity × occurrence × detectability. The severity of failures on
the availability of the wind turbine system is very high, because any failures can easily
cause the loss of its primary function during a long period and can even result in
unsafe operation. For this reason, manufactures are forced to keep the occurrence and
detectability numbers low (low detectability number means almost certain detection),
and models (and also control) play an important role to achieve this. This role will
increase as the risk of failure grows due the trend in rotor size and the relocation
offshore.
1.3.1
First-principles modelling
Currently, the laws of physics are used to describe a relevant number of properties of
the wind turbine in a mathematical language. These so-called first-principles models
are extensively used to determine the static and dynamic loads under extreme conditions, to predict the lifetime of components by calculating the damage equivalent
loads, to optimize the structural design, to tune and verify controllers, and condition
monitoring. In the wind energy community there is a wide variety of different
16
Chapter 1: Introduction
modelling codes. Most of these modelling codes are very complex and have been
especially developed to deal with design calculations and time-domain simulations.
For detailed information and a comparison the reader is referred to Molenaar (2003).
The main advantage of first-principles modelling is that you can have a model
before the actual wind turbine is built and consequently the model can be used for
its design. For the purpose of lifetime prediction, controller tuning and verification,
and condition monitoring, these first-principles models needs to be calibrated and
validated to the real system. One of the reasons is that the amount of detail in the
model is normally the choice of the designer, who normally tends to “overmodel” the
system to make sure to capture all the dynamics (van Wingerden, 2008). Further,
it is hard to have exact knowledge of the material and aerodynamic properties,
and small differences may produce significant different dynamics (Witteveen et al.,
2007). The calibration and validation of these models is very time consuming and
labour intensive process, therefore only a small part of the modelling codes are
verified and even a smaller part are validated (Molenaar, 2003). This means that
during the commissioning of the production wind turbine often requires a complete
retuning of controllers and models due to differences between the model and the true
system (van der Veen et al., 2011). A common approach for calibration is to do actual
measurements at a number of different operation positions and to compute the Power
Spectral Density (PSD) diagrams and the Campbell diagram to indicate the periodic
disturbances and time-varying dynamics (Rossetti et al., 2008).
The main disadvantage of first-principles models is that they are not tailored for
model-based control synthesis, because they typically contain irrelevant dynamics
and are highly non-linear. Although there are a number of dedicated design tools
that have the capability to derive models for model-based controller synthesis, it is
still far from trivial how to isolate properties belonging to the dynamics of the wind
turbine that are needed for model-based controller synthesis. For optimizing the
controller it is more interesting to directly model the dynamics between the actuators,
disturbances, and sensors because in these signals gain and phase information is
present, which is basically the information needed for controller synthesis (van
Wingerden, 2008). System identification can be used to create a more direct up-todate model than the existing one by using measurement data. System identification is
well established, but only a few applications are reported in the wind energy (Bongers
and van Baars, 1991; Font et al., 2010; Iribas, 2006; van Baars and Bongers, 1994),
mostly for the purpose of the validation of first-principles models.
1.3.2
Data-driven modelling
With the introduction of new rotor concepts with more sensors and actuators, modelbased controller design becomes more important (maybe even necessary) for the
wind energy community. Model-based controller synthesis is more suitable for
multivariable control problems, but its effectiveness largely depends on the derivation
of accurate and suitable models (preferably with an uncertainty description) and a
characterization of the noises and disturbances acting on the system. The academic
wind energy community has already a strong focus towards modern model-based
control, for example in Bianchi et al. (2007); Bongers (1994); Østergaard (2008);
17
1.3 Introduction to data-driven modelling and control
Steinbuch (1989); Stol (2001), but with current wind turbine designs the industry is
not necessitated to use these model-based methods in practice, as the loopshaping
methods suffice the requirements. Nevertheless, as wind turbines become larger and
more flexible, it is possible that model-based control methods, perhaps in conjunction
with these new rotor concepts, will increasingly find a way towards industry.
The trend is to use system identification not only as an update step to have
a more accurate first-principles model, but to use identification methods directly
in the (control) design of a wind turbine system. This so-called data-driven or
experimental modelling directly obtains an identified model of the true system and
avoids the non-trivial controller retuning, see Figure 1.9. Obviously, the quality of
the identified model depends on the information in the data. More “informative”
data can be generated by adding additional excitation to the inputs. If a more
accurate model is obtained around the bandwidth, a less conservative model-based
controller can be synthesized and implemented. To further improve the performance,
the steps in the design of the excitation signals and the model-based controller can
be repeated Gevers (2005); Hjalmarsson (2005). Although the number of significant
advantages, system identification can not totally replace first-principles modelling.
Apart that first-principles models are needed for the general design of the system
and probably the synthesis of an initial base-line controller, they will keep playing
an important role in the data-driven wind turbine modelling methodology to evaluate
the designed excitation signal for excessive loads and to verify the behaviour of any
synthesized model-based controller under extreme conditions.
excitation design
and verification
closed-loop
data acquisition data
model identification
and verification
output
input
time
model magnitude
reconfiguration
and validation
phase
frequency
controller embedding
and verification
model
controller synthesis
controller
Figure 1.9: Control design cycle with data-driven modelling.
Apart that the identification problem is normally multivariable, there are a number
of additional challenges with the identification of wind turbine systems. In the specific
case of wind turbine systems, the identification has to be performed with taking the
18
Chapter 1: Introduction
existing stabilizing controller into consideration. Most identification techniques for
multivariable systems are all based on the open-loop setting and will give biased
results in the closed-loop setting, see Chapter 2. The resulting identified models are
also only valid around one operating point. For a fixed operation point of a wind
turbine, the dynamics can be assumed to be linear time invariant, such that wellestablished identification methodologies can be used to obtain a model for control.
Identification methods for the estimation of time-varying dynamics are available
in the literature, for example the LPV identification methods in Tóth (2008); van
Wingerden and Verhaegen (2009); van Wingerden et al. (2009). Mainly due to the
“curse of dimensionality” they are not yet suited for realistic systems (van Wingerden,
2008). Further, the disturbances in the wind turbine are periodic in nature, therefore
ideas similar to those in van Baars et al. (1993) (periodic basis functions) and in Kanev
(2008) (Coleman transformation) needs to be applied for compensation of these
periodic disturbances. Contributions in the form of identification algorithms that can
deal with the above mentioned problems should be very general, therefore they could
also be used outside the field of wind energy. Possible applications are active magnetic
bearings (Balini, 2011; Blom, 2011), and adaptive optics (Hinnen, 2006; Song, 2011).
1.3.3
Data-driven control
The control system itself has the second largest annual failure rate for wind turbines
(number one is the electrical system), although the annual downtime for this component is relatively low compared with faults in mechanical components (Faulstich et al.,
2011; Ribrant and Bertling, 2007). One of the possible problems is the less straightforward integration with the supervisory control during changes in operational conditions and the safety system to reduce extreme actuator and structural loads. If
model-based controllers are going to be used, this integration with the supervisory
control system requires even further sophistication than with classical controllers,
where adjustments such as the addition of notch filters, saturation with anti-windup,
gain scheduling, and even startup/shutdown are more straightforward. A promising
methodology under research is the so-called data-driven control design (Dong, 2009).
In the data-driven control methodology, the intention is to use reliable data-driven
methods that can update the controllers and the monitoring filters with low and
intuitive supervision by detecting changes in the data. Basically, the data-driven
control approach is to let the wind turbine speak for itself (through data), perhaps
by asking first (through excitation signals), and adapt the control signals accordingly.
Similar to the control design based on data-driven modelling, the data-driven
control methodology also enables that the controller is designed using data from
the actual system to be controlled. In contrast, the data-driven controller constantly
optimizes the closed-loop performance criterion based on actual data from the
system, instead of on a fixed identified model of that system under nominal operating
conditions. The data-driven control design can evade the need to accurately model
the system and the need to be robust to all hypothetical disturbances, parasitic
dynamics and operating conditions (the worst-case scenario in Woodley (2001)), and
can consider only those that actually and most probably occur (the cautious scenario
in Dong and Verhaegen (2009)). Not only for higher performance, but also to prevent
1.4 Problem formulation
19
“impossible” choices for the controller, especially during the very uncertain transient
periods. By taking additional signal constraints in the control objective, it is possible
to account for input saturation, fatigue loads (in terms of signal variances), sensor
and actuator faults (if redundant), and the lack of information in the data through the
optimisation of the excitation signal (Dong, 2009; Hallouzi, 2008).
Besides the computational complexity of the algorithms involved, which may
require large online computations, it is difficult to allow industrial implementation
of such control systems. Most steps of the control design cycle depicted in Figure 1.9
will be automated in the data-driven control methodology, therefore it is very hard
to verify each step individually before online implementation. The acceptance of a
data-driven controller depends heavily on the global stability without cumbersome
assumptions on the system and disturbances. Although some data-driven control
design methods incorporate some sort of stability proof (Kulcsár and Verhaegen,
2011), they generally rely on system knowledge like the order of the plant, or they
require sufficiently exciting input signals of some order. Additionally, the closed-loop
is tuned to specific conditions and the effects of a change in these conditions on the
controller and its adaptation are unclear in general, therefore the tuning happens not
in a very insightful way. As the current theory and simulations are limited, research
experiments will play an important role to get more experience, thrust, and knowledge
of the adaptation and stability behaviour in reality. The way towards data-driven
control may lie in experiments with data-driven methods for condition monitoring
and fault detection of wind turbines (Wei and Verhaegen, 2008; Wei et al., 2009, 2010),
thus without closing the feedback loop to overcome the stability requirement.
1.4
Problem formulation
Further developments towards control techniques for the load reduction of modern
wind turbines can achieve an increased lifetime of components and make the scaling
to larger rotor diameters possible, and therefore improve the cost effectiveness of wind
turbines. Developments in active control technologies played an important role in
the history of load reduction for wind turbines, and this is probably also the case in
the future. For example, in Section 1.2.3 new rotor concepts were described for future
wind turbines. The success of these designs will heavily depend on new developments
in active control technology.
This thesis focuses on the development of data-driven modelling and modelbased control algorithms for the load reduction of modern wind turbines, where the
definition of modern wind turbines include the new developments in rotor concepts.
The focus will be on novel algorithms for the rejection of periodic disturbances. The
load disturbances seen by an individual rotor blade are to a large extent periodic and
repetitive; for instance, tower shadow, wind shear, yawed error, and gravity are depending on the azimuth angle and rotational speed, and will change slowly over time.
The contribution of these periodic disturbances will become even larger in the future,
due to the increasing length and mass of the rotor blades and the possible downwind
location of a two-bladed rotor. By compensating only the periodic disturbances, it is
expected that the required control action would not create too much fatigue damage
20
Chapter 1: Introduction
on the actuators, as they are much smoother than the turbulence induced vibrations.
To summarize, the main problem formulation is:
Problem Description 1.1 (Periodic disturbance rejection). Develop control algorithms,
suited for real-time implementation, for the rejection of periodic load disturbances in
modern wind turbines, and verify/validate the feasibility of these algorithms for this
purpose.
In pursuit of this goal, the novel control synthesis would require a mathematical
model description of the system to be controlled. As discussed in Section 1.3, the way
to obtain these models is from measured data using identification techniques. Being
that modern wind turbines are not asymptotically stable in open-loop (only closedloop experiments are supported in reality), that new rotor concepts would require an
increasing number of actuators and sensors, and that the operational conditions of
modern wind turbines slowly change over time, makes it necessary to formulate a
second problem as:
Problem Description 1.2 (Data-driven modelling). Develop identification algorithms
that do not require any controller related information to consistently identify a linear
model from systems with multi inputs and multi outputs, and are able to track the
dynamics under slowly changing operational conditions.
1.5
Main contributions and outline
This section provides first a brief overview of the outline and then the main contributions of this thesis. This thesis contains six chapters, where the first is this
introduction. Each of the subsequent chapters, except for Chapter 6, are basically extended versions of papers that have been published in or accepted to wellestablished journals and conferences in the field of wind energy, system identification,
and control. Chapters 2 and 3 are dedicated to the identification problem of modern
wind turbine systems. Chapters 4 and 5 are dedicated to the load reduction control
problem of modern wind turbine rotor concepts. Chapter 4 follows the data-driven
control design cycle using the identification methods in Chapter 2. The main contributions and the detailed content in these main chapters are given in the following
subsections. The summary of the chapter goals, the means of which algorithms and
tools they are achieved with, and the proof of concept of these algorithms and tools
are schematically given in Table 1.1. The main references where the content of each
chapter has been published are reported in a local bibliography as well. Chapter 6
concludes this dissertation, summarizes the main conclusions that can be drawn, and
will provide suggestions for future research.
Chapter 2: Predictor-Based Subspace Identification for the ModelBased Control of Wind Turbine Systems
In this chapter, the “optimized” Predictor-Based Subspace IDentification (PBSIDopt )
algorithm will be presented for closed-loop wind turbine model estimation (Houtzager et al., 2009, 2010). Further, a novel VARMAX-based PBSIDopt method is developed
Bibliography
21
that relaxes the requirement that a certain window has to be large for asymptotically
consistent estimates. The PBSIDopt methods are very attractive for the wind power
community, since it provides the user with the ability to identify LTI systems with
multiple inputs and multiple outputs, and does not require any controller related
information. The PBSIDopt methods are verified on a data source from controllerin-the-loop simulations of a modern wind turbine. The developed algorithms have
been implemented by the author in software, which are contributed in the PBSIDToolbox (Houtzager, 2010).
Bibliography
I. Houtzager. Predictor-Based Subspace Identification Toolbox version 0.4. Website,
2010. URL http://www.dcsc.tudelft.nl/~datadriven/pbsid/.
I. Houtzager, J. W. van Wingerden, and M. Verhaegen. VARMAX-based closed-loop
subspace model identification. In 48th IEEE Conference on Decision and Control,
Shanghai, China, 2009.
I. Houtzager, B. A. Kulscár, J. W. van Wingerden, and M. Verhaegen. System identification for the control of wind turbine systems. In 3rd Conference on the Science of
Making Torque from Wind, Heraklion, Crete, Greece, 2010.
Chapter 3: Recursive Identification with Application to the Real-Time
Closed-Loop Tracking of Flutter
In this chapter, a novel Recursive Predictor-Based Subspace IDentification (RPBSIDpm )
algorithm will be presented to identify LTI systems with multiple inputs and multiple
outputs (Houtzager et al., 2009). The algorithm is suited for real-time implementation
and will provide consistent estimates from data gathered in open loop or closed
loop. The real-time implementation and the ability to work with MIMO systems
operating in closed loop makes this approach suitable for online estimation of
unstable dynamics. The ability to do so is demonstrated by the detection of flutter
on a “smart” rotor test section (Houtzager et al., 2011). The developed algorithms
have been implemented by the author in software, which are contributed in the
PBSIDToolbox (Houtzager, 2010).
Bibliography
I. Houtzager. Predictor-Based Subspace Identification Toolbox version 0.4. Website,
2010. URL http://www.dcsc.tudelft.nl/~datadriven/pbsid/.
I. Houtzager, J. W. van Wingerden, and M. Verhaegen. Fast-array recursive closed-loop
subspace model identification. In 15th IFAC Symposium on System Identification,
Saint-Malo, France, 2009.
22
Chapter 1: Introduction
I. Houtzager, J. W. van Wingerden, and M. Verhaegen. Recursive predictor-based
subspace identification with application to the real-time closed-loop tracking of
flutter. Accepted for IEEE Transactions on Control System Technology, 2011.
Chapter 4: Rejection of Periodic Wind Disturbances on an Experimental Smart Rotor Test Section
In this chapter, an extended lifted Repetitive Control (RC) algorithm will be presented
for periodic wind disturbance rejection of linear systems with multiple inputs and
multiple outputs and with both repetitive and non-repetitive disturbance components. This extended RC algorithm can reject periodic wind disturbances in modern
wind turbines, and this is demonstrated on an experimental “smart” rotor test section (Houtzager et al., 2011a,b). The developed algorithms have been implemented by
the author in software, which are contributed in the LRCToolbox (Houtzager, 2011a,
LRCToolbox) and the SSSToolbox (Houtzager, 2011b).
Bibliography
I. Houtzager. Lifted Repetitive Control Toolbox version 0.2. Website, 2011a. URL
http://www.dcsc.tudelft.nl/~datadriven/lrc/.
I. Houtzager. Sequentially Semi-Separable Matrix Toolbox version 0.8. Website, 2011b.
URL http://www.dcsc.tudelft.nl/~datadriven/sss/.
I. Houtzager, J. W. van Wingerden, and M. Verhaegen. Rejection of periodic wind disturbances on a smart rotor test section using lifted repetitive control. Provisionally
accepted for IEEE Transactions on Control System Technology, 2011a.
I. Houtzager, J. W. van Wingerden, and M. Verhaegen. Rejection of periodic wind disturbances on an experimental “smart” rotor section using lifted repetitive control.
In IEEE Multi-Conference on Systems and Control, Denver, CO, USA, 2011b.
Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic
Load Disturbances
In this chapter, a novel lifted Repetitive Control (RC) algorithm that can reject
the periodic load disturbances for modern fixed-speed wind turbines and modern
variable-speed wind turbines operating above-rated equipped with individually pitch
controlled blades. For verification, the load reduction capabilities of RC compared
with typical IPC are investigated in the operation of the UPWIND 5MW wind turbine
in aerolastic modelling software (Houtzager et al., 2011). The developed algorithms
have been implemented by the author in software, which are contributed in the
LRCToolbox (Houtzager, 2011b) and the DISCON tool (Houtzager, 2011a).
Bibliography
23
Bibliography
I. Houtzager. External controller design tool for GH BLADED and SIMULINK. Website,
2011a. URL http://www.dcsc.tudelft.nl/~datadriven/discon/.
I. Houtzager. Lifted Repetitive Control Toolbox version 0.2. Website, 2011b. URL
http://www.dcsc.tudelft.nl/~datadriven/lrc/.
I. Houtzager, J. W. van Wingerden, and M. Verhaegen. Wind turbine load reduction
by rejecting the periodic load disturbances. Provisionally accepted for Wind Energy,
2011.
24
Table 1.1: Scheme of the outline of this thesis
papers
methods
software tools
proof of concept
Chapter 2
Problem 1.1: Datadriven modelling
Houtzager
et al. (2010);
Houtzager
et al. (2009b)
PBSIDopt
PBSIDToolbox
(Houtzager, 2010)
simulations with ECN’s SUSCON,
TURBU, and PHATAS modelling
software (and experiments with
“smart” rotor test section in
Chapter 3 and 4)
Chapter 3
Problem 1.1: Datadriven modelling
Houtzager
et al. (2009a);
Houtzager
et al. (2011a)
RPBSIDpm
(recursive version of
PBSIDopt )
PBSIDToolbox
Houtzager (2010)
experiments with “smart” rotor
test section
Chapter 4
Problem 1.2: Rejection of periodic disturbances
Houtzager
et al. (2011d);
Houtzager
et al. (2011b)
RC (using lifted
model descriptions)
PBSIDToolbox
(Houtzager,
2010); LRCToolbox
(Houtzager, 2011b);
SSSToolbox
(Houtzager, 2011c)
experiments with “smart” rotor
test section
Chapter 5
Problem 1.2: Rejection of periodic disturbances
Houtzager
et al. (2011c)
RC (with extension
to periodic LTV)
LRCToolbox
(Houtzager, 2011b);
DISCON
(Houtzager, 2011a)
simulations with GH BLADED
modelling software
Chapter 1: Introduction
problem
CHAPTER
2
Predictor-Based Subspace Identification
for the Model-Based Control of Wind
Turbine Systems
The chapter presents the optimized predictor-based subspace identification
method and shows the effectiveness for the estimation of wind turbine
models in closed loop. This identification technique does not require any
controller related information, consequently the identified model becomes
consistent no matter the wind turbine operates with or without controller in
the loop. Since the wind turbine is not asymptotically stable in open-loop,
only closed-loop experiments are supported in reality. This fact makes the
proposed method very attractive for the wind power community. Further,
a novel extension to the optimized predictor-based subspace identification
method is proposed that relaxes the requirement that the past window has
to be large for asymptotically consistent estimates. For this purpose, a finite
description of the input-output relation is formulated that can be solved
efficiently using the extended least squares recursion. The effectiveness and
robustness of the proposed methods are discussed in a simulation study,
where the algorithms are used on a data source based on controller-in-theloop simulations of a typical modern wind turbine.
2.1
F
Introduction
the wind energy community, model-based controller design becomes more
and more important. Model-based controller synthesis necessitates a nominal
description of the real plant. Nominal description of the plant can either be derived
from physical principles or using measured data, respectively. Therefore, the latter
is considered as a preliminary phase on the way towards a model-based controller
design. System identification of wind turbines is not only important at the design of
OR
25
26
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
a new wind turbine setup, but also when existing devices have to be re-identified in
order to create a more up-to-date and accurate model than the existing one. In the
specific case of wind turbine systems, the identification has to be performed in taking
consideration the existing stabilizing controller as well. Further, the disturbances in
the wind turbine are periodic in nature, therefore ideas similar to those in van Baars
et al. (1993) (periodic basis functions) and in Kanev (2008) (Coleman transformation)
needs to be applied for compensation.
Subspace IDentification (SID) methods are known to be computationally efficient
methods to identify Linear Time-Invariant (LTI) state-space models from Multi-Input
and Multi-Output (MIMO) measurements of a dynamic system. In Van Overschee
and De Moor (1996), an unified methodology is suggested in which many of the SID
methods fall, such as Canonical Variate Analysis (CVA) (Larimore, 1990), ORThogonal
decomposition (ORT) (Katayama, 2005), N4SID (Van Overschee and De Moor, 1996),
and MOESP (Verhaegen and Verdult, 2007) methods. These methods store input and
output data in structured block Hankel matrices, such that it is possible to retrieve
certain subspaces that are related to the system matrices. These methods are very
successful for offline batchwise identification, because the key linear algebra steps are
an RQ factorization, an SVD, and the solution of a linear least-squares problem, therefore the problem of forming a non-linear optimization is circumvented. Especially the
use of an RQ factorization resulted in computationally efficient implementations (Verhaegen and Verdult, 2007). The disadvantage with these traditional SID methods is
that they give biased results when the system to be modelled operates in closed loop,
because the future inputs are correlated with the noise, due to the feedback controller.
The traditional SID methods have been updated to give unbiased results in closedloop operation, but these methods can only be applied if the controllers and/or the
external signals are known in advance (Oku et al., 2006; Verhaegen, 1993), or by an
iterative procedure to the optimal instrumental variables (Gibson and Mercère, 2006).
The first alternative to SID are the Prediction Error Methods (PEM) (Ljung, 1999).
These methods use a pre-defined parametrization of the model, for example an Auto
Regressive with eXogenous inputs (ARX) model structure, where the parameters are
obtained by minimizing a quadratic cost function. PEM can provide asymptotical
consistent estimates in closed loop if the model structure and order fits the dynamics
of the true system (Van den Hof and Schrama, 1995), but the order is difficult to estimate in practice, the methods are computational expensive for MIMO systems, and
the model structures other than ARX require the forming of a non-linear optimization
problem. The second alternative to SID is the Observer/Kalman-filter IDentification
(OKID) method (Juang and Pappa, 1985; Phan et al., 1992). This method obtains first
Markov parameters, by estimating an Vector Auto-Regressive with eXogenous inputs
(VARX) model, which is in the case of SISO measurements similar to a high-order
ARX model. VARX models, with high order, can also provide asymptotically consistent
estimates even on closed-loop data if there is sufficient excitation from an external
signal or a controller of sufficiently high order, and that the system to be modelled or
the feedback controller does not contain direct feedthrough (Chiuso and Picci, 2005).
Then with the Markov parameters, a Hankel matrix is constructed. Finally, the system
matrices of the state-space model are directly estimated from the Hankel matrix by
using the Eigensystem Realization Algorithm (ERA) (also known as the Ho-Kalman
2.1 Introduction
27
decomposition). To obtain a good estimate of the system matrices, a large number of
Markov parameters are needed to construct a sufficiently large Hankel matrix, making
OKID normally less efficient and accurate than SID in open loop.
Recently, a number of significant advances have been presented to identify LTI
state-space models from measurements of dynamic systems operating in closed loop.
The SSNEW (Ljung and McKelvey, 1996; Qin and Ljung, 2003), SSARX (Jansson, 2003),
and Predictor-Based Subspace IDentification (PBSID) (Chiuso, 2007) methods are all
extensively utilizing the Vector Auto-Regressive with eXogenous inputs (VARX) model
parametrized by Markov parameters. The Markov parameters are used to construct a
Toeplitz matrix instead, where from multiplication with past input and output data,
and an SVD, an estimation of the state sequence can be obtained. With the state sequence, it is straightforward to recover the system matrices. The proposed algorithm
is based on the optimized version of the Predictor-Based Subspace IDentification
(PBSID) method, the so-called PBSIDopt method (Chiuso, 2007). All these methods
have a lot of common characteristics, and it is shown in Chiuso (2006, 2007); Chiuso
and Picci (2005) that the asymptotic variance of most these methods can be a unified
up to a particular weight. Furthermore, it was shown that the performance of PBSIDopt
compares favourably with PBSID and the other SID algorithms.
The main contributions of this chapter are twofold. The first contribution is the
implementation of the PBSIDopt to identify LTI models from data measured in open
or closed loop. A slightly modified description is used then in the original work
of Chiuso (2007), to improve the algorithm for practice. The second contribution
is a novel improvement on the first part of the PBSIDopt method; the estimation of
the Markov parameters. Inspired by Phan et al. (1995), an Vector Auto-Regressive
Moving Average with eXogenous inputs (VARMAX) model is utilized, which enables
consistent estimates of the model parameters of the finite-order input-output model.
This relaxes the asymptotic consistency results in the case of VARX models, resulting
in a very large past window in practice. A large past window has a number of
disadvantages. First is that the increasing number of parameters to be fit gives an
increase in the variance of the estimate. Second is that the matrix with stacked input
and output data becomes more ill conditioned, because normally in practice the input
signals do not persistently excite the system enough, which means that the estimate
becomes very sensitive to perturbations on the measurement data. Third is that the
memory usage increases considerably for increasing past window. The estimation of
a VARMAX model comes at the cost of a non-linear problem; however most of the
times it can be estimated efficiently. Instead of the residual whitening operations in
Phan et al. (1995), we propose the use of the Extended Least Squares (ELS) recursion
described in Ljung (1977, 1999) to solve the estimation problem.
The outline of this chapter is as follows. In Section 2.2 the problem formulation, the assumptions made, and some notations are presented. In Section 2.3
the theoretical framework is presented for the VARX-based predictor-based subspace
identification problem and a batchwise algorithm for LTI systems is presented. In
Section 2.4, the theoretical framework is presented for the VARMAX-based predictorbased subspace identification problem and a batchwise algorithm for LTI systems is
proposed. In Section 2.5, the effectiveness of the methods are shown by simulation
studies with modern wind turbine models. Finally, the conclusions are presented.
28
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
2.2
System description, assumptions and notations
In this section the system description, the assumptions made, and some notations are
presented.
2.2.1
System description and assumptions
The dynamics of the system S to be modelled can be written as the following statespace model:
¨
x k +1 = Ax k + Bu k + w k ,
(2.1)
S
y k = C x k + Du k + v k ,
where x k ∈ Rn , u k ∈ Rr , y k ∈ Rℓ , w k ∈ Rn , and v k ∈ Rℓ , are the state, input, output,
process noise, and measurement noise vectors. The state-space matrices A ∈ Rn ×n ,
B ∈ Rn ×r , C ∈ Rℓ×n , and D ∈ Rℓ×r are also called the state, input, output, and direct
feedthrough matrices, respectively.
We consider the case that the system S to be modelled operates in closed loop,
see Figure 2.1. In Assumption 2.1, we adopt mild conditions commonly used for the
direct closed-loop state-space identification problem (Aling, 1990).
Assumption 2.1.
• The process noise w k and measurement noise v k are zero-mean white or coloured
Gaussian sequences.
• The external excitation rk is a zero-mean sequence uncorrelated with the process
noise w k and measurement noise v k , and is sufficiently persistently exciting (Ljung,
1999).
• The feedback loop contains at least one sample delay, i.e., the system S or the
controller C has no direct feedthrough.
rk
−
6
C
+ uk
+ ?
-
wk
?
S
vk
+
+ ?
-
yk
-
Figure 2.1: Block diagram of a system operating in closed loop
2.2.2
Innovation form descriptions
Using Kalman filter theory, we can reformulate (2.1) to the innovation form (Ljung,
1999; Verhaegen and Verdult, 2007):
¨
x̂ k +1 = A x̂ k + Bu k + K e k ,
(2.2)
SI
y k = C x̂ k + Du k + e k ,
2.2 System description, assumptions and notations
29
where x̂ k ∈ Rn denotes the predicted state vector with E x k − x̂ k = 0 and E x k −
T x̂ k x k − x̂ k
minimal, and e k ∈ Rℓ denotes the white innovation sequence with
T
E e k e k = R, and K ∈ Rn ×ℓ denotes the Kalman gain matrix. We can also rewrite the
system SI in (2.2) in the so-called predictor form as:
¨
SP
x̂ k +1
yk
= Ã x̂ k + B̃ u k + K y k ,
= C x̂ k + Du k + e k ,
(2.3)
with the matrix à = A − K C asymptotically stable, and the matrix B̃ = B − K D. Similar
as in Phan et al. (1995), we can also introduce another deadbeat observer matrix
M that will create additional freedom for the optimizer later on. By considering a
deadbeat gain matrix M , which is expected to exist as only an observable part of the
system can be identified from the measurement data, we can rewrite (2.2) in the socalled deadbeat predictor form as:
¨
SD
= Ā x̂ k + B̄ u k + M y k + K̄ e k ,
= C x̂ k + Du k + e k ,
x̂ k +1
yk
(2.4)
with the matrix Ā = A − M C asymptotically stable and nilpotent, i.e. Ā j = 0 for j ≥ n,
the matrix B̄ = B − M D, and the matrix K̄ = K − M .
The retrieval of the system matrices (including the Kalman gain) is the goal of
the identification procedure. We can only estimate the observable and controllable
part (to both the deterministic and stochastic inputs) of the system S , therefore we
assume that this state-space description is a minimal realization. It is noted that for
coloured noise sequences this includes the noise filter dynamics, and therefore the
state vector can be larger than the order of the deterministic part only. Further, it
is well-known that an invertible linear transformation of the state vector does not
change the input-output behaviour of a state-space system. Therefore, we can only
determine the system matrices up to an unknown similarity transformation T ∈ Rn ×n :
T −1 AT , T −1 B , T −1 K , and C T .
2.2.3
Lifted system description and notations
We define a past window denoted by p ∈ N+ and a future window denoted by f ∈ N+ ,
where n/ℓ ≤ f . These windows are used to define the following stacked vectors:



yk
y k −p




 y k +1 
y k −p +1 



.
=  .  , ȳ k ,f =  . 
. 
.
 . 
 . 
y k + f −1
y k −1

ȳ k −p,p
The stacked vectors ū k −p,p , ū k ,f , ē k −p,p , and ē k ,f are defined in a similar way. For the
batchwise case, i.e., the case when a batch of N data is available, we can also define
the stacked matrix Y :
”
—
Y = y p , · · · , y N −1 .
(2.5)
30
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
The stacked matrices U , X are defined in a similar way. Further, we can define the
stacked matrix Ȳp :
—
”
(2.6)
Ȳp = ȳ 0,p , · · · , ȳ N −p,p .
Again, we can obtain in a similar way the stacked vectors Ūp .
Using the definition of the stacked vectors, we can “lift” the system in (2.2) to the
following lifted description:
¨
= A p x̂ k −p + Lū k −p,p + Kē k −p,p ,
= Γx̂ k + G ū k ,f + Hē k ,f ,
x̂ k
ȳ k ,f
(2.7)
where L ∈ Rn ×p r and K ∈ Rn ×p ℓ are the extended controllability matrices, and are
given by:
”
—
”
—
L = A p −1 B · · · A B B , K = A p −1 K · · · AK K ,
and G ∈ R f ℓ× f r and H ∈ R f ℓ× f ℓ are the impulse matrices with a lower block triangular
structure, Γ ∈ R f ℓ×n is the extended observability matrix, and are given by:


D
C




 CA 
 CB

Γ=
 ..  , G = 
..

.


.

C A f −1
C A f −2 B


I


 CK
H=
..

.

C A f −2 K
0
D
..
.
C A f −3 B
0
I
..
.
C A f −3 K
···
..
.
..
.
···
···
..
.
..
.
···

0
.. 

.
,

0
D

0
.. 

.
.

0
I
Note that I is used to represent an identity matrix. Similarly, the system in (2.3) can
be “lifted” to the following description:
¨
x̂ k
ȳ k ,f
= Ã p x̂ k −p + L̃ū k −p,p
€ + K̃
Š ȳ k −p,p ,
= Γ̃x̂ k + G̃ ū k ,f + I − H̃ ȳ k ,f + ē k ,f ,
(2.8)
where the lifted matrices L̃ ∈ Rn ×p r , K̃ ∈ Rn ×p ℓ , Γ̃ ∈ R f ℓ×n , G̃ ∈ R f ℓ× f r , and H̃ ∈ R f ℓ× f ℓ
˜ For the matrices L̃, K̃, Γ̃, and G̃ the
are defined in an almost similar way without (·).
only difference is that the generator matrices A and B are replaced with à and B̃ . The
matrix H̃ is defined as:


I
0
··· 0

.
..

. .. 
I
 −C K

H̃ = 
(2.9)
.
..
..
..


.
. 0
.

−C Ã f −2 K −C Ã f −3 K · · · I
More interesting is the relation between the innovation and predictor form given in
Lemma 2.1.
31
2.2 System description, assumptions and notations
Lemma 2.1. It can be shown that H̃ is invertible, and the following transformation
holds:
”
—
”
—
Γ G H I = H̃−1 Γ̃ G̃ I H̃ .
(2.10)
Proof. Proof follows through straightforward multiplications of the Markov parameters (matrix block elements).
Similarly, the system in (2.4) can be “lifted” to the following description:
¨
= Ā p x̂ k −p + L̄ū k −p,p + P̄ ȳ k −p,p + K̄ē k −p,p ,
= Γ̄x̂ k + Ḡ ū k ,f + I − F̄ ȳ k ,f + H̄ē k ,f ,
x̂ k
ȳ k ,f
(2.11)
where the lifted matrices L̄ ∈ Rn ×p r , P̄ ∈ Rn ×p ℓ , K̄ ∈ Rn ×p ℓ , Γ̄ ∈ R f ℓ×n , Ḡ ∈ R f ℓ× f r ,
¯ For the
F̄ ∈ R f ℓ× f ℓ , and H̄ ∈ R f ℓ× f ℓ are defined in an almost similar way without (·).
matrices L̄, K̄, Γ̄, and Ḡ the only difference is that the generator matrices A, B , and K
are replaced with Ā, B̄ , and K̄ , respectively. The matrix P̄ , F̄ , and H̄ are defined as:
”
—
P̄ = Ā p −1 M · · · ĀM M ,

I
0


I
 −C M
F̄ = 
..
..

.
.

f
−2
f
−C Ā M −C Ā −3 M

I
0
···

..

.
I
 C K̄
H̄ = 
..
..
..

.
.
.

C Ā f −2 K̄ C Ā f −3 K̄ · · ·

··· 0
.
..
. .. 

.
..

. 0
··· I

0
.. 

.
.

0
I
(2.12)
(2.13)
More interesting is the relation between the innovation, predictor, deadbeat predictor
form given in Lemma 2.2 and Lemma 2.3.
Lemma 2.2. It can be shown that F̄ is invertible, and the following transformation holds:
”
Γ
G
H
”
—
I = F̄ −1 Γ̄
Ḡ
H̄
—
F̄ .
(2.14)
Proof. Proof follows through straightforward multiplications of the Markov parameters (matrix block elements).
Lemma 2.3. It can be shown that H̄ is invertible, and the following transformation holds:
”
Γ̃
G̃
I
—
”
H̃ = H̄−1 Γ̄
Ḡ
H̄
—
F̄ .
(2.15)
Proof. Proof follows through straightforward multiplications of the Markov parameters (matrix block elements).
32
2.3
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
Batchwise identification with VARX-based PBSIDopt
In this section, the theoretical framework for the VARX-based predictor-based subspace identification method to identify LTI state-space models from measured data
is presented. The goal is to obtain the predicted state sequence, because if this is
obtained the state-space system matrices can easily be estimated by solving two linear
problems (Van Overschee and De Moor, 1996; Verhaegen and Verdult, 2007).
2.3.1
VARX predictor with finite past window
We define the one-step-ahead Vector Auto-Regressive with eXogenous inputs (VARX)
predictor as:
p
p
X
X
Ξ̃(yk −i ) y k −i ,
(2.16)
Ξ̃(u k −i ) u k −i +
ŷ k |k −1 =
i =0
i =1
where ŷ k |k −1 is the predicted output for time instant k using the inputs of time
instants k , . . . , k − p and using the outputs of time instants k − 1, . . . , k − p . Further,
Ξ̃ ∈ Rℓ×p (r +ℓ)+r is the set of VARX parameters to be estimated:
”
—
Ξ̃ ¬ Ξ̃(u k −p ) · · · Ξ̃(u k ) Ξ̃(yk −p ) · · · Ξ̃(yk −1 ) .
(2.17)
In Chiuso and Picci (2005) it was shown that estimated VARX predictors can
provide consistently estimated Markov parameters of the system SP in (2.3) even
with closed-loop data. To see this implicit relation between the Markov parameters
and the VARX predictor, we first need to introduce an approximation for the state.
The contribution of the initial state x̂ k −p in (2.8) can be made arbitrarily small by
making p large, because the estimated system SP in (2.3) is asymptotically stable
by definition. It is noted that in the case of coloured noise, the Kalman predictor is
asymptotically stable when the poles of the stochastic part which are not shared with
the deterministic part are asymptotically stable and zeros of the stochastic part are
not on the unit disk (Anderson and Moor, 2005). We assume that the past window is
chosen large enough such that the contribution of the initial state can be neglected,
in other words the matrix à is nilpotent.
Assumption 2.2 (Nilpotency). The transition matrix is deadbeat with degree p , i.e., the
matrix à j = 0 for all j ≥ p , see Chiuso and Picci (2005).
In a number of closed-loop SID methods it is well known to make this approximation, see Chiuso (2007); Jansson (2003). Now the following implicit relation is
proposed between the Markov parameters and the VARX predictor.
Proposition 2.1. Under Assumptions 2.1 and 2.2, consider the linear estimation of (2.17)
asymptotically with an infinite data sequence (N → ∞), then the VARX parameters
in (2.17) are related to the Markov parameters in (2.8), and described by:
¨
D,
if i = 0
(u k −i )
Ξ̃
=
, Ξ̃(yk −i ) = C Ã i −1 K .
C Ã i −1 B̃ , if i > 0
2.3 Batchwise identification with VARX-based PBSIDopt
33
Proof. See Appendix A and Chiuso and Picci (2005).
It is noted that there are two cases for the VARX predictor where a direct implicit
relation for a finite past window exists without considering an approximation. The
first case is if the system SI in (2.2) is of the ARX type, which is not common in
practical situations, then the transition matrix becomes nilpotent for all p ≥ n. The
second case is if the system S in (2.1) has no process noise, i.e., w k = 0, then the
feedback gain K in the system SP in (2.3) becomes a deadbeat gain M , which also has
a nilpotent transition matrix if p ≥ n.
2.3.2
The relation between the VARX predictor and the state
To extract the predicted state sequence using the estimated Markov parameters, we
introduce in this procedure again the approximation for the state equation in (2.8).
With Assumption 2.2, the state x k is given by:
x̂ k = L̃ū k −p,p + K̃ȳ k −p,p .
With the approximation given in (2.18), we can rewrite the system in (2.8) as:
Ý
Ý
ȳ k ,f = Γ
Lū k −p,p + Γ
Kȳ k −p,p + G̃ ū k ,f + I − H̃ ȳ k ,f + ē k ,f ,
(2.18)
(2.19)
Ý
Ý
where Γ
L ∈ R f ℓ×p r and Γ
K ∈ R f ℓ×p ℓ are the products between the extended observability and the extended controllability matrices, and are given by:

 (u )
Ξ̃ k −p Ξ̃(u k −p +1 ) · · · Ξ̃(u k −p + f −1 ) · · · Ξ̃(u k −1 )


Ξ̃(u k −p ) · · · Ξ̃(u k −p + f −2 ) · · · Ξ̃(u k −2 ) 
 0

Ý
ΓL =  .
.. 
..
..
..
..
,
.
.
.
.
. 
.
 .
0
···
0
Ξ̃(u k −p )
· · · Ξ̃(u k − f )
(2.20)

 (y )
Ξ̃ k −p Ξ̃(yk −p +1 ) · · · Ξ̃(yk −p + f −1 ) · · · Ξ̃(yk −1 )


Ξ̃(yk −p ) · · · Ξ̃(yk −p + f −2 ) · · · Ξ̃(yk −2 ) 
 0
Ý
Γ
K=
.. 
..
..
..
..
.
 ..
.
.
.
. 
.
 .
0
···
0
Ξ̃(yk −p )
· · · Ξ̃(yk − f )
These are upper block triangular matrices, because the introduced zeros follow
from Assumption 2.2. When f ≤ p , this implies that in the asymptotic case the
Ý
Ý
approximation of the matrices Γ
L and Γ
K can be fully constructed by the Markov
parameters Ξ̃ obtained from the least squares problem given in (2.16). Observe that
the product between the state and the extended observability matrix is given by:
Ý
Ý
Γ̃x̂ k = Γ
Lū k −p,p + Γ
Kȳ k −p,p ,
(2.21)
and this observation is the key idea behind the PBSIDopt method. To summarize, after
Ý
Ý
the construction of the matrices Γ
L and Γ
K, we obtain a product of the extended
observability matrix and the state sequence. Both of them can be estimated by solving
a low-rank approximation problem. Generally, the relation in (2.21) can be premultiplied with any pre-defined weight matrix W ∈ R f ℓ× f ℓ from the left side. Applying
a weight leads to a weighted version of the PBSIDopt method and also to a weighted
version of the SSNEW method in Ljung and McKelvey (1996), see also Chiuso (2007).
34
2.3.3
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
The VARX-based PBSIDopt algorithm
In this subsection the VARX-based predictor-based subspace identification method
for LTI systems operating in open or closed loop is presented. The batchwise system
identification problem can be formulated as:
Problem Description 2.1 (Batchwise system identification).
Given the input sequence u k , output sequence y k over a time k = {0, . . . , N −1} find, if they
exist, system matrices A, B , C , D, and K up to an unknown similarity transformation T .
Solving the problem with the theory of the previous subsections, results in the socalled “optimized” predictor-based subspace identification (PBSIDopt ) method which
was originally presented in Chiuso (2007). The algorithm can be divided in three steps,
where in each step linear problems need to be solved. A slightly modified description
is given in Algorithm 2.1. First, the innovation sequence needed for the estimation of
the system matrices in Step 3 is estimated using the relation Ê = Y − C X̂ − DU . In
the original recipe the estimated VARX model was used using the relation Ê = Y − Ξ̃Ψ,
—T
”
where Ψ = ŪpT U T ȲpT . Second, the weighting matrix W = I ⊗R 1/2 is not applied.
This weight was originally applied to get similar weighting as the CCA subspace
algorithm (Chiuso, 2010). The experience with multiple systems in practice (especially
if the innovation is not perfectly white) is that both suggestions are not beneficial for
the quality of the estimate, although they work well on small simulation models. The
last modification is that a guaranteed stabilizing Kalman matrix K is estimated using
the Riccati equation, as described in Verhaegen and Verdult (2007). For validation
purposes, it is important to have a stable Kalman predictor of the identified model.
In Assumption 2.2 an approximation is made which requires the past window
to be chosen as large as possible. However, in practical experiments it is most of
the time needed to design the excitation signal such that it does not exceed the
load specifications and ensures that the system to be identified operates around a
particular operation point. In this case, especially with a low-order controller in the
feedback loop, the matrix Ψ becomes ill conditioned, because the input signals do not
persistently excite the system. Apart from just selecting a lower past window size, a
number of other solutions exist to get a well-conditioned matrix Ψ for the least squares
estimation. The first possible solution is generating more persistently exciting control
signals by changing the design properties of the controller. For example automatically
generated constraints can be added to a Model Predictive Control (MPC) design
problem when the matrix Ψ becomes ill conditioned, see Dong (2009). The second
possible solution is utilizing a VARMAX model, for which the input-output description
relaxes the need for a very large past window needed due to the formulation of a
deadbeat predictor, see Section 2.4. The third and well-used solution to the ill-posed
least squares problem is the inclusion of a regularization quantity. In the batchwise
case, Tikhonov regularization from (Tikhonov and Arsenin, 1977) can be included by
replacing the least squares estimation of the Markov parameters in Algorithm 2.1 with:
−1
Ξ̃ = Y ΨT ΨΨT + µI
,
(2.22)
where µ is the regularization parameter. The Tikhonov regularization parameter µ
can be determined from a batch of data obtained around an operation point using
the L-curve or Generalized Cross Validation (GCV) criterion, see Hansen (1992).
2.3 Batchwise identification with VARX-based PBSIDopt
Algorithm 2.1 (The VARX-based PBSIDopt algorithm).
input u , y , p , f , W .
require f ≤ p
Step 1: The estimation of the Markov parameters
Construct the matrices Y , U , Ūp , and Ȳp , see (2.5) and (2.6).
Solve the linear problem in (2.16) using least squares:
—T
”
Ξ̃ = Y Ψ+ , where Ψ = ŪpT U T ȲpT .
Step 2: The estimation of the state sequence
Ý
Ý
Construct the matrices Γ
L and Γ
K in (2.20).
Ý
Ý
Ó =W Γ
LŪp + Γ
KȲp .
Calculate the sequence ΓX
Solve the low-rank
– approximation
™ – ™ problem using the SVD:
”
— Σn 0 Vn
Ó = Un U⊥
,
ΓX
0 Σ V⊥
where the system order n is determined by detecting a gap
that separates the n largest singular values.
1/2
Calculate the state sequence: X̂ = Σn Vn
Step 3: The estimation of the system matrices
Solve linear problem using least squaresa :
™+
–
”
—
X̂ (:,1:N −p −1)
C D = Y (:,1:N −p −1)
.
U (:,1:N −p −1)
Compute the innovation sequence Ê = Y − C X̂ − DU .
Solve linear problem using least squaresa :

+
X̂ (:,1:N −p −1)
”
—


A B K = X̂ (:,2:N −p ) U (:,1:N −p −1) .
Ê (:,1:N −p −1)
Step 3a: Calculate the stabilizing Kalman gain
a
Compute
™
™ :–
™–
™ – the residuals
–
A B
X̂ (:,1:N −p −1)
X̂ (:,2:N −p )
Ŵ
.
−
=
Y (:,1:N −p −1)
C D U (:,1:N −p −1)
V̂
Estimate
– ™ matrices:
™the covariance
–
—
Ŵ ” T
Q S
1
Ŵ
V̂ T .
= N −p
T
V̂
S
R
Calculate the solution of the Riccati equation:
−1
T
P = APA T − S + APC T R + C PC T
S + APC T +Q.
Estimate a stabilizing Kalman gain:
−1
K = S + APC T R + C PC T
.
a For simplicity
MATLAB™ notation is used.
35
36
2.4
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
Batchwise identification with VARMAX-based PBSIDopt
In this section, the theoretical framework for the VARMAX-based predictor-based
subspace identification method to identify LTI state-space models from measured
data is presented. The estimation of a VARMAX model comes at the cost of a nonlinear problem; however most of the times it can be estimated efficiently.
2.4.1
VARMAX predictor with finite past window
We define the one-step-ahead Vector Auto-Regressive Moving Average with eXogenous
inputs (VARMAX) predictor as:
p
X
ŷ k |k −1 =
i =0
Ξ̄(u k −i ) u k −i +
p
X
i =1
Ξ̄(yk −i ) y k −i +
p
X
i =1
Ξ̄(e k −i ) ê k −i ,
(2.23)
where ŷ k |k −1 is the predicted output for time instant k using the inputs of time
instants k , . . . , k − p and using the outputs and estimated innovation of time instants
k −1, . . . , k −p . By substituting ê k −i = y k −i − ŷ k −i |k −i −1 in the former gives an expression
which resembles the well-known ARMAX model structure as:
p
X
i =0
Ξ̄(e k −i ) ŷ k −i |k −i −1 =
p
X
i =0
Ξ̄(u k −i ) u k −i +
p
X
€
i =1
Š
Ξ̄(e k −i ) + Ξ̄(yk −i ) y k −i ,
(2.24)
with Ξ̄(e k ) = I , and Ξ̄ ∈ Rℓ×p (r +2ℓ)+r is the set of VARMAX parameters to be estimated:
”
—
Ξ̄ ¬ Ξ̄(u k −p ) · · · Ξ̄(u k ) Ξ̄(yk −p ) · · · Ξ̄(yk −1 ) Ξ̄(e k −p ) · · · Ξ̄(e k −1 ) .
(2.25)
Although the one-step-ahead predictor does not have the property that it is a linear
function in the Markov parameters, the computation of the solution can still be done
efficiently using the extended least squares recursion, see also Subsection 2.4.3.
In Subsection 2.3.1, an approximation for the state was introduced to get an
implicit relation between the Markov parameters and the VARX predictor. We had to
assume that the past window is chosen large enough such that the contribution of the
initial state can be neglected, in other words the matrix à is nilpotent. In the model
description in (2.4), an additional freedom in the form of a deadbeat observer matrix
M was introduced on purpose to get the nilpotent or deadbeat property during the
estimation VARMAX parameters. From pole-placement theory, for example by placing
all the eigenvalues of the matrix Ā in the origin, a deadbeat matrix M of degree p ≥ n,
i.e. Ā p = 0, is known to exist if the system is observable. As only an observable part
of the system can be identified from the measurement data, this is expected to exist.
With this reasoning, a very large past window is not required and a direct implicit
relation between the Markov parameters and the VARMAX predictor can be proposed.
Proposition 2.2. Under Assumptions 2.1, consider the linear estimation of (2.25)
asymptotically with an infinite data sequence (N → ∞) and p ≥ n, then the VARMAX
parameters in (2.25) are related to the Markov parameters in (2.11), and described by:
¨
D,
if i = 0
(u k −i )
Ξ̄
=
, Ξ̄(yk −i ) = C Ā i −1 M , Ξ̄(e k −i ) = C Ā i −1 K̄ .
C Ā i −1 B̄ , if i > 0
2.4 Batchwise identification with VARMAX-based PBSIDopt
37
Proof. See Appendix A.
If the system S in (2.1) has no process noise, i.e. w k = 0, then the feedback gain
K = 0 in the system SD in (2.4), and basically becomes the system SP in (2.4) with K
replaced by M , which can be related to a VARX predictor with finite past window.
2.4.2
The relation between the VARMAX predictor and the state
Almost similar as in Subsection 2.3.2, the predicted state sequence can be estimated
using the estimated Markov parameters and the estimated innovation sequence. With
p ≥ n, the state x k in (2.11) is given by:
x̂ k = L̄ū k −p,p + P̄ ȳ k −p,p + K̄ē k −p,p .
With the deadbeat property, we can also rewrite the system in (2.11) as:
ö
ö
ö
ȳ k ,f = Γ
Lū k −p,p + Γ
P ȳ k −p,p + Γ
Kē k −p,p + Ḡ ū k ,f + I − F̄ ȳ k ,f + H̄ē k ,f ,
(2.26)
(2.27)
ö
ö
ö
where Γ
L ∈ R f ℓ×p r , Γ
P ∈ R f ℓ×p ℓ and Γ
K ∈ R f ℓ×p ℓ are the products between the
extended observability and the extended controllability matrices, and are given by:
 (u )

Ξ̄ k −p Ξ̄(u k −p +1 ) · · · Ξ̄(u k −p + f −1 ) · · · Ξ̄(u k −1 )


Ξ̄(u k −p ) · · · Ξ̄(u k −p + f −2 ) · · · Ξ̄(u k −2 ) 
 0
ö
Γ
L=
..
.. 
..
..
..
 ..
,
.
.
.
.
. 
 .
0
···
0
Ξ̄(u k −p )
· · · Ξ̄(u k − f )
 (y )

Ξ̄ k −p Ξ̄(yk −p +1 ) · · · Ξ̄(yk −p + f −1 ) · · · Ξ̄(yk −1 )


Ξ̄(yk −p ) · · · Ξ̄(yk −p + f −2 ) · · · Ξ̄(yk −2 ) 
 0

ö
Γ
P = .
(2.28)
..
.. 
..
..
..
.
..
.
.
.
.
. 

0
···
0
Ξ̄(yk −p )
· · · Ξ̄(yk − f )
 (e )

Ξ̄ k −p Ξ̄(e k −p +1 ) · · · Ξ̄(e k −p + f −1 ) · · · Ξ̄(e k −1 )


Ξ̄(e k −p ) · · · Ξ̄(e k −p + f −2 ) · · · Ξ̄(e k −2 ) 
 0

ö
ΓK =  .
.. 
..
..
..
..
.
.
.
.
.
. 
.
 .
0
···
0
Ξ̄(e k −p )
· · · Ξ̄(e k − f )
These are upper block triangular matrices, because the introduced zeros follow from
ö
ö
ö
the deadbeat property. When f ≤ p , this implies that the matrices Γ
L, Γ
P and Γ
K can
be fully constructed by the Markov parameters Ξ̄ obtained from the model estimation
problem given in (2.23). Observe that the product between the state and the extended
observability matrix is given by:
ö
ö
ö
Γ̄x̂ k = Γ
Lū k −p,p + Γ
P ȳ k −p,p + Γ
Kē k −p,p .
(2.29)
ö
ö
ö
To summarize, after the construction of the matrices Γ
L, Γ
P and Γ
K, we obtain
a product of the extended observability matrix and the state sequence. Both of
them can be estimated by solving a low-rank approximation problem. Generally,
the relation in (2.29) can also be pre-multiplied with any pre-defined weight matrix
W ∈ R f ℓ× f ℓ from the left side.
38
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
2.4.3
The VARMAX-based PBSIDopt algorithm
In this subsection the VARMAX-based predictor-based subspace identification method
for LTI systems operating in open or closed loop is proposed. The batchwise system
identification problem can be formulated similar as in Problem Definition 2.1 stated
in Section 2.3.3. Solving the problem with the theory of the previous subsections
results in the so-called VARMAX-based PBSIDopt in Algorithm 2.3. The algorithm can
again be divided in three steps, where in the first step a non-linear problem needs to
be solved instead.
The VARMAX model estimation problem can not be solved directly using ordinary
least squares, because the innovation elements in (2.23) are not known in advance. In
fact this becomes a non-linear problem, which can be solved for example by applying
the residual whitening iterations (Phan et al., 1995) or the recursive Extended Least
Squares (ELS) method (Ljung, 1977, 1999), where the recursive scheme is preferred by
experience due to its simplicity, computational complexity, and better convergence
properties. The ELS algorithm is basically an extended version of the exponentiallyweighted regularized RLS scheme, and given in Algorithm 2.2. It is recommended
to employ such an exponentially-weighted and regularized cost function, because
it ensures that during the initial iterations of the scheme a local solution exists
and converges. It is possible to apply the exponentially weighting only during
initial calculations and set it later close to one, which means no forgetting. For
clarification, the iteration shows the conventional implementation of the recursive
ELS scheme. For practical use it is recommended to use square-root or array-type
implementations, for better numerical robustness against round-off errors, see for
more discussion (Houtzager et al., 2011a).
Algorithm 2.2 (The Extended Least Squares (ELS) algorithm).
input Ξ̄−1 , λ, µ
require 0 ≪ λ ≤ 1, µ > 0
init ē −p,p = 0, P−1 = µ1 I p (r +2ℓ)+r
for k = 0, 1, 2, . . .
input u k , y k
Shift:
”
z̄ k −p,p = ū kT−p,p
u kT
Update RLS:
ȳ kT−p,p
ē kT−p,p
—T
.
−1
G k = z̄ kT −p,p Pk −1 λI + z̄ kT −p,p Pk −1 z̄ k −p,p
,
€
Š
Ξ̄k = Ξ̄k −1€ + y k − Ξ̄k −1 z̄ k −p,p G
Š k,
Pk = λ−1 Pk −1 − Pk −1 z̄ k −p,p G k .
Estimate innovation:
ê k = y k − Ξ̄k z̄ k −p,p .
2.4 Batchwise identification with VARMAX-based PBSIDopt
Algorithm 2.3 (The VARMAX-based PBSIDopt algorithm).
input u , y , p , f , W .
require f ≤ p
Step 1: The estimation of the Markov parameters
Construct the matrices Y , U , Ūp , and Ȳp , see (2.5) and (2.6).
Solve the non-linear problem in (2.23) using ELS in Algorithm 2.2.
Construct the matrix Ē p (similar as in (2.6)) using the estimated innovation.
Step 2: The estimation of the state sequence
ö
ö
ö
Construct the matrices Γ
L, Γ
P and Γ
K in (2.28).
Ó =W Γ
ö
ö
ö
Calculate the sequence ΓX
LŪp + Γ
P Ȳp + Γ
K Ē p .
Solve the low-rank
– approximation
™ – ™ problem using the SVD:
”
— Σn 0 Vn
Ó = Un U⊥
,
ΓX
0 Σ V⊥
where the system order n is determined by detecting a gap
that separates the n largest singular values.
1/2
Calculate the state sequence: X̂ = Σn Vn
Step 3: The estimation of the system matrices
Solve linear problem using least squaresa :
™+
–
”
—
X̂ (:,1:N −p −1)
C D = Y (:,1:N −p −1)
.
U (:,1:N −p −1)
Compute the innovation sequence Ê = Y − C X̂ − DU .
Solve linear problem using least squaresa :

+
X̂ (:,1:N −p −1)
”
—


A B K = X̂ (:,2:N −p ) U (:,1:N −p −1) .
Ê (:,1:N −p −1)
Step 3a: Calculate the stabilizing Kalman gain
a
Compute
™
™ :–
™–
™ – the residuals
–
A B
X̂ (:,1:N −p −1)
X̂ (:,2:N −p )
Ŵ
.
−
=
Y (:,1:N −p −1)
C D U (:,1:N −p −1)
V̂
Estimate
– ™ matrices:
–
™the covariance
—
Ŵ ” T
Q S
1
Ŵ
V̂ T .
= N −p
T
V̂
S
R
Calculate the solution of the Riccati equation:
−1
T
P = APA T − S + APC T R + C PC T
S + APC T +Q.
Estimate a stabilizing Kalman gain:
−1
K = S + APC T R + C PC T
.
a For simplicity
MATLAB™ notation is used.
39
40
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
The practical experience with different models is that the global minimum is
usually found without too much problem, especially if the initial Ξ̄−1 is close to the
global minimum, for example by first estimating Ξ̃ by linear techniques and substitute
(y )
(u )
(e )
Ξ̄−1 and Ξ̄−1 by Ξ̃(u ) and Ξ̃(y ) and setting Ξ̄−1 to zero. If the solution still has not
converged to the minimum yet, then the iteration can be repeated with as initial
solution the previous. Although not common, the method is known to diverge if the
following positive real condition:
n
€
Š−1 o 1
K̄ > , ∀ω ∈ R,
Re I + C e i ω I − Ā
2
(2.30)
is not fulfilled, see for proof and possible solutions (Ljung, 1977, 1999).
2.5
Simulation study
In this section, the effectiveness of the VARMAX-based PBSIDopt method in comparison with the VARX-based PBSIDopt method are shown by simulation studies on
modern wind turbine models. Three modern wind turbine models are considered
to be identified; namely, the model from the SUSCON project (van Engelen et al.,
2007), a model in TURBU software (van Engelen, 2007), and a model in PHATAS
software (Lindenberg, 2005). Since, the section focuses on the application of the
PBSIDopt algorithms to the provided benchmark data of Kanev et al. (2009), the full
physical first-principle modelling concepts of the models are omitted, but a clear distinction can be made with respect to the increasing model complexity. The SUSCON
wind turbine model has simplified LTI dynamics, and the noise and excitation signals
fulfil the Assumption 2.1. The TURBU wind turbine model has approximated LTI
dynamics of high-order and the Coleman transformation is applied to remove the first
harmonic component of the periodic disturbances. The PHATAS wind turbine model
has a full non-linear description, where the Coleman transformation is also used to
linearise the periodic dynamics as far as possible (at least up to the second rotational
frequency). In both cases, the process noise resembling the turbulence are created by
a turbulence generator, and the excitation signals are designed, to conform to a real
wind turbine identification scenario, therefore the assumptions in Assumption 2.1 of
white Gaussian noise sequences and of persistent excitation are not perfectly fulfilled.
2.5.1
Identification results with the SUSCON wind turbine model
In this subsection, the SUSCON wind turbine model is used to demonstrate the
effectiveness of the algorithms with a small past window size. The simplified SUSCON
wind turbine model is of seventh-order and is described in van Engelen et al. (2007).
Despite the small order of this model, which is useful for fast calculations, it still has
the relevant features of individually pitch-controlled blades, main rotation, fore-aft
and sidewards tower bending mode, shaft torsion mode, and controllable generator
torque. The blades are considered to be rigid. The model describes thus the most
relevant rotational dynamics of a modern wind turbine around a particular operating
point and has a constant state matrix while the input and output matrices strongly
2.5 Simulation study
41
depend on the azimuth angle. For this reason, this model has been used before
in van Wingerden et al. (2009) to demonstrate the closed-loop subspace LPV system
identification algorithm. Similarly as in Selvam et al. (2009); van Engelen et al. (2007),
the Coleman transformation can be used to transform the wind turbine model to LTI.
The full description of the LTI state-space model is given in Wei and Verhaegen (2009).
The LTI SUSCON wind model is used to obtain the input and output sequences
for the identification experiments. For this purpose, the equations are converted to
discrete time using a zero-order hold discretization method with a sample time of
0.1 s. The wind turbine system is not asymptotically stable, as it has an pure integrator.
Therefore, controllers are added in feedback loops to the system for stabilization. The
descriptions of these controllers and additional filters used can be found in Selvam
et al. (2009); van Engelen et al. (2007). For consistent closed-loop estimation, it
is important that there is sufficient excitation from an external excitation signal or
a controller of sufficiently high order, see Ljung (1999); Van den Hof and Schrama
(1995). As the controllers are not sufficiently high order, we take an additional zero
mean white noise with var θi ,k = 1 deg with i = 1, 2, 3, which is added to the control
signal of the collective pitch controller. As additional excitation
Š for the generator
€ input
torque we take also a zero-mean white noise signal with var Tge,k = 1 · 106 Nm. The
wind disturbance signal is also zero-mean white noise with var Vi ,k = 1 m/s, but this
signal is assumed to be unknown.
The collected data of u k and y k from the simulations are used in the identification
experiments. For the identification experiments we used N = 10000, p = f = 10,
and λ = 0.99 for the forgetting in the ELS algorithm. To emphasize the difference
with small windows, the VARX model estimation is also carried out with the same
small window size. To investigate the sensitivity of the identification algorithm with
respect to the wind disturbances, a Monte-Carlo Simulation (MCS) with 100 runs was
carried out. For each of the 100 MCS, a different realization of the input u k and
wind disturbance Vk is used. The performance of the identified system is evaluated
Ó , and comparing the frequency response
by looking at the singular values of matrix ΓX
functions of the identified models to the real model.
Ó , including the error bounds for 100
In Figure 2.2 the singular values of matrix ΓX
MCS are given using VARX/VARMAX-based model identification. As expected, the
singular values of VARMAX-based identification show a large gap after the first seven
largest singular values, which equals the order of the system. This is not the case with
the VARX-based method, because the past window is too short to successfully model
the system. Note, that 19 singular values are added to the tail of the 7 singular values
of the system; this corresponds precisely to the 19 states of the controller. Figure 2.3
and 2.4 show the Bode diagrams of a selected number of inputs and outputs. The
VARMAX-based method gives more consistent identifications results for the wind
turbine system. For both methods, the identified natural frequencies are very close
to the true €natural frequencies,
although a zero, which is clearly visible in the bode
Š
diagram of Tge → Ωge , is not estimated with the VARX-based method.
Further simulations showed that increasing the value of the past window size,
which was possible due to the additional white noise excitation signal, improves
the model obtained from VARX-based identification considerably, and hardly any
improvement was visible in the model obtained from VARMAX-based identification.
42
105
105
100
100
10−5
10−10
10−15
0.9
Singular values
Singular values
0.8
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
0
10
20
30
40
10−5
10−10
10−15
50
0
10
20
30
40
50
Number of values
Number of values
Ó with error bounds for 100 MCS using
Figure 2.2: The singular values of the matrix ΓX
VARX (left) and VARMAX (right).
Magnitude (dB)
-50
-100
-150
10−2
0
Magnitude (dB)
-100
100
10−1
Frequency (Hz)
Bode diagram (θcol → ẋ fa )
-100
-50
-100
10−2
10−1
100
Frequency (Hz)
101
€
Š
Bode diagram Tge → Ωge
-150
-200
10−2
101
Magnitude (dB)
Magnitude (dB)
0
€
Š
Bode diagram θcol → Ωge
10−1
100
Frequency (Hz)
101
€
Š
Bode diagram Tge → ẋ sw
-150
-200
10−2
10−1
100
Frequency (Hz)
101
Figure 2.3: Bode diagrams of the original transfer functions (dashed) and the identified transfer functions of the experiment with the best fit (bold) using VARX-based
identification. The other 99 MCS are within the grey region.
43
2.5 Simulation study
€
Š
Bode diagram θcol → Ω g e
-50
-100
-150
10−2
10−1
100
Frequency (Hz)
Magnitude (dB)
-100
-50
-100
10−2
10−1
100
Frequency (Hz)
-150
-200
10−2
101
€
Š
Bode diagram θcol → ẋ f a
0
Magnitude (dB)
-100
Magnitude (dB)
0.1
Magnitude (dB)
0
101
€
Š
Bode diagram Tge → Ωge
10−1
100
Frequency (Hz)
101
€
Š
Bode diagram Tge → ẋ sw
-150
-200
10−2
100
10−1
Frequency (Hz)
101
Figure 2.4: Bode diagrams of the original transfer functions (dashed) and the identified transfer functions of the experiment with the best fit (bold) using VARMAX-based
identification. The other 99 MCS are within the grey region.
As expected from the theory in Section 2.3, the VARX-based identification method
gives for large past window sizes (p > 30) almost similar results as the VARMAX-based
identification results with past window size p = 10. In the next subsection, the VARXbased and VARMAX-based PBSIDopt algorithm will be applied on data from a more
realistic wind turbine model, because it is expected that under these conditions the
VARX-based method can perform better than the VARMAX-based method.
2.5.2
Identification results with the TURBU wind turbine model
In this subsection, an aero-elastic wind turbine model created with the ECN software
TURBU (van Engelen, 2007) is used to demonstrate the effectiveness of the closedloop subspace LTI system identification algorithms. The model describes the rotational dynamics of a wind turbine around a particular operating point with mean wind
speed V = 18 m/s. The multi-body model contains around 100 states, representing
the degrees of freedom in the tower, drivetrain, blades and the pitch servo actuators.
The input signals to the model are three reference blade pitch angles θi and the
reference generator torque Tge . The outputs are defined as the generator speed
Ωge , the tower top fore-aft velocity ẋ fa and tower top sidewards velocity ẋ sw . The
disturbance signals are the three blade effective wind signals Vi . The wind turbine
44
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
system is not asymptotically stable, as it contains a pure integrator. Therefore, a
collective pitch controller and a generator torque controller are added in the feedback
loop of the system for stabilization. The collective pitch controller, used to regulate
the rotational speed, is based on a PI compensator, and the torque controller on a
P compensator. Due to the restrictions from a third commercial party, most of the
results in this chapter are normalized. This means that the frequencies, amplitudes,
and other values illustrated in the figures are scaled.
Following the benchmark description in Kanev et al. (2009), the TURBU simulator
is used to generate input and output data sequences for the identification algorithms.
In Kanev et al. (2009), a proper excitation signal was selected by carefully inspecting
the effect of additional loads on the drivetrain and on the tower in the fore-aft and
sidewards directions. The excitation signals for the collective pitch and generator
torque inputs were created by filtering a Pseudo Random Binary Signal (PRBS). In
Figure 2.5, the power spectral densities of a realization for both excitation signals
are given. It is clearly visible that filtering was applied at the higher frequencies
(for a smoother signal) and also in the case of the collective pitch around the first
natural frequency of the fore-aft displacement of the tower. In Kanev et al. (2009)
the data length was chosen to be N = 5000 with a sampling time of h = 0.4s, which
corresponds to 33 min. Longer data lengths will normally give better results, however
in reality the wind turbine stays only in an operation region for a limited amount of
time, depending on the wind. Two different turbulence realizations have been used;
one for identification and one for validation. The turbulence was generated by the
ECN software SWIFT and is of the Von Karman type (Kanev et al., 2009). Figure 2.5(c)
shows the frequency content of a turbulence realization in the rotational frame. The
turbulence spectrum has a low-pass nature with a peak at the rotational frequency.
To investigate the sensitivity of the identification algorithm with respect to excitation, Monte-Carlo Simulations (MCS) with 100 runs are carried out. For each simulation a different realization of the filtered PRBS excitation signal is used. Before the
identification is applied, all the input and output signals are individually scaled to an
equal variance, to improve the conditioning of the identification problem. Afterwards,
the scaling is reversed on the identified state-space model. The identified models are
analysed both in the time and frequency domains. Time-domain analysis consists of
the computation of the Variance-Accounted-For (VAF) on a data set different from the
data set used for determining the model. The VAF resembles the percentage of the
output variation that is explained by the model. The VAF is defined as:
var y k − ŷ k
, 0 · 100%,
vaf y k , ŷ k = max 1 −
var y k
where ŷ denotes the output signal obtained by simulating the identified model, y is
the measured output signal, and var denotes the variance of a quasi-stationary signal.
Frequency-domain validation of the identified models covers the comparative evaluation €of the most relevant
channel-wise frequency functions: two from
the collective
Š
€
Š
pitch θcol → Ωge , ẋ fa , and two transfers from the generator torque Tge → Ωge , ẋ sw .
When the “optimized” predictor-based subspace identification methods are applied, the state dimension n and the past and future window size p and f need
to be tuned. Theoretically, the model behind the TURBU simulator has almost 100
45
0
0
-10
-10
-20
-20
Magnitude (normalized) [dB]
Magnitude (normalized) [dB]
2.5 Simulation study
-30
-40
-50
-60
-70
-80
-30
-40
-50
-60
-70
-80
-90
-90
-100 −2
10
-100 −2
10
10−1
Frequency (normalized) [Hz]
100
(a)
10−1
Frequency (normalized) [Hz]
100
(b)
50
Magnitude (normalized) [dB]
40
30
20
10
0
-10
-20
-30
-40
-50 −2
10
10−1
Frequency (normalized) [Hz]
100
(c)
Figure 2.5: Spectral densities of the excitation signals on the pitch (a) and the
generator torque (b) inputs, and of the turbulence in the rotational frame (c).
states. Not all of these states are relevant or visible during identification, because
their contribution are small compared with the noise or their dynamics hare to fast
and therefore they are beyond the Nyquist frequency. Overall, a reasonable gap in
the singular values is detected at n = 16, so a considerable lower-order model is
identified. It is hard to quantify the effect for a finite past window size p . First, there
is the effect of the truncation error for the VARX predictor, which equals to the sum
of the remaining tail of Markov parameters. Second, is that the matrix Ψ becomes ill
conditioned for large past windows, because normally the designed excitation signals
do not persistently excite the system. To quantify this effect, we use the so-called
condition number, which equals the ratio between the largest and smallest non-zero
singular value. In Figure 2.6, the trade-off between the truncation error and condition
number is illustrated. Tikhonov regularization is used for the VARX-based method in
order to overcome possible numerical problems, such as singularity of the regression
problem. The selection of the future window size f is even more difficult. It heavily
depends on the input spectrum and system properties. It is suggested in Chiuso
(2010) that a large future window is often better for identification experiments when
the Signal-to-Noise Ratio (SNR) is low (more averaging effect). Otherwise for higher
SNR, the optimal future window size has normally a smaller finite value.
46
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
101
1010
109
100
107
Condition number
Truncation error
108
106
10−1
105
10−2
0
5
10
15
20
25
30
35
40
45
104
50
Past window size p
Figure 2.6: The truncation error of the VARX predictor and the condition number of
ΨΨT as a function of the past window size p .
With the definition of the VAF, the influence of the past and future window sizes p
and f , which are indispensable for the PBSIDopt method, can be investigated. Computationally, the only constraint on the window sizes is to consider p ≥ f . Figure 2.7
gives the mean VAF percentage for different past and future window sizes. This shows
clearly the nature of the VARX-based and VARMAX-based model structures. For the
VARX-based methods, more accurate mean VAF values are provided when the past
window size is large. For the VARMAX-based method, more accurate mean VAF
values are provided with relatively small past window sizes. In both experiments, the
future horizon can be relatively small. It is noted that Tikhonov regularization must
be used for the VARX-based method to obtain consistent VAF results to reflect the
characteristics of the truncation error. These facts motivate the choice of the following
windows for the TURBU model identification: p = 50 and f = 15 for the VARX-based
with Tikhonov regularization, and p = 15 and f = 15 for the VARMAX-based method.
Using the two identification techniques, the four relevant input-output frequency
responses are verified. Figures 2.8 and 2.9 show the comparative analysis of the
frequency responses based on the pre-selected model order and past and future window sizes. Note that MIMO system identification is performed and Bode magnitude
diagrams are plotted. As a generic conclusion, despite the short data length and the
low SNR due to the turbulence intensity, one can see accurate identification of the
TURBU model around the natural frequencies. The transfer functions from the pitch
angle are more accurate for the TURBU model than those from the generator torque,
because more excitation was allowed on this channel resulting in an higher SNR. The
lower accuracy in the low frequencies due to the restricted number of repetitions of
these frequencies in the short data sequence. In contrast to the previous subsection,
the VARX-based method with the pre-selected variables performs better in terms of
47
2.5 Simulation study
50
50
45
93.45
45
93.4
93.5
40
35
93
30
92.5
25
20
past window size f
past window size f
40
93.3
93.25
30
93.2
25
93.15
93.1
20
92
93.05
15
10
10
93.35
35
15
91.5
15
20
30
25
35
40
future window size p
45
10
10
50
(a)
93
15
20
25
30
40
35
future window size p
45
50
92.95
(b)
50
30
40
20
30
10
Magnitude (normalized) [dB]
Magnitude (normalized) [dB]
Figure 2.7: The mean VAF as a function of the past window and future window sizes
for the identified TURBU model, where (a) is VARX-based with Tikhonov regularization and (b) is VARMAX-based. Note that the plots have different scales. (n = 16)
20
10
0
-10
-20
-30
0
-10
-20
-30
-40
-50
-40
-60
-50 −2
10
-70 −2
10
10−1
Frequency (normalized) [Hz]
100
50
30
40
20
30
10
20
10
0
-10
-20
-30
-40
-50 −2
10
100
(b) From Tge to Ωge
Magnitude (normalized) [dB]
Magnitude (normalized) [dB]
(a) From θcol to Ωge
10−1
Frequency (normalized) [Hz]
0
-10
-20
-30
-40
-50
-60
10−1
Frequency (normalized) [Hz]
(c) From θcol to ẋ fa
100
-70 −2
10
10−1
Frequency (normalized) [Hz]
100
(d) From Tge to ẋ sw
Figure 2.8: Bode diagrams of the TURBU model (dashed) and obtained using VARXbased PBSIDopt with the highest VAF value (bold). The other 99 MCS are within the
grey region. (n = 16, p = 50, f = 15)
48
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
the mean VAF in Figure 2.7 and the error bounds in Figures 2.8 and 2.9 than the
VARMAX-based method. Although the extended least squares problem had enough
persistent excitation due to the smaller past window size, it turns out that for the
VARMAX-based method it is more difficult to accurately estimate a low-order model,
especially the damping of the drivetrain; see the peak of the second natural frequency.
30
40
20
30
10
Magnitude (normalized) [dB]
Magnitude (normalized) [dB]
eplacements
50
20
10
0
-10
-20
-30
0
-10
-20
-30
-40
-50
-40
-60
-50 −2
10
-70 −2
10
10−1
Frequency (normalized) [Hz]
100
50
30
40
20
30
10
20
10
0
-10
-20
-30
-40
-50 −2
10
100
(b) From Tge to Ωge
Magnitude (normalized) [dB]
Magnitude (normalized) [dB]
(a) From θcol to Ωge
10−1
Frequency (normalized) [Hz]
0
-10
-20
-30
-40
-50
-60
10−1
Frequency (normalized) [Hz]
(c) From θcol to ẋ fa
100
-70 −2
10
10−1
Frequency (normalized) [Hz]
100
(d) From Tge to ẋ sw
Figure 2.9: Bode diagrams of the TURBU model (dashed) and obtained using
VARMAX-based PBSIDopt with the highest VAF value (bold). The other 99 MCS are
within the grey region. (n = 16, p = 15, f = 15)
2.5.3
Identification results with the PHATAS wind turbine model
In this section, the same wind turbine as in the previous section is going to be modelled in the ECN software PHATAS instead. The non-linear aero-elastic wind turbine
model is used to demonstrate the closed-loop subspace LTI system identification
algorithms. The model describes the rotational dynamics of a wind turbine with
mean wind speed V = 17m/s. Due to the introduction of a wind shear correction
profile, the mean wind speed is lower than in the previous subsection. The model
contains the degrees of freedom in the foundation, tower, drive train, blades and
2.5 Simulation study
49
the pitch servo actuators. The input signals to the model are three reference blade
pitch velocities θ̇i (not angles as before) and the reference generator torque Tge . The
outputs are again defined as the generator speed Ωge , the tower top fore-aft velocity
ẋ fa and tower top sidewards velocity ẋ sw . The disturbance signals are the three blade
effective wind signals Vi . A non-linear identification is not considered in the sequel,
but a dedicated Coleman transformation to fixed-frame coordinates is used to provide
linear-like behaviour of the wind turbine dynamics.
The identification of the PHATAS model is performed under almost the same
conditions as with the identification with the TURBU model, see also the benchmark
description in Kanev et al. (2009). The first difference is that the PHATAS simulator
uses the pitch rate command signal instead, therefore the additional excitation signal
for the pitch angle is differenced to obtain the pitch rate. The second difference
is that due to the costly (in terms of time and availability) PHATAS simulations
the identification and its verification can only be performed on one data sequence,
therefore no Monte Carlo experiments can be applied. At last, we do not possess
a linearised model of the non-linear PHATAS model. Consequently, the frequency
response verification is done using the spectral analysis method (see Appendix B), as
the true frequency response is not exactly unknown. Let us emphasize again that the
validation/verification results are based on a single data realization.
The influence of the past and future window sizes p and f , which are indispensable for PBSIDopt method, is again investigated using the VAF. Computationally, the
only constraint on the window sizes is to consider p ≥ f . Figure 2.10 gives the VAF
percentage for different past and future window sizes. This shows again clearly the
nature of the VARX-based and VARMAX-based model structure. For the VARX-based
methods, more accurate mean VAF values are provided when the past window size is
large. For the VARMAX-based method, more accurate mean VAF values are provided
with relatively small past window sizes. In both experiments, the future horizon can
be relatively small. It is noted again, that Tikhonov regularization must be used for the
VARX-based method to obtain consistent VAF results to reflect the characteristics of
the truncation error. These facts motivate the choice of the following windows for the
PHATAS model identification: p = 50 and f = 15 for the VARX-based with Tikhonov
regularization, and p = 15 and f = 15 for the VARMAX-based method.
Using the two identification techniques, the four relevant input-output frequency
responses are verified using spectral analysis. Figures 2.11 and 2.12 show the comparative analysis of the frequency responses based on the pre-selected model order
and past and future window sizes. Note that MIMO system identification is performed
and Bode magnitude diagrams are plotted. Similar to the TURBU identifications, one
can see accurate identification of the PHATAS model around the natural frequencies.
The transfer functions from the pitch rate are more accurate for the PHATAS model
than those from the generator torque, because more excitation was allowed in this
pitch channel resulting in a higher SNR. The lower accuracy in the low frequencies
is due to the restricted number of repetitions of these frequencies in the short data
sequence. Similar to the previous subsections, the performance of VARX-based
method have again been increased relative to the VARMAX-based method in terms of
the VAF in Figure 2.10 and the results in Figures 2.11 and 2.12. With the increasing
complexity of the system to be modelled, it turns out that for the VARMAX-based
50
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
50
50
45
96.05
35
96
30
past window size p
past window size p
45
96.1
40
95.4
95.395
40
95.39
35
30
95.385
95.95
25
95.9
20
10
10
15
20
30
25
35
future window size f
40
45
50
95.38
20
95.85
15
25
95.8
95.375
15
10
10
15
20
25
35
30
future window size f
40
45
50
(b)
(a)
40
30
30
20
20
10
Magnitude (normalized) [dB]
Magnitude (normalized) [dB]
Figure 2.10: The mean VAF as a function of the past window and future window sizes
of the identified PHATAS model, where (a) is VARX-based with Tikhonov regularization
and (b) is VARMAX-based. Note that the plots have different scales. (n = 16)
10
0
-10
-20
-30
-40
-50
-60 −2
10
0
-10
-20
-30
-40
-50
-60
10−1
Frequency (normalized) [Hz]
-70 −2
10
100
30
30
20
20
10
10
0
-10
-20
-30
-40
0
-10
-20
-30
-40
-50
-50
-60
-60 −2
10
-70 −2
10
10−1
Frequency (normalized) [Hz]
(c) From θ̇col to ẋ fa
100
(b) From Tge to Ωge
40
Magnitude (normalized) [dB]
Magnitude (normalized) [dB]
(a) From θ̇col to Ωge
10−1
Frequency (normalized) [Hz]
100
10−1
Frequency (normalized) [Hz]
100
(d) From Tge to ẋ sw
Figure 2.11: Bode diagrams of the PHATAS model obtained using spectral analysis
(grey line) and obtained using VARX-based PBSIDopt (black line). (n = 16, p = 50,
f = 15)
51
2.5 Simulation study
40
30
30
20
20
10
Magnitude (normalized) [dB]
Magnitude (normalized) [dB]
method it is more difficult to accurately estimate a low-order model, especially at the
natural frequency of the drivetrain which is in turn very important to determine the
stability of the collective pitch controller.
10
0
-10
-20
-30
-40
0
-10
-20
-30
-40
-50
-50
-60
-60 −2
10
-70 −2
10
10−1
Frequency (normalized) [Hz]
100
40
30
30
20
20
10
10
0
-10
-20
-30
-40
-50
-60 −2
10
100
(b) From Tge to Ωge
Magnitude (normalized) [dB]
Magnitude (normalized) [dB]
(a) From θ̇col to Ωge
10−1
Frequency (normalized) [Hz]
0
-10
-20
-30
-40
-50
-60
10−1
Frequency (normalized) [Hz]
(c) From θ̇col to ẋ fa
100
-70 −2
10
10−1
Frequency (normalized) [Hz]
100
(d) From Tge to ẋ sw
Figure 2.12: Bode diagrams of the PHATAS model obtained using spectral analysis
(grey line) and obtained using VARMAX-based PBSIDopt (black line). (n = 16, p = 15,
f = 15)
It can be concluded from these identification experiments (and also with other
models and experiments), that the VARMAX-based PBSIDopt is less robust to small
violations in the zero-mean white Gaussian assumption on the noise sequences and
to the dynamics that are not represented in the identified state-space model of lower
order. These violations will normally cause the estimated innovation sequence to be
not perfectly white and/or Gaussian. This heavily impacts the course of the following
steps in the VARMAX-based algorithm and therefore will influence the final quality of
the model estimate more than in the case of the VARX-based algorithm.
Finally, it must be emphasized that the identification of the TURBU and PHATAS
models are performed under extreme (worst case) turbulence conditions. With
normal turbulence conditions, it is expected that the accuracy of the identified models
will improve considerably. If still a more accurate model is required, a possible
52
Chapter 2: PBSID for the Model-Based Control of Wind Turbines
additional step is to further optimize the individual state-space parameters using the
Prediction-Error Method (PEM) (Ljung, 1999). Even for the parameter estimation
of LTI state-space models, it normally involves solving a non-convex optimization
problem. It may therefore be difficult to guarantee that the global optimum will
be found. It is considered to be very likely that an identified model from subspace
identification lies in the region of attraction of the global optimum (Lyzell et al., 2009).
If the predictor form of the identified model is stable, and the parametrization of the
state-space model is carefully selected to avoid numerical problems (Gevers and Li,
1993), the prediction-error estimate can then be obtained by a local search starting at
this initial estimate.
2.6
Conclusion
In this chapter, the optimized predictor-based subspace identification method is
presented and its effectiveness for the estimation of a wind turbine model in closed
loop is shown. This identification technique does not require any controller related information, consequently the identified model becomes consistent no matter
whether the wind turbine operates with or without a controller in the loop. Since the
wind turbine is not asymptotically stable in open loop, only closed-loop experiments
are supported in reality. This fact makes the proposed method very attractive for
the wind power community. Further, a novel extension to the optimized predictorbased subspace identification method is developed that relaxes the requirement that
the past window size has to be large for asymptotically consistent estimates. For
this purpose, a finite description of the input-output relation is formulated, and is
solved efficiently using the extended least squares recursion. The effectiveness and
robustness of the proposed methods are discussed in a simulation study, where the
algorithms are used on a data source based on controller-in-the-loop simulations of a
typical modern wind turbine. It can be concluded that the novel method performs
very well with a small past window under perfect conditions, however the novel
method is less robust than the original method to the increasing violations that occur
in the noise assumptions or the model order in practice. Despite the short data
length due to the limited amount of time in an operation region, and the low signalto-noise ratio due to the high turbulence intensity, an accurate identification of the
SUSCON/TURBU/PHATAS model is obtained for both methods, especially around the
natural frequencies of the wind turbine.
CHAPTER
3
Recursive Identification with
Application to the Real-Time
Closed-Loop Tracking of Flutter
A novel recursive predictor-based subspace identification method is presented to identify linear time-invariant systems with multiple inputs and
multiple outputs. The method is implemented in real time and is able to
operate in open loop or closed loop. The recursive identification is performed
via the subsequent solution of only three linear problems, which are solved
using recursive least squares. The recursive implementation of the method
is not only able to identify linear time-invariant models from measured
data, but can also be used to track slowly time-varying dynamics if adaptive
filters are used. The computational complexity is reduced by exploiting the
structure in the data equations and by using array algorithms to solve the
main linear problem. This results in a fast recursive predictor-based subspace
identification method suited for real-time implementation. The real-time
implementation and the ability to work with multi-input and multi-output
systems operating in closed loop makes this approach suitable for online
estimation of unstable dynamics. The ability to do so is demonstrated by the
detection of flutter on an experimental 2D-airfoil system.
3.1
T
Introduction
literature on recursive identification is almost entirely based on parametrized
transfer functions. Since many control design methods have been derived for
state-space models, a state-space approach is a useful extension to the existing
results on recursive identification for systems operating in both open loop and closed
loop. The PBSIDopt method from the previous chapter is becoming a well-established
technique to identify LTI state-space models from real systems in practice, see Balini
HE
53
54
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
et al. (2010); van Wingerden et al. (2011). However, extending this approach to deal
with time-varying dynamics is still a general problem. One possible extension to
PBSIDopt is the identification of affine Linear Parameter-Varying (LPV) or bilinear
models (van Wingerden and Verhaegen, 2009; van Wingerden et al., 2009). The timevarying dynamics in these linear model descriptions are considered linear dependent
on known scheduling signals. The challenge is to make this extension suitable for
practical applications, which is difficult because of the very large memory requirements due to the so-called curse of dimensionality (van Wingerden, 2008). Recursive
Subspace IDentification (RSID) methods have much lower memory requirements
than their batchwise counterparts. In this chapter we will not include the extension
with scheduling signals, but we will introduce and improve the recursive implementation of the PBSIDopt method. If formulated in the right way and by using adaptive
filters, the recursive implementation can be used to track slowly varying dynamics by
introducing forgetting.
RSID methods are intended for real-time implementation, such that the system
can be identified online. When dealing with high sampling rates and high dimensional
systems the limiting computational complexity is an additional challenge. It was
suggested in Chiuso et al. (2010); Wu et al. (2008) that PBSIDopt can be adapted to
a recursive implementation with the algorithms used in a number of successful RSID
methods (Gustafsson, 1998; Lovera, 2003; Lovera et al., 2000; Oku and Kimura, 2002).
However, these algorithms are based on the Projection Approximation Subspace
Tracking (PAST) method (Yang, 1995). As remarked in Mercère et al. (2004), the
disadvantage of PAST is that the identified state-space model does not converge to
a constant state basis; the basis slowly changes over time. This observation presents
a problem for the recursive PBSIDopt implementation, especially if it is used for
change detection, like flutter detection. Instead, the use of the Propagator Method
(PM) is proposed in Mercère and Lovera (2007); Mercère et al. (2008). Together
with the inclusion of a weight to the algorithm, it results in the novel recursive
PBSIDopt implementation. The main advantage of the weighted PM over the previous
concept is that if the dynamics are time-invariant, the recursively estimated statespace realization converges after a short transient period to a constant state-basis.
The real-time implementation and the ability to work with systems operating in
closed loop make this approach suitable for online estimation of unstable dynamics,
like flutter. Flutter is the most feared aerodynamic instability, where wing mode
oscillations extract energy from the airstream and introduce excessive aerodynamic
forces that can lead to structural failure (Mukhopadhyay, 2003). The 2D-airfoil system
is commonly used for research in flutter suppression (Rivera et al., 1992). The main
issue is that the 2D-airfoil dynamics are rather complex and strongly dependent on
the free-stream wind speed. A common way to describe the aeroelastic response
in a mathematical model is to optimize a large number of parameters of a firstprinciples model, obtained for example from the theory in Theodorsen (1935). Due to
the size and the non-linearity of the parameter identification problem, the suggested
optimisation methods in Bieniawski (2005); Cavagna et al. (2009); De Gaspari et al.
(2009) require hours of offline computation, and for a satisfactory result a good
initial estimate is needed. Faster identification methods based on the Eigensystem
Realization Algorithm (ERA) have been successfully applied offline in De Marqui et al.
3.2 Recursive identification with RPBSIDpm
55
(2006); McEver et al. (2007). These methods do not require a priori knowledge of
the LTI model structure and work in closed loop, but are computationally difficult
to implement online and to be made adaptive. The use of adaptive RSID methods
has been explored with open-loop simulations in De Cock et al. (2006); Goethals
et al. (2004). Although they have the potential to be implemented in real time, these
algorithms still have problems with identifying systems operating in closed loop.
The main contributions of this chapter are fourfold. The first contribution is the
recursive implementation of the PBSIDopt method using PM (the so-called RPBSIDpm
method) to identify LTI models from data measured in open or closed loop. The
RPBSIDpm method can also be used to track slowly time-varying dynamics if adaptive
filters are used. The second contribution is that we show that a certain weight is
required to minimize variations of the state basis. The third contribution is that
computational complexity is reduced by exploiting the structure in the data equations
and by using array algorithms to solve the main linear problem, such that a real-time
implementation of the proposed method can be obtained. The final contribution is
that we show that we can detect the onset of flutter. Moreover, we can track the
natural frequencies and modal damping. We will provide guidelines on how to pick
the free parameters in the proposed algorithm.
The outline of this chapter is as follows. In Section 3.2 the recursive implementation of the VARX-based predictor-based subspace identification method is presented
and discussed. In Section 3.3 the practical implementation issues are discussed and
some modifications to the algorithm are presented. In Section 3.4 the closed-loop
performance of the proposed RSID method to existing methods are evaluated by
a simulation study. In Section 3.5 the effectiveness of the proposed algorithm is
emphasized with an experimental study on flutter detection of a 2D-airfoil system
operating in closed loop and some tuning guidelines are provided. In the final section
we present the conclusions in this chapter.
3.2
Recursive identification with RPBSIDpm
In this section, the recursive identification and tracking with predictor-based subspace methods of systems with slowly time-varying dynamics operating in either open
loop or in closed loop is presented. The recursive system identification problem can
be formulated as:
Problem Description 3.1 (Recursive system identification). Given the system matrices
A k −1 , B k −1 , C k −1 , D k −1 , and K k −1 up to an unknown similarity transformation T , and
given the new input sample u k , and output sample y k find, if they exist, system matrices
A k , B k , C k , D k , and K k in a similar state basis.
3.2.1
Step 1: Adaptive filtering of the Markov parameters
Adaptive filters are used to estimate the Markov parameters, because of their ability
to track slow variations in the underlying signal statistics. This is handled in a natural
56
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
way by assigning less weight to older measurements that are no longer representative
for the system. Overviews of the properties of different adaptive filters can be found
in Diniz (2008); Ljung and Söderström (1983); Sayed (2008).
Let’s rewrite (2.16) to the following linear regression:
(3.1)
y k = Ξ̃k ϕk + e k ,
—
T
where ϕk = ū kT−p,p u kT ȳ kT−p,p . The adaptive filters that can be considered are
described by the following structure:
Ξ̃k = Ξ̃k −1 + y k − Ξ̃k −1 ϕk ϕkT Pk ,
(3.2)
”
with different choices of the error covariance matrix Pk . For the Least Mean Squares
(LMS) filter, the error covariance matrix Pk = µ1 I is used, where the scalar µ1 > 0 is a
step-size parameter that must be sufficiently small to prevent instability of the update
process but large enough to adopt any changes in the dynamics. For the Recursive
Least Squares (RLS) filter, the error covariance matrix is started with P−1 = ρ1 I , and
1
iterated as:
−1 T
1
ϕk Pk −1 ,
Pk −1 − Pk −1 ϕk λ1 I + ϕkT Pk −1 ϕk
Pk =
λ1
where the scalar 0 ≪ λ1 ≤ 1 is the forgetting factor to weight past data less (or equal
if λ = 1) than more recent data. The initial regularization value ρ1 > 0 makes sure
the problem is well conditioned in the first iterations. For the Kalman filter, the error
covariance matrix is started with P−1 = ρ1 I , and iterated as:
1
Pk = Pk −1 − Pk −1 ϕk R + ϕkT Pk −1 ϕk
−1
ϕkT Pk −1 +Q,
where the covariance matrix Q > 0 denotes the variations of the Markov parameters.
The values of these parameters have to be chosen by insight of the user. In this chapter
the focus is on the RLS filter. The parameter λ1 of the RLS filter is much easier in
practice to tune than the parameters R and Q of the Kalman filter. In general the
RLS filter has much better convergence properties of the estimate during changes in
parameters compared with the LMS algorithm, at the cost of more computations for
each update. However it is noted that there exist models for which the LMS filter
performs better than the RLS filter, see Sayed (2008).
3.2.2
Step 2: The selection of the weight and state vector basis
From Section 2.3.2 it is given that after the estimation of the Markov parameters, the
Ý
Ý
matrices Γ
L and Γ
K are constructed, and the predicted state x k can be estimated from
a low-rank approximation using the relation:
Ý
Ý
Γ̃x̂ k = Γ
Lū k −p,p + Γ
Kȳ k −p,p .
(3.3)
The following modification becomes extremely relevant when we consider recursive
identification. Using Lemma 2.1, it is possible to obtain a product between the state
and the extended observability matrix, given by:
Ý
Ý
H̃−1 Γ̃x̂ k = H̃−1 Γ
Lū k −p,p + H̃−1 Γ
Kȳ k −p,p
Γx̂ k = ΓL̃ū k −p,p + ΓK̃ȳ k −p,p .
(3.4)
57
3.2 Recursive identification with RPBSIDpm
Generally the relation in (3.3) can be pre-multiplied with any pre-defined weight
matrix W ∈ R f ℓ× f ℓ from the left side. In the batchwise, the SVD method gives the
best low-rank approximation. In online identification, it is important to update the
model during one sampling period. In spite of the existence of updating methods for
the SVD, it is still difficult to make them adaptive and to implement these algorithms
online due to the computational load and numerical stability. The main difficulty with
adaptive SVD methods is to keep the left and right subspace orthogonal, the property
which makes the SVD unique. Consequently, researchers try to find alternative
methods to apply the subspace concept in an adaptive framework.
The transformation in Lemma 2.1 leads also to an alternative way to obtain the
system matrices. Like in most traditional SID methods, the matrices A and C are
obtained from the observability matrix Γk as1 :
A = Γ(1:( f −1)ℓ,:)+ Γ(ℓ+1:f ℓ,:),
C = Γ(1:ℓ,:),
where (·)+ is the pseudo inverse. This alternative method plus the recursive estimation
of B and K can be quite costly to compute for large window sizes, therefore the
calculation of the system matrices through the state sequence is preferred.
Most RSID methods use alternative methods to obtain a low-rank approximation.
If the order n is known, it is possible to design a selection matrix S k ∈ Rn ×ℓ f , such that:
Ý
Ý
Lk ū k −p,p + Γ
x̂ k = S k z k = S k Wk Γ
Kk ȳ k −p,p ,
(3.5)
−1/2
where with an SVD update, the selection matrix becomes S k = Σn ,k UnT,k . At the cost
of the addition of another linear problem, it is also possible to estimate the selection
matrix S k using subspace trackers, like the Projection Approximation Subspace Tracking (PAST) method in Yang (1995) or the gradient type method in Oku and Kimura
(2002). The update of the PAST method is given by:
S Tk = S Tk −1 + z k − S Tk −1S k −1 z k z kT PS,k ,
(3.6)
which is very similar to the filter update in (3.2) and also has similar choices for
the error covariance matrix PS,k . The disadvantage of subspace trackers is that the
estimated state vector x̂ k = Tk x k = S k Γ̃x k does not converge to a constant basis,
because the estimated S k varies slowly over time due to forgetting during the updating
of PS,k (Mercère et al., 2004).
More recently, the Propagator Method (PM) has been put forward in Mercère et al.
(2008). As the system to be modelled S in (2.1) is observable, the observability matrix
Γ has at least n linearly independent rows. If the order n is known, it is possible to
design a stationary selection matrix S ∈ Rn ×ℓ f such that:
x̂ k = S ΓL̃k ū k −p,p + ΓK̃k ȳ k −p,p .
(3.7)
The main advantage of this Propagator Method (PM) over the previous concept is
that as long as the linear dynamics are time invariant, the recursively estimated state
vector converges after a short transient period to a constant state basis: Tk → T = SΓ.
1 For simplicity
MATLAB™ notation is used.
58
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
However this advantage is not yet completely valid for the proposed method with
the observability matrix in the predictor form, because the Kalman gain in Γ̃ has the
tendency to vary over time when forgetting is introduced even when the deterministic
system dynamics are constant.
In recursive identification with a moving horizon, for example using a sliding
window or by the inclusion of forgetting, it is well known and expected that the
Kalman gain K becomes non-stationary, and varies when the Kalman predictor is
updated with new data (and downdated with data which falls out of the window).
As a result, the estimated state vector x̂ k = Tk x k = S Γ̃k x k does not converge to a
constant basis, because Γ̃k is perturbed over time by the non-stationary Kalman gain.
Fortunately the transformation in Lemma 2.1 can transform Γ̃k into Γ. By selecting
the weight matrix in (3.5) to be Wk = H̃k−1 , the propagator method can be applied. The
(y )
matrix H̃k can be constructed from the estimated Markov parameters Ξ̃k .
3.2.3
Step 3: Adaptive filtering of the system matrices
For clarification, the last two linear problems for the estimation of the system matrices
are written in short as:
(y )
y k = Θk φk + e k ,
(x )
x̂ k +1 = Θk ψk ,
(3.8)
(3.9)
—T
”
, ψk = x̂ kT u kT ê kT , and Θk denotes the set of state-space
”
—
”
—
(y )
(x )
parameters with Θk = C k D k and Θk = A k B k K k of the corresponding
state-space equations. The innovation noise sequence, which is needed for the
second linear problem in (3.8), can be obtained from the solution of the first linear
(y )
problem (3.8) by ê k = y k − Θk φk . For recursive solvers of these two linear problems,
similar adaptive filters can be used as with the estimation of the Markov parameters
in Subsection 3.2.1. For a recursive implementation of the calculation of stabilizing
Kalman gain (Step 3a in 2.1) the forward Riccati recursion can be considered, but due
to the larger computational cost this is not applied in the recursive implementation.
”
where φk = x̂ kT
3.2.4
u kT
—T
The RPBSIDpm algorithm
Using the recursive methods mentioned in the previous subsections, the PBSIDopt
method in Algorithm 2.1 can be converted to the recursive method in Algorithm 3.1,
the so-called RPBSIDpm method. By comparing the recursive and batchwise method,
two clear differences can be remarked. The first difference is that for the recursive
method the three steps are calculated in a sample cascaded manner, i.e., after new
available input-output data the three steps are updated directly, and not in a batchwise cascaded manner. Thus the last two adaptive filters are updating their estimate
with data containing possible errors from the preceding adaptive filter. Therefore
the information, especially in the initial transient period, is somewhat misused. The
second difference is that the system order is not estimated online and therefore needs
to be given as an input to the algorithm.
3.3 Practical implementation in real time
59
An advantage of the recursive predictor-based subspace method over most other
RSID methods is that the estimated state vector (and also the system matrices) directly
becomes available and not after f steps of delay. For example, with MOESP-PO (Verhaegen and Verdult, 2007) based methods the data u k − f +1 . . . u k and y k − f +1 . . . y k are
used as instrumental variables to eliminate the influence of noise. This results that at
time step k the state x k − f is estimated instead, see Lovera et al. (2000); Mercère et al.
(2008).
Algorithm 3.1 (The RPBSIDpm algorithm).
input p , f , n, S, Θ−1 , λ1,2,3 , ρ1,2,3
require n > 0, n/ℓ ≤ f ≤ p , 0 ≪ λ1,2,3 ≤ 1, ρ1,2,3 > 0
init P−1 = ρ1 I , M −1 = ρ1 I , N −1 = ρ1 I , Ξ̃−1 from Θ−1
1
2
3
for k = 0, 1, 2, . . .
input u k , y k
Step 1: Update the Markov parameters
Update RLS:
−1 T
ϕk Pk −1 ,
Pk = λ1 Pk −1 − λ1 Pk −1 ϕk λ1 I + ϕkT Pk −1 ϕk
1
1
Ξ̃k = Ξ̃k −1 + y k − Ξ̃k −1 ϕk ϕkT Pk .
Step 2: Estimate the state vector
Ý
Ý
Construct the matrices Γ
Lk and Γ
Kk in (2.20).
e
Construct the matrix Hk in (2.9).
Ý
Ý
e −1 Γ
Compute: x̂ k = S H
Lk ū k −p,p + Γ
Lk ȳ k −p,p .
k
Step 3: Update the system matrices
Update RLS:
−1
M k = λ1 M k −1 − λ1 M k −1 φk −1 λ2 I + φkT−1 M k −1 φk −1 φkT−1 M k −1 ,
2
2
(y )
(y )
(y )
Θk = Θk −1 + y k −1 − Θk −1 φk −1 φkT−1 M k .
(y )
Compute e k −1 = y k −1 − Θk φk −1 .
Update RLS:
−1
N k = λ1 N k −1 − λ1 N k −1 ψk −1 λ3 I + ψTk −1 N k −1 ψk −1 ψTk −1 N k −1 ,
3
3
(x )
(x )
(x )
Θk = Θk −1 + x k − Θk −1 ψk −1 ψTk −1 N k .
3.3
Practical implementation in real time
In this section, practical implementation issues are discussed and solutions are
presented. In the first subsection, regularization is introduced to overcome the typical
ill-conditioned Markov estimation problem. In the second subsection, faster and
more numerical stable algorithms are presented for real-time implementation.
60
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
3.3.1
Regularization of the Markov estimation problem
In Assumption 2.2 an approximation is made which requires the past window to
be chosen as large as possible. However, in practical experiments it is most of
the time needed to design the excitation signal such that it does not exceed the
load specifications and ensures that the system to be identified operates around a
particular operation point. In this case, especially with a low-order controller in the
feedback loop, the matrix Pk becomes ill conditioned, because the input signals do
not persistently excite the system. Apart from just selecting a lower past window size,
a number of other solutions exist to get a well-conditioned matrix Pk for the least
squares update in (3.2). The first possible solution is generating more persistently
exciting control signals by changing the design properties of the controller. For
example automatically generated constraints can be added to the control design
problem when the matrix Pk becomes ill conditioned, see Dong (2009). The second
possible solution is utilizing a VARMAX model, for which the input-output description
relaxes the need for a very large past window due to the formulation of a deadbeat
predictor, see (Houtzager et al., 2009b). However, a non-linear problem is formulated
which can be solved recursively by the Extended Least Squares (ELS) method. The
third and well-used solution to the ill-posed least squares problem is the inclusion of
a regularization quantity in the cost to be minimized. This solution has been used for
the experimental study in Section 3.5.
In the batchwise case, Tikhonov regularization from (Tikhonov and Arsenin, 1977)
can be included by replacing the least squares estimation of the Markov parameters
in Algorithm 2.1 with:
−1
Ξ̃ = Y ΨT ΨΨT + µI
,
(3.10)
where µ is the regularization parameter. The Tikhonov regularization of the information matrix corresponds in the recursive case to the normalization of the covariance
matrix Pk . When we let the RLS filter iterate for k = 0 . . . N − 1, the covariance matrix
at time step N is described by:
PN−1−1 = λN
1 ρ1 I +
N
−1
X
λ1N −k +1 ϕk ϕkT ,
(3.11)
k =0
where the affect of the initial regularization ρ1 fades away when N increases. To
keep the covariance matrix normalized over time, we want a non-fading regularization
as (Tsakiris et al., 2010):
N
−1
X
(3.12)
λ1N −k +1 ϕk ϕkT .
P̄N−1−1 = µI +
k =0
When the effect of the initial regularization is neglected, the normalized covariance
matrix is related to the non-normalized covariance matrix by:
P̄k = Pk I + µPk
−1
.
(3.13)
So Tikhonov regularization can be included by replacing Pk with P̄k in (3.2).
However the computation is very costly for each iteration and still needs the illconditioned matrix Pk to be built. Although faster calculations are possible by
61
3.3 Practical implementation in real time
approximations, modification of the covariance matrix update equation is preferred
instead. Consider the following update of the information matrix:
T
P̄k−1 = λ1 P̄k−1
−1 + ϕk ϕk + 1 − λ1 µI .
By taking the inverse of (3.14) and by starting with P̄−1 =
covariance matrix as (Gunnarsson, 1994):
P̄k =
(3.14)
1
I,
µ
we can iterate the
−1 T
1
ϕk P̄k −1 +Q 1 ,
P̄k −1 − P̄k −1 ϕk λ1 I + ϕkT P̄k −1 ϕk
λ1
where
−1
Q 1 = − 1 − λ1 µP̄k I − 1 − λ1 µP̄k
P̄k .
From this iteration it is obvious that the addition of Q 1 directly affects the variation
of the estimated Markov parameters. The introduction of Tikhonov regularization not
only effects the conditioning of the covariance matrix but also the tracking ability of
the algorithm, see also Gunnarsson (1996). It is however still a problem that Q 1 is
given by an implicit relation.
To have the computational benefits of low-rank updates, we can update the
equation in (3.14) by adding (1 − λ1 )µ in m = p (ℓ + r ) + r low-rank updates (Gay,
1995). If we consider only one additional low-rank update at each iteration, we get
an approximated covariance matrix update as follows:
T
T
P̄k−1 = λ1 P̄k−1
−1 + ϕk ϕk + ξk ξk .
where
ξk = · · ·
0
p
(1 − λm
1 )µ
|
{z
}
0
(3.15)
···
T
.
(k mod m )+1th column
By taking the inverse of (3.15) and by starting with P̄−1 =
covariance matrix as:
1
I,
µ
we can iterate the
−1 T
1
ϕk P̄k −1 ,
P̄k −1 − P̄k −1 ϕk λ1 I + ϕkT P̄k −1 ϕk
λ1
−1 T
P̄k = P̂k − P̂k ξk I + ξTk P̂k ξk
ξk P̂k .
P̂k =
Using only one low-rank update at each iteration causes the regularization value to
vary periodically with period m . The shape of the variation becomes a sawtooth wave
with an amplitude of (1 − λm
1 )µ. If the forgetting factor λ < 1 is chosen such that
m ≪ 2/(1 − λ1 ), one can obtain a reasonably steady amount of regularization.
3.3.2
Computational cost
For simplicity, Algorithm 3.1 shows the conventional implementation of the RLS filter.
In Verhaegen (1989) it is recommended for practical use to implement a square-root
type of implementation instead of the referred conventional RLS filter implementation
due to better numerical robustness against round-off errors. The array methods are
62
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
powerful variants of square-root RLS that perform the computations in a reliable manner using a sequence of elementary Givens and/or hyperbolic rotations, see for stable
implementations (Chandrasekaran and Sayed, 1996; Sayed, 2008). Another advantage
is that array methods can be used to exploit any shifting structure in the data. This
can become very beneficial if you have to select a large past window size. The fastarray RLS scheme in Algorithm 3.2 exploits the shifting structure of the data vector
ū k −p and ȳ k −p in (2.19). This results for the main linear problem in a computational
load in the order of O((r + ℓ)2 p ), instead of O((r + ℓ)2 p 2 ). As Algorithm 3.2 shows,
it is possible to implement r + ℓ low-rank regularization updates in a similar shifted
structure at the cost that the computational load becomes O((r + ℓ)3 p ). Further, to
fit the Tikhonov regularization in the shifted structure, the iteration starts with the
regularization matrix in the form P̄−1 = µ1 diag λp −1 I , λp −2 I , . . . , I instead of P̄−1 = µ1 I .
Algorithm 3.2 (The fast-array RLS filter with regularization).
input Ξ−1 , λ1 , µ
require 0 ≪ λ1 ≤ 1, µ > 0
–
init γ−1 = 1, χ j ,−1 = 1, L̄ −1 =
1
p
µ
0
p p
λ1
0
···
I
™T
0 ··· 0
for k = 0, 1, 2, . . .
input u k , y k
—T
”
Shift: ϕk = ϕkT−1 (r +ℓ+1:(r +ℓ)p,:) y kT−1 u kT .
Find a J -orthogonal matrix Φ using (hyperbolic) Givens rotations,
with
J = diag (1, −I , I ), such 
that: 


−1/2
γk −1
–
™
−1/2
 g
 k −1 γk −1
0
p1 ϕ T L̄ k −1
λ1 k

−1/2
– γk
™
Φ = 
0

p1 L̄ k −1 
−1/2
λ1
g k γk
−1/2
1/2 T
Ξ̃k = Ξ̃k −1 + y k − Ξ̃k −1 ϕk
g k γk
γk
.
if λ1 < 1
for j = 1, 2, . . . , r + l
if k mod p + 1 = p
Æ
p
(1 − λ1 )µ
ǫ j ,k = · · · 0
|
{z
}
0
···
0
.
L̄ k 
j th column
else
”
ǫ j ,k = 0
···
—
0
—T
”
Shift: ξ j ,k = ξTj,k −1 (r +ℓ+1:(r +ℓ)p,:) ǫ j ,k .
Find a J -orthogonal matrix Φ using (hyperbolic) Givens rotations,
with J = diag (1, −I , I ), such
 that:



−1/2
– χ j ,k −1 ™
−1/2
 h
 j ,k −1 χ j ,k −1
0
−1/2
ξTj,k L̄ k 
– χ j ,k
™
Φ = 
0

L̄ k 
−1/2
h j ,k χ j ,k
0
.
L̄ k 
63
3.3 Practical implementation in real time
The recommended inclusion of the transformation in Lemma 2.1 increases the
computational cost of the method considerably, especially when the future window
size f is large. Using Gauss-Jordan elimination for calculating the inverse of the
matrix H̃, the computational complexity of the multiplication becomes O(ℓ3 f 2 p ).
Fortunately, a faster algorithm can be obtained by exploiting the structure of the
Markov parameters in the lifted matrices. In Dong et al. (2008), a fast algorithm is
described to apply this transformation on (3.3) which is given by:
ΓL̃ ΓK̃
(i )
j (i ) X (y
Ý
Ý
Ξ̃ k −i +j ) ΓL̃ ΓK̃ ( ) ,
L Γ
K +
= Γ
i −1
j =1
where
with i = 1, . . . , f denotes the i th block row. The computational load is then
only O(ℓ2 f p ), which is a reduction compared with Gauss-Jordan elimination.
[·](i )
A comparison of the computational time needed for six RSID schemes to compute the system matrices with different past (and future) window sizes is given in
Figure 3.1. The methods EIVPM and EIVsqrtPM (square-root version of the previous)
are found in Mercère et al. (2008). The proposed method is denoted by RPBSIDpm
where the solid line is the square-root implementation and the dashed line is the
implementation with the fast-array method. Further we denote RPBSIDpast as the
method where the PM method is replaced with the PAST method to estimate the
weight. From the figure, it is clearly visible that the proposed RPBSIDpm method with
fast-array RLS has the smallest computation time, especially for large past window
sizes. The RPBSIDpast method is a bit slower due to the implementation of an
additional adaptive filter.
EIVPM
EIVsqrtPM
RPBSIDpast
2.5
RPBSIDpast,fast−array
RPBSIDpm
RPBSIDpm,fast−array
time [s]
2
1.5
1
0.5
5
10
15
20
25
30
35
40
45
50
past window size p
Figure 3.1: Computational time of EIVPM, EIVsqrtPM, RPBSIDpast , and RPBSIDpm
(dashed line is with fast-array RLS) for different past window sizes. N = 1000
64
3.4
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
Simulation study
In order to evaluate and compare the closed-loop performance of the proposed RSID
method with the existing methods EIVPM (Mercère et al., 2008) and VPC (Wu et al.,
2008), the state-space system from Mercère et al. (2008) is used:

0.8

x k +1 =  0
0
–
0.5
yk =
0
−0.4
0.3
0
0.5
0




0.2
0
0
0.055




−0.5 x k +  0 −0.6 u k +  0
0.5
0.5
0
0
™
™
–
0.025
0
0
v ,
x +
0
0.03 k
1 k
0
0.05
0

0

0  wk ,
0.045
(3.16)
where v k and w k are white noise sequences of variance 1. To create a closed-loop
system, a time-varying state-feedback control law is applied as described in Ogata
(1994) that stabilizes the above system over the whole trajectory. For sufficient
excitation, a white noise sequences of variance 1 is added to the input signal, therefore
no regularization is required. The EIVPM (Mercère et al., 2008) method is a fast
recursive version of the MOESP (Verhaegen and Verdult, 2007) method, where the QR
factorization of the data matrix is updated by low-rank updates, the A and C matrix are
obtained through an estimated observability matrix using the PM algorithm, and the B
and D matrix can be obtained using an additional RLS algorithm. The VPC (Wu et al.,
2008) method is a recursive version of the SSARX (Jansson, 2003) method, where the
VARX parameters are obtained by the RLS algorithm, and the A, B, C, and D matrix are
obtained through an estimated state vector using the PAST algorithm. Two practical
situations are considered to evaluate and compare the tracking performance: a slowly
time-varying case and an abrupt-change case.
3.4.1
Slowly time-varying case
After a time-invariant phase of 665 samples, the following state matrix is used:
€
A + diag −0.3,
−0.5,
Š exp (−(k − 665)/2000) − 1
0.2
.
exp (−1) − 1
(3.17)
Thus the poles of the system drift from {0.3, 0.5, 0.8} during the next 1335 samples
and then becomes time invariant again. The estimated pole trajectories averaged over
the 100 MCS are displayed in Figure 3.2. As expected the open-loop EIVPM method
gives biased results, because the system to be modelled operates in closed-loop. The
EIVPM method cannot handle the problem that the future inputs are correlated with
past noises. The VPC and the presented RPBSIDpm method follow the eigenvalue
trajectories much better. From the averaged responses, it visible that VPC has more
difficulties and sometimes gives some biased results after the change, because the
PAST algorithm changes the state basis. Further, it was observed from the simulations
that the variance around the mean trajectories is much larger for the VPC method
compared with the RPBSIDpm method.
65
3.4 Simulation study
0.8
0.7
True and estimated poles
0.6
0.5
0.4
TRUE
EIVPM
VPC
RPBSIDpm
0.3
0.2
0.1
0
-0.1
-0.2
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
Samples
Figure 3.2: Trajectories of the estimated poles using EIVPM, VPC, and RPBSIDpm in
a slowly changing environment. The following parameters are used: λ1 = λ2 = 0.98,
f = p = 5, SNR = 25dB, and MCS = 100.
0.8
True and estimated poles
0.7
0.6
TRUE
EIVPM
VPC
RPBSIDpm
0.5
0.4
0.3
0.2
200
300
400
500
600
700
800
900
1000
1100
1200
Samples
Figure 3.3: Trajectories of the estimated poles using EIVPM, VPC, and RPBSIDpm in an
abrupt time-varying environment. The following parameters are used: λ1 = λ2 = 0.98,
f = p = 5, SNR = 25dB, and MCS = 100.
66
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
3.4.2
Abrupt-change case
After a time-invariant phase of size 665 samples, the following abrupt change A(3, 3) =
0.65 in the state matrix is applied. Thus the pole 0.5 shifts to 0.65. The estimated pole
trajectories averaged over the 100 Monte-Carlo Simulations (MCS) are displayed in
Figure 3.3. Similar as in Subsection 3.4.1, the open-loop EIVPM method gives biased
results. In the figure, VPC shows a faster convergence than RPBSIDpm in spite of some
bias after the change. The PAST algorithm makes this method more adaptive directly
after the abrupt change. This comes at the cost that the change in the state basis
causes afterwards some loss in performance. Further, it is shown in simulation that
the individual responses of RPBSIDpm are again much smoother.
3.5
Experimental study on 2D-airfoil system
In this section we compare the accuracy of the models from the proposed RPBSID
method with different settings on a 2D-airfoil system operating in closed loop. The
2D-airfoil system can become unstable when the wind speed reaches a certain wind
speed; the so-called flutter limit. First we present the experimental setup used to show
the feasibility of flutter detection with the advanced identification method. Second,
we describe the batchwise identification experiment to obtain an accurate model for
the verification and control synthesis. Third, the proposed RPBSID method is applied
on the 2D-airfoil system where the wind speed reaches the flutter limit.
Figure 3.4: Photo of the 2D airfoil in front of the wind generator
3.5 Experimental study on 2D-airfoil system
3.5.1
67
Description of the experiment
The experimental setup mainly consists of the following components: wind generator,
blade, actuator and sensors, and real-time environment. The wind generator blows air
with considerable turbulence into a tunnel section with the 2D airfoil hanging inside.
The diameter of the outlet of the wind generator is 350mm and the tunnel section is
400mm by 400mm. The maximum attainable air speed is around 15m/s.
The 2D airfoil that we use for our experimental validation is an airfoil with at the
trailing edge a control surface, the so-called trailing-edge flap. An illustration of the
blade with trailing-edge flap is given in Figure 3.4. The blade is at the top connected
to a half meter long aluminium plate with the other end fixed to a rigid frame. As
stated in the name “2D airfoil” the blade has two degrees of freedom. The plate
allows the blade only to move in the flapwise and torsional direction. The flutter limit
of the 2D airfoil is around 12.5m/s. The exact flutter limit varies from day to day
with 0.5m/s depending on the temperature and the density of the air. In Figure 3.5
and 3.6, schematic representations of the 2D-airfoil cross section and sideview are
provided. The airfoil is shaped from soft foam and cut in two halves, where each halve
is attached to a side of an aluminium plate and covered with a plastic skin to provide
a smooth surface. The shape is based on the NACA 0012 airfoil with a chord of 200mm
and a span of 300mm. The NACA 0012 airfoil is a symmetrical, low-drag airfoil, and
12% thick (thickness divided by chord). The total mass of the blade is 0.7kg. The
dimensions of the aluminum plate are 500×60×1mm. The flap also spans the 300mm
and covers one third of the chord length (67mm). The active part of the trailing-edge
flap consists of a THUNDER™ TH-7R actuator (FACE International). These are piezoelectric based benders which deflect several millimeters under the application of a
voltage ranging from -300V to 700V. However, for safety reasons we limit the maximum
voltage from -250V to 250V. The actual defection depends on the structure around the
piezo-electric based bender and the aerodynamic loading.
For control purposes, the 2D airfoil is equipped with sensors which measure
the dynamic behaviour of the blade. Since the final goal for this experiment is to
detect flutter without destabilizing the 2D-airfoil system, two Macro Fiber Composite
(MFC) (Smart Material) patches are adhered to the root at the backside of the half
meter long aluminium plate under an angle of 45 degrees to measure the high strains
associated with the first flapwise and torsional bending mode. Thus two strain sensors
will measure approximately the same strain under flapwise bending, but will have
an opposite sign under torsional bending. The main advantage of an MFC is that
no amplification is required to have a good signal-to-noise ratio. However, with the
MFC it is not possible to do static measurements due to the capacitance behaviour
of the MFC. This high-pass behaviour is desirable for this experiment, as we want
to control the dynamic behaviour of the system, rather than the static deformations.
The trailing-edge flap does not stabilize the 2D-airfoil system when there is no control
enabled. The controller intelligence and data acquisition capability are added with a
dSPACE™ system. The controller and data acquisition scheme are fully developed in
the MATLAB™ (Mathworks, a) and SIMULINK™ (Mathworks, b) environment and then
compiled to the dSPACE™ (dSPACE) chip. On a separate computer, all the signals are
monitored using CONTROL DESK™ (dSPACE) and also the control and identification
parameters can be adjusted in real-time.
68
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
L
M
α
kα
c.g.
β
kh
h
Figure 3.5: Schematic representation of the 2D airfoil cross section
60
front
back
45o
500
70
300
67
200
Figure 3.6: Schematic representation of the 2D airfoil sideview. The dimensions are
given in millimetres.
3.5 Experimental study on 2D-airfoil system
3.5.2
69
Experimental modelling and control
Due the instability of the 2D-airfoil system, we have to apply feedback control. The
main issue is that the 2D-airfoil dynamics (the state matrix A and input matrix
B) are rather complex and strongly dependent on the free-stream wind speed and
will necessitate model-based controller design. A model for modern model-based
controller design is a mathematical model normally governed by (preferably linear)
differential equations. For controller synthesis this model should only contain the
relevant dynamics between the input and the outputs, and should be accurate over
the bandwidth of the controller. Analytical modelling has been performed in the
design stage using the theory in Theodorsen (1935), which strongly depends on a large
number of parameters that determine the dynamic behaviour of the system. Most
of these parameters can be roughly estimated or calculated. Still, a large amount of
uncertainty is present; this makes it difficult to design a stable feedback controller
based on such models. This motivates that predictor-based subspace identification
can become a necessary building block for high bandwidth controller design.
The control loops in the prototyped 2D-airfoil system are the transfer functions
between the trailing-edge flap (u ), and the two backside MFC sensors (y ). Identification can be done for different operating points. First, we performed open-loop
batchwise identification experiments for six stable operating conditions from 6m/s
to 12.5m/s. The identified model with the operating point nearest to the flutter
limit of 12.5m/s is used as an initial model for the model-based controller design
in the unstable region. For this purpose, this state-space model of seventh order is
converted to the so-called modal form, where the pole locations of the torsion mode
are manually adjusted (by changing its frequency and damping value) to the expected
unstable pole locations just outside the unit circle. Second, we performed the
batchwise identification experiments in closed loop, including the unstable operating
conditions 12.7m/s and 13m/s. With these newly identified and unstable models the
model-based controller design is redone to obtain better stability and performance
properties. The identified pole locations and Bode diagrams in the discrete time for
a wind speed ranging from 9m/s to 13m/s are given in Figure 3.7 and Figure 3.8,
respectively. The first resonance frequency is located between 0.7–0.8Hz and is related
to the 1st flapwise bending mode. The second resonance frequency is located between
2.4–3.3Hz and is related to the 1st torsional bending mode. This is the mode that
becomes unstable above 12.5m/s. The third resonance frequency is located at 4.5Hz
and is related to the 2nd flapwise bending mode.
For the controller synthesis we use the so-called H2 method which is illustrated
in the well-known generalized plant setting in Figure 3.9. The goal of the feedback
controller is to suppress the unknown disturbances as much as possible with the
requirement that the system should remain stable and that the control input signal
is bounded between −250V and +250V. The method does this by minimizing the H2
norm between e k and {z k , y k } and is mathematically given by:
WU (I + CG WL )−1 H 2
,
(3.18)
min C
(I + G WL C )−1 H 2
where
G (z ) = D + C (z I − A)−1 B , H (z ) = R 1/2 + C (z I − A)−1 K R 1/2 ,
70
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
1
imaginary axis
0.8
0.6
0.2
0.7
0.6
0.5
0.6
0.7
0.8
real axis
0
-0.2
imaginary axis
imaginary axis
0.4
0.8
-0.4
-0.6
-0.8
-1
0
0.5
0.25
0.2
0.15
0.85
1
0.9
0.95
1
real axis
real axis
Figure 3.7: Identified pole locations, for a wind speed ranging between 9m/s (light
grey) and 13m/s (black). The boxes to the right show a magnification of two pole
locations. For PBSIDopt , closed loop, N = 15000.
from: u k
0
to: y 1,k
-20
-40
magnitude (dB)
-60
-80
0
to: y 2,k
-20
-40
-60
-80
10−1
100
frequency (Hz)
101
Figure 3.8: Identified Bode diagrams, for a wind speed ranging between 9m/s (light
grey) and 13m/s (black). For PBSIDopt , closed loop, N = 15000.
71
3.5 Experimental study on 2D-airfoil system
where the covariance and system matrices of the innovation form in (2.2) can be obtained using the Algorithm 2.1. In these objective functions we can embed manually
the loop shaping ideas in the weighting filters WL and WU . We choose them in such a
way that the resulting controller will not act on high frequencies in the error signal by
weighting the controller sensitivity by a filter with high gains at high frequencies. The
H2 controller is very effective at stabilizing, but it frequently encounters the actuator
saturation when operating in the unstable region. During input saturation the closedloop system can even become unstable. To overcome this problem, a so-called antiwindup feedback loop is designed, see Åström and Wittenmark (1997) and Figure 3.10,
which takes action when the control signal saturates.
dk
-
−
-
C
6
WL
zk
-
WU
H
uk
-
yk
+
+ ?
-
G
-
Figure 3.9: Block diagram of generalized plant to synthesize the H2 controller
−
6
?
C
6
+ −
uk
_
- \_
u ks a t
-
S
yk
-
Figure 3.10: Block diagram of closed-loop system with anti-windup loop
3.5.3
Experimental results
In this subsection, the main results of the experiments on the 2D-airfoil system
are presented. We present five cases that will discuss and show some features of
the proposed recursive predictor-based subspace identification method. When the
recursive predictor-based subspace identification method is applied on a system in
real-time, a large number of parameters have to be tuned. We have the past and future
window p and f , the selection matrix S, the forgetting factors λ1 , λ2 , and λ3 , and the
regularization parameter µ. In the following five cases some tuning guidelines are
provided by analysing and discussing the effects of these tuning parameters.
Effect of the past window size p
It is difficult to quantify the effect for the finite past window size p . First, there is
the effect of the truncation error for the VARX predictor, which equals to the sum of
72
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
the remaining tail of Markov parameters. Second, the matrix becomes ill conditioned
for large past window sizes, because normally in practice the input signals do not
persistently excite the system. To quantify this effect, we use the so-called condition
number of matrix ΨΨT , which equals the distance between the largest and smallest
nonzero singular value. In Figure 3.11, the tradeoff between the truncation error
and condition number is illustrated. Third, the number of Markov parameters to be
estimated grows linearly with the past window size. It is well known that an increasing
number of parameters to be estimated will increase the variance of the solution.
Lastly, the computational time of the Markov estimation increases considerably when
the past window becomes larger.
Effect of the future window size f and the weight matrix
The selection of the future window size f is even more difficult. The effect of this
parameter on the identification is known to heavily depend on the input spectrum
and system properties. In Chiuso (2010), it is suggested through simulation studies
that a large future window is often better for identification experiments when the
Signal-to-Noise Ratio (SNR) is low, due to the so-called averaging effect. Otherwise for
large SNR, the optimal future window size is normally the smallest value. However,
from our experiments it is observed that the future window size does not only depend
on the above properties, but also heavily on whether the transformation in Lemma 2.1
is applied.
From practical experience with different systems it is found that applying the
transformation first with the weight matrix Wk = H̃k−1 gives two advantages. First,
applying the weight allows the user to select a smaller future window, which has
a positive effect on the computational time. Second, the estimation of the system
matrices behaves more smoothly over time, as the system matrices are estimated
in a more constant state basis, especially when a lot of forgetting is applied. The
construction of a predefined selection matrix S can be a challenge if the matrix Γ is
not known in advance. As discussed in Mercère et al. (2008), for simple MISO systems
the matrix S can easily be chosen as the identity matrix, such that the first n rows
of SΓ are linearly independent, as the one output observes all the states. For MIMO
systems this is generally not the case, because it is possible that one of the outputs
does not observe a particular state. A solution is given in Mercère et al. (2008), where
the selection matrix S is constructed in such a way that it replaces for each future
step one of the outputs by a combination of the outputs. As described in Mercère
and Bako (2011), this solution will bring the estimated state-space model close to
canonical form. It is well known that the canonical form is very sensitive to numerical
errors, therefore this solution was not very reliable for the system in practice. For the
experiment, we construct the matrix ΓL̃Ūp + ΓK̃Ȳp from a batch of data and applied
a rank-revealing QR decomposition. The row permutations show which rows are the
most important. An alternative is to apply, as suggested in Chiuso et al. (2010), an SVD
and take the first n columns of the left nullspace as: S = UnT .
In Figure 3.12 the effect of the future window size on the accuracy of the identification is illustrated for both the batchwise and the recursive case. The accuracy of the
identification is evaluated by looking at the Variance-Accounted-For (VAF) on a data
73
3.5 Experimental study on 2D-airfoil system
truncation error
4
2
condition number
4 ×106
6
2
0
0
10
20
30
40
50
60
70
80
0
100
90
past window size p
Figure 3.11: The truncation error of a VARX predictor and the condition number of
matrix ΨΨT against the past window size. N = 5000, V = 11.7
60
55
VAF [%]
50
45
40
35
30
5
10
15
20
25
30
35
40
45
50
future window size f
Figure 3.12: The VAF against the future window size. Solid lines are for PBSIDopt
and dashed lines are for RPBSIDpm . The light grey is without and dark grey with
transformation H̃−1 applied. N = 5000, V = 11.7
74
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
set different from the data set used for determining the model. The VAF is defined as:
vaf y k , ŷ k
var y k − ŷ k
, 0 · 100%,
= max 1 −
var y k
where ŷ denotes the output signal obtained by simulating the identified model, y
is the measured output signal, and var denotes the variance of a quasi-stationary
signal. It is observed that when the transformation is not applied, the best accuracy
for both the batchwise and the recursive case is obtained when the future window
size is chosen close to the past window size. If the transformation is applied, then the
optimal future window size is normally the smallest value. Moreover, in the recursive
case the transformation with the smallest future window gives the best accuracy in
most cases. For the batchwise case it is the other way around; the largest future
window without transformation gives the best accuracy in most cases.
Effect of the forgetting and the regularization
The choice of the forgetting factor is conceptually simple. Select it so that the covariance matrix contains more recent data that are relevant for the current dynamical
2
representation of the system. The data within a window length of N = (1−λ)
accounts
for approximately 90% of the data contained in the covariance matrix, and therefore
this relation can be used as an initial guess for the forgetting factor. The trade-off is
between the tracking capability, which benefits from more forgetting, and the noise
sensitivity, which improves with less forgetting. It can be noticed that the last two
adaptive filters are updating their estimate with data containing possible errors from
the adaptive filter before them. It is well known that least squares estimates are very
sensitive to possible errors in the variables. Therefore the data is somewhat misused,
especially in the initial transient period, and should therefore be less weighted than
later measurements of which the information contents are more accurate. Further,
we consider a constant forgetting factor for each adaptive filter. An adaptive choice of
λ could be implemented, although it should also take into account that the changes
in λ directly influence the size of the regularization.
If regularization is necessary, as in our experimental study, the Tikhonov regularization parameter µ can be determined from a batch of data obtained around an
operation point using the L-curve or Generalized Cross Validation (GCV) criterion,
see Hansen (1992). When applying regularization it is strongly advised to scale the
input and output signals, especially with MIMO systems. We have scaled for example
the signals before the algorithm such that var (u k ) = 1 and var (y k ) = 1. For recursive
estimation, especially when the SNR changes considerably over time, as is the case
in flutter experiment, the optimal regularization parameter changes considerably
over time and should preferably be made adaptive. In Figure 3.13, the trend of the
regularization parameter µ using the GCV criterion is illustrated during the flutter
detection experiment. Further, the regularization parameter can be lowered during
estimation to increase the sensitivity to any changes, therefore it can also be used to
alter the tracking capability of the algorithm.
75
3.5 Experimental study on 2D-airfoil system
Batch vs. recursive identification
We compare the recursive identification scheme with the original batchwise identification scheme by looking at the accuracy of the estimated models. The identification
is done with a Generalized Binary Noise (GBN) signal with an amplitude of 200V on
the flap actuator. The data of 5000 samples is acquired with a sampling rate of 20Hz.
The accuracy of the identification is evaluated by looking at the Variance-AccountedFor (VAF) on a data set different from the data set used for determining the model.
The VAF values are lower for the higher wind speeds. This is caused by the fact that
with similar excitation the contribution of the turbulence is larger in the measured
signals and consequently the SNR is lower, which has a direct effect on the VAF.
The results of batchwise and recursive identification experiments in the stable
operating region can be found in Table 3.1. We see, as expected, that the batchwise
identification scheme is superior to the recursive identification scheme for all the
wind velocities. By comparing the experiments with and without transformation, we
can see that applying the weight gives better results in the recursive case. In the
batchwise case it is the opposite, without transformation gives the best VAF results.
Thus the benefits of applying the transformation of Lemma 2.1 are only observed
for the recursive predictor-based subspace identification method. The explanation is
that applying the weight in the recursive algorithm causes the system matrices to be
estimated in a constant basis, therefore giving a positive effect on the accuracy. This
effect is not applicable for the batchwise algorithm.
Table 3.1: Variance-Accounted-For (VAF) for the stable operating points. N=5000
VAF[%]
V [m/s] SNR
9.1
19.6
10.6
12.8
11.2
10.4
11.7
5.6
12.2
3.6
12.5
2.1
PBSIDopt
RPBSIDpm
f =5,W =H̃−1
f =50
f =5,W =H̃−1
f =50
88.5
76.4
66.0
52.9
30.3
17.4
88.7
76.6
66.2
53.7
35.0
17.5
87.9
76.1
65.8
47.8
30.1
11.5
87.3
75.1
65.2
46.9
29.3
9.4
Tracking of the flutter dynamics
Introducing forgetting in the recursive predictor-based subspace identification problem makes the recursive method suitable for detecting small and slow changes in
the flutter dynamics. We show the effectiveness of the proposed algorithm to this
application. With the wind generator a gust of wind is created according to the
wind-speed profile with period of T = 900s and specified in IEC (2004), see also the
measured air velocity given in Figure 3.14. As the wind speed crosses the flutter
limit of 12.5m/s, the identification experiment is done in closed-loop. Using the
modern model-based controller synthesis presented in Subsection 3.5.2, a robustly
stabilizing controller is designed for the operating region between 11m/s and 13m/s.
The identification experiment is done with an additional external GBN signal rk with
an amplitude of 100V, see Figure 2.1. The data is resampled to a sampling rate of 20Hz.
76
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
400
regularization parameter µ
350
300
250
200
150
0
100
200
300
400
500
600
700
800
900
1000
time [s]
Figure 3.13: Regularization parameter µ over time using the GCV criterion. The value
of the regularization parameter is updated every 2.5s (50 samples).
13.5
wind speed [m/s]
13
12.5
12
11.5
11
0
100
200
300
400
500
600
700
800
900
1000
time [s]
Figure 3.14: Measured wind-speed profile (light grey) and filtered wind-speed profile
(dark grey) of the applied wind gust. The dashed line resembles the flutter limit. Wind
speeds higher then the limit will bring the airfoil to flutter.
3.6 Conclusion
77
The torsional bending is the mode of interest, because this mode becomes unstable during flutter. In Figure 3.15, we can see pole locations of the 1st torsional
bending mode during the wind gust using the RPBSIDpm algorithm compared with the
pole locations obtained from batchwise identification experiments around multiple
operating points. From this figure we can see that the recursively estimated pole
locations follow the expected pole trajectory quite closely. In Figure 3.16, we can
see the estimated natural frequencies and modal damping values during the wind
gust. The torsional bending frequency is smoothly estimated with a delay between
50–100s compared with the expected frequencies values. The RPBSIDpm method
estimates the damping values less smoothly, but the damping values take as expected
a negative value, again with a delay between 50–100s, when the wind speed crosses
the flutter limit. Thus after some delay, the RPBSIDpm algorithm successfully detects
the instability when the 2D airfoil is operating in flutter. The large delay time was
expected, because of the low SNR a high value for the forgetting factor was selected
to overcome the sensitivity to the noise. It is also noted that the delay time builds
up in threefold, because the three RLS filters are in cascade. The delay time can be
reduced considerably by selecting a lower forgetting factor, or in smaller amount by
selecting a lower regularization value, at the cost of a less accurate estimation. To
lower the delay time, one could also investigate the use of flutter detection algorithms
where the damping values are not directly calculated from state space, but rather a
statistical threshold is derived to determine if the system is in flutter, for example
as done in Mevel et al. (2003). As described in Basseville et al. (2007), these flutter
detection algorithms have considerable smaller delay time compared with online
subspace identification, however less accuracy and robustness to turbulent excitation.
3.6
Conclusion
In this chapter we presented a novel recursive predictor-based subspace identification
method which is a recursive version of the ”optimized” predictor-based subspace
identification method for batch data. The recursive identification is performed via the
subsequent solution of only three linear problems, which are solved using recursive
least squares. The recursive implementation of the method is not only able to
identify linear time-invariant models from measured data in open or closed loop,
but can also be used to track slowly time-varying dynamics if adaptive filters are
used. The computational complexity is reduced by exploiting the structure in the
data equations and by using array algorithms to solve the main linear problem, such
that the proposed method can be implemented in real-time. We have shown in
the experiments that a certain weight with transformation is required to minimize
variations of the particular state used and provided some guidelines on how to pick
the free parameters in the algorithm. Moreover, the chapter suggests to use Tikhonov
regularization in order to overcome possible numerical problems in practice (and
therefore give better results in some sense), such as the singularity of the regression
problem. The real-time implementation and the ability to work with multi-input and
multi-output systems operating in closed loop make this approach suitable for online
estimation of unstable dynamics. Flutter detection is one of the applications of this
approach and we have demonstrated this on an experimental 2D-airfoil system. We
can track after some delay the natural frequencies and modal damping values.
78
Chapter 3: Recursive ID with Application to the RT Closed-Loop Tracking of Flutter
0.8
0.75
V = 11.2
imaginary axis
V = 11.7
V = 12.2
V = 12.5
V = 13
V = 12.7
0.7
0.65
0.6
0.65
0.7
real axis
Figure 3.15: The light-grey dots are the pole locations of the 1st torsional bending
mode during the wind gust with RPBSIDpm , W = H̃−1 , f = 5, λ1 = 0.9993, λ2,3 = 0.9992.
The dark-grey dots are the pole locations from the PBSIDopt experiments around
multiple operating points in Figure 3.7.
frequency [Hz]
2.7
2.6
2.5
0
100
200
300
400
0
100
200
300
400
500
600
700
800
900
1000
500
600
700
800
900
1000
damping
0.02
0.01
0
-0.01
time [s]
Figure 3.16: The dark-grey lines are the frequencies (top) and damping values
(bottom) of the 1st torsional bending mode during the wind gust with RPBSIDpm ,
W = H̃−1 , f = 5, λ1 = 0.9993, λ2,3 = 0.9992. The light-grey lines are the interpolated
frequencies (top) and damping values (bottom) calculated from the pole locations
from the PBSIDopt experiments in Figure 3.7.
CHAPTER
4
Rejection of Periodic Wind
Disturbances on an Experimental
Smart Rotor Test Section
A repetitive control method is presented that is implemented in real time for
periodic wind disturbance rejection for linear systems with multiple inputs
and multiple outputs and with both repetitive and non-repetitive disturbance
components. The novel repetitive controller can reject the periodic wind
disturbances for fixed-speed wind turbines and variable-speed wind turbines
operating above-rated and we will demonstrate this on an experimental
“smart” rotor test section. The “smart” rotor is a rotor where the blades
are equipped with a number of control devices that locally change the lift
profile on the blade, combined with appropriate sensors and controllers. The
rotational speed of wind turbines operating above-rated will vary around a
defined reference speed, therefore methods are given to robustify the repetitive controllers for a mismatch in the period. The design of the repetitive
controller is formulated as a lifted linear stochastic output-feedback problem
on which the mature techniques of discrete linear control may be applied.
For real-time implementation, the computational complexity can be reduced
by exploiting the structure in the lifted state-space matrices. With relatively
slow changing periodic disturbances it is shown that this repetitive control
method can significantly reduce the structural vibrations of the “smart” rotor
test section. The cost of additional wear and tear of the “smart” actuators
are kept small, because a smooth control action is generated as the controller
mainly focuses on the reduction of periodic disturbances.
4.1
T
Introduction
load disturbances acting on an individual wind turbine blade are to a large
extent deterministic; for instance, tower shadow, wind shear, yawed error, unbalance, and gravity, and they depend on the rotation angle and speed. The contribution
HE
79
80
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
of these periodic disturbances is becoming even larger, due to the increasing length
and mass of the rotor blades. Repetitive controllers consist of a periodic signal
generator, enabling rejection of only these periodic disturbances. Repetitive Control
(RC) methods for discrete Multiple-Input and Multiple-Output (MIMO) systems that
can potentially expand the application to wind turbines greatly have been developed
in the time-domain, like in de Roover et al. (2000), and in the trial domain using lifted
models, see Lee et al. (2001). A recent overview of past literature in RC can be found
in Wang et al. (2009). RC methods can achieve perfect reduction in the deterministic
case, when excluding non-repetitive disturbances. In Rice and Verhaegen (2010b), a
lifted RC method is presented for MIMO linear systems with both repetitive and nonrepetitive disturbance components. This design method is formulated as a lifted linear
stochastic output-feedback problem on which the mature techniques of discrete
linear control may be applied. The computational complexity is even considerably
reduced by exploiting the structure in the lifted state-space matrices.
An important drawback of RC is that a small mismatch between the controller
period and the actual period of the disturbance signal can decrease the performance
of RC substantially. Several approaches have been proposed in the literature to
improve robustness and are summarized in Gupta and Lee (2006). These approaches
can be divided into two groups, depending if the true rotor speed is accurately
measured or not known. In this paper, we focus on the latter as this is the case with
the “smart” rotor test section. Although it is expected that RC performs better with
the approaches that require the rotor speed to be measured. In Rice and Verhaegen
(2010b), a weight is included in the formulated lifted output-feedback problem to
specify the amount of variation in the periodic wind disturbances. In Gupta and
Lee (2006); Pipeleers et al. (2008); Steinbuch (2002); Steinbuch et al. (2007) the RC is
extended with more memory loops. This so-called high-order RC can be made more
robust for small changes in period time, but this comes at the cost of the amplification
of noise at the non-harmonic frequencies. Using these developments, the application
of RC to wind turbines with small variations in rotational speed, such as fixed-speed
and variable-speed operating above-rated with a pitch controller that will keep the
rotational speed of the wind turbine close to the reference speed, becomes more
practical.
The contributions of this chapter are as follows. First, we propose a RC method
which can deal with periodic wind disturbances, and leave the non-periodic disturbances (turbulence) mostly unaffected. This will generate a smooth control action
that will keep the additional wear and tear on the “smart” actuators small. Second,
we implement and study two extensions (random walk weight, multiple memory
loops) in the lifted output-feedback problem to robustify the performance against
period mismatch. Third is the experimental verification of RC on a “smart” rotor test
section with periodic and turbulent wind disturbances generated by a wind generator.
In addition, we verify the capability of RC to operate with an additional feedback
controller, which is present in modern wind turbines for stabilization. To the author’s
knowledge, this is the first experiment with an implemented lifted repetitive controller
to reduce the periodic wind disturbances (with small variations in the period time)
in a turbulent wind field, although repetitive controllers in the time-domain for
disturbance rejection have been implemented in real time before in Pipeleers et al.
(2009); Steinbuch et al. (2007); Tammi et al. (2007).
4.2 System description, assumptions and notations
81
The outline of this chapter is as follows. In Section 4.2 the problem formulation,
the assumptions made, and some notations are presented. In Section 4.3 the theoretical framework is presented for the RC problem and an algorithm for MIMO Linear
Time-Invariant (LTI) systems is given. In Section 4.4 the effectiveness of the proposed
algorithm is shown with an experimental study of disturbance rejection on a “smart”
rotor test section. Moreover, some tuning guidelines are derived. In the final section
we present the conclusions of this chapter.
4.2
System description, assumptions and notations
In this section the system description, the assumptions made, and some notation is
presented.
4.2.1
System description and assumptions
The dynamics of the discrete-time system S to be controlled can be written as the
following state-space model:
¨
x k +1 = Ax k + Bu k + F d k + w k ,
(4.1)
S
y k = C x k + Du k + v k ,
where x k ∈ Rn , u k ∈ Rr , y k ∈ Rℓ , d k ∈ R f , w k ∈ Rn , and v k ∈ Rℓ , are the state,
input, output, periodic disturbance, process noise, and measurement noise vectors.
The state-space matrices A ∈ Rn ×n , B ∈ Rn ×r , F ∈ Rn × f , C ∈ Rℓ×n , and D ∈ Rℓ×r are
also called the state, input, periodic input, output, and direct feedthrough matrices,
respectively. For simplification and clarification, we consider an LTI system. However
the lifted repetitive control method described in this chapter can be extended to
periodic time-varying systems.
The process and
noise
are considered Gaussian noise
” measurement
—
” disturbances
—
sequences with E v k v kT = R v and E w k w kT = R w , respectively. In the case of
coloured Gaussian noise sequences, the state-space system S can be augmented with
the noise filter dynamics. The periodic disturbance may consist of an integrated
“random walk” component as (Chin et al., 2004; Cho et al., 2005; Lee et al., 2001):
d k = d k −p + n k ,
(4.2)
”
—
where n k is a Gaussian noise sequence with E n k n Tk = R n , and p ∈ N+ is the trial
window size (or period). This is a quite versatile noise formulation; the disturbance
contains not only a stochastic and a periodic component, but also a random walk
nature, so the “periodic disturbance” can vary between trials. In addition, the noise
and periodic disturbances can be correlated in time.
We consider also the possibility that the system S already operates in closed loop
with a feedback controller for stabilization or load reduction due to turbulence. A
condition is that the feedback controller does not have poles on the unit disk, thus
the feedback controller should let the repetitive controller deal with the repetitive
82
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
disturbances. In this chapter, we consider the repetitive control input u k as an
additional input signal fed to the system, see Figure 4.1. As discussed in De Cuyper
(2006), the repetitive control input u k could also be fed as a reference signal to
the feedback controller, but this can require a redesign of the feedback controller
to include reference tracking capabilities. For a successful RC design, the statespace description in (4.1) should in both cases be replaced by its closed-loop statespace description. Thus the system S should also be augmented with a state-space
description of the feedback controller CFB .
dk
uk
−
6
CFB
+
+ ?
-
-
?
S
wk
?
vk
+
+ ?
-
yk
-
Figure 4.1: Block diagram of the system S operating in closed loop
4.2.2
Lifted system description and assumptions
With the trial window size p , the following stacked vector can be defined as:
”
—T
ȳ k ,p = y kT y kT+1 · · · y kT+p −1 .
(4.3)
The stacked vectors ū k ,p , d¯k ,p , n̄ k ,p , w̄ k ,p , and v̄ k ,p are defined in a similar way. Using
the definition of the stacked vectors, we can “lift” the system in (4.1) to the following
lifted description:
¨
x k +p = A p x k + Lū k ,p + F d¯k ,p + P w̄ k ,p ,
S̄
(4.4)
ȳ k ,p = Γx k + G ū k ,p + J d¯k ,p + Hw̄ k ,p + v̄ k ,p ,
where L ∈ Rn ×p r , F ∈ Rn ×p f , and P ∈ Rn ×p n are the extended controllability matrices,
and are given by:
”
—
”
—
”
—
L = A p −1 B · · · A B B , F = A p −1 F · · · AF F , P = A p −1 · · · A I ,
and G ∈ Rp ℓ×p r , J ∈ Rp ℓ×p f , and H ∈ Rp ℓ×p n are the impulse matrices with a lower
block triangular structure, Γ ∈ Rp ℓ×n is the extended observability matrix, and are
given by:




0
0
··· 0
C



.
..

 CA 
. .. 
C
0



Γ=
,
 ..  , H = 
.
.
.

.
..
. . 0
 . 
 .

C A p −1
C A p −2 C A p −3 · · · 0




D
0
··· 0
0
0
··· 0


.
.
..
..


. .. 
. .. 
D
0
 CB

 CF

G =
, J = 
.
..
..
..
..
..
..




.
. 0
.
. 0
.
.


C A p −2 B C A p −3 B · · · D
C A p −2 F C A p −3 F · · · 0
83
4.3 Repetitive control with lifted LQG design
These lifted system matrices are considered trial invariant. Note that I is used to
represent an identity matrix of appropriate dimensions.
In Assumption 4.1, we adopt the mild conditions commonly used for the lifted
repetitive control problem (Rice and Verhaegen, 2010b).
Assumption 4.1.
• The system S̄ is asymptotically stable (if not, stabilize with feedback control),
controllable on (A p , L), observable on (A p , Γ), and does not contain any zeros at
+1.
• The process noise w k and measurement noise v k are zero-mean white or coloured
(if so, augment dynamics to system) Gaussian sequences.
• The “random walk” noise n k is a zero-mean white Gaussian sequence.
When the number of inputs is smaller than the number of outputs (r < ℓ), the
controllability/observability condition in Assumption 4.1 is normally violated. In this
case the matrix G + Γ (I − A p )−1 L does not have full row rank. It is proposed to only
reduce the error that lies in the column space of that matrix, thus we try not to achieve
perfect rejection for all disturbance signals (or all the harmonics). To achieve this, we
use the Singular Value Decomposition (SVD) as:
p −1
G + Γ (I − A )
”
L = Us
–
— Σs
U⊥
0
0
Σ
™–
™
Vs
,
V⊥
(4.5)
where the order s is determined by detecting a gap that separates the s largest singular
values from the remaining ones. Now, we can replace the output equation of the lifted
system in (4.4) by a projected output equation given by:
UsT ȳ k ,p = UsT Γx k + UsT G ū k ,p + UsT J d¯k ,p + UsT Hw̄ k ,p + UsT v̄ k ,p .
(4.6)
Furthermore, we can select the number of singular values s not only on the controllability/observability condition, but also to further reduce the size of the RC outputfeedback problem and let the lifted repetitive controller focus on the main harmonic
components and therefore robustify the controller performance in some sense; see
also Subsection 4.4.3.
4.3
Repetitive control with lifted LQG design
In this section the theoretical framework for the repetitive control method to reject
periodic disturbances is presented. The goal is to formulate the RC design problem
into an output-feedback problem. After the formulation, the state-space matrices of
the lifted repetitive controller can be calculated by solving two Riccati equations.
84
4.3.1
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
Output-feedback formulation
To let the repetitive controller respond to periodic disturbances, a “non-stochastic”
output from (4.4) is defined as:
ỹ k ,p = ȳ k ,p − Hw̄ k ,p − v̄ k ,p = Γx k + G ū k ,p + J d¯k ,p ,
(4.7)
and a “non-stochastic” error is defined, which is independent of the noise sequence
from trial k (w̄ k ,p , v̄ k ,p ) as:
ẽ k ,p = r̄k ,p − ỹ k ,p ,
(4.8)
where r̄k ,p is considered trial invariant. The change between two trials is denoted
by the ∆ operator, such that for example the difference between the vector of the
input from time k and the vector of the input from time k − p is defined as ∆ū k ,p =
ū k ,p − ū k −p,p . Using this definition, it follows that the “non-stochastic” tracking error
is defined as:
ẽ k ,p = ẽ k −p,p − Γ∆x k − G ∆ū k ,p − J n̄ k ,p .
(4.9)
An expression for the current “stochastic” tracking error based on the previous “nonstochastic” error can be obtained and is given by:
ē k ,p = ẽ k ,p − Hw̄ k ,p − v̄ k ,p .
(4.10)
By using the ∆ operator again on the state as ∆x k = x k − x k −p , the difference in the
initial conditions of S̄ between trials is derived from (4.7), and (4.2) as:
∆x k +p = A p ∆x k + L∆ū k ,p + F n̄ k ,p + P ∆w̄ k ,p .
(4.11)
Now we can combine (4.9), (4.10), and (4.11) into a stochastic linear system description as:
¨
ξ′k +p = A′ ξ′k + B ′ ∆ū k ,p + BW′ ν̄k ,p ,
′
S̄RC
(4.12)
ē k ,p = C ′ ξ′k + D∆ū k ,p + DW ν̄k ,p ,
where ′ denotes the system is differenced once, the vectors ξ′k ∈ Rp (ℓ+n )+n , and ν̄k ,p ∈
Rp ( f +n +ℓ) are given by:
”
ξ′k = ∆x kT
ẽ kT−p,p
w̄ kT−p,p
—T
”
, ν̄k ,p = n̄ Tk ,p
w̄ kT,p
v̄ kT,p
—T
,
and the system matrices A′ ∈ Rp (ℓ+n )+n ×p (ℓ+n )+n , B ′ ∈ Rp (ℓ+n )+n ×p r , C ′ ∈ Rp ℓ×p (ℓ+n )+n ,
D ∈ Rp ℓ×p r , BW′ ∈ Rp (ℓ+n )+n ×p ( f +n +ℓ) , and DW ∈ Rp ℓ×p ( f +n +ℓ) are given by:


 
A p 0 −P
L
”


 
0  , B ′ = −G  , C ′ = −Γ
A′ = −Γ I
0 0
0
0


F
P 0
”
—


BW′ = −J 0 0 , DW = − J H I .
0
I 0
I
—
0 , D = −G ,
85
4.3 Repetitive control with lifted LQG design
4.3.2
Multiple memory loops
As described in Pipeleers et al. (2008); Steinbuch (2002); Steinbuch et al. (2007), additional memory loops can be introduced to make the RC more robust for small changes
in period time or less sensitive to disturbances at non-periodic frequencies. This highorder RC output-feedback problem can be formulated in an almost similar state-space
system as in (4.12). In Gupta and Lee (2006) it was shown that the multiplication
of the multiple memory loops with the repetitive controller is commutative in the
lifted setting. This means that an additional memory loop can be introduced in the
output-feedback problem by double-differencing the input. Let’s denote the doubledifferenced input as:
∆2 ū k ,p = ū k ,p − θ1 ū k −p,p − θ2 ū k −2p,p ,
where θ1 ∈ R and θ2 ∈ R with θ1 + θ2 = 1. For example, in Steinbuch et al. (2007) it was
shown that θ1 = 2 and θ2 = −1 gives the best performance against period mismatch,
and that θ1 = 2/3 and θ2 = 1/3 gives the lowest sensitivity to disturbances at nonperiodic frequencies. The equations in (4.9), and (4.11) are differenced again as:
ẽ k ,p = θ1 ẽ k −p,p + θ2 ẽ k −2p,p − Γ∆2 x k − G ∆2 ū k ,p − J n̄ k ,p − θ2 J n̄ k −p,p ,
(4.13)
∆2 x k +p = A p ∆2 x k + L∆2 ū k ,p + F n̄ k ,p + θ2 F n̄ k −p,p + P ∆2 w̄ k ,p .
(4.14)
and
Similarly as the system in (4.12) with single-differenced input, the system with doubledifferenced input can also be combined into two matrix equations. This results in the
following double-differenced (or second-order) RC output-feedback problem:
¨
ξ′′k +p = A′′ ξ′′k + B ′′ ∆2 ū k ,p + BW′′ ν̄k ,p ,
′′
S̄RC
(4.15)
ē k ,p = C ′′ ξ′′k + D∆2 ū k ,p + DW ν̄k ,p ,
where ′′ denotes the system with double-differenced input, the vector ξ′′k ∈ Rp (2ℓ+2n )+n
is given by:
”
—T
ξ′′k = ∆2 x kT ẽ kT−p,p ẽ kT−2p,p n̄ Tk −p,p w̄ kT−p,p w̄ kT−2p,p ,
and the system matrices A′′ ∈ Rp (2ℓ+2n + f )+n ×p (2ℓ+2n + f )+n , B ′′ ∈ Rp (2ℓ+2n + f )+n ×p r ,
′′
∈ Rp (2ℓ+2n + f )+n ×p ( f +n +ℓ) , and C ′′ ∈ Rp ℓ×p (2ℓ+2n + f )+n are given by:
BW

 


F
P
L
Ap
0
0
θ2 F
−θ1 P −θ2 P

 


0
0 
−J 0
−G 
−Γ θ1 I θ2 I −θ2 J

 


0
I
0
0
0
0 
 0
 0 
 0
′′
=
A′′ = 
 , B ′′ =   , BW
0
0
0
0
0
0 
 I
 0 
 0

 


I
0
0
0
0
0 
 0
 0 
 0
0
0
0
0
0
0
0
I
0
”
—
C ′′ = −Γ θ1 I θ2 I −θ2 J 0 0 .

0

0

0
,
0

0
0
The above computations can be repeated to design a period-robust structure based on
a higher number of memory loops. For example, the high-order RC output-feedback
problem for m ∈ N+ memory loops would have a m -times differenced input as:
∆m ū k ,p = ū k ,p −
m
X
i =1
θi ū k −i p,p ,
m
X
i =1
θi = 1.
(4.16)
86
4.3.3
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
The lifted LQG solution
In this subsection the repetitive control method for LTI systems with both repetitive
and non-repetitive disturbances is presented. Let’s replace ′ and ′′ by a more general
notation of ∗ to denote the system is differenced with an arbitrary number. Now, the
high-order RC problem can be formulated as:
Problem Description 4.1 (Lifted repetitive control). Given the system in the form of
∗
S̄RC
, like in (4.12) or (4.15), and θ in (4.16), design a strictly-proper dynamic repetitive
∗
controller C¯RC
.
uk
+
+
6
yk
- S
-
CFB ”
··· ,
|
u kT ,
{z
···
−
?
+
ek
rk
?
”
—T
··· ,
|
}
6
- z −p
y kT ,
{z
···
—T
}
ȳ k ,p
ū k −p,p
+
?
+
ū k ,p
∆ū k ,p
−
?
′
C¯RC
+
ē k ,p
r̄k ,p
Figure 4.2: Block diagram of the closed-loop system with the RC loop with one
memory loop.
uk
+
+
6
yk
- S
-
CFB ”
··· ,
|
u tT ,
{z
···
−
?
”
—T
6
- z −p
- z −p
?
rk
··· ,
|
}
ū k −p,p
θ1
ū k −2p,p
?
?
y kT ,
{z
···
—T
}
ȳ k ,p
θ2
+
?
ū k ,p
+
ek
+
+
?
+ 2
∆ ū
k ,p
−
′′
?
C¯RC
ē k ,p +r̄k ,p
Figure 4.3: Block diagram of the closed-loop system with the RC loop with two
memory loops.
The controller is strictly proper, because current trial measurements are assumed
unavailable for feedback. The block diagrams in Figure 4.2 and 4.3 illustrate how
4.3 Repetitive control with lifted LQG design
87
the lifted repetitive controllers can be implemented with one memory loop and two
memory loops, respectively.
Given Assumption 4.1, the infinite-horizon discrete-time LQG solution is well
known, see Åström and Wittenmark (1997); Ogata (1994), and it is desirable in our
∗
situation, as it stabilizes the system S̄RC
and minimizes the expectation of the loss
function:


N
X
2
∗
2

kē k +i p,p ||2 + ρk∆ ū k +i p,p ||2 .
J = E lim
N →∞
i =0
Compared with the minimum-variance criterion in Rice and Verhaegen (2010b), we
introduced in the LQG criterion a weight to penalize the differenced input, and therefore we are able to make a trade-off between the variance of the differenced input and
∗
∗
output signals. By requiring that C¯RC
stabilizes S̄RC
, it can be shown to get exponential
convergence of the RC algorithm. This does not imply that the RC algorithm has
monotonic decay (kẽ k +p,p k2 ≤ kẽ k ,p k2 ) as recommended in Longman (2000), but the
experience is that the introduction of a weight on the differenced input could give the
engineer some means to prevent for the large overshoots conventionally seen with
some RC laws with non-monotonic decay in the initial transient periods, see also
Subsection 4.4.3. Using a steady-state approximation of the optimal LQG solution,
∗
the lifted controller C¯RC
is given by:
¨
ξ̂∗k +p = AC ξ̂∗k + LC ē k ,p ,
∗
¯
CRC
(4.17)
∗
∆ ū k ,p = −KC ξ̂∗k ,
where
AC = A∗ − B ∗ K C − L C C ∗ + L C D ∗ K C ,
−1
∗
,
+ A∗ ZC ∗ T R∗W + C ∗ ZC ∗ T
L C = SW
−1
T
T
KC = R + B ∗ X B ∗
B ∗ X A∗ + S T .
The matrices X and Z are the positive-definite stabilizing solutions to the Discrete
Algebraic Riccati Equations (DARE):
X = Q − A∗ T X B ∗ + S K C + A∗ T X A∗ ,
Z = Q∗W − LC C ∗Z A∗ T + S ∗W + A∗Z A∗ T ,
(4.18)
where R = ρI +D∗ T D∗ , Q = C ∗ T C ∗ , S = C ∗ T D∗ , and the disturbance covariance matrices
are calculated as:
–
– ™
™ – ™
i D T
∗ T
DW h
R∗W SW
W
T
E
=
ν̄
ν̄
.
(4.19)
k ,p k ,p
∗
∗
∗
BW
BW
SW
Q∗W
In many cases, the solvers for the Riccati equations in (4.18) have difficulties to
determine the solution numerically. For example the QZ-based DARE solver in
MATLAB™ will fail, because the Hamiltonian or sympletic matrix has eigenvalues on
and/or close to the unit circle. To overcome this problem, the learning rate should be
lowered by adding forgetting or leakage. With the inclusion of forgetting,P
the weights
m
of the m times differenced input in (4.16) should be constrained with i =1 θi = λ,
where the scalar 0 ≪ λ < 1 is the so-called forgetting factor.
88
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
The biggest challenge of implementing this RC description of the output-feedback
problem is that the lifted matrices are enormous, and it requires large memory storage
and will take a long time to solve the DARE. These issues can be resolved by exploiting
the matrix structure in (4.18). For matrices in dense form, the computational com
plexity for DARE solvers using the Schur method is O ℓ3 p 3 . In Rice and Verhaegen
(2010b), efficient methods for solving Riccati equations using the so-called Sequentially Semi-Separable (SSS) matrix structure (Chandrasekaran et al., 2003) and its
properties are discussed. An SSS matrix toolbox for MATLAB™ has been developed that
includes iterative algorithms to check matrix stability, to solve Lyapunov and Riccati
equations, and to compute LQG controllers in O ℓ3 p computational complexity.
The matrices A∗ , B ∗ , C ∗ , D, R W∗ , Q ∗W , and S ∗W can be put into this structure for fast
computation of the similar structured LQG solutions K and L to the RC output∗
feedback problem. Further, the lifted repetitive controller C¯RC
in Figures 4.2 and 4.3
can be implemented real time using three SSS structured matrix-vector computations
in O ℓ2 p instead of O ℓ2 p 2 . In Figure 4.4, the m-function of the SSS structured
matrix-vector product is compared with the inbuilt dense matrix-vector product in
computation time. Even in m-code the SSS matrix-vector product is faster for p >
400. Although not directly beneficial for the “smart” rotor section experiment (with
p = 60), the SSS matrix-vector algorithms are used to verify their capability for realtime computation. For controller synthesis, which requires the offline computation
of two Riccati matrix equations, the SSS Riccati solver using sign iterations is also
considerably faster for p > 250, see Rice and Verhaegen (2010b). It is noted that the
SSS Riccati solver requires after some iterations a model reduction step, therefore the
result will be an approximation of the true solution up to a given tolerance. Increasing
the tolerance value will considerably speed up the computation of the DARE solution.
1.5
time [ms]
1
0.5
0
0
100
200
300
400
500
600
700
800
900
1000
period length p
Figure 4.4: Computation time averaged over 10 runs of matrix-vector product. Light
grey line is with dense matrix structure and dark grey line with SSS matrix structure.
4.4 Experimental study on “smart” rotor section
4.4
89
Experimental study on “smart” rotor section
In this section we compare the periodic wind disturbance rejection of the proposed
RC method with different settings on a “smart” airfoil section with trailing-edge flap
actuator. First, we present the experimental setup used to show the feasibility of
periodic wind disturbance rejection with the repetitive control method. Second, we
describe the identification experiment to obtain an accurate model for the RC design.
Third, the proposed RC method is applied on the “smart” rotor test section, where the
periodic and non-periodic disturbances are created using a wind generator.
4.4.1
Description of the experiment
As in Chapter 3, the experimental setup mainly consists of the following components:
wind generator, blade, actuator and sensors, and real-time environment. The wind
generator blows air with considerable turbulence into a tunnel section with a blade
section hanging inside. The diameter of the outlet of the wind generator is 350mm
and the tunnel section is 400mm by 400mm. The blade section that we use for our
experimental verification is an airfoil with at the trailing edge a control surface, the
so-called trailing-edge flap. The blade section is at the top connected to a half meter
long aluminium plate with the other end fixed to a rigid frame and has two degrees
of freedom. The plate allows the blade only to move in the flapwise and torsional
direction. A full description and schematic illustrations of the experimental setup are
given in Chapter 3, where it has already successfully been used to study the recursive
predictor-based subspace identification method.
For control purposes, the “smart” rotor section is equipped with sensors which
measure the dynamic behaviour of the blade. Since the final goal for this experiment
is to reduce vibrations of the structure, one of the three Macro Fiber Composite
(MFC) (Smart Material) patches that are adhered to the root at the frontside of the half
meter long aluminium plate is used to measure the high strains associated with the
first flapwise bending mode. The main advantage of an MFC is that no amplification
is required to have a good Signal-to-Noise Ratio (SNR) ratio. However, with the MFC
it is not possible to do static measurements due to the capacitance behaviour of the
MFC. This high-pass behaviour is desirable for this experiment, as we want to control
the dynamic behaviour of the system, rather than the static deformations.
The trailing-edge flap can reduce the structural vibrations of the “smart” rotor
section when this repetitive controller is enabled. The controller intelligence and data
acquisition capability are added with a dSPACE™ (dSPACE) system. The controller
and data acquisition scheme are fully developed in the MATLAB™ (Mathworks, a) and
SIMULINK (Mathworks, b) environment and then compiled to the dSPACE™ (dSPACE)
chip. On a separate computer, all the signals are monitored using CONTROL DESK™
(dSPACE) and also the control parameters can be adjusted in real time. The input
to the wind generator is a reference wind speed profile with a given period and is
generated by a separate system. For this reason, the actual period of the periodic
wind disturbances is not directly available to the repetitive controller.
90
4.4.2
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
Experimental modelling and control
A model for modern model-based controller design is a mathematical model normally
governed by (preferably linear) differential equations. For controller synthesis this
model should only contain the relevant dynamics between the input and the outputs
and should be accurate over the bandwidth of the controller. Analytical modelling
has been performed in the design stage using the theory in Theodorsen (1935), which
strongly depends on a large number of parameters that determine the (aero) dynamic
behaviour of the system. Most of these parameters can be roughly estimated or
calculated. Still, a large amount of uncertainty is present; this makes it difficult to
design a stable controller based on such models. This motivates that identification
can become a necessary building block for model-based controller design.
The control loops in the prototyped “smart” rotor section are the transfer functions
between the trailing-edge flap (u ), and the first frontside MFC sensor (y ). Using the
Predictor-Based Subspace IDentification (PBSID) method for systems with periodic
disturbances as described in van Wingerden et al. (2011), we identify the system to be
controlled in the innovation form as:
¨
x̂ k +1 = A x̂ k + Bu k + F d˜k + K e k ,
(4.20)
Ŝ
y k = C x̂ k + Du k + e k ,
where x̂ k ∈ Rn denotes the predicted state vector with E x k − x̂ k = 0 and E x k −
T x̂ k x k − x̂ k
minimal, d˜k ∈ R f contains the basis functions needed to express the
periodic disturbances, and e k ∈ Rℓ denotes the innovation sequence with E e k e kT =
R, and K ∈ Rn ×ℓ denotes the Kalman gain matrix. The identified Bode diagram of the
fifth-order SISO model is given in Figure 4.5. The first resonance frequency is located
at 0.75Hz and is related to the 1st flapwise bending mode. The second resonance
frequency is located at 4.5Hz and is related to the 2nd flapwise bending mode. The
resonance frequency located at 3Hz related to the 1st torsional bending mode is
hardly observable due to the sensor position and therefore is not identified, however
in Figure 4.6 a small resonance peak is still visible at that location.
The subspace identification is done with a pink (1/f ) Generalized Binary Noise
(GBN) signal with a bandwidth of 50Hz and with an amplitude of 200V on the flap
actuator. The data is filtered and resampled to a sampling rate of 20Hz (h = 0.05s).
The periodic wind disturbance is created by changing the speed of the fan in the wind
generator. The reference signal given to the wind generator fan is based on the bilinear
windshear model and is described by:
(
sin k h p2πh , if Vref,k >= 10.5,
Vref,k = 10.5 +
1.5 sin k h p2πh , if Vref,k < 10.5.
As the wind disturbance profile is not purely a sinusoid, multiple harmonics are
expected to be excited. In Figure 4.6, the square-root of the power spectral density
with and without GBN excitation is illustrated. With no excitation, we clearly see
peaks at the 1P–6P (Per revolution) frequencies due to the wind disturbance with a
velocity ranging between 9 and 11.5 m/s, period of p h = 3s (p = 60, h = 0.05s), and
no variation on the period time: std(p h) = 0. When we excite the system, we see that
91
4.4 Experimental study on “smart” rotor section
From: u k To: y k
magnitude (dB)
-30
-40
-50
-60
phase (deg)
-70
90
0
-90
-180
10−1
100
frequency (Hz)
101
Figure 4.5: Identified Bode diagram. For N = 15000.
20
1P
10
3P
magnitude (dB)
0
-10
2P
4P
5P
-20
6P
-30
-40
-50
10−1
100
frequency (Hz)
101
Figure 4.6: Square-root of the PSD of the output signal. The light grey line is with GBN
excitation and the dark grey line is without excitation. For V = 9−11.5m/s, p = 60 and
std(p h) = 0.
92
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
especially at lower frequencies a good SNR is obtained. However, the 1P is dominantly
present and that is the main reason we use an identification scheme that can handle
these periodic disturbances. Further, we can conclude from Figure 4.6 that there is a
good control authority at the 1P-6P frequencies, because these harmonics are in the
output range in terms of amplitude and bandwidth of the actuator.
For the feedback control synthesis we use the so-called H2 controller synthesis
method which is illustrated in the well-known generalized plant setting in Figure 4.7.
The goal of the feedback controller is to suppress the unknown disturbances as much
as possible with the requirement that the system should remain stable and that the
control input signal is bounded between −250V and +250V. The method does this by
minimizing the H2 norm between e k and {z k , y k } and is mathematically given by:
WU (I + CFBG WL )−1 H 2
,
min (4.21)
CFB
(I + G WL CFB )−1 H 2
where
G (z ) = D + C (z I − A)−1 B , H (z ) = R 1/2 + C (z I − A)−1 K R 1/2 .
In these objective functions we can embed manually the loop shaping ideas in the
weighting filters WL , WU , and H . The wind turbulence spectrum H is obtained
from identification. To let the controller not act on the 1st torsion and the 2nd
flapwise bending mode frequencies in the error signal, the input sensitivity function
is weighted by a first-order filter WU with high gains at those high frequencies. A
second-order low-pass filter WL is added in series with the controller, such that the
final feedback controller will roll-off with increasing frequency.
ek
-
−
6
CFB
-
WL
uk
-
zk
-
WU
H
G
+
+ ?
-
yk
-
Figure 4.7: Block diagram of generalized plant to synthesize the H2 controller.
4.4.3
Experimental results
In this subsection, the main results of the experiments on the “smart” rotor test
section are presented. We present four cases that will show the features of the
proposed lifted RC algorithm. During the design of the proposed lifted RC method, a
large number of parameters have to be tuned. We have the singular value order s , the
disturbance covariance matrices R w , R v , and R n , and the weight ρ on the differenced
input, and the number of memory loops m and its weighting factors θ (and λ). In the
following four cases some tuning guidelines are provided by analysing the effects of
these tuning parameters.
93
4.4 Experimental study on “smart” rotor section
Effect of the singular value order s
In Subsection 4.2.2, we proposed to select the number of singular values s in (4.5)
not only on the controllability/observability condition, but also to further reduce the
size of the RC output-feedback problem. It turns out that for even lower values of s ,
such that the error to be reduced is in a smaller column space, the lifted repetitive
controller will focus more on the main harmonic components and therefore robustify
the controller performance in some sense. To analyse the effect, we consider the
frequency response of the lifted repetitive controller CRC . From Figure 4.2 and Figure 4.3, we can derive the following frequency-domain representation of the complete
repetitive controller as:
€
Š
”
CRC e j ω , k = − e (1−k )j ω I
×
1
1−
m
P
···
θi e −i p j ω
—
e (p −k ) j ω I
€
Š−1
K C e p j ω I − AC
LC
i =1
”
× e (−p −k +1) j ω I
···
e −k j ω I
—T
.
The lifted repetitive controller CRC is in fact a periodic linear time-varying system,
thus this transformation to the frequency domain does not generally hold. However
during the design is observed that for LTI systems, the lifted controller CRC behaves
almost as a very high-order LTI system. This is observed from the frequency responses
CRC e j ω , k that are very similar to each other for every step k within the period.
Plotting one of the Bode diagrams of the sensitivity function is then also a useful
engineering tool for lifted RC design.
In Figure 4.8, the Bode diagrams of two noise sensitivity functions with s = 12 and
s = 26 are given. The two noise sensitivity functions are almost similar for the main
(1P–4P) harmonic components. Increasing the number of singular values to s = 26
introduces also notches at the (5P–8P) harmonic components. Basically, reducing
the number of singular values lets the lifted repetitive controller focus on the main
harmonic components. In our case we selected the number s = 12, because first the
repetitiveness of the higher harmonics (5P–8P) are questioned (especially if the period
time p h varies over time) and second the system should not be excited around 2-3Hz,
due to the badly damped (unobservable) torsion mode.
For the SISO case, an almost similar Bode diagram of the noise sensitivity function
can be created using H ∞ control design synthesis with inverse notches as weights.
If the frequency response closely matches each other, the asymptotic behaviour will
be almost similar. However, the transient behaviour will be very different. The
feedback controller based on inverse notches will immediately react on any harmonic
variations in the error signal. Instead with lifted RC, the input signals are averaged
over previous periods and are not directly influenced by the current error signal,
therefore the input signal will be considerably smoother, but it will take longer to
converge.
94
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
10
magnitude (dB)
0
-10
-20
-30
-40
10−1
100
frequency (Hz)
101
Figure 4.8: Bode diagram of the two noise sensitivity functions S = (I + G CRC )−1 H .
Dark grey line (partly under light grey) is designed with s = 26 and light grey line is
designed with s = 12. Dotted line is the noise sensitivity function H . For ρ = 0.05,
R n = 0.001, m = 1, and λ = 0.99.
Effect of parameter ρ
As the identified state-space model Ŝ is considered to be a close estimate of the
system S to be controlled, the disturbance covariance matrices can be estimated from
the identification as: R̂ w = K R K T , R̂ v = R. The strategy to pick the parameter ρ is
very similar to the time-domain LQG problem, see Åström and Wittenmark (1997).
It is common practice to pick the parameter ρ such that a smooth control signal
is obtained and a large overshoot is avoided. In Figure 4.9, the initial responses of
the input kū k ,p k2 for three different weights are given. In Longman (2000) it is even
recommended to have a monotonic decrease of kẽ k ,p k2 , which is almost (apart from
the initial differenced state) directly related to the monotonic increase of kū k ,p k2 ). By
increasing the weight ρ, the overshoot over the trials can be lowered in most cases
and improve the smoothness of the generated control signal at the cost that it takes
longer to converge to steady state. As shown in Figure 4.9, an overshoot during the
initial response of kū k ,p k2 is clearly visible with ρ = 0.01, and monotonic increase is
visible with the higher values of ρ = 0.05 and ρ = 0.1.
RC with variations in period time
In this case we show the performance of solely the repetitive controller using the
results of the experiments on the “smart” rotor test section. We have stochastic
95
4.4 Experimental study on “smart” rotor section
500
400
kū k ,p k2
300
200
100
0
0
50
100
150
200
250
number of trials
Figure 4.9: Initial responses of the input kū k ,p k2 . Dark grey line is designed with ρ =
0.01, grey line is designed with ρ = 0.05, and light grey line is designed with ρ = 0.1.
For s = 12, R n = 0.001, m = 1, and λ = 0.99.
disturbances coming from the wind (turbulence) and periodic disturbances due to
generated wind speed profile. Experimentally the implications of variations in period
time to the performance with respect to the periodic disturbance rejection and the
smoothness of control signal are investigated. For this reason, the experiments
have been repeated with different settings for the deviation of the period time. The
considered maximum deviation std p h = 0.03s (1%) is based on 10 min simulations
with the UPWIND 5MW wind turbine (see Bossanyi (2009); Jonkman et al. (2008))
in above-rated conditions and with a turbulence intensity of 12% (IEC, 2004). The
maximum difference in rotational speed was about 3%.
As suggested in the introduction, two ways (or better: a combination of the
two) are proposed to improve the performance of the repetitive controller in the
case there is a small mismatch between the controller period and the actual period
of the periodic disturbance signal. If the period is not measured it is common
practice to formulate a high-order repetitive control problem with multiple memory
loops. The selection of the number of memory loops m and its weighting factors
θ (and λ) is not an easy task. In Pipeleers et al. (2008); Steinbuch et al. (2007) it
is shown that there is a clear trade-off between improved suppression of periodic
disturbances and amplification of non-periodic noise. It was shown that θ = (2, −1)
and θ = (3, −3, 1) gives the best performance against period mismatch and that
θ = (2/3, 1/3) and θ = (3/6, 2/6, 1/3) gives the lowest sensitivity to non-periodic noise
for two and three memory loops respectively. The design of these parameters highly
depends on the periodic and non-periodic disturbance characteristics. In Pipeleers
et al. (2008); Steinbuch et al. (2007) periodic and non-periodic performance criterion
96
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
are used for designing time-domain high-order repetitive controllers. As shown in
Subsection 4.3.2, multiple memory loops can be introduced in the lifted RC problem
at the cost of larger matrices. However the design of the weighting factors θ becomes
even more difficult and in this chapter we have even tuned manually, because they
are implicitly related with the output-feedback problem formulation in (4.15).
Compared with the time-domain RC methods in Pipeleers et al. (2008); Steinbuch
et al. (2007), the lifted RC problem allows a more versatile disturbance formulation. It
includes a periodic component with a random walk nature n k , such that the periodic
disturbance can vary between trials. In addition, the noise and periodic disturbances
can be even correlated and varying in time. In Figure 4.10, the Bode diagram of
the noise sensitivity function for three different weights R n are illustrated. With an
increasing value R n the inverted notches at 1P and 2P frequencies become wider, but
less deep. It can be concluded that the increasing weight provides better disturbance
rejection at frequencies near the harmonics at the cost of performance degradation
at the exact repetitive frequency and its harmonics. When the system is identified
(without exact period time) in the innovation form in (4.20), where basis functions
in d˜k are used to express the periodic disturbances, it is even possible to relate a
lifted covariance matrix R̄ n with the variance of the period time p h by a first-order
approximation as:
T
∂ d¯k ,p p h
∂ d¯k ,p p h
ˆ
var p h
,
R̄ n =
∂ ph
∂ ph
where d¯k ,p is the stacked vector of d˜k . As the lifted covariance matrix R̄ n has a timevarying structure, the start of the repetitive controller should be matched with the
phases of the basis functions.
In Table 4.1, the manually tuned values for the parameters of R n and θ are given
with an increasing value for the deviation of the period time and an increasing value
for the number of memory loops m . These values have been used during the experiments on the “smart” rotor test section; see the results in the next subsections. The
parameters have been tuned by looking at the cost J = var y k +var ∆2 u k /∆h 2 ·10−4
after the control signal has converged to steady state. By including an approximation
for the acceleration of the generated input signal to the cost, we try to achieve a
smooth input signal which should reduce the wear and tear of the actuator. From
the table it is observed that with increasing deviation of the period-time the weight
R n should be increased, although at a slower rate if more memory loops are used. The
memory loop weights θ are moving clearly in the direction of the values θ = (2, −1)
and θ = (3, −3, 1) which should give the best performance against period mismatch for
two and three memory loops respectively. It is noted that in most wind turbines and
in the rotating “smart” rotor in van Wingerden et al. (2011), the period time could be
estimated from the measured rotational speed or from the measured azimuth angles
of the blades. For future research it would be interesting to formulate a real-time RC
output feedback problem, where we can make the parameters R n and/or θ adaptive
to changes in the measured data.
Table 4.2 summarizes the variance values of the output in percentage with respect
to the output without any control J y = 100% · (1 − var y k / var y 0,k ) and the approx
imated input acceleration J u = var ∆2 u k /∆h 2 after the control signal has converged
97
4.4 Experimental study on “smart” rotor section
15
10
magnitude (dB)
5
0
-5
-10
-15
10−0.6
10−0.5
10−0.4
10−0.3
frequency (Hz)
10−0.2
10−0.1
Figure 4.10: Bode diagram of the noise sensitivity function S = (I + G CRC )−1 H at the
1P and 2P frequencies. Dark grey line is designed with R n = 0.01, grey line is designed
with R n = 0.005, and light grey line is designed with R n = 0.001. For s = 12, ρ = 0.05,
m = 1, and λ = 0.99.
Table 4.1: Values for R n and θ for different deviations of period time. For s = 12, and
λ = 0.99 (multiply θ with λ).
std p h
0%
1/3%
1/2%
2/3%
1%
m =1
Rn
0.001
0.003
0.006
0.016
0.022
Rn
0.001
0.002
0.003
0.008
0.015
m =2
θ1
1.25
1.54
1.55
1.63
1.7
m =3
θ2
-0.25
-0.54
-0.55
-0.63
-0.7
Rn
0.001
0.002
0.003
0.006
0.014
θ1
1.3
1.45
1.55
1.75
2.25
θ2
-0.4
-0.6
-0.75
-1
-2
θ3
0.1
0.15
0.2
0.25
0.25
Table 4.2: Reduction of variance for output and approximated input acceleration for
different deviations of period time. For N = 6000, s = 12, and λ = 0.99.
std p h
0%
1/3%
1/2%
2/3%
1%
m =1
Jy
Ju
41.8%
48
40.9%
57
38.2%
82
36.3% 110
34.0% 116
m =2
Jy
Ju
42.2%
44
41.2%
54
38.7%
80
37.9% 102
34.9% 105
m =3
Jy
Ju
42.7%
42
41.5%
52
39.5%
79
38.2%
98
35.5% 101
98
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
to steady state. The variance of the corresponding signals can be seen as a rough
measure for fatigue. We see that we have a reduction in variance of at least 34% if only
repetitive disturbance signals are compensated. This increases to a variance reduction
of around 42% when the actual period of the wind profile matches the period of the
controller. The use of more memory loops gives an increase between 1–2%, which
is a small increase compared with the improvements made in earlier experiments
with high-order time-domain repetitive controllers in Pipeleers et al. (2009); Steinbuch
et al. (2007). The tuning of the weight R n already considerably improves the robust
performance for variations in period time, such that incorporating more memory
loops will not make that much difference anymore. However with more memory
loops, the smoothness of the input signal is most of the time much better, probably
due to the averaging effect over multiple periods, at the cost that the convergence is
considerably slower.
The steady-state performance of any repetitive controller is bounded by an inherent trade-off between the suppression of periodic disturbances and the amplification
of non-periodic inputs (Pipeleers et al., 2009). In Figures 4.11 and 4.12 the squareroot of the Power-Spectral Density (PSD) is given for the system output without any
control and with the proposed repetitive control for both 0% and 1% deviation in
period time, respectively. In both figures, we clearly see an improvement with respect
to the uncontrolled case. For the 0% period variation case, the spikes at the 1P–4P
harmonic frequencies are almost completely removed, and the noise amplifications
at some lower and higher intermediate frequencies are small. For the 1% period
variation case, the spikes at the 1P–4P harmonic frequencies are only partly reduced,
and this reduction comes even at the cost that some amplifications are visible at some
intermediate frequencies and higher harmonic frequencies.
RC with feedback control
In this case we show the performance of the repetitive controller and feedback
controller together using the results of the experiments on the “smart” rotor test
section. For this case, the repetitive controller has been redesigned with the model
of the closed loop system in Figure 4.1. Table 4.3 again summarizes the deviation
values of the output and approximated input acceleration after the control signal has
converged to steady state. We see that we have an improved reduction in deviation
of at least 64% on the sensor channel if both the feedback and the repetitive control
is active. This increases to a reduction of around 72% when the period of the wind
profile matches the period of the controller. The use of more memory loops gives
again an increase between 1–2%. The introduction of feedback makes the variance
of the input acceleration 4-8 times larger than with RC only. Thus the input signal
is considerably less smooth and can cause in practice a considerable reduction in
actuator lifetime. Most of this increase is caused by the feedback controller.
In Figures 4.13 and 4.14 the square-root of the Power-Spectral Density (PSD) are
given for the system output without any control and with the feedback and repetitive
control together for both 0% and 1% deviation in period time. In both figures,
we again see a clear improvement with respect to the uncontrolled case. For both
cases, there is an additional 10dB reduction at the first natural frequency caused by
99
4.4 Experimental study on “smart” rotor section
20
10
magnitude [dB]
0
-10
-20
-30
-40
-50
10−1
100
frequency [Hz]
101
Figure 4.11: Square-root of the PSD of the output signal. Dark grey line is the output
spectrum with RC and light grey line is without. For V = 9 − 11.5m/s, m = 3, p = 60
and std(p h) = 0s.
20
10
magnitude [dB]
0
-10
-20
-30
-40
-50
10−1
100
frequency [Hz]
101
Figure 4.12: Square-root of the PSD of the output signal. Dark grey line is the output
spectrum with RC and light grey line is without. For V = 9 − 11.5m/s, m = 3, p = 60
and std(p h) = 0.03s.
100
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
20
10
magnitude [dB]
0
-10
-20
-30
-40
-50
10−1
100
frequency [Hz]
101
Figure 4.13: Square-root of the PSD of the output signal. Dark grey line is the output
spectrum with RC+FB and light grey line is without. For V = 9−11.5m/s, m = 3, p = 60
and std(p h) = 0.
20
10
magnitude [dB]
0
-10
-20
-30
-40
-50
10−1
100
frequency [Hz]
101
Figure 4.14: Square-root of the PSD of the output signal. Dark grey line is the output
spectrum with RC+FB and light grey line is without. For V = 9−11.5m/s, m = 3, p = 60
and std(p h) = 0.03s.
101
4.5 Conclusion
the feedback control action. This comes at the cost by some small amplification
of the lower frequencies and the frequencies between the two natural frequencies.
Compared with RC only, the spikes at the 2P–3P frequencies are reduced further to
the same level of the surrounding frequencies. Thus feedback and repetitive control
could be used together for the reduction of both harmonic and natural frequencies, if
the dynamics of the feedback controller are taken into the RC design.
Table 4.3: Reduction of variance for output and input acceleration (and RC only) for
different deviations of period time described in percentage of period time. For N =
6000, s = 14, and λ = 0.99.
std p h
0%
1/3%
1/2%
2/3%
1%
4.5
m
Jy
71.8%
70.0%
67.9%
66.5%
64.1%
=1
Ju
378(26)
402(92)
463(102)
475(106)
484(150)
m
Jy
71.9%
70.4%
68.4%
67.1%
65.2%
=2
Ju
375(22)
401(87)
446(94)
461(102)
475(140)
m
Jy
72.0%
70.8%
69.2%
67.6%
65.6%
=3
Ju
375(21)
398(81)
442(90)
455(98)
470(139)
Conclusion
In this chapter we presented a novel repetitive control method that is implemented
in real time for periodic wind disturbance rejection in linear systems with multiple
inputs and multiple outputs and with both repetitive and non-repetitive disturbance
components. The design of the repetitive controller is formulated as a lifted linear
stochastic output-feedback problem on which the mature techniques of discrete-time
linear control may be applied. The formulation of the lifted repetitive control problem
can be made more robust to small changes in period time by using multiple memory
loops, but also by including a periodic component with a random walk nature, so
that the periodic disturbance can vary between trials. Efficient algorithms exist for
controller synthesis and for real-time implementation to reduce the computational
complexity and memory usage by exploiting the structure in the lifted state-space
matrices. Moreover, the chapter provides some guidelines on how to pick the
free parameters in the algorithm. The novel repetitive controller could learn the
periodic wind disturbances for fixed-speed wind turbines and variable-speed wind
turbines operating above-rated and we have demonstrated this on an experimental
“smart” rotor test section. For relatively slow changing periodic and turbulent wind
disturbances created by a wind generator it was shown that this repetitive control
method could reduce the variance of the load signals of the “smart” rotor test section
up to 42%. The cost of additional wear and tear of the “smart” actuators are kept
small, because a smooth control action is generated as the controller mainly focuses
on the reduction of periodic disturbances.
102
Chapter 4: Rejection of Periodic Wind Disturbances on a Smart Rotor Section
CHAPTER
5
Wind Turbine Load Reduction by
Rejecting the Periodic Load
Disturbances
To decrease the cost per kWh, the trend in offshore wind turbines is to
increase the rotor diameter as much as possible. The increasing dimensions
have led to a relative increase of the loads on the wind turbine structure, thus
it is necessary to react to disturbances in a more detailed way; for example,
each blade separately. The disturbances acting on an individual wind turbine
blade are to a large extent deterministic; for instance, tower shadow, wind
shear, yawed error, and gravity are depending on the rotational speed and
azimuth angle, and will change slowly over time.
This chapter aims to contribute to the development of individually pitch
controlled blades by proposing a lifted repetitive controller that can reject
these periodic load disturbances for modern fixed-speed wind turbines and
modern variable-speed wind turbines operating above-rated. The performance of the repetitive control method is evaluated on the UPWIND 5MW wind
turbine model and compared with typical individual pitch control. Simulation results indicate that for relatively slow changing wind disturbances this
lifted repetitive control method can significantly reduce the vibrations in the
wind turbine structure with considerable less high-frequent control action.
5.1
T
Introduction
chapter investigates the load reduction capabilities of Repetitive Control (RC)
with individual pitch actuators, and in the operation of the UPWIND 5MW wind
turbine in the aerolastic modelling software GH BLADED 3.85 (Garrad Hassan). A
lifted RC method is presented for MIMO linear systems with both repetitive and nonrepetitive disturbance components, which already have been verified experimentally
HIS
103
104 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
in previous chapter for its load reduction capabilities on a “smart” rotor test section
with periodic disturbances generated by a wind generator. The advantage of using RC
instead of a typical Individual Pitch Control (IPC) is that the Coleman transformation
is not needed. Therefore, the RC method is more generic, for example to extend to
rotors with more actuators and sensors, or to other rotor designs; like, two-bladed
rotors. We also address an important drawback of RC, which is a small mismatch
between the controller period and the actual period of the disturbance signal that
can decrease the performance of the repetitive controller substantially. However,
the application of RC to wind turbines with small variations in rotational speed,
such as fixed-speed and variable-speed operating above-rated with a collective pitch
controller that will keep the rotational speed of the wind turbine close to the given
reference speed, is still considered competitive. Therefore, the relative performance in
terms of load reduction and control action of RC is compared with the typically-used
IPC method with Coleman transformation and with different turbulence intensities
and consequently different variations in the rotational rotor speed.
The outline of this chapter is as follows. In Section 5.2, the simulation model of the
UPWIND 5MW wind turbine is briefly described including the baseline controllers,
and the wind conditions. In Section 5.3, the model linearisation and the theoretical
framework is presented for the lifted RC design problem and an algorithm for MIMO
periodic Linear Time-Varying (LTV) systems is given. In Section 5.4, the effectiveness
of the proposed RC algorithm is evaluated and compared with typical IPC in an
simulation study on the UPWIND 5MW wind turbine operating in a periodic wind
field and some tuning guidelines are provided. In the final section, we present the
conclusions of this chapter.
5.2
Description of the simulations
This section describes the specifications of the reference wind turbine, the simulation
environment which simulates the operation of the wind turbine equipped with the
load reduction control, the baseline controllers for above-rated operation, a typical
IPC for comparison, and the wind conditions during the simulation runs.
5.2.1
Specifications of the reference wind turbine
The reference wind turbine used for this simulation is the UPWIND 5MW (also
referred to as the NREL 5MW) wind turbine described in Jonkman et al. (2008).
The UPWIND 5MW wind turbine is fairly representative for typical commercial wind
turbines in its class, although it does not represent any particular commercial wind
turbine (Bossanyi, 2009). For this reason, this wind turbine is commonly used as
a baseline model for the research in state-of-the-art controller design, for example
with IPC in Bossanyi (2009) and SRC in Lackner and van Kuik (2010a). The wind
turbine under consideration is onshore, upwind, variable-speed, three-bladed, and
pitch-controlled. Its design specifications are given in Table 5.1, and a schematic
representation is given in Figure 1.4.
105
5.2 Description of the simulations
Table 5.1: UPWIND 5 MW wind turbine specifications
description
rated power
rotor diameter
hub height
cut in wind speed
rated wind speed
cut out wind speed
rated rotational rotor speed
gearbox ratio
pitch rate limit
5.2.2
symbol
Prated
d ro
h hub
v cutin
v rated
v cutout
Ωro
ν
θ̇limit
value
5 MW
126 m
90 m
4 m/s
11.3 m/s
25 m/s
2π/4.95 rad/s
97
8 deg/s
Simulation environment
There is a wide variety of different model simulation codes that can be used to model
the dynamic behaviour of wind turbines, however only a small part is verified and
even a smaller part is validated (Molenaar, 2003). For this case, the simulations of
the UPWIND 5MW wind turbine are carried out in the proven aerolastic modelling
software GH BLADED™ 3.85 (Garrad Hassan). The GH BLADED™ software provides
the following important features in their model representation (Lackner and van Kuik,
2010a):
• The aerodynamics are calculated using the well-known blade element momentum approach. Dynamic inflow and dynamic stall models are incorporated to
represent the turbine wake and deal with unsteady aerodynamic conditions.
• The structural dynamics of the turbine model are calculated using a limited
degree of freedom modal model. Note, that models with limited order are
normally preferred for model-based control design.
• The dynamics of the power train (shaft, gearbox and generator) are modelled.
• The external wind conditions can be generated, including three-dimensional (3D) turbulent wind fields, wind shear, tower shadow effects and prescribed gusts.
• Control of the turbine can be accomplished using either internal controllers
provided by GH BLADED™, or by external controllers created by the user.
• The loads on the various components of the turbine and the turbine performance can be calculated, analysed and exported.
The capability to use external controllers using the shared libraries is a very
necessary requirement, because the proposed controllers can not be represented by
the standard controllers in GH BLADED™. The building of these so-called Dynamically
Linked Libraries (DLL) is not an easy task. First, the controllers have to be written in
the code Fortran or C. Second, the code has to be compiled correctly using a suitable
compiler (with Microsoft VISUAL StUDIO™ C++ (Microsoft)) to a DLL. To automate
106 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
the process of DLL building and to ease the algorithm writing in an environment
already familiar for most control engineers, the Mathworks Real-Time Workshop
(RTW) toolbox for Simulink™ (Mathworks, b) has been used to automatically generate executable code from implemented controllers in Simulink diagrams. For this
purpose the generic Real-Time Target in the RTW toolbox is adjusted following the
specifications in (GH , 2010b). The specially adjusted Real-Time Target files have been
made available to the readers through our website.1
5.2.3
Baseline control for above-rated operation
The UPWIND 5MW wind turbine operates at variable speeds, which has the advantage
that at lower wind speeds than the rated wind speed, called below rated, the rotor
speed can vary to maintain an optimal power production. At low wind speeds the
reduced rotor speed also reduces the aerodynamically generated acoustic noise and
the stress on the mechanical structure. For wind speeds higher than the rated wind
speed, called above rated, the rotational speed is regulated at rated rotational speed to
keep the rotational speed (and indirectly the power) within the design specifications
of the power train, and also to limit the stresses on the mechanical structure.
The baseline controllers for the UPWIND 5MW wind turbine including the logic
for switching between below-rated and above-rated operation using only the measurements of the generator torque Tge and the generator speed Ωge (and not directly
from the wind speed) are described in Bossanyi (2009). Because the simulations with
lifted repetitive control are intended for above-rated operation only, we will describe
the following implemented baseline controllers for above-rated operation: Collective
Pitch Controller (CPC), additional Non-linear Pitch Controller (NPC) to overcome
large wind gusts, and the Torsional Vibration Damping (TVD) filter for torsional drive
train vibrations. Note, that the filter for damping the tower In Bossanyi (2009) is
not implemented to better assess the load reduction capabilities of the proposed
controller. For comparison, a typical IPC with the Coleman transformation is implemented. Because the IPC was not fully specified in Bossanyi (2009), we describe
in the next subsection our interpretation/implementation of the IPC following the
recipe in Bossanyi (2009). All the baseline controllers are described in a continuoustime representation. For implementation, each filter is discretized separately with
the Tustin approximation, where the feedback control loop runs at the sample rate
of 100Hz.
In below-rated operation, it is expected that the larger variations in rotational
speed will reduce the load reduction performance of the repetitive controller considerably and therefore it is not considered. In above-rated operation, the CPC will keep
the rotational speed of the wind turbine close to the rated rotational speed. Thus a
well performing CPC can increase the load reduction performance of the RC. For an
honest comparison, we will use in the GH BLADED™ simulations the following CPC as
1
http://www.dcsc.tudelft.nl/~datadriven/discon/
107
5.2 Description of the simulations
specified in Bossanyi (2009):
ω2L
Ki
× 2
CCPC (s ) = K s (θ ) K p +
s
s + 2ζL ωL s + ω2L
|
{z
}
{z
}
|
gain-scheduled PI controller
×
s 2 + 2ζ
A,1 ωA s
s 2 + 2ζA,2 ωA s
{z
|
low-pass filter L
+ ω2A
+ ω2A
notch filter A
×
s 2 + 2ζ
B,1 ωB s
+ ω2B
s 2 + 2ζB,2 ωB s + ω2B
} |
{z
}
,
notch filter B
where the parameters are specified in Table 5.2. This PI controller with additional
filters was designed to generate a pitch position demand θ from the generator speed
error Ωge without causing to much excitation of the fore-aft tower vibrations, because
adjusting the pitch influences not only the aerodynamic torque but also the rotor
thrust. The pitch rate was limited to 8 deg/s. Furthermore, the control response at
3P and 6P (Per revolution) was reduced. For these reasons, two notch filters at around
0.6Hz (3P) and 1.3Hz (6P) were added to the feedback loop. An low-pass filter with
a bandwidth of 1.6Hz was also added is to prevent unnecessary high frequency pitch
action. Close to rated, the sensitivity of aerodynamic torque to pitch angle is very
small. Instead at higher wind speeds, a small change in pitch angle can have a large
effect on the aerodynamic torque. To compensate for this non-linear aerodynamic
characteristic, gain scheduling was added to the PI controller by a factor which varies
linearly from 1 at fine pitch (0 deg) to 3.5 at 25 deg.
Table 5.2: Parameters of collective pitch control
description
symbol
value
gain-scheduled PI controller
proportional gain
Kp
0.0135
integral gain
Ki
0.00453
180
θ
scheduling gain
Ks
1 + 10π
low-pass filter L
frequency
ωL
10 rad/s (≈ 1.6 Hz)
damping
ζL
1
notch filter A
frequency
ωA
3.8 rad/s (≈ 0.6 Hz)
damping 1
ζA,1
0.01
damping 2
ζA,2
0.15
notch filter B
frequency
ωB
8.2 rad/s (≈ 1.3 Hz)
damping 1
ζB,1
0.01
damping 2
ζB,2
0.2
In addition to the CPC, a further contribution to the pitch position demand was
recommended in Bossanyi (2009) to increase the response to sudden gusts. This
additional Non-linear Pitch Control (NPC) term depends on both the generator speed
error and its rate of change, which was obtained by differentiation preceded by a firstorder low-pass filter with time constant of τ = 0.05 to prevent overreaction to noise.
108 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
Both the generator speed error and its rate of change was normalised by a scale factor
c 1 = 25 and c 2 = 10, respectively. When the sum of scaled signals is over 1, then any
excess over 1 was multiplied by a gain factor of K n = 0.15 to give a contribution to the
pitch rate demand as follows:
¨
0,
if β ≤ 1,
CNPC (s ) = K n
β − 1, if β > 1.
where β = c 1 + c 2 s τs1+1 . In the discretized case, multiplying this pitch rate demand
by the sample time h = 0.01s gives the increase in pitch position demand. In normal
turbulent circumstances, this additional pitch demand remains zero. The logic of the
NPC will only give additional pitch action in an extreme case; when the generator
speed error is large, positive, and increasing.
In variable-speed wind turbines the torsional vibration modes of the drive train
are usually very lightly damped, because the rotor blades provide very little aerodynamic damping in the in-plane direction, therefore the structural damping and any
mechanical damping from the gearbox or generator will be small. In Bossanyi (2009),
it was recommended to provide very significant damping by modifying the constant
generator torque demand with an additional term which adds a small ripple at the
appropriate frequencies (1st drivetrain torsion mode and a sidewards tower bending
mode). This was obtained by passing the measured generator speed through the
following Torsional Vibration Damping (TVD) filter:
CTVD (s ) = K 1
|
2ζ1 ω1 s
s 2 + 2ζ
1 ω1 s
{z
+ ω21
+ K2
} |
inverse notch filter 1
2ζ2 ω2 (1 + τ2 ) s
s 2 + 2ζ2 ω2 s + ω22
{z
}
,
inverse notch filter 2
where the parameters are specified in Table 5.3.
Table 5.3: Parameters of torsional vibration damping filter
description
symbol
value
inverse notch filter 1
gain
K1
1560
frequency
ω1
24.2 rad/s (≈ 3.85 Hz)
damping
ζ1
0.132
inverse notch filter 2
gain
K2
1625
frequency
ω2
8.998 rad/s (≈ 1.4 Hz)
damping
ζ2
0.5041
offset
τ2
0.0138
5.2.4
Individual pitch control for load reduction
Wind turbine simulations with IPC using the Coleman transformation are considered
for comparison with the RC simulations in load reduction performance and control
109
5.2 Description of the simulations
action. The goal of this IPC is to reduce the out-of-plane bending vibrations in the
blade root of each of the three blades (M y ,1 , M y ,2 and M y ,3 ), by adjusting the blade
pitch angles θ1 , θ2 and θ3 . The major challenge of implementing load reduction
control is that the blades are rotating, and so the equations of motion that relate
M y ,1 , M y ,2 and M y ,3 and θ1 , θ2 and θ3 are dependent on the azimuth angle ϕ and
the rotational speed Ωro . The result is that a model obtained trough linearisation is
normally a periodic LTV system, and it is much more challenging to design controllers
for periodic LTV systems than LTI systems.
The typical solution is the use of the Coleman transformation (also referred to as
the Park or the Multiple Blade Coordinate (MBC) transformation), which maps the
variables of each blade in the rotating reference frame to a fixed reference frame of
the nacelle-tower. For the application of the Coleman transformation to variablespeed wind turbines, it is assumed that the azimuth angle of the rotor blades are
measured accurately. It is not assumed that the rotor blades have to be identical
structurally or aerodynamically, however they have to be equally spaced around the
rotor azimuth (Bir, 2008). Applying the Coleman transformation on complex models
of wind turbines, such as the non-linear models used in GH BLADED, does not result
in a completely decoupled LTI system. However, a Coleman transformation designed
for the 1P frequency, will at least decouple the periodic signal components at that
1P frequency, such that SISO LTI feedback control techniques like loop-shaping can
be used to reduce the 1P loads. Loads at higher harmonic frequencies can be
reduced using the solutions in Bossanyi (2009); van Engelen (2006), where parallel IPC
feedback loops are placed to the existing 1P-harmonic individual pitch controller, see
Figure 5.1. In practice, IPC with Coleman transformation are primarily designed for
reducing the 1P loads only, and sometimes 2P is considered. The remaining periodic
components are ignored, because current solutions require at these higher harmonics
considerable more pitch control action, due to the larger compensation needed for
the larger amount of phase lag (delay) in the input-output dynamics, and from the
more complicated MIMO controllers to overcome the increasing coupling between
the transformed measurement channels.
The IPC was not fully specified in Bossanyi (2009), but by following the recipe
in the same reference we have obtained a competitive implementation with almost
similar performance results. Before the Coleman transformation, the out-of-plane
bending moments measured in the blade root are scaled and filtered with a notch
filter tuned to the j P frequency. The Coleman transformation is applied to decouple
the loads in the direction of two axes, that for the 1P case may be thought as the
asymmetrical load components representing the horizontal (yaw) and vertical (tilt)
directions. The Coleman transformation for the j P harmonic frequency can be written
as:


–
™
M y ,1
2
M j P,yaw


= P jTP ϕ M y ,2  ,
M j P,tilt
3
M y ,3
with

Pj P
cos
jϕ
€
Š

cos € j ϕ + 2π
ϕ =
3 Š

cos j ϕ + 4π
3

sin
j
ϕ
€
Š 

sin € j ϕ + 2π
3 Š  ,
4π
sin j ϕ + 3
110 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
CNPC
+ +
− Ωge,rated
+
CCPC
CTVD
Tge,rated +
+
+
+
+
+
+
Tge
Ωge
θ1
θ2
θ3
M y ,1
M y ,2
M y ,3
S
+
+
L
+ +
L
+ +
L
+
θ1P,yaw
P1P(ϕ+δ1P)
θ2P,yaw
L
L
L
θ1P,tilt
K i ,1P
s
K i ,1P
s
K i ,2P
s
M 1P,yaw
M 1P,tilt
K i ,2P
s
M 2P,tilt
1P
1P
M 2P,yaw
P2P(ϕ+δ2P)
θ2P,tilt
1P
2 T
P (ϕ)
3 1P
2P
2 T
P (ϕ)
3 2P
2P
2P
Figure 5.1: Closed-loop wind turbine system S for above-rated operation with the
baseline controllers (CPC, NPC, TVD) and 1P and 2P IPC feedback loops in detail.
where ϕ is the azimuth angle of the first rotor blade, and M j P,tilt and M j P,yaw are
the transformed loads. After the loads are transformed, each channel is decoupled
and is integrated using I controllers, as recommended in Kanev (2009). The gains
for the I controllers are tuned to be same for both yaw and tilt directions, as the
case in Bossanyi (2009). However, for each rotational frequency different gains are
applied. All the I controllers are subjected to limits of 5 deg on their outputs,
which means that after reverse transformation the maximum amplitude of the nearsinusoidal individual pitch action is limited. For controllers with only integral action,
this saturation limit is easily implemented, but for complicated controller designs
the difficulties with saturation should be taken care of by using anti-windup loops
as described in Kanev (2009). The corresponding reverse Coleman transformation is
given by:
 
–
™
θ1
θ j P,yaw
 
θ
.
=
P
ϕ
 2
jP
θ j P,tilt
θ3
After the reverse transformation, the signals are filtered again by a low-pass filter with
a bandwidth of 1.6Hz (similar low-pass filter is used with CPC) to obtain a smooth
control signals. The complete IPC controller for the 1P frequency can be represented
111
5.2 Description of the simulations
as:
C IPC,1P (s ) =
ω2L
s 2 + 2ζ
|
L ωL s
{z
+ ω2L
I 3×3 P1P ϕ + δ1P
}
– K i ,1P,yaw
s
0
0
K i ,1P,tilt
s
™
2 T
P1P ϕ
3
low-pass filter L
× K 1P
|
2ζ1P ω1P s
s 2 + 2ζ
1P ω1P s
{z
+ ω21P
}
I 3×3 ,
inverse notch filter 1P
where the parameters, including the parameters for the 2P harmonic frequency, are
specified in Table 5.4.
The performance of these IPC controllers can be reduced considerably by having a
large phase lag (delay) between the controller and the pitch actuator. It was suggested
in Bossanyi (2009), that a small offset added to the azimuth angle in the reverse
Coleman transformation can be used to compensate the delay. Moreover, as the
control signals from the 1P and 2P harmonics are near sinusoidal it is also possible
to compensate for the complete phase lag of the open-loop dynamic system. In our
case, by looking at the Bode diagram of the linearised plant with filters, this is 25 deg
and 33 deg for the 1P and 2P harmonic frequencies, respectively. With this additional
phase compensation and the non-linear transformation using the azimuth angle, one
could say that the IPC with Coleman transformation becomes and behaves more as an
adaptive feedforward controller than a feedback controller. Although, this becomes
considerable more challenging for higher harmonics.
Table 5.4: Parameters of individual pitch control
description
symbol
value
I controller
integral gain K i ,1P,yaw
0.01
integral gain
K i ,1P,tilt
0.01
integral gain K i ,2P,yaw
0.002
integral gain
K i ,2P,tilt
0.002
low-pass filter L
frequency
ωL
10 rad/s (≈ 1.6 Hz)
damping
ζL
1
inverse notch filter 1P
scaling gain
K 1P
10−6
frequency
ω1P
2π/4.95 rad/s (≈ 0.2 Hz)
damping
ζ1P
0.5
inverse notch filter 2P
scaling gain
K 2P
10−6
frequency
ω2P
4π/4.95 rad/s (≈ 0.4 Hz)
damping
ζ2P
0.3
reverse Coleman transformation
phase offset
δ1P
25π/180 rad
phase offset
δ2P
33π/180 rad
112 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
5.2.5
Wind conditions during the simulations
A variety of simulations are performed in GH BLADED in order to evaluate the
effectiveness of the load reduction controllers; the baseline controllers CPC, NPC,
and TVD with either IPC-1P, IPC-2P, or RC. It has been shown in Kühn (2001), that
the wind phenomenon can be separated in a low and high frequent part, where
the low frequent part is the undisturbed mean wind speed and the high frequent
part is the turbulence. The external wind conditions used in the simulations, or
load cases, are derived from the International Electrotechnical Commission (IEC)
standards, specified in IEC (2004). Following the standards, the simulations are
performed with at least of a length of 600 s, the mean wind speed values of 16
and 20 m/s, and with 3D turbulent data generated using the Von Karman spectrum
model in GH BLADED. For each mean wind speed, eight load cases are generated
in ascending order of turbulence intensity, up to the extreme turbulence intensity
of I T = 0.16. The details of the eight load cases in the turbulence intensities for all
three the directions are summarized in Table 5.5. For the purpose of fatigue damage
calculation, the IEC standard considers that the turbulence intensity of I T = 0.16 is
“high”, I T = 0.14 is “medium”, and I T = 0.12 is “normal”. However, in Nino and Eecen
(2001) it was concluded from measurements at multiple offshore locations that the
common longitudal turbulence intensities for a wind speed interval from 9 m/s to 18
m/s at 50 m height is in the range between 8% and 10%.
Table 5.5: Turbulence intensities for a reference wind speed of 15 m/s generated by
the Von Karman model, and the related standard deviations of the period time.
turb. intensity longitudinal
lateral
vertical std p h
IT = 0
−
−
−
−
I T = 0.04
4.05 %
3.27 %
2.46 %
0.024 s
I T = 0.06
6.01 %
4.79 %
3.52 %
0.030 s
I T = 0.08
8.01 %
6.35 %
4.62 %
0.036 s
I T = 0.10
10.01 %
7.90 %
5.70 %
0.041 s
I T = 0.12
12.04 %
9.48 %
6.81 %
0.047 s
I T = 0.14
14.01 %
11.01 %
7.88 %
0.056 s
I T = 0.16
16.01 %
12.57 %
8.97 %
0.060 s
In the lower layers of the atmosphere, the mean wind speed is influenced by the
friction with the surface of the earth. The roughness of the landscape, like buildings
and trees, could reduce the wind speed considerably and increase turbulence. Two
important periodic wind effects created by these obstructions in the airflow near
the wind turbine are considered in the simulations: wind shear, and tower shadow.
Wind shear is the variation in height of the average undisturbed wind velocity, see
Figure 5.2. The selected exponential model relates the local wind speed disturbed by
the wind shear effect at height z above the ground level to the wind speed at hub
height v (z 0 ) as (GH , 2010a):
α
z
,
v w s (z ) = v (z 0 )
z0
113
5.3 Lifted Repetitive Control Design
where α = 0.2 is the wind shear exponent. Tower shadow is the distortion of wind
speed caused by the presence of a support structure in the wind flow, see Figure 5.3.
The potential flow model relates modifies the local wind speed using the assumption
of incompressible laminar flow around a cylinder of diameter D as (GH , 2010a):
vt s
‚
y 2 − x 2 (κD)2
x,y = 1 −
2
4
y 2 +x2
Œ
vw s x , y ,
where κ = 1.2 is the tower diameter correction factor.
y
x
z
Figure 5.2: The left sideview of a wind
turbine, showing the wind shear effect.
5.3
Figure 5.3: The topview of a cross section
from the wind turbine tower, showing the
tower shadow effect.
Lifted Repetitive Control Design
In this previous section the baseline controllers for above-rated operation, and a
typical IPC, were described for comparison. Now the theoretical framework for the
repetitive control method to reject periodic disturbances is presented. First, the
system descriptions, the assumptions made, and some notations are presented. Then
the goal is to formulate the RC design problem into an output-feedback problem.
After the formulation, the state-space matrices of the lifted repetitive controller can be
calculated by solving two large Riccati equations. For this purpose, the linearisation
procedure for obtaining the state-space matrices are described. At last, the extensions
to RC for the uncertain period time are discussed and some tuning guidelines are
provided.
114 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
5.3.1
System descriptions and assumptions
The dynamics of the linearised system S to be controlled can be written by the
following state-space model:
¨
x k +1
yk
S
= A k x k + B k u k + Fk d k + w k ,
= C k x k + Dk u k + vk ,
(5.1)
where x k ∈ Rn , u k ∈ Rr , y k ∈ Rℓ , d k ∈ R f , w k ∈ Rn , and v k ∈ Rℓ , are the state,
input, output, periodic disturbance, process noise, and measurement noise vectors.
The state-space matrices A k ∈ Rn ×n , B k ∈ Rn ×r , Fk ∈ Rn × f , C k ∈ Rℓ×n , and D k ∈
Rℓ×r are also called the state, input, periodic input, output, and direct feedthrough
matrices, respectively. In our case, the input vector contains the individual pitch angle
demands, the output vector contains the individual blade-root bending moments, and
the disturbance vector contains the wind disturbances as:
”
u k = θ1,k
θ2,k
θ3,k
—T
”
, y k = M y 1,k
M y 2,k
M y 3,k
—T
”
d k = v 1,k
,
v 2,k
v 3,k
—T
.
For simplification and clarification, we consider an LTV system of fixed order. However, the lifted repetitive control method described in this chapter can be extended to
LTV systems of varying order. The linear time-varying dynamics are also considered
to be periodic, thus A k = A k −p , B k = B k −p , Fk = Fk −p , C k = C k −p , and D k =
D k −p . The process and” measurement
noise
are considered Gaussian
—
” disturbances
—
noise sequences with E v k v kT = R v and E w k w kT = R w , respectively. In the case
of coloured Gaussian noise sequences, for example with a random walk model to
represent the wind turbulence dynamics as described in Selvam et al. (2009), the statespace system S should be augmented with the noise filter dynamics.
With the trial window size p ∈ N+ , the following stacked vector can be defined as:
”
ȳ k ,p = y kT
y kT+1
···
y kT+p −1
—T
.
The stacked vectors ū k ,p , d¯k ,p , n̄ k ,p , w̄ k ,p , and v̄ k ,p are defined in a similar way. Using
the definition of the stacked vectors, we can “lift” the system in (5.1) to the following
lifted description:
¨
S̄
x k +p
ȳ k ,p
= Φk ,p x k + Lū k ,p + F d¯k ,p + P w̄ k ,p ,
= Γx k + G ū k ,p + J d¯k ,p + Hw̄ k ,p + v̄ k ,p ,
(5.2)
where Φk ,p = A k +p −1 . . . A k +1 A k is the state transition matrix, and L ∈ Rn ×p r , F ∈ Rn ×p f ,
and P ∈ Rn ×p n are the extended controllability matrices, and are given by:
”
L = Φk ,p −1 B k +p −1 · · · A k B k +1
”
—
P = Φk ,p −1 · · · A k I ,
—
”
B k , F Φk ,p −1 Fk +p −1
···
A k Fk +1
—
Fk ,
and G ∈ Rp ℓ×p r , J ∈ Rp ℓ×p f , and H ∈ Rp ℓ×p n are the impulse matrices with a
lower block triangular structure, Γ ∈ Rp ℓ×n is the extended observability matrix, and
are given in (5.3). These lifted system matrices are trial invariant, because of the
115
5.3 Lifted Repetitive Control Design
assumption that the time-domain dynamics are periodic. Note, that I is used to
represent an identity matrix of appropriate dimensions.


Dk
Ck




C
A


k +1
k
C k +1 B k
, G = 
Γ=

..


..

.


.

C k +p −1 Φk +1,p −1
C k +p −1 Φk +1,p −1 B k


0
0
··· 0

.
..

. .. 
C k +1
0


H=
,
..
..
..


. 0
.
.

C k +p −1 Φk +1,p −1 C k +p −1 Φk +2,p −1 · · · 0

0
0
···

..

.
C k +1 Fk
0

J =
..
..
..

.
.
.

C k +p −1 Φk +1,p −1 Fk C k +p −1 Φk +2,p −1 Fk −2 · · ·

0
D k +1
..
.
C k +p −1 Φk +2,p −1 B k −1
···
..
.
..
.
···
0
..
.
0
D k +p −1




,


(5.3)

0
.. 

.
.

0
0
The stacked periodic disturbance may consist of an integrated “random walk”
component as (Chin et al., 2004; Cho et al., 2005; Lee et al., 2001):
d¯k ,p = d¯k −p,p + n̄ k ,p ,
(5.4)
where n̄ k ,p is a Gaussian noise sequence with E n̄ k ,p n̄ Tk ,p = R̄ n . This is a quite versatile noise formulation; the process disturbance contains not only a stochastic and a
periodic component, but also a random walk nature, so the “periodic disturbance” can
vary between trials. In addition, the noise and periodic disturbances can be correlated
in time.
In Assumption 5.1, we adopt the mild conditions commonly used for the lifted
repetitive control problem (Rice and Verhaegen, 2010b).
Assumption 5.1.
• The system S̄ is€ asymptotically
stable (if
Š
€ not, Šstabilize with feedback control),
controllable on Φk ,p , L , observable on Φk ,p , Γ , and does not contain any zeros
at +1.
• The process noise w k and measurement noise v k are zero-mean white or coloured
(if so, augment dynamics to system) Gaussian sequences.
• The “random walk” noise n̄ k ,p is a zero-mean white Gaussian sequence.
When the number of inputs is smaller than the number of outputs (r < ℓ), the
controllability/observability condition in Assumption 5.1 is normally violated. In this
Š−1
€
L does not have full row rank. It is proposed to only
case the matrix G + Γ I − Φk ,p
reduce the error that lies in the column space of that matrix, thus we try not to achieve
116 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
perfect rejection for all disturbance signals (or all the harmonics). To achieve this, we
use the Singular Value Decomposition (SVD) as:
–
™– ™
Š−1
”
— Σs 0
€
Vs
,
(5.5)
L = Us U⊥
G + Γ I − Φk ,p
0 Σ V⊥
where the order s is determined by detecting a gap that separates the s largest singular
values from the remaining ones. Now, we can replace the output equation of the lifted
system in (5.2) by a projected output equation given by:
UsT ȳ k ,p = UsT Γx k + UsT G ū k ,p + UsT J d¯k ,p + UsT Hw̄ k ,p + UsT v̄ k ,p .
(5.6)
Furthermore, we can select the number of singular values s not only on the controllability/observability condition, but also to further reduce the size of the RC outputfeedback problem and let the lifted repetitive controller focus on the main harmonic
components and therefore robustify the controller performance in some sense.
5.3.2
Lifted repetitive control by LQG design
In this subsection the repetitive control method for periodic LTV systems with both
repetitive and non-repetitive disturbances is presented. To let the repetitive controller
respond to periodic disturbances, a “non-stochastic” output from (5.2) is defined as:
ỹ k ,p = ȳ k ,p − Hw̄ k ,p − v̄ k ,p = Γx k + G ū k ,p + J d¯k ,p ,
(5.7)
and a “non-stochastic” error independent of the noise disturbances from trial k (w̄ k ,p ,
v̄ k ,p ) as:
ẽ k ,p = r̄k ,p − ỹ k ,p ,
(5.8)
where r̄k ,p is considered trial invariant. The change between two trials (or periods) is
denoted by the ∆ operator, such that for example the difference between the vector of
the input from time k and the vector of the input from time k −p is defined as ∆ū k ,p =
ū k ,p − ū k −p,p . Using this definition, it follows that the “non-stochastic” tracking error
is defined as:
ẽ k ,p = ẽ k −p,p − Γ∆x k − G ∆ū k ,p − J n̄ k ,p .
(5.9)
An expression for the current “stochastic” tracking error based on the previous “nonstochastic” error can be obtained and is given by:
ē k ,p = ẽ k ,p − Hw̄ k ,p − v̄ k ,p .
(5.10)
By using the ∆ operator again on the state as ∆x k = x k − x k −p , the difference in the
initial conditions of S̄ between trials is derived from (5.7), and (5.4) as:
∆x k +p = Φk ,p ∆x k + L∆ū k ,p + F n̄ k ,p + P ∆w̄ k ,p .
(5.11)
Now we can combine (5.9), (5.10), and (5.11) into an stochastic linear system description as:
¨
ξk +p = Aξk + B ∆ū k ,p + BW ν̄k ,p ,
(5.12)
S̄RC
ē k ,p = C ξk + D∆ū k ,p + DW ν̄k ,p ,
117
5.3 Lifted Repetitive Control Design
where the vectors ξk ∈ Rp (ℓ+n )+n , and ν̄k ,p ∈ Rp ( f +n +ℓ) are given by:
”
—T
”
—T
ξk = ∆x kT ẽ kT−p,p w̄ kT−p,p , ν̄k ,p = n̄ Tk ,p w̄ kT,p v̄ kT,p ,
and the system matrices A ∈ Rp (ℓ+n )+n ×p (ℓ+n )+n , B ∈ Rp (ℓ+n )+n ×p r , C ∈ Rp ℓ×p (ℓ+n )+n , D ∈
Rp ℓ×p r , BW ∈ Rp (ℓ+n )+n ×p ( f +n +ℓ) , and DW ∈ Rp ℓ×p ( f +n +ℓ) are given by:


 
Φk ,p 0 −P
L
”
—


 
0  , B = −G  , C = −Γ I 0 , D = −G ,
A =  −Γ I
0
0
0
0


F
P 0
”
—


B W =  − J 0 0 , D W = − J H I .
0
I 0
The lifted RC problem can now be formulated as:
Problem Description 5.1 (Lifted repetitive control). Given the system in the form of S̄RC
in (5.12), design a strictly-proper dynamic lifted repetitive controller C¯RC .
Given Assumption 5.1, the infinite-horizon discrete-time LQG solution is well
known, see Åström and Wittenmark (1997), and it is desirable in our situation, as it
stabilizes the system S̄RC and minimizes the expectation of the cost function:


N
X
2
2
(5.13)
kē k +i p,p ||2 + ρk∆ū k +i p,p ||2  .
J = E  lim
N →∞
i =0
Compared with the minimum-variance criterion in Rice and Verhaegen (2010b), we
introduced in the LQG criterion a weight to penalize the differenced input, and therefore we are able to make a trade-off between the variance of the differenced input and
output signals. By requiring that C¯RC stabilizes S̄RC , it can be shown to get exponential
convergence of the RC algorithm. This does not imply that the RC algorithm has
monotonic decay (kẽ k +p,p k2 ≤ kẽ k ,p k2 ) as recommended in Longman (2000), but the
experience is that the introduction of a weight on the differenced input could give the
engineer some means to prevent for the large overshoots conventionally seen with
some RC laws with non-monotonic decay in the initial transient periods. Using a
steady-state approximation of the optimal LQG solution, the lifted controller C¯RC is
given by:
¨
ξ̂k +p = AC ξ̂k + LC ē k ,p ,
¯
(5.14)
CRC
∆ū k ,p = −KC ξ̂k ,
where
AC = A − BKC − LC C + LC DKC ,
−1
LC = SW + AZC T RW + CZC T
,
−1
KC = R + B T X B
BT X A + S T .
The matrices X and Z are the positive-definite stabilizing solutions to the Discrete
Algebraic Riccati Equations (DARE):
X = Q − AT X B + S K C + AT X A,
(5.15)
Z = QW − LC CZ AT + S W + AZ AT ,
118 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
where R = ρI + DT D, Q = C T C , S = C T D, and the disturbance covariance matrices are
calculated as:
–
™ – ™ h
i –D ™T
RW S W T
DW
W
T
.
(5.16)
=
E ν̄k ,p ν̄k ,p
BW
SW Q W
BW
In many cases, the solvers for the DARE in (5.15) have difficulties to determine the
solution numerically. For example the QZ-based DARE solver in MATLAB™ will fail,
because the Hamiltonian or sympletic matrix has eigenvalues on and/or close to the
unit circle. To overcome this problem, the learning gain should be lowered by adding
forgetting or leakage. With the inclusion of forgetting, the input difference is defined
as ∆ū k ,p = ū k ,p − λū k −p,p , where the scalar 0 ≪ λ < 1 is the forgetting factor.
The block diagram in Figure 5.4 illustrates how the lifted repetitive controller
including the memory loop can be implemented. The output measurements are p times sampled during the previous trial and buffered into a stacked vector. For the
start of the next trial, the lifted repetitive controller is evaluated and an updated
stacked input vector is returned. According to the sample time, the elements of the
calculated stacked input vector are unbuffered during the current trial and given
as input to the pitch actuator of the wind turbine system. Note, that the lifted
repetitive controller is strictly proper, because the current trial output measurements
are assumed unavailable for feedback.
yk
uk
- S
”
··· ,
|
u kT ,
{z
···
-
”
—T
··· ,
|
}
6
- λz −p
···
—T
}
ȳ k ,p
ū k −p,p
+
?
ū k ,p
?
y kT ,
{z
+
∆ū k ,p
C¯RC −
?
ē k ,p
+
r̄k ,p
Figure 5.4: Block diagram of the wind turbine system S (CPC and TVD included in
block) with the RC loop with one memory loop.
The biggest challenge for the implementation of this RC description is that the
lifted matrices are enormous, and it requires large memory storage and can take a
long time to calculate the solution. A possible solution that can be used to resolve the
computational complexity issue is by exploiting the matrix structure. In Appendix C,
efficient linear algebra computations using the so-called Sequentially Semi-Separable
(SSS) matrix structure are proposed. The lifted repetitive controller could be implemented real time using three SSS structured matrix-vector computations such that
the complexity is O ℓ2 p instead of O ℓ2 p 2 .
Another challenge for lifted RC is that in practice the inputs are normally constrained to certain limits. All smart actuators currently in development have operational limits on the input signal; for example, piezo-electric flaps have always a
5.3 Lifted Repetitive Control Design
119
certain limitation in deflection, and MEM-tabs can only be switched on or off. If not
taken care of correctly, it is well known that RC will destabilize the closed-loop system
if the inputs are saturated. Since Model Predictive Control (MPC) can in principle
handle the above features, our formulation can be converted to a repetitive MPC
problem as have been done in Gupta and Lee (2006); Lee et al. (2001). The difficulty
is that normally an optimization have to be performed online to solve the quadratic
problem with inequality constraints. The problem with input saturation can occur
for example when the wind turbine leaves the above-rated region. In the current
RC implementation, the learning capability of RC is simply set to a hold when the
collective pitch angle reaches 2 deg or lower. This means that the stacked input vector
is not updated any more in the below-rated region. Moreover, the stacked input values
gradually converge to zero due to the forgetting. If the wind turbine enters the aboverated operation region after a long leave, it would take several periods to arrive at
steady-state.
5.3.3
Model linearisation for the lifted RC design
The use of models has a long tradition in control engineering. Most modern control
techniques, including the proposed RC method, are model based and require a mathematical representation of the system under consideration. Thus a model describing
the relevant number of properties of the wind turbine in a mathematical language is
needed to tune and even design the lifted repetitive controller. The linearisation tool
in GH BLADED is used for this purpose, to obtain a linear model in a state-space system
form suited for the proposed RC design. During the linearisation, the following inputs
are selected: the variations in the generator torque Tge , the pitch angles θi , and the
wind speed disturbance v i on each of the three rotor blades (i = 1, 2, 3). Further, the
following outputs are selected: the variations in generator speed Ωge , the out-of-plane
blade-root bending moments M y ,i of each rotor blade (i = 1, 2, 3). In Figure 1.4, a
schematic representation of the wind turbine with the selected inputs, disturbances
and outputs are given.
The equations of motion that relate M y ,1 , M y ,2 and M y ,3 and the blade pitch angles
θ1 , θ2 and θ3 are dependent on the azimuth angle ϕ and the rotational rotor speed
Ωro , therefore a periodic LTV system through linearisation is acquired. In our case,
the azimuth is divided in 99 sections, thus for each increment of 3.636 deg(or 0.05s
with rated rotational rotor speed) an LTI state-space system is obtained. The lifted
RC loop is intended to run at the sample rate of 20Hz, i.e sample time of h = 0.05s,
and the LTI system is discretized accordingly using the zero-order-hold method. The
obtained linear models from the linearisation tool in GH BLADED are still in open-loop
form and do not contain the dynamics of the baseline controllers. These controller
dynamics have been included afterwards by closing the feedback loops, for example
from the generator speed Ωge to the the pitch angle θi of each rotor blade with the
CPC transfer description. Furthermore, we apply a second-order high-pass filter with
bandwidth from 0.1Hz in series with the output of the blade-root bending moments
M y ,i of each rotor blade. The dynamic transfer function of the high-pass filter is given
120 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
by:
H (s ) =
s 2 + 2ζH,1 ωH s
s 2 + 2ζH,2 ωH s + ω2H
,
where the frequency is ωH = 0.63 rad/s and damping values are ζH,1 = 0.7 and ζH,2 =
1. This high-pass behaviour is desirable for this simulation experiment, as we want
to control the dynamic behaviour of the system, rather than the static deformations.
This behaviour is normally present in the open-loop system, but then in the form
of sensors dynamics, like with Macro Fiber Composites (MFC) sensors from Smart
Material have typical high-pass behaviour. At last, a low-pass filter similar to the one
in CPC and IPC is used to smooth the control signals from the RC and before they are
send to the pitch actuator.
For the model-based control design, the model should only contain the relevant
dynamics between the inputs, and the outputs and should be accurate over the
bandwidth of the controller; for RC design, these are the main harmonic frequencies.
Model reduction could be applied (is not used in this chapter) to obtain a model
of sufficiently small order with a relevant number of properties, which is normally
preferred for model-based control design (Molenaar, 2003). In this chapter, we consider models based on the analytical first-principles modelling through the simulation
environment of GH BLADED. The equations behind the software strongly depends
on a large number of parameters that determine the (aero) dynamic behaviour of
the wind turbine system. Most of these parameters can be roughly estimated or
calculated. Still, a large amount of uncertainty can be present and this makes it
difficult in practice to design a stable controller based on such models, because it
is known from Doyle (1978) that the LQG design is sensitive to uncertainties. More
accurate models could be devised from measured data, for example using the periodic
LPV system identification technique in van Wingerden et al. (2009). It is suggested that
identification becomes a necessary building block for the model-based RC design in
practice.
5.3.4
Extensions to RC for uncertain period time
An important drawback of RC is that a small mismatch between the controller period
and the actual period of the disturbance signals can decrease the performance of
RC substantially. This drawback makes the application of RC less practical for
wind turbines with small variations in the rotational rotor speed, in spite of a pitch
controller that will keep (during above-rated operation) the rotational speed of the
wind turbine close to the defined reference rotational speed. Several approaches
have been proposed in the literature to improve robustness of RC in the presence of
possible variations in the period-time of p h = 4.95 s. For the proposed lifted RC, these
possible approaches can be divided into two groups, depending if the true period time
or the true rotational rotor speed is accurately measured.
An important issue for RC designed with lifted periodic LTV model compared with
lifted LTI models is that the true azimuth angle of the wind turbine should correspond
to the index in the sequence of LTV state-space matrices during lifting, otherwise
the linearised LTV dynamics will not match with the true dynamics of the wind
5.3 Lifted Repetitive Control Design
121
turbine system. If not taken care of correctly, the variations in the period time will
cause an increasing mismatch that can eventually lead to a considerable reduction
in performance and even instability. To keep the period mismatch as minimal as
possible during the operation, the buffering of the input and output samples over
the period will be done according an increment of 3.636 deg in the azimuth angle of
the first blade, which approximately equals the sampling time of h = 0.05 s when the
rotational rotor speed is rated. The azimuth angle of the first blade is assumed to be
measured accurately, as is the case with IPC. In other words, we allow the sample rate
to vary instead, because the sampler is designed and implemented using the angular
position instead of time, so that the disturbance period is always exactly one rotation
regardless of speed variations. On the other hand, sample time adaptation of a lifted
repetitive controller where the sample rate of the underlying model is unchanged, may
also result in a decrease in performance and even instability. The experience of the
authors, and also the experience in Tsao et al. (2000); Wen and Longman (2001), is
that the use of this approach affects the robustness of the closed-loop system in more
favourable way, but a robustness analysis as given in Costa-Castelló et al. (2011); Olm
et al. (2011) will be very challenging, and is therefore not explored.
If it is not possible to measure the period time accurately, it is common practice to
formulate a high-order RC problem with multiple memory loops. in Pipeleers et al.
(2008); Steinbuch (2002); Steinbuch et al. (2007) the RC problem is extended with
more memory loops. This so-called high-order RC can be made more robust for small
changes in period time, but this comes at the cost of the amplification of noise at
the non-harmonic frequencies. As shown in Gupta and Lee (2006), multiple memory
loops can be introduced in the lifted RC problem at the cost of larger matrices.
Compared with the time-domain RC methods in Pipeleers et al. (2008); Steinbuch
et al. (2007), the lifted RC problem in Rice and Verhaegen (2010b) allows a more
versatile disturbance formulation to specify the amount of variation in the periodic
wind disturbances. With the introduction of a random walk nature n k in (5.4),
the RC formulation allows the periodic disturbance to vary between trials. When
the periodicity of the disturbances are known, for example with the identification
techniques in van Baars et al. (1993); van Wingerden et al. (2011), it is even possible
to relate the lifted covariance matrix R̄ n with the variance of the period time p h by a
first-order approximation as:
T
∂ d¯k ,p p h
∂ d¯k ,p p h
ˆ
var p h
.
R̄ n =
∂ ph
∂ ph
where the basis functions in d˜k are used to express the periodic disturbances. Both
approaches perform for small variations in period time, however the latter approach is
preferred in this case. First, the addition of more memory loops comes at the cost of a
considerable increased computational burden. Second, the selection of the number of
memory loops and its weighting factors is not an easy task. At last, it is noted that the
combination of the approaches will probably only give 1–2% additional performance,
as was experimentally shown in Chapter 4.
122 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
5.3.5
Guidelines for selection of the parameters
In this subsection, some guidelines are given on how to pick some initial values for
the free parameters in the algorithm. During the design of the proposed lifted RC
method, a large number of parameters have to be selected. We have the singular value
order s , the disturbance covariance matrices R w , R v , and R̄ n , and the weight ρ on the
differenced input, and the forgetting factor λ.
An initial guess for the number singular values s that have to be taken into
account can be determined by detecting gaps that separates groups of singular values
in (5.5). It turns out that for even lower values of s , the lifted repetitive controller will
focus more on the main harmonic components and therefore robustify the controller
performance in some sense. In our case, we selected the values s = 18, s = 24, and
s = 30 for further investigation.
The values of the process and measurement noise covariance matrices R w , R v
must be chosen according to the level of measurement and process noise present in
the controlled system. A close estimate of these covariance matrices could be found
from an identification of the system to be controlled. In our simulations, apart of the
turbulence, no additional process or measurement noise is considered, therefore we
selected these values to be small: R w = 0.01I , and R v = 0.01I .
The choice of the non-periodic disturbance covariance matrix R̄ n can be very
challenging. First, the value depends on the level of non-periodic turbulence included
in the wind disturbance. Second, the value can be increased even more to make the
lifted repetitive controller more robust against period-time mismatch. In our case, the
covariance matrix is chosen to be R̄ n = ηI , where η is a scalar to be selected. If the
contributions of the weights R w , R v to the cost function in (5.13) can be considered
small, such that it can be neglected compared with the contribution of the weight R̄ n ,
then the expected performance of the RC will only be a trade-off between η and ρ. So
let η = 10, then we can only adjust ρ until a satisfactory response.
The strategy to pick the parameter ρ is very similar to the time-domain LQG
problem, see Åström and Wittenmark (1997). It is common practice to pick the
parameter ρ such that a smooth control signal is obtained and a large overshoot is
avoided. in Longman (2000) it is even recommended to have a monotonic decrease
of kẽ k ,p k2 , which is almost (apart from the initial differenced state) directly related to
the monotonic increase of kū k ,p k2 ). By increasing the weight ρ, the overshoot over
the trials can be lowered in most cases and improve the smoothness of the generated
control signal at the cost that it takes longer to converge to steady state and is less
adaptive to any changes in the periodic disturbance.
The updated stacked input vector for the next period is a combination of the
previous input vector times a forgetting factor and a small correction obtained from
the lifted repetitive controller. Thus, the updated stacked input vector consists
for large part by a component which is averaged over the past trials (or periods).
Forgetting will introduce less weight on the past trials than more recent trials, such
that the more recent trials are more relevant for the current periodic disturbance
2
= 200 periods in
representation. With λ = 0.99, the stacked input vectors up to (1−λ)
the past, will approximately contribute for 90% to the averaged stacked input vector,
5.4 Simulation results and analysis
123
therefore this relation can be helpful for an initial selection. Further, the value λ = 0.99
for the forgetting factor is normally more than sufficient to overcome the problem
with DARE solvers.
5.4
Simulation results and analysis
In this section, the main results of the different simulation studies on the UPWIND
5MW wind turbine operating in a periodic wind field are presented. The effectiveness
of the proposed RC algorithm is analysed and compared with the IPC algorithms with
Coleman transformation. First, we will show the results of the IPC with Coleman
transformation. Second, we will compare them with the results of the proposed lifted
RC. It should be emphasized that the relative performance of RC and IPC detailed
here depends strongly on both the selection of the free parameters, for example less
aggressive integrator gains would lower the pitch rates considerably and would likely
lead to lower load reductions. A truly equal comparison is difficult to be pursued,
however some important differences can be concluded from evaluating the trends.
The simulations are performed for two different mean wind speed values of
16 and 20 m/s, and for eight turbulent cases given in Table 5.5. To evaluate the
performance in load reduction, the results of the blade-root bending moment M y ,
the hub tilt moment M tilt , and the hub yaw moment M yaw are calculated in the
form of the standard deviation (std) or in percentage of the baseline control results.
The 1Hz damage equivalent loads were also calculated, however these results are
more uncertain than the standard deviation, and therefore not considered. With the
damage equivalent loads, it was much more difficult to compare the load reduction
results between the different load cases, although similar trends can be shown in the
results of equal turbulent cases. The cost of additional wear in the pitch actuators
is quantified by an increase in the standard deviation of the pitch rate std(θ̇ ). The
output of the more representative pitch forces or the related pitch accelerations are
currently not supported in GH BLADED. It is expected that by examining the pitch rate
the high-frequent pitch actions would be less dominant, therefore the results are less
favourable than when the pitch acceleration would be used.
5.4.1
Performance of individual pitch controller
The simulation results of the percent load reduction and pitch action of IPC compared
with the baseline control case are summarized in Tables 5.6 and 5.7. Four different IPC
cases are considered; namely IPC-1P, IPC-1P-25deg, IPC-2P, and IPC-2P-33deg. These
names represent the IPC approaches as discussed in Subsection 5.2.4. IPC-1P contains
only the 1P feedback loop with a phase offset of δ1 = 0 deg. IPC-1P-25deg contains
only the 1P feedback loop with a phase offset of δ1 = 25 deg. IPC-2P contains both
1P and 2P feedback loops with δ1,2 = 0 deg. IPC-2P-33deg contains both 1P and 2P
feedback loops with δ1 = 25 deg and δ2 = 33 deg. All the IPC cases produce sizeable
reductions in standard deviation of the loads in all turbulent cases and the two mean
wind speeds. This comes at the cost that the pitch rates are considerably larger; for
the higher turbulent cases, it is between two and four times higher.
124 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
Several observations can be made regarding the different IPC cases. The introduction of the 2P feedback loop is only beneficial in load reduction, if the phase offset of
the reverse Coleman transformation is tuned correctly. For the high turbulent cases,
this will give 1-2% extra load reduction at the cost of an increase 5% in pitch rate. In
Figure 5.5, the square-root Power Spectral Density (PSD) of the blade-root bending
moments are given for the baseline (light grey), IPC-1P-25deg (blue), and IPC-2P33deg (black) controlled system. The PSD plots are very similar for both the 1P and
2P cases, except that the 2P frequency is unaffected in the 1P case.
107
1P
magnitude (Nm)
2P
106
105
10−1
100
frequency (Hz)
Figure 5.5: Square-root PSD of the blade-root bending moment M y ,1 . Black line is
with IPC-1P-25deg, blue line is with IPC-2P-33deg, and light grey line is with baseline
control. For I T = 0.12, v = 16m/s.
Tuning the phase offset of the reverse Coleman transformation is clearly beneficial
for both the 1P and 2P case for high turbulent cases. Especially, in the form of pitch
rate needed to obtain similar or better load reduction results. For the 1P case, a
reduction of 5% in pitch rate is observed, and for the 2P case even 10%. In the low
turbulent cases, the results are very similar. In the cases without turbulence, the
individual control signals for the tilt and yaw directions are almost a perfect sine
and cosine wave, respectively. It is observed that the combined pitch signal is also
a sine wave, but with a compensated phase shift constructed by a combination of the
individual sine and cosine waves. With increasing turbulence intensity, the individual
control signals become less sinusoidal, and a combination will not lead to perfectly
corrected phase and will create signals with larger amplitudes. In Figure 5.6, the
square-root PSD of the first blade-root bending moments are given for the baseline
(light grey), IPC-2P (blue), and IPC-2P-33deg (black) controlled system. By comparing
the PSD plots, the 0 deg case shows, compared with 25-33 deg case, an amplification
at frequencies close to right of the 1P and 2P frequencies. This causes an additional
125
5.4 Simulation results and analysis
amplification of a natural frequency of an in-plane mode, which explains the worse
load reduction results obtained with IPC-2P.
107
magnitude (Nm)
2P
NF
106
105
10−1
100
frequency (Hz)
Figure 5.6: Square-root PSD of the blade-root bending moment M y ,1 . Black line is
with IPC-2P-33deg, blue line is with IPC-2P, and light grey line is with baseline control.
For I T = 0.12, v = 16 m/s.
5.4.2
Performance of lifted repetitive controller
The simulation results of the percent load reduction and pitch action of RC compared
with the baseline control case are summarized in Tables 5.6 and 5.7. Three different
RC cases are considered; namely RC-18s, RC-24s, and RC-30s. These names represent
to the RC approaches as discussed in Section 5.3. RC-18s, RC-24s, and RC-30s reduces
the error that lies in a column space of the largest 18, 24, and 30 singular values.
The simulation results in the tables are obtained by improving the load reduction
results by tuning the parameter ρ, and by keeping the parameter ρ is constant for the
value tuned for I T = 0.12. All the RC cases produces sizeable reductions in standard
deviation of the loads in all turbulent cases and the two mean wind speeds. Compared
with IPC, RC performs better in the low turbulent cases; like, I T = 0 and I T = 0.04.
From I T = 0.10 and larger, IPC will outperform RC. This was expected, because the
relative performance of RC depends heavily on the ratio between amount of periodic
and stochastic disturbances. The reason is that RC focus mainly on the rejection
periodic disturbances, where IPC also acts on turbulence by feedback. Further,
more turbulence will generate more variations in rotational speed that can affect the
performance of RC. That RC mainly focuses on the periodic components, is visible in
the values for the pitch rate. For the increasing turbulence intensity, these pitch-rate
values hardly increase when compared relative with the baseline results. We have 15%
126 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
to 30% less pitch rate action than for IPC at high turbulent cases. In Figure 5.7, the
square-root of the PSD of the pitch angles of the first blade are given for the baseline
(light grey), RC-30s (blue), and IPC-2P-33deg (black) controlled system. The PSD plots
shows that compensation of the pitch actuator for RC is mainly focused on the 1P,
2P, 3P, 4P, and 5P frequencies, where IPC compensates the 1P, 2P and intermediate
frequencies instead.
magnitude (rad)
10−1
10−2
10−3
10−4
100
10−1
frequency (Hz)
Figure 5.7: Square-root PSD of the pitch angle θ1 . Black line is with IPC-2P-33deg,
blue line is with RC-30s with ρ = 3 · 105 , and light grey line is with baseline control.
For I T = 0.12, v = 16 m/s.
Several observations can be made regarding the different RC cases. The introduction of larger column space, such that RC compensates more harmonic components,
does not give much performance improvements in terms of the standard deviation.
The dominance of the 1P and 2P peak makes it difficult to investigate the load
reduction capabilities of RC at higher harmonic frequencies. In Figure 5.8, the squareroot PSD of the first blade-root bending moments are given for the baseline (light
grey), RC-18s (light blue), RC-24s (blue), and RC-30s (black) controlled system. The
reduction of the 4P and 5P peaks can be viewed in the spectrum, when the number
singular values are increased: the RC-24s case rejects the 4P frequency, and the
RC-30s case also the 5P frequency. In Figure 5.9, the variation in pitch angle θ1
during one period is given. It shows that the addition of the 4P and 5P frequencies
are translated to the time domain as a better fit to compensate the tower shadow
effect. In Figure 5.10, the square-root PSD of the first blade-root bending moments
are given when the turbulence intensity is increased from I T = 0.04 to I T = 0.12.
The contribution of the 4P and 5P frequencies are now almost neglectable compared
with the surrounding noise level, therefore the additional harmonics are probably not
worthwhile to compensate for.
127
5.4 Simulation results and analysis
107
1P
magnitude (Nm)
2P
3P
106
4P
5P
baseline
105
10−1
100
frequency (Hz)
Figure 5.8: Square-root PSD of the blade-root bending moment M y ,1 . Black line is
with RC-30s, blue line is with RC-24s, light blue line is with RC-18s, and light grey line
is with baseline control. For I T = 0.04, v = 16 m/s.
0.21
baseline
IPC-1P-25d
IPC-2P-33d
RC-18s
RC-30s
0.2
pitch angle (rad)
0.19
0.18
0.17
0.16
0.15
0
π/3
2π/3
π
4π/3
5π/3
2π
azimuth angle (rad)
Figure 5.9: The variation in pitch angle θ1 during one period. Averaged over 600 s. For
I T = 0, v = 16 m/s.
128 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
107
1P
magnitude (Nm)
2P
3P
4P
106
5P
105
10−1
100
frequency (Hz)
Figure 5.10: Square-root PSD of the blade-root bending moment M y ,1 . Black line is
with RC-30s, blue line is with RC-24s, light blue line is with RC-18s, and light grey line
is with baseline control. For I T = 0.12, v = 16 m/s.
Other observations can be made regarding the tuning of the parameter ρ. From
the Tables 5.6 and 5.7, it can be determined that increasing the weight ρ improves
the smoothness of the generated pitch signal at the cost that it is less adaptive to
any changes in the periodic disturbance and takes longer to converge. Selecting a
low-valued weight ρ can be beneficial for the load reduction in the low turbulent
cases; like, I T = 0 and I T = 0.04. It is not surprising, that for larger turbulence
intensities the weight ρ should be increased to force the RC to design the pitch angle
vector over more periods. When selecting a constant ρ (tuned for I T = 0.12), the
results in Table 5.6 show only small reduction in load reduction performance. In
Figure 5.11, the square-root PSD of the first blade-root bending moments are given
baseline (light grey), ρ = 106 (light blue), ρ = 105 (blue), and ρ = 104 (black) weights.
It is visible in the figure, that small weights cause more unwanted amplifications at
the intermediate frequencies, which explains the decreased load reduction at large
turbulence intensities. The selection of the parameter ρ should not only be done on
the basis of constant load cases. The RC can adapt the pitch action to the changing
periodic loads, and probably a lower weight ρ is better to faster adapt to any changes.
For this purpose, a sinusoidal wind direction (yawed error) transient with amplitude
of 10deg is added to the simulations. In Figure 5.12, the percentage reduction in
the blade-root bending moment is given for the cases of no transient (no wave), a
slow transient (half wave), and a fast transient (full wave). It is shown that a better
load reduction can be obtained by selecting a smaller weight to follow the adaptive
direction of the wind.
129
5.4 Simulation results and analysis
107
1P
magnitude (Nm)
2P
3P
4P
106
5P
105
10−1
100
frequency (Hz)
Figure 5.11: Square-root of the PSD of the blade-root bending moment M y ,1 . Black
line is with ρ = 104 , blue line is with ρ = 105 , light blue line is with ρ = 106 , and light
grey line with baseline control. For I T = 0.04, v = 16 m/s.
−16
no wave
percentage load reduction (%)
−17
half wave
full wave
−18
−19
−20
−21
−22
−23
−24
104
105
weight ρ
106
Figure 5.12: The weight ρ to the percentage reduction in the blade-root bending
moment mean std M y ,(1,2,3) . For RC-18s, I T = 0.12, v = 16 m/s.
130 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
5.5
Conclusion
This chapter contribute to the development of wind turbines with individually pitch
controlled blades or/and with the smart rotor control by proposing a novel lifted
repetitive controller that can learn to reject periodic wind disturbances. The main
advantage of using lifted repetitive control instead of typical individual pitch control
is that the Coleman transformation is not needed. Instead a lifting technique is
used, for which a periodic linear-time-varying model can be used with multiple
inputs and multiple outputs and with both repetitive and non-repetitive disturbance
components. Therefore, the lifted repetitive control method is more generic, for
example to extend to rotors with more actuators and sensors, or to other rotor designs;
like, two-bladed rotors. An important drawback of lifted repetitive control is also
addressed, which is that a small mismatch between the controller period and the
actual period of the disturbance signal can decrease the performance substantially.
The formulation of the lifted repetitive control problem can be made more robust
for small changes in period time by designing a sampler based on the angular
position instead of time, and by including a periodic component with a random
walk nature, so that the periodic disturbance can vary between trials. With these
additions, the lifted repetitive controller is still considered competitive for fixed-speed
and variable-speed wind turbines operating above-rated with a pitch controller, that
will keep the rotational speed of the wind turbine close to the given rated rotor speed.
Moreover, the chapter provides some guidelines on how to pick the free parameters
in the algorithm. The load reduction capabilities of the lifted repetitive controller are
investigated on the UPWIND 5MW wind turbine in the aerolastic modelling software
GH BLADED. Therefore, the relative performance in terms of load reduction and
control action is compared with individual pitch control with Coleman transformation
and with different turbulence intensities and consequently different variations in the
rotational rotor speed. Simulation results indicate that for relatively slow changing
periodic wind disturbances this repetitive control method can significantly reduce the
vibrations in the wind turbine structure with significant less high-frequent control
action.
v0 =
IT =
16
0
0.04
0.06
0.08
20
0.10
0.12
0.14
0.16
0
0.04
0.06
0.08
0.10
0.12
0.14
0.16
(Nm)
1.226
51.8
52.7
55.5
57.1
1
56.4
56.5
56.7
30
55.2
55.5
55.7
1.288
49.3
50.2
51.8
54.1
3
49.5
49.6
49.7
30
48.3
48.5
48.7
1.379
41.5
42.9
42.3
45.6
7
38.7
38.7
38.8
30
38.5
38.6
38.7
1.460
35.0
36.0
32.8
38.3
10
32.0
32.1
32.2
30
32.0
32.1
32.2
1.554
28.8
30.4
25.6
32.0
30
27.3
27.4
27.5
30
27.3
27.4
27.5
1.648
24.2
25.5
20.4
26.8
70
22.4
22.4
22.5
30
22.3
22.4
22.5
1.786
20.2
21.3
16.2
22.1
100
18.2
18.2
18.3
30
18.1
18.1
18.2
(rad/s)
1.856
6.756
6.683
7.337
7.206
1
7.438
7.733
7.837
30
6.284
6.418
6.550
2.093
6.983
6.823
7.702
7.419
3
7.306
7.512
7.688
30
6.315
6.450
6.580
2.294
7.213
6.895
8.007
7.495
7
7.055
7.230
7.421
30
6.394
6.529
6.659
2.776
7.310
7.159
8.590
7.817
10
7.047
7.240
7.395
30
6.533
6.669
6.787
3.094
7.802
7.346
9.064
8.043
30
6.748
6.887
7.004
30
6.748
6.887
7.004
3.472
7.896
7.449
9.609
8.230
70
6.399
6.499
6.554
30
6.967
7.113
7.155
3.938
8.135
7.768
9.920
8.569
100
6.401
6.487
6.531
30
7.255
7.407
7.423
5.5 Conclusion
Table 5.6: IPC and RC (with variable and constant µ) results in standard deviation or in percentage of the baseline control results.
−6
baseline
IPC-1P [%]
IPC-1P-25deg [%]
IPC-2P [%]
IPC-2P-33deg [%]
µ = ×104
RC-18s [%]
RC-24s [%]
RC-30s [%]
µ = ×104
RC-18s [%]
RC-24s [%]
RC-30s [%]
0.244
5.162
5.161
5.526
5.526
1
5.660
5.765
5.916
30
5.106
5.215
5.336
1.057
52.9
53.7
56.2
58.0
1
54.9
55.0
55.2
30
53.7
54.0
54.2
1.208
5.627
5.586
6.102
5.989
1
6.150
6.317
6.487
30
5.154
5.269
5.382
1.065
49.0
50.8
50.8
53.6
3
48.5
48.7
49.9
30
46.7
46.9
47.1
1.436
5.830
5.759
6.430
6.213
3
6.005
6.180
6.343
30
5.173
5.279
5.395
1.181
36.0
37.1
36.1
39.6
7
31.6
31.5
31.7
30
31.4
31.4
31.5
1.544
6.095
5.829
6.898
6.414
7
5.862
6.036
6.145
30
5.243
5.355
5.457
1.278
29.6
30.8
27.2
32.8
10
26.4
26.4
26.5
30
26.4
26.4
26.5
1.827
6.258
6.031
7.463
6.632
10
5.788
5.939
6.058
30
5.305
5.414
5.516
1.437
21.6
22.9
18.5
24.3
30
19.6
19.6
19.7
30
19.6
19.6
19.7
mean std M y ,(1,2,3) × 10
1.597
1.780
1.099
17.8
14.1
73.2
18.7
14.8
73.5
15.0
11.6
85.6
19.6
15.7
85.6
70
100
1
14.7
10.6
90.9
14.7
10.3
91.6
14.8
10.3
93.9
30
30
30
14.7
10.4
85.3
14.7
10.3
86.2
14.7
10.3
86.8
2.136
6.684
6.480
8.088
7.150
30
5.419
5.526
5.616
30
5.419
5.526
5.616
mean std θ̇y ,(1,2,3) × 102
2.387
2.698
0.395
7.348
7.768
6.135
7.044
7.550
6.135
8.793
9.166
6.593
7.594
8.099
6.593
70
100
1
5.048
4.878
6.783
5.107
4.946
6.917
5.158
4.956
7.066
30
30
30
5.528
5.591
6.148
5.651
5.706
6.283
5.696
5.739
6.419
131
baseline
IPC-1P
IPC-1P-25deg
IPC-2P
IPC-2P-33deg
µ = ×104
RC-18s
RC-24s
RC-30s
µ = ×104
RC-18s
RC-24s
RC-30s
0.900
74.4
74.7
86.4
86.4
1
90.7
91.5
93.7
30
85.2
85.9
86.5
v0 =
IT =
16
0
0.04
0.06
0.08
20
0.10
0.12
0.14
0.16
0
0.04
0.06
0.08
0.10
0.12
0.14
0.16
std M tilt × 10
(Nm)
1.779
1.552
1.676
30.0
71.5
58.1
32.4
71.8
59.4
22.1
85.1
63.9
34.7
85.1
66.5
100
1
1
21.7
88.3
64.5
21.6
89.4
65.0
20.5
91.5
65.3
30
30
30
20.0
86.6
63.8
19.9
89.6
64.5
19.7
89.5
64.8
1.794
53.6
54.8
57.0
60.3
3
53.9
54.2
54.4
30
53.3
53.7
53.9
1.864
48.2
50.6
49.3
54.7
7
44.7
44.8
45.1
30
44.8
45.1
45.2
1.934
41.7
43.4
38.8
47.1
10
37.6
37.6
38.0
30
38.0
38.2
38.3
2.006
38.3
40.7
33.2
43.5
30
35.2
35.3
35.4
30
35.2
35.3
35.4
2.096
32.1
34.9
26.5
37.5
70
30.6
30.6
30.7
30
30.1
30.1
30.2
2.198
28.5
30.9
22.3
32.9
100
25.7
25.8
25.9
30
25.4
25.5
25.5
M yaw × 10−6 (Nm)
1.739
1.550
1.658
32.0
70.8
57.1
34.0
71.1
58.6
24.9
85.0
62.9
37.1
85.1
65.4
100
1
1
23.2
88.3
63.5
23.2
90.6
63.8
23.3
91.9
64.0
30
30
30
23.1
85.8
63.1
23.1
88.9
63.6
23.2
89.7
63.8
1.775
51.7
53.3
55.1
58.5
3
52.4
52.6
52.8
30
52.0
52.2
52.3
1.866
47.5
49.7
48.5
54.0
7
44.5
44.8
44.9
30
44.4
44.5
44.6
1.937
42.7
44.5
39.0
48.0
10
39.4
39.5
39.5
30
39.1
39.3
39.3
2.001
35.3
38.9
30.0
41.5
30
34.9
35.0
35.1
30
34.9
35.0
35.1
2.072
33.0
35.6
25.9
37.3
70
30.6
30.7
30.7
30
30.6
30.7
30.7
2.188
30.0
31.9
22.4
33.0
100
26.5
26.6
26.6
30
26.5
26.5
26.5
−6
baseline
IPC-1P [%]
IPC-1P-25d [%]
IPC-2P [%]
IPC-2P-33d [%]
µ = ×104
RC-18s [%]
RC-24s [%]
RC-30s [%]
µ = ×104
RC-18s [%]
RC-24s [%]
RC-30s [%]
baseline
IPC-1P [%]
IPC-1P-25d [%]
IPC-2P [%]
IPC-2P-33d [%]
µ = ×104
RC-18s [%]
RC-24s [%]
RC-30s [%]
µ = ×104
RC-18s [%]
RC-24s [%]
RC-30s [%]
1.257
71.6
71.9
84.4
84.4
1
89.1
90.1
92.0
30
87.4
89.2
90.4
1.257
71.2
71.5
84.3
84.4
1
88.9
90.8
92.0
30
87.7
89.6
90.6
1.344
60.3
61.7
65.7
69.0
1
66.0
66.3
66.7
30
65.3
65.8
66.1
1.344
60.2
61.5
65.8
68.9
1
65.9
66.4
66.6
30
65.3
65.8
66.1
1.432
55.3
56.8
57.9
62.5
3
55.2
55.5
55.8
30
54.1
54.5
54.7
1.493
55.2
56.5
58.1
62.5
3
55.5
55.7
55.8
30
54.1
54.4
54.5
1.481
49.4
51.9
49.3
56.7
7
42.6
42.9
43.0
30
41.2
41.4
41.5
1.484
49.5
52.2
49.8
57.0
7
43.5
43.5
43.6
30
43.3
43.4
43.5
1.555
44.1
46.4
39.0
50.7
10
38.6
38.7
38.8
30
38.5
38.7
38.7
1.551
43.4
45.9
38.7
50.4
10
38.6
38.6
38.7
30
38.2
38.3
38.4
1.597
38.6
41.6
30.1
44.6
30
34.9
35.0
34.7
30
34.9
35.0
34.7
1.687
35.6
38.9
27.5
41.2
70
27.8
27.9
27.9
30
27.8
27.9
27.8
1.606
39.6
42.9
32.5
47.2
30
35.4
35.6
35.7
30
35.4
35.6
35.7
std
1.686
36.5
39.1
29.8
42.8
70
30.0
30.1
30.4
30
30.0
30.1
30.1
132 Chapter 5: Wind Turbine Load Reduction by Rejecting the Periodic Load Disturbances
Table 5.7: IPC and RC (with variable and constant µ) results in standard deviation or in percentage of the baseline control results.
CHAPTER
6
Conclusions and Recommendations
In this thesis two research goals have been addressed: one that deals with the
development of data-driven modelling methods using system identification
and one that deals with the development of a repetitive controller to reject
periodic disturbances. This final chapter has two sections. In the first section,
the main conclusions that can be drawn from the research presented in this
thesis are summarized. In the second section, a number of suggestions and
recommendations for further research will be made.
6.1
T
Conclusions
achieve the two research goals, identification and control algorithms are developed for the rejection of periodic disturbances in modern wind turbines.
These developments for the load reduction of modern wind turbines can achieve
an increased lifetime of components and make the scaling to larger rotor diameters
possible, and therefore improve the cost effectiveness of modern wind turbines. Also
the success of future rotor designs will heavily depend on these new developments in
identification and control algorithms. The following main conclusions can be drawn
from the research in the main chapters.
O
In Chapter 2, the optimized Predictor-Based Subspace IDentification (PBSIDopt )
algorithm is proposed for the closed-loop wind turbine model estimation. The PBSID
method is very attractive for the wind power community, as it is able to identify
LTI systems with multiple inputs and multiple outputs, and does not require any
controller related information. Further, a novel VARMAX-based PBSIDopt method
is developed that relaxes the requirement that the past window has to be large for
asymptotically consistent estimates. For this purpose, a finite description of the inputoutput relation is formulated at the cost of a non-linear problem, but this is solved
efficiently using the extended least squares recursion. From simulations, it can be
concluded that the novel VARMAX-based method performs very well with a small past
window under perfect conditions, however the novel method is less robust than the
133
134
Conclusions and Recommendations
original method to small violations that can occur in the noise assumptions or the
model order. Despite this, an accurate identification of the TURBU model is obtained
for both methods, especially around the natural frequencies of the wind turbine.
In Chapter 3, a novel Recursive Predictor-Based Subspace IDentification (RPBSIDpm )
algorithm is developed to identify LTI systems with multiple inputs and multiple
outputs. The algorithm is suited for real-time implementation and will provide consistent estimates from data gathered in open loop or closed loop. The computational
complexity is reduced by exploiting the structure in the data equations and by using
array algorithms to solve the main linear problem, such that the proposed method can
be implemented in real time. The ability to do so is demonstrated by the detection of
flutter on an experimental “smart” rotor test section. In the experiments it is shown
that a certain weight with transformation is required to minimize variations of the
particular state used. Moreover, the chapter suggests to use Tikhonov regularization
in order to overcome possible numerical problems in practice (and therefore give
better results in some sense), such as the singularity of the regression problem. With
these updates, the algorithm is successfully able to track after some delay the natural
frequencies and modal damping values.
In Chapter 4, a novel lifted Repetitive Control (RC) algorithm for periodic wind
disturbance rejection of linear systems with multiple inputs and multiple outputs
and with both repetitive and non-repetitive disturbance components. The design of
the repetitive controller is formulated as a lifted linear stochastic output-feedback
problem on which the mature techniques of discrete-time linear control may be
applied. The formulation of the lifted repetitive control problem can be made more
robust to small changes in period time by using multiple memory loops, but also
by including a periodic component with a random walk nature, so that the periodic
disturbance can vary between trials. Efficient algorithms are developed for controller
synthesis and for real-time implementation to reduce the computational complexity
and memory usage by exploiting the structure in the lifted state-space matrices. For
relatively slow changing periodic and turbulent wind disturbances created by a wind
generator it was shown that this repetitive control method could reduce the variance
of the load signals of the “smart” rotor test section up to 42%. The cost of additional
wear and tear of the “smart” actuators are kept small, because a smooth control
action is generated as the controller mainly focuses on the reduction of periodic
disturbances.
In Chapter 5, the novel lifted Repetitive Control (RC) algorithm is extended for
the use on modern fixed-speed wind turbines and modern variable-speed wind
turbines operating above-rated equipped with individually pitch controlled blades.
The repetitive control method is very generic, and can be extended to rotors with
more actuators and sensors, or to other rotor designs; like, two-bladed rotors. An
important drawback of repetitive control is also addressed, which is that a small
mismatch between the controller period and the actual period of the disturbance
signal can decrease the performance substantially. The formulation of the lifted
repetitive control problem can be made more robust for small changes in period
time by designing a sampler based on the angular position instead of time, and by
including a periodic component with a random walk nature, such that the periodic
disturbance is allowed to vary between trials. With these additions, the repetitive
6.2 Recommendations
135
controller is competitive for modern wind turbines operating above-rated with a
pitch controller that will keep the rotational speed of the wind turbine close to the
given rated rotor speed. The relative performance in terms of load reduction and
control action is compared with typical individual pitch control and with different
turbulence intensities and consequently different variations in the rotational rotor
speed. The simulation results indicate that for relatively slow changing disturbances
this method can significantly contribute to the reduction of vibrations in the wind
turbine structure.
As an additional conclusion, it is noted that all the developed algorithms are very
general, therefore the developed algorithms can also be used outside the field of wind
energy. Further, the developed algorithms have been implemented by the author in
software and are made available in the form of toolboxes. Also examples have been
included in the toolboxes to make the tools easily applicable. For these reasons,
the toolboxes have been extensively used in the application of active magnetic bearings (Balini et al., 2010, 2011), adaptive optics (Doelman et al., 2009a,b; Song, 2011),
biomechanics (Kind et al., 2010; van Eesbeek et al., 2011), and fault detection (Wei
et al., 2009).
6.2
Recommendations
Even though the results presented in this thesis have clearly demonstrated the opportunities and advantages of the identification and repetitive control algorithms for
modern wind turbines, the research within this area is far from finished. In view of
experience and insights gained over the course of time, the following suggestions and
recommendations are made for future research:
• The proposed VARX-based predictor-based subspace identification method requires harder excitation conditions than required for the identification of innovation state-space systems, see Aling (1990). The selection of the past window
strongly affects the excitation requirement, and therefore indirectly the quality
of the estimated model. It has been suggested many times to use ℓ1 -regularized
methods for solving the main linear problem. The idea is to select a very large
past window and let the optimizer introduce zeros into the solution, which
are most likely the smallest Markov parameters. As the optimizer requires
the selection of the ℓ1 -regularization parameter which in turn replaces the
selection of the past window, basically nothing is gained. However, in the future
these methods can become useful when the ℓ1 -regularization parameter can
automatically be determined, for example with a more robust version of the
method in presented Rojas and Hjarmarsson (2011).
• From the identification experiments with the VARMAX-based predictor-based
subspace identification method it was concluded that identification methods
can be very sensitive to small violations that can occur in the noise assumptions
or the model order. These robustness issues in the form of model reduction or
linearisation effects of these identification methods on the system under study
are hardly analysed or researched in literature. A number of suggestions can
136
Conclusions and Recommendations
be made to improve the robustness of the VARMAX-based method. First, it is
suggested to remove the dependence on the innovation in the following steps
by using Lemma 2.3. This transformation can be used on the VARX Markov
parameters from the VARMAX Markov parameters, than the state sequence can
be obtained through a multiplication with the input and output sequence, and
not with the innovation sequence any more. Second, it has been suggested
many times to use more robust methods to estimate the VARMAX model;
like instrumental variable methods. However, the selection of the correct
instrumental variables in closed-loop operation is not very trivial and requires
multiple iterations (and therefore even more parameters need to be tuned).
• In Chapter 5, the linear period time-varying model was obtained by linearising
the first-principle non-linear model of the wind turbine. It was not possible to
obtain an accurate model using (quasi) periodic LTV/LPV system identification
methods, for example with the method in van Wingerden et al. (2009). The
problem is that current methods require too much memory, due to the curse
of dimensionality in the computational complexity of the algorithm. Further
research is required to exploit the special structures in the data matrix. Specific
tensor computations seems to have very similar structures, and if the structure
fit, these computations could reduce the computational complexity and memory storage. Another solution is the use of special computational hardware; like
distributed or parallel computing.
• Apart from the system identification itself, the data-driven modelling process
requires the design of an excitation sequence and the verification/validation of
the identified models for controller design. The experience (gained with the
experiments) is that these tasks are very challenging and time-consuming if the
system is MIMO and operates in closed-loop. Apart from the VAF, the prediction error, and the spectral analysis, it currently lacks of available verification
and validation methods especially in contrast of the MIMO controller to be
designed. A first step that can be helpful is the determination of the first-order
variance bounds of the state-space model parameters for the proposed method.
Also the development of methods for the analysis of the excitation sequence
during closed-loop identification would be very useful, for example a closedloop version of the method in Hallouzi (2008).
• With the predictor-based subspace identification of MIMO systems, it regularly
occurs that identified poles and zeros from one transfer channel are parasitically
coupled with the other transfer channels. A first step to a solution could be
to generalize the proposed predictor-based subspace identification methods to
estimate more generalized state-space models (the so-called Box-Jenkins and
ARARMAX structures in (Reynders and De Roeck, 2009)), such that any parasitic
couplings between the stochastic and deterministic dynamics are removed. For
e Ȳp , and with a
this purpose, the Lemma 2.1 can be used to obtain ΓLeŪp and ΓK
Canonical Correlation Analysis (CCA) between these sequences the state transformation (or weight matrix) can be obtained that separates the state sequence
in a stochastic only, a deterministic only, and a combined state sequence.
6.2 Recommendations
137
• Further research to reliable methods for selecting a weight matrix is recommended, especially in the recursive case. The additional freedom to choose a predefined weight matrix, can give opportunities to improve model estimate under
certain conditions. The optimization of a pre-defined weight matrix such that
the low-rank approximation error is minimized by considering a weighted lowrank approximation problem is very non-linear and very hard (NP) to compute.
However, other measures such as the nuclear norm can be used to determine
the weight matrix and improve the model estimate, see Gebraad et al. (2011).
• Another suggestion is the development of identification algorithms that directly
identify lifted state-space models with SSS-structured system matrices. These
models can then be directly and efficiently used for lifted repetitive control or
distributed control; for example with the optimization of floating wind farms. A
first step in the development has been made by Rice and Verhaegen (2011), but
the non-injective (at both global and generator level) parametrization of the SSS
system matrices in this method is known to give convergence issues. Further,
the linear computational complexity (in N ) of the algorithm is not proven, but
only observed in simulations.
• An important restriction of the proposed repetitive controller is that it can only
be applied for above-rated operation. The following suggestion is a wild idea
to cope with the large changes in the period time for lifted repetitive and also
iterative learning controllers. It is known that for a lifted state-space system,
the individual system matrices with SSS matrix structure can be represented
by a sequence of discrete-time mixed-causal LTV state-space systems, or 1D
heterogeneous distributed system (Rice, 2010). It is also observed that for a
synthesized lifted state-space controller, the individual system matrices have a
SSS matrix structure. According to a measured azimuth angle or measured rational speed the generator subsystems can be efficiently be splitted and merged
with neighbouring subsystems, concatenated and separated with additional
subsystems, or replaced with other subsystems. Off course the difficulty is to
show that these applied changes in the sequence of generator systems improve
the performance, robustness, and stability of the global closed-loop system.
• In the current research, the fatigue in mechanical parts have been be expressed
in terms of signal variances of the measured accelerations or moments. Although it is expected that signal variances give a reasonable initial impression of
the load reduction, the accurate description of the fatigue damage can contain
some important differences in the results. The computation of the damage
equivalent loads itself is straightforward. It is not only an implementation of
rainflow counting and applying Miner’s rule. The problem is that the fatigue
damage results are very sensitive to the selected parameters during its calculation; namely, the data length, the number of bins, the width of the bins, the
material characteristics in the S/N curve (Veldkamp, 2006). The tuning of these
parameters such that the estimated damage equivalent loads represents reality
is very challenging. To get consistent and accurate fatigue damage results,
and for honest comparison between “smart” rotor controllers and concept, it
is suggested that a predefined and verified benchmark for the UPWIND (or
similar) 5MW turbine is designed.
138
Conclusions and Recommendations
• The computations with (multiple level) SSS matrices can also be very useful
for solving and simulating discretized PDE’s, for example to speed up the
simulations of distributed wind turbines in a wind farm. Note, that this
purpose was even the intended use of the first developers. With finite-difference
discretization the matrices have normally a lot of zeros, and sparse matrix
computations are in this case much faster. However, with the use of spectral
methods or Green functions for the discretization of the PDE’s the matrices are
normally much smaller and dense, in this case the use SSS matrices can be
competitive to sparse matrices (Chandrasekaran et al., 2003). Also, the current
MATLAB implementation has a lot of overhead in the form of the calculation of
the sizes of the generators. This overhead can considerably be reduced if the
functions are implemented in C (or better in object-oriented C++, see the SSS
matrix-vector implementation in the toolbox).
APPENDIX
A
On the Consistency of VARX Model
Estimation in Closed Loop
In Chapter 2 and 3, an implicit relation between the VARX model and the
Markov parameters of the state-space model in predictor form was proposed.
Following the work of Chiuso and Picci (2005), some additional details with
respect to the consistency of the VARX model estimation and its relation to
the closed-loop conditions are presented in this appendix.
A.1
Consistency when the true system is the VARX model
C
that the dynamics of the system SV to be modelled can be represented
by the following Vector Auto-Regressive with eXogenous inputs (VARX) model:
(
n
n
X
X
(yk −i )
(u
)
Ξ̃0
y k −i + e k ,
(A.1)
Ξ̃0 k −i u k −i +
SV y k =
ONSIDER
i =1
i =0
where e k is a zero-mean white noise sequence, n is the order, and Ξ̃0 ∈ Rℓ×n (r +ℓ)+r is
the set of true VARX parameters. The following consistency result is well known.
Proposition A.1. Let SV with n ≤ p be the data generating system with an infinite data
sequence (N → ∞) to Assumptions 2.1, then the estimates of the VARX model parameters
in (2.17) are consistent: Ξ̃ → Ξ̃0 .
Proof. See Ljung (1999); Van den Hof (2007).
Thus an unique and consistent VARX model is obtained despite the presence
of feedback. It is not required that the excitation sequence rk is measured, only
the presence of a signal rk is sufficient. In certain cases, the persistently excitation
condition of the sequence rk can be weakened if the applied feedback controller has
a significantly high order (or is non-linear).
139
140
A.2
The Consistency of VARX Model Estimation under Closed Loop
Consistency when the true system is the innovation
state-space model
That the true system is of the VARX type is not common in practical situations, and
is also not very useful for controller synthesis. Proposition 2.2 suggest that Assumption 2.2 is sufficient to relate the VARX parameters with the Markov parameters of the
state-space model in predictor form SP in (2.3). To simplify and clarify the relation,
we first introduce the controller description, and some assumptions are made.
A.2.1
Controller description and assumptions
It is considered that the controller has LTI dynamics and can be represented by the
following state-space model:
¨
s k +1 = A c s k + B c r1,k − y k ,
(A.2)
C
u k = C c s k + D c r1,k − y k + r2,k ,
where s k ∈ Rc , r1,k ∈ Rℓ , and r2,k ∈ Rr , are the controller state, reference, and
excitation vectors. For the closed-loop identification problem, the reference and
excitation sequences are unknown and are considered as noise sequences. The statespace matrices A c ∈ Rc ×c , B c ∈ Rc ×ℓ , C c ∈ Rr ×n , and D c ∈ Rr ×ℓ , are also called the
controller state, input, output, and feedthrough matrix, respectively. To fulfil the
Assumptions 2.1, the noise sequences w k and v k are considered to be uncorrelated
with the reference and excitation sequences r1,k and r2,k , and the direct feedthrough
terms are considered to obey DD c = 0 and D c D = 0.
The reference and excitation sequences r1,k and r2,k are considered to be zeromean white Gaussian noise, such that the controller can be represented in innovation
form CI and in the predictor form CP . These controller forms are defined in a similar
way as for the system with the predicted controller state sequence sˆk ∈ Rs , the
innovation sequence rk ∈ Rr , the Kalman gain K c ∈ Rs ×ℓ , the matrix B̃ c = B c − K c C c ,
and matrix à c = A c − K c C c asymptotically stable. By joining the innovation system
in (2.2) and the innovation controller together by feedback, the system J for the joint
process can be obtained in the joint innovation form as:

 –q̂k +1
™ = Aq̂k + Kξk ,
yk
(A.3)
JI

= Cq̂k + ξk ,
uk
where the structure of the state and system matrices are denoted by:
™
– ™
™
–
–
x̂ k
C
DC c
A + B Dc C
BC c
.
q̂k =
, C=
, A=
Dc C
Cc
Bc C
A c + B c DC c
sˆk
and the structure of the Kalman gain matrix and innovation is denoted by:
–
™– ™
–
™
I
D ek
K
B̃
ξk =
, K=
.
rk
Dc I
B̃ c K c
A.2 Consistency when the true system is the innovation state-space model
A.2.2
141
The relation between the VARX and Markov parameters
The predictor form (output only) of the innovation model JI in (A.3) is denoted as:
¨
q̂k +1|k ,p +1 = Ãq̂k |k −1,p + B̃u k + K̃y k ,
JP
(A.4)
ŷ k |k −1,p = C̃q̂k |k −1,p + Du k + e k ,
where the structure of the system matrices are denoted by:
–
™
– ™
– ™
”
à 0
B̃
K
à =
, B̃ =
, K̃ =
, C̃ = C
0 Ã c
Kc
B̃ c
—
0 .
The key observation is that the controller states sˆk of the predictor JP in (A.4) are
not observable, and that after calculating the minimal realization, a predictor form
similar to the system JP = SP in (2.3) remains. As the Kalman predictor SP is
the optimal (causal) predictor that minimizes the prediction error, its infinite power
series description can in the asymptotic case (N → ∞) directly be related to the
estimated VARX predictor with p → ∞. The lack of observability depends solely
on the Assumptions 2.2, therefore the additional assumptions in Subsection A.2.1,
namely that the controller is LTI and the reference and excitation sequence is white
Gaussian noise, are not necessary but made the observation much clearer. Note, that
the Assumption 2.2 is not yet required, because p → ∞ is assumed.
A.2.3
The asymptotic behaviour for a finite VARX predictor
In the case that the past window is finite, the Assumption 2.2 is required to obtain
a steady-state Kalman gain K. To see this, consider the availability of infinite data
(N → ∞), such that the deterministic part of the system JI in (A.3) is consistently
estimated, and the joint noise covariance matrix is defined as:
 


w k w kT
0
w k v kT
0
B
™ I B D c B D c
–
 
T
T 
r1,k r1,k
0
r2,k r1,k
Bc
Bc
Bc D   0
Q S
0

=
E




0
I
D   v k w kT
0
v k v kT
0 
ST R
0
T
T
0 Dc
Dc
I
0
r1,k r2,k
0
r2,k r2,k
T

I B Dc B Dc
B


Bc
Bc
Bc D 
0
(A.5)
×
 .
0
I
D 
0
0 Dc
Dc
I
From filtering theory, it is given that the minimum variance prediction of the output
can be computed using the steady-state Kalman filter. The steady-state Kalman gain
b j for j = 1, . . . , p as:
K can be computed by updating the matrix K
€
Š€
Š−1
b j = S + APbj −1 CT R + CPbj −1 CT
K
,
where the update equation for the state-error covariance matrix Pbj is given by the
forward Riccati difference equation:
€
Š
b j S + APbj −1 CT T + APbj −1 AT + Q,
Pbj = K
142
The Consistency of VARX Model Estimation under Closed Loop
”
—
and the initial state-error covariance matrix is expected to be Pb0 = E qk −p qkT−p , because of the finite past window and of the neglected initial state during the estimation.
It is well known that the rate of convergence of this Kalman filter gain iteration
depends on the converge rate of à j for a increasing past window (Anderson and Moor,
2005; Verhaegen and Verdult, 2007). As the matrix à is decoupled, and the Kalman
filter is known to be asymptotically stable, the limit for p → ∞ will bring the steadyb j → K, and will converge at the same rate as à j → 0. Note, that the
state solution K
matrix à = A − K C represents the transition matrix of the true Kalman filter and
not the estimated. With Assumption 2.2, i.e., the matrix à j = 0 for all j ≥ p , the
solution converges to the steady-state solution within at least p steps. This means
that the decoupling in Subsection A.2.2 still takes place, and the Proposition 2.2 with
this additional assumption is valid for a finite past window.
b as a perturbed version of the
When we consider the estimated Kalman gain K
steady-state Kalman gain K, then the state matrix of the predictor JP is denoted by:
–
™ –
™–
™
b − KbCb
A + B Dc C
BC c
Kb
B
C
DC c
b̃
b
A = A − KC =
− b
Bc C
A c + B c DC c
Dc C
Cc
B c − Kbc Cbc
Kbc
€
Š
€€
Š
€
ŠŠ ™
–
b Dc C
b − Kb D − D
b Cc
A − KbCŠ + B €− B
B−B
Š
ŠŠ
€
.
(A.6)
= €€
b
b
b
b
b
B c − B c − K c D c − D c C A c − K c C c + B c − B c DC c
Thus in reality with a finite past window, the estimated VARX predictor has a coupling
with the feedback controller as given by:
(
D,
if i = 0,
(u k −i )
b̃ i −1 K.
b̃
i −1
, Ξ̃(yk −i ) = C̃A
Ξ̃
=
b̃
b̃
C̃A B if i > 0
If the past window is chosen large enough, a contribution of the controller in the
estimated VARX predictor is still present but small. These remaining dynamics of
the controller are normally removed in second step by a sort of model reduction. By
selecting only the singular values to the number of states of the system, which are
probably the most dominant, the parasitic dynamics of the controller can be removed.
Remark A.1. In open loop there is off course no coupling with the controller states,
but with a finite past window the estimated Kalman gain Kbp is still expected to be a
perturbed version of the steady-state Kalman gain K . Unless the system is of the VARX
type or there is no process noise.
”
—
Remark A.2. In a different asymptotic case with infinite data (N → ∞) and E wk wTk =
0, that is no process noise, also an implicit relation can be formed between the Markov
parameters and the power series description of a deadbeat predictor, which is also
nilpotent if p ≥ n (Houtzager et al., 2009b).
Remark A.3. If the VARX predictor with p → ∞ is estimated in a moving horizon, for
b j is
example using a sliding window or by forgetting, the expected Kalman filter gain K
also non-stationary, and varies when the predictor is updated (and downdated) with
data. This is easily visible when the true covariance matrix in (A.5) is replaced with a
estimated time-varying covariance matrix based on a finite data sequence.
B
APPENDIX
Spectral Analysis for the Verification of
Models in Closed Loop
The spectral analysis method can estimate from inputs and outputs measurements the frequency response of the system to be modelled. However, it
should be performed with care, because the conventional frequency response
estimate will give a biased estimate under closed-loop. When we do not
posses any exact frequency response of the model, this frequency response
estimate can be used to verify that the identified model is a successful and
accurate approximation of the real system under investigation.
B.1 Transfer function description and assumptions
L
ET
the system to be modelled be LTI and denoted by the transfer functions:
y k = G (z ) u k + H (z ) e k ,
(B.1)
with
G (z ) = D + C (z I − A)−1 B ,
H (z ) = I + C (z I − A)−1 K ,
and e k ∈ Rℓ is a zero-mean white noise sequence with covariance R. Let the LTI
system to be modelled be operating in closed-loop and denoted by:
y k = S (z )G (z ) rk + S (z ) H (z ) e k ,
(B.2)
where the sensitivity and input-sensitivity functions are defined by:
S (z ) = (I ℓ + G (z )C (z ))−1 ,
U (z ) = (I r + C (z )G (z ))−1 .
The assumptions for the frequency response estimation are similar as given in Assumption 2.1.
143
144
Appendix B: Spectral Analysis for the Validation of Models under Closed Loop
B.2 Consistency with conventional spectral analysis
When only u k and y k are available from measurements, then the following closedloop descriptions of the input and output signals are obtained:
u k = U (z ) (rk − C (z ) H (z ) e k ) ,
y k = S (z ) (G (z ) rk + H (z ) e k ) .
Now, the spectral densities Φy u (ω) and Φu (ω) can be defined as:
€
Š
€
Š €
Š
€
ŠT €
ŠT €
ŠT
Φu (ω) = U e j ω Φr (ω) + C e j ω H e j ω RH e j ω C e j ω
U e jω ,
€
Š €
€
Š
€
ŠT €
Š
ŠT €
ŠT
Φy u (ω) = S e j ω G e j ω Φr (ω) − H e j ω RH e j ω C e j ω
U e jω .
The conventional SPectral Analysis (SPA) estimate is defined as:
€
Š
Gb e j ω = Φy u (ω)Φu (ω)−1 .
(B.3)
In order to see that (B.3) is not consistent, notice that:
€
Š
€
Š €
Š
€
Š
€
ŠT €
ŠT Gb e j ω = S e j ω G e j ω Φr (ω) − H 0 e j ω RH e j ω C e j ω
€
Š €
Š
€
ŠT €
ŠT −1 €
Š−1
U e jω
.
× Φr (ω) + C e j ω H e j ω RH e j ω C e j ω
It can be easily seen that the estimated frequency response coincides with the true
dynamics only in case of R = 0, thus noise-free measurements, or for C e j ω = 0. In
all of the other cases, the estimate of the transfer function becomes biased.
B.3 Spectral analysis for closed-loop estimation
An unbiased alternative is to use the cross-spectral density between the input/output
signals and the external excitation signal rk (Akaike, 1967). Hence, the SPA closed-loop
estimate is defined as:
€
Š
(B.4)
Gb e j ω = Φy r (ω) Φu r (ω)−1
In order to see that (B.4) is consistent, notice that:
€
Š−1
€
Š
€
Š €
Š
Gb e j ω = S e j ω G e j ω Φr (ω) Φr (ω)−1 U e j ω
Š−1
Š €
Š €
€
=G e jω U e jω U e jω
Š
€
=G e jω .
Note, that the quality depends heavily on how the additional excitation signal rk
excite the system to be modelled over the frequency range. Additional excitation on
natural frequencies is not always allowed due to load restrictions. To get a smoother
spectral analysis estimate, averaging in the frequency domain is normally applied,
see (Van den Hof, 2007). The spectrum is smoothed locally in the region of the
target frequencies, as a weighted average of the values to the right and left of a target
frequency.
APPENDIX
C
Sequentially Semi-Separable Matrix
Descriptions and Computations
Computations with Sequentially Semi-Separable (SSS) matrices enables you
to perform very fast calculations for systems of linear equations with a
sequentially semi-separable structure. Operations are available for most matrix computations; for example, addition, multiplication, division, and most
matrix decompositions. To get you started, this Appendix will explain the
current available matrix operations, where the sequentially semi-separable
structure is exploited. For more information, see the examples and the
descriptions of the functions in the SSS matrix toolbox (Houtzager, 2011c).
C.1
C
Sequentially semi-separable matrix descriptions
ONSIDER
the following discrete-time mixed-causal LTV state-space system:

 x k +1 = R k x k +Q k u k , with x 1 = 0,
z
= Wk z k + Vk u k , with z N = 0,
S
 k −1
y k = Pk x k + Uk z k + D k u k ,
(C.1)
where k = 1 . . . N , and x k ∈ Cn k , and z k ∈ Cm k , u k ∈ Crk , y k ∈ Cℓk , are the time-varying
causal state, anti-causal state, input, and output vectors. The state-space matrices
R k ∈ Cn k +1 ×n k , Wk ∈ Cm k −1 ×m k , Q k ∈ Cn k +1 ×rk , Vk ∈ Cm k −1 ×rk , Pk ∈ Cℓk ×n k , Uk ∈ Cℓk ×m k ,
and D k ∈ Cℓk ×rk are also called the time-varying causal state, anti-causal state, causal
input, anti-causal input, causal output, and anti-causal output matrices, respectively.
Note, that the input size, output size, and system orders can be time-varying and can
even be zero valued.
With the window N , the following stacked vector can be defined as:
”
ȳ N = y 1T
y 2T
145
···
y NT
—T
,
(C.2)
146
Appendix C: SSS Matrix Computations for RC
PN
with ȳ N ∈ Cℓ̄×1 where n̄ N = k =1 n k . The stacked vector ū N ∈ Cr̄N ×1 , r̄N , n̄ N , and m̄ N
are defined in a similar way. Using the definition of the stacked vectors, we can “lift”
the system in (C.1) to the following lifted description:
ȳ N = S̄ ū N ,
(C.3)
where the SSS matrix S̄ ∈ Rℓ̄N ×r̄N is defined by:





S̄ = 



D1
P2Q 1
U1 V2
D2
P3 R 2Q 1
P3Q 2
P4 R 3 R 2Q 1
PN R N −1 . . . R 2Q 1
P4 R 3Q 2
PN R N −1 . . . R 3Q 3
U1 W2 V3
U2 V3
..
.
..
.
...
U1 W2 W3 V4
U2 W3 V4
..
.
D N −1
PN Q N −1

U1 W2 . . . WN −1 VN

U2 W3 . . . WN −1 VN 

..

.
.


UN −1 VN

DN
In the remainder of this appendix, an SSS matrix is denoted by the bar symbol¯. The
global SSS matrix is not stored in the dense matrix form, but by its local generators
D k , Uk , Vk , Wk , Pk , Q k , and R k instead, or better known by state-space matrices. The
SSS generator realization is not unique, a different SSS generator realization can be
obtained by the similarity transformations Tk ∈ Cn k +1 ×n k and S k ∈ Cm k −1 ×m k : Uk S k ,
S k −1 Vk , S k −1 Wk S k , Pk Tk , Tk +1Q k , and Tk +1 R k Tk .
The individual blocks of the SSS matrix (for i =
reconstructed to a dense matrix as:

if
 Di ,
Ui Wi +1 . . . Wj −1 Vj , if
S̄ i j =

Pi R i −1 . . . R j +1Q j ,
if
1 . . . N and j = 1 . . . N ) can be
i =j
j >i .
j <i
(C.4)
It is not recommended to convert the SSS matrix to the dense matrix form completely,
as it can require a lot of memory usage for large N . The number of parameters to
describe the SSS matrix is given by:
N
X
ℓk r k +
PSSS =
k =1
N
−1
X
k =1
N
X
ℓk m k +
k =2
m k −1 rk +
N
−1
X
k =2
N
X
m k −1 m k +
ℓk n k +
k =2
N
−1
X
n k +1 rk +
k =1
N
−1
X
n k +1 n k ,
k =2
and the number of parameters for the dense matrix form is described by:
N
X
N
X
Pdense =
k =1
ℓk ×
rk .
k =1
Note, when the sizes of the generators are considered constant, than for an increasing
N the number of parameters of the SSS matrix increases linearly instead of quadratically for the dense matrix form. Further, if a lifted LTI state-space realization is
considered (or large part of the sequence is LTI), then the SSS matrix becomes “almost
Toeplitz” (Rice, 2010), and the computational complexity and memory storage can be
decreased again up to a factor N .
C.2 Sequentially semi-separable matrix computations
C.2
147
Sequentially semi-separable matrix computations
Numerous operations with sequentially semi-separable matrices (also called quasiseparable or low-rank Hankel block matrices) can be found in the literature, however
most exist on paper only. A nice overview of the most important SSS matrix computations can be found in Chandrasekaran et al. (2003, 2005); Rice (2010). Further,
the book Dewilde and van der Veen (1998) describes various operations that are
possible on time-varying systems in great detail, including the efficient application
of orthogonal transformations. As shown in the previous subsection, the SSS matrix
computations are strongly related to operations with LTV state-space systems.
In Table C.1 the currently available SSS matrix operations are summarized. Operations are available for the most commonly applied matrix computations; for
example, addition, multiplication, division, and most matrix decompositions (except
Hessenberg and the ones that require pivoting). In addition, conversion operations
between SSS and dense/sparse matrices are implemented in the SSS matrix toolbox (Houtzager, 2011c). The operations are divided into two levels: 1, and 2.
Level 1 This level contains the efficient SSS matrix operations based on direct methods. These operations can make use of other level 1 operations, but do not use
level 2 operations. If the SSS matrix is stable, most of these operations (except
the inverse and the determinant) are backward stable to numerical errors.
Level 2 This level contains SSS matrix operations based on iterative methods originally used for dense matrices. The iterative methods are adjusted to preserve
the SSS matrix structure. For this purpose, these operations make extensive
use of level 1 operations. The computational complexity, numerical stability,
and convergence properties depends heavily on successful low-order approximations of the generators during the structure preserving iterations.
To avoid numerical errors in the computations, it is required that the SSS matrix
is stable. An SSS matrix is stable if kWk k2 ≤ 1 and kR k k2 ≤ 1 (Chandrasekaran et al.,
2003). Commonly with the manual construction of an SSS matrix it occurs that the
SSS matrix representation is only weakly stable, that is if kW2 W3 . . . WN −1 k2 ≤ 1 and
kR N −1 . . . R 1 k2 ≤ 1. However, they can be made stable by using the fast order reduction
algorithm for the generators, which estimates a stable low-order approximation of the
weakly stable SSS matrix.
Essential for efficient computations with SSS matrices is that the off-diagonal
blocks have low numerical rank. With efficient computations it is meant that operations have linear computational complexity in the number of generators N . In other
words, if n k ≪ N , m k ≪ N , rk ≪ N , and ℓk ≪ N , than the computational complexity
and memory storage of SSS matrices is more efficient than with dense matrices.
Compared with sparse matrices, the SSS matrices are normally more efficient if the
number of zeros in the matrix is very low. Some level 1 operations can increase the
numerical rank of the off-diagonal blocks. The specific operations are denoted with
the symbol † in Table C.1. After calling the specific level 1 operation in a level 2
operation, it is required to check if the generators do not grow unbounded. If this
is the case an low-order approximation of the generators is necessary.
148
Table C.1: Operations with SSS matrices
description
mathematical description (example)
algorithm (reference)
comments
SSS-dense/sparse matrix conversion
SSS matrix construction
Ā = construct (A, ℓk , rk )
dense matrix in Chandrasekaran et al.
(2003), banded matrix in Rice (2010),
sym. pos. definite in Gu et al. (2010)
complexity O N 2 with p
non-zero bands O N p 3 , see
also remark C.1
dense matrix reconstruction
A = full (A) = Ā I
matrix-vector multiplication is used
for exact complexity see ref.
transpose
B̄ = Ā T
complex conjugate
B̄ = conj Ā

Level 1 operations
shuffled concatenation† and
separation
addition† and subtraction†
low-rank update of
matrix†
C̄ = Ā + B̄ , and C̄ = Ā − B̄
B̄ = Ā
+uvT

U1 (W2 W3 ) V4

U2 W3

V4 
U3

D4
in Chandrasekaran et al. (2003)
in Rice (2010)
shuffled by known permutation T to keep SSS structure
in Chandrasekaran et al. (2003)
in Chandrasekaran et al. (2003)
Hadamard low-rank product
B̄ = Ā ⊙ u v T = diag (u ) Ādiag (v )
matrix-scalar multiplication
B̄ = αĀ
matrix-vector multiplication
b = Āx
in Chandrasekaran et al. (2003)
matrix-matrix multiplication†
C̄ = Ā B̄
in Chandrasekaran et al. (2003)
Kronecker identity product†
B̄ = Ā ⊗ I¯, and B̄ = I¯ ⊗ Ā
in Chandrasekaran et al. (2003)
in Chandrasekaran et al. (2003)
also called the Schur product
for exact complexity see ref.
for exact complexity see ref.
also called the Tensor product
Appendix C: SSS Matrix Computations for RC
splitting and merging
”
—
D1
U1 V2 W2 V3

D2
U2 V3
P2

Q1
S̄ (N − 1) = 
P”3Q 2
D3 —
 P3 R 2
P4 (R 3 R 2 )Q 1 P4 R 3Q 2 Q 3
”
—
”
—
C̄ T = Ā B̄ T = A B
mathematical description (example)
algorithm (reference)
comments
Level 1 operations (continued)
− Āx k22
matrix-vector solver
x = arg min kb
matrix-matrix solver
X̄ = arg min k B̄ − Ā X̄ k2F
determinant
det Ā
= Ā −1 ,
with the QR/URV decomp. Dewilde
and van der Veen (2001); Eidelman and
Gohberg (2001); Sun (1997), or the fast
solver in Chandrasekaran et al. (2003)
for exact complexity see ref.
with the QR/URV and backward
substitution
in
Chandrasekaran
et al. (2003), or the superfast solver
in Chandrasekaran et al. (2003)
in Rice (2010)
(pseudo) inverse
B̄
Cholesky decomposition
H̄ = R̄ R̄ T , where H̄ is a positive-definite Hermitian matrix
LU decomposition
Ā = L̄Ū
QR/RQ/LQ/QL decomp.
Ā = Q̄ R̄ with Q̄ T Q̄ = I¯ and R is an upper triangular matrix
see matrix-vector solver
very similar to the URV/ULV
decomposition.
JQR/JRQ/JLQ/JQL decomp.
Ā = Q̄ R̄ with Q̄ T J¯1Q̄ = J¯2 and R is an upper triangular matrix
tr H̄ , where H̄ is a symmetric matrix
basically, replace QR with JQR locally
also called the hyperbolic QR
Frobenius matrix norm
kĀkF
in Rice (2010)
order reduction of the generator realization
ē ≤ tol
kĀ − Ak
∗
with the fast model reduction method
in Chandrasekaran et al. (2003), or
the balanced truncation in Rice (2010),
or the Hankel norm approximation
in van der Veen and Dewilde (1994)
matrix trace
and B̄
= Ā +
with the QR/URV and backward substitutions with identity, or directly with
Riccati recursions in Rice (2010)
C.2 Sequentially semi-separable matrix computations
description
in Rice (2010)
149
150
description
mathematical description (example)
algorithm (reference)
comments
Level 2 operations
matrix second norm
kĀk2
with bisection in Rice (2010)
matrix square-root
B̄ = Ā 1/2
with the sign iteration in Rice (2010), or
with the Denman-Beavers iteration
matrix balancing
B̄ = D̄ Ā D̄ −1 where D̄ is a diagonal scaling matrix
with the EBE iteration (a modified Osborne’s method) in Rice (2010)
matrix sign decomposition
L
0 −1
−I¯L
0
V̄ −1 , where X = P
sign X̄ = V̄
P has
0 R
0
I¯R
−
an Jordan decomposition with λ (L) ∈ C and λ (R) ∈ C+
in Rice (2010)
used for low-rank approx.
Ā = Q̄ Ū Q̄ T with Q̄ T Q̄ = I¯ and U is an upper triangular matrix
with unsorted eigenvalues on the diagonal
with the explicit QR iteration
in Chandrasekaran et al. (2007);
Eidelman et al. (2005), or the LR
iteration in Bevilacqua et al. (2010)
see also Remark C.2
SVD decomposition
Ā = Ū S̄ V̄ T with Ū T Ū = I¯ and V̄ T V̄ = I¯ and S̄ is a diagonal
matrix with the singular values unsorted
see also Remark C.3
see also Remark C.2
Lyaponuv equation
Ā X̄ + X̄ Ā T +Q = 0, and X̄ = Ā X̄ Ā T + Q̄ = 0
for continuous-time the sign iteration
in Rice (2010), and for discrete-time
the squared Smith iteration
square-root Lyaponuv eq.
Ā R̄ R̄ T + R̄ R̄ T Ā T + B̄ B̄ T = 0, and R̄ R̄ T = Ā R̄ R̄ T Ā T + B̄ B̄ T
for continuous-time in Larin and
Aliev (1993), and for discrete-time
in Fernando (1984)
CARE and DARE (algebraic
Riccati equation)
Ā T X̄ + X̄ Ā − X̄ B̄ +S̄ R̄ −1 B̄ T X̄ +S̄ T +Q̄ = 0, and X̄ = Ā T X̄ Ā −
T
−1 T
T
T
Ā X̄ B̄ + S̄ B̄ X̄ B̄ + R̄
Ā X̄ B̄ + S̄ + Q̄
for continuous-time the sign iteration
in Rice (2010), for discrete-time the
doubling iteration, or the Newton iteration, or the Kleinman iteration
Appendix C: SSS Matrix Computations for RC
Schur/eigenvalue decomp.
C.2 Sequentially semi-separable matrix computations
151
Remark C.1 (SSS matrix construction). The SSS matrix construction operation proceed
from the top-left to the bottom-right blocks of the dense matrix. The methods always
try to compress the first block row, and then merge it with the second-block row. This
is normally a sub-optimal choice. For example, after compressing the first blockrow, and before merging with the second block-row, it might be wise to compress
the second-block row first. This has the advantage of compressing a block-row with
potentially much smaller rank. Of course, one can extend this idea to consider all the
block-rows. It is recommended to find the best ordering before we start the algorithm.
The reordering is difficult to apply in practice, as automated ordering algorithms are
not (yet) suitable with the sequentially semi-separable matrix representation.
Remark C.2 (Schur/Eigenvalue decomposition). The iterative QR algorithm is the
most used method to obtain the Schur/Eigenvalue decomposition. To let the QR
algorithm successfully converge to the Schur/Eigenvalue decomposition the following
well known extensions are required:
• the initial conversion to Hessenberg matrix form and its invariance during the
QR iterations
• clever shifts strategies (for example double Wilkinson shifts) for complex eigenvalues of real matrices
• the implicit implementation of shifts
It is not fully theoretically proven that these extensions guarantee the convergence of
the so-called implicit QR algorithm under all conditions. However, in practice (for
example the dense matrix implementation in LAPACK) these extensions proved to be
successful in almost all cases. The problem is that these extensions can currently only
be applied on SSS matrices with m k = n k = l k = rk = 1 (Bella et al., 2008). For the
more general case, the basic explicit QR algorithm (with single or complex shifts)
can be used at the cost of slower convergence or even in some cases divergence.
Further, the eigenvalues are not ordered in any way, because a column pivotation
is not possible in the sequentially semi-separable matrix representation. Also the
calculation of the left- and right-orthogonal matrices requires a lot of multiplications
and the related generator reductions. For these reasons, it is recommended to use
the Schur/Eigenvalue decomposition only for the calculation of the eigenvalues, and
not for solving the low-rank matrix approximation problem, as the calculation of
the orthogonal matrices is very inefficient and the unsorted eigenvalues makes the
separation difficult. When the eigenvalues are obtained, the sign decomposition or
the hyperbolic QR decomposition can be used to obtain the low-rank approximation
of the matrix under study, see Rice (2010).
Remark C.3 (SVD decomposition). To obtain the singular values of a matrix, the wellknown relation between the singular value decomposition (SVD) and the eigenvalues
of the so-called symmetric embedding is used. Consider that A = USV T is the SVD of
matrix A, then for i = 1 : rank (A) the following eigenvalue decomposition exists:
–
™
–
™
”
—
u iT
0 A
u
±v
,
=
±σ
i
i
i
±v iT
AT 0
where u i = U (:, i ), v i = V (:, i ), and σi = S (i , i ). In the case of SSS matrices, it is noted
that the concatenation shuffles the rows and columns.
152
C.3
Appendix C: SSS Matrix Computations for RC
SSS matrix construction for efficient repetitive control synthesis
The SSS matrix operations can not only be used to speed up the LTV controller
synthesis, but also for the synthesis of lifted ILC (Dijkstra, 2004; Tousain and Bosgra, 2001), lifted RC (Rice and Verhaegen, 2010b), 1D heterogeneous/homogeneous
distributed systems (Rice and Verhaegen, 2009), and rationally parametric LPV systems (Rice and Verhaegen, 2010c). Compared with the other systems, the lifted RC
formulation in Chapter 4 and 5 requires for an efficient SSS matrix implementation
a reordering of the state basis for the lifted state-space models in (4.12), (4.15),
and (5.12). See also the Remark C.1 on the construction operation for the SSS
matrices.
Basically, an efficient SSS representation can be obtained if the state basis and also
the noise vector basis is shuffled as follows (as a deck of cards):




nk
∆x k


 wk 


e
 



k



 vk 


n̄ k ,p
∆x k


 wk 



..
 = w̄ k ,p 
 = Tξ  ẽ k −p,p  = Tξ ξ′ , 

 Tν = ν̄k ,p Tν .
..
.
k




.




v̄
w̄
k
,p
k
−p,p
 n k +p −1 




 e k +p −1 
w k +p −1 
w k +p −1
v k +p −1
An explicit calculation of the transformations Tξ and Tν is not required. If the shuffled
concatenation in Table C.1 is used to build the lifted state-space matrices in (4.12),
(4.15), and (5.12) from the individual lifted blocks, the required shuffled state basis is
automatically obtained in a SSS matrix representation with N = p + 1. Afterwards
the individual blocks can even be splitted, such that for one memory loop a SSS
matrix representation with N = 2p + 1 is obtained. If the time-domain state-space
model should be augmented with the noise model or/and the feedback controller
description, it is also be worthwhile to first lift these state-space systems separately,
and than augment the systems in the lifted domain using concatenation and splitting
operations. This will result (for one memory loop) in a SSS matrix representation with
N = 4p + 3.
153
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List of Abbreviations
1P
2D
2P
3D
3P
ARMAX
ARX
CCA
CVA
DARE
ELS
ERA
FB
FF
FMEA
GBN
GCV
HAWT
ILC
IPC
LMS
LPV
LQG
LTI
LTV
MCS
MEM
MFC
MIMO
MPC
MW
OKID
ORT
PAST
PBSID
Once-per-revolution
Two-dimensional
Twice-per-revolution
Three-dimensional
Thrice-per-revolution
Auto-Regressive Moving Average with eXogenous inputs
Auto-Regressive with eXogenous inputs
Canonical Correlation Analysis
Canonical Variate Analysis
Discrete-time Algebraic Riccati Equation
Extended Least Squares
Eigensystem Realization Algorithm
Feedback
Feedforward
Failure Modes and Effects Analysis
Generalized Binary Noise
Generalized Cross Validation
Horizontal-Axis Wind Turbine
Iterative Learning Control
Individual Pitch Control
Least Mean Squares
Linear Parameter-Varying
Linear Quadratic Gaussian
Linear Time-Invariant
Linear Time-Varying
Monte-Carlo Simulations
Micro-Electro-Mechanical
Macro Fiber Composite
Multiple-Input and Multiple-Output
Model Predictive Control
Megawatt
Observer/Kalman-filter Identification
Orthogonal Decomposition
Projection Approximation Subspace Tracking
Predictor-Based Subspace Identification
169
170
PBSIDopt
PEM
PM
PRBS
PSD
RLS
RPM
RSID
SID
SISO
SNR
SPA
SRC
SSS
STD
SVD
RC
RPBSID
VAF
VAR
VARMAX
VARX
List of Abbreviations
“optimized” Predictor-Based Subspace Identification
Prediction Error Method
Propagator Method
Pseudo Random Binary Signal
Power Spectral Density
Recursive Least Squares
Revolutions-per-minute
Recursive Subspace Identification
Subspace Identification
Single-Input and Single-Output
Signal-to-Noise Ratio
Spectral Analysis
“Smart” Rotor Control
Sequentially Semi-Separable
Standard Deviation
Singular Value Decomposition
Repetitive Control
Recursive Predictor-Based Subspace Identification
Variance-Accounted-For
Variance
Vector Auto-Regressive Moving Average with eXogenous inputs
Vector Auto-Regressive with eXogenous inputs
Summary
Towards Data-Driven Control for Modern Wind Turbines
Ivo Houtzager
Further developments in data-driven control techniques for the load reduction of
modern wind turbines can achieve an increased lifetime of components and make the
scaling to larger rotor diameters possible, and therefore improve the cost effectiveness
of modern wind turbines. Also the success of future rotor designs will heavily depend
for their operation on new developments in active control technologies. This thesis
proposes a novel control algorithm for the rejection of periodic disturbances. The
load disturbances seen by an individual rotor blade are to a large extent periodic
and repetitive; for instance, tower shadow, wind shear, yawed error, and gravity are
depending on the azimuth angle and rotational speed, and will change slowly over
time. The contribution of these periodic disturbances will become even larger in the
future, due to the increasing length and mass of the rotor blades and the possible
downwind location of a two-bladed rotor. By compensating only the smooth periodic
disturbances, the control action would not create to much fatigue damage on the
actuators.
In pursuit of this goal, the novel control synthesis requires a mathematical model
description of the system to be controlled. The proposed way to obtain these
models is from measured data using identification techniques. For this purpose, the
“optimized” predictor-based subspace identification methods are proposed and their
benefits for the estimation of modern wind turbine models are shown. These identification techniques do not require any controller related information, consequently
the identified model becomes consistent no matter the wind turbine operates with or
without controller in the loop. A modern wind turbine is not asymptotically stable
in open-loop, and only closed-loop experiments are supported in reality. Further,
the identification algorithms can efficiently deal with multi-input and multi-output
measurements. This makes these methods very attractive for the new rotor concepts
that require an increasing number of actuators and sensors to operate. Despite
the short length of the measurement data, due to the limited amount of time in
an operation region, and the low signal-to-noise ratio, due to the high turbulence
intensity, the results show that an accurate batchwise identification of a realistic
modern wind turbine model is possible, especially around the natural frequencies of
the wind turbine.
The operational conditions of modern wind turbines will change slowly over time,
due to changes in the mean wind speed. A novel recursive predictor-based subspace
171
172
Summary
identification algorithm is developed to track the slowly time-varying dynamics. The
algorithm is suited for real-time implementation and is also able to operate in open
loop or closed loop. The real-time implementation and the ability to work with
systems operating in closed loop make this approach suitable for online estimation of
unstable dynamics. Flutter detection is one of the applications of this approach and is
demonstrated this on an experimental “smart” rotor test section. The “smart” rotor is
a new rotor concept where the blades are equipped with a number of control devices
that locally change the lift profile on the blade, combined with appropriate sensors.
The algorithm is successfully able to track after some delay the natural frequencies
and modal damping values.
The experimental “smart” rotor test section is also used to demonstrate that
the novel repetitive controller can reject the periodic disturbances in a turbulent
wind field. The repetitive control method is implemented in real-time for periodic
disturbance rejection on linear systems with multi-inputs and multi-outputs and with
both repetitive and non-repetitive disturbance components. For real-time implementation, the computational complexity can be reduced by exploiting the structure
in the lifted state-space matrices. For relatively slow changing periodic and turbulent
wind disturbances created by a wind generator it is shown that this repetitive control
method could reduce the variance of the load signals of the “smart” rotor test section
up to 42%. The cost of additional wear and tear of the “smart” actuators are kept
small, because a smooth control action is generated as the controller mainly focuses
on the reduction of periodic disturbances.
The load reduction capabilities of the repetitive controller using individually pitch
controlled blades is verified on the UPWIND 5MW wind turbine in the aerolastic
modelling software GH BLADED. The repetitive control method is very generic, and
can be extended to rotors with more actuators and sensors, or to other rotor designs;
like, two-bladed rotors. An important drawback of repetitive control is also addressed,
which is that a small mismatch between the controller period and the actual period of
the disturbance signal can decrease the performance substantially. The formulation
of the repetitive control problem is made more robust for small changes in period
time. With these additions, the repetitive controller is still considered competitive for
modern fixed-speed and modern variable-speed wind turbines operating above-rated
with a pitch controller that will keep the rotational speed of the wind turbine close to
the given rated rotor speed. The relative performance in terms of load reduction and
control action is compared with typical individual pitch control and with different
turbulence intensities and consequently different variations in the rotational rotor
speed. The simulation results indicate that for relatively slow changing disturbances
this method can significantly contribute to the reduction of vibrations in the wind
turbine structure.
Samenvatting
Naar op data gebaseerde besturing voor moderne windturbines
Ivo Houtzager
Nieuwe op data gebaseerde regeltechnische ontwikkelingen voor de belastingvermindering van moderne windturbines kunnen de levensduur van componenten verlengen en maken het schalen naar grotere rotordiameters mogelijk, en verbeteren dus de
kosteneffectiviteit van moderne windturbines. Ook het succes van toekomstige rotor
ontwerpen zullen sterk voor hun operatie afhankelijk zijn van deze nieuwe ontwikkelingen in de actieve regeltechnologie. Dit proefschrift stelt een nieuwe regelalgoritme
voor om de periodieke verstoringen te compenseren. De belastingverstoringen op
een individueel rotorblad zijn voor een groot deel periodiek en repetitief, bijvoorbeeld
de torenschaduw, de windschering, de fout in gieren, en de zwaartekracht zijn allen
afhankelijk van de azimut en de rotatie snelheid, en zal langzaam veranderen in de
tijd. De bijdrage van deze periodieke verstoringen zullen nog groter worden in de
toekomst, door het gevolg van de toenemende lengte en massa van de rotorbladen
en door de mogelijke plaatsing van een tweebladige rotor met de wind mee. Door
de periodieke verstoringen te compenseren, zullen de regelacties niet leiden tot veel
vermoeidheidsschade aan de actuatoren.
Ter verwezenlijking van dit doel heeft de nieuwe regelaar ontwerpmethode een
wiskundig model benodigd van het geregelde systeem. De voorgestelde manier om
deze modellen te verkrijgen is via de gemeten data met behulp van identificatietechnieken. Hiervoor zijn geoptimaliseerde op voorspeller gebaseerde subruimte
identificatie methoden voorgesteld en zijn hun voordelen voor het schatten van
moderne windturbine modellen aangetoond. Deze identificatietechnieken vereisen
geen regelaar gerelateerde informatie, dus ook het geïdentificeerde model wordt
consistent geschat ongeacht of de windturbine werkt met of zonder regelaar in de
lus. Een moderne windturbine is niet asymptotisch stabiel in de open lus, dus alleen
experimenten in de gesloten lus kunnen in de werkelijkheid worden ondersteund.
Verder kunnen de identificatie algoritmen efficiënt omgaan met meerdere ingangsen uitgangsmetingen. Dit maakt deze methoden zeer aantrekkelijk voor de nieuwe
rotor concepten die een toenemend aantal actuatoren en sensoren vereisen voor hun
bediening. Ondanks de korte lengte van de meetgegevens, als gevolg van de beperkte
hoeveelheid tijd in een operatiegebied, en de lage signaal-ruis verhouding, als gevolg
van de hoge turbulentie-intensiteit, laten de resultaten zien dat een nauwkeurige
batchgewijs identificatie van een realistische moderne windturbinemodel mogelijk is,
vooral rond de natuurlijke frequenties van de windturbine.
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Samenvatting
De operationele omstandigheden van moderne windturbines veranderen langzaam in de tijd, als gevolg van veranderingen in de gemiddelde windsnelheid. Een
nieuwe recursieve op voorspeller gebaseerde subruimte identificatie algoritme is ontwikkeld om het langzame tijdvariërend dynamische gedrag te volgen. Het algoritme
is geschikt voor real-time implementatie en is ook in staat om te werken in zowel
open lus of gesloten lus. De real-time implementatie en het vermogen om te werken
met systemen die actief in gesloten lus zijn, maakt deze benadering geschikt voor
het online schatten van instabiel dynamisch gedrag. Fladderdetectie is één van de
toepassingen van deze benadering en dit wordt gedemonstreerd op een experimentele
“slimme” rotor testsectie. De “slimme” rotor is een nieuw rotorconcept waarbij de
bladen zijn voorzien van een aantal apparaten die lokaal op het blad het liftprofiel kan
veranderen, gecombineerd met passende sensoren. Het algoritme heeft met succes de
natuurlijke frequenties en modale dempingwaarden na enige vertraging gevolgd.
De experimentele “slimme” rotor testsectie is ook gebruikt om aan te tonen dat
de nieuwe repetitieve regelaar ook de periodieke verstoringen kan compenseren in
een turbulent windveld. De repetitieve regelaar ontwerpmethode is toegepast in
real-time voor de compensatie van periodieke verstoring in lineaire systemen met
meerdere ingangen en uitgangen, en met zowel repetitieve en niet-repetitieve verstoringscomponenten. Voor de real-time implementatie kan de berekeningscomplexiteit
worden verminderd door gebruik te maken van de structuur in de matrices van de
vergrote toestandsruimte. Voor relatief langzaam variërende periodieke en turbulente
windverstoringen, gemaakt door een windgenerator, is aangetoond dat deze repetitieve regelaar de variatie van de belastingsignalen van de “slimme” rotor testsectie
tot 42% kan verminderen. De kosten van extra slijtage van de “slimme” actuatoren
zijn beperkt, omdat de regelactie gegenereerd door de regelaar zich vooral richt op de
vermindering van de periodieke verstoringen.
De belastingverminderingscapaciteiten van de repetitieve regelaar met gebruik
van individuele pitch geregelde bladen is geverifieerd op de UPWIND 5 MW windturbine in de aerolastische modelleringsoftware GH BLADED. De repetitieve regelmethode is zeer generiek en kan worden uitgebreid naar rotoren met meer actuatoren en
sensoren of naar andere rotorontwerpen, zoals de tweebladige rotoren. Een belangrijk
nadeel van repetitieve regelaars is ook behandeld, namelijk dat een kleine verschil
tussen de periode van de regelaar en de werkelijke duur van het verstoringssignaal
de prestaties aanzienlijk kunnen verminderen. De formulering van het repetitieve
regelprobleem is meer robuust gemaakt voor kleine veranderingen in de periodetijd.
Met deze toevoegingen kan de repetitieve regelaar nog steeds als concurrerend worden beschouwd voor de moderne vaste-snelheid en de moderne variabele-snelheid
windturbines opererend in het bovenwerkgebied met een pitchregelaar, die de rotatiesnelheid van de windturbine dichtbij de opgegeven nominale rotorsnelheid zal
houden. De relatieve prestaties in termen van belastingvermindering en regelactie
is vergeleken met een typische pitchregelaar en met verschillende turbulentie intensiteiten, en daaruitvolgend ook verschillende variaties in de rotatiesnelheid. De
simulatie resultaten geven aan dat voor de relatief langzame veranderlijke storingen
deze methode aanzienlijk kan bijdragen tot de vermindering van trillingen in de
windturbineconstructie.
Curriculum Vitae
I
HOUTZAGER was born on the 6th of January, 1982 in Krimpen aan den IJssel, The
Netherlands.
VO
In 1999 he graduated from high school (HAVO) at the Krimpernerwaard College in
Krimpen aan den IJssel, The Netherlands.
In 2003 he received the B.Eng. degree in mechanical engineering with speciality in
design and production from the Rotterdam University of Applied Sciences in Rotterdam, The Netherlands. His bachelor project entitled “Vierrollen rondbuigmachine:
Een nieuw concept voor het coilen” was carried out at Fontijne Grotness B.V. in
Vlaardingen, The Netherlands.
In 2007 he received the M.Sc. degree in mechanical engineering with speciality in
systems and control from the Delft University of Technology in Delft, The Netherlands.
His master project entitled “Adaptive learning control for wind turbines equipped with
smart rotor: Applied to a NM80 wind turbine model” was carried out at Delft Center
of Systems and Control (DCSC) in Delft, The Netherlands.
In 2007 he started pursuing a Ph.D. degree in systems and control engineering on the
project “Data-Driven Control for Modern Wind Turbines” at Delft Center of Systems
and Control (DCSC) and Delft University Wind Energy Research Institute (DUWIND)
under the supervision of Michel Verhaegen and Jan-Willem van Wingerden. During
his Ph.D. project, he participated in several international conferences, workshops,
and graduate courses, for which he received a Dutch Institute of Systems and Control
(DISC) certificate.
As of August 2011 he works as embedded software developer of the motion control
facilities at ASML in Veldhoven, The Netherlands.
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Curriculum Vitae
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