mscThesis_TJKoomen_1384791.
General Control Framework for
Shape Memory Alloy Based Actuators
A Phase Transformation Approach
Master of Science Thesis
T.J. Koomen
Delft Center for Systems and Control
General Control Framework for
Shape Memory Alloy Based
Actuators
A Phase Transformation Approach
Master of Science Thesis
For the degree of Master of Science in Mechanical Engineering at
Delft University of Technology
T.J. Koomen
June 12, 2015
A15.015.1206
Faculty of Mechanical, Maritime and Materials Engineering (3mE) · Delft University of
Technology
Copyright © Delft Center for Systems and Control (DCSC)
All rights reserved.
Abstract
Shape Memory Alloy (SMA)-based actuators are versatile actuators with a high powerto-weight ratio and they are easy to miniaturize. These actuators are ideal for applications such as smart wings and smart rotors. However, SMA-based actuators are generally
difficult to control, due to a low bandwidth and a hysteretic and non-linear response.
A general control framework for SMA-based actuators is proposed. This new control
framework deals with some of the challenges that are generally faced when working
with SMA-based actuators. The biggest challenge that was set was that there is no
information on the some of the states of the SMA actuators, such as the temperature
or the stress in the material. The general control framework eliminates the need for
this information and also proposes a simple method to control multiple inputs to the
actuator based on a single output of the system. The main goal of the general control
framework is to provide control with increased bandwidth applicable to all SMA-based
actuators with control parameters which are easy to tune.
The general control framework, or Phase Transformation Approach, focuses on fundamental properties of the SMA material. By using some of the fundamental properties to
construct four control laws, the general control framework ensures that is effective for all
SMA-based actuators. The control principles were tested and showed that the proposed
control framework was effective. However, the parametrization of the control framework
was not straightforward. It seemed to have a strong dependency on the amplitude and
frequency of the reference signal, also due to the still non-linear response. The individual control principles were effective at achieving what they were designed for but their
hierarchy within the general control framework has to be rearranged.
Master of Science Thesis
T.J. Koomen
ii
T.J. Koomen
Master of Science Thesis
Contents
Acknowledgements
xiii
1 Introduction
1-1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
1-2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
1-4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Theory
5
I
2 Shape Memory Alloy
7
2-1 SMA properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2-2 Modeling SMAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2-2-1
Thermodynamical approach . . . . . . . . . . . . . . . . . . . . . . . 11
2-2-2
Mechanical approach . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2-3 SMA actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2-3-1 Functional Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2-3-2
Modeling of SMA temperature . . . . . . . . . . . . . . . . . . . . . 14
2-4 SMA-based actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2-4-1 Modeling of the SMA-based actuator . . . . . . . . . . . . . . . . . . 18
2-4-2 Model Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2-5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Master of Science Thesis
T.J. Koomen
iv
Contents
3 Control
3-1 Problems with SMA control . . . . . . . . . . . .
3-2 Requirements for the General Control Framework
3-3 State-of-the-art Control . . . . . . . . . . . . . .
3-4 Phase Transformation Approach . . . . . . . . .
3-5 Control Principles . . . . . . . . . . . . . . . . .
3-5-1 Input Pairing . . . . . . . . . . . . . . .
3-5-2 Response Threshold . . . . . . . . . . . .
3-5-3 Linearization of SMA response . . . . . .
3-5-4 Bandwidth Optimization . . . . . . . . .
3-5-5 Feed Forward Stabilizer . . . . . . . . . .
3-6 Summary . . . . . . . . . . . . . . . . . . . . . .
II
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Practice
39
4 Controller Implementation
4-1 Hardware . . . . . . . . . . . . . . . .
4-2 Control Principles . . . . . . . . . . .
4-2-1 Hysteresis . . . . . . . . . . .
4-2-2 Linearization . . . . . . . . . .
4-2-3 Bandwidth Optimization . . .
4-2-4 Feed Forward Stabilizer . . . .
4-3 Control Framework Performance . . .
4-3-1 Periodical Reference Signal . .
4-3-2 Non-periodical reference signal
4-3-3 Parameterization . . . . . . .
4-3-4 Influence of Active Cooling . .
4-3-5 Hardware Performance . . . .
4-4 Summary . . . . . . . . . . . . . . . .
III
.
.
.
.
.
.
.
.
.
.
.
23
23
26
26
27
28
29
31
32
33
35
36
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Conclusions and Recommendations
41
41
44
44
46
46
49
51
51
56
57
57
59
59
61
5 Conclusions
63
IV
67
Appendix
A Hardware
A-1 SMA-based Actuator
A-2 Setup . . . . . . . . .
A-3 Power Supplies . . . .
A-4 Valves . . . . . . . .
T.J. Koomen
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
69
69
70
73
73
Master of Science Thesis
Contents
v
B Control Framework Parameters
75
Bibliography
81
Glossary
85
List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Master of Science Thesis
T.J. Koomen
vi
T.J. Koomen
Contents
Master of Science Thesis
List of Figures
2-1 Illustration of different phase transformations of the SMA in the Shape Memory Effect cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2-2 Graph depicting the Shape Memory Effect (SME) cycle and corresponding
stress, strain and temperature . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2-3 Expected number of life cycles as a function of stress and strain during cycles
(© SEAS Getters). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2-4 Transformation temperatures for SMA wires as a function of the internal stress 10
2-5 Example of the use of an SMA actuator subjected to a constant load. . . . . 14
2-6 Response for a new SMA wire and an SMA wire that has been subjected to
5000 full cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2-7 Resistivity of SMA material (source: SAES Getters) . . . . . . . . . . . . . . 15
2-8 Schematic representation of the SMA-based actuator. . . . . . . . . . . . . . 17
2-9 SMA-based actuator showing upwards and downwards deflection. . . . . . . . 17
2-10 Modeling approach for SMA-based actuator using superposition principle . . 18
2-11 Temperature response of a single SMA actuator and the simulated temperature response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3-1 Bode plot of SMA actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3-2 SMA-based actuator measured deflection rate per amplitude of the input current 25
3-3 Temperature response of an SMA actuator when a constant current is switched
on and off at intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3-4 Linearized SMA-based actuator in feedback configuration . . . . . . . . . . . 26
3-5 SMA-based actuator represented by three subsystems . . . . . . . . . . . . . 27
3-6 Schematic showing the inside of the linearizer of the proposed control framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Master of Science Thesis
T.J. Koomen
viii
List of Figures
3-7 Heat transfer coefficient as a function of the valve aperture Fi . . . . . . . . 33
3-8 Filter used to increase bandwidth and filter out high frequency noise . . . . . 34
3-9 Simulation of the effects of the filter on the evolution of the temperature of
a SMA actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3-10 Control framework output values for the four inputs as function of the error
and its derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3-11 Union of the control inputs for heating and cooling as function of the error
and its derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4-1 Picture of the actuator with the connections indicated. . . . . . . . . . . . . 42
4-2 Illustration depicting the total hardware setup and connections. . . . . . . . . 42
4-3 Linearized SMA-based actuator in feedback configuration . . . . . . . . . . . 44
4-4 Deflection rate of the SMA-based actuator to different step inputs on the
input of the linearizer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4-5 Tracking of sinusoidal signal at 1 Hz for PD controller with and without
threshold value for SMA-based actuator. . . . . . . . . . . . . . . . . . . . . 45
4-6 Graph depicting the effect of increase the threshold value for the current Ii
by 0.1 amperes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4-7 Empirical Transfer Function Estimate (ETFE) obtained frequency response
of SMA-based actuator with proposed control framework. . . . . . . . . . . . 48
4-8 ETFE frequency response obtained from experiments. . . . . . . . . . . . . . 49
4-9 Graph depicting the effects of increasing the derivative gain on the error for
the current Ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4-10 Example of the input for the stabilizer current testing. . . . . . . . . . . . . 50
4-11 Actuator response for two different initial currents and different stabilizer
currents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4-12 Tracking performance for 4 different frequencies and a 2 millimeter amplitude. 53
4-13 Response from three experiments with identical parameterization and initial
conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4-14 Erratic actuator response to a step input with a constant current of I1 = 1.45
ampere which is depicted in the bottom graph. . . . . . . . . . . . . . . . . 54
4-15 Mean squared error of the actuator response for tracking a sinusoidal reference
signal with different frequencies and amplitudes. . . . . . . . . . . . . . . . . 54
4-16 Maximum of the mean squared error that were found during the experiments
for random parameterization. . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4-17 Tracking performance of dual sinusoid for different controller parametrizations. 56
4-18 Response to a step input as reference for three experiments with identical
control parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4-19 Actuator response to sudden opening of the valves showing oscillatory behavior. 58
A-1 Picture of the actuator with the connections indicated. . . . . . . . . . . . . 69
A-2 An Arduino Mega 2560 board. . . . . . . . . . . . . . . . . . . . . . . . . . 70
A-3 Illustration depicting the total hardware setup and connections. . . . . . . . . 71
T.J. Koomen
Master of Science Thesis
List of Figures
ix
A-4 Mirco-Epsilon optoNCDT 1401 laser displacement sensor. . . . . . . . . . . . 72
A-5 Bode plot estimate for the analog low pass filter. . . . . . . . . . . . . . . . 73
A-6 Elektra-Automatik EA-PS 8032-20. . . . . . . . . . . . . . . . . . . . . . . . 74
A-7 Proportional valve and a valve controller from Burkert. . . . . . . . . . . . . 74
Master of Science Thesis
T.J. Koomen
x
T.J. Koomen
List of Figures
Master of Science Thesis
List of Tables
2-1 Four inputs of the SMA-based actuator . . . . . . . . . . . . . . . . . . . . 18
3-1 Example of a gain matrix for a system with two inputs and two outputs . . . 29
4-1 Controller parameters used for testing solutions for the hysteretic and nonlinear response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4-2 Best parameter values for the proportional gain KP ,I per frequency and amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4-3 Best parameter values for the threshold value Ii per frequency and amplitude. 58
A-1 SMA wire properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
A-2 Relevant specifications of the Arduino Mega 2560 . . . . . . . . . . . . . . . 71
A-3 Specifications of the Elektra-Automatik model EA-PS 8032-20 . . . . . . . . 73
B-1 Best parameter values for the threshold value for the current Ii per frequency
and amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
B-2 Best parameter values for the threshold value for the valve aperture Fi per
frequency and amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
B-3 Best parameter values for the proportional gain KP ,I per frequency and amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
B-4 Best parameter values for the proportional gain KP ,F per frequency and
amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
B-5 Best parameter values for the derivative gain KD,I per frequency and amplitude. 77
B-6 Best parameter values for the derivative gain KD,F per frequency and amplitude. 78
B-7 Best parameter values for the stabilizer current IF F per frequency and amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Master of Science Thesis
T.J. Koomen
xii
List of Tables
B-8 Best parameter values for the proportional gain KF F for the stabilizer current
IF F per frequency and amplitude. . . . . . . . . . . . . . . . . . . . . . . . 79
T.J. Koomen
Master of Science Thesis
Acknowledgements
First I would like to sincerely thank my supervisors for their trust and the interesting
discussions that we have had. The idea for the subject for my thesis was suggested by
one of my supervisors, Adrián Lara-Quintanilla from the Faculty of Aerospace Engineering (AE) and made possible by my other supervisors Jan-Willem van Wingerden and
Sachin Navalkar from my own faculty of Mechanical, Maritime and Materials Engineering (3mE). I have learned a great deal from my graduation project and not only from
the theoretical and scientific perspective, but also about the application of the knowledge
that I have gained during my master courses to actual systems. The experience once
again confirms the words of Einstein: In theory, theory and practice are the same. In
practice, they are not.
I would also like to thank Kees Slinkman and Will van der Geest, the technical staff of
the Delft Center for Systems and Control (DCSC) labs. They were always there to help
when I needed equipment or with the implementation of said equipment. If it would not
have been for them it would have taken a lot longer to build the set-up.
My thanks also go out to Hans Hellendoorn for accepting to be on my MSc. defense
committee alongside my supervisors. Finally I would like to thank many of my fellow
students and friends who have either been a there to listen to me rattle on about my
theories or for the interesting discussions that we have had.
Delft, University of Technology
June 12, 2015
Master of Science Thesis
Teun J. Koomen
T.J. Koomen
xiv
T.J. Koomen
Acknowledgements
Master of Science Thesis
“Everything should be made as simple as possible, but not simpler.”
— A. Einstein
Chapter 1
Introduction
This report presents the research of my MSc thesis project. The aim of the research is to
investigate the possibility of a general control framework for a specific type of actuator;
the Shape Memory Alloy (SMA)-based actuator.
1-1
Background
Within the field of wind turbines, as well as the field of aeronautics, a significant amount
of research has been done on the so called smart rotors and smart wings [1, 2, 3]. Smart
wings have the ability to change their shape and thereby change their aerodynamic
properties. These smart wings can counteract fatigue loads and thereby extend the
lifetime. Extending the lifespan or increasing the maintenance interval will in the end
reduce the overall costs for wind turbines.
The lifespan is a very important factor in the design of a wind turbine and it is influenced
by many factors. The negative influences of the wind and wind gusts on the lifespan of
the wind turbine are minimized as much as possible by optimizing the structural design
of the wind turbine. However, the wear can be minimized even further by means of
active control. Certain wind speeds or wind gusts can excite structural modes of the
turbine blades or of the turbine tower. This excitation can be prevented by actively
controlling the aerodynamic loading for each turbine blade. This can either be achieved
by individual pitch control of the turbine blades or by changing the camber of the turbine
blade. The first option can already be applied since all the required hardware is present
in modern wind turbines. However, it leads to a higher wear on the bearings and motors.
A better option would be to control the camber of the blades by introducing one or more
actuators in each of the turbine blades.
Master of Science Thesis
T.J. Koomen
2
Introduction
One possible actuator that can be used to manipulate the camber of the turbine blades
is an SMA-based actuator[4]. This actuator uses SMA wires that can individually be
contracted. This allows for the SMA wires to bend the polymer beam that they are
embedded in. The exact working of the actuator will be explained later on. Though the
focus is primarily on the application to a turbine blade, the SMA-based actuator could
also be applied to a fixed wing aircraft.
1-2
Problem Statement
Although these SMA-based actuators have been researched for over two decades now,
they still remain unused in common applications. This is mostly due to that they are
not easy to control, since their response is very hysteretic and non-linear.
Most research in to the operation and control of SMA-based actuators is very case specific
and a general framework seems to be missing[5]. A lot of problems with the control of
SMA actuators are only solved for a specific application, such as controlling the system
with a trained Neural Network system. These kind of solutions will most likely not be
effective for other SMA-based actuators.
The most important factor for all control principles that will be used in this thesis is
that they have to be based on fundamental properties of the SMA material. This is a
way to ensure that the control framework will be valid for other SMA-based actuators.
Also, further research into the effects of the multi-input control of SMA-based actuators
is required. The thesis objective can be summarized as:
To develop a parameterized control framework for SMA-based actuators
which linearizes the response using control laws that are based on fundamental properties of the SMA material.
To review the effectiveness of the general control framework, the following questions are
formulated:
1. Can the proposed control framework be parameterized such that it cancels out the
hysteretic and non-linear response of the actuator?
2. What is the maximal bandwidth that can be achieved with this control framework
with a phase delay smaller than 30 degrees?
3. What is the influence of different initial conditions of the SMA actuators on its
performance?
1-3
Methodology
This thesis will really focus on the fundamental properties of the SMA actuator itself.
If a fundamental property that hinders the control of the actuator can be identified, a
T.J. Koomen
Master of Science Thesis
1-4 Outline
3
new control strategy can be devised to counter this. This way, the control strategy will
be applicable for all SMA-based actuators as long as the right material properties are
used. This research focuses on a system which also uses compressed air to actively cool
the actuator, whereas most systems only use active heating. The introduction of the
two air inputs to the system not only complicates the modeling of the subsystems, but
it also complicates the control of the actuator since the air exerts a force from within
the actuator.
A portion of the research focuses on the deployment of the hardware that is used to
control and power the SMA-based actuator. A new setup was constructed and its performance will shortly be discussed. The goal is to construct a control framework with a
low level computational requirement. When the control algorithm is not computationally intensive it is easy to combine multiple SMA-based actuator modules to form larger
wing surfaces of which the modules can individually be controlled.
The performance of the general control framework is tested for use with a SMA-based
actuator. The performance is reviewed for each of the challenges that are generally faced
during controller design for an SMA-based actuator.
1-4
Outline
Chapter 2 elaborates on the principles of the SMA material and their application as
SMA actuators. Then the design and principles of the SMA-based actuator as designed
by my supervisor are discussed. Chapter 2 looks at some of the mechanics and modeling
methods of the SMA-based actuator. In Chapter 3, the challenges of controlling these
SMA-based actuators are briefly discussed and a proposal for a new control strategy
based on theory as discussed in Chapter 2 is made. In Chapter 4 the proposed control
strategy will be tested and compared to other control strategies. This thesis concludes
with conclusions and recommendations in Chapter 5.
Appendix A contains detailed specifications of the hardware that was used for the experiments. Appendix B contains more information on the results from the parameterization
of the control framework from the experiments. It also lists the best and worst performing parameters for the control framework that were encountered during the experiments.
Master of Science Thesis
T.J. Koomen
4
T.J. Koomen
Introduction
Master of Science Thesis
Part I
Theory
Master of Science Thesis
T.J. Koomen
Chapter 2
Shape Memory Alloy
Shape Memory Alloy (SMA) is an alloy which exhibits very distinctive behavior when it is
deformed and then heated. In order to obtain a proper understanding of the behavior of
SMA, the properties of this material will first be discussed. Once the material properties
have been discussed, the modeling methods that are relevant for this thesis work are
reviewed. At the end of this chapter, the knowledge of the SMA material and the
modeling methods will be applied to the SMA-based actuator. This will allow for a
better understanding of its operation and control challenges.
2-1
SMA properties
SMA exhibits a solid-state phase transformations when it goes through a specific temperature cycle. The phase transformations and corresponding crystalline structures that
present themselves in the different stages of the cycle are depicted in Figure 2-1. When
the material is heated to a certain temperature, which depends on the internal stress,
a phase transformation is triggered. During this phase transformation the crystalline
structure of the material completeley changes. For a wire made of SMA it means that
it will contract. If the material has been deformed beforehand, this process is capable
of recovering the material to its original shape. This phenomenon is called the Shape
Memory Effect (SME).
Shape Memory Effect
The SME cycle consists of two phase transformations; the forward transformation when
the SMA is cooled and the reverse transformation when the SMA is heated. The graph
in Figure 2-2 shows the full SME cycle for SMA material which has no initial stress
Master of Science Thesis
T.J. Koomen
8
Shape Memory Alloy
Figure 2-1: Illustration of different phase transformations of the SMA in the Shape Memory
Effect cycle (source: Creative Commons (CC)). Bottom left corner depicts the detwinned
martensite and the bottom right depicts the twinned martensite
and an initial temperature of zero degree Celsius. The material starts in point B when a
force is applied to the material, which causes stress and strain to develop in the material,
following the curve in the graph from point B to point C. The twinned martensite has now
been transformed into detwinned martensite, which has the same cubic system structure,
but the molecules have been rearranged. The load is now removed from the material,
causing the stress to drop from point C to point D in the graph. Though the force is
removed, the strain remains present in the material. When the detwinned martensite is
heated to a certain temperature, a solid state phase change occurs where the crystalline
structure changes into an austenite structure. During this transformation, the material
recovers its original shape. As the SMA cools down to zero degrees Celsius, another
solid state phase change occurs and the material is transformed into twinned martensite
crystalline structure again. The forward transformation is an exothermic process, thus
it causes the SMA wire to develop some heat during the cooling process [6].
This remarkable feature of the SMA is that this cycle can be repeated many times,
as much as ten million times depending on the specific SMA material, the continuous
stress that the SMA material is subjected to and the recovered strain [7]. The maximum
recoverable strain typically lies somewhere in the range of 5% to 8%, which is high
for metal alloys. The expected lifetime of the wires used for this thesis is depicted in
Figure 2-3, where it is clear to see that high levels of strain and stress reduce the expected
number of lifecycles of the SMA material.
SMA temperatures
The phase transformations of the material are triggered when certain temperatures are
reached. These temperatures depend on the stress level in the material; the higher the
stress, the higher the temperature needs to be. This stress and temperature correlation
T.J. Koomen
Master of Science Thesis
2-1 SMA properties
9
Figure 2-2: Graph depicting the SME cycle and corresponding stress, strain and temperature
Figure 2-3: Expected number of life cycles as a function of stress and strain during cycles
(© SEAS Getters).
Master of Science Thesis
T.J. Koomen
10
Shape Memory Alloy
Figure 2-4: Transformation temperatures for SMA wires as a function of the internal stress.
The dotted line is an example of the strain during a SME cycle for an SMA wire. The dots
from the legend that are connected by the small dotted lines indicate the respective starting
or finishing temperatures for the different transformations for different stress levels. (source:
SAES Getters)
is depicted in Figure 2-4. There are four important transformation temperatures for the
SMA material, namely the temperatures for martensite finish Mf , martensite start Ms ,
austenite start AS and austenite finish Af . They indicate the temperature where the
transformations start or finish. The grey dotted line in the figure depicts the process
for a wire with a constant stress of approximately 150 MPa. The colored dots in the
graph indicate the different temperatures for the starting and finishing points for varying
internal stress. From this graph, it already becomes clear that both temperature and
stress play a significant role when determining the phase of the material. Therefore these
factors are also really important in determining the amount of deformation of the SMA
material. The graph in Figure 2-4 can basically be summarized into 3 regions:
Region I :
T
Region II : Ms (σ) < T
Region III : As (σ) < T
T.J. Koomen
< Ms (σ) → Forward transformation of SMA
< As (σ) → No transformation of SMA
→ Reverse transformation of SMA
(2-1)
Master of Science Thesis
2-2 Modeling SMAs
2-2
11
Modeling SMAs
Modeling of SMA is not straightforward due to the presence of three different crystalline
phases in the material at different stages during the SME cycle [8]. Because the phase
changes during the SMA cycle, the material properties are discontinuous. A property
like the convective heat transfer coefficient h is dependent on the phase as well as the
temperature [9, 10]. As a results of this, some of the properties of the SMA wire are
governed by discontinuous functions. In this section two constitutive modeling methods
will be discussed; the first one is based on a thermodynamical approach and the second
one is a more mechanical approach. These two approaches are the two main model types
found in literature that are used to describe the behavior of SMA material .
2-2-1
Thermodynamical approach
The constitutive model, or free-energy model, tries to resolve the difficulties of modeling
by combining the properties of the three phase fractions into a single model [11, 12, 13].
Three phase fractions xM + , xM − and xA are introduced, which are governed by the
evolution of the martensitic fractions ((2-2) and (2-3)). Since the sum of fractions needs
to equal one, the austenite phase fraction can be written as (2-4).
ẋM + (t) = −p+A xM + (t) + pA+ xA (t)
(2-2)
ẋM − (t) = −p−A xM − (t) + pA− xA (t)
(2-3)
xA (t) = 1 − xM + (t) − xM − (t)
(2-4)
Here xA is the austenite phase fraction, xM + is the tension-induced martensitic phase
faction, xM − is the compression-induced martensitic phase fraction and pij are the phase
transition probabilities. Since an SMA wire can only experience tension forces it is
assumed that at low temperatures the material is composed of detwinned martensite
(M+ ) at high stresses and twinned martensite (M+/− , which is a blend of both martensite
phases) at low stress.
The phase transition probabilities are determined by examining the barriers in the Gibb’s
energy landscape. The exact details on this can be found in the Phd dissertation by
Heintze [14]. The non-linear stress-strain relation for an SMA tendon is given by:
σ(t) =
ε(t) − εT (xM + (t) − xM − (t))
xA (t)
EA
+
xM + (t)+xM − (t)
EM
(2-5)
Here ε(t) is the strain of the SMA wire, εT is the transformation induced strain, EA
is the Young’s modulus of the austenite phase and EM is the Young’s modulus of the
martensite phases. Equation (2-5) divides the strain which is not a direct result of the
Master of Science Thesis
T.J. Koomen
12
Shape Memory Alloy
transformation, by a phase fraction dependent elasticity modulus to yield the stress in
the SMA. The thermodynamic behavior of the SMA is described by
mcṪ (t) = −hAs (T (t) − T∞ ) + H ẋM + (t) + H ẋM − (t) + j(t)
| {z } |
{z
} |
{z
} |{z}
Energy
flux
Energy dissipation by
(natural) convection
(2-6)
Heat
input
Latent heat from
transformations
Here m is the mass of the SMA wire, c is the specific heat capacity, h is the convective
heat transfer coefficient between the SMA wire and ambient air at temperature T∞ , As
is the surface of the SMA wire, H is the latent heat of phase transformation and j(t)
is the Joule heating input. Equation (2-2) to (2-6) together describe the constitutive
behavior of a SMA wire.
2-2-2
Mechanical approach
The other approach that is considered is that of Arai et al. [15] with the improvement
made by Wang and Yan [16]. This model is derived from a more mechanical than
thermodynamic perspective and describes the relationship between the temperature to
the development of strain and stress in the SMA wire. It achieves this by first looking
at the mechanical energy per unit length of the wire (2-7), where c0 , c1 , c2 and c∗ are
constants, T is the temperature of the wire and ε is the strain of the wire.
1
1
U = c0 + ε2 + c2 ε4 − c∗ T ε
2
4
(2-7)
According to the relationship (2-8), where c is defined by the current phase state of
the SMA, the generalized temperature (T − cσ) can be used instead of T to describe
the stress dependence of the transformation temperatures. A viscous coefficient γ is
introduced, which will account for the viscous damping which occurs during the phase
transformations. Equation (2-9) represents the relation between the elastic and the
viscous force of the SMA.
∂σ
1
=
∂T
c
(2-8)
∂U
+ γ ε̇ = 0
∂ε
(2-9)
Combining Equations (2-7) through (2-9) yields the generalized formula as in (2-10),
where the origin (T0 , ε0 ) can be chosen at any place
γ̂ ε̇ + ĉ1 (ε − ε0 ) + ĉ2 (ε − ε0 )3 = T − T∞ − cσ
T.J. Koomen
(2-10)
Master of Science Thesis
2-3 SMA actuator
13
Wang and Yan [16] adapted this formula in such a way that it can also describe the
hysteresis curve of the SMA. Hence, the previous formula mainly focuses on the strain
and temperature relation, and not the stress-strain. A non-elastic term is added to
(2-10), which contains a certain error function as defined in (2-12). The error function
is active in both the loading and the unloading phase of the SMA cycle, which allows
for more accurate modeling of the super-elastic hysteresis curve of the SMA.
i
h
γ̂ ε̇ + ĉ1 (ε − ε0 ) + ĉ2 (ε − ε0 )3 = T − T∞ − c σ − ĉ3 (σ − σ0 )ĉ4 ferr (a (σ − σ0 )) σ0 (2-11)
2
ferr = √
π
Z
x
2
e−t dt
(2-12)
0
A special case arises according to Wang and Yan [16] when ε and ε̇ are equal to zero
(ε0 can take any positive value). In that particular case, the stress that develops in the
wire is defined by (2-13), where c is a constant of the SMA.
c(σ − σ0 ) = T − T∞
(2-13)
Given the fact that the stress that develops in the wire when the phase transformation
is triggered actually causes the austenite starting temperature As to increase, then it
might results in a (piece-wise) linear relation. However, no proof was found that this
will be the case for the SMA actuators that were used this thesis work, since both the
strain ε and the strain rate ε̇ will not exactly be equal to zero. This property of SMA
material will be reviewed in Chapter 3.
2-3
SMA actuator
SMAs in the form of wires, springs or beams are also often referred to as SMA actuator
whenever it is used to deliver a force or displacement. SMA actuators have the highest
power-to-weight ratio among light-weight actuation technologies [17, 18]. This makes
it an ideal actuator for miniaturization or applications where the actuators needs to be
integrated, or retrofitted, into an existing design. The SMA actuator is an ideal actuator
for situations where miniaturization or clean and silent operation is required.
These SMA actuators can be applied in many ways, but the most general one is as
a muscle-like actuator. An example is depicted in Figure 2-5, where an SMA wire is
subjected to a constant load from a mass with weight m. Due to the mass, the SMA
wire has been elongated by length change dL. When the circuit closes and a current
passes through the SMA actuator it heats up. As the temperature of the wire reaches
As , it starts to transform into the austenite crystalline structure which causes the wire to
contract. the transformation is complete when the entire wire is heated to a temperature
Master of Science Thesis
T.J. Koomen
14
Shape Memory Alloy
of Af . At this point, the wire has contracted a length of dL. When the circuit is opened
again, the wire slowly cools down. When the temperature of the wire reaches Ms , it
slowly starts to elongate again due to the mass that is suspended from the wire. When
the temperatures passes Mf the transformation is complete and the wire has returned
to its previous length L0 .
Figure 2-5: Example of the use of an SMA actuator subjected to a constant load.
2-3-1
Functional Fatigue
Functional fatigue is a phenomenon which occurs during the first several thousand lifecycles of new wires. The SMA material slowly develops a permanent elongation (strain)
which is unrecoverable and the maximum recoverable strain decreases. The permanent
elongation is approximately 0.2 percent after 50.000 cycles for the type of wire that is
used in the SMA-based actuator [19]. The effect of the functional fatigue becomes apparent when the responses of a new wire and a wire that has been subjected to 5000 full
cycles is plotted in Figure 2-6. The initial full stroke for the new wire is 5.8 %. After it
has been cycled 5000 times, a full stroke of 5.2 % remains. The cycling effects the micro
structural features and the actuation performance of the actuator.
This response degradation and development of permanent elongation does not continue
through the entire lifespan of an SMA actuator. Eventually, after a number of cycles,
which ranges from 10.000 to 60.000, the behavior of the actuator converges [19]. It
is deemed sufficient that SMA wires are used that have been conditioned in order to
ensure consistent results. No further attempt was made to incorporate the response
degradation of the SMA actuator in the modeling methods as it would hardly improve
the functionality of the SMA actuator.
2-3-2
Modeling of SMA temperature
The temperature of an SMA wire itself can best be modeled by the simple equation
(2-14).
dT
Q̇in − Q̇out
=
(2-14)
dt
m cp
T.J. Koomen
Master of Science Thesis
2-3 SMA actuator
15
Figure 2-6: Response for a new SMA wire and an SMA wire that has been subjected to
5000 full cycles (source: SAES Getters)
Figure 2-7: Resistivity of SMA material (source: SAES Getters)
The terms Q̇in and Q̇out are defined by (2-15) and (2-16), where I is the current through
the SMA wire, RSMA is the resistivity for the SMA material and h(Fi ) is the convective
heat transfer coefficient as a function of the valve aperture i. The convective heat
transfer coefficient is almost linear in the valve aperture, but the resistance of the wire
is a non-linear function of the temperature and the phase of the SMA (Figure 2-7). The
function for the resistance also show a similar hysteresis loop as the description for the
phase transformation of the SMA.
Q̇in = I 2 R = I 2 RSMA (T (t))
L
ASMA
Q̇out = Aconv h (F ) (T (t) − T∞ )
(2-15)
(2-16)
The temperature response of the SMA wire can be defined as (2-17), where the latent
heat that develops during the cooling process has been neglected. The temperature
Master of Science Thesis
T.J. Koomen
16
Shape Memory Alloy
response of the SMA will behave like a damped system. At t = 0 when a current is sent
through the SMA wire, the term Q̇in will drastically increase while the term Q̇out will
only slowly increase as the temperature T (t) rises. At a certain time, Q̇out will approach
the value of Q̇in and the equilibrium temperature Teq will be reached. An interesting
thing to observe is that both the terms Q̇in and Q̇in , and the term preceding the integral
sign contain the length L of the SMA wire. These terms cancel each other out which
implies that the temperature of the SMA wire as a function of I and F is independent
of its length.
1
T (t) =
ASMA L ρ cp
Zt
2
I RSMA (T (t))
L
ASMA
− πdL h (F ) (T (t) − T∞ ) dt
(2-17)
0
Equation (2-17) will be very difficult to solve analytically since it contains a number
of discontinuous and/or non-linear terms. Therefore, a description for the equilibrium
temperature as function of a constant I and F is defined in (2-18), where t → ∞. This
description will be used for control purposes in Chapter 3.
Teq = lim T (I, F )
t→∞
2-4
(2-18)
SMA-based actuators
SMA-based actuators are actuators which use one or multiple SMA actuators in the
form of wires, springs or even beams [20] in order to achieve actuation. The SMA-based
actuator that is used for this thesis work is depicted in a sketch in Figure 2-8a. It shows
the side view of the actuator where the tip of the actuator can deflect in an upwards or
downwards motion, corresponding to a positive or negative x direction. The beam in
which the wires are placed is made out of a flexible nylon polymer, allowing for deflection
of the beam at reasonably low stress levels in the wire. A cross section of the beam is
shown Figure 2-8b, where the dashed line indicates the neutral axis of the beam.
The wires that are used in the actuator have been prestrained to 4 % and are then
clamped in the actuator under tension. This means that the wires have an initial stress of
σ0 and an initial strain of 0 ≈ 4%. When the wire is heated to at least the corresponding
As it will start to transform into the austenite crystalline structure, causing it to recover
the initial strain. Contraction of the wire is very limited since it has been clamped in
the actuator. Since contraction is very limited, the tension in the wire increases as the
phase transformation progresses. In Figure 2-9a and Figure 2-9b the extra tension in
the dotted wire, which is heated, causes the beam to bend. This deflection of the tip of
the actuator is enhanced when the other wire, the solid wire, is cooled to a temperature
below Ms or preferably Mf . This causes the wire to return to the martensite phase
T.J. Koomen
Master of Science Thesis
2-4 SMA-based actuators
(a)
17
(b)
Figure 2-8: Schematic representation of the SMA-based actuator, with (a) showing neutral
axis x0 and the deflection in positive and negative x direction and (b) the cross section
showing the air channels and SMA wires.
and maximize strain and minimize stress in that wire. So when the upper wire of the
actuator is heated and the lower wire is cooled as in Figure 2-9a the actuator will exhibit
an upward deflection. When the heating and cooling of the wires is switched the actuator
will bend downwards as in Figure 2-9b.
(a)
(b)
Figure 2-9: SMA-based actuator showing an (a) upwards and (b) downwards deflection.
A dotted wire indicates the heating of the wire, which causes it to contract. A solid wire
indicates the cooling of the wire, causing it to elongate.
Inputs and Outputs
From a control perspective, the SMA-based actuator has four inputs; separate heating
and cooling control for two wires. The heating is achieved through Joule heating and
the cooling by convective heat transfer which can be controlled by increasing the airflow through the air channels which house the SMA wires. Joule heating is the simple
principle of a current I that passes through a wire with a resistance R, which causes the
wire to heat up. Most SMA-based actuators only use active heating of the SMA wire or
spring. This actuator however, also uses active cooling control by means of compressed
air that is fed through the channels that house the SMA wire. The advantage of also
actively controlling the cooling process of the SME cycle is that the cooling of the SMA
takes significantly longer than the heating of the wire. Thus by increasing the rate of
cooling, the speed and hence the bandwidth of the actuator can potentially be increased
. The four inputs are defined as in Table 2-1, where the first two inputs are the currents
Master of Science Thesis
T.J. Koomen
18
Shape Memory Alloy
in amperes and the last two inputs are the valve apertures where zero indicates a fully
closed and 1 indicates a fully opened valve. It is important to remark that all inputs
have a minimum of zero, i.e. the inputs are always greater or equal than zero and can
thus not be negative.
Table 2-1: Four inputs of the SMA-based actuator
Input Designation
Current
Valve aperture
Symbol
Unit
Range
I1
Ampere
<0,20>
I2
Ampere
<0,20>
F1
(-)
<0,1>
F2
(-)
<0,1>
The only output of the system is an external signal measuring the deflection at the tip
of the actuator. This measurement is obtained from a laser displacement sensor. At any
given time, the stress in both wires and the exact temperature of the wires are unknown.
This complicates the use of the modeling methods since they are primarily based on the
SMA’s temperature and stress. The input is defined as (2-19), where the range of the
laser sensor is [1, 5] Volts, which translates to a distance of [20, 40] millimeters from the
laser sensor itself. The output of the laser sensor is shown in millimeters and the starting
position will always be marked as x = x0 = 0.
x (mm)
2-4-1
(2-19)
Modeling of the SMA-based actuator
The modeling methods described in Section 2-2 only describe the behavior of the individual SMA actuators. A model needs to be conceived to be able to predict the influence
of these individual actuators to the SMA-based actuator. Modeling of the SMA-based
actuator that is used for this thesis work can best be approached by considering a beam
with two wires that are under a certain stress σ1 and σ2 . The stress will cause the
actuator to compress in longitudinal direction, proportional to the Young’s modulus of
the nylon polymer. The difference in stress between the two wires can be rewritten to
a resulting torque at the end of the actuator. The resulting torque M can be used to
calculate a deflection at the tip of the actuator. This superposition principle is depicted
in Figure 2-10.
Figure 2-10: Modeling approach for SMA-based actuator using superposition principle
T.J. Koomen
Master of Science Thesis
2-4 SMA-based actuators
19
The compression dL of the beam is neglected since it has no influence on the deflection
dx and this is the only deformation that we are interested in. The deflection of the tip
of the actuator is defined by (2-20).
dx =
M L2
2EI
(2-20)
M = 2 α ASMA (σ1 − σ2 )
(2-21)
Here L is the length of the beam, E is the Young’s modulus of the nylon polymer and
I is the moment of inertia of the beam. The torque is defined by (2-21), where α is the
distance from the wire to the neutral axis of the actuator, ASMA is the surface of the
cross section of the wire. The factor 2 in (2-21) is due to the fact that the wires run
back and forth over the length of the actuator per side. Therefore the effective number
of wires per side is two, thus twice the force. Using Equation (2-13), rewritten as (2-22),
and the assumption that the strain is equal to zero, one could consider the response of
the actuator to be linear. This will only be possible when the parameter c describing the
relation between the temperature and stress is linear and that it is feasible to keep the
two wires at constant temperatures. Equation (2-20) can then be rewritten as (2-23).
Since the initial stress is assumed to be equal in both wires and T∞ is also equal for both
wires the equation can be reduced to (2-24) where it is clear to see that the deflection is
linear in the difference in temperatures of the wires.
σi =
dx =
dx =
2-4-2
Ti − T∞
c
2 α ASMA
(2-22)
T1 −T∞
c
+ σ0 −
2EI
2 α ASMA (T1 − T2 ) L2
2EI c
T2 −T∞
c
+ σ0
L2
(2-23)
(2-24)
Model Choice
Two modeling methods have been discussed in Section 2-2; the first method with the
thermodynamical approach and the second method with a mechanical approach. Both
models predict the behavior of the SMA material as a function of the temperature.
Unfortunately, these models are not very useful for controlling the actuator since the
exact temperatures of the SMA actuators are unknown. The model for the temperature
of the SMA actuators can only model the ideal equilibrium temperature and it fails to
predict the erratic temperature fluctuations which occur as can be seen in Figure 2-11.
Extensive identification for six different data sets have been performed in an attempt to
find parameters which could accurately describe the temperature response as a function
Master of Science Thesis
T.J. Koomen
20
Shape Memory Alloy
of the current I and the valve aperture F . Unfortunately they have not been successful
in simulating the erratic temperature fluctuations.
Figure 2-11: Temperature response of a single SMA actuator measured at two points
(solid lines; T1 and T2 ) and the simulated minimum temperature response (dotted line) for
a constant current of 1.5 amperes being switched on and off.
Another interesting observation from Figure 2-11 is that the temperature for the second
heating process is slightly higher on average. This is due to the slow heating of the SMA
actuator housing. This heating also complicates modeling of the SMA-based actuator
and could serve as a motivation to use model free control strategies. However, when
the specific heat capacity cp is correctly identified the temperature models are able to
provide an accurate estimate for the minimum temperature of the SMA actuator. The
presence of reverse transformation in the SMA material can then be ensured if that
minimum temperature exceeds As (σ0 ).
2-5
Summary
SMA actuators have some very promising advantages such as the high power to weight
ratio. Unfortunately they also suffer from a few disadvantages, such as the hysteresis
in the SME cycle. However, if these disadvantages can be overcome, the SMA actuator
will be very useful for many applications such as smart rotors for wind turbines. Their
T.J. Koomen
Master of Science Thesis
2-5 Summary
21
small size and the ability to be shaped into almost any possible shape makes them very
versatile.
The modeling methods for the SMAs that were discussed seemed to be able to yield
useful information for control design purposes. Unfortunately since the temperatures
of the SMA actuators in the SMA-based actuator are unknown the models lose their
usefulness. The temperature model can still provide a minimum equilibrium temperature
for a certain current and valve aperture, which is going to be an important piece of
information for solving the hysteresis challenge in Chapter 3. The stress-temperature
relation (2-13) from the mechanical modeling approach plays a vital role in attaining
a constant stress in the SMA actuators in order to keep the SMA-based actuator at a
fixed deflection.
Control strategies have to be pursued that do not employ models that require knowledge
on the exact temperature of the SMA actuators. However, the temperature cannot
simply be dismissed since the temperature governs the behavior of the SMA. Instead of
focusing on an exact temperature, the focus needs to be on a temperature region. To
be more precise it has to focus on the phase transformation temperature regions since
these regions dictate how the SMA actuators respond.
Master of Science Thesis
T.J. Koomen
22
T.J. Koomen
Shape Memory Alloy
Master of Science Thesis
Chapter 3
Control
This chapter investigates the challenges that are generally faced when designing controllers for Shape Memory Alloy (SMA)-based actuators. After these problems have
been discussed, the requirements for a general control framework for SMA-based actuators are established. A short review is then be presented on state-of-the-art control
strategies that have been applied to similar actuators. Then the proposal for the new
general control framework is introduced. Section 3-5 addresses the control principles
that are used in the new control framework and show how they have been established.
This chapter is concluded with a summary.
3-1
Problems with SMA control
This section discusses the problems that are generally encountered when working with
an SMA actuator. Four challenges are faced with the use of SMA-based actuators:
• Low bandwidth
• Hysteresis
• Non-linear response
• Multi Input, Single Output (MISO) system
The last mentioned challenge is an important factor for the general control framework
since the extra inputs are required to increase the bandwidth of the actuator [21]. In
the next four paragraphs all challenges of the SMA-(based) actuators are presented.
Master of Science Thesis
T.J. Koomen
24
Control
Low Bandwidth One of the disadvantagess of the SMA actuator is that it has a low
bandwidth. This means that the response of the actuator is slow and it will not be
able to accurately track a reference that changes rapidly. The actuator behaves like an
overdamped system, i.e. a system with a damping coefficient ζ greater than 1. The Bode
plot in Figure 3-1 depicts the frequency response of the SMA actuator. In literature it
was found that many actuators, of which only one uses active cooling, have a bandwidth
which does not exceed 1 Hz [17, 4, 22, 23, 24]. For many applications this bandwidth is
not sufficient and therefore, increasing the bandwidth will need to be addressed by the
control framework. The desired actuation bandwidth for wind turbine applications of
the SMA-based actuator would be 12 Hz [2]. This frequency would be required if the
actuator needs to compensate for all of the structural modes of the wind turbine. If only
the first mode has to be addressed, 2 Hz will suffice [2].
Figure 3-1: Bode plot of SMA actuator
Hysteresis Another disadvantage of the SMA-based actuator is that its behavior is hysteretic. For the actuator to respond, it requires a certain threshold value to be exceeded
on the input. This is true for the reverse transformation. The forward transformation
can occur as long as the temperature is below Ms , so even when all inputs are equal to
zero. The required threshold value can be related to the phase transformation temperatures (2-4). The exact threshold value also depends on the direction of the process, so in
the case of an SMA actuator it is different for the forward and reverse transformation.
This results in a hysteresis loop, which is visible in the graph in Figure 2-6, where the
actuator barely responds until a certain temperature is reached. Ideally, an actuator
responds as soon as an input is activated.
Non-Linearity The actuator also has a non-linear response, which is depicted in Figure 3-2. This non-linear response could be accredited to the Joule heating term Q̇in
which is non-linear in the current Ii or the resistance of the wire which is non-linear
in the temperature. The graph in Figure 3-3 shows the response of the temperature,
measured at two positions, to a constant current being switched on an off. It is clear to
see that the temperature shows some erratic fluctuations once it approaches the steady
state, or equilibrium, temperature. This could be caused by numerous reasons; heating
of the actuator encasing, accuracy of the sensor, phase transformations in the SMA etc.
It is logical to assume that, if the temperature is not constant the phase transformation
T.J. Koomen
Master of Science Thesis
3-1 Problems with SMA control
25
Figure 3-2: SMA-based actuator measured deflection rate per amplitude of the input
current. This deflection rate was experimentally determined by applying a constant current
for a duration of 1 second at 0.1 ampere intervals.
Figure 3-3: Temperature response of an SMA actuator when a constant current of 1.5
amperes is switched on and off at intervals. The temperature response was recorded with
an infrared thermal imaging camera [25].
rate is not constant and that the deflection rate of the actuator is therefore not constant.
This would mean that the response is non-linear and could even be quite erratic. An
approach which linearizes the system as much as possible is required.
Multi Input, Single Output System Two cooling inputs have been added to the SMAbased actuator in order to maximize the bandwidth of the actuator. The presence of four
inputs with only one output, rendering the system a MISO system, not only complicates
control but also modeling of the actuator. A relative gain analysis can be performed
for each subsystem to determine the influence of each input on the response of the total
system. Finally, an approach will be required which translates the single available output
of the system into the four inputs for the system. This approach may not destabilize
the actuator and also not interfere with any of the compensation methods that are used
Master of Science Thesis
T.J. Koomen
26
Control
for the other challenges that have been discussed.
3-2
Requirements for the General Control Framework
The goal for the general control framework is to linearize the error e(t) to the output
x(t) in a feedback configuration as depicted in Figure 3-4. The control framework will
also need to solve the problems as described in the previous section. The objectives for
the control framework can be defined as:
• Control both heating and cooling for two wires using only one measured output
(MISO system)
• Compensate for hysteresis
• Compensate for non-linear response
• Maximize bandwidth
Additionally, two conditions are set to ensure a safe operation of the actuator and to
enable the modular actuator to be extended easily. These conditions are:
• Constrain power output to prevent wire or nylon encasing from melting
• Limit computational complexity of control framework

+
−
r (t)

I1
 I2 


 F1 
F2
e(t)
Linearizer
SMA-actuator
x(t)
Figure 3-4: Linearized SMA-based actuator in feedback configuration
3-3
State-of-the-art Control
A review on state of the art controllers that have contributed to the development of
this general control framework was performed during the literature survey [5]. The
state-of-the-art include several types of control strategies such as:
•
•
•
•
Pulse Width Modulation (PWM)
Proportional, Integral and Derivative (PID) Control
Iterative Learning Controller (ILC)
Adaptive Control
T.J. Koomen
Master of Science Thesis
3-4 Phase Transformation Approach
27
• Fuzzy Logic Control
• Sliding Mode Control (SMC)
A lot of the control strategies employ models, or an approximation of the model in the
form of a trained Neural Networks (NN) [26]. Due to the complexity of the modeling
methods that have been discussed in Chapter 2, most of those control strategies have
to be eliminated as candidates for a general control framework. The control strategies
seem to use methods which are not based on the fundamental properties of the SMA.
To establish a general control framework which is valid for all SMA-based actuators this
is a strict requirement.
Furthermore, in the literature that has been reviewed only one control strategy was
found that actively controlled the cooling [4] This control strategy uses a fuzzy logic
controller to calculate a proportional gain for the four inputs of the system. It achieves
a bandwidth of approximately 1 Hz which is higher than the other control strategies
have achieved with a similar SMA.
It is unclear which control strategy is best suitable as a general control framework for a
MISO SMA-based actuator since most of them are not usable for a multi-input systems
or they use very actuator specific models or control strategies.
3-4
Phase Transformation Approach
A difficulty in controlling SMA-based actuators is that they consist of a number of
complex subsystems, as depicted in Figure 3-5. The actuator can be split up into three
subsystems:
G1 : Subsystem that converts the current and airflow into two temperatures
G2 : Subsystems that converts the two temperatures T1 and T2 into stress of the SMA
wires
G3 : Subsystems that converts the stress of the SMA wires into the deflection of the
actuator


I1
 I2 


 F1 
F2
G1
T1
T2
G2
σ1
σ2
G3
x(t)
SMA-based actuator
Figure 3-5: SMA-based actuator represented by three subsystems
For these three subsystems models are available. However, the question is: how accurate
and reliable are these models? The temperature model for the first subsystem G1 was
Master of Science Thesis
T.J. Koomen
28
Control
tested on 6 data sets, but it was unable to predict the erratic temperature fluctuations
as depicted in 3-3. It was only sufficient for providing an estimation of the temperature
of the SMA wires. The temperature of the SMA is very important for determining the
development of stress in the SMA wires. This is because if the temperature is even 1
degrees Celsius below the austenite start temperature As (σ), no phase transformation
will occur. Since the first model was unable to provide an accurate estimate of the
temperature of the SMA, all the models have been deemed insufficient for use in this
control framework. Instead of using these models, the phase transformation approach is
proposed.
The phase transformation approach focuses on triggering the desired phase transformation for a SMA actuator when it is required. This means that the focus is not on an
exact temperature, but on a certain temperature region. These regions are defined as
(3-1). If the control framework controls the inputs as such that the temperature of the
SMA wires are safely within the required regions it can guarantee the corresponding
transformations. The most important condition is that the temperature needs to either
exceed As (σ) or be lower than Ms (σ) if, respectively, the stress of the SMA actuator
needs to increase or decrease.
Region I :
T
Region II : Ms (σ) < T
Region III : As (σ) < T
3-5
< Ms (σ) → Forward transformation of SMA
< As (σ) → No transformation of SMA
→ Reverse transformation of SMA
(3-1)
Control Principles
This section will elaborate on the different principles that are used to implement the
phase transformation approach and how they compensate for the problems of SMAbased actuators as discussed in Section 3-1. Section 3-5-1 will investigate the relative
gain of the four inputs. These four inputs will become a function of the error e(t) and
the reference r(t) as depicted in Figure 3-4. When the four inputs have been paired, the
next two control laws are introduced. These control laws will calculate the values for the
appropriate inputs. They contain a threshold value to compensate for the hysteresis, a
proportional gain and a derivative gain. Since these control laws are not active all the
time, a Feed Forward (FF) control law is devised that ensures that the actuator stays
stable at a certain position once it reaches it and the error becomes zero. The structure
of the general control framework is depicted in Figure 3-6. The hierarchy of the control
elements is defined within the input pairing block.
T.J. Koomen
Master of Science Thesis
3-5 Control Principles
29
Linearizer
r (t)
IF F
FF
e(t)
Threshold
Threshold
+
+ +
Ii
Fj
KP (e)
KP
d
dt
KD
de
dt

I1 (t)
 I2 (t) 


 F1 (t) 
F2 (t)

Input Pairing
Figure 3-6: Schematic showing the inside
of the linearizer of the proposed control frame
work. The blocks KP (e) and KD de
represent
the functions for the current and the valve
dt
aperture as a function of respectively the error e(t) or the derivative of the error de/dt.
3-5-1
Input Pairing
The linearizer block as depicted in Figure 3-4 will determine the required inputs for the
system based on the error e(t). This requires the pairing of the inputs which can be
achieved by considering the gain matrix of the whole system. The gain matrix consists
of entries which indicate the influence of an input to a certain output. An example is
given in Table 3-1. The entries in the gain matrix will not be represented by numbers
but by either a positive sign, a zero or a negative sign since the analysis is done for the
general configuration and that the exact value is not yet important at this point. For the
analysis, the subsystems will be analyzed in the reverse order. The gain matrix analysis
for subsystem G3 and G2 are valid for certain conditions which describe the mechanics
of the nylon polymer beam and the phase transformations of the SMA.
Table 3-1: Example of a gain matrix for a system with two inputs and two outputs
Inputs
u1
Outputs
u2
y1
K11 =
∆y1 ∆u1 u
y2
K21 =
∆y2 ∆u1 u
∆y1 ∆u2 u
K22 =
∆y2 ∆u2 u
2
2
K12 =
1
1
Subsystem G3 Subsystem (3-2) has the inputs σ1 and σ2 and has one output x(t). The
inputs are either greater than or equal to zero since the SMA wires can only subjected
to a tension. The gain matrix analysis yields the results in (3-3), which are derived from
the conditions of the movement of the actuator as defined in (3-4). It is clear that the
two inputs have the opposite effects on the output.
Master of Science Thesis
T.J. Koomen
30
Control
x(t) = G3
σ1
σ2
(3-2)
KG3 = + 
 > 0, {σ1 | σ1 > 0 and σ1 > σ2 }
dx(t) 
= 0, {σ1 , σ2 | σ1 = σ2 }
dt 

< 0, {σ2 | σ2 > 0 and σ2 > σ1 }
(3-3)
(3-4)
Subsystem G2 The subsystem (3-5) represents the phase transformation of the SMA
and the stress in the SMA actuators. It has two inputs, T1 and T2 and has two outputs,
σ1 and σ2 . The gain matrix of subsystem G2 with conditions as defined in (3-7) is defined
as (3-6). This gain matrix is only valid under the assumption that the SMA enclosure
provides sufficient insulation to prevent T1 from influencing T2 and vice versa.
σ1
σ2
T1
T2
= G2
+
0
KG2 =
0
+

 > 0, {Ti | Ti > As (σi )}
dσi (t) 
= 0, {Ti | Ms (σi ) < Ti < As (σ)}
dt 

< 0, {Ti | Ti < Ms (σ)}
(3-5)
(3-6)
(3-7)
Subsystem G1 The subsystem (3-8) has four inputs, I1 I2 F1 F2 and two outputs T1 T2 . The conditions for this system are stated in (3-10). They basically
state that the temperature Ti will change as long as the equilibrium temperature Ti,eq has
not been reached, or that the temperature will increase or decrease when, respectively,
Ii or Fi are increased.


I1
 I2 
T1

= G1 
 F1 
T2
F2
+ 0 KG1 =
0 + 0

 > 0, {Ii , Fi
dTi (t) 
= 0, {Ii , Fi
dt 

< 0, {Ii , Fi
T.J. Koomen
(3-8)
0
-
(3-9)
dIi
dt > 0}
i dFi
and dI
dt , dt
i
or dF
dt > 0}
| Ti < Ti,eq (Ii , Fi ) or
| Ti = Ti,eq (Ii , Fi )
| Ti > Ti,eq (Ii , Fi )
= 0}
(3-10)
Master of Science Thesis
3-5 Control Principles
31
Linearizer subsystem The linearizer subsystem, as introduced in Figure 3-4, will transform the error e(t) into the correct inputs for the SMA-based actuator. This requires
a pairing of the four inputs and representing them as a function of e(t). Since the subsystems are subject to some discontinuous conditions, the resulting pairing function will
also be discontinuous. The gain matrix for the complete system as defined by (3-11) is
evaluated, resulting in the gain matrix (3-12). The gain matrix clearly indicates that
there are input pairs; the pair I1 and F2 which have a positive gain and the pair I2 and
F1 which have a negative gain.

I1
 I2 

x(t) = G3 G2 G1 
 F1 
F2

KG1 ,G2 ,G3 = KG3 KG2 KG1 =
(3-11)
+ - - +
(3-12)
Switching between the two input pairs can be achieved by looking at the sign of the
error e(t). If the error is positive the inputs I1 and F2 should be used and if the error
is negative then the inputs I2 and F1 should be used. The values that have to be used
either for Ii or Fj is determined in the subsections.
To prevent the control framework from switching at a very fast rate between the two
inputs, a boundary eboundary is defined. This will ensure that the controller only switches
from the input pair when the absolute value of the error exceeds the error boundary value
and if the sign of the error changes.
3-5-2
Response Threshold
The next problem of the SMA-based actuator that is addressed is the response threshold.
The problem with most of the control strategies used for SMA-based actuator is that
they do not consider the fact that for the SMA-actuator to respond either condition
(3-13) or condition (3-14) must be met, depending on which transformation of the SMA
is required. If these conditions are met, the controller can ensure that the SMA actuator
responds.
Ti > As (σ0 ),
(Reverse transformation)
(3-13)
Ti < Ms (σ0 ), (Forward transformation)
(3-14)
The condition can be extended to calculate the corresponding values for the current
(3-15) and the valve aperture (3-16). The equation for the threshold value for the current,
Ithreshold , is obtained by rewriting (2-18). It is a function of the desired temperature
Master of Science Thesis
T.J. Koomen
32
Control
and an airflow of F = 0 and the reference temperature needs to comply to Ti,ref ≥
As (σ0 ). Determining the threshold for the valve aperture is a more difficult case since
temperature is unknown. The threshold value for the valves needs to be chosen such that
they result in a quick drop of the temperature of the SMA to a temperature below the
martensitic starting temperature. Since the phase transformation state of the actuator,
i.e. in what temperature region it is, is unknown there is no definitive value to be
determined. The only knowledge available is that the temperature of the SMA actuator
is within one of the three temperature regions. However, since the requirement states
that the temperature must be lower than Ms (σi ), any arbitrary value for Fthreshold can
be set. Any value larger than zero ensures that the SMA actuator starts to cool down
and that the stress is minimized. This value can also be coupled to a minimal value for
the airflow to start to flow through the valve, but this depends on the hardware that is
used. The threshold values are the minimum value for the current and valve aperture
when control is active, i.e. |e(t)| ≥ eboundary .
s
Ii ≥ Ithreshold =
mcp (Ti,ref − T∞ ) + ASMA h(F ) (Ti,ref − T∞ )
R (Ti,ref )
Fj ≥ Fthreshold ≥ Fvalve,min
3-5-3
(3-15)
(3-16)
Linearization of SMA response
Now that the hysteresis effect has been compensated for, the next step is to linearize
the inputs to the change in temperature. The control inputs influence the change in
temperature in the terms Q̇in (2-15) and Q̇out (2-16). The term Q̇in is quadratic in Ii .
To linearize this term, the square root of the absolute value of the error must be used
for the calculation of the proportional gain KP ,I as in (3-17). This should result in an
equilibrium temperature that is proportional to the current Ii (3-18).
Ii,KP = KP ,I
p
|e(t)|
Ti,eq ∝ Ii
(3-17)
(3-18)
The term Q̇out is slightly non-linear in Fi since the heat transfer coefficient h(Fi ) is
non-linear in Fi as can be seen in Figure 3-7. However, in the range of the control
variable Fi the non-linearity is not apparent, hence a linear approximation is used. Any
proportional gain for valve aperture is therefore calculated by just using the absolute
value of the error (3-19).
Fj,KP = KP ,F |e(t)|
T.J. Koomen
(3-19)
Master of Science Thesis
3-5 Control Principles
33
Figure 3-7: Heat transfer coefficient as a function of the valve aperture Fi over its full
input range of <0,1>, obtained from experimental data [4].
This linearization is valid under the assumption that the phase transformation process
is linear in the power input and output to the SMA actuators, i.e. the terms Q̇in and
Q̇out . Experiments have to determine whether that assumption is correct.
3-5-4
Bandwidth Optimization
The bandwidth of an SMA actuator is generally low, somewhere in the order of 0.1 to
0.3 Hz. If these actuators are to be used in applications such as wind turbines, the
bandwidth needs to be increased. Increasing the bandwidth of the SMA-based actuator
is achieved with two complementary control strategies.
1. Pole/zero cancellation
The Bode plot in 3-1 of the SMA actuator indicates the presence of at least one slow
pole at approximately 0.8 rad/s. By feeding the error signal through a filter that
has a zero at that frequency, the slow pole is canceled. Since applying a single zero
would mean that the gain in that filter would be unlimited for high frequencies,
another pole needs to be added. Since noise is usually present at higher frequencies
in real systems (noise from sensors etc.), an additional pole is added at a higher
frequency. This extra pole will reduce the gain for high frequencies and thus filter
out the effects of the noise on the system. The resulting filter is defined as (3-20)
with an example Bode plot as depicted in Figure 3-8.
H1 (s) =
1/z1 s
+1
(1/p1 s + 1)(1/p2 s + 1)
(3-20)
The effects of the filter can be demonstrated by applying them to the temperature
model of the SMA actuator. The filter increases the initial input which results
Master of Science Thesis
T.J. Koomen
34
Control
Figure 3-8: Filter used to increase bandwidth and filter out high frequency noise
Figure 3-9: Simulation of the effects of the filter on the evolution of the temperature of a
SMA actuator
in a faster increasing temperature, as can be seen in Figure 3-9. The equilibrium
temperature is reached faster and since the phase transformation rate and therefore
also the actuator response rate is relative to this temperature, the bandwidth of
the actuator increases.
2. Derivative control
To further increase the bandwidth and also the performance of the actuator in
terms of reference tracking capabilities, a derivative control element is added to the
linearizer control law as depicted in 3-4. This derivative control element increases
or decreases the gain according to the magnitude and sign of the derivative. The
resulting derivative control laws are defined as (3-21) and (3-22).
T.J. Koomen
Master of Science Thesis
3-5 Control Principles
35
Ii,KD
Fj,KD
3-5-5
de
de
= sign
(t) KD,I | (t)|
dt
dt
de
de
= sign
(t) KD,F | (t)|
dt
dt
(3-21)
(3-22)
Feed Forward Stabilizer
Depending on the application of the SMA-based actuator, it may be desirable to attain
a certain position x with the actuator, where x 6= x0 . Keeping the actuator at fixed
deflection means that after the actuator reaches that position, the stresses σ1 and σ2
must be kept constant. It also means that the strain in the SMA actuators is constant,
and thus that the derivative of the strain ε̇(t) is equal to zero. This permits the use
(2-13) to compose a FF stabilizer strategy. First, the term ∆x is defined as the offset of
x(t) to the neutral axis of the SMA-based actuator (3-23).
∆x = x(t) − x0
(3-23)
The stabilizer feed forward current is determined as (3-24), where IFF is a constant and
KFF,I is a proportional gain on the absolute value of the offset of x(t) to the neutral
axis of the SMA-based actuator. The term IFF is related to the hysteresis problem.
It ensures a certain temperature where the stress will be greater than σ0 . The second
term, KFF,I will be proportional to the term c in (2-13). This proportionally increases
the stress σi as ∆x increases.
Istabilizer = IFF + KFF,I |∆x|
(3-24)
The FF stabilizer is only active when the absolute value of the error is within the error
boundary, i.e. |e(t)| ≤ eboundary . The sign of ∆x determines which of the two inputs,
I1 or I2 , is used. The current Istabilizer is smaller than the current that will be used for
control when the error exceeds the error boundary.
Control of the airflow will only be used by the FF stabilizer when the reference equals
the neutral position x. Then both valves are opened allowing for cooling of both SMA
actuators. This will results in the stresses σ1 and σ2 returning to σ0 . When the reference
is not equal to x0 , the control of the airflow will be performed by the other control
strategies as discussed in this subsection. By switching from the normal control laws to
only the FF control law, the energy input to the system already decreases. Since the
energy input decreases, the temperature should decrease. This already ensures that no
further phase transformation occurs and that the stress in the SMA actuator remains
constant.
Master of Science Thesis
T.J. Koomen
36
Control
3-6
Summary
The governing control law is obtained by combining the separate control elements as
described in the Section 3-5, rendering them to (3-25) and (3-26). The minimum values
are defined as (3-27) and (3-28). The threshold value and proportional gain are switched
by the condition |e(t)| ≥ Ithreshold , but the derivative gain is not switched by that
condition. This means that the derivative gain is active when the error is within the
error boundary. This results in a damping effect of the actuation when the actuator has
reached its reference position where e(t) = 0. The stabilizer current and airflow will also
be active when the error is within the error boundary, where the current and the airflow
are a function of ∆x.
Ii = Ithreshold + Ii,KP + Ii,KD
(3-25)
Fj = Fthreshold + Fi,KP + Fi,KD
(3-26)
min(Ii ) =
min(Fj ) =
Istabilizer , if: |e(t)| < eboundary
Ithreshold , if: |e(t)| ≥ eboundary
(3-27)
Fstabilizer , if: |e(t)| < eboundary
Fthreshold , if: |e(t)| ≥ eboundary
(3-28)
All of these control laws generate the control inputs as depicted in Figure 3-10. The
gains that are used for these surfaces are not definitive values, they can be adjusted to
decrease or increase aggressiveness of the controller. The steepness of the surface in the
direction of the error can be increased by increasing KP ,I or KP ,F . In the same way, the
steepness of the surface in the direction of the derivative of the error can be increased by
increasing KD,I or KD,F . To increase the overall level of the entire surface, the threshold
value can be increased. This might be necessary when the tension in the SMA actuators
rise.
When the union of the control surfaces in Figure 3-11 is considered, a point with a
minimum controller output can clearly be observed. This is the center point where both
the error and its derivative are smaller than the error boundary value. The union of the
control surfaces seems to mimic a surface that describes the energy for a system with a
global energy minimum.
T.J. Koomen
Master of Science Thesis
3-6 Summary
37
Figure 3-10: Control framework output values for the four inputs as function of the error
and its derivative
Master of Science Thesis
T.J. Koomen
38
Control
Figure 3-11: Union of the control inputs for heating and cooling as function of the error
and its derivative
T.J. Koomen
Master of Science Thesis
Part II
Practice
Master of Science Thesis
T.J. Koomen
Chapter 4
Controller Implementation
This chapter will investigate the validity of the theories and the performance of the proposed general control framework as discussed in Chapter 3. It starts with an introduction
to the Shape Memory Alloy (SMA)-based actuator and the rest of the hardware that
was used for the experiments. The first experiments investigate the effectiveness and
reliability of the individual control strategies for the challenges for control of SMA-based
actuators, which were:
• Hysteresis
• Non-linear response
• Multi Input, Single Output (MISO) system
• Low bandwidth
4-1
Hardware
The SMA-based actuator that was used is depicted in Figure 4-1. It has a common
ground connection terminal for the two SMA actuators and two separate positive connection terminals. Hollow bolts were used to connect the air hoses from the proportional
valves. The valves have their own Pulse Width Modulation (PWM) signal generator and
power supply. The current for the Joule heating of the SMA actuator was supplied by
two controllable power supplies. The current that was supplied was controlled via an
analog signal. Both the power supplies and the PWM signal generators were controlled
using an Arduino board.
The schematic in Figure 4-2 illustrates how the hardware was configured. This section
shortly elaborates on the hardware that was used for the experiments. More details of
the hardware can be found in Appendix A.
Master of Science Thesis
T.J. Koomen
42
Controller Implementation
Figure 4-1: Picture of the actuator with the connections indicated. The base indicates
where the actuator is clamped into the setup and the tip of the actuator is where the
deflection is measured.
Figure 4-2: Illustration depicting the total hardware setup and connections.
T.J. Koomen
Master of Science Thesis
4-1 Hardware
43
I/O Interface
An Arduino Mega 2560 board had been selected to facilitate the input output interface
and to run the controller algorithm. Motivation for the use of this board is that it
is cheap, readily available and offers a good possibility for easy scaling for use with
multiple modules. It was interesting to investigate the functionality and reliability of
these reasonably new and upcoming low-cost systems.
All experiments were performed with a sampling rate of 50 Hz, allowing for identification
of dynamics of the system up to a frequency of 25 Hz according to Shannon’s sampling
theorem [27]. The Arduino was connected to a computer running Matlab and Simulink.
The computer itself was running Windows, which has a default USB sampling rate of
125 Hz, which is sufficient for the required sampling rate for the experiments of 50 Hz.
Arduino boards do not have an actual analog output, but they simulate analog outputs
by using Pulse Width Modulation (PWM). Since the goal was to keep the currents as
stable as possible, pin 4 and 13 are used for controlling the power supplies given that
these pins run at a higher PWM frequency of 980 Hz instead of the standard 490 Hz. A
higher PWM frequency will result in a smoother analog signal. The PWM signals are
then fed through an analog low pass filter before they reach the power supplies.
The Arduino was programmed using Matlab and Simulink, which compiles the code to
C or C++ and uploads it to the Arduino. The Arduino then runs the algorithm and
sends data in real-time to Matlab for analysis. Initiating the compilation of the code to
run externally on the Arduino could only be done manually, which drastically increased
the time required to perform the experiments. An average experiment of 30 seconds
required almost 90 seconds from compilation to the saving of the data.
Power Supply
The power supplies that were used are from Elektro Automatik (Type EA PS-803220T). These power supplies were configured as current sources, with a range from zero
to a maximum of 20 Amperes. The current was controlled by applying a voltage on the
analog input on the power supply.
Airflow Control
Control of the airflow through the channels in the SMA-based actuator was provided by
valve controllers (Type 8605) and proportional valves (Type 2821) from Burkert. The
PWM signal generators are required since the valves operate at 24 Volts and the Arduino
board can only supply a 5 Volts PWM signal. The valve controllers from Burkert serve
as an amplifier of the PWM signal from the Arduino.
Master of Science Thesis
T.J. Koomen
44
Controller Implementation

I1
 I2 


 F1 
F2

e(t)
Linearizer
SMA-actuator
x(t)
Figure 4-3: Linearized SMA-based actuator in feedback configuration
Displacement Sensor
The sensor that was used to measure the deflection of the SMA-based actuator was the
Micro Epsilon optoNCDT 1401. It is a laser position sensor with a range of 20 millimeters
and is configured such that the center position of the measurement range corresponds to
the neutral position x0 of the actuator. The maximum observable displacement in both
negative and positive x direction was 10 millimeter.
4-2
Control Principles
In Section 3-1 some of the difficulties with control of SMA actuators were discussed. In
this section, the effectiveness of the individual control laws as proposed in Section 3-5
are investigated before testing the Phase Transformation Approach as a whole.
4-2-1
Hysteresis
The hysteresis for the actuator has already been shown in Figure 3-2. It shows that
the actuator does not (or barely) respond for currents lower than approximately 1.2
Amperes. For currents that exceed this value the actuator does deflect.
An experiment was carried out where different step inputs were directly put on the input
of the linearizer (e(t)). The controller parameters that were used for this experiment
are listed in Table 4-1. The results are shown in Figure 4-4. It can be observed that the
actuator responds for any input that is larger than zero. This means that the hysteresis
of the SMA actuators is no longer a problem.
Table 4-1: Controller parameters used for testing solutions for the hysteretic and non-linear
response.
Ii
Fi
T.J. Koomen
Threshold
1.25
0.5
KP
8
0.5
KD
1
0
Master of Science Thesis
4-2 Control Principles
45
Figure 4-4: Deflection rate of the SMA-based actuator to different step inputs on the input
of the linearizer. The error e(t) correspond to the voltage difference that is measured by
the laser, where 0.1 Volts equals 0.5 millimeters. The input range on this graph therefore
corresponds to an error of 0 to 2 millimeters.
The controller with the threshold value is compared to a similarly structured controller
without the threshold value, which is a simple Proportional and Derivative (PD) controller. The performance of the simple PD controller is shown in Figure 4-5. It is clear
from this figure that there is a delay in the response for the PD controller. The PD
controller with the threshold values has a better response, though it is very sensitive
to the proportional control parameter. If the proportional gain is too high, overshoot
occurs when the reference signal changes direction (small peak at 8.2 seconds).
Figure 4-5: Tracking of sinusoidal signal at 1 Hz for PD controller with and without
threshold value for SMA-based actuator. Both controllers use the same structure except for
the threshold value.
For some threshold values the controller shows good performance, but when the frequency or amplitude is increased or if the threshold value is slightly increased, the
performance drastically decreases. This effect can be observed in 4-6, where the threshMaster of Science Thesis
T.J. Koomen
46
Controller Implementation
Figure 4-6: Graph depicting effect of increase the threshold value for the current Ii by 0.1
amperes. By increasing the threshold value (dash-dotted line), the controller becomes too
aggressive causing the actuator to overshoot at the point where the reference signal peaks.
old value for the current is slightly increased by 0.1 amperes. This causes the actuator
to overshoot at the point where the reference signal peaks. If the threshold value is
further increased or if the frequency or amplitude of the reference signal is increased,
the overshoot also increases.
4-2-2
Linearization
From Figure 3-2 it can be observed that once the input exceeds the threshold value that
the increase of the actuator deflection rate is not linearly proportional to the increase in
the current. To see if the response of the actuator is now linear in the input, Figure 4-4
needs to be examined. For the first part, for an input up till 0.3 it can be observed
that the response is linear. Though from inputs greater than 0.3 the actuator response
increases non-linear. The proposed control law for the linearization does not achieve
a linear response over the possible input range, which is even larger than the depicted
input range in Figure 4-4. The linear regime can defined as inputs from zero up until
0.3. Beyond this point the response seems to be linear again, but not with the same
gradient. Inputs larger than 0.4 were not tested since they saturated the output within
less than half a second.
4-2-3
Bandwidth Optimization
The optimization of the bandwidth was achieved by means of two control principles; a
bandwidth filter and a gain on the derivative of the error. The effects of the bandwidth
filter are first discussed, which affects the whole Single Input, Single Output (SISO)
system of e(t) to x(t). The derivative gain applies to the subsystems, of which the
effects are hard to identify.
T.J. Koomen
Master of Science Thesis
4-2 Control Principles
47
Bandwidth Increasing Filter
To identify the frequency response of the SMA-based actuator with the proposed control
framework, an experiment was designed using the control structure as in 4-3. The system
is tested with and without the filter that was designed to increase the bandwidth of the
actuator. For the data collection for the identification of the dynamics of the set-up, a
Random Binary Signal (RBS) is chosen as an the input signal u(t) (4-1), where w(t) is
stochastic white noise and d is the amplitude of the RBS.
u(t) = d · sign [w(t)]
(4-1)
This signal was selected since it has the following advantages for this system:
• u(t) covers the wide frequency spectrum of a white noise. For a frequency response
identification it is desired that the obtained output data contains information on
the system over a sufficiently wide frequency range. In order to obtain data that
complies to that requirement, the input signals have to be sufficiently exciting, and
thus have a spectrum that covers a wide frequency range.
• u(t) is bounded in amplitude. This is favourable since it can therefore be set to
only excite the linear dynamics. In 4-2-2 a linear regime of the system was found.
• u(t) is binary. This is desirable since the power will therefore be maximal under
an amplitude bound, and thus the signal-to-noise ratio will be high at the output.
The actual frequency response was obtained by means of an Empirical Transfer Function
Estimate (ETFE), defined as the expression (4-2). It represents the frequency response
and is equal to the quotient of the Fourier transforms of the input signal u(t) and the
output signal y(t).
ĞN (eiω ) :=
YN (ω)
UN (ω)
(4-2)
• Without filter
The ETFE yields the frequency response as depicted in 4-7a. The dots represent
the raw data obtained from the experiment and the solid line represents the resampled and smoothed data. The frequency response shows a gain margin of 3.67
dB (19.89 Hz) and a phase margin of 96.7 degrees (2.46 Hz). This means that at
a frequency of 2.46 Hz, the response suffers from a delay of approximately 0.11
seconds. This suggests that the actuator is stable and that the bandwidth has
been increased to 2.46 Hz. However, this is only valid for inputs that are within
the linear regime as defined in Section 4-2-2.
Master of Science Thesis
T.J. Koomen
48
Controller Implementation
(a) Without bandwidth filter
(b) With bandwidth filter
Figure 4-7: ETFE obtained frequency response of SMA-based actuator with proposed
control framework without the bandwidth filter. The dots represent the ETFE for the raw
and unsampled data and the solid line represents ETFE for the resampled and smoothed
data.
T.J. Koomen
Master of Science Thesis
4-2 Control Principles
49
Figure 4-8: ETFE frequency response obtained from experiments.
• With filter
The ETFE for the system with bandwidth filter is depicted in 4-7b. A comparison
between the two identified systems is shown in the bode plot in Figure 4-8. It
shows that the bandwidth is increased, from approximately 2.46 Hz to 2.83 Hz.
Derivative Gain
The effects of the derivative gain are hard to represent in a frequency response. This
happens because the gains apply to the subsystem G2 where the inputs I1 , I2 , F1 and F2
yield the output T1 and T2 . Since the temperatures are unavailable, no direct frequency
response identification can be performed and therefore the effects on the bandwidth are
not discussed. However, the effects on the performance can be observed from some of
the experiments.
The derivative gain stabilizes the response of the actuator at lower frequencies (f ≤ 1
Hz), but it causes an increase of the delay at higher frequencies. In Figure 4-9, the effects
of increasing the derivative gain for the current Ii can be observed. For a derivative gain
of KD,I = 0.25 the overshoot is clearly present, but for when this gain is increased to
0.5 the overshoot is significantly reduced.
4-2-4
Feed Forward Stabilizer
The graphs in Figure 4-11 shows the experiments that have been performed to identify
the stabilizing currents. The experiment consisted of an initial step input which was
followed by different values for the stabilizing current. The initial step input drives
the output of the actuator to a certain deflection. Then the stabilizer current replaces
the initial step input and the output of the actuator can remain constant, which is the
Master of Science Thesis
T.J. Koomen
50
Controller Implementation
Figure 4-9: Graph depicting the effects of increasing the derivative gain on the error for
the current Ii . An increase of the derivative gain from 0.25 to 0.5 significantly reduces the
overshoot.
Figure 4-10: Example of the input for the stabilizer current testing. For the time between
1 and 3 seconds the initial step input will drive the output of the system to a certain state.
The function of the stabilizer current is to keep the output of the system constant at that
state.
desired response, or it can decrease or increase. The response for two different initial
currents is depicted in Figure 4-11.
From Figure 4-11 it is clear to see that for certain stabilizing current, the actuator
deflection decreases or increases for t > 3 seconds. For a number of stabilizing currents,
the actuator attains the same deflection, though be it with minor fluctuations. For
an initial step input of 1.7 amperes (Figure 4-11a), only the stabilizing current of 1.10
amperes is able to attain a relative constant deflection. For an initial step input of 1.8
amperes (Figure 4-11a), the stabilizing currents 1.15 and 1.20 amperes are both able to
maintain a relative constant deflection. The experiment was also done for initial step
inputs of 1.4, 1.5 and 1.6 amperes. These experiments combined yielded estimates for
the stabilizer current gains IFF and KFF,I .
These results seem to indicate that the stress-temperature relation (2-13) for SMA actuators is valid for the SMA-based actuator. This because the deflection rate is equal
to zero, the strain rate is also zero. The reason why the deflection is not perfectly constant is assumed to be caused by the erratic temperature behavior. Unfortunately this
assumption could not be confirmed since the temperature data was unavailable.
An interesting thing to notice is that deflection of the actuator at t = 3 seconds is not
always the same for an identical initial step input (Figure 4-11b). This inconsistency
T.J. Koomen
Master of Science Thesis
4-3 Control Framework Performance
(a) I1 (1 ≤ t ≤ 3) = 1.7 A
51
(b) I1 (1 ≤ t ≤ 3) = 1.8 A
Figure 4-11: Actuator response for two different initial currents and different stabilizer
currents. The initial step input for 1 ≤ t ≤ 3 seconds was (a) 1.7 and (b) 1.8 amperes, for
t > 3 seconds the input was equal to the value of the stabilizer currents as defined in the
legends of the graphs.
proves to be quite a problem for the tuning of the control parameters. The cause of
the inconsistency is thought to be different initial conditions, such as a higher initial
temperature from residual heat from previous experiments. The thermal history of the
SMA proves to be of significant importance for future experiments. A great effort was
made to prevent the lingering of residual heat by venting both valves of the actuator
between experiments to bring the temperature of the actuator back to room temperature.
4-3
Control Framework Performance
The results from the individual control principles have all been discussed but an overall
performance has yet to be determined. The performance of the general control framework
on the SMA-based actuator is based on an analysis on the tracking performance of
a single and dual sinusoidal reference over a relevant frequency range and for several
amplitudes. Once the performance has been discussed, a short review on the control
parameters selection procedure is performed.
4-3-1
Periodical Reference Signal
Several hundred experiments were performed were the actuator was set to follow the
reference signal (4-3), where A is the amplitude and B is the frequency in Hz. Figure 412 show the response to four different frequencies at the same amplitude. The tracking
performance up to a frequency of 1.5 Hz is good, though for increasing frequency the
delay slightly increases. The median of the delay for frequencies below 1 Hz is 4 samples,
which is equal to a delay of 0.08 seconds. It is uncertain if the delay isn’t a result of the
limited computational power of the Arduino board, since there was already a problem
Master of Science Thesis
T.J. Koomen
52
Controller Implementation
with missing samples. The phase delay for the different responses shown in Figure 4-12
are (a) 20 degrees, (b) 28 degrees, (c) 65 degrees and (d) 74 degrees.
ri (t) = A sin(2πBt)
(4-3)
Inconsistency of the Response
The SMA-based actuator showed an inconsistent response for experiments with identical
parameterization and initial conditions. An example of this phenomena is shown in Figure 4-13. Experiment 1 shows a decent tracking performance with a smooth response.
The second and third experiment both have some large fluctuations. The cause of the
fluctuations for experiment 2 and experiment 3 is unknown. The experiments were executed with a few minutes between them and the valves were vented between experiments
to remove any residual heat.
This phenomenon indicates a great challenge in controlling SMA-based actuators, which
is the erratic response. This erratic response is thought to be caused by both the erratic
temperature response and the non-linear relation between the temperature and the phase
transformation rate. The erratic response to a constant current is depicted in the graph
in Figure 4-14. It is clear to see that the response is not constant. This could be caused
by the erratic temperature response, an unstable current from the power supplies or a
non-linear relation between the temperature and the phase transformation rate.
From Figure 4-15 is can be observed that both increasing the frequency as increasing
the amplitude of the reference signal increases the mean squared error. Both these
parameters influence the required deflection rate. Their influence is expected to be equal.
However this is not the case. The graph shows that the influence of the amplitude on
the performance is much more dominant than the frequency. At a frequency of 1.2 Hz,
the mean squared error for an amplitude of 3.0 millimeters is almost 7 times larger than
for an amplitude of 1.0 millimeters. Even the difference between the performance for an
amplitude of 2.5 and 3.0 millimeters is a factor 2.
In theory, the required deflection rate for a reference signal with a frequency of 0.5 Hz
and an amplitude of 2 millimeter is equal to that for a reference signal with a frequency
of 1 Hz and an amplitude of 1 millimeter. The only difference that remains between
the required deflection rate is the speed at which they change, that is, the derivative
of the deflection rate. Proof for this claim is mathematically expressed by (4-4) and
(4-5), where the amplitudes of the two derivatives are equal to each other. This theory
suggests that the mean squared error for a reference signal r1 (t) = sin(2πt) should be
equal to that of r2 (t) = 2 sin(πt), but this is not the case. This leads to the conclusion
that the change in deflection rate is not the bottleneck, but that the deflection rate
itself is hard to control. This is because the error is significantly larger for reference
signals with large amplitudes than reference signals with a low amplitudes (≤ 2 mm) at
high frequencies (f ≥ 1 Hz). The increase of the amplitude of the input signal causes
T.J. Koomen
Master of Science Thesis
4-3 Control Framework Performance
53
(a) f1 = 0.7 Hz, average delay of 4 samples, td = 0.08 seconds.
(b) f2 = 1 Hz, average delay of 4 samples, td = 0.08 seconds.
(c) f3 = 1.5 Hz, average delay of 6 samples, td = 0.12 seconds.
(d) f4 = 1.7 Hz, average delay of 6 samples, td = 0.12 seconds.
Figure 4-12: Tracking performance for 4 different frequencies and a 2 millimeter amplitude.
Master of Science Thesis
T.J. Koomen
54
Controller Implementation
Figure 4-13: Response from three experiments with identical parameterization and initial
conditions.
Figure 4-14: Erratic actuator response to a step input with a constant current of I1 = 1.45
ampere which is depicted in the bottom graph.
Figure 4-15: Mean squared error of the actuator response for tracking a sinusoidal reference
signal with different frequencies and amplitudes.
T.J. Koomen
Master of Science Thesis
4-3 Control Framework Performance
55
Figure 4-16: Maximum of the mean squared error that were found during the experiments
for random parameterization.
so much degradation of the tracking performance that it requires a decrease of the gain
values and more damping. This means that the system still has a non-linear response. A
frequency response identification would therefore only be valid for inputs with an equal
amplitude.
d
[A sin(Bt) ] = AB cos(Bt)
dt
d A
sin(2Bπt) = AB cos(2Bt)
dt 2
(4-4)
(4-5)
Figure 4-16 shows the mean squared error for the worst combinations of parameter that
were encountered during the experiments. These combinations had the highest mean
squared error per frequency and amplitude. For frequencies up to 0.5 Hz we can see
that the performance is still reasonable. The error for all amplitudes does not exceed
25 %, except for the peak at 0.2 Hz for an amplitude of 3.0 millimeter. Even for bad
combinations of parameters the control framework still has a reasonable performance.
When the frequency increase this performance really decreases, especially when the
amplitude is also increased. This yet again indicates the difficulty of working with a
system that still has a non-linear response and an inherently low bandwidth.
Dual Sinusoid
A sinusoidal reference signal was constructed using two sinuses with different frequencies
to test the tracking performance for a signal with a average value that is not equal to
the neutral position x0 of the SMA-based actuator. The graph in Figure 4-17 shows the
actuator response for a number of different parametrizations. It shows an interesting
response to the increase of the derivative gain which seems less damped than for the lower
derivative gain (dash-dotted line). By decreasing the proportional gain, the tracking
performance increases again (dotted line). The graph shows that the system response
destabilizes when the proportional gain is selected too high. It causes a fast switching
Master of Science Thesis
T.J. Koomen
56
Controller Implementation
Figure 4-17: Tracking performance of dual sinusoid for different controller parametrizations. Reference signal defined by r(t) = sin(πt) + sin(0.5πt).
and oscillatory response as the controller is unable to maintain a constant deflection.
This is especially true in combination with a poor parametrization of the feed forward
controller. If the feed forward controller is not able to maintain a constant deflection, the
feedback controller is constantly switched on and off. If this controller is overpowered it
means that the sign of the error switches very fast.
4-3-2
Non-periodical reference signal
To fully test the performance of the feed forward control law, the response to a step
input was tested. The response to a step input determines if the control law is able to
maintain a constant deflection once the feedback control law reaches the final reference
deflection position. Figure 4-18 shows the response of three experiments with identical
parametrization of the controllers to a step input with an amplitude of 1.5 millimeters.
The average settling time is approximately 0.5 seconds which is reasonable for SMAbased actuators. The delay from the system is approximately 2 samples, which is equal
to 0.04 seconds. This delay is thought to be caused by the hardware of the setup itself
and not the SMA-based actuator.
The deflection for t ≥ 2 seconds is not very smooth, though the average deviation from
the set-point is less than 10 %. This inability to maintain a constant deflection can be
accredited to two things; either a poorly parametrization of the feed forward controller of
the inability of the controller to maintain a constant temperature in the SMA actuators.
The latter has already been mentioned and provides a good explaination for the already
complicated parametrization of the both the feed forward and feedback control laws. It
is impossible to validate the stress-temperature relation of the SMA since temperature
data is unavailable and a constant constant temperature of the SMA can therefore not
be guaranteed.
T.J. Koomen
Master of Science Thesis
4-3 Control Framework Performance
57
Figure 4-18: Response to a step input as reference for three experiments with identical
control parameters.
4-3-3
Parameterization
The control parameters that were used for testing the performance of the general control
framework were randomly selected within a certain range. Due to the high number of
parameters, there was no possibility to test the performance for each possible combination. Instead, the parameters were intuitively chosen to optimize performance for every
specific frequency and amplitude. The expectation was that a pattern would emerge
from the measured data.
However, no significant pattern for the selection of control parameters emerged from the
measured data. The quality of each parameterization was based on its mean squared
error, where a lower error represents a better combination of parameters. The results
for the best values for the proportional gain KP ,I and for the threshold value for Ii are
shown in Table 4-2 and 4-3. There does seem to be a pattern for the proportional gain
and the threshold value; the proportional gain seems to decrease for higher frequencies
and the threshold value seems to increase for increasing amplitudes. However, since not
all combinations of control parameters have been tested it is impossible to determine a
real pattern. Listings of the controller parameterizations with the lowest and highest
mean squared error can be found in Appendix B.
Bandwidth Filter
During experiments it was also found that the bandwidth filter with one zero and two
poles was not optimal. By removing the second pole at the high frequency the performance increased. The second pole resulted in more delay and thus a decrease in
performance especially for frequencies higher than 1 Hz.
4-3-4
Influence of Active Cooling
An experiment was designed to investigate the influence of the airflow and sudden valve
opening to the excitation of the SMA-based actuator. It was suspected that the sudden
Master of Science Thesis
T.J. Koomen
58
Controller Implementation
Table 4-2: Best parameter values
for the proportional gain KP ,I per
frequency and amplitude.
Frequency
(Hz)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.0
12
18
18
18
18
18
14
12
12
10
8
8
Amplitude
1.5 2.0
12 12
8
8
10
8
10 12
12 10
8
10
10 10
10
8
8
10
10 14
8
8
8
6
(mm)
2.5
12
8
10
12
8
8
10
10
8
14
8
8
Table 4-3: Best parameter values for the
threshold value Ii per frequency and amplitude.
3.0
12
8
8
10
10
14
10
14
10
8
8
8
Frequency
(Hz)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.0
1.40
1.20
1.20
1.20
1.20
1.20
1.50
1.50
1.25
1.25
1.40
1.25
Amplitude (mm)
1.5
2.0
2.5
1.40 1.40 1.40
1.40 1.40 1.40
1.50 1.40 1.50
1.25 1.25 1.25
1.40 1.50 1.25
1.25 1.50 1.75
1.25 1.50 1.50
1.25 1.25 1.25
1.50 1.25 1.50
1.50 1.25 1.25
1.25 1.50 1.50
1.25 1.25 1.50
3.0
1.40
1.40
1.50
1.25
1.50
1.00
1.50
1.00
1.50
1.50
1.50
1.50
opening of the valves caused some oscillation which in turn caused an unstable response
of the actuator. Figure 4-19 shows the response and corresponding input values of the
actuator for one of these experiments. At t = 2 seconds, the valve is fully opened,
resulting in more oscillations and a deflection of the actuator. The maximum deflection
that is caused by the opening of the valves is almost 0.1 millimeters.
The airflow is required to reduce the time required for the Shape Memory Effect (SME)
cycle, but there does seem to be a downside of using air to actively cool the SMA
actuators. They cause small oscillations which, if they are not filtered properly, can
destabilize the system. This is especially true for systems with a high proportional and
derivative gain combined with a small delay. The fact that the sudden opening of the
Figure 4-19: Actuator response to sudden opening of the valves showing oscillatory behavior.
T.J. Koomen
Master of Science Thesis
4-4 Summary
59
valves causes oscillations suggests that it might be beneficial to never fully close the
valves and possibly add a low-pass filter to smoothen the control of the valves.
4-3-5
Hardware Performance
It was found, for increasing sampling frequencies, that the data package loss rate increased. This means that for higher sampling frequencies, more samples were lost. For
the sampling frequency of 50 Hz an average of approximately 10 percent of the samples
was missing. It is unsure however if this is due to the low computational power of the
Arduino or that it is purely a communication problem between the Arduino and the PC.
However, Windows features a default 125 Hz sampling rate on the USB ports. Therefore
it is suspected that the computational power of the Arduino board is too low to facilitate
a sufficient sampling rate. However, since the data package loss was not that high, the
data could still be used. A simple algorithm was used to calculate the missing samples
using linear interpolation.
Another problem that was encountered with the Arduino board was that the voltage of
the board dropped by approximately 5% to 10% when the PWM analog outputs were
active at a PWM duty cycle of 100 %. When a PWM signal is operating at a duty cycle
of 100 % it means that the output is at the maximum voltage. When there is a voltage
drop on the Arduino board, the result is that all of the outputs have a lower maximum
output voltage. So when a analog output is set to its maximum, which is 5 Volts, it
will only go as high as 90 % or 95 % of this value. This means that the accuracy of the
control signal drops by the same percentage as the voltage drop.
4-4
Summary
The four control elements of the proposed control framework were individually tested
and the results suggests that they had positive effects on the performance of the SMAbased actuator. The control framework was able to solve three of the four challenges
that had been posed. The threshold values for the current and the valve aperture ensures
that the correct phase transformation is triggered. However, in some cases for certain
parametrizations it was found that the combination of the different control elements
provided a too aggressive controller output for which it was hard to stop the phase
transformations once they had been triggered. This resulted in sharp peaks instead of
the smooth peaks from a sinusoidal reference signal.
The challenge that was not solved by the proposed control framework was the non-linear
response. The response was found to be close to linear for a certain input range, but
that once the input exceeds this range that the response would increase non-linearly.
Because the response was still non-linear, the frequency response identification could
not be performed since it would only capture the linear dynamics. The linear regime of
the input on e(t) was between 0 and 0.3. When the error exceeds this value the response
Master of Science Thesis
T.J. Koomen
60
Controller Implementation
rapidly increases. This resulted in overshoot for reference signals with a amplitude larger
than 2.0 millimeter.
The overall performance of the proposed control framework was good for frequencies
below 1 Hz. The response of the actuator still suffers from increasing delay for frequencies exceeding 1.2 Hz. When the amplitude of the reference signal is increased, the
performance decreases much faster than for an increase in frequency. This is due to the
non-linear response of the SMA-based actuator. For low amplitudes the proposed control
framework can achieve good tracking performance even for higher frequencies, though
be it with an increase in delay that is proportional to the increase of the frequency.
The combination of the threshold value, the proportional and derivative gain and the
feed forward control law sometimes resulted in an overpowered control system which
caused some oscillatory behavior. Even with a very large variation of parameters that
were tested on the control framework, no significant pattern for the parameterization
emerged from all of the experiments. Two important control parameters were analysed
for the presence of a pattern for the parameterization. The analysis showed a pattern
where the best value for the proportional gain for the current KP ,I decreases as the
frequency increases and that the best value for the threshold current Ithreshold increases
as the amplitude increases.
The use of compressed air has led to minor oscillations when the valves are opened. This
can potentially destabilize the system and severely decrease the performance if the output
signal from the laser is not sufficiently filtered. During the experiments it was found that
for each experiment, a number of samples were missing from the data that came from the
Arduino. Therefore, the Arduino was deemed to have insufficient computational power
to facilitate a constant sample rate. Another undesired phenomenon of the Arduino was
a significant voltage drop which was detected when some of the PWM outputs were used
at full duty cycle. The drop in voltage results in a loss in accuracy for all outputs of the
Arduino board.
T.J. Koomen
Master of Science Thesis
Part III
Conclusions and Recommendations
Master of Science Thesis
T.J. Koomen
Chapter 5
Conclusions
In this thesis, a general control framework for Shape Memory Alloy (SMA)-based actuators was proposed. The general control framework, or Phase Transformation Approach,
uses fundamental properties of the SMA material to construct four control laws. These
control laws were specifically designed to solve one of the four challenges that were
defined in Chapter 3.
The three questions were formulated in the introduction of the thesis to assess the
effectiveness of the control framework will now be answered.
• Can the proposed control framework be parameterized such that it cancels out the
hysteretic and non-linear response of the actuator?
No, the proportional controller that is proposed in the control framework
is only able to compensate for the hysteretic response with the correct
threshold value. Overall the SMA-based actuator still has a non-linear
response. The SMA-based actuator only shows a linear response for
a small input range from zero to 0.3 millimeters. So as long as the inputs, and thus the amplitudes of the reference signals, are constrained to
this maximum value of the linear regime the control framework will ensure linear operation of the SMA-based actuator. The individual control
laws indicate that they are successful in accomplishing their designated
objective, although for the linearization objective it is only partly successful. So the general structure and hierarchy of the control framework
in their current state are not yet capable of providing linear control of
SMA-based actuators, but a with a rearrangement of this hierarchy the
linearization may be improved.
Master of Science Thesis
T.J. Koomen
64
Conclusions
• What is the maximal bandwidth that can be achieved with this control framework
with a phase delay smaller than 30 degrees?
The maximal achievable bandwidth is dependent on the amplitude of
the input signal. For amplitudes that do not exceed 1.5 millimeters,
the bandwidth can go up to approximately 1.5 Hertz. For amplitudes
greater than 1.5 millimeters the bandwidth is limited to approximately
1 Hertz.
• What is the influence of different initial conditions of the SMA actuators on its
performance?
The initial conditions such as the temperature and stress are very important for the response of the actuator. If the actuator starts at a higher
initial temperature the response is much more aggressive since the transformation temperature is reached faster. A difference in initial stress of
the two wires causes an asymmetrical response of the actuator. One of
the wires has a lower stress, thus also a lower austenite starting temperature and therefore it reacts faster and more aggressively than the other
wire. The fact that there is no real-time information available on the
temperature and stress complicates control efforts for the SMA-based
actuator.
Discussion
The most challenging issue has been the inconsistent response from the SMA-based
actuator. At random times the response would spike or show a really slow response.
The real cause for the inconsistent response could not be identified since only one system
output was available. If measurements of the temperature or the stress in the SMA wires
would have been available, the cause could perhaps have been determined. Some of the
fluctuations and inconsistency in the response can possibly be accredited to the different
initial conditions. However, since the venting procedures between experiments have been
done very consistently, there most likely is another reason for the inconsistent behavior.
It is more likely that the erratic behavior is a caused by fluctuation temperatures or the
non-linear relation between the phase transformation rate and the temperature.
The practical work was performed on only one of the actuators that was available.
This is due to that this actuator is more reliable than the other actuator that had been
supplied and that it therefore produces more consistent output data. All SMA wires that
have been used were 0.4 millimeter SmartFlex® wires manufactured by SEAS Getters.
Unfortunately no tests with alternative SMA wires from other manufacturers have been
performed.
Something that was not within the scope of this thesis is the design of the SMA-based
actuator. The method for clamping the wires in to the actuator was not optimal. A
T.J. Koomen
Master of Science Thesis
65
constant tension on the SMA wires could therefore not be guaranteed. There was no
time for me or my supervisor to improve the design and rebuild the actuator to solve
this problem. This is also one of the main reasons why the newer actuator was not used
for testing, since for this actuator it was even more difficult to tension the wires. In
some of the graphs depicting the response of the actuator it can be observed that the
response is not completely symmetrical. This unsymmetrical response was caused by
the difference in initial tension in the two SMA wires.
Only a limited amount of parameter combinations have been tested for the proposed control framework. If all combinations had to be tested for reasonable parameter intervals
and 5 amplitudes and the 12 frequencies that were now tested, at least 5000 experiments
would have had to been performed. This would have taken weeks of brainlessly executing
experiments and running optimization scripts. Due to time constraints it was therefore
not possible to perform an optimization for the parameteriziation or to identify patterns
from the data since the lack of combinations would render them invalid.
Most of the hardware that was used was deemed sufficient. However, the computational
capabilities and the internal power regulator of the Arduino board were deemed insufficient. There was a loss of samples and during the experiments, the average voltage
of the board sometimes dropped by as much as 10%. This decreases the accuracy and
therefore also the reliability of the outputs of the Arduino board.
Recommendations
Throughout this thesis, a number of aspects have been observed that could improve the
process of constructing a general control framework for SMA-based actuators. These
observations have lead to the following recommendations for future research.
• Control Framework Structure
Some elements of the proposed control framework proved to work for their designated purpose, such as the threshold value and the feed forward stabilizer. However, when some of the elements were combined their controller output often proved
too aggressive or out of balance. This suggests a rearrangement of the hierarchy of
the control framework. The current hierarchy is that the threshold value, proportional gain and derivative gain are the "main" controller and that the feed forward
stabilizer is only active when the main controller is not active. The best threshold
values that were found during experiments showed that for an increase in amplitude of the input higher threshold value resulted in better performance. This can
be solved by integrating the threshold value and the feed forward stabilizer. The
function of the feed forward controller has been to maintain a certain tension in
each of the SMA wires. This tension dictates the austenite staring temperature
that determines the threshold value. Therefore by integrating these two control
elements, the controller ensures that the temperature will approach the austenite
Master of Science Thesis
T.J. Koomen
66
Conclusions
starting temperature. When the temperature is already at the austenite starting
temperature, any additional control input causes the stress to increase in the SMA
wires causing the actuator to respond. This additional control input can either be
achieved by the current proportional and derivative control structure or any other
suitable controller. The proportional gain should however be proportional to the
amplitude of the input to further linearize the response of the SMA-based actuator. To prevent excitation of the structural modes of the SMA-abased actuator, it
is recommended to never fully close the valves and possibly add a low-pass filter
to smoothen the control of the valves. This will minimize the oscillations that
were observed during the experiments when the valves were suddenly opened and
closed.
• Sensors
The absence of measurements of temperature and stress in the SMA wires was
really problematic. Therefore it is advised that more effort should be put in embedding one or more sensors in the actuator. An attempt was made to connect a
thermocouple to one of the SMA actuators in the SMA-based actuator. However,
a thermocouple has to be welded on to the surface of the material of which it has
to measure the temperature. This presents a huge problem for a SMA wire which
only has a diameter of 0.4 millimeters, which is approximately the same diameter
of the wires of the thermocouple itself. Real time data of the temperatures and/or
stress of the SMA wires allows for better control using the models that have been
discussed. A better analysis of the cause of the fluctuations is then also possible.
• I/O interface
The input and output interface and the all the computational work should not be
processed by an Arduino board. It is recommended that a more powerful system
is used to improve both the accuracy of the controller outputs but also to improve
the reliability of the data acquisition.
• Clamping mechanism
A better clamping mechanism is required to ensure a constant tension and also
the possibility of fine tuning of the tension. During the experiments, some asymmetrical behavior was noticed which can only be caused by a difference in initial
tension in the SMA wires. A better clamping mechanism ensures a smoother and
more consistent response of the SMA-based actuator.
• Modeling
If there is no possibility of obtaining more sensory information such as the temperature or the stress, another option would be to construct a more detailed model
that describes the mechanics of the actuator. If the deflection can be expressed
as a function of the stress, this stress can then possibly be coupled to a certain
input. It would still be very complicated to incorporate the influences of the valve
aperture in this approach.
T.J. Koomen
Master of Science Thesis
Part IV
Appendix
Master of Science Thesis
T.J. Koomen
Appendix A
Hardware
A-1
SMA-based Actuator
The Shape Memory Alloy (SMA)-based actuator that was used for this thesis was designed an manufactured by Adrian Lara-Quintanilla [4]. The actuator is shown in Figure A-1.
SMA-actuator
The SMA actuator that were used in the SMA-based actuator were from the SmartFlex®
which are manufactured by SEAS Gettter. The properties of the wire are shown in
Table A-1.
Figure A-1: Picture of the actuator with the connections indicated. The base indicates
where the actuator is clamped into the setup and the tip of the actuator is where the
deflection is measured.
Master of Science Thesis
T.J. Koomen
70
Hardware
Table A-1: SMA wire properties
SmartFlex®
Diameter
400 µm
Density
6450 kg/m3
Specific Heat Capacity1
700 J/kgK
Figure A-2: An Arduino Mega 2560 board.
A-2
Setup
The illustration in Figure A-3 depicts the configuration of the used hardware and the
connections between them. The device that facilitated the input and output interface
and also runs the control framework was a board from Arduino. The Arduino Mega
2560 had been selected because it offers standalone operation and real-time tuning and
monitoring in the Simulink.
Arduino Mega 2560
A picture of the type of Arduino that was used is shown in Figure A-2. The relevant
specifications of the Arduino board that was used are listed in Table A-2. The Arduino
was connected via USB and was power both by USB and an external power supply at 12
volts and 2 ampere. The internal power regulator automatically selects the best suitable
power source.
Displacement sensor
The displacement sensor that was used was the optoNCDT 1401 from Micro-Epsilon
(Figure A-4). The laser sensor has a range of 20 millimeters. The laser sends an analog
output signal, where 1 volt corresponds to the beginning of the range at a distance of 30
millimeter and 5 volts corresponds to the end of the measurement range at a distance
of 50 millimeters. The sensor has a minimal resolution of 5 micrometer.
T.J. Koomen
Master of Science Thesis
A-2 Setup
71
Table A-2: Relevant specifications of the Arduino Mega 2560
Clock speed:
16 MHz
SRAM:
8 KB
Flash memory:
256 KB (248 KB available)
Digital I/O:
54 (of which 15 provide PWM output)
DC current per pin:
40 mA
Analog Inputs:
16
Resolution Analog Inputs:
10 bits
Voltage Resolution Analog Inputs:
≈ 5 millivolt
Special:
Standalone Matlab and Simulink execution
Interactive tuning and monitoring
Figure A-3: Illustration depicting the total hardware setup and connections.
Master of Science Thesis
T.J. Koomen
72
Hardware
Figure A-4: Mirco-Epsilon optoNCDT 1401 laser displacement sensor.
PWM Low-Pass Filters
An analog low pass filter was used for the Pulse Width Modulation (PWM) outputs
of the Arduino to the analog inputs of the power supplies. A resistor of 2.4 KΩ and
a capacitor of 3.3 µF were used to build the analog low pass filter. The theoretical
response of the filter is shown in Figure A-5.
Power Supply Power Relay
During early experiments the Arduino board had some problems with initializing the
code from Simulink. During the initialization it would reset the analog outputs but this
sometimes resulted in very high outputs on these ports. For a few experiments this
resulted in an output from the power supply of over 10 ampere. It caused the wire to
heat up beyond the melting temperature of the nylon of the actuator, which caused the
SMA wires to melt into the nylon. To prevent this from occurring, a simple power relay
was constructed. The relay was set remotely operate the power supplies, setting it to
standby by default and to "on" for when Simulink was done with the compiling and the
Arduino was done with the initialization.
T.J. Koomen
Master of Science Thesis
A-3 Power Supplies
73
Figure A-5: Bode plot estimate for the analog low pass filter.
Table A-3: Specifications of the Elektra-Automatik model EA-PS 8032-20
A-3
Output Current:
0-20 A
Output Voltage:
0-32 V
Output Power:
640 W
Regulation (10-100 % load):
< 2 ms
Stability (0-100% load):
< 0.15%
Power Supplies
Two power supplies from Elektra-Automatik were used to supply the current to the two
SMA wires in the SMA-based actuator (Figure A-6). The specifications for the model
EA-PS 8032-20 are listed in Table A-3.
A-4
Valves
For the airflow control, two proportional valves (Type 2821) from Burkert were used
(Figure A-7a). The valves were rated to a maximum pressure of 6 bar. The valves were
controlled by valve controllers (Type 8605) from Burkert (A-7b). A power supply from
National Instruments (FP-PS-4) was used to power the valve controllers. The FieldPoint
power supplies delivers +24 volts to both valve controllers.
Master of Science Thesis
T.J. Koomen
74
Hardware
Figure A-6: Elektra-Automatik EA-PS 8032-20.
(a)
(b)
Figure A-7: Proportional valve (a) and a valve controller (b) from Burkert.
T.J. Koomen
Master of Science Thesis
Appendix B
Control Framework Parameters
This appendix list the control parameters that were found to have the lowest mean
squared error per frequency and amplitude of a sinusoidal reference signal. The mean
squared error is calculated with Equation B-1, where n is the number of samples of the
output y(k) and reference signal r(k).
n
MSE =
1X
(y(k) − r(k))2
n
(B-1)
i=1
Table B-1: Best parameter values for the threshold value for the current Ii per frequency
and amplitude.
Frequency
(Hz)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Master of Science Thesis
1.0
1.40
1.20
1.20
1.20
1.20
1.20
1.50
1.50
1.25
1.25
1.40
1.25
Amplitude (mm)
1.5
2.0
2.5
1.40 1.40 1.40
1.40 1.40 1.40
1.50 1.40 1.50
1.25 1.25 1.25
1.40 1.50 1.25
1.25 1.50 1.75
1.25 1.50 1.50
1.25 1.25 1.25
1.50 1.25 1.50
1.50 1.25 1.25
1.25 1.50 1.50
1.25 1.25 1.50
3.0
1.40
1.40
1.50
1.25
1.50
1.00
1.50
1.00
1.50
1.50
1.50
1.50
T.J. Koomen
76
Control Framework Parameters
Table B-2: Best parameter values for the threshold value for the valve aperture Fi per
frequency and amplitude.
Frequency
(Hz)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.0
0.50
0.35
0.35
0.35
0.50
0.50
0.75
0.75
0.50
0.50
0.50
0.25
Amplitude (mm)
1.5
2.0
2.5
0.50 0.50 0.50
0.50 0.50 0.50
0.50 0.50 0.50
0.50 0.50 0.50
0.50 0.75 0.50
0.50 0.75 0.75
0.40 0.50 0.50
0.50 0.50 0.50
0.50 0.50 0.50
0.50 0.50 0.50
0.25 0.50 0.50
0.25 0.25 0.50
3.0
0.50
0.50
0.50
0.50
0.75
0.60
0.50
0.60
0.50
0.75
0.50
0.50
Table B-3: Best parameter values for the proportional gain KP ,I per frequency and amplitude.
Frequency
(Hz)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
T.J. Koomen
1.0
12
18
18
18
18
18
14
12
12
10
8
8
Amplitude (mm)
1.5 2.0 2.5 3.0
12 12 12 12
8
8
8
8
10
8 10
8
10 12 12 10
12 10
8 10
8 10
8 14
10 10 10 10
10
8 10 14
8 10
8 10
10 14 14
8
8
8
8
8
8
6
8
8
Master of Science Thesis
77
Table B-4: Best parameter values for the proportional gain KP ,F per frequency and amplitude.
Frequency
(Hz)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.0
0.50
2.00
2.00
2.00
1.00
1.00
4.00
4.00
0.50
1.00
0.50
0.50
Amplitude (mm)
1.5
2.0
2.5
0.50 0.50 0.50
0
0
0
0
0 0.25
0 0.25 0.50
0.25 2.00 1.00
1.00 2.00 4.00
0.50 0.50 0.50
1.00 1.00 1.00
4.00 0.50 4.00
0.50 2.00 2.00
0.50 0.50 0.50
0.50 0.50 0.50
3.0
0
0
0
0
4.00
1.00
0.50
1.00
4.00
4.00
0.50
0.50
Table B-5: Best parameter values for the derivative gain KD,I per frequency and amplitude.
Frequency
(Hz)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Master of Science Thesis
1.0
1.50
0.20
0.20
0.15
0.50
0.50
1.50
1.50
0.20
0.25
1.00
1.00
Amplitude
1.5
2.0
1.00 1.00
0.75 0.75
1.00 0.75
1.00 1.00
0.50 1.50
0.25 1.50
1.50 0.50
1.25 1.25
1.25 1.50
0.50 1.50
0.75 0.50
1.25 1.25
(mm)
2.5
3.0
1.00 1.0000
1.00
1.50
1.00
1.00
1.00
1.00
1.25
1.00
1.25
0.30
0.50
0.50
1.25
0.30
1.25
0.25
1.25
1.00
0.50
0.50
0.50
0.50
T.J. Koomen
78
Control Framework Parameters
Table B-6: Best parameter values for the derivative gain KD,F per frequency and amplitude.
Frequency
(Hz)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.0
1.50
0.20
0.20
0.15
0.50
0.50
1.50
1.50
0.20
0.25
1.00
1.00
Amplitude
1.5
2.0
1.00 1.00
0.75 0.75
1.00 0.75
1.00 1.00
0.50 1.50
0.25 1.50
1.50 0.50
1.25 1.25
1.25 1.50
0.50 1.50
0.75 0.50
1.25 1.25
(mm)
2.5
3.0
1.00 1.0000
1.00
1.50
1.00
1.00
1.00
1.00
1.25
1.00
1.25
0.30
0.50
0.50
1.25
0.30
1.25
0.25
1.25
1.00
0.50
0.50
0.50
0.50
Table B-7: Best parameter values for the stabilizer current IF F per frequency and amplitude.
Frequency
(Hz)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
T.J. Koomen
1.0
1.00
1.00
1.00
1.00
1.10
1.10
0.80
0.80
0.80
0.80
1.00
0
Amplitude (mm)
1.5
2.0
2.5
1.00 1.00 1.00
1.00 1.00 1.10
0 1.00
0
0
0
0
1.00 0.80
0
0.80 0.80 0.80
0 0.80 0.80
0
0
0
0.25
0 0.25
0.80 0.25 0.25
0 0.80 0.70
0
0 0.70
3.0
0
0
0
0
1.00
0.80
0.80
0.80
0.25
1.00
0.70
0.70
Master of Science Thesis
79
Table B-8: Best parameter values for the proportional gain KF F for the stabilizer current
IF F per frequency and amplitude.
Frequency
(Hz)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Master of Science Thesis
1.0
2.50
0.75
0.75
0.75
0.60
0.60
0.50
0.50
0.50
0.50
2.00
0
Amplitude (mm)
1.5
2.0
2.5
2.50 2.50 2.50
2.50 2.50 2.00
0 2.50
0
0
0
0
2.50 0.25
0
0.50 0.25 0.25
0 1.50 1.50
0
0
0
0
0
0
2.00 0.25 0.25
0 2.00 2.00
0
0 2.50
3.0
0
0
0
0
0.50
0.50
1.00
0.50
0.25
0.50
2.00
2.50
T.J. Koomen
80
T.J. Koomen
Control Framework Parameters
Master of Science Thesis
Bibliography
[1] J. Kudva, “Overview of the darpa smart wing project.” Journal of Intelligent Material Systems and Structures, vol. 15, pp. 261–267, 2004.
[2] T. Barlas and G. van Kuik, “Review of state of the art in smart rotor control
research for wind turbines,” Progress in Aerospace Sciences, vol. 46, pp. 1–27, 2010.
[3] ——, “State of the art and prospectives of the smart rotor control for wind turbines,”
Journal of Physics, 2007.
[4] A. Lara-Quintanilla, A. W. Hulskamp, and H. E. Bersee, “A high-rate shape memory alloy actuator for aerodynamic load control on wind turbines,” Journal of Intelligent Material Systems and Structures, vol. 24, no. 15, pp. 1834–1845, 2013.
[5] T. Koomen, Control of Shape Memory Alloy Based Actuators: A Review on Control
Strategies and the Potential of a General Framework, Delft University of Technology
Std., November 2014.
[6] R. Stalmans and J. Van Humbeeck, “Shape memory alloys: Functional and smart,”
in Smart Materials and Technologies - sensors, control systems and regulators, 1995.
[7] D. C. Lagoudas, Shape Memory Alloys - Modeling and Engineering Applications.
Springer.
[8] J. H. Crews and G. D. Buckner, “Design optimization of a shape memory alloyactuated robotic catheter,” Journal of Intelligent Material Systems and Structures,
vol. 23, no. 5, pp. 545–562, 2012b.
[9] M. K. Stanford, “Thermophysical properties of 60-nitinol for mechanical component
applications,” National Aeronautics and Space Administration, 2012.
[10] S. Getters, “Smartflex wire,” SAES Getters, Tech. Rep., 2009.
Master of Science Thesis
T.J. Koomen
82
BIBLIOGRAPHY
[11] L. Brinson, “One-dimensional constitutive behavior of shape memory alloys:
Thermomechanical derivation with non-constant material functions and redefined
martensite internal variable,” Journal of Intelligent Material Systems and
Structures, vol. 4, no. 2, pp. 229–242, 1993. [Online]. Available: http:
//jim.sagepub.com/content/4/2/229.abstract
[12] S. Seelecke, “Modeling the dynamic behavior of shape memory alloys,” International
Journal of Non-Linear Mechanics, vol. 37, no. 8, pp. 1363 – 1374, 2002. [Online].
Available: http://www.sciencedirect.com/science/article/pii/S0020746202000306
[13] S. Seleecke and I. Müller, “Shape memory alloy actuators in smart structures: Modeling and simulation,” Applied Mechanics Reviews, vol. 57, no. 1, pp. –, 2004.
[14] O. Heintze, “A computationally efficient free energy model for shape memory alloys
- experiments and theory,” Phd, 2004.
[15] K. Arai, K. Aramaki, and K. Yanagisawa, “Continuous system modelling of shape
memory alloy (sma) for control analysis,” 1994.
[16] W. Wang and S. Yan, “A new continuous model of shape memory alloys,” Applied
Mechanics and Materials, vol. 226-228, pp. 2467–2470, 2012.
[17] A. Nespoli, S. Besseghini, S. Pittaccio, E. Villa, and S. Viscuso, “The high potential
of shape memory alloys in developing miniature mechanical devices: A review on
shape memory alloy mini-actuators,” Sensors and Actuators A: Physical, vol. 158,
no. 1, pp. 149–160, 2010.
[18] K. Ikuta, M. Tsukumoto, and S. Hirose, “Shape memory alloy servo acttuator system with electric resistance feedback and application for active endoscope,” 1988.
[19] L. Toia, A. Coda, G. Vergani, and A. Mangioni, “Functional characterization of
sma wires in actuation conditions,” SEAS Getters, Tech. Rep.
[20] D. J. Hartl and D. C. Lagoudas, “Aerospace applications of shape memory
alloys,” Proceedings of the Institution of Mechanical Engineers, Part G: Journal
of Aerospace Engineering, vol. 221, no. 4, pp. 535–552, 2007. [Online]. Available:
http://pig.sagepub.com/content/221/4/535.abstract
[21] A. Lara-Quintanilla and H. Bersee, “Improvement of the attainable working frequency of sma wires by means of active cooling and working strain-ratios,” ASME
2014 Conference on Smart Materials, Adaptive Structures and Intelligent Systems,
2014.
[22] G. Song and N. Ma, “Robust control of a shape memory alloy wire actuated flap,”
Smart Materials and Structures, vol. 16, no. 6, pp. N51–N57, 2007.
[23] C. Bil, K. Massey, and E. J. Abdullah, “Wing morphing control with shape memory
alloy actuators,” Journal of Intelligent Material Systems and Structures, vol. 24,
no. 7, pp. 879–898, 2013.
T.J. Koomen
Master of Science Thesis
BIBLIOGRAPHY
83
[24] F. Peng, X.-X. Jiang, Y.-R. Hu, and A. Ng, “Actuation precision control of sma actuators used for shape control of inflatable sar antenna,” Acta Astronautica, vol. 63,
no. 5-6, pp. 578–585, 2008.
[25] A. Lara-Quintanilla, “Manufacturing and control of shape memory alloy based actuators,” Master Thesis, 2010.
[26] G. S. Batur, V. Chaudhry, and C., “Precision tracking control of shape memory
alloy actuators using neural networks and a sliding-mode based robust controller,”
Smart Materials and Structures, vol. 12, no. 2, p. 223, 2003.
[27] K. J. Aström and B. Wittenmark, Computer Controlled Systems. Theory and Design, T. Kailath, Ed. Prentice Hall, 1997, page 244.
Master of Science Thesis
T.J. Koomen
84
T.J. Koomen
BIBLIOGRAPHY
Master of Science Thesis
Glossary
List of Acronyms
3mE
Mechanical, Maritime and Materials Engineering
AE
Aerospace Engineering
DCSC
Delft Center for Systems and Control
MISO
Multi Input, Single Output
PWM
Pulse Width Modulation
SISO
Single Input, Single Output
SMA
Shape Memory Alloy
SMC
Sliding Mode Control
SME
Shape Memory Effect
DCSC
Delft Center for Systems and Control
ETFE
Empirical Transfer Function Estimate
FF
Feed Forward
ILC
Iterative Learning Controller
NN
Neural Networks
PID
Proportional, Integral and Derivative
PD
Proportional and Derivative
RBS
Random Binary Signal
Master of Science Thesis
T.J. Koomen
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement