Download Controller Analysis with Inverted Pendulum

DEGREE PROJECT, IN BACHELOR'S THESIS IN MECHATRONICS , FIRST
LEVEL
STOCKHOLM, SWEDEN 2015
Controller Analysis with Inverted
Pendulum
REGULATORANALYS MED INVERTERAD
PENDEL
FILIP STENBECK ARON NYGREN
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF INDUSTRIAL ENGINEERING AND MANAGEMENT
Controller Analysis with Inverted Pendulum
FILIP STENBECK
ARON NYGREN
Bachelor’s Thesis in Mechatronics
Supervisor:
Examiner:
Approved:
Martin Edin Grimheden
Martin Edin Grimheden
2015-05-20
TRITA MMK 2015:21 MDAB074 2015-06
Abstract
The aim of this thesis is to examine if feedback of the angle from an inverted
pendulum is sufficient to control its angle at an unstable equilibrium with statics and force impulses, and through different approaches and choice of controller
find the most suitable one for these types of applications. The controllers that
was tested was, the PID regulator and the state space regulator. The results
would show that the mathematical approach to find a controller is difficult and
time consuming, and it is often better to use a trial and error approach to find
a regulator if repeated test on the system is possible.
The core of the thesis lies in the mathematical approximations of the mechanical and electrical system, the analysis of the controller and the choice and
usage of components. Analysis of the combined electrical and mechanical systems were made in Simulink and Matlab and was then generated to mechanical
code to an micro controller controlling the voltage to a dc-motor.
The system is non linear but can be linearised around the equilibrium point
that we want to maintain, which is a good approximation for small angles.
This thesis describes the electrical and mechanical components used to build a
rotary inverted pendulum and how to produce an effective controller in detail.
iii
Sammanfattning
Regulatoranalys med Inverterad Pendel
Målet med examensarbetet är att utvärdera om återkoppling av vinkeln från
en inverterad pendel är tillräcklig för att kontrollera denna kring en instabil
jämviktspunkt med störningar samt pålagda kraftimpulser, och genom val av
olika tillvägagångssätt och regulatorer finna den mest lämpliga för dessa typer
av tillämpningar. De regulatorer som användes i projektet var PID-regulatorn
samt state space regulatorn. Resultaten kom att visa att ett matematisk tillvägagångsätt att skapa en regulator är svårt och tidskrävande, och det är ofta
mer effektivt att testa sig fram till en regulator om systemet tillåter.
Kärnan i arbetet ligger i de matematiska approximationerna av de mekaniska
och elektriska system, analysen av kontrollern och valet samt tillämpningen av
komponenter. Analysen av det kombinerade elektriska och mekaniska systemen
gjordes i Simulink och Matlab och var därefter genererad till mekanisk kod till
en mikro-kontroller för att regulera spänningen till en likströmsmotor.
Den inverterade pendeln är ett olinjärt system men kan med god approximation och litet fel linjariseras runt dess instabila jämnviktspunkt.
Detta examensarbete kommer i huvudsak handla om hur man konstruerar en
regulator genom simulering samt analys av systemet. Alla komponenter såväl
elektriska som mekaniska kommer att beskrivas i detalj.
v
Preface
During our project we have received lots of assistance, all from building the prototype to modelling the system. We like to thank Staffan Qvarnström, Tomas
Östberg and Jonas Stenbeck for inputs and usage of tools to construct the prototype, Sebastian Quiroga for valuable information of the software and control theory
and Martin Edin Grimheden along with our fellow students taking the course for
guidance writing the thesis.
Filip Stenbeck, Aron Nygren
Stockholm, May, 2015
vii
Contents
Abstract
iii
Sammanfattning
v
Preface
vii
Contents
ix
Nomenclature
xi
1 Introduction
1.1 Background
1.2 Purpose . .
1.3 Scope . . .
1.4 Method . .
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2 Theory
2.1 Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Demonstrator
3.1 Problem Formulation . .
3.2 Analysis of the System .
3.3 Pole Placement . . . . .
3.4 Friction Approximation
3.5 Rotary Encoder . . . . .
3.6 Sensor Values . . . . . .
3.7 Hardware . . . . . . . .
3.8 Software . . . . . . . . .
3.8.1 Arduino . . . . .
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4 Results
4.1 Comparing Mathematical Controllers . . . . . . . . . . . . . . . . . .
4.2 Comparing Controllers On Demonstrator . . . . . . . . . . . . . . .
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5 Discussion and conclusions
5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Recommendations and future work
6.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
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Appendices
A Additional information
A.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B Proofs
B.1 Derivation of the systems transfer function .
B.1.1 Mechanical system . . . . . . . . . . .
B.1.2 Electrical system . . . . . . . . . . . .
B.1.3 Combined systems in Laplace domain
B.2 Pendulum Friction Approximation . . . . . .
C References
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x
Nomenclature
Symbols
Symbols
Description
I1
l1
m
θ1
lcrank
θ2
θ¨2
Icrank−arm
ẍ
Mmotor
Kt
Ke
Ia
La
Ra
Vin
VC
VR
I
R
C
ωc
UA
URa
ULa
Inertia of the pendulum (kg m2 )
Length of the pendulum (m)
Mass of the pendulum (kg)
Angle of the pendulum (rad)
Length of the crank-arm (m)
Angle of the crank-arm (rad)
Angle acceleration of the crank-arm (rad/s2 )
Inertia of the crank-arm (kg m2 )
Acceleration in x direction (m/s2 )
Torque from the motor (N m)
Torque constant (N m/A)
Back emf constant(V /rpm)
Electrical current (A)
Inner inductance (H)
Inner resistance (Ω)
Voltage from sensor (V )
Voltage across capacitor (V )
Voltage across resistor (V )
Current (A)
Resistance (Ω)
Capacitance (F )
cut-off frequency (Hz)
Voltage from H-bridge (V )
Voltage across motor resistance (V )
Voltage across motor inductor (V )
xi
Symbols
Description
Ub
Te
Tω̇
Tω
Ja
B
Back electromotive force voltage (V )
Electromagnetic torque (N m)
Acceleration torque(N m)
Velocity torque(N m)
Inertia of the rotor (kg m2 )
Motor viscous friction (N ms)
Abbreviations
Abbreviation Description
LQ
PWM
Matlab
CAD
PZ
EMF
Linear Quadratic Optimal Control
Pulse Width Modulation
Matrix Laboratory
Computer Aided Design
Pole Zero
Electromotive force
xii
Chapter 1
Introduction
We have all tried to balance something like a pole or a pen on our finger tip or seen
a clown balancing plates on a pole. From this we know that we have to adjust and
change the direction of our finger to maintain balance of the pen. This is done by
motion tracking of our eyes and through computation in our brain evaluate which
output signal should be sent to our muscles. An electronic apparatus balancing
an object needs to have these hardware and software as well. For this application
we can correlate the eyes of the human as a sensor, the brain as software and the
muscles as the motor. To balance an object does not often seem like a difficult
achievement, but in theory it is a complex task. In our daily lives inverted pendulums often appears and it is essential to understand the mechanics and to control
it in an upright position. The computation that combines the input signal with the
output signal is a delicate operation. Depending on how the output signal is computed, different properties of the system is obtained. To control the output signal
to the muscle some sort of a regulator is needed, this calculates what acceleration of
the pendulum has to be applied to keep it from falling. A regulator can be more or
less advanced and take different aspects in consideration. For example not just the
angle of the pendulum but also its velocity and the position of the finger. All people are not equally skilled at balancing objects. This is often not due to differences
within hardware, as eyesight or muscle strength but more within the computation
of the signal to the muscle. This project goes in depth of how to produce a regulator
and discusses which methods are preferable for different applications.
This chapter describes in more detail the background, purpose, limitations and
methods used in the project.
1.1
Background
The inverted pendulum in general is a well known and basic mechatronic problem
that focuses much on modelling and control of non linear differential equations.
1
CHAPTER 1. INTRODUCTION
This problem occurs in other areas such as the Segway and the SpaceX program
and is often used as an example of showing the usage of control theory.
The rotary inverted pendulum was first invented by Katsuhisa Furuta, and has
since then been called the Furuta pendulum [14]. The Furuta pendulum is a simple
design and compared with a one dimension motion control of a pendulum reduces
the number of moving parts and friction within the system, this is naturally better
for the project since it makes the mathematical model easier and with less approximations.
A regulator for a system can be produced in a number of ways, all with different
advantages. The most common is the PID regulator that has been used since the
beginning of the 20th century [2]. A newer regulator is the state space regulator
that can take other things than the output signal in consideration to calculate the
input signal. These two controllers generally have diverse approaches to generate
the controllers. The PID controller is often used as a trial and error method to
create a working controller [2]. The state space is usually used when more information than the output signal is needed to control a system, with this method it is
necessary to have a good mathematical model of the system which is then used to
extract a state space controller.
1.2
Purpose
What is the best approach to produce a controller with only respect to an output
signal?
Since only one output signal is measured, this is equivalent to "How to build the
best PID-regulator?". To create a regulator is not a fundamental process, therefore
knowledge of which controller and how to generate that controller is necessary to
be an efficient mechatronics technician. In this project two of the most essential
and wide spread regulators are compared and analysed. The conclusions within
this thesis are meaningful for many companies involved with control design. The
purpose of this project is to answer the research question stated above and to gain
knowledge in automatic control and in the use of mathematical models.
1.3
Scope
To evaluate the properties of the mechanical system, one must first try to explain
the system in mathematical terms that as well as possible reflects the real prototype. This was the most challenging phase during the project which involves lots of
electronic and mechanical theory. Nature is often non linear and many used equations in this report involves approximations based on assumptions. The mechanical
model is difficult to evaluate without linearisation around its equilibrium point, and
by doing this, approximate a complex non linear system to a slightly less complex
linear system. Approximations within the motor model are that the inner friction
and inner inductance are neglected. It is difficult to clarify and examine all approx2
1.4. METHOD
imations made and to determine how well the approximation represents reality.
The state space model has no more information than what the PID controller has,
which in this case is only the angle of the pendulum and its derivatives. This makes
the state space model control the system on the same premises and therefore a very
specific way of comparing the two controller design methods.
1.4
Method
The research question is about how to produce efficient regulators in both functionality and development. to answer this question some sort of evaluation of regulators
must be made. To compare the regulators, certain properties of the system under
impulses is analysed, such as rise time, overshoot and settling time. Investigating
these criteria is a way to evaluate the speed of the controlled system.
After the system was derived as seen in appendix B Matlab was used to make a
controller using the LQ-method in state space form and a regular PID. The state
space model was derived through equations in the derivation of the transfer function
whilst the PID was constructed using the transfer function of the system and the
built in PID tuning within Matlab. The software Matlab and Simulink was used
to evaluate the theoretical system and then the two regulators was tested on the
prototype for a final validation.
3
Chapter 2
Theory
In order to develop and evaluate regulators controlling a pendulum some theory
in electronics and control theory must be implemented. For development of the
regulators one must first understand how they work and how to generate them
through the mathematical model of the system. Many electrical components are
used in the project and this chapter reviews what had to be accounted for, such
as how the PWM frequency affects the signal and how to reduce statics within the
sensor measurements.
2.1
Control Theory
A mechanical system can be described with differential equations [2]. From these
equations the transfer function can be derived using Laplace transform, and thus
convert it to the complex plane. The transfer function is a function that describes
the system, in this case the correlation between input voltage of the motor to angle
of the pendulum.
Pole Zero plot
A pole zero (PZ) plot is a graphical representation of the transfer function. From the
PZ plot certain properties can be derived, such as stability and speed of the system.
The poles of a transfer function corresponds to the roots of the denominator and
the zeroes are the roots of the numerator. The pole closest to the imaginary axis is
the dominating pole. Imaginary poles indicates oscillations in the system [2].
PID Regulator
A regulator is a device that from a given input calculates a control signal,u, and
transmits it to a system. [2] When the system receives this control signal it will
respond and give an output y. The output can be used to again calculate the control
signal, this is called feedback. When feedback is obtained one can implement a PID
5
CHAPTER 2. THEORY
controller. The PID uses the error e
e(t) = r − y
(2.1)
which is the difference between the output and a desired value/reference value r, in
the following way [2]
u(t) = KP e(t) + KI
Z t
0
e(τ ) dτ + KD
d
e(t)
dt
(2.2)
where KP , KI and KD are design parameters. The design parameters can be determined by following this heuristic technique. [2]
KP increases the response time of the system but worsens the stability.
KI eliminates consistent errors, but also worsens the stability.
KD improves the stability.
State Space Regulator
In a state space model the internal states x of the system are used to predict a
future output, y(t). Therefore y is no longer only depending on the input but also
on the internal states of the system.
A state space model can be written as
ẋ = A x + B u
(2.3)
where A is a matrix that that describes how the system is depending on the internal
states and B is how the system depends on the input
y = Cx + Du
(2.4)
where C and D are matrices that describes how the output is depending on the
internal states respective input.
The feedback is introduced as
u = −Lx + r∗
(2.5)
this inserted in 2.3 gives the state space feedback as
ẋ = (A − BL) x + B r∗
where r∗ is connected to the reference signal.
6
(2.6)
2.2. ELECTRONICS
Linear Quadratic Regulator
A Linear quadratic regulator minimizes the deviations, i.e in this case the deviations
from the desired angle. This is done by minimizing the cost function, which is a
sum of the deviation of the measured angle from its desired value [3]. The cost
function for a continuous-time linear system is
J=
Z ∞
0
(xT (Q + AT P + P A − P B B T P ) x + (u + B T P x)2 )dt
(2.7)
where Q is a diagonal matrix. The diagonal elements of Q indicates how fast you
want the corresponding x component to stabilize [2]. The u that minimizes the cost
is
u = −L x = −B T P x
(2.8)
where P can be calculated from the Reccati equation
Q + AT P + P A − P B B T P = 0
(2.9)
When P is solved and inserted in 2.8 the following system with feedback is
2.2
ẋ = ( A − B L ) x + B u
(2.10)
y=Cx
(2.11)
Electronics
Low Pass Filter
The system is made stable around an unstable equilibrium point by controlling
the output from a dc-motor with regard to the angle of a rotary encoder. This
encoder gives an analogue output signal depending on the angle of the pendulum.
This signal contains lots of statics, and without filtering the output signal, a stable
system cannot be generated. Statics occurs often in higher frequencies and with a
first order low pass filter the influence of these statics can be decreased significantly.
A resistor-capacitor circuit in series was chosen as the low pass filter, this is a brief
derivation of how a RC-filter lowers the affect from statics.
Kirchhoff’s voltage law on figure 2.1 and transformation into the Laplace domain
gives the following
Vin − VC − VR = 0
(2.12)
where VC and VR are [15]
VR = RI → I = VR R
Z
i
VC =
dt
C
(2.13)
(2.14)
the above equation 2.14 in the Laplace domain results in
VC =
I
→ I = VC Cs
Cs
7
(2.15)
CHAPTER 2. THEORY
Figure 2.1: RC-Circuit.
the equations 2.14 and 2.13 can be combined and inserted in 2.12 to derive the
transfer functions
VC
1
= HC (s) =
(2.16)
Vin
1 + RCs
and
VR
RCs
(2.17)
= HR (s) =
Vin
1 + RCs
With substitution of the imaginary Laplace variable s to iw the transfer functions
can be analysed for different frequencies w.
For small w
(
HC (iω) → 1
ω→0
HR (iω) → 0
And for high frequencies
HC (iω) → 0
HR (iω) → 1
(
ω→∞
So if ω → 0 all of the voltage drop is over the capacitor and nothing over the resistor.
The other way if ω → ∞ then all the voltage drop is over the resistor. Thus for a
low pass filter the output signal should be measured over the capacitor and for a
high pass filter the voltage over the resistor should be measured.
The cut-off frequency is defined as when HC (s) = HR (s) = √12 Solving this will
generate the cutoff frequency ωc as
ωc =
1
.
RC
(2.18)
Pulse Width Modulation
Pulse width modulation is a way to create what can be interpreted as an analogue
signal but is actually a digital signal. This is done by setting the voltage to high
and low at high frequencies. For example, if the digital signal is high 50 percent
of the time and low 50 percent then the average voltage is one half of the high
voltage. When the signal is high 50 percent of the time it is called 50 percent duty
8
2.2. ELECTRONICS
cycle. In order to use this method properly the frequency of the PWM should be so
high that the driven device cannot differentiate the variations between high and low.
The micro-controller used in this project [6] has 3 timers. These timers counts
from 0 to a set value over and over again.[7] The timers also have two outputs compare registers each. The output compare registers check for a value set by the user
while the timers are counting. When the value is found it toggles the output, thus
creating the PWM width of the signal. For example if the compare register value
is set to 180 the duty cycle of the output is 180
256 = 0.70, i.e 70 percent.
The timers have two main modes, these are called fast PWM and phase-correct
PWM. In the fast PWM the counters count from 0 to 255 and the phase-correct
PWM also counts from 0 to 255 but then back to 0 again. The main differences
between these two modes are that in phase-correct mode the output is turned off
both when the timer hits the compare register value on the way from 0 to 255
and from 255 to 0, the fast PWM mode only turns off the output when the timer
hits the compare register value when counting from 0 to 255. This will give the
phase-correct PWM mode about half the frequency of the fast PWM mode.
PWM Frequency
The output of the PID- controller updates every 3 milliseconds. Therefore a high
PWM frequency is required, because you want as many periods as possible between
every update to get a continuous torque to the motor. If a high frequency is not used
there is a chance that the PWM signal to the motor is not able to complete a full
period and thus the duty cycle is changed from its given value from the controller.
Figure 2.2: PWM signal 500 Hz with 3 ms uppdate.
As seen in figure 2.2 if a 50 percent duty cycle on a 500 Hz frequency with 3 ms
update the signal is cut in the middle of a period and thus creating a new duty
cycle of cirka 67 percent.
9
Chapter 3
Demonstrator
This chapter is about the components of the demonstrator and how it is used to
analyse the system and answer the research question.
3.1
Problem Formulation
What is the best approach to produce a controller with only respect to an output
signal?
Answering this question with a good basis can be done through a number of ways,
the path chosen in this project was to develop a prototype with one input signal
and establish two separate regulators to draw conclusions from. When comparing
the controllers the outcome depends on how well the mathematical model reflects
the prototype and it is then crucial to make it as accurate as possible.
3.2
Analysis of the System
The transfer function of the system is derived through equation B.23. With the
approximation to ignore the influence of the inner friction of the motor and inner
inductance the following transfer function is obtained.
−0.1646s
s3 − 1.044s2 − 41.84s + 0.001771
(3.1)
This transfer function should approximate the mechanics of the prototype, to investigate the properties of the system PZ plot is used. As seen in figure 3.1a the
position of the pole on the right hand side of the imaginary axis indicates that this
system is unstable. To make the system stable and move the unstable pole to the
left half plane a negative feedback from a regulator can be implemented.
Some systems can be unstable in open loop but stable in closed loop configuration
[1]. To investigate if the system can be controlled with negative feedback of only
the angle a root locus can be drawn, figure 3.1b. The pole on the right hand side
11
CHAPTER 3. DEMONSTRATOR
Pole zero map for uncompensated system
Root locus for open loop system
1
3
0.8
2
Imaginary Axis (seconds−1)
Imaginary Axis (seconds−1)
0.6
0.4
0.2
0
−0.2
1
0
−1
−0.4
−0.6
−2
−0.8
−1
−6
−4
−2
0
2
4
6
−3
−30
8
Real Axis (seconds−1)
−20
−10
0
10
20
30
Real Axis (seconds−1)
(a) 3.1 Pole Zero map.
(b) 3.1 Root Locus.
of the imaginary axis in the s-plane has a branch that never passes the imaginary
axis, and thus the system cannot be stable using a closed loop a configuration with
loop gain of K [2]. Controlling this system require a more advanced controller. To
move the poles of the PZ plot two methods were used. The first choice was the LQ
method. This involves expressing the system in state space form and from this find
a regulator sufficient for stabilisation. The second choice was using Matlab’s built
in application PID Tuning.
3.3
Pole Placement
State Space Model
The state space was made based on the equations B.8 and B.24 and from those the
following block diagram could be manufactured. The internal states x were chosen
Figure 3.2: Block diagram for state space modelling.
to Θ1 , θ1 and θ˙1 . Here θ1 is the angle of the pendulum and the only data measured.
12
3.3. POLE PLACEMENT
Θ1 and θ˙1 was obtained through integration and derivation of the angle.
In matrix form the states are
θ1
Θ1
˙
 
x =  θ1 ẋ = θ1  .
θ˙1
θ¨1


 
The state space model is defined through 2.3 and 2.4 as the following.
ẋ = A x + B u
y = Cx + Du
With the specified states and the block diagram 3.2 the state space model could
be generated. The first two values of ẋ are already defined within the x vector,
however θ¨1 have to be computed through the block diagram and be expressed with
the states in x and input signal Ua . The output signal is defined as the angle and
second state θ1
0
1.0000
0
0
h
i
h i




0
1.0000 B= 0  C= 0 1 0 D= 0
A= 0
1.0435 41.8448 0.0018
0.1646




With the state space model it is then possible to construct a regulator if the system is
controllable.
This is determined
by looking at the control matrix S, that is defined
as S= B AB ... An−1 B . If the system only has one input signal, then this
matrix is quadratic and controllable if the determinant of S 6= 0 [2]. The control
matrix used in the project is
S = −0.0045 6= 0
(3.2)
thus controllable. The system seen in figure 3.1a is clearly unstable and it is needed
to move the unstable pole to the left hand plane. This is done with negative feedback
of the states and to calculate a sufficient regulator for this project a LQ regulator
was chosen.
Linear Quadratic Regulator
Matlab was used to solve the Ricatti equation with the built in function care, using
the generated state space model. Except from the solution to the Ricatti equation
the care function also returns the gain matrix L and the eigenvalues of the closed
loop system, i.e the poles. The gain matrix is the feedback needed to control the
13
CHAPTER 3. DEMONSTRATOR
system. Where the input signal is determined through the gain matrix and states
as the following
u = Lx
(3.3)
h
i
where the gain matrix is calculated as L= 12.6818 529.3612 80.2195 . This is the
feedback combining the states of the system with the input signal in volts, though
the input is controlled with a resolution of maximum 12 volts in 8 bits. This can
12
the be accounted for with multiplication of this factor ( 255
). hThe controlleri is
now directly comparable with a PID controller with the values of Ki Kp Kd =
h
i
0.5968 24.9111 3.7750 .
PID Tuning
Matlab has an built in application that allow users to design a PID controller by
importing their plant and choosing different parameters, such as speed, robustness
and aggressiveness to optimize their controller. Using the transfer function B.25
and tuning the parameters to suitable characteristics, i.e low overshoot, fast rise
time and settling time, the resulting PID became
• Kp=651.3617
• Ki=616.3995
• Kd=84.0449
This can interpreted for the controller used in Arduino with the same multiplication
12
factor as for LQR, ( 255
) as
• Kp=30.6523
• Ki=29.0070
• Kd=3.9551
3.4
Friction Approximation
A difficult task is to approximate friction within a system, it is often non linear but
can be approximated as linear in many cases. Friction of the axis of the pendulum
and within the sensor was approximated as bθ̇ and could be experimentally estimated with the model of the system. The differential equations for the pendulum
seen in equation B.2 implemented in Matlab [16] gives the following plot.
Comparing figure 3.3a with figure 3.3b for the measured values with the same initial
conditions almost the same result is achieved. Here the vertical axis is the angle
of the pendulum in radians and the horizontal axis is time in seconds. This shows
that the mathematical models share mechanical properties and are comparable to
each other.
14
3.5. ROTARY ENCODER
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0
2
4
6
8
10
12
14
16
0
(a) 3.3 Matlab simulation.
3.5
2
4
6
8
10
12
14
16
(b) 3.3 Measured values.
Rotary Encoder
A rotary encoder is used to measure the angle of the pendulum. It is a noncontacting sensor with a 360 degrees position feedback, which in this project is
beneficial because the friction should contribute as little as possible to the control
of the pendulum. 360 degrees position feedback is useful to measure the angle
both for a potential swing-up phase and the balancing phase as the angle should be
measured at all time.
The rotary encoder has a 12 bit resolution which gives an angle accuracy of 12 degree,
which in this case is a sufficient accuracy for the pendulum’s angle.
3.6
Sensor Values
The rotary encoder used gave an analogue output signal for any given angle of the
pendulum, but with accurate examination of the output signal it was determined
that the signal contained statics. The statics are harmful for the PID-controller
due to that fluctuation of a small error is amplified with the derivative part of the
controller. This was first tried to be avoided with a digital filter, the digitalSmooth
function from the Arduino library was tested. This function saves a certain number
of previous values of the output signal, sorts it by value, discards the extreme values
and then the final output is the mean of the remaining values.[9]
This will to a certain extent eliminate the effects of the static within the signal but
the drawback is that it introduces a large delay in the output signal(approximately
6 [ms]) thus impairing the controller.
1
Instead an analogue RC-filter was chosen, where frequencies ω < RC
leaves the
output unaffected, thus reducing the effect of noise. The analogue low pass filter
will not result in a delay of the output signal and is therefore much better than a
15
CHAPTER 3. DEMONSTRATOR
digital filter in this instance. The cut-off frequency chosen was 140 [Hz] and was
determined through experiments adequate. The result of the RC-filter can be seen
in the following figure
Figure 3.4: Filtering of sensor values.
where the green signal is the unfiltered measurements and the yellow signal is the
filtered measurements. As seen in figure 3.4 the statics are drastically reduced.
3.7
Hardware
The mechanical construction consists mainly of three parts, the base on which the
motor is attached, the crank arm on which the rotary encoder and housing of the
bearing are attached and the axis where the sensor and pendulum are connected.
The electrical components used in the demonstrator are the motor driver board
for the L298n H-bridge [11], the Arduino microcontroller board [10] and the DC
motor from Faulhaber, model 028c in the 2842 series [8]. For recreation purposes
important features of the components can be found in appendix A.
Figure 3.5: Blockscheme of electronics.
16
3.8. SOFTWARE
3.8
Software
Simulating the transfer function and state space of the system can be accomplished
through software. To generate working regulators for the system Matlab was used.
Though Matlab has many working functions for analysing a system, some things
are not taken in consideration. Simulink is a program more suitable for analysing
system and has been developed for control theory.
With Simulink statics, discrete signal and sample time can be simulated. These
are concepts that are not shown in the mathematical model analysed in Matlab.
Simulink was used to simulate the system B.23 and the controller seen in the Matlab
code [16]. Each block represent a differential equation for each system. An impulse
is used at the beginning to simulate a angle of the pendulum that is not zero.
Negative feedback is used. With the two scopes the angle of the pendulum and the
voltage given to the DC motor can be shown in a graph.
(a) Simulink Model
(b) Impulse Response
3.8.1
Arduino
Arduino has a build in PWM call function, analogW rite(P IN, DU T Y Y CLE),
which use the fast PWM mode, where the timers count from 0 to 255 with a
frequency of 500 Hz. With a sample frequency of 330 Hz, this made the built
in PWM frequency inadequate and instead pins on the ATmega328 was activated
manually to increase the PWM frequency.
17
Chapter 4
Results
To compare the two controllers the stability of the system is first analysed. This
is done by investigate the position of the poles and zeroes of the system with a PZ
plot. Next a impulse response of the controlled system will be looked at to determine
the rise time, settling time and overshoot. The controllers were also tested on the
demonstrator to examine if they correspond to the mathematical model.
4.1
Comparing Mathematical Controllers
The Linear quadratic regulator and the PID controller successfully moved the poles
so that the system is stable.
Pole Zero plot with PID
4
0.6
3
0.4
2
Imaginary Axis (seconds−1)
Imaginary Axis (seconds−1)
Pole Zero Map with LQ Regulator
0.8
0.2
0
−0.2
1
0
−1
−0.4
−2
−0.6
−3
−0.8
−7
−6
−5
−4
−3
−2
−1
−4
−7
0
−1
−6
−5
−4
−3
−2
−1
0
−1
Real Axis (seconds )
Real Axis (seconds )
(a) 4.1 LQR.
(b) 4.1 PID.
As seen in figure 4.1a and 4.1b all poles are on the left hand side of the imaginary
axis and thus the feedback system is stable.
By analysing a impulse response of the stable system, characteristics such as rise
time, overshoot and settling time can be obtained. The vertical axis in the plots
corresponds to the calculated angle and the horizontal axis is time in seconds. In
19
CHAPTER 4. RESULTS
the plots the systems are not affected by the same impulse thus not generating the
same magnitude in peak value, this will be discussed later in discussions. Though
all characteristics comparing the two systems are unaffected. The impulses are simulated with a built in function in Matlab. From figure 4.2a and 4.2b the following
Impulse Response
−3
Impulse Response with PID
x 10
10
14
12
8
10
6
Amplitude
Amplitude
8
4
6
4
2
2
0
0
−2
0
1
2
3
4
5
6
Time (seconds)
7
8
9
−2
10
0
0.5
(a) 4.2 LQR.
1
1.5
2
2.5
3
Time (seconds)
3.5
4
4.5
5
(b) 4.2 PID.
results were collected
Characteristics
Rise time
Settling time
Overshoot
4.2
LQR
0.77
0.929
0.008
PID
0.0261
1.95
18.4
Unit
s
s
%
Comparing Controllers On Demonstrator
These controllers were used on the demonstrator and the result can be seen in the
following figures.
In figures 4.3a, 4.3b and 4.4 the Digital value corresponds to angle of the pendulum.
The pendulum was placed at a stable position with zero angle velocity before the
regulator started. As seen in figure 4.3a the angle was kept stable for approximately
3.5 seconds. In figure 4.3b the angle was never kept stable. The reason for this
behaviour will be discussed in discussions and conclusions.
A controller was also made by trial and error method until the angle was shown
stable [17]. A short measurement showing the stable system is seen in figure 4.4.
20
4.2. COMPARING CONTROLLERS ON DEMONSTRATOR
ADC Testing
ADC Testing
10
3
2
Digital Value
Digital Value
5
1
0
0
-1
-2
-5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
0.2
0.4
Time in seconds
0.6
0.8
1
(b) 4.3 PID.
ADC Testing
25
20
15
10
5
0
-5
-10
-15
-20
-25
0
0.5
1.2
Time in seconds
(a) 4.3 LQR.
Digital Value
0
1
1.5
2
2.5
3
Time in seconds
Figure 4.4: Own PID.
21
3.5
4
4.5
1.4
1.6
1.8
Chapter 5
Discussion and conclusions
5.1
Discussion
The results when comparing the two regulators was that even though both were
stable and had promising properties by analysing the model was when implemented
on the prototype shown to not be stable. This must depend on that the model is
not a good enough representation of the system. This chapter will try to answer
how the analysis of the model diverges from the testing of the prototype.
Sample Time, Sensor Values and Statics
The development of the regulators was done in Matlab. Here the LQ-method produces a regulator that minimizes the cost function of the system and thus creating
the state space regulator and through PID Tuning a regulator was created with
desired properties. A large flaw in this way of producing a regulator is that Matlab
only takes the transfer function in consideration. This is equivalent to a system
were the measured values are continuous and all states has infinite high refresh rate
and no statics occur.
Another way to produce the controller would be to analyse with Simulink and with
a trial and error approach finding a suitable regulator. With Simulink sample time,
discrete values and statics can be generated to better simulate reality. The most
critical property for the prototype is probably that the resolution of the sensor values are only 0.5 degrees and to approximate this with a continuous function is far
from correct. Even though a filter was implemented small statics from the sensor
data occurred. With a low resolution from the sensor combined with statics results
in that the derivative part of the controller fluctuates drastically and therefore the
controller acts much more unpredictable than what is shown when simulating in
Matlab.
23
CHAPTER 5. DISCUSSION AND CONCLUSIONS
Mechanical Modelling
When computing the transfer function of the system approximations are inevitable.
For all parameters within the model estimations are made. Estimations should be
as accurate as possible and have to be in the right magnitude of the correct value.
One of the few model validations that are made is shown when approximating the
friction within bearing and sensor 3.3a, 3.3b. It is shown that the model is similar
to the prototype but there are distinct differences in the period time and the shape
of the transients. This might be because the approximation of the friction with bθ˙1
[16] is surely not an accurate approximation due to that the friction has a dynamic
part and static part and those are not accounted for.
Another reason to why the period time is incorrect is the approximation of moment
of inertia for the rod. This is approximated with
I0 =
ZZZ
r2 ρ dV =
V
(ρAl)l2
3
(5.1)
where A is the base area, l is the length and ρ is the density of the rod. This is only
for the pendulum and not the axis which the pendulum is attached to and thus a
fault is generated.
The Euler method approximates the derivatives and the smaller the step length the
better estimation of the differential equation. In the Matlab code a step value of
0.001 seconds was chosen and the influence of this is negligible.
So many approximations are made by estimating parameters and simulating systems. Though they are approximated they correspond well to the prototype and it
is believed that the mechanical model is not the reason that the Matlab simulation
does not correlate to the prototype.
The two plots 4.2a and 4.2b was noted not to have the same magnitude in peak
value. This was said to be due to different magnitude in impulses, but was never
motivated. One could argue that this may in fact be due to that the regulators
are not the same for the two systems. This is unlikely due to that the parameters
Kp , Ki , Kd and the speed of the systems are all in the same magnitude. The difference between peak values in the plots are by a factor of 103 and therefore most
probable affected by different impulses.
The plots are comparable due to, if the impulse is scaled so is the input to the
motor, and since a max value of the motor is not determined in Matlab the scaled
impulse would not generate a different results in rise time and settling time, and the
overshoot is in percent which also is unaffected. In other words, only the amplitude
axis is scaled with a larger impulse.
5.2
Conclusions
Constructing a regulator is not a simple procedure. Before choosing a regulator
and work flow one must first understand which approach is the most effective for a
24
5.2. CONCLUSIONS
certain application.
In general a trial and error approach for constructing a PID regulator is recommended, as seen in figure 4.4. This is due to that for most applications it is possible
to test its way towards a working regulator, without taking in consideration the mechanical model and simulations. This is often an easy and efficient way to develop
a controller that is functional.
With this said there are instances when analysing a system might be better. Suppose that a system that is extremely difficult to regulate must be controlled or that
a regulator can not be generated through trial and error. Then the LQ method
might be a much better way to generate a controller. Assuming that the system
can be well approximated with a state space model and it is controllable, then the
solution of the Riccati equation always makes the system stable. Finding this solution through calculations makes it possible to make a working regulator for any
given system that is controllable, though it is difficult and time consuming.
The same system can be simulated through PID-tuner which is an easy and user
friendly way to generate a working regulator. The drawback is that it is also based
on trial and error but for the transfer function and if the system is to difficult to
control it might be a more time consuming approach than the LQ method.
25
Chapter 6
Recommendations and future work
6.1
Recommendations
It is recommend to try and minimize the moment of inertia of the crank arm and
pendulum. We believe that when the motor changes polarity the inertia of moving
components generates a high current that can exceed the maximum allowed current
of driver board.
6.2
Future work
To try controlling the pendulum with also measuring the angle or angle velocity of
the crankshaft and then comparing the performance between this and the regulator
that only measures the angle of the pendulum.
Construct a working swing-up method for the prototype. During the project it was
tried to be achieved with a negative D regulator. This was proved effective but to
hazardous for the driver board to continue. This can be done in other ways as well
that might be more interesting in a control theory perspective and involve a very
accurate mathematical model of the system.
From the beginning of the project it was first desired to control a non stiff pendulum.
This could possibly be done with analysis of mechanics of materials to determine
the bow of the pendulum and therefore the angle to the center of mass given the
angle of the pendulum and torque from the motor. It is a non linear problem that
would be interesting to examine whether it would be able to control or not.
27
Appendix A
Additional information
A.1
Hardware
Figure A.1: mount for the motor, is connected to the planet gear.
1
APPENDIX A. ADDITIONAL INFORMATION
Figure A.2: Crank arm
Figure A.3: The axis of the pendulum and rotary sensor.
2
A.1. HARDWARE
Figure A.4: Crank arm
3
Appendix B
Proofs
B.1
B.1.1
Derivation of the systems transfer function
Mechanical system
Figure B.1: Figure of the crank-arm.
This paragraph is how the transfer function of the system is obtained, step by
step it is shown what approximations are made and how we combine the mechanical
and electrical systems.
y
O:
I2 θ¨2 = Mmotor − F lcrank
(B.1)
Torque equation of the crank-arm seen in figure Figure B.1 with regard to the
motor axis (right figure). Where I2 , θ2 , lcrank is the inertia, angle, half the length
of the crank-arm. F is the force from the pendulum on the crank-arm and Mmotor
is the torque from the motor. The approximation that the friction from bearings
within the motor is insignificant is made.
5
APPENDIX B. PROOFS
Figure B.2: Figure of the pendulum.
2
F = m ẍ + m l1 θ¨1 cos θ1 − m l1 θ˙1 sin θ1
(B.2)
B.2 is the force equation of the pendulum in horizontal direction seen in figure
Figure B.2, m, θ1 , l1 is the mass, angle and half the length of the pendulum shown
in the left figure. From equation B.1 and B.2 the following expression is obtained.
I2 θ¨2 = Mmotor − lcrank m (ẍ + l1 θ¨1 cos θ1 − l1 θ˙1 sin θ1 )
2
With substitution as in B.3 the following equation for the torque of the motor
is derived.
ẍ = θ¨2 lcrank
(B.3)
Substitution of the acceleration to polar coordinates.
2
Mmotor = θ¨2 (I2 + lcrank
m) + lcrank m (ẍ + l1 θ¨1 cos θ1 − l1 θ˙1 sin θ1 )
2
Torque equation for the pendulum with regard to the center of mass witch gives the
following expression. Here the friction from the bearing is estimated with b θ˙1 .
y
mg:
I1 θ¨1 = −l1 N sin θ1 − l1 F cos θ1 + b θ˙1
(B.4)
This gives us the following expression for N sin θ1 + F cos θ1 .
N sin θ1 + F cos θ1 =
6
−I1 θ¨1 + b θ˙1
l1
(B.5)
B.1. DERIVATION OF THE SYSTEMS TRANSFER FUNCTION
The sum of the forces perpendicular to the pendulum and substitution with equation
B.3.
N sin θ1 + F cos θ1 − m g sin θ1 = m l1 θ¨1 + m θ¨2 lcrank cos θ1
(B.6)
With equations B.5 and B.6 combined results in.
−I1 θ¨1 + b θ˙1
− m g sin θ1 = m l1 θ¨1 + m θ¨2 lcrank cos θ1
l1
Due to that the pendulum will be balanced at an angle π + θ where θ is small.
linearisation with small angle approximations are made.
θ≈0→


sin(π + θ) = −θ

cos(π + θ) = −1


θ˙2 = 0
These approximations results in the following equations for the mechanical system.
B.1.2
2
Mmotor = θ¨2 (I2 + lcrank
m) − lcrank m l1 θ¨1
(B.7)
−I1
b θ˙1
θ¨1 (
− m l1 ) +
+ m g θ1 = −m lcrank θ¨2
l1
l1
(B.8)
Electrical system
This is the derivation how the voltage to the motor transitions into mechanical
torque.
Figure B.3: Electrical circuit of DC motor.[4]
Using Kirchoff´s voltage law on the circuit above following equations can be derived
[5]
Ua − URa − ULa − Ub = 0
(B.9)
where Ua is a voltage source, i.e the voltage across the coil of the armature, URa is
the voltage across the resistor R, ULa is the voltage across the inductor L and Ub
7
APPENDIX B. PROOFS
is the back emf voltage.
With Ohm’s law URa can be derived as
URa = Ia Ra
(B.10)
where Ra is the resistance of of R and Ia is the armature current. The voltage
across the inductator is
ULa = La I˙a
(B.11)
where La is the inductance of the armature coil. The back emf Ub can be written
as
Ub = Ke θa
(B.12)
where Ke is the velocity constant and θa is the angular velocity of the armature, i.e
the angular veocity of the motor axis. Equations B.10, B.11 and B.12 inserted in
B.9 gives the following expression
Ua − Ia Ra − La I˙a − Ke θa = 0
(B.13)
To get the sum of the torques a energy balance is done. And the following expression
is derived
Te − Mmotor − Tω̇ − Tω = 0
(B.14)
where Te is the electromagnetic torque, Tω̇ is the torque due to acceleration of the
rotor, Tω is the torque due to the velocity of the rotor.
The electromagnetic torque can be written as
Te = Kt Ia
(B.15)
where Kt is the motor torque constant. Tω̇
Tω̇ = Ja θ¨a
(B.16)
where Ja is the inertia of the rotor and θ¨a is the angular acceleration of the rotor.
Tω
Tω = B θ˙a
(B.17)
where B is the motor viscous friction constant and θ˙a is the angular velocity of the
rotor.
Equations B.15, B.16 and B.17 in B.14 gives the following expression
Kt Ia − Mmotor − Ja θ¨a − Bθa = 0
(B.18)
Substituting Ia from B.13 gives the following expression
Ia =
Ua − La I˙a − Ke θ˙a
Ra
8
(B.19)
B.1. DERIVATION OF THE SYSTEMS TRANSFER FUNCTION
Substituting Mmotor and B.19 inserted in B.18
Mmotor =
Kt (Ua − La I˙A − Ke θ˙a )
− Ja θ¨a − B θ˙a
Ra
(B.20)
As the DC motor used in this project also contains a gear θ˙a and θ¨a is not the same
as θ˙2 and θ¨2 , and thus need to be converted to θ˙2 and θ¨2 via
θ˙2
θ˙a =
14
(B.21)
θ¨2
θ¨a =
14
(B.22)
θ˙2 is divided by 14 because the gear ratio is 14:1.
B.1.3
Combined systems in Laplace domain
Now combine the mechanical and electrical systems and from B.7 and B.20 and
derive an expression of how the input signal (voltage) affects the output signal
(angle)
θ˙2
Kt (Ua − La I˙A − Ke 14
)
θ¨2
θ˙2
2
− Ja − B
= θ¨2 (I2 + lcrank
m) − lcrank m l1 θ¨1 (B.23)
Ra
14
14
If we do not want to control the system with respect to the angle of the crank-arm
we could transform B.8 to the Laplace domain and solve θ2 as a function of θ1 .
L → θ2 (s) =
1
s2 ( −I
l1 − m l 1 ) +
−m lcrank
bs
l1
s2
+mg
θ1 (s)
where we define the transfer function Gi (s)
Gi (s) =
1
s2 ( −I
l1 − m l 1 ) +
−m lcrank
bs
l1
s2
+mg
(B.24)
From equation B.23 and B.24 the following expression is obtained for the combined
system with regard to all motor parameters. L →
(−Ua + La I˙A )
kt
Ra (s2 (− Ja G14i (s)
2
+ lcrank ml1 − Gi (s)(I2 + lcrank
m)) − s( BG14i (s) +
Kt Ke Gi (s)
))
14Ra
If now we choose a motor where the material parameters La and B is small we can
approximate the systems transfer function as the following.
G(s) =
−kt
Ra (s2 (lcrank ml1
2
− Gi (s)(I2 + lcrank
m+
9
Ja
14 ))
−
sGi (s)Kt Ke
)
14Ra
(B.25)
= θ1 (s)
APPENDIX B. PROOFS
B.2
Pendulum Friction Approximation
It is important that the mathematical model represent the physical system as good
as possible. To do this it is needed to estimate the friction from in this case the
bearing and rotary sensor attached to the axis of the pendulum.
To approximate the friction the pendulum is dropped from a determined angle and
then compared to the mathematical model of the system. The most optimal value
of the friction constant is then through experiments chosen when comparing the
transients of the model and system. To simplify the system the crank arm was in a
fixed position. Now the differential equation for the angle of the pendulum is given
by B.26.
y
mg:
I1 θ¨1 = −l1 N sin θ1 − l1 F cos θ1 + b θ˙1
(B.26)
Since the crank arm is in a fixed position the force F applied from the crank arm
to the pendulum is 0. With the initial conditions the we drop the pendulum from
a fixed angle the differential can be solved.
−l1 N sin θ1 (t) − b θ˙1 (t)
θ¨1 (t) =
I1
π
θ1 (0) =
8
θ˙1 (0) = 0
θ¨1 (0) = 0
This is a second order differential equation and is solved through Euler method
in two stages. The Euler method is a way to approximate an ordinary differential
equation and since this is a second order we can formulate two first order differential
equations. First the variables are substituted.
θ1 (t) = x1 (t)
θ˙1 (t) = x2 (t)
θ¨1 (t) = x3 (t)
Euler method approximates the angle and angle velocity
x1 (t + ∆t) ≈ x2 (t)∆t + x1 (t)
x2 (t + ∆t) ≈ x3 (t)∆t + x2 (t)
and the approximation of the angle acceleration are
x3 (t) =
−b x2 (t) − l1 N sin (x1 (t))
.
I1
Now the angle as a function of time be calculated by step-wise calculating x3 and x2
to approximate the angle x1 given the initial conditions. The angle is then plotted
as a function of time for a specific b, a friction value is chosen as the one most
corresponding to the actual oscillation of the pendulum [16].
10
Appendix C
References
1. "Design and Implementation of Control System for Inverted Pendulum” [Raheel Tariq,Saad Rasheed,Sajid Ali] avaliable at http://sajidpak.weebly.com/
uploads/1/8/4/7/18473948/final_report_ip-1.pdf
2. Torkel Glad and Lennart Ljung. Reglerteknik grundläggande teori. 4:th edition. Demigraf Poland 2014. Studentlitteratur AB.
3. [Kwakernaak, Huibert and Sivan, Raphael] (1972). Linear Optimal Control
Systems. First Edition. Wiley-Interscience. ISBN 0-471-51110-2
4. [Ivan Virgala, Peter Frankovský, Mária Kenderová]American Journal of Mechanical Engineering. 2013, avaliable at http://pubs.sciepub.com/ajme/1/1/1/figure/
3
5. 24.509 Lecture Notes by Dr. J. R. White, UMass-Lowell (Spring 1997) avaliable
at http://www.profjrwhite.com/system_dynamics/sdyn/s6/s6fmathm/s6fmathm.
html
6. Datasheet Atmega [2009 Atmel Corporation] avaliable at https://www.sparkfun.
com/datasheets/Components/SMD/ATMega328.pdf
7. Secrets of Arduino PWM [Ken Shirriff] avaliable at http://www.arduino.cc/
en/Tutorial/SecretsOfArduinoPWM
8. [MicroMo Electronics, Inc] DC-Micromotors data sheet avaliable at http:
//www.me.mtu.edu/~wjendres/ProductRealization1Course/Motor_Specs.pdf
9. DigitalSmooth [Paul Badger](2007) avaliable at http://playground.arduino.
cc/Main/DigitalSmooth
10.Arduino reference page avaliable at http://www.arduino.cc/en/Main/ArduinoBoardUno
11.L298N Motor Driver Board reference page avaliable at http://www.geeetech.
com/wiki/index.php/L298N_Motor_Driver_Board
12.[Piher] MTS-360 data sheet avaliable at https://www1.elfa.se/data1/wwwroot/
assets/datasheets/MTS360_series_eng_tds.pdf
13.[faulhaber] planatary gearheads data sheet avaliable at https://fmcc.faulhaber.
com/resources/img/EN_23-1_FMM.PDF
11
APPENDIX C. REFERENCES
14.F uruta, K., Y amakita, M.andKobayashi, S.(1992)ıSwing−upcontrolof invertedpendulumusingpseud
statef eedback, Journalof SystemsandControlEngineering.
15. Hans Johansson.Elektroteknik. 2013 års upplaga. Instutionen för Maskinkonstruktion Mekatronik 2013. KTH
16.Dropbox project:"inverted pendulum"[Filip Stenbeck] (2015) avaliable at https:
//www.dropbox.com/sh/dugu1enhohp0mie/AACE_windUtKmPTc8eorxcFJa?dl=0
17.["IVP-BOT", Filip Stenbeck, Aron Nygren](2015) avaliable at https://www.
youtube.com/watch?v=wZ82Sgmk4pw
12
TRITA MMK 2015:21 MDAB074
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