UNIVERSIT ` A DEGLI STUDI DI ROMA TOR VERGATA FACOLT ` A DI INGEGNERIA CORSO DI LAUREA MAGISTRALE IN INGEGNERIA DELL’AUTOMAZIONE A.A. 2010/2011 Tesi di Laurea Modeling and nonlinear control for MAST tokamak RELATORE CANDIDATO Dott. Daniele Carnevale Antonio De Paola CORRELATORE Dott. Luigi Pangione
92
Embed
UNIVERSITA DEGLI STUDI DI ROMA` TOR VERGATAcontrol.disp.uniroma2.it/carnevale/archivio/Tesi/antoniodepaola/... · tween ”Universit`a degli studi di Roma Tor Vergata” and ”Culham
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
This thesis has been developed in the context of the scientific research on controlled
thermonuclear fusion. The final aim of the scientists devoted to this field is to achieve
the necessary knowledge to create a thermonuclear fusion reactor. This device would
allow commercial production of net usable power by a nuclear fusion process. This
source of energy, with respect to nuclear fission energy production, is cleaner and
safer. More specifically, this work has been realized through the collaboration be-
tween ”Universita degli studi di Roma Tor Vergata” and ”Culham Centre for Fusion
Energy” that runs MAST experiment, the tokamak considered for this thesis. The
professionals who have made this collaboration possible are the professors Luca Za-
ccarian and Daniele Carnevale from ”Dipartimento di Ing. Informatica, sistemi e
produzione”, doctor Luigi Pangione and Graham McArdle from ”Culham Center for
Fusion Energy”. The subject of this work has been the design of a nonlinear system,
to be added to the existing shape controller of MAST, which would avoid current sat-
uration on the electric circuits of the poloidal coils. In order to do so, a model of the
plant has been realized using as a starting point the precious work of the CREATE
team, that has developed the linearized model of MAST, and of Graham McArdle,
which has created a model of the plasma shape controller (PCS). This preliminary
modeling phase has made possible to design the nonlinear control and to test its
performances in a simulative environment. The first chapter of this work represents
Abstract 1
Abstract
a general introduction to the physical principles of thermonuclear fusion, it also de-
scribes the purposes of magnetic confinement and how this confinement is performed
by tokamaks. There is also a general overview of spherical tokamaks and a presenta-
tion of the MAST experiment. Chapter 2 contains the description of the CREATE-L
model, the linearized model which has been used in the creation of the simulation
environment. The mismatchings and the problems that have been experienced during
its implementation, together with the proposed solutions, are discussed. In Chapter 3
there is a detailed description of the plasma shape controller (PCS) which is used at
the moment on MAST: the control law is analyzed and the saturation phenomena that
are desired to be avoided are discussed. There is also the description of the system
used to model the PCS, which has been implemented in the simulation environment
and tested. Chapter 4 addresses the problem of the currents saturation on the coils
and describes the proposed solution: a nonlinear subcompensator which allocates the
inputs in order to minimize a cost function and achieve a trade-off between output
performances and input allocation. Different versions of the allocator are described
and tested through the simulation environment and their performances are compared
and discussed. In the last chapter the results are summarized and possible future
developments and applications for the present work are considered.
Abstract 2
Chapter 1
Nuclear fusion and MAST
1.1 Nuclear fusion
Nuclear fusion is, in a sense, the opposite of nuclear fission. Fission, which is a mature
technology, produces energy through the splitting of heavy atoms like uranium in
controlled energy chain reactions. Unfortunately, the by-products of fission are highly
radioactive and long lasting. In contrast, fusion is the process by which the nuclei
of two light atoms such as hydrogen are fused together to form a heavier (helium)
nucleus, with energy produced as by-product. This process is illustrated in Figure 1.1
where two isotopes of hydrogen (deuterium and tritium) combine to form a helium
nucleus plus an energetic neutron.
Figure 1.1: The process of nuclear fusion.
3
Cap. 1 Nuclear fusion and MAST §1.1 Nuclear fusion
In this reaction a certain amount of mass changes form to appear as the kinetic
energy of the products, in agreement with the equation E = ∆mc2. Fusion produces
no air pollution or greenhouse gases since the reaction product is helium, a noble gas
that is totally inert. The primary sources of radioactive by-products are neutron-
activated materials (materials made radioactive by neutron bombardment) which can
be safely and easily disposed of within a human lifetime, in contrast to most fission
by-products which require special storage and handling for thousands of years. The
primary challenge of fusion is to confine the plasma, a state of matter similar to gas
in which most of the particles are ionized, while it is heated and its pressure increases
to initiate and sustain fusion reaction. There are three known ways to do so:
• Gravitational confinement: the method used by the stars. The gravitational
forces compress matter, mostly hydrogen, up to very large densities and tem-
peratures at the star-centers, igniting the fusion reaction. The same gravita-
tional field balances the enormous thermal expansion forces, maintaining the
thermonuclear reactions in a star, like the sun, at a controlled and steady rate.
Unfortunately huge gravitational forces, not available on Earth, are required.
• Inertial confinement: a fuel target, typically a pellet containing a mixture of
deuterium and tritium, is compressed and heated through high-energy beams of
laser light to initiate the nuclear fusion reaction. This method has not reached
the efficiency and the results that were expected in the 1970s but new approaches
and techniques are currently experimented in some research centers such as the
NIF (National Ignition Facility) in California and the Laser Megajoule in France.
• Magnetic confinement: hydrogen atoms are ionized, so that magnetic fields can
4
Cap. 1 Nuclear fusion and MAST §1.2 Tokamak
exert a force on them, according to the Lorentz law, and confine them in the
form of a plasma.
The magnetic confinement is the most promising technique and it is worth spend-
ing a few words to describe it in more detail. In normal conditions the gas is unconfined
and free to move, if the gas is ionized and subject to a magnetic field the forces im-
posed by the field cause the ions to travel along the magnetic fields lines with a radius
known as the Larmor radius. Ions and electrons have opposite charges, these particles
move in opposite directions along the field lines under the influence of an electric field.
Since positively charged ions are more massive than electrons, the positive ions rotate
in a much larger radius circle. The number of rotations per second at which the ions
and electrons rotate around the field lines are the ion cyclotron frequency and electron
cyclotron frequency, respectively.
Figure 1.2: The trajectory of ionized gas subject to a magnetic field.
1.2 Tokamak
The most promising device for magnetic confinement of plasma is the tokamak (Rus-
sian acronym for ”Toroidal chamber with axial magnetic field”), a device shaped as
a torus (or doughnut) that has been originally designed in Russia during the 1950s.
The general structure of the device is shown in Figure 1.3.
5
Cap. 1 Nuclear fusion and MAST §1.2 Tokamak
Figure 1.3: General structure of the tokamak.
The main problem with the magnetic confinement described in the previous section
is that the particles remain confined by the magnetic field until the field lines end or
dissipate, contrary to the desire of keeping them confined. To solve this problem, the
tokamak bends the field lines into a torus so that these lines continue forever. The
magnetic fields that create and confine the plasma in the tokamak are generated by
electric coils which can be located outside the chamber, such in JET and most of the
tokamak, or inside, as in MAST experiment. Since the plasma is ionized and confined
inside the toroidal chamber, it can be considered as a coil circuit, the secondary side
of a coupled circuit whose primary side is the central solenoid. Figure 1.4 displays
the currents and fields that are present inside the tokamak.
All existing tokamak are pulsed devices, that is, the plasma is maintained within
the tokamak for a short time: from a few seconds to several minutes. There is no
agreement yet among fusion scientist on whether a fusion reactor must operate with
truly steady-state (essentially infinite length) pulses or just operate with a succession
of sufficiently long pulses. The main reason for this limitation is that, in order to
6
Cap. 1 Nuclear fusion and MAST §1.2 Tokamak
Figure 1.4: Currents and magnetic fields of the tokamak.
sustain constant values of plasma current, the derivative of the current on the central
solenoid must be constantly ramping up (or down), rapidly reaching a structural lim-
it on the coil which cannot be exceeded. To avoid this limitation, different methods
to sustain the plasma current have been studied and introduced, such as LH/ECRH
antennas or neutral beams injectors, currently used at MAST. All tokamak produce
plasma pulses (also referred to as shots) with approximatively the same sequence of
events. Time during the discharge is measured relative to t=0: the time when the
physical experiment starts after all the preliminary operations. The toroidal field coil
current is brought up early to create a constant magnetic field to confine the plasma
when this is initially created. Just prior to t=0 deuterium is puffed into the interior of
the torus and the ohmic heating coil (primary coil in Figure 1.4) is brought to its max-
imum positive current, in preparation for pulse initiation. At t=0 the primary coil is
driven down to produce a large electric field within the torus. This electric field accel-
erates free electrons, which collide with and rip apart the neutral gas atoms, thereby
7
Cap. 1 Nuclear fusion and MAST §1.3 Spherical tokamaks and MAST
producing the ionized gas or plasma. Since plasma consists of charged particles that
are free to move, it can be considered as a conductor. Consequently, immediately after
plasma initiation, the primary coil current continues its downward ramp and operates
as the primary side of a transformer whose secondary is the conductive plasma. At
the end of the downward ramp of the primary coil the plasma current is gradually
driven to zero and the shot moves towards its conclusion. The separate time intervals
in which the plasma current is increasing, constant and decreasing are referred to,
respectively, as ramp-up, flat-top and ramp-down phase of the shot. At the moment
the tokamak technology has reached a point such as the quantity of energy produced
by these devices is almost as much as the one used in heating and confining the plas-
ma. The next step is the construction and operation of the proposed International
Thermonuclear Experimental Reactor (ITER) which, supported by an international
consortium of governments, will provide major advancements in fusion physics and
constitute a testbed for developing technology to support high fusion levels.
1.3 Spherical tokamaks and MAST
MAST (Mega Amp Spherical Tokamak) is the fusion energy experiment, based at
Culham Centre for Fusion Energy, which has been used for the present thesis. Its
main difference from a classical tokamak is the shape: since the origin of tokamak
in the 1950s, research is mainly concentrated on machines that hold the plasma in a
doughnut-shaped vacuum vessel around a central column. MAST belongs to a differ-
ent category of tokamak, named spherical tokamak, which presents a more compact,
cored apple shape and a lower aspect ratio.
Spherical tokamak hold plasmas in tighter magnetic fields and could result in more
economical and efficient fusion power for many reasons:
8
Cap. 1 Nuclear fusion and MAST §1.3 Spherical tokamaks and MAST
• plasmas are confined at higher pressures for a given magnetic field. The greater
the pressure, the higher the power output and the more cost-effective the fusion
device.
• The magnetic field needed to keep the plasma stable can be a factor up to ten
times less than in conventional tokamak, also allowing more efficient plasmas.
• Spherical tokamaks are cheaper, since they do not need to be as large as con-
ventional machines and superconducting magnets, which are very expensive, are
not required.
Spherical tokamaks, at the moment, are at a very early stage of development and
they will not be used for the first nuclear fusion power plants but they can be very
useful for component test facilities and they are providing insight into the way changes
in the characteristic of the magnetic field affect plasma behaviour. These informa-
tions have been very useful for the development of ITER, the advanced experimental
tokamak which is being built in France. MAST, along with NSTX at Princeton, is
one of the world’s two leading spherical tokamak.Table 1.1 and Figure 1.5 give an
idea of its dimension, structure and technical specifications.
Plasma Vacuum vesselCurrent 1, 300, 000 amps Height 44.4mCore up to Diameter 4mtemperature 23, 000, 000◦CPulse length up to 1 second Material Stainless steel 304LNPlasma 8m3 Toroidal field 24 turns, 0.6 teslavolume @ 0.7m radiusDensity 1020 particles/m3 Total mass 70 tonnes
Table 1.1: Technical specifications of MAST experiment.
9
Cap. 1 Nuclear fusion and MAST §1.3 Spherical tokamaks and MAST
Figure 1.5: Section of MAST.
A cross-section of the MAST vessel and the position of the six PF (poloidal field)
coils is shown in Figure 1.6.
Since the present thesis has focused on the control system on the PF coils which
confine and shape the plasma, it is worth describing them in more detail:
• Solenoid (P1): Provides the magnetizing field used to control plasma current,
it is analogous to the primary winding of a transformer, where the plasma itself
acts as a single-turn secondary winding. It is composed of four layers (152 turns
per layers), 2.7 meters long. Its power supply (P1PS) is four quadrant and it
normally drives current in the range [−45kA, +45kA] although its maximum
current range is [−55kA, +55kA]
• Divertor coil (P2): It is composed of two independent windings in each coil pack,
it can be used to achieve the desired plasma configuration and compensate the
10
Cap. 1 Nuclear fusion and MAST §1.3 Spherical tokamaks and MAST
Figure 1.6: Cross-section of the MAST vessel and position of the six PF coils.
stray field from the solenoid. Its power supply (EFPS) has a single direction,
although this direction can be reversed during pulse. The maximum current
value that can be driven is 27 kA.
• Start-up coil (P3): It is a capacitor bank used for the pre-ionization of the
plasma. It has no power supply or feedback, just a switch that starts the
discharging of the capacitor hence it cannot really be considered an actuator
from the plasma shape controller point of view.
• Vertical field/shaping coils (P4 and P5): Both coils contribute to the main
vertical field for radial position control. The shape and elongation depend both
on the plasma internal profile and on how the total vertical field current is
divided between P4 and P5. Each of them is driven by a bank which provide
the rapid initial vertical field rise and by power supplies (respectively SFPS
11
Cap. 1 Nuclear fusion and MAST §1.3 Spherical tokamaks and MAST
and MFPS), which provide controlled flat-top current. Both power supplies can
drive current in a single direction. The maximum value of the current is 17kA
for P4 and 18 kA for P5.
• Vertical position coil (P6): There are actually two coils in one can, each of them
with two turns. These coils provide the radial field for vertical position control.
Since the vertical dynamics are much faster than the time scale of the existing
MAST PCS, they are independently driven by a separate analogue controller.
The time behaviour of the currents on the PF coils for a standard shot, together
with the associated value of the plasma current, is shown in Figure 1.7
Figure 1.7: Typical PF current evolution.
12
Chapter 2
CREATE model
2.1 General description
The first step for the realization of a simulation environment has been the choice of
the model of MAST. The model that has been adopted is the CREATE-L model,
developed by the CREATE team. This model, which has already been successfully
tested on various tokamaks (TCV, FTU and JET), is a linearized model about an
equilibrium point. It is obtained from the following set of equations:
dΨ
dt+ RI = U Circuit equations
[Ψ, Y ]T = η(I,W ) Grad-Shafranov constraint
(2.1.1)
I Poloidal field (PF) circuit currents and plasma current Ip
Ψ Fluxes linked with the above circuitsU Applied voltagesR Resistance matrixW Poloidal beta (βp) and internal inductance (li)Y Most remaining quantities of interest
(plasma shape descriptors and current moments)
Table 2.1: List of phisical quantities in eq. 2.1.1.
The Grad-Shafranov constraint is the equilibrium equation in ideal magnetohy-
drodynamics (MHD) for a two dimensional plasma. This set of equations is linearized
13
Cap. 2 CREATE model §2.1 General description
using incremental ratios or Jacobian matrix and the result is the eq. 2.1.2 (L∗ is
an inductance matrix modified by the presence of the plasma which, differently from
many similar models, is not included in the state space).
L∗di
dt+ Ri = u − L∗
E
dw
dt
y = Ci + Fw
(2.1.2)
with
L =∂Ψ
∂ILE =
∂Ψ
∂WC =
∂Y
∂IF =
∂Y
∂W
From the equation 2.1.2 it is quite straightforward to obtain a state-space form of
the modeldx
dt= Ax + Bu + E
dw
dt
Y = Cx + Fw
(2.1.3)
with
x = i A = −(L∗)−1R B = (L∗)−1 E = −(L∗)−1L∗E
In the starting configuration of the model, the signal of interests are the following:
Figure 2.10: Measured plasma current (blue) and simulated plasma current(red) whenthe parameter apl is modified in order to minimize the quadratic error.
25
Cap. 2 CREATE model §2.3 Feedforward currents simulations
2.3 Feedforward currents simulations
After the model has been modified in the way described in the previous sections, it
is possible to test it running the first simulations. Currents on the coils are retrieved
from the database and fed in the model, the results are then compared with the
correspondent measured signals. The currents of the shot n. 24542 during the flat-
top phase which are used as inputs are shown in Figure 2.11, while in Figures 2.12,
2.13 and 2.14 there is the comparison between measured and simulated signals. It can
be observed that the outputs of the CREATE-L model, especially the plasma current,
have a good fitting with the correspondent measured signals. A mismatching can be
noticed in the fields measurements but the relative error is below 5% and it has been
considered acceptable.
0.3 0.32 0.34 0.36 0.38−3.6
−3.5
−3.4
−3.3
−3.2
−3.1
−3x 10
4
time [s]
P1
curr
ent [
A]
0.3 0.32 0.34 0.36 0.38500
1000
1500
2000
2500
3000
3500
time [s]
P2
curr
ent [
A]
0.3 0.32 0.34 0.36 0.38−9200
−9100
−9000
−8900
−8800
−8700
−8600
time [s]
P4
curr
ent [
A]
0.3 0.32 0.34 0.36 0.38−6650
−6600
−6550
−6500
−6450
−6400
−6350
−6300
time [s]
P5
curr
ent [
A]
Figure 2.11: Measured currents on the coils P1,P2,P4 and P5 for the shot n.24542during the flat-top phase.
26
Cap. 2 CREATE model §2.3 Feedforward currents simulations
The matrices Rc and Lc that have been used to test the model have been retrieved,
for each shot, from the database of the controller which uses them to convert its
currents requests in voltages. Voltages measurements from a certain shot are used as
input of the model and the resulting currents are compared with the measured ones.
The results are shown in Figure 2.15.
−0.1 0 0.1 0.2 0.3−4
−2
0
2
4
6x 10
4
time [s]
P1
curr
ent [
A]
meas. dataest. data
−0.1 0 0.1 0.2 0.3−5000
0
5000
10000
15000
time [s]
P2
curr
ent [
A]
meas. dataest. data
−0.1 0 0.1 0.2 0.3−10000
−8000
−6000
−4000
−2000
0
2000
time [s]
P4
curr
ent [
A]
meas. dataest. data
−0.1 0 0.1 0.2 0.3−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
time [s]
P5
curr
ent [
A]
meas. dataest. data
Figure 2.15: The measured current for the shot n. 24542 (blue) are compared withthe estimation of the R-L model (green).
It is clear from the graph that, although a certain fitting of the currents is achieved,
there are still some inaccuracies. The main ones are thought to be the following:
• The discharge of the bank of capacitors on P3 at t = 0 causes an induction
29
Cap. 2 CREATE model §2.4 Model of the coils
effect which cannot be ignored: this is likely the cause of the increase of error
in the current estimation error at that time.
• the presence of the plasma (and its induction effect) is not considered.
In order to achieve a better estimation of the currents, a model error has been
introduced:
yem = ζ(uem) (2.4.3)
The vector uem includes the voltages on the six coils (as to take in account the
inductive phenomena between P3, P6 and the other four coils) and the plasma current
(in order to consider the presence of the plasma) while yem is a vector composed by
the current error estimation on P1, P2, P4 and P5 and the current on P3 and P6.
This black-box system has then been identified through the Matlab Identification
Toolbox, using the data of four shots retrieved from the database (n. 24532, 24533,
24534 and 24538). Iterative prediction-error minimization and subspace method have
been tried as well as different orders of the system. On the basis of the simulative
results, the model chosen to identify the estimation current error has been a tenth-
order state-space model obtained with the subspace method and described by the
following equations:xem = Aemxem + Bemuem
yem = Cemxem
(2.4.4)
The model obtained through the identification has been tested with a validation
shot (n.24542): in Figure 2.16 the error of the first coil model is compared with the
estimation performed by the error model and in Figure 2.17 the new error on the
current estimation is compared with the original one. Other simulations have been
run for different shots (specifically n. 24552, 24567, 24568 an 24572) showing the
same performances for the error identification model.
30
Cap. 2 CREATE model §2.4 Model of the coils
−0.1 0 0.1 0.2 0.3 0.4−4000
−3000
−2000
−1000
0
1000
2000
3000
time [s]
P1
erro
r [A
]
meas. errorest. error
−0.1 0 0.1 0.2 0.3 0.4−3000
−2000
−1000
0
1000
2000
time [s]
P2
erro
r [A
]
meas. errorest. error
−0.1 0 0.1 0.2 0.3 0.4−2000
−1500
−1000
−500
0
500
1000
time [s]
P4
erro
r [A
]
meas. errorest. error
−0.1 0 0.1 0.2 0.3 0.4−3000
−2500
−2000
−1500
−1000
−500
0
500
time [s]
P4
erro
r [A
]
meas. errorest. error
Figure 2.16: Estimation errors of the currents on the coil P1,P2,P4 and P5 (blue) andnew estimate of the current errors(red).
−0.1 0 0.1 0.2 0.3 0.4−4000
−3000
−2000
−1000
0
1000
2000
time [s]
P1
erro
r [A
]
−0.1 0 0.1 0.2 0.3 0.4−3000
−2000
−1000
0
1000
2000
time [s]
P2
erro
r [A
]
−0.1 0 0.1 0.2 0.3 0.4−2000
−1500
−1000
−500
0
500
1000
time [s]
P4
erro
r [A
]
−0.1 0 0.1 0.2 0.3 0.4−3000
−2500
−2000
−1500
−1000
−500
0
500
time [s]
P5
erro
r [A
]
Figure 2.17: Estimation errors on the currents using the first model of the coils (blue)and the second (green).
31
Cap. 2 CREATE model §2.4 Model of the coils
The system described by the eq. 2.4.4 is used to improve the current estimation
subtracting from it the estimated error:
Icm2 = Icm1 − yem (2.4.5)
This version of the coils model, represented in Figure 2.18, leads to a general
improvement of the results, as can be seen in Figure 2.19.
Error model
R-L model
1 cm V -
+
pl I
3 V
6 V
1 cm I 2 cm I
em y _
Figure 2.18: Scheme for the coils model with current error estimation.
−0.1 0 0.1 0.2 0.3−4
−2
0
2
4
6x 10
4
time [s]
P1
curr
ent [
A]
meas. dataest. data
−0.1 0 0.1 0.2 0.3−5000
0
5000
10000
15000
time [s]
P2
curr
ent [
A]
meas. dataest. data
−0.1 0 0.1 0.2 0.3−10000
−8000
−6000
−4000
−2000
0
2000
time [s]
P4
curr
ent [
A]
meas. dataest. data
−0.1 0 0.1 0.2 0.3−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
time [s]
P5
curr
ent [
A]
meas. dataest. data
Figure 2.19: The measured current for the shot n. 24542 (blue) are compared withthe results of the coils model which includes the error estimation (green).
32
Cap. 2 CREATE model §2.5 Feedforward voltages simulations
2.5 Feedforward voltages simulations
Once the model of the coils has been created, it has been possible to run feed-forward
simulations of the cascade coils-CREATE model in order to validate its behaviour
in the final model. Since these simulations are only run during the flat-top phase,
whose time interval will be henceforth expressed as [tin, tfin], it has been necessary
to correctly set the initial conditions on the coils model. If only the first part of
the model had been used, the initial value of its states would have been the value
of the measured coil currents at tin. It is slightly more complicated to set the initial
conditions if the error model is used: its ten states have been obtained through a
black-box identification and do not correspond to any physical parameter. To set
them correctly, a feed-forward simulation of the coils is preemptively run: the value
of xem at t = tin is used as initial condition of the cascade simulation and the initial
condition for the R-L model are such that the estimation error of the currents at
t = tin is equal to 0:
Icm1(tin) = I(tin) + Cemxem(tin) (2.5.1)
It is now possible to properly run the simulation of the cascade, whose results are
shown in Figures 2.20, 2.21, 2.22, 2.23 and 2.24. The simulated values of the currents,
compared in Figure 2.21 with the measured ones, can be considered satisfactory: the
highest error is on the coil P5 and it is lesser than 5%. The plasma current in Figure
2.22 shows a good fit with the actual one if the measurement noise is not considered
while the differences on fields and fluxes are considered acceptable.
33
Cap. 2 CREATE model §2.5 Feedforward voltages simulations
0.3 0.32 0.34 0.36 0.38−750
−700
−650
−600
−550
−500
−450
time [s]
volta
ge o
n P
1 [V
]
0.3 0.32 0.34 0.36 0.38−200
−150
−100
−50
0
50
100
150
time [s]
volta
ge o
n P
2 [V
]
0.3 0.32 0.34 0.36 0.38−200
−100
0
100
200
300
400
500
time [s]
volta
ge o
n P
4 [V
]
0.3 0.32 0.34 0.36 0.38−200
−100
0
100
200
300
400
time [s]
volta
ge o
n P
5 [V
]
Figure 2.20: Measured voltages on the coils P1,P2,P4 and P5 for the shot n.24542during the flat-top phase.
0.3 0.32 0.34 0.36 0.38−3.6
−3.5
−3.4
−3.3
−3.2
−3.1
−3x 10
4
time [s]
P1
curr
ent [
A]
meas. datasim. data
0.3 0.32 0.34 0.36 0.38500
1000
1500
2000
2500
3000
3500
time [s]
P2
curr
ent [
A]
meas. datasim. data
0.3 0.32 0.34 0.36 0.38−9400
−9200
−9000
−8800
−8600
time [s]
P4
curr
ent [
A]
meas. datasim. data
0.3 0.32 0.34 0.36 0.38−7000
−6900
−6800
−6700
−6600
−6500
−6400
−6300
time [s]
P5
curr
ent [
A]
meas. datasim. data
Figure 2.21: Measured currents on the coils P1,P2,P4 and P5 (blue) and simulatedcurrents for the shot n.24542 during the flat-top phase.
34
Cap. 2 CREATE model §2.5 Feedforward voltages simulations
Figure 3.5: Measured plasma current (blue) and simulated plasma current(red) forthe shot n. 24542.
44
Cap. 3 PCS: Plasma control system §3.3 Simulations
0.3 0.32 0.34 0.36 0.380.19
0.195
0.2
0.205
0.21
time [s]
ccbv
11 fi
eld
[T]
meas. datasim. data
0.3 0.32 0.34 0.36 0.380.33
0.34
0.35
0.36
0.37
time [s]
ccbv
16 fi
eld
[T]
meas. datasim. data
0.3 0.32 0.34 0.36 0.380.38
0.39
0.4
0.41
0.42
0.43
time [s]
ccbv
20 fi
eld
[T]
meas. datasim. data
0.3 0.32 0.34 0.36 0.380.33
0.34
0.35
0.36
0.37
0.38
time [s]
ccbv
24 fi
eld
[T]
meas. datasim. data
Figure 3.6: Measured fields (blue) and simulated ones (red) for the shot n. 24542.
0.3 0.32 0.34 0.36 0.38
−0.28
−0.27
−0.26
−0.25
−0.24
−0.23
time [s]
flcc0
3 flu
x [W
b]
meas. datasim. data
0.3 0.32 0.34 0.36 0.38−0.27
−0.26
−0.25
−0.24
−0.23
−0.22
time [s]
flcc0
7 flu
x [W
b]
meas. datasim. data
0.3 0.32 0.34 0.36 0.38
−1.16
−1.14
−1.12
−1.1
−1.08
−1.06
time [s]
flp4u
4 flu
x [W
b]
meas. datasim. data
0.3 0.32 0.34 0.36 0.38−1.26
−1.24
−1.22
−1.2
−1.18
−1.16
−1.14
time [s]
flp4l
4 flu
x [W
b]
meas. datasim. data
Figure 3.7: Measured fluxes (blue) and simulated ones (red) for the shot n. 24542.
45
Chapter 4
The input allocator
4.1 General introduction
The input allocation is a technique which provides input variations generated by a
given controller in order to achieve additional performances, preserving closed-loop
properties. In the case of MIMO systems with input redundancy, it is possible to
design a control system which performs a suitable allocation of the actuators without
affecting the plant dynamics or at least the steady-state of the plant outputs. If the
system has more outputs than inputs, the allocator can be considered as a way to
trade some output performances, for example zero steady-state tracking error, for a
more desirable input allocation. Given the canonical scheme of a feed back controlled
linear system, the input allocation is realized by adding a subcompensator, shown in
Fig. 4.1, between the controller output yc and the plant input u, according to the
following equations:
uc = y − P ∗ya
u = yc + ya(4.1.1)
The inputs of the allocator are the controller output yc and the steady-state vari-
ation δy introduced by the subcompensator on the outputs of the plant. Since the
model of the plant is linear, it is possible to obtain δy as the product of ya by the
transfer function P (s) of the plant evaluated for s = 0. For the sake of clarity, all
46
Cap. 4 The input allocator §4.1 General introduction
Controller Plant
Allocator P*
+
- + -
+
r
c u c y u
d
y
a y
y d
Figure 4.1: General scheme of the plant with allocator.
steady-state signals and transfer functions will be henceforth denoted with an asterisk,
therefore the steady-state gain matrix of the plant will be P ∗. It is worth underlining
that the signal δy is subtracted to the output of the plant feedbacked to the controller
as to hide the intervention of the allocator to the controller and, consequently, keep
unchanged the steady-state value y∗c . The trade-off between the modified steady state
value u∗ and the associated input modification δy∗ can be measured by a continuously
differentiable cost function J(u∗, δy∗).
The dynamics of the allocator are described by the relations:
w = −ρK
(OJ
[IP ∗
]B0
)′
ya = B0w
(4.1.2)
where K is a symmetric positive definite matrix, B0 is a suitable full column rank
matrix and OJ denotes the gradient of function J . It can be shown that (see [1] for
more details), if the following holds:
• J(u∗, δy∗) is continuous differentiable and, for any fixed value of y∗c , is radially
unbounded and strictly convex.
47
Cap. 4 The input allocator §4.2 Design of the allocator
• Defined J(w∗) as follows:
J(w∗).= J(y∗
c + B0w∗, P ∗B0w
∗)
there exist positive constant c, k1, k2 and k3 such that, if V1(w).= J(w) −
min(J(s)
), the following holds:
k1|w − w∗|c ≤ V1(w) ≤ k2|w − w∗|c,
OV1(w)w ≤ −k3|w − w∗|c
then, for any y∗c , the system 4.2.4 has exactly one globally exponentially stable
equilibrium w∗ which is the minimizer of J(u∗, δy∗) with respect to every steady state
change in u and y that the allocator can introduce. Furthermore, if the transfer
function P (s) from u to y has no pole at s=0, it can be shown that there exists
a ρ > 0 such that for any ρ ∈ (0, ρ) the input allocated closed loop in Figure 4.1
is globally exponentially stable and, with constant exogenous signals r and d, its
response converges to a constant steady state value minimizing J(u∗, δy∗).
4.2 Design of the allocator
In some shots it has been observed that the currents on the coils, during the flat-top
phase, are close to their saturation limit as can be seen in Figure 4.2 where an example
for the current on P4 during the shot n. 24552, used to test the allocator, is shown.
This is obviously a case that should be avoided for many reasons: the controller
may not be able to recover the system in case of an unexpected disturbance, the
mechanical and electric structure of the system are under stress, there is a larger
consumption of energy because the power dissipated for resistive effect on the circuits
is proportional to the square of the current. This problem is currently taken into ac-
count by the PCS in the following way: saturation levels which are more conservative
48
Cap. 4 The input allocator §4.2 Design of the allocator
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
−12000
−10000
−8000
−6000
−4000
−2000
0
2000
time [s]
P4
curr
ent [
A]
meas. datasat. limit
Figure 4.2: Example of P4 current close to the saturation limit during the flat-topphase for the shot n. 24552.
than the hardware limits are set in the software, the controller will issue a warning if
the requests set for the shot are thought to cause currents with higher values than the
software limit. The control law, at the moment, does not operate to avoid saturation
on the currents.
4.2.1 Allocation on the currents
The allocator described in the previous section can be usefully applied to keep the
values of the currents in a safe range. Let us first consider for our system the scheme
in Figure 4.3.
Given the cost function J(u, δy), u will represent the steady state value of the
currents, which are inputs of the CREATE-L model, while δy∗ will denote the steady
state variation introduced by the allocator on the outputs. The first choice for the
49
Cap. 4 The input allocator §4.2 Design of the allocator
PCS Coils
model CREATE model
Allocator P*
+
+
+ -
u r
y
y d a y
c u
d
V I
Figure 4.3: First implementation of the allocator in the plant.
cost function J has been the following:
J =
ncoil∑i=1
aiξ2i (ui, k) +
ny∑j=1
bj(δyj)2 (4.2.1)
The function ξi(ui, k) has the following expression:
ξi(ui, k) =
{ui − THRi if ui > k · THRi
0 if ui ≤ k · THRi
(4.2.2)
The value of the currents is penalized in a quadratic way only if it is in the range
[k · THRi, THRi] where THRi is the software saturation threshold for the i-th coil
recovered from the PCS database and k is a design parameter which belongs to the
interval [0, 1]. The parameters a and b can be used to achieve the desired trade-off
between input allocation and tracking performances. Once the cost function has been
defined, it is necessary to design the other parameters in the eq. 4.2.4. The matrix B0
is selected considering that each of its columns corresponds to an allocation direction.
Therefore, it can be used to leave unchanged a certain number of scalar outputs if it
holds the following:
Im(B0) = ker [SyP∗] (4.2.3)
50
Cap. 4 The input allocator §4.2 Design of the allocator
where Sy is a selection matrix obtained by selecting from a ny × ny identity matrix
the rows corresponding to the outputs that must be left unchanged. The eq. 4.2.3 has
been used to design a B0 whose columns are linearly independent vectors that belong
to the null space of the row of P ∗ which corresponds to the ‘ZIP’ signal. In this way
the allocator will not affect the vertical position signal ‘ZIP’, which has been used
to preemptively stabilize the CREATE-L model (see chapt. 2). It should be pointed
out that, in general, it is possible for the allocator to distribute the input without
introducing steady-state variations on the outputs if the kernel of P ∗ is not empty.
In systems with a number of outputs much larger than the number of inputs, like
the CREATE-model, this is rarely the case. The parameter ρ is used to determine
how fast and aggressive is the desired behaviour of the allocator: low values of ρ will
cause gradual changes in the input of the system but on the other hand will cause to
achieve later the new steady state. If ρ is too high, though, the variation of the input
may be too abrupt and also some overshooting may be introduced. A comparison
between input allocations with different values of ρ is shown in Figures 4.4 and 4.5.
It can be seen how the current on P4 is driven away faster if the value of ρ is equal
to 10−2. On the other hand, if the values of ρ is too low (for example equal to 10−4)
the allocator does not reach its steady state value before the end of the flat-top phase
and the variation on the currents is not satisfactory.
The final implementation that has been adopted includes a saturation with thresh-
old on the derivative of the allocator state. It is possible, especially if the starting value
of the inputs is close to saturation, that the allocator would request steep variations of
the inputs that may be dangerous for the system. The saturation, parameterized by
the coefficient h, allows to achieve a trade-off between the promptness of the allocator
and the safety of its input variations. With the introduction of the saturation, the
51
Cap. 4 The input allocator §4.2 Design of the allocator
0.24 0.26 0.28 0.3 0.32−3.6
−3.4
−3.2
−3
−2.8
−2.6x 10
4
time [s]
P1
curr
ent [
A]
meas. datasim. data (no all.)sim. data (all. ro=1e−2)sim. data (all. ro=1e−3)sim. data (all. ro=1e−4)
0.24 0.26 0.28 0.3 0.320.8
1
1.2
1.4
1.6
1.8
2x 10
4
time [s]
P2
curr
ent [
A]
meas. datasim. data (no all.)sim. data (all. ro=1e−2)sim. data (all. ro=1e−3)sim. data (all. ro=1e−4)sat. limit
0.24 0.26 0.28 0.3 0.32−1.2
−1.15
−1.1
−1.05
−1x 10
4
time [s]
P4
curr
ent [
A]
meas. datasim. data (no all.)sim. data (all. ro=1e−2)sim. data (all. ro=1e−3)sim. data (all. ro=1e−4)sat. limit
0.24 0.26 0.28 0.3 0.32−8000
−7000
−6000
−5000
−4000
−3000
time [s]
P5
curr
ent [
A]
meas. datasim. data (no all.)sim. data (all. ro=1e−2)sim. data (all. ro=1e−3)sim. data (all. ro=1e−4)sat. limit
Figure 4.4: Allocation of the currents for different values of ρ
0.24 0.26 0.28 0.3 0.320
100
200
300
400
500
600
time [s]
w(1
)
sim. data (all. ro=1e−2)sim. data (all. ro=1e−3)sim. data (all. ro=1e−4)
0.24 0.26 0.28 0.3 0.32−100
0
100
200
300
time [s]
w(2
)
sim. data (all. ro=1e−2)sim. data (all. ro=1e−3)sim. data (all. ro=1e−4)
0.24 0.26 0.28 0.3 0.32−500
−400
−300
−200
−100
0
100
time [s]
w(3
)
sim. data (all. ro=1e−2)sim. data (all. ro=1e−3)sim. data (all. ro=1e−4)
0.24 0.26 0.28 0.3 0.320
100
200
300
400
500
600
time [s]
w(4
)
sim. data (all. ro=1e−2)sim. data (all. ro=1e−3)sim. data (all. ro=1e−4)
Figure 4.5: States of the allocator for different values of ρ
52
Cap. 4 The input allocator §4.2 Design of the allocator
allocator can be described by the following equations:
w = −SATh
(ρK
(OJ
[IP ∗
]B0
)′)ya = B0w
(4.2.4)
The allocator in Figure 4.3 has been implemented in the simulation environment
and tested for the shot n. 24552. It is important to point out that, once the parameters
of the allocator have been properly set in order to achieve the desired trade-off, there
is no necessity to change said parameters for other shots. The values of the currents
are shown in Figure 4.6: it is possible to notice how the allocator drives away the
current on the coil P4 introducing changes on the other currents which are considered
acceptable. The Figure 4.7, which shows the voltage output of the PCS, points out
that the current variations introduced by the allocator are not properly hidden to
the PCS, which changes noticeably its outputs but keeps them, nonetheless, in an
acceptable range. This is due do the length of the flat-top phase of the shot: the
system reaches the steady state only towards the end of the flat-top phase, until that
moment the constant matrix P* is just an approximation of the transfer function of
the system. This problem will be addressed and solved in a later section of the work.
The references and the controllable variables are shown in Figure 4.8 and it can be
seen that the tracking error introduced by the allocator is minimal and noticeable only
in the reference of P5-P4. The figures 4.9 and 4.10 show, respectively, the variation
introduced by the allocator on the current inputs and on some outputs of the system:
the vertical position ’ZIP’ signal is kept unchanged, as requested during the design
phase, while the current variations reach the steady state value correspondent to the
equilibrium point of the allocator with a speed that is set through the parameters ρ
and h.
53
Cap. 4 The input allocator §4.2 Design of the allocator
0.24 0.26 0.28 0.3 0.32−3.5
−3.4
−3.3
−3.2
−3.1
−3
−2.9
−2.8
−2.7x 10
4
time [s]
P1
curr
ent [
A]
0.24 0.26 0.28 0.3 0.320.8
1
1.2
1.4
1.6
1.8
2
x 104
time [s]
P2
curr
ent [
A]
0.24 0.26 0.28 0.3 0.32−1.22
−1.2
−1.18
−1.16
−1.14
−1.12
−1.1
−1.08
−1.06x 10
4
time [s]0.24 0.26 0.28 0.3 0.32
−8000
−7000
−6000
−5000
−4000
−3000
−2000
time [s]
P5
curr
ent [
A]
meas. datasim. data (all.)sim. data (no all.)saturation
meas. datasim. data (all.)sim. data (no all.)saturation
meas. datasim. data (all.)sim. data (no all.)saturation
meas. datasim. data (all.)sim. data (no all.)
Figure 4.6: Measured currents on the coil (blue), results of the simulation withoutallocator (black) and with allocator (red) for the shot n. 24552.
54
Cap. 4 The input allocator §4.2 Design of the allocator
0.24 0.26 0.28 0.3 0.32−850
−800
−750
−700
−650
−600
−550
−500
−450
time [s]
V1
[V]
0.24 0.26 0.28 0.3 0.320
10
20
30
40
50
60
70
80
time [s]
V2
[V]
0.24 0.26 0.28 0.3 0.32−350
−300
−250
−200
−150
−100
−50
0
time [s]
V4
[V]
0.24 0.26 0.28 0.3 0.32−200
−150
−100
−50
0
50
time [s]
V5
[V]
meas. datasim. data (all.)sim. data (no all.)
meas. datasim. data (all.)sim. data (no all.)
meas. datasim. data (all.)sim. data (no all.)
meas. datasim. data (all.)sim. data (no all.)
Figure 4.7: Measured voltages on the coil (blue), results of the simulation withoutallocator (black) and with allocator (red) for the shot n. 24552.
55
Cap. 4 The input allocator §4.2 Design of the allocator
0.24 0.26 0.28 0.3 0.327.8
7.9
8
8.1
8.2
8.3
8.4x 10
5
time [s]
plas
ma
curr
ent [
A]
0.24 0.26 0.28 0.3 0.321.5
1.55
1.6
1.65
1.7
1.75x 10
4
time [s]
P2−
K*P
1 [A
]
0.24 0.26 0.28 0.3 0.326200
6400
6600
6800
7000
7200
time [s]
P5−
P4
[A]
0.24 0.26 0.28 0.3 0.32−0.2
−0.18
−0.16
−0.14
−0.12
−0.1
−0.08
time [s]
dPsi
[W]
set−pointmeas. datasim. data (all.)sim. data (no all.)
set−pointmeas. datasim. data (all.)sim. data (no all.)
set−pointmeas. datasim. data (all.)sim. data (no all.)
set−pointmeas. datasim. data (all.)sim. data (no all.)
Figure 4.8: References for the controlled outputs (green), measured output (blue),simulated output without allocator (black), simulated output with allocator (red) forthe shot n. 24552.
56
Cap. 4 The input allocator §4.2 Design of the allocator
0.24 0.26 0.28 0.3 0.320
200
400
600
800
1000
1200
1400
time [s]
delta
ip1 [A
]
0.24 0.26 0.28 0.3 0.32−100
0
100
200
300
400
500
600
time [s]
delta
ip2 [A
]
0.24 0.26 0.28 0.3 0.320
100
200
300
400
500
time [s]
delta
ip4 [A
]
0.24 0.26 0.28 0.3 0.32−50
0
50
100
150
200
250
300
time [s]
delta
ip5 [A
]
Figure 4.9: Variations on the coil currents imposed by the allocator for the shot n.24552.
57
Cap. 4 The input allocator §4.2 Design of the allocator
0.24 0.26 0.28 0.3 0.32−8
−6
−4
−2
0
2x 10
4
time [s]
delta
ipl [A
]
0.24 0.26 0.28 0.3 0.32−1
−0.5
0
0.5
1
time [s]
delta
ZIP
[A]
0.24 0.26 0.28 0.3 0.32−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
time [s]
delta
r2 [A
]
0.24 0.26 0.28 0.3 0.32−100
−80
−60
−40
−20
0
time [s]
delta
r3 [A
]
Figure 4.10: Variation introduced by the allocator with respect to the plasma current,the ZIP signal and the two current references for the shot n. 24552.
58
Cap. 4 The input allocator §4.2 Design of the allocator
4.2.2 Allocation on the voltages
The design of the allocator in the previous section assumes that it is possible to
directly change the values of the currents in the system. Unfortunately, as explained
in previous chapters, this is not the case: the actuators of the plant are actually the
voltages, therefore it is necessary to implement a dynamic system which converts the
current requests of the allocator into voltages to be applied at the coils, which is to
say an inverse model of the coils. The result of the scheme in Figure 4.3 after this
modification is the one in Figure 4.11.
PCS Coils
model CREATE model
Allocator P* + -
u r
y
y d
c u
d
V I +
a y
Coils inverse model
+
Figure 4.11: Scheme of the plant with allocator and inverse model of the coils.
It is important to underline that all the the properties of the allocator described
in section 4.1 are still valid: the allocator has exactly one globally exponentially
stable equilibrium w∗ which is the minimizer of the cost function J(u∗, δy∗) and the
input allocated closed loop is globally exponentially stable. This can be easily shown
through a different representation of the system which is equivalent to the scheme in
fig. 4.1, for which all these properties hold. In the fig. 4.12 the block P1 represents
the coils model, P1−1
is the inverse coil model which is used to convert the current
59
Cap. 4 The input allocator §4.2 Design of the allocator
requests in voltages and d1 represents the mismatch between the used inverse model
P1−1
and P−11
PCS CREATE model
Allocator P* + -
u r
y
y d
c u
d
I
+ +
1 P
1 d
+ +
+
Figure 4.12: Equivalent representation of the plant wit the allocator and inverse modelof the coils.
Static model
The first model which has been used for the conversion of the current requests is
static: the voltages are calculated as the product of the currents by the estimated
resistance of the coils retrieved from the PCS database:
Vc = RcmIc (4.2.5)
The equation 4.2.5 introduces an error since the current variation in the system is
different from the one requested by the allocator. On the other hand, this happens
only during the transitory: in the steady state the current variations requested by the
allocator are constant therefore they are correctly converted in voltages by the static
equation.
60
Cap. 4 The input allocator §4.2 Design of the allocator
Dynamic model
In order to achieve better performances, the inverse model of the coil has been mod-
ified to take in account the inductive phenomena which were ignored in the previous
implementation. The new equation of the inverse model is the following:
Vc = RcmIc + LcmIc (4.2.6)
The main problem which arises in the implementation of the 4.2.6 is the calculation
of the derivative of Ic. The proposed solution is to make use of the following transfer
function to calculate the derivative estimation ¯Ic:
¯Ic =s
εs + 1Ic (4.2.7)
It has been observed that the performances of the allocator are strongly dependent
from the value of the ε parameter: low values of ε give a better approximation of
the derivative but they cause abrupt changes in the voltage signals. If ε is set to a
higher value, the higher frequency components are attenuated: the evolution of the
signals is smoother but the estimation of the derivation is less precise. In the Figures
4.13, 4.14,4.15 and 4.16 the results of the simulations with a voltage allocator with a
static inverse model of the coils are compared with the ones obtained when a dynamic
inverse model for different values of ε is used. From the Figure 4.13 it can be seen that
very low values of ε, in this case 10−4, cause abrupt variations on the coils P4 and P5
(almost 300V on P5) therefore a higher value of ε has to be chosen. The Figure 4.14
shows, coherently with what expected, that when a static inverse model of the coils is
used, the variations introduced by the allocator are not properly hidden through the
signal dy and this introduces a consistent difference in the voltages requested by the
PCS, especially on the coils P4 and P5. This influences the currents: it is possible
61
Cap. 4 The input allocator §4.2 Design of the allocator
to notice from Figure 4.15 how the current on P4 which is close to the saturation
is driven away more slowly. Also the tracking of the controlled variables is slightly
worse: it is possible to notice, for example, the bigger overshoot of the plasma current
in Figure 4.16.
0.24 0.26 0.28 0.3 0.32−900
−800
−700
−600
−500
−400
−300
time [s]
V1
[V]
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
0.24 0.26 0.28 0.3 0.320
20
40
60
80
100
time [s]
V2
[V]
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
0.24 0.26 0.28 0.3 0.32−350
−300
−250
−200
−150
−100
−50
0
time [s]
V4
[V]
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
0.24 0.26 0.28 0.3 0.32−400
−300
−200
−100
0
100
time [s]
V1
[V]
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
Figure 4.13: Voltages on the coils for the simulation with an allocator with a staticand dynamic model of the coils with different values of ε for the shot n. 24552.
62
Cap. 4 The input allocator §4.2 Design of the allocator
0.24 0.26 0.28 0.3 0.32−900
−800
−700
−600
−500
−400
−300
time [s]
V1
[V]
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
0.24 0.26 0.28 0.3 0.32−20
0
20
40
60
80
time [s]
V2
[V]
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
0.24 0.26 0.28 0.3 0.32−350
−300
−250
−200
−150
−100
−50
0
time [s]
V4
[V]
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
0.24 0.26 0.28 0.3 0.32−200
−150
−100
−50
0
50
time [s]
V5
[V]
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
Figure 4.14: Voltages requested by the PCS for the simulation with an allocator witha static and dynamic model of the coils with different values of ε for the shot n. 24552.
63
Cap. 4 The input allocator §4.2 Design of the allocator
0.24 0.26 0.28 0.3 0.32−3.4
−3.3
−3.2
−3.1
−3
−2.9
−2.8
−2.7x 10
4
time [s]
P1
curr
ent [
A]
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
0.24 0.26 0.28 0.3 0.321
1.2
1.4
1.6
1.8
2
x 104
time [s]
P2
curr
ent [
A]
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
sat. limit
0.24 0.26 0.28 0.3 0.32−1.22
−1.2
−1.18
−1.16
−1.14
−1.12
−1.1
−1.08
−1.06x 10
4
time [s]
P4
curr
ent [
A]
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
sat. limit
0.24 0.26 0.28 0.3 0.32
−8000
−7000
−6000
−5000
−4000
−3000
time [s]
P5
curr
ent [
A] sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
sat. limit
Figure 4.15: Currents on the coils for the simulation with an allocator with a staticand dynamic model of the coils with different values of ε for the shot n. 24552.
64
Cap. 4 The input allocator §4.2 Design of the allocator
0.24 0.26 0.28 0.3 0.327.8
7.9
8
8.1
8.2
8.3
8.4
8.5x 10
5
time [s]
plas
ma
curr
ent [
A]
set−point
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
0.24 0.26 0.28 0.3 0.321.5
1.55
1.6
1.65
1.7
1.75x 10
4
time [s]
P1−
KP
2 [A
]
set−point
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
0.24 0.26 0.28 0.3 0.326000
6200
6400
6600
6800
7000
7200
time [s]
P5−
P4
[A]
set−point
sim. data (no all.)
sim. data (stat. all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
0.24 0.26 0.28 0.3 0.32−0.2
−0.18
−0.16
−0.14
−0.12
−0.1
−0.08
time [s]
dPsi
[Wb]
set−point
meas. data
sim. data (no all.)
sim. data (all. eps=1e−2)
sim. data (all. eps=1e−3)
sim. data (all. eps=1e−4)
Figure 4.16: Controlled outputs for the simulation with an allocator with a static anddynamic model of the coils with different values of ε for the shot n. 24552.
65
Cap. 4 The input allocator §4.3 Design of the allocator on the closed-loop system
4.3 Design of the allocator on the closed-loop sys-
tem
Alternatively to the solution proposed and described in the previous section, where
the allocator was applied to the CREATE model and directly changed the input cur-
rents or the correspondant input voltages, a different implementation of the allocator
has been tested. One of the features of the previous version that is not completely
satisfying is the length of the transitory: regardless the time that the allocator needs
to reach its equilibrium point, the system requires almost the whole flat-top phase to
reach the new expected steady-state value. During this time the matrix P ∗ is only
an approximation of the transfer function P (s) of the CREATE model, the variations
of the allocator are not properly hidden to the PCS which changes noticeably its
outputs. In order to avoid this, the configuration shown in Figure 4.17 is proposed:
PCS coils
model CREATE model
V ff r
y
d
+ -
fb r d
fb r +
Allocator
e I
Figure 4.17: Scheme of the plant with allocator on the closed loop system.
In the scheme in Figure 4.17 the allocator is applied to the closed loop system
which includes the PCS, the coils model and the CREATE model that, for the sake
of simplicity, will henceforth be named C, P1 and P2. This different configuration
has been chosen because the closed loop system is expected to have better dynamic
66
Cap. 4 The input allocator §4.3 Design of the allocator on the closed-loop system
performances, such as a shorter transitory, and to be more robust for parametric
variations of the system.
In this implementation the cost function J will have the following expression:
J =
ncoils∑i=1
aiξ2i (ui) +
ny∑j=1
bj(δrj)2 (4.3.1)
where u is the vector of the three currents who are subject to saturation limits,
ξi is the same function defined in eq. 4.2.2, a and b are design parameters and δrj
represents the variations introduced by the allocator on the references of the controller.
Since the allocator is applied to the closed loop system and not directly on the plant,
its dynamics are slightly different from the ones described by the eq. 4.2.4 and are
the following:
w = −ρK
(OJ
[H∗
W ∗
]B0
)′
ya = B0w
(4.3.2)
Defined H(s) and W (s) as the closed-loop transfer functions between references
and currents and between references and controlled outputs, they have the following
expression:
H = (I + P1CP2)−1P1C
W = (I + P2P1C)−1P2P1C = P2H(4.3.3)
The matrix H∗ and W ∗ represents the transfer functions H e W evaluated for
s=0, that is to say the matrices of the steady-state gains. The matrix C(s) has
been calculated from the equations in chapter 3 while P1 is directly derived from
the coil equation 2.4.2. The calculation of P2 has requested a preliminary reduction
of the order of the system: the CREATE state-space model has 101 states and the
67
Cap. 4 The input allocator §4.3 Design of the allocator on the closed-loop system
relative transfer function is difficult to evaluate and would introduce computational
issues. For this reason a balanced realization of the CREATE model and the relative
matrix of Hankel singular values have been obtained. The positive eigenvalue has
been considered independent and preemptively stabilized by the vertical controller so
it has been excluded. Among the remaining stable eigenvalues, the ones with a higher
Hankel singular value, which retain the most important input-output characteristics of
the original system, have been used to create the reduced model. It has been necessary
to consider a trade-off between the accuracy and the computational requirements that
a high order system would introduce: a tenth-order system has proven to be a good
approximation of the original model and it has kept calculation relatively simple.
Once the transfer functions C, P1 and P2 have been obtained, it has been possible to
calculate H(s) and W (s) and their steady state value. As a partial confirmation of
the correctness of the calculation, the matrix obtained for W ∗ has been almost equal
to an identity matrix (the expected steady state value for a transfer function between
references and controlled output), with an error on each element lower than 1%.
Once the steady-state transfer functions have been calculated, it has been possible
to implement this different version of the allocator in the simulation environment in
order to compare its performances with the previous version. The parameters a and
b in 4.3.1, analogously to the allocator directly applied on the process, have been set
empirically, considering that a normalization on the variation of the references needs
to be introduced since, for example, the plasma current reference is seven orders of
magnitude greater than the dΨ reference. The shot used for the simulations is the n.
24552 and the results are shown if Figures 4.18, 4.19 and 4.20. It is possible to notice
from Figure 4.19 that the intervention of the closed-loop system allocator on the coil
currents is very similar to the open-loop version which is shown in Figure 4.6. On
68
Cap. 4 The input allocator §4.3 Design of the allocator on the closed-loop system
the other hand, the voltage requests in Figure 4.18 are substantially different from
the correspondent requests of the open-loop allocator in Figure 4.7: in this case the
controller is made aware of the intervention of the allocator through the change in
the references and it reacts consequently, requesting smoother voltages that are less
different from the ones without the allocator.
0.24 0.26 0.28 0.3 0.32−900
−800
−700
−600
−500
−400
−300
time [s]
V1
[V]
meas. datasim. data (all.)sim. data (no all.)
0.24 0.26 0.28 0.3 0.32−10
0
10
20
30
40
50
60
70
time [s]
V2
[V]
meas. data
sim. data (all.)
sim. data (no all.)
0.24 0.26 0.28 0.3 0.32−350
−300
−250
−200
−150
−100
−50
0
time [s]
V4
[V]
meas. datasim. data (all.)sim. data (no all.)
0.24 0.26 0.28 0.3 0.32−250
−200
−150
−100
−50
0
50
100
time [s]
V5
[V]
meas. data
sim. data (all.)
sim. data (no all.)
Figure 4.18: Voltages on the coils for the simulation with an allocator on the closed-loop system for the shot n. 24552.
69
Cap. 4 The input allocator §4.3 Design of the allocator on the closed-loop system
0.24 0.26 0.28 0.3 0.32−3.5
−3.4
−3.3
−3.2
−3.1
−3
−2.9
−2.8
−2.7x 10
4
time [s]
P1
curr
ent [
A]
meas. data
sim. data (all.)
sim. data (no all.)
0.24 0.26 0.28 0.3 0.32
1
1.2
1.4
1.6
1.8
2
x 104
time [s]
P2
curr
ent [
A]
meas. data
sim. data (all.)
sim. data (no all.)
saturation
0.24 0.26 0.28 0.3 0.32−1.22
−1.2
−1.18
−1.16
−1.14
−1.12
−1.1
−1.08
−1.06x 10
4
time [s]
P4
curr
ent [
A]
meas. data
sim. data (all.)
sim. data (no all.)
saturation
0.24 0.26 0.28 0.3 0.32
−8000
−7000
−6000
−5000
−4000
−3000
time [s]
P5
curr
ent [
A]
meas. data
sim. data (all.)
sim. data (no all.)
saturation
Figure 4.19: Currents on the coils for the simulation with an allocator on the closed-loop system for the shot n. 24552.
70
Cap. 4 The input allocator §4.3 Design of the allocator on the closed-loop system
0.24 0.26 0.28 0.3 0.327.8
7.9
8
8.1
8.2
8.3
8.4x 10
5
time [s]
plas
ma
curr
ent [
A]
set−point (no all.)
set−point (all.)
meas. data
sim. data (all.)
sim. data (no all.)
0.24 0.26 0.28 0.3 0.321.45
1.5
1.55
1.6
1.65
1.7
1.75x 10
4
time [s]
P2−
K*P
1 [A
]
set−point (no all.)
set−point (all.)
meas. data
sim. data (all.)
sim. data (no all.)
0.24 0.26 0.28 0.3 0.326000
6200
6400
6600
6800
7000
7200
time [s]
P5−
P4
[A]
set−point (no all.)
set−point (all.)
meas. data
sim. data (all.)
sim. data (no all.)
0.24 0.26 0.28 0.3 0.32−0.2
−0.18
−0.16
−0.14
−0.12
−0.1
−0.08
−0.06
time [s]
dPsi
[W]
set−point (no all.)
set−point (all.)
meas. data
sim. data (all.)
sim. data (no all.)
Figure 4.20: Controlled variables for the simulation with an allocator on the closed-loop system for the shot n. 24552.
71
Cap. 4 The input allocator §4.3 Design of the allocator on the closed-loop system
In order to test and compare the robustness of the systems with the two different
implementations of the allocator, some simulations have been run. The A matrix of
the CREATE model, which has been previously diagonalized for the elimination of
the eigenvalue described in Chapter 2, has been modified in the following way:
A = A · (I + ∆AK) (4.3.4)
The diagonal matrix ∆A contains random elements in the range [-0.5,0.5], K has
been set equal to 0.2 and 0.4, therefore considering variations on the diagonal elements
equal to 10% and 20%. The open and closed-loop allocator implementations have
been tested running simulations on the system with the modified A matrix. The
Figures 4.21 and 4.22 show the voltages on the coils respectively for the open-loop and
closed-loop implementation of the allocator: the most evident difference underlined
by the figures is the lesser variation of the voltages in the closed-loop implementation,
especially on P4 and P5. For variation up to the 20% on A, there is a difference in
the transitory respectively equal to 150 and 200 V. This can be explained by the fact
that in this case the allocator is applied to the closed-loop system and the controller is
able to preemptively reduce, albeit partially, the error introduced by the parametric
variations.
72
Cap. 4 The input allocator §4.3 Design of the allocator on the closed-loop system
0.24 0.26 0.28 0.3 0.32−900
−800
−700
−600
−500
−400
−300
time [s]
V1
[V]
meas. data
sim. data
sim. data (10% error)
sim. data (20% error)
0.24 0.26 0.28 0.3 0.320
20
40
60
80
100
120
time [s]
V2
[V]
meas. data
sim. data
sim. data (10% error)
sim. data (20% error)
0.24 0.26 0.28 0.3 0.32−350
−300
−250
−200
−150
−100
−50
0
time [s]
V4
[V]
meas. data
sim. data
sim. data (10% error)
sim. data (20% error)
0.24 0.26 0.28 0.3 0.32−400
−300
−200
−100
0
100
200
time [s]
V5
[V]
meas. data
sim. data
sim. data (10% error)
sim. data (20% error)
Figure 4.21: Voltages on the coils for the simulation with the open-loop allocator andperturbed matrix A for the shot n. 24552.
0.24 0.26 0.28 0.3 0.32−900
−800
−700
−600
−500
−400
−300
time [s]
V1
[V]
meas. data
sim. data
sim. data (10% error)
sim. data (20% error)
0.24 0.26 0.28 0.3 0.32−20
0
20
40
60
80
time [s]
V2
[V]
meas. data
sim. data
sim. data (10% error)
sim. data (20% error)
0.24 0.26 0.28 0.3 0.32−350
−300
−250
−200
−150
−100
−50
0
time [s]
V4
[V]
meas. data
sim. data
sim. data (10% error)
sim. data (20% error)
0.24 0.26 0.28 0.3 0.32−250
−200
−150
−100
−50
0
50
100
time [s]
V5
[V]
meas. data
sim. data
sim. data (10% error)
sim. data (20% error)
Figure 4.22: Voltages on the coils for the simulation with the closed-loop allocatorand perturbed matrix A for the shot n. 24552.
73
Cap. 4 The input allocator §4.4 Comparison between the two allocators
4.4 Comparison between the two allocators
For a better understanding of the differences between the allocator on the process
and the one on the closed-loop system, it has been decided to apply them to simpler
systems: in this way it is easier to notice and analyze their intervention, less hidden
by the considerable number of dynamics of the CREATE-model. The analysis has
been quantitative: the allocators have been tested in particular situations aimed at
underline their dynamic performances. The first test has been carried out on a very
simple system with two states, two inputs and two outputs whose eigenvalues are both
equal to -0.1 and consequently its settling time is approximatively equal to 35 seconds.
This system has been controlled in feedback with a very aggressive PI controller which
achieves tracking for constant references and a much shorter settling time of about 0.4
seconds; in doing so it drives inputs on the plant, during the transitory, that are 100
times higher than their steady state value. The next step has been the introduction
of the two different allocators in order to test and compare their behaviours if the
saturation limit on the inputs is set to their steady state values and the controller
violates these limits during the transitory. The cost function used for the allocators is,
in both cases, the eq. 4.2.1 and the parameters a,b,K and B0 have been chosen equal
for both subcompensators. Increasing values of ρ, the parameter that sets the speed of
the allocator, have been tried for the two versions: in both cases there is no noticeable
difference for values greater than 103 (the value used for the simulations shown below)
and integration issues in the simulations are experienced for values greater than 109.
The results of the simulation are shown in the Figures 4.23, 4.24, 4.25, 4.26 and 4.27.
It can be seen that the two allocators have very similar behaviours: they promptly
reduce the high initial value of the inputs (see Figure 4.23), which are far beyond the
74
Cap. 4 The input allocator §4.4 Comparison between the two allocators
set saturation limits and then they slowly drive the system towards its new steady-
state value. It is evident that the settling time has increased, pretty much by the same
amount in both cases, since high values of the inputs are now strongly penalized. The
main differences in the two implementations of the allocator are the different steady
state values for inputs and outputs (see Figures 4.24 and 4.25) and slightly better
performances of the closed loop allocator during the transitory: it does not introduce
undershoot on the input n. 1 and it reaches sooner the steady-state (see Figure 4.24).
0 0.5 1 1.5 2
−10
−5
0
5
10
15
20
25
30
time [s]
inpu
t #1
no all.o.l. all.c.l. all.
0 1 2 3 4 50
2
4
6
8
10
12
14
16
18
time [s]
inpu
t #2
no all.o.l. all.c.l. all.
Figure 4.23: Transitory of the inputs for the system with and without the allocatoron the process and on the closed-loop system.
75
Cap. 4 The input allocator §4.4 Comparison between the two allocators
70 80 90 1000.049
0.05
0.051
0.052
0.053
0.054
0.055
0.056
time [s]
inpu
t #1
no all.o.l. all.c.l. all.
70 80 90 1000.175
0.18
0.185
0.19
0.195
0.2
0.205
time [s]
inpu
t #2
no all.o.l. all.c.l. all.
Figure 4.24: Steady state of the inputs for the system with and without the allocatoron the process and on the closed-loop system.
0 10 20 30 40 50 60 70−1
0
1
2
3
4
time [s]
outp
ut #
1
no all.o.l. all.c.l. all.reference
0 10 20 30 40 50 60 700
0.5
1
1.5
2
2.5
time [s]
outp
ut #
2
no all.o.l. all.c.l. all.reference
Figure 4.25: Outputs for the system with and without the allocator on the processand on the closed-loop system.
76
Cap. 4 The input allocator §4.4 Comparison between the two allocators
0 20 40 60 80 100−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
time [s]
delta
inpu
t #1
(o.l.
all.
)
0 20 40 60 80 100−0.2
−0.18
−0.16
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
time [s]
delta
inpu
t #2
(o.l.
all.
)
Figure 4.26: Variations introduced on the inputs by the allocator on the process.
0 20 40 60 80 100−2.5
−2
−1.5
−1
−0.5
0
time [s]
delta
ref
. #1
(c.l.
all.
)
0 20 40 60 80 100−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
time [s]
delta
ref
. #2
(c.l.
all.
)
Figure 4.27: Variations introduced on the references by the allocator on the closed-loop system.
77
Cap. 4 The input allocator §4.4 Comparison between the two allocators
Another test that has been carried out regards the robustness of the two different
kinds of subcompensator: a simple system with 5 states, two inputs and two outputs
has been controlled in feedback through the H-infinity technique, choosing shaping
functions that achieve different sensitivity and robustness with respect to additive
uncertainties. Simulations have been run with both kind of allocators on the nominal
system and on the same system with an additive variation. The aim of this simulations
is to verify how the allocator influences the robustness of the system and if there is
any difference between the two versions. The results of the simulations are shown
in Figures 4.28,4.29,4.30 and 4.31. The most evident result is the general similarity
between the original system and the input-allocated ones: in the system with the
first H-infinity controller, the one which is less subject to the additive variation, the
variations after the perturbation have the same order of magnitude. In the system
controlled with the second H-infinity controller some oscillations can be noticed when
the additive variation is applied but the amplitude of these oscillations does not change
considerably when the allocator is introduced. If the behaviours of the system with
the two different allocators are compared, we can notice how no significant variation
is present: the allocator appear not to influence the robustness of the system.
78
Cap. 4 The input allocator §4.4 Comparison between the two allocators
0 10 20 30 40 50 601
1.5
2
2.5
3
time [s]
Input n. 1,Hinf control bis
ol. all.ol. all. pert.cl. all.cl. all. pert.no all.no all. pert.
0 10 20 30 40 50 601
1.5
2
2.5
3
time [s]
Input n. 1,Hinf control
ol. all.ol. all. pert.cl. all.cl. all. pert.no all.no all. pert.
Figure 4.28: Input n. 1 of the system (nominal and perturbed) with and without theallocator on the process and on the closed-loop system.
0 10 20 30 40 50 60 70 800.5
1
1.5
2
2.5
time [s]
Input n. 2,Hinf control bis
ol. all.ol. all. pert.cl. all.cl. all. pert.no all.no all. pert.
0 10 20 30 40 50 60 70 800.5
1
1.5
2
2.5
time [s]
Input n. 2,Hinf control
ol. all.ol. all. pert.cl. all.cl. all. pert.no all.no all. pert.
Figure 4.29: Input n. 2 of the system (nominal and perturbed) with and without theallocator on the process and on the closed-loop system.
79
Cap. 4 The input allocator §4.4 Comparison between the two allocators
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
3
time [s]
Output n.1,Hinf control bis
ol. all.ol. all. pert.cl. all.cl. all. pert.no all.no all. pert.reference
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
3
time [s]
Output n.1,Hinf control
ol. all.ol. all. pert.cl. all.cl. all. pert.no all.no all. pert.reference
Figure 4.30: Output n. 1 of the system (nominal and perturbed) with and withoutthe allocator on the process and on the closed-loop system.
0 10 20 30 40 50 60 70 80 90 100−3
−2
−1
0
1
time [s]
Output n.2,Hinf control bis
ol. all.ol. all. pert.cl. all.cl. all. pert.no all.no all. pert.reference
0 10 20 30 40 50 60 70 80 90 100−3
−2
−1
0
1
time [s]
Output n.2,Hinf control
ol. all.ol. all. pert.cl. all.cl. all. pert.no all.no all. pert.reference
Figure 4.31: Output n. 2 of the system (nominal and perturbed) with and withoutthe allocator on the process and on the closed-loop system.
80
Chapter 5
Conclusions
The present work is the result of the collaboration between Universita di Roma Tor
Vergata and the Culham Centre for Fusion Energy and should be considered in the
framework of the thermonuclear fusion research. The thesis has addressed the problem
of the shape control in the tokamak experiments and more specifically in the MAST
spherical tokamak. The purpose of the work has been the realization of a simulation
environment for MAST, which has required modeling on the different components
of the plant, and the design of a subcompensator to be added on the actual shape
controller in order to prevent saturation on the actuators of the system. To favor
an actual implementation of the controller, a solution which is not invasive has been
chosen: the allocator described in Chapter 4 can be directly added to the existing
controller which does not require any modifications. The work has been based on
the research papers [1] and [2] that have addressed the saturation problem for output
redundant plans and whose conclusions have already been tested through simulations
for the JET (Joint European Torus) shape control. A very important tool for all
this work has been the CREATE-L model of MAST: it has been the basis for the
simulation environment which has been created to validate the designed controller.
Said simulations have shown that the allocator, in both versions described in Chap-
81
Cap. 5 Conclusions §5.1 Possible future developments
ter 4, effectively drives away the currents on the coils from their saturation limits,
introducing variation on the voltages that are considered acceptable. Furthermore,
the number of parameters that need to be tuned for an actual implementation of the
subcompensator are limited (only a,b and ρ of the eq. 4.2.4 and 4.2.1) and the porting
in C language should be straight-forward as long as the cost function described by
the eq. 4.2.1 is used, since only sums and multiplications have to be performed.
5.1 Possible future developments
At the moment the simulation environment used throughout the present work is be-
ing validated using the PCS in simulative mode: the real controller is interfaced with
the simulink model, which provides the necessary outputs and receives the relative
inputs. The first tests show a substantial fitting of the simulation with the real data
and confirm that the allocator can actually be implemented on the plant, eventually
in one of the next experimental campaigns. It should be pointed out that the simu-
lation environment can easily be used as a testbed for any proposed modification of
the control law of the PCS, whose behaviour can be simulated before applying the
changes on the actual plant. There are also possible applications for the allocator on
MAST-U, the upgraded MAST tokamak which is currently under construction: in
this case the available coils will be nine and the shape control will not be limited to
the only external radius. From a theoretical point of view, it would be interesting to
analyze the behaviour of the system if a different cost function is chosen, for example
introducing priorities if all the coils are close to the saturation limit, or adding addi-
tional constraints on the actuators. Furthermore, the introduction of an anti-wind up
controller could be considered, in order to take in account the saturation phenomena
during the transitory phase.
82
List of Figures
1.1 The process of nuclear fusion. . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The trajectory of ionized gas subject to a magnetic field. . . . . . . . 5
1.3 General structure of the tokamak. . . . . . . . . . . . . . . . . . . . . 6
1.4 Currents and magnetic fields of the tokamak. . . . . . . . . . . . . . . 7