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UNIVERSITA DEGLI STUDI DI ROMA
TOR VERGATA
FACOLTA 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
Contents
Abstract 1
1 Nuclear fusion and MAST 3
1.1 Nuclear fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Spherical tokamaks and MAST . . . . . . . . . . . . . . . . . . . . . 8
2 CREATE model 13
2.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Change of coordinates . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Vertical stability . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Order reduction of the model . . . . . . . . . . . . . . . . . . 20
2.2.4 Plasma current parameter . . . . . . . . . . . . . . . . . . . . 23
2.3 Feedforward currents simulations . . . . . . . . . . . . . . . . . . . . 26
2.4 Model of the coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Feedforward voltages simulations . . . . . . . . . . . . . . . . . . . . 33
3 PCS: Plasma control system 37
3.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
CONTENTS I
CONTENTS
3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 The input allocator 46
4.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Design of the allocator . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.1 Allocation on the currents . . . . . . . . . . . . . . . . . . . . 49
4.2.2 Allocation on the voltages . . . . . . . . . . . . . . . . . . . . 59
4.3 Design of the allocator on the closed-loop system . . . . . . . . . . . 66
4.4 Comparison between the two allocators . . . . . . . . . . . . . . . . . 74
5 Conclusions 81
5.1 Possible future developments . . . . . . . . . . . . . . . . . . . . . . . 82
List of figures 83
Bibliography 88
CONTENTS II
Abstract
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
of load assemblyDiameter approximatively 3m Neutral beam 5, 000, 000 watts
heating power @ 75, 000 volts
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:
Inputs: - Active PF circuit voltages
Disturbances: - Poloidal beta- Internal inductance
Outputs: - Active PF circuit currents- Passive PF circuite currents- Plasma shape descriptors- Magnetic signals- Plasma current moments
State Variables: - Active and passive PF circuit currents
14
Cap. 2 CREATE model §2.2 Implementation
2.2 Implementation
The CREATE team has developed a graphic interface which makes very easy to obtain
the desired model. Initially the number of the shot is chosen and all the data needed
by the tool to generate the model are downloaded from the database. In order for
the linearization performed by the CREATE tool to be reliable, the chosen shot must
have a long flat-top phase and no important nonlinearities which may be caused, for
example, by plasma disruptions. It has to be considered that the MAST top-flat
phase lasts at most 0.3 seconds and the signals are sampled at 2Khz so only about
700 measurements are available for the modeling of the experiment. The next step is
the choice of the settings of the model: it is possible to create models for plasmaless
shots, take in account the presence of eddy currents, choose a double null or limiter
configuration. In the limiter configuration the border of the confined region of the
plasma (LCFS) is limited by inserting a barrier a few centimetres into the plasma, in
the double null configuration there is a different shaping of the plasma which leads to
the formation of two poloidal field nulls, above and below the plasma column. In the
first simulations used to test the CREATE-model, a plasma model with eddy currents
and plasma in double null configuration of the shot n. 24542 (a standard shot with a
long top-flat phase) has been used. The tool returns the matrices L,R andLE of the
eq. 2.1.2 but it is necessary to do some preliminary modifications, described in the
next sections, in order to correctly run the simulations.
15
Cap. 2 CREATE model §2.2 Implementation
2.2.1 Change of coordinates
The first step towards an implementation of this model has been the change of its
inputs. The eq. 2.2.1 is an equivalent representation of the state-space model:[L11 L12
L21 L22
] [x1
x2
]+
[R1 00 R2
] [x1
x2
]=
[0S2
]U −
[LE1
LE2
]W (2.2.1)
The components of the state vector x can be divided in x1 (passive currents gen-
erated by inductive phenomena) and x2 (active currents on the coils). A problem
experienced by the CREATE team when the linear model of MAST has been realized
is that the voltages signals (the original inputs of the model) are too noisy hence they
cannot be used in the simulations. The problem has been solved in the following way:
the dynamics of the coil circuits have been removed from the model, considering the
currents on the coils as new inputs. If the new state vector p1 is defined as follows:
p1 = L11x1 + L12x2 + LE1w (2.2.2)
it holds the following:
p1 = L11x1 + L12x2 + LE1w
x1 = L−111 p1 − L−1
11 L12x2 − L−111 LE1w
(2.2.3)
and it is straightforward to obtain a new set of state-space equations where p1 is
the new state variable:
p1 = −R1L−111 p1 +
[R1L
−111 L12 R1L
−111 LE1
] [x2
w
]y = C1L
−111 p1 +
[−C1L
−111 L12 + C2 −C1L
−111 LE1 + F
] [x2
w
] (2.2.4)
which can be easily rewritten in state-space form, considering the vector ξ =
[x2 w], containing the currents on the poloidal coils and the disturbances, as the new
input:
16
Cap. 2 CREATE model §2.2 Implementation
p1 = Ap1 + Bξ
y = Cp1 + Dξ(2.2.5)
This new model has a lower order than the original one (p1 has the same dimension
of x1) because it totally ignores the electric dynamics on the poloidal coils and assumes
that the value of the currents can be arbitrarily imposed. A model of the electric
circuits of the coils which receives the applied voltages and returns the correspondent
values of currents has been created and will be described in a later section.
2.2.2 Vertical stability
The next step in the implementation of the CREATE model has been its closed-loop
stabilization. In the model, as well as in the plant, there is an unstable mode relative to
the vertical instability of the plasma. During the shot the plasma is elongated, pulled
along its vertical direction by the magnetic field generated by the coils: in elongated
plasma it is easier to achieve higher values of current and better performances. The
more the plasma moves in one direction, the bigger is the attraction towards that very
same direction and the smaller in the opposite one: the plasma, if adequate control
is not applied, crashes on the wall of the vessel and it disrupts. To avoid this, the
coil P6 is used to generate a magnetic field which balances the vertical displacement
of the plasma: the signal ‘ZIP’, which represent the z position of the plasma current
centroid, is used as a controlled variable for a PD controller that returns the value of
the voltage to be applied on P6. As pointed out before, the vertical controller, shown
in Figure 2.1, is not included in the PCS and it is implemented separately through
analogue circuits since very fast time response is needed to achieve stability.
In order to run any kind of simulation, it is necessary to preemptively stabilize the
model. In the simulations run by the CREATE team the A matrix of the model has
17
Cap. 2 CREATE model §2.2 Implementation
Figure 2.1: Diagram of the vertical control on MAST.
been diagonalized, the stable modes have been normally simulated forward while the
unstable mode has been simulated backward in order to grant stability. In this way
the unstable model exponentially converges to zero. This solution is inadequate for
the purpose of this work since it is not feasible for feedback simulations. The method
that has been adopted instead is conceptually similar to the one actually implemented
on MAST: a feedback on the ‘ZIP’ signal and the implementation of a PD controller.
The first step has been the identification of the electric circuits of the P6 coil; it has
to be kept in mind that at the moment our model receives currents as inputs but the
controller returns voltages. An initial attempt has considered the coil n.6 as an R-L
circuit described by the equation 2.2.6.
VP6 = RP6IP6 + LP6dIP6
dt(2.2.6)
which is equivalent to the first order system described by 2.2.7
IP6 =KP6
τP6s + 1VP6 (2.2.7)
Given the low number of parameters used to describe the system, a manual tun-
ing on their values has been used to achieve a satisfactory fitting between the mea-
sured values of currents and the output of 2.2.7 when the correspondent voltages are
inputted. After a few attempts, the results shown in Figure 2.2 have been obtained.
18
Cap. 2 CREATE model §2.2 Implementation
0.28 0.3 0.32 0.34 0.36 0.38−1000
−500
0
500
time [s]
curr
ents
[A]
meas. dataest. data
Figure 2.2: Results of the identification on P6 coil.
The coil identification has been considered acceptable and the next step has been
the design of the PD controller and the tuning of its two parameters. The reference
has been the filtered (low-pass) reference of the actual controller, in order to avoid
abrupt changes in the output voltage signal due to initial high value of the tracking
error. The ‘ZIP’ signal, the voltage on P6 and the resultant current on P6 are shown
in fig. 2.3, 2.4 and 2.5.
There is a mismatch between the measured signals and the results of the simulation
and it is mostly caused by measurement noise and nonlinear phenomena which are
not considered by the CREATE-model. It has to be underlined, though, that there
is a decoupling between the unstable mode of the vertical position of the plasma and
the other dynamics: they are actually controlled by two independent controllers and
the only purpose of the implemented vertical controller is to stabilize the system and
make simulations possible.
19
Cap. 2 CREATE model §2.2 Implementation
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38−6000
−5000
−4000
−3000
−2000
−1000
0
1000ZIpl MAST shot #24542
time [s]
ZIP
[m *
A]
exp. datasim. data
Figure 2.3: ’ZIP’ signal for the simulation of shot n.24542.
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38−40
−20
0
20
40
time [s]
P6
volta
ge [V
]
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38−800
−600
−400
−200
0
200
400
600
time [s]
P6
curr
ent [
V]
Figure 2.4: Measured current and voltage on P6 for the shot n.24542.
2.2.3 Order reduction of the model
Once the system has been stabilized, it has been possible to run feed-forward simula-
tions in order to test the performances and the reliability of the new model. Currents
20
Cap. 2 CREATE model §2.2 Implementation
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38−20
−15
−10
−5
0
5
10
time [s]
P6
volta
ge [V
]
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38−40
−20
0
20
40
60
time [s]
P6
curr
ent [
V]
Figure 2.5: Current and voltage on P6 for the simulation of shot n.24542.
on the coils for a certain shot are retrieved from the database and are used as inputs
of the CREATE-model, whose outputs are compared with the measured data. The
output of these simulations fit the experimental data but they show a high-frequency
oscillation which is not present in the input. This has been thought to be caused by a
very low eigenvalue in the A matrix of the eq. 2.2.5 which would also explain the long
time needed for the simulation, since Matlab has to shorten the integration time in
order to simulate the fast dynamic relative to this eigenvalue. To solve this problem
a change of coordinates has been performed on the model and a new matrix A, diag-
onal, has been obtained. The eigenvalue which was thought to cause the oscillation
has been removed from the state dynamics and simulations have been repeated. In
Figure 2.6 the plasma current for the two cases, during the whole flat top phase, is
shown: the values of the signal appear to be identical. If a shorter time interval is
considered, as in Figure 2.7, the oscillation on the output signal of the model without
reduction and its total absence in the new model are evident. The model reduction
by truncation has allowed to achieve a ten times shorter simulation time and the
21
Cap. 2 CREATE model §2.2 Implementation
elimination of the high frequency component on the outputs, without introducing any
error in the simulation results.
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.389.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9x 10
5
time [s]
plas
ma
curr
ent [
A]
ord. red.=0ord. red.=1
Figure 2.6: Plasma current during the flat top phase for the original model (blue) andfor the reduced one (red).
0.2879 0.2879 0.288 0.288 0.2881
9.136
9.1361
9.1362
9.1363
9.1364
9.1365
9.1366
x 105
time [s]
plas
ma
curr
ent [
A]
ord. red.=0ord. red.=1
Figure 2.7: Plasma current for the original model (blue) and for the reduced one (red)during a shorter time interval.
22
Cap. 2 CREATE model §2.2 Implementation
2.2.4 Plasma current parameter
Once all the procedures described above had been implemented, there still were dif-
ferences between the measured and the simulated signals, especially in the plasma
current. In the attempt to understand the source of the problem, the row of C rela-
tive to that output in the eq. 2.2.5 has been analyzed and it has been noticed that
the plasma current is, with a good approximation, dependant from just one state
which, furthermore, is independent in its evolution from the others. Basically in the
CREATE model, considering Ipl0 as the value measured at the starting time of the
simulation, the expression of the plasma current can be approximated as follows:
xpl = aplxpl + Bplu
Ipl = cplxpl + Ipl0
(2.2.8)
If we consider the value of apl initially set by the CREATE model for the shots
taken into account, it is usually in the range [-0.3 -0.1]. This means that, if u is set
to 0, xpl converges exponentially to 0 and the plasma current remains constant. It
is known this is not the case since the plasma current presents a resistive effect that
makes it decay if the coils are not powered. This explains why, in the simulations,
if currents measured from a shot are used as input of the model, the plasma current
increases instead of being constant, as can be seen in Figure 2.8
In order to solve this inaccuracy of the model, the value of the parameter apl has
been changed and tested with new simulations, trying to minimize the least square
error between the measured plasma current and the simulated one. It has been noticed
that, for all the analyzed shots, the error has only one minimum with respect to the
value of apl as it can be seen in figure 2.9.
It is worth saying that the new parameter apl which minimizes the error is a
23
Cap. 2 CREATE model §2.2 Implementation
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.388.8
8.9
9
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8x 10
5
time [s]
plas
ma
curr
ent [
A]
meas. datasim. data
Figure 2.8: Measured plasma current (blue) and simulated plasma current(red) whenno correction on the plasma parameter is applied.
−0.5 0 0.5 1 1.5 20
1
2
3
4
5
6
7x 10
6
apl
quad
ratic
err
or
Figure 2.9: Value of the quadratic error between measured and simulated plasmacurrent with respect to the parameter apl.
positive number: this means that in the modified model the plasma current decreases
exponentially (cpl is negative) if currents are not applied on the coils. Simulations
run with the modified parameter apl have led to an improvement of the fitting of the
24
Cap. 2 CREATE model §2.2 Implementation
simulated data, especially for the plasma current, as can be seen in figure 2.10.
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.388.8
8.85
8.9
8.95
9
9.05
9.1
9.15
9.2x 10
5
time [s]
plas
ma
curr
ent [
A]
meas. datasim. data
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
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.388.85
8.9
8.95
9
9.05
9.1
9.15
9.2x 10
5
time [s]
plas
ma
curr
ent [
A]
meas. datasim. data
Figure 2.12: Measured plasma current (blue) and simulated plasma current(red) forthe shot n. 24542.
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 2.13: Measured fields (blue) and simulated ones (red) for the shot n. 24542.
27
Cap. 2 CREATE model §2.4 Model of the coils
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
−1.14
−1.12
−1.1
−1.08
−1.06
−1.04
time [s]
flp4u
4 flu
x [W
b]
meas. datasim. data
0.3 0.32 0.34 0.36 0.38−1.24
−1.22
−1.2
−1.18
−1.16
−1.14
−1.12
time [s]
flp4l
4 flu
x [W
b]
meas. datasim. data
Figure 2.14: Measured fluxes (blue) and simulated ones (red) for the shot n. 24542.
2.4 Model of the coils
The model whose simulative results have been shown in the previous section assumes
that any value of the coil currents can be obtained instantaneously. This is not the
case: in reality a voltage signal is applied to the electric circuits of the coils and then
the resultant value of current is measured. Since the dynamics of the coils have been
excluded from the CREATE model because of the inaccuracy of the voltage signals,
it is now necessary to independently model them. It should be pointed out, though,
that the model will involve only the coils P1,P2,P4 and P5 which are the ones used
for the feedback control, P6 has been pre-emptively modelled (eq. 2.2.6) and P3 is
used in feed-forward. The first adopted approach has been to consider the coils as
R-L circuits described by the following equations:
Vcm1 = Rcm1Icm1 + Lcm1dIcm1
dt(2.4.1)
28
Cap. 2 CREATE model §2.4 Model of the coils
which can be easily expressed in the state-space form:
dIcm1
dt= −L−1
cm1Rcm1Icm1 + L−1cm1Vcm1 = Acm1Icm1 + Bcm1Vcm1 (2.4.2)
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
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.388.85
8.9
8.95
9
9.05
9.1
9.15
9.2x 10
5
time [s]
plas
ma
curr
ent [
A]
meas. datasim. data
Figure 2.22: Measured plasma current (blue) and simulated plasma current(red) forthe shot n. 24542.
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
0.44
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 2.23: Measured fields (blue) and simulated ones (red) for the shot n. 24542.
35
Cap. 2 CREATE model §2.5 Feedforward voltages simulations
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 2.24: Measured fluxes (blue) and simulated ones (red) for the shot n. 24542.
36
Chapter 3
PCS: Plasma control system
3.1 General description
The plasma control system (PCS) is the device which is used to control and configure
the plant. It can be roughly schematized in two main sections which operate in
different times and perform different operations. The first section is a configuration
tool that allows to set the parameters and the waveforms to be used during the
shots and synchronizes with the Machine Control System, the machine that actually
runs the shot. The second part operates in real-time: it gets data from the plant
via analogue and digital inputs and, on the basis of the data it receives and the
parameters and waveforms which have been set before the starting of the shot, it
makes control decisions and drives the plant via analogue and digital outputs. For
the purpose of this thesis, we will focus our attention on this last part: in particular
on the controlled variables and the way the output of the PCS (voltages driven on
the coils) is calculated. The general scheme of the plant is represented in fig. 3.1.
The PCS receives as input the value of the currents (I), a feedforward reference
(Iff ), the error on the current feedback reference (Iref −SM · I) and the error on the
plasma current and on the flux Ψ. Each input is used to calculate a different vector of
37
Cap. 3 PCS: Plasma control system §3.1 General description
PCS Coils
model CREATE model
V I + -
- +
SM
ref Ip ref y ff I
ref I
Ip y
MAST
Figure 3.1: General scheme of the plant.
voltage requirements on the coils which are then summed to calculate the final output
of the controller. It is now described in more detail how this is done:
• Resistive term: the drops on the voltages due to resistive effect have to be
taken in account. A matrix RPCS which describes the resistance of the coils
is stored as a PCS parameter and used to calculate the voltages required to
compensate the resistive losses:
Vres = RPCSI (3.1.1)
• Current Feedforward: at the beginning and the ending of the shot, respec-
tively when plasma is created and ramped down, a reliable model of the system
is not currently available and it is difficult to design a feed back controller. That
is why in these phases of the shot, and in minor part during the flat-top phase,
the value of currents are preemptively calculated and driven in feed forward.
The requested derivative on the currents are set offline in the PCS (Iff ). The
corresponding values of voltages are calculated as follows:
Vff = Iff · L · FFcoeff (3.1.2)
In the expression above L is the matrix which describes the mutual inductance
between the coils (in a similar way to the R described in the previous point)
38
Cap. 3 PCS: Plasma control system §3.1 General description
while FFcoeff is used to take in account a reduction of the mutual inductances
caused by the presence of the plasma.
• Current Feedback: it is necessary, in order to drive the plant, a feed back
component in the controller which compensates disturbances and model un-
certainties. References are set offline in the PCS on P1, P2-0.5P1 and on the
sum and difference of P4 and P5. The currents on P4 and P5 are expressed in
sum/difference terms because of the strong mutual coupling between the two
coils. The feedback control on each coil can be enabled or disabled at any time
during the shot setting offline the relative gain function. The voltages that aim
at correct the error on the current references have the following expression:
Vfb = Ierr · τ−1 · L (3.1.3)
Ierr is the tracking error vector, L is the mutual inductance matrix of the coils
and τ is a diagonal matrix which contains the time constant for each coil (design
parameter).
• Plasma current: the plasma current is controlled in feedback with a PI
controller.
The tracking error and its integral (respectively Iperr and IpIerr) are calculated
and the voltages are obtained as follows:
Vpl =
IpIerr
τIpInt+ Iperr
τIp
·(− Lpl
Mpl−sol
)· L +
cfac
τBv
dΨerr (3.1.4)
The first factor calculates the required value of Ipl which is then converted in
the correspondent required value of IP1 by the inductance ratio and finally the
voltages are obtained through the inductance matrix L. The second term of
39
Cap. 3 PCS: Plasma control system §3.1 General description
the sum is used to take in account the shape of the plasma which influences the
amount of current that needs to be driven.
• Radial position feedback: Since the difficulties to reconstruct the radial
position of the plasma in real-time, the control of the radius uses flux signals.
Two isoflux lines are considered: one on CC, supposedly close to the isoflux
line on the plasma boundary (therefore representing a good estimate of the
flux in that point), and the other at a chosen control point RC . The value
of the flux reference dΨref is calculated multiplying the radius reference dRref
by the factor ∂Ψ∂R
, which is estimated through a linear combination of magnetic
measurements. The controlled output is calculated through a similar linear
combination of measurements and the expression of the tracking error is the
following:
dΨerr = dΨref − dΨ =
(nm∑i=1
aimi
)dRref −
nm∑j=1
bimi
The voltages are calculated similarly to the previous cases:
VR = dΨerr · KΨ · 1
τPsi
· L +Sh
τIp
Iperr (3.1.5)
The flux error is converted through the gain KΨ into a current error which is
then multiplied by a time costant and by the inductance matrix. The second
term of the sum represents correction estimated by the Shafranov equation. It
is worth saying that, during real shots, the radial position of the plasma can
be measured through a camera positioned inside the chamber, converted in a
magnetic measurement multiplying it by the factor ∂Ψ∂R
and used as an alternative
estimation of dΨ. Unfortunately it is not possible to implement this case in the
simulations because the CREATE-model does not return the camera signal.
40
Cap. 3 PCS: Plasma control system §3.2 Implementation
Once the four voltages term, each one relative to a different input of the PCS, are
calculated, their values are summed to obtain the voltage output of the PCS:
V = Vres + Vff + Vfb + Vpl + VR
3.2 Implementation
The PCS has been implemented in the simulation environment using a Simulink model
developed by G. McArdle which accurately reproduces the control laws described
above. It also takes in account and models the power supplies which are schematized
by a clipping function followed by a one-pole low-pass filter. The model has been
tested with feedforward simulations, throughout the whole length of the shot and
not only in the top-flat phase: input data of the PCS from previous shots have been
retrieved from the database and used as input of the model. The results of the
simulation have then been compared with the measured output of the PCS as can be
seen in Figure 3.2
This simulation has been crucial for two reasons:
1. It proves that the PCS model correctly reproduces the actual control signals,
there is only some marginal disagreement in the preliminary phase. As we can
see from the graphs the measured data (blue) fit with the simulated data (red)
apart from the considerable measurement noise.
2. It provides all the parameters that are needed to initialize correctly the simu-
lations with the CREATE model. Since these simulations are only run in the
top-flat phase, the initial values of all the plant outputs are initialized with the
measured values at the starting time of the simulation. If the initial conditions
41
Cap. 3 PCS: Plasma control system §3.3 Simulations
−0.1 0 0.1 0.2 0.3−1500
−1000
−500
0
500
1000
1500
2000
time [s]
P1
volt.
[A]
meas. datasim. data
−0.1 0 0.1 0.2 0.3−200
−100
0
100
200
300
400
time [s]
P2
volt.
[A]
meas. datasim. data
−0.1 0 0.1 0.2 0.3−400
−200
0
200
400
600
800
time [s]
P4
volt.
[A]
meas. datasim. data
−0.1 0 0.1 0.2 0.3−400
−200
0
200
400
600
800
time [s]
P5
volt.
[A]
meas. datasim. data
Figure 3.2: Comparison between the measured (blue) and simulated (red) outputvoltages of the PCS for the shot n.24552.
of the controller and of the power supplies (integral term of the plasma current
error in the PCS, initial input voltage on the power supplies) are not initialized
properly, abrupt changes in the signals are experienced. A correct initializa-
tion of the parameters grants smoother signals which are more similar to the
measured ones.
3.3 Simulations
Once the initial conditions of the PCS are properly set, it has been possible to run feed-
back simulations: given a certain shot, the references are retrieved from the database
and inputted in the PCS which drives the tensions on the coils. As explained also in
Chapter 2, the CREATE model considers βp and li as disturbances. Since there is no
model for this signals, the actual values calculated offline, after the shot, are used: it
42
Cap. 3 PCS: Plasma control system §3.3 Simulations
is safe to assume that this choice does not introduce any kind of misrepresentation or
error in the simulations.
The results for the shot n.24552 are shown in the figures 3.3, 3.4, 3.5,3.6 and 3.7.
It is possible to compare these figures with the ones in the previous chapter for the
feed forward simulations and see how the introduction of the model of the PCS does
not substantially change the overall behaviour of the system.
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
]
meas. datasim. data
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
]
meas. datasim. data
0.3 0.32 0.34 0.36 0.38−500
−400
−300
−200
−100
0
100
200
time [s]
volta
ge o
n P
4 [V
]
meas. datasim. data
0.3 0.32 0.34 0.36 0.38−400
−300
−200
−100
0
100
200
time [s]
volta
ge o
n P
5 [V
]
meas. datasim. data
Figure 3.3: Measured voltages on the coils P1, P2, P4 and P5 (blue) and simulatedvoltages for the shot n.24542 during the flat-top phase.
43
Cap. 3 PCS: Plasma control system §3.3 Simulations
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
4000
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 0.38−6800
−6700
−6600
−6500
−6400
−6300
time [s]
P5
curr
ent [
A]
meas. datasim. data
Figure 3.4: Measured currents on the coils P1, P2, P4 and P5 (blue) and simulatedcurrents for the shot n.24542 during the flat-top phase.
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.388.85
8.9
8.95
9
9.05
9.1
9.15
9.2x 10
5
time [s]
plas
ma
curr
ent [
A]
meas. datasim. data
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
1.5 Section of MAST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Cross-section of the MAST vessel and position of the six PF coils. . . 11
1.7 Typical PF current evolution. . . . . . . . . . . . . . . . . . . . . . . 12
2.1 Diagram of the vertical control on MAST. . . . . . . . . . . . . . . . 18
2.2 Results of the identification on P6 coil. . . . . . . . . . . . . . . . . . 19
2.3 ’ZIP’ signal for the simulation of shot n.24542. . . . . . . . . . . . . . 20
2.4 Measured current and voltage on P6 for the shot n.24542. . . . . . . . 20
2.5 Current and voltage on P6 for the simulation of shot n.24542. . . . . 21
2.6 Plasma current during the flat top phase for the original model (blue)
and for the reduced one (red). . . . . . . . . . . . . . . . . . . . . . . 22
2.7 Plasma current for the original model (blue) and for the reduced one
(red) during a shorter time interval. . . . . . . . . . . . . . . . . . . . 22
2.8 Measured plasma current (blue) and simulated plasma current(red)
when no correction on the plasma parameter is applied. . . . . . . . . 24
83
LIST OF FIGURES LIST OF FIGURES
2.9 Value of the quadratic error between measured and simulated plasma
current with respect to the parameter apl. . . . . . . . . . . . . . . . 24
2.10 Measured plasma current (blue) and simulated plasma current(red)
when the parameter apl is modified in order to minimize the quadratic
error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.11 Measured currents on the coils P1,P2,P4 and P5 for the shot n.24542
during the flat-top phase. . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.12 Measured plasma current (blue) and simulated plasma current(red) for
the shot n. 24542. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.13 Measured fields (blue) and simulated ones (red) for the shot n. 24542. 27
2.14 Measured fluxes (blue) and simulated ones (red) for the shot n. 24542. 28
2.15 The measured current for the shot n. 24542 (blue) are compared with
the estimation of the R-L model (green). . . . . . . . . . . . . . . . . 29
2.16 Estimation errors of the currents on the coil P1,P2,P4 and P5 (blue)
and new estimate of the current errors(red). . . . . . . . . . . . . . . 31
2.17 Estimation errors on the currents using the first model of the coils
(blue) and the second (green). . . . . . . . . . . . . . . . . . . . . . 31
2.18 Scheme for the coils model with current error estimation. . . . . . . . 32
2.19 The measured current for the shot n. 24542 (blue) are compared with
the results of the coils model which includes the error estimation (green). 32
2.20 Measured voltages on the coils P1,P2,P4 and P5 for the shot n.24542
during the flat-top phase. . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.21 Measured currents on the coils P1,P2,P4 and P5 (blue) and simulated
currents for the shot n.24542 during the flat-top phase. . . . . . . . . 34
84
LIST OF FIGURES LIST OF FIGURES
2.22 Measured plasma current (blue) and simulated plasma current(red) for
the shot n. 24542. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.23 Measured fields (blue) and simulated ones (red) for the shot n. 24542. 35
2.24 Measured fluxes (blue) and simulated ones (red) for the shot n. 24542. 36
3.1 General scheme of the plant. . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Comparison between the measured (blue) and simulated (red) output
voltages of the PCS for the shot n.24552. . . . . . . . . . . . . . . . . 42
3.3 Measured voltages on the coils P1, P2, P4 and P5 (blue) and simulated
voltages for the shot n.24542 during the flat-top phase. . . . . . . . . 43
3.4 Measured currents on the coils P1, P2, P4 and P5 (blue) and simulated
currents for the shot n.24542 during the flat-top phase. . . . . . . . . 44
3.5 Measured plasma current (blue) and simulated plasma current(red) for
the shot n. 24542. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Measured fields (blue) and simulated ones (red) for the shot n. 24542. 45
3.7 Measured fluxes (blue) and simulated ones (red) for the shot n. 24542. 45
4.1 General scheme of the plant with allocator. . . . . . . . . . . . . . . . 47
4.2 Example of P4 current close to the saturation limit during the flat-top
phase for the shot n. 24552. . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 First implementation of the allocator in the plant. . . . . . . . . . . . 50
4.4 Allocation of the currents for different values of ρ . . . . . . . . . . . 52
4.5 States of the allocator for different values of ρ . . . . . . . . . . . . . 52
4.6 Measured currents on the coil (blue), results of the simulation without
allocator (black) and with allocator (red) for the shot n. 24552. . . . 54
85
LIST OF FIGURES LIST OF FIGURES
4.7 Measured voltages on the coil (blue), results of the simulation without
allocator (black) and with allocator (red) for the shot n. 24552. . . . 55
4.8 References for the controlled outputs (green), measured output (blue),
simulated output without allocator (black), simulated output with al-
locator (red) for the shot n. 24552. . . . . . . . . . . . . . . . . . . . 56
4.9 Variations on the coil currents imposed by the allocator for the shot n.
24552. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.10 Variation introduced by the allocator with respect to the plasma cur-
rent, the ZIP signal and the two current references for the shot n.
24552. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.11 Scheme of the plant with allocator and inverse model of the coils. . . 59
4.12 Equivalent representation of the plant wit the allocator and inverse
model of the coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.13 Voltages on the coils for the simulation with an allocator with a static
and dynamic model of the coils with different values of ε for the shot
n. 24552. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.14 Voltages requested by the PCS for the simulation with an allocator
with a static and dynamic model of the coils with different values of ε
for the shot n. 24552. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.15 Currents on the coils for the simulation with an allocator with a static
and dynamic model of the coils with different values of ε for the shot
n. 24552. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.16 Controlled outputs for the simulation with an allocator with a static
and dynamic model of the coils with different values of ε for the shot
n. 24552. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
86
LIST OF FIGURES LIST OF FIGURES
4.17 Scheme of the plant with allocator on the closed loop system. . . . . 66
4.18 Voltages on the coils for the simulation with an allocator on the closed-
loop system for the shot n. 24552. . . . . . . . . . . . . . . . . . . . . 69
4.19 Currents on the coils for the simulation with an allocator on the closed-
loop system for the shot n. 24552. . . . . . . . . . . . . . . . . . . . . 70
4.20 Controlled variables for the simulation with an allocator on the closed-
loop system for the shot n. 24552. . . . . . . . . . . . . . . . . . . . . 71
4.21 Voltages on the coils for the simulation with the open-loop allocator
and perturbed matrix A for the shot n. 24552. . . . . . . . . . . . . . 73
4.22 Voltages on the coils for the simulation with the closed-loop allocator
and perturbed matrix A for the shot n. 24552. . . . . . . . . . . . . . 73
4.23 Transitory of the inputs for the system with and without the allocator
on the process and on the closed-loop system. . . . . . . . . . . . . . 75
4.24 Steady state of the inputs for the system with and without the allocator
on the process and on the closed-loop system. . . . . . . . . . . . . . 76
4.25 Outputs for the system with and without the allocator on the process
and on the closed-loop system. . . . . . . . . . . . . . . . . . . . . . . 76
4.26 Variations introduced on the inputs by the allocator on the process. . 77
4.27 Variations introduced on the references by the allocator on the closed-
loop system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.28 Input n. 1 of the system (nominal and perturbed) with and without
the allocator on the process and on the closed-loop system. . . . . . . 79
4.29 Input n. 2 of the system (nominal and perturbed) with and without
the allocator on the process and on the closed-loop system. . . . . . . 79
87
LIST OF FIGURES LIST OF FIGURES
4.30 Output n. 1 of the system (nominal and perturbed) with and without
the allocator on the process and on the closed-loop system. . . . . . . 80
4.31 Output n. 2 of the system (nominal and perturbed) with and without
the allocator on the process and on the closed-loop system. . . . . . . 80
88
References
[1] G.De Tommasi - S. Galeani - A. Pironti - G. Varano - L.Zaccarian, “Nonlinear dy-
namic input allocator for optimal input/output performance trade-off: application
to the JET Tokamak shape controller”.
[2] G. Ambrosino - G. De Tommasi - S. Galeani - A. Pironti - G. Varano - L. Zac-
carian and JET-EFDA Contributors “On dynamic input allocation for set-point
regulation of the JET Tokamak plasma shape”.
[3] G. Artaserse, F. Maviglia “Porting of XSCTools on MAST fusion device”.
[4] A. Pironti, M. Walker “Control of Tokamak Plasmas”.
[5] G. Cunningham, J. Lister “Exploring the MAST vertical control system using
RZIP”.
[6] G. McArdle “MAST plasma control system”.
[7] R. Martin “SL Training - Introduction to MAST: TF and PF coil set, vertical
position control and control parameters”.
89