universita degli studi di roma` tor...

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


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


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


• 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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


−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

]


−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]

P4

erro

r [A

]


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


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


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


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


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


35


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


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


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


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


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


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


44


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


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


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


−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


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


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


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]


0.24 0.26 0.28 0.3 0.32−8000

−7000

−6000

−5000

−4000

−3000

time [s]

P5

curr

ent [

A]


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

)


0.24 0.26 0.28 0.3 0.32−500

−400

−300

−200

−100

0

100

time [s]

w(3

)


0.24 0.26 0.28 0.3 0.320

100

200

300

400

500

600

time [s]

w(4

)


Figure 4.5: States of the allocator for different values of ρ

52


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


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.)

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


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]





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


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.)




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


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


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


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


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


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


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)



0.24 0.26 0.28 0.3 0.320

20

40

60

80

100

time [s]

V2

[V]

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]

sim. data (no all.)





0.24 0.26 0.28 0.3 0.32−400

−300

−200

−100

0

100

time [s]

V1

[V]

sim. data (no all.)





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


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.)





0.24 0.26 0.28 0.3 0.32−20

0

20

40

60

80

time [s]

V2

[V]

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]

sim. data (no all.)





0.24 0.26 0.28 0.3 0.32−200

−150

−100

−50

0

50

time [s]

V5

[V]

sim. data (no all.)





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


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.)





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.)





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.)





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.)





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


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.)





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.)





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.)





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.)




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


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


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


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]


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]


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


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


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 (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 (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 (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


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


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)


0.24 0.26 0.28 0.3 0.320

20

40

60

80

100

120

time [s]

V2

[V]

meas. data

sim. data



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



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



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



0.24 0.26 0.28 0.3 0.32−20

0

20

40

60

80

time [s]

V2

[V]

meas. data

sim. data



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



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



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


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


Figure 4.23: Transitory of the inputs for the system with and without the allocatoron the process and on the closed-loop system.

75


70 80 90 1000.049

0.05

0.051

0.052

0.053

0.054

0.055

0.056

time [s]

inpu

t #1


70 80 90 1000.175

0.18

0.185

0.19

0.195

0.2

0.205

time [s]

inpu

t #2


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


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


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


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


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


0 10 20 30 40 50 60 70 800.5

1

1.5

2

2.5

time [s]

Input n. 2,Hinf control


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


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


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


0 10 20 30 40 50 60 70 80 90 100−3

−2

−1

0

1

time [s]

Output n.2,Hinf control


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



the shot n. 24542. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35



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


the shot n. 24542. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44



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


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


n. 24552. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.16 Controlled outputs for the simulation with an allocator with a static


n. 24552. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

86


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-


4.20 Controlled variables for the simulation with an allocator on the closed-


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


87


4.30 Output n. 1 of the system (nominal and perturbed) with and without


4.31 Output n. 2 of the system (nominal and perturbed) with and without


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

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