Saint-Petersburg State UniversityV.I. Zubov Institute of Computational Mathematics and Control Processes

Макеев

Иван

Владимирович

Mathematical Methods of Plasma Position, Current and Shape Stabilization in Modern Tokamak

A.P. Zhabko, I.V. Makeev, D.A. Ovsyannikov, A.D. Ovsyannikov, E.I. Veremey, N.A. Zhabko

Joint Meeting of

The 3rd IAEA Technical Meeting on Spherical Tori and The 11th International Workshop on Spherical Torus,

3 to 6 October 2005

Introduction

2H

Tokamaks represent an interesting field of applied investigations in modern control theory. The central problem is plasma position, current and shape control.

We widely use the mathematical methods of stabilizing control design based on modern optimization theory. One of the modern approaches is to use and -optimization methods.

As it is known, the plasma shape and current stabilization systems in tokamaks function in conditions of essential influence of a various kind of uncertainties both in relation to a mathematical model of controlled plant, and in relation to exterior perturbations. In this connection, it represents doubtless interest to use the various approaches to the analysis and synthesis of stabilization systems of plasma with the account of uncertainties. The particular interest is represented by the problem of estimation of a measure of robust stability and robust performance of the closed-loop system.

H

Main purposes:

Definition of stabilizing controller synthesis problem for the MAST plasma vertical feedback control system.

Stabilizing controllers synthesis using several different approaches and dynamical features analysis for designed controllers.

Robust features analysis for obtained closed-loop systems on the basis of frequency approach.

The MAST Tokamak Control SystemCS

P2

P2

P4

P4

P5

P5

P6

P6

X

Czc

rc

zx

rx

Outrout

In

Central Point

Vertical Stabilisation

Coils

For vertical stabilization needs MAST uses the P6 coils and the measurements of central point vertical displacement.

xCy

uBxAx

hv

hvhv

12550 ,, EyEuEx

1149 ,, EEEx yucx

bAxx

y

u

nhx )0(

cmzc 5.1

As an initial conditions the following vector was chosen, which corresponds to the only unstable matrix A eigenvalue and ensures a 1.5 cm initial plasma vertical displacement:

100 if,100

100 if,)(

100 if,100

u

uypK

u

u

The P6 coil voltage is bounded by 100 volt limit :

Nominal LTI System:

We obtain a full controllable LTI system after usage of special approach for excluding of the bad controllable mode from the model equations:

Ikykyku 321

6321 104.0,20,1 kkk

PD-Controller

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.005

0.01

0.015

0.02

0.025

Time (sec)

Ver

tical

dis

plac

emen

t, m

PD-Controller

The first controller was designed in a PD-form. Corresponding analysis had shown that the best dynamical features of closed-loop system are achieved with the following coefficients values. This figure illustrates the transient process in the closed-loop system with such controller.

0

20

211 )(),( dt(t)uctyHKII

0

20

22

2122 )()(),( dt(t)uctyntyn HKII

LQG-Optimal Controllers

Kz

czHbAzz

=

)(

u

yu

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.005

0.01

0.015

0.02

Time (sec)

Ver

tical

Dis

plac

emen

t, m

LQG Controllers

Fast LQG

"Classic" LQG

The purpose of LQG-optimal synthesis is the construction of controller in the following form. The choice of H and K matrixes should ensure a minimum of certain mean-square functional. We consider the task of LQ-optimization with these two functionals, where second one has an auxiliary component of measurement derivative. This fact provides an additional flexibility in controller synthesis with help of coefficients varying. As a result, plot below illustrates transient processes for mentioned controllers. It is clear, that the second controller is better.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.0180

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Time (sec)

Ver

tical

Dis

plac

emen

t, cm

Fast LQG

PD

sec004.0sec008.0 LQGPD TT

Controllers Comparison

The obtained LQG-controller two times faster than nominal PD-controller.

LQG-Controller Reduction

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-0.005

0

0.005

0.01

0.015

0.02

0.025

Time (sec)

Vert

ical D

ispla

cem

ent, m

Full LQG

Reduced 3rd

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-0.005

0

0.005

0.01

0.015

0.02

0.025

Time (sec)

Vert

ical D

ispla

cem

ent, m

Full LQG

Reduced 4th

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-0.005

0

0.005

0.01

0.015

0.02

0.025

Time (sec)

Vert

ical D

ispla

cem

ent, m

Full LQG

Reduced 5th

3rd order

4th order5th order

There are essential difficulties in practical implementation of mentioned fast LQG-controller. The point is that this controller has an order of 49. With the help of the Schur balanced model reduction approach it is possible to reduce controller’s order. The following figures represent the transient process in closed-loop system with LQG-controller reduced to corresponding order. One can see that the fifth order reduced controller compares well with full order LQG controller.

1. Fast LQG-optimal Controller

0

20

22

21 )()( ucynynuJ

1,7000000 21 nn

3. Reduced LQG Controllers of 3rd, 4th and 5th order

2. PD Controller

Ikykyku 321 400000,20,1 321 kkk

Robust Features Analysis

For robust features analysis the following controllers were used. The analysis was being carried out on the basis of the frequency approach. It allows to construct so-called frequency robust stability margins for different controllers and compare them because the wider are robust stability margins, the better is corresponding controller. According to this approach the figure shows constructed frequency robust stability margins for all specified controllers. It is clear that the LQG-optimal controller provides much wider robust stability area than nominal PD-controller does. Despite the fact that order reduction adversely affects robust features, reduced controllers of fourth and fifth order keep essential advantage in their robust features concerning nominal PD-controller. So we obtain controller that is better than PD in both dynamics and robustness.

Results

Several different controllers were designed for the MAST plasma vertical feedback control system on the basis of LQG-optimal theory.

The obtained controllers were used for comparison on their dynamical and robust features. The best one was determined.

It was shown, that mathematical model of the best LQG-controller may be reduced keeping its characteristics at admissible level.

These facts allow us to recommend this controller as a good possible alternative for the plasma vertical stabilization problem in the MAST machine.