Saint-Petersburg State University
V.I. Zubov Institute of Computational Mathematics and Control Processes
A.P. Zhabko, I.V. Makeev, D.A. Ovsyannikov,
A.D. Ovsyannikov, E.I. Veremey, N.A. Zhabko
Mathematical Methods of Plasma Position,
Current and Shape Stabilization in Modern Tokamak
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
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 H 2 and H  -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.
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 System
P2
CS
P4
rx
X
P6
zx
rc
P5
C
Vertical
Stabilisation
Coils
zc
In
rout
Out
Central Point
P6
P4
P2
P5
For vertical stabilization
needs MAST uses the P6
coils and the
measurements of central
point vertical
displacement.
Nominal LTI System:
x  A hv x  B hv u
x  E50 , u  E5 , y  E12
y  C hv x
We obtain a full controllable LTI system after usage of special approach for excluding of
the bad controllable mode from the model equations:
x  Ax  bu
y  cx
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:
x(0)  h n
zc  1.5 cm
x  E49 , u  E1 , y  E1
The P6 coil voltage is bounded by 100 volt
limit :
  100, if u  100

u   K ( p) y, if u  100
 100, if u  100

PD-Controller
The first controller was designed in a PDform. Corresponding analysis had shown
that the best dynamical features of closedloop system are achieved with the following
coefficients values. This figure illustrates the
transient process in the closed-loop system
with such controller.
u  k1 y  k2 y  k3 I
k1  1, k2  20, k3  0.4  106
PD-Controller
0.025
Vertical displacement, m
0.02
0.015
0.01
0.005
0
0
0.001
0.002
0.003
0.004
0.005 0.006
Time (sec)
0.007
0.008
0.009
0.01
LQG-Optimal Controllers
z  Az  bu  H( y  cz)
u = Kz
I1  I1 (K , H)   y 2 (t )  c0u 2(t)dt

0

I 2  I 2 (K , H)   n1 y 2 (t )  n2 y 2 (t )  c0u 2(t)dt
0
LQG Controllers
Fast LQG
"Classic" LQG
Vertical Displacement, m
0.02
0.015
0.01
0.005
0
0
0.001
0.002
0.003
0.004
0.005 0.006
Time (sec)
0.007
0.008
0.009
0.01
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 LQoptimization 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.
Controllers Comparison
Fast LQG
0.02
PD
0.018
Vertical Displacement, cm
0.016
0.014
The obtained LQGcontroller two times
faster than nominal
PD-controller.
0.012
0.01
0.008
0.006
0.004
0.002
0
0
0.002
0.004
0.006
TPD  0.008 sec
0.008
0.01
0.012
Time (sec)
0.014
0.016
0.018
TLQG  0.004 sec
LQG-Controller Reduction
0.025
0.025
Full LQG
Full LQG
Reduced 4th
0.02
0.02
0.015
0.015
Vertical Displacement, m
Vertical Displacement, m
Reduced 5th
0.01
0.005
0
-0.005
0.01
0.005
0
0
0.001
0.002
0.003
0.004
0.005 0.006
Time (sec)
5th
0.007
0.008
0.009
0.01
order
-0.005
0
0.001
0.002
0.003
0.004
0.005 0.006
Time (sec)
0.007
0.008
0.009
0.01
4th order
0.025
Full LQG
Reduced 3rd
Vertical Displacement, m
0.02
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.
0.015
0.01
0.005
0
-0.005
0
0.001
0.002
0.003
0.004
0.005 0.006
Time (sec)
3rd order
0.007
0.008
0.009
0.01
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.
1. Fast LQG-optimal Controller

J (u )   (n1 y 2  n2 y 2  c0u 2 )
0
n1  7000000, n2  1
2. PD Controller
u  k1 y  k2 y  k3 I
k1  1, k 2  20, k3  400000
3. Reduced LQG Controllers of
3rd, 4th and 5th order
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.