2 OPTIMAL CONTROL PROBLEM FORMULATIONVERTICAL VELOCITY OF THE PLASMA...
2.2 Optimal control problem formulationvertical velocity of the plasma current centroid.The desired controller must stabilize vertical velocity of the plasma current centroid. One ofSince the order of this linear model is very high, an order reduction is desirable to simplifythe approaches to control synthesis is based on the optimal control theory. In this framework,the controller synthesis problem. The standard Matlab functionschmrwas used to performplasma vertical stabilization problem can be stated as follows. One needs to find a feedbackmodel reductionfrom 58th to 3rd order. As a result, we obtain a transfer function of thecontrol algorithmu
=
u(
t,y)
that provides a minimum of the quadratic cost functionalreduced SISO model (from inpututo outputy)∞
J=
J(
u) =
0
(
y2
(
t) +
λu2
(
t))
dt, (7)P(
s) =
1.732·10−6
(
s−121.1)(
s+
158.2)(
s+
9.641)
(
s+
29.21)(
s+
8.348)(
s−12.21)
. (5)subject to plant model (5) and constraints (6), and guarantees closed-loop stability. Hereλis aThis transfer function has poles which dominate the dynamics of the initial plant. The un-constant multiplier setting the trade-off between controller’s performance and control energycosts.stable pole corresponds to vertical instability. It is natural to assume that two other polesSpecifically, in order to find an optimal controller, LQG-synthesis can be performed. Such aare determined by the virtual circuit dynamic related to the most significant elements in thecontroller has high stabilization performance in the unconstrained case. However, it is per-tokamak vessel construction. The quality of the model reduction can be illustrated by thecomparison of the Bode diagram for both initial and reduced models. Fig. 1 shows the Bodehaps not the best choice in the presence of constraints.diagrams for initial and reduced 3rd
order models on the left and for initial and reduced 2nd
Contrary to this, the MPC synthesis allows to take into account constraints. Its basic schemeorder model on the right. It is easy to see that the curves for initial model and reduced 3rd
implies on-line optimization of the cost functional (7) over a finite horizon subject to plantorder model are actually indistinguishable, contrary to the 2nd
order model.model (5) and imposed constraints (6).Plasma stabilization system design on the base of model predictive control 201The working capacity and effectiveness of the MPC algorithms is demonstrated by the exam-Bode Diagram
ple of ITER-FEAT plasma vertical stabilization problem. The comparison of the approaches is−70
done.−80