Page 277 - Fundamentals of Magnetic Thermonuclear Reactor Design
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256 Fundamentals of Magnetic Thermonuclear Reactor Design
and the equation describing the plasma geometry becomes
∂ g ∂ g ∂ g
gt=∂g∂IδI+∂g∂IpδIp+∂g∂ξδξ. gt () = I δ + I δ p + δξ . (8.23)
I ∂ I ∂ p ξ ∂
Substitution δI from Eq. (8.22) into Eqs (8.21) and (8.23), we obtain the
p
desired MLE that accounts for possible disturbances of plasma parameters:
d d
L I δ ( )+ M δξ ( ) + RI = δ U
∗
δ
L∗dδIdt+Mξdδξdt+RδI=δUg=CδI+Fδξ, dt ξ dt (8.24)
δ
=
gC I + Fδξ ,
where
∂Ψ ∂Ψ ∂Ψ
M ≡ − M p ,
ξ
Mξ≡∂Ψ∂ξ−Mp∂Ψ¯∂ξ/∂Ψ¯∂Ip, ξ ∂ ξ ∂ ∂ I p
and
∂ g ∂ g ∂Ψ ∂Ψ
F ≡ − .
F≡∂g∂ξ−∂g∂Ip∂Ψ¯∂ξ/∂Ψ¯∂Ip. ξ ∂ I ∂ p ξ ∂ I ∂ p
In the theory of controller synthesis, the equations for a control object are
routinely written in the A, B, C, D form. From Eq. (8.24) it follows that in our
case they will take the form
d
d I δ ( ) = A I δ + B U + E ( )
δ
δξ
ddtδI=AδI+BδU+Eddtδξg=CδI+DδU+Fδξ, dt dt (8.25)
gC I δ= + D U + Fδξ ,
δ
–1
*
*
ξ
M
A≡−L*−1R
*
(
ξ
E≡− B L ≡ − L 1 *)-1 where A ≡− L ( ) − 1 R , B ≡ L(*) , D is the zero matrix, and E ≡− L ( ) − 1 M .
(d/dt)δξ Excluding variable ddt(/ ) δξ ( ) by substitution x δ≡ I − Eδξ we, bring
Eq. (8.25) to final form
x≡δI−Eδξ
AxB U +
+
x =
δ
x˙=Ax+BδU+AEδξg=Cx+DδU+CE+Fδξ. gCxD U ( AEδξ ) (8.26)
+
= + δ + CEF δξ .
Now let us describe the procedure for calculating matrix coefficients of a
linear model for a given plasma configuration. Fig. 8.1 shows an example of
a plasma configuration and positions of points on separatrix that are used as
targets to control the plasma shape. The passive structure is represented sche-
matically by a combination of inner and outer axisymmetric rings inside the
vacuum vessel and by the vessel’s double shell. As for the control coils, in the
linear model they are described by 11 circuits. The inner and the outer rings are
segmented into three loops each. The inner and the outer shells of the vacuum
vessel are segmented into 25 axisymmetric loops each. As a result, the linear
model comprises 67 states (circuits).
