Page 287 - Fundamentals of Magnetic Thermonuclear Reactor Design

266 Fundamentals of Magnetic Thermonuclear Reactor Design

l norm H :
H∞ ∞
( ),
H∞=maxw∈0,∞σ¯(w), H ∞ = max σω (8.39)
, 0 )
∈
ω [ ∞
H(jΩ) where ()σ ω is the maximum singular value of matrix Hj( Ω ), that is, the square
σ¯(w)
T
Ω
J )
(
HT (−JΩ) H(JΩ) root of the maximum eigenvalue of hermitian matrix H (−Ω HJ ).
For the SISO problem
ω
H∞=maxw∈0,∞H(jw); H ∞ = max H( ) ; (8.40)
, 0 )
ω [ ∞
∈
HS 12 l weighted norm HS 1 2
1 ∞

Hjω) (
T
T
= ∫ tr S (− jω) H (− jω) ( Sjω)   dω
HS 1 2  1 1
HS =12π∫−∞∞trS1T−jwHT−jwHjwS jwdw 2π −∞
1
12
1 ∞

(
T
= ∫ tr Hjω) S 1 ( jω) S (− jω) H (− jω)   d ,
T
ω

=12π∫−∞∞trHjwS jwS1T−jwHT−jw dw, 2π −∞ 1
1
1 ∞
S ω ()
(
T
ω
HS =12π∫−∞∞trHjwSνwHT−jw dw , HS 1 2 = 2π ∫ −∞ tr   Hjω) ν H (− jω)   d , (8.41)
12
ω
T
ω
(
Sν(w)=S (jw)S1T(−jw) where S ()ω = Sj ) S (− j ). For the SISO problem
ν
1
1
1
1 ∞ 2
(
ω
2
HS =12π∫−∞∞Hjw Svw dw, HS 1 2 = 2π ∫ −∞ Hjω) S v ω () d , (8.42)
12
where weighted norm HS
HS ∞ 1 ∞
1
HS ∞=maxw∈0,∞σ¯νw HS 1 ∞ = max σω () (8.43)
ν
1
∞
ω∈  0, )
(
S ω
σ¯vw where σ ω () is the maximum singular value of matrix Hjω) () .
1
v
8.6 PLASMA START-UP PHASE
HjwS w 8.6.1 Dynamics of Tokamak Electromagnetic Processes
1
Simply put, the tokamak plasma start-up includes the breakdown of the fuel gas
mixture, subsequent ramp-up of the generated plasma current and formation of
closed magnetic flux surfaces. The start-up phase is paid particular attention to
at the design of the tokamak power supply and control systems.
First, the vortex electric field and acceptable stray magnetic fields required
for plasma breakdown specify the critical operational conditions for the power
supply of poloidal field coils (PFCs). Second, at the start-up phase the ionisa-
tion and radiation ‘barriers’ should be overcome. For this purpose the auxiliary
heating (electron cyclotron resonance power) can be used. These challenges
make the transient plasma control problem a special case of mathematical
modelling.