Page 286 - Fundamentals of Magnetic Thermonuclear Reactor Design

Plasma Control System Chapter | 8 265

alised gain matrix of the ‘external input disturbances d − generalised output
vector e’ loop. As external disturbances lead to deviation of the plasma’s dy-
namic parameters from their equilibrium values, the stabilisation effectiveness
depends on how well the control system suppresses them. The lower the H(s, K)
gain matrix the higher the stabilisation effectiveness.
In this connection, it makes sense to solve a problem of the maximum
degree of suppression of external disturbances through minimisation of the
H(s,K) gain matrix by optimal choice of transfer matrix of the K(s) control-
ler. To accomplish this, it is necessary to formalise the requirement that the
H(s, K) transfer matrix should be ‘small.’ To this end, we can use a matrix
(
norm Hs,K) as a functional to be minimised. Then, a generalised prob- Hs,K
lem of the maximum degree of suppression of external disturbances may be
expressed as

(
()
I= IK =H s,K) → inf, (8.36) I=IK=Hs,K→infK∈Ω,
K∈Ω
where Ω is an ensemble of transfer matrices with rational fraction components,
for which the characteristic polynomial of the (8.33), (8.35) closed-loop system
is unambiguously Hurwitzian.
The choice of specific norm in Eq. (8.36) is associated with different classes
of problems dealing with the optimised synthesis of a stabilising controller. Cur-
rently, problems that have to be dealt with most often include the following [6]:

l   H 2 norm minimisation problem (typically represented by the LQG- H 2
optimal synthesis problem),
l   H ∞ norm minimisation problem (the H -optimal synthesis problem), and H∞
∞
l the problem of minimisation of the first two norms for ‘weighted’ transfer
matrices HS , where S (s) is a given weighted matrix function; examples in-
1
1
clude root mean square optimal synthesis problems and ensuring controller
synthesis problems.
Going forward, we assume that all the roots of the characteristic polynomial
of the closed-loop system, mathematically expressed by Eqs (8.33) and (8.35),
lie in the open left semiplane. We also assume that for all transfer matrices,

−
T
H(s), discussed later, expression trH ( ) ()is a proper rational fraction. trHT−sHs
sH s


Then, the given norms are introduced through
l norm H : H 2
2
1 ∞

ω
T
H 2 = 2π ∫ −∞ tr H (− jω) (  d . (8.37) H =12π∫−∞∞trHT−jwHjwdw.
Hjω) 

2
For example, for the single-input single-output (SISO) d/e problem
1 ∞ 2
ω
(
2
H 2 = 2 ∫ −∞ Hjω) d ; (8.38) H =12π∫−∞∞Hjw dw;
2