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Plasma Control System Chapter | 8 261

problems. Such approaches are premised on formalised ideas about a con-
trol system’s performance. This formalisation implies that there exists a set
of quantitative characteristics describing performance, whose values are
determined by design solutions. Such characteristics may include various
functionals specified in relevant metric spaces. If, for example, the spaces of
sought-for elements are normalised, relevant norms may be used as quality
functionals.
In dealing with a closed-loop stabilising system, the ‘dynamic process
quality’ is reduced to the use of matrix norms for a quantitative evaluation of
the transfer matrices. This evaluation allows us to understand how large output
signals are relative to certain classes on input signals. If the input signals are
disturbances making plasma’s dynamic parameters deviate from equilibrium,
the stabilisation quality depends on how well the control system suppresses
them. The suppression effectiveness, in turn, depends on the values of transfer
matrix norms: the smaller the norm the higher the suppression effectiveness.
In this context, the optimal stabilisation problem may be interpreted as the
choice of a feedback controller capable of minimising a relevant transfer ma-
trix norm.

8.5.2 Problem Generalisation

Let us assume that a plasma column numerical simulation is described by a
system of ordinary nonlinear differential equations
=
ϕ
 x Fx u(, ,), (8.27) x˙ = F(x, u, ),
n
defined on positive time semiaxis t ∈ (0, ∞). Here, x ∈ E is the vector of the
l
m
object's state, u ∈ E is the vector of control actions, and φ ∈ E is the vector
of disturbances acting on the object under control. We assume that the F vec-
tor function in the right-hand side satisfies the conditions of the existence and
uniqueness of the solution of the Cauchy problem for Eq. (8.27).
We further introduce vector functions x = x (t), u = u (t) and φ = φ (t) that
p
p
p
determine the controlled dynamics of the object and satisfy Eq. (8.27)
, (
 x = F tx ,u ,ϕ ) .
p
p
p
p
x˙p=Ft,xp,up,p.
For the purposes of our control system, dynamics of the plasma’s equilib-
rium position is assumed to be a controlled movement. We denote deviations
of respective variables from a controlled movement as x = x(t), u = u(t), and
φ = φ(t), where x(t) = x (t) + x(t), u(t) = u (t) + u(t), and φ(t) = φ (t) + φ(t). Sub-
p
p
p
stitution of these relations into the original Eq. (8.27) gives the equation for a
disturbed plasma movement, expressed as deviations from equilibrium position:
(
 x = Gt xu,ϕ) , (8.28) x˙=Gt,x,u,,
,,
where Gt xu(, ,, )ϕ = Ft x + x u + u,ϕ + ϕ )− Ft xu ,ϕ p ). G(t,x,u,)=F(t,xp+x,up+u,p+)−F(t,xp
(
,
,
,
(,
p
p
p
p
p
,up,p).