Page 283 - Fundamentals of Magnetic Thermonuclear Reactor Design

262 Fundamentals of Magnetic Thermonuclear Reactor Design

We further introduce a linear mathematical model of the controller:
=
()
u=WsCx, uW sCx, (8.29)
where C is a constant matrix under a given measurement law, and W(s) is the
transfer matrix of the controller. We also introduce a linearised system (8.28),
expressed as deviations from the equilibrium position:

ϕ
+
+
x˙=Ax+Bu+H,  x = AxBuH , (8.30)
where matrices A, B and H have constant components. The (8.29) controller is
considered to be stabilising the zero equilibrium position of system (8.28) or
the system’s controlled movement (8.27) under the assumption that the zero
equilibrium position of systems (8.28) and (8.29) is asymptotically stable in the
sense of Lyapunov.
We know that controlled movement x = x (t), u = u (t) and φ = φ (t) (or the zero
p
p
p
equilibrium position of system (8.28)) can be made asymptotically stable in the
sense of Lyapunov using the feedback control given by Eq. (8.29). To accomplish
+
 Gt xu ,,0) −
−
()
()
x
u
Gt, x, u, 0−Atx−Btu≤θx+u this, it is necessary that condition  (, At xB tu ≤ ( θ  ) ,
x
u
θ→0,x →0, u→0 (at φ(t) ≡ 0, and where θ → 0, → 0, → 0) is satisfied, and that the uncon-
trolled part of the (8.30) linear approximation is stable.
The second requirement is certainly met if controllability, in the sense of
2
Kalman, exists, that is, rank (B AB A B…A n−1 B) = n. If the (8.29) controller
ensures that the roots of characteristic polynomials of the (8.29), (8.30) closed-
loop system lie in the open left semiplane, this controller is considered to be
stabilising the zero equilibrium state of system (8.28) or the system's controlled
movement (8.27).
Infinitely many controllers stabilising a given controlled movement may
be involved in satisfying the stated conditions. It is therefore logical to intro-
duce the quantitative characteristics of stabilisation quality. To this end, on
the movements of a closed-loop system, expressed through Eqs (8.27) and
(8.29) as
  x = F tx tu t),ϕ p t () ϕ+ t ()]
,(
()
[,
=
)(
x˙=Ft,xt,ut,pt+tu=upt+Wpxt−  uu t()+ W px t() = x ], (8.31)
)[

xp, p p
where p = d/dt is a differential operator, we specify some non-negative
functional
,
x tu t ].
IH=[xt,ut]. I = [ () ()
H
With the (8.31) closed-loop system’s initial conditions being similar, and
the φ(t) functional being equal, the I functional depends upon the choice of
H
transfer matrix W(s) of the (8.29) controller. Hence,
I = { x tW pu tW p()]} = I [ Wp)] = I ( ).
[,
(
[,
)],
(
W
IH=xt,Wp,ut,Wp=IHWp=IHW. H H H