polynomial controller design based on flatness - Kybernetika

Polynomial Controller Design Based on Flatness 579
By denoting R*(p
x ) = 1 + Q*(p x ), this control can be written in the RST form:
As a remark, from (50) and (19), we get:
R*(p~ 1 ) u(t) = K(p) z d (t) - S*(p- X ) y(t). (50)
R*(p' 1 ) A(p) z(t) = K(p) z d (t) - S*^- 1 ) B(p) z(t). (51)
After some manipulations, we deduce R*(p~ x ) A(p)-{-S*(p~ 1 ) B(p) = K(p), and if we
notice that A(p) and K(p) are of the same degree, the expression A(p~ l ) R*(p~ 1 ) +
B(p~ 1 )S*(p~ 1 ), can be written as:
Afø-^ДҶp- 1 ) +B(p- 1 )S*(p- 1 ) =p' n K(p), (52)
which gives the relationship of the poles of the RST <strong>controller</strong> with the tracking
dynamics.
A second remark can be done here. Namely, it follows also:
n-1
z(t) = h I •_4 n - 1 p-< n - 1 )z + Yl A'^Bp-'u \ ,
І=l
(53)
where h is previously defined. Thus:
Z(t) = h {
ІП-l/Q-l
o (A,C)
p-("-l)y
p~( n ~ 2 îy n-1
+ '—]A i - 1 Bp- i i
г=l
AП
°(A,C) M (A,-9,C)
!-(^-2) 7i
= ^(p-^yW+D*^- 1 )^^,
p
l U
(54)
which defines the flat output in terms of the proper operator p _1 .
6. DISTURBANCE REJECTION
In order to reject a static perturbation, an integral action must be added to the
model. The proposed methodology is then applied to the following augmented
model:
ŽW =ÃŽ + Bй,
y = ÕŽ,
(55)