68
been constructed in agreement with equation (4-44). Since asymptotic
tracking and disturbance rejection will occur, there must be a steady-
state solution to (4-44). Consequently, if both w(t) and r(t) are
applied to the system at the time t = 0 then there exist initial states
x(0) = xQ and n(0) = n0 such that no transients appear in the state
trajectory and the tracking error is zero. We denote this trajectory by
the pair [xs$(t), n$s(t)]. In terms of this notation, (4-44) becomes
s
F
o'
\s(ti
+
G
uss(t> +
E '
w(t) +
0
."ssW
-BH
A
."ss(t)_
0
* .
0
B
uss(t) = -KjXss(t) -K2nss(t) (4-45)
yss(t) = r(t) = Hxss(t)
If (4-45) is subtracted from (4-44) the following equation results:
(t)
F 0
x(t)"
G
as
+
n(t)
-BH A
(t)
0
where
x(t) = x(t) xss(t)
n(t) n(t) nss(t)
u(t) = u(t) us$(t) = -Kjxt) K2(t)
(4-47)