Saunder January 22, 2014 10:43 K15053˙Book
164 Disturbance Observer-Based Control: Methods and Applications
then, it follows from (10.19) that the output can be represented by
Y(s ) = g
n
(s )e
−θ
n
s
U(s ) + D
l
(s ). (10.22)
As shown in Figure 10.2, the control law is
U(s ) = C(s ) −
ˆ
D
f
(s ), (10.23)
where
ˆ
D
f
(s ) = Q(s )g
−1
n
(s )Y(s ) − Q(s )e
−θ
n
s
U(s ), (10.24)
Substituting Equation (10.22) into Equation (10.24) yields
ˆ
D
f
(s ) = Q(s )g
−1
n
(s )D
l
(s ), (10.25)
Define
˜
D
l
as the error between the real value and the estimated value of lumped
disturbance, i.e.,
˜
D
l
(s ) = D
l
(s ) − g
n
(s )e
−θ
n
s
ˆ
D
f
(s )
= [1 − Q(s )e
−θ
n
s
]D
l
(s ), (10.26)
According to the final-value theorem, one obtains from Equation (10.26) that
˜
D
l
(∞) = lim
t→∞
˜
D
l
(t)
= lim
s →0
s
˜
D
l
(s )
= lim
s →0
[1 − Q(s )e