
Saunder January 22, 2014 10:43 K15053˙Book
152 Disturbance Observer-Based Control: Methods and Applications
Sliding motion
Convergence of
sliding mode surface
Convergence of FDO
State trajectory
d
ˆ
(t)
–d
ˆ
(t) = –d(t)
–d
ˆ
(t)
–
x
2
x
1
–
x(t
0
)(s(t
0
)> 0)
––
s(t)(t
0
≤ t< t
f
)
–– ––
s(t)(t
≥ t
f
)
x(t
0
)(s(t
0
) < 0)
––
x(t
f
)
–
x(t
f
+ t
r
)
–
x(t
f
+ t
r
)
–
s(t)(t
0
≤ t < t
f
)
–– – –
x(t
f
)
–
–
Figure 9.6 Phase plot of the system under the proposed FDO-NTSMC method.
for t ≥ t
f
+ t
r
. With the chosen control parameters, system (9.42) is finite-time
stable, which implies that there is a time constant t
s
> 0 such that y (t) = x
1
(t) = 0
and x
2
(t) =−d(t) for t ≥ t
f
+ t
r
+ t
s
.
The phase plane ...