Saunder January 22, 2014 10:43 K15053˙Book
264 Disturbance Observer-Based Control: Methods and Applications
where
ˆ
d
lx
is the estimate of the lumped disturbance on the states d
lx
, z is an auxiliary
vector, and L is the observer gain matrix to be designed.
Lemma 15.1 [12]: Consider system (15.7) under the lumped disturbances that satisfy
Assumption 15.1. The estimates of the disturbance observer (15.12) asymptotically tracks
the lumped disturbances if the observer gain matrix L is chosen such that −L is a Hurwitz
matrix.
Proof The disturbance estimation error of the DOB (15.12) is defined as
e
d
lx
=
ˆ
d
lx
− d
lx
. (15.13)
Combining (15.7), (15.12) with (15.13) gives
˙
e
d
lx
=
˙
ˆ
d
lx
−
˙
d
lx
=
˙
z +L
˙
x −
˙
d
lx
=−L
ˆ
d
lx
− L(A
n
x +B
n
u)
+L(A
n
x +B
n
u +d
lx
) −
˙
d
lx
=−L(
ˆ
d
lx
− d
lx
) −
˙
d