Hardware Reference
In-Depth Information
transforming state vector p into a new state vector x as,
∙
¸
∙
¸∙
¸
x
1
x
2
10
0 T
S
p
1
p
2
x =
=
(2.46)
where, x
1
and x
2
are state variables in units of tracks and tracks/sector, re-
spectively. The sampling interval T
S
is the same as the time between two servo
sectors in HDD. Then the transformed state equation is,
∙
¸
∙
¸
dx
dt
=
0
T
S
00
0
aT
S
x +
u,
y =[1 0]x,
(2.47)
Corresponding discrete-time state space model is
x(k +1) = Φx(k)+Γu(k),
y(k)=Hx(k).
(2.48)
This nominal model is good enough for designing an observer that can estimate
the two states, position and velocity, required by the PTOS algorithm. But we
also want to estimate the input disturbance. Augmenting the nominal state
space model to include the disturbance as an additional state is a common
practice. In the case of HDD actuator, the input disturbance is assumed
constant during the linear range of operation. If w is the input disturbance,
then w(k +1)=w(k). Combining this with the nominal discrete state space
model of equation 2.48 we get,
∙
¸
∙
¸∙
¸
∙
¸
x(k +1)
w(k +1)
ΦΓ
00
x(k)
w(k)
Γ
0
=
+
u(k)
(2.49)
∙
¸
x(k)
w(k)
y(k)=[H 0]
or
z(k +1)=Φ
a
z(k)+Γ
a
u(k);
y(k)=H
a
z(k).
(2.50)
∙
¸
∙
¸
ΦΓ
00
Γ
0
Here z =[x
T
w]
T
is the augmented state vector, Φ
a
=
, Γ
a
=
,
and H
a
=[H 0 ]. The prediction observer using this model is
z(k +1)=Φ
a
z(k)+Γ
a
u(k)+L
p
[y(k) −H
a
z(k)],
(2.51)
where z is the estimate of the augmented state vector z. The observer gain
L
p
must be chosen to satisfy the stability condition, i.e., all eigenvalues of
Φ
a
− L
p
H
a
are inside the unit circle. There are many standard methods to
select the observer gains such as Ackerman's formula.
2.6.3 Rejection of Input Disturbance
Integral control is the most widely acknowledged solution for eliminating steady
state error. The state-space design presented earlier in conjunction with PTOS
