Hardware Reference
In-Depth Information
3.4.2 H
2
Optimal Control: Discrete-time Case
Consider a stabilizable and detectable linear time-invariant system Σ with a
proper output feedback controller Σ
c
, show in Fig 3.34 where,
⎨
x(k +1) = Ax(k)+B
1
w(k)+B
2
u(k),
y(k) = C
1
x(k)+D
11
w(k),
z(k) =C
2
x(k)+D
21
w(k)+D
22
u(k),
Σ :
(3.53)
⎩
with x ∈
n
the state, u ∈
n
the control input, w ∈
l
the disturbance input,
y ∈
p
the measured output (the measured PES in case of HDD), and z ∈
q
the output to be controlled (the true PES in this case).
The controller in the form
½
x
c
(k +1) = A
c
x
c
(k)+B
c
y(k),
u(k) =C
c
x
c
(k)+D
c
y(k),
Σ
c
:
(3.54)
||Φ
zw
||
2
<µare parameterized by LMI (Linear Matrix Inequal-
such that
ity) [41]:
⎨
⎩
trace(W) <µ,
⎡
⎣
⎤
⎦
WC
2
X + D
22
LC
2
+ D
22
RC
1
∗
−PI + S
−J
X + X
> 0,
−H
∗
∗
Y + Y
⎡
⎣
PJAX+ B
2
LA+ B
2
RC
1
B
1
+ B
2
RD
11
⎤
(3.55)
∗ H
Q
A + UC
1
YB
1
+ UD
11
⎦
∗∗X −X
−PI
S
−J
0
> 0,
Y + Y
−H
∗∗ ∗
0
∗∗ ∗
∗
I
wher
⎨
⎩
D
c
= R,
C
c
= L−RC
1
X)Λ
−1
,
B
c
= Ξ
−1
(U −YB
2
R),
A
c
= Ξ
−1
[Q−Y (A + B
2
D
C
C
1
)X −ΞB
c
C
1
X −YB
2
C
c
Λ]Λ
−1
,
(3.56)
and Ξ and Λ are nonsingular with ΞΛ = S −YX.
For the case of disk drive, augmenting the plant mode with process distur-
