Hardware Reference
In-Depth Information
If the following conditions are satisfied:
1. D
2
is injective, i.e., D
2
is of maximal column rank,
2. the subsystem (A, B, C
2
,D
2
) has no invariant zeros on the imaginary
axis,
3. D
1
is surjective, i.e., D
1
is of maximal row rank,
4.thesubsystem(A, E, C
1
,D
1
) has no invariant zeros on the imaginary
axis,
then the H
2
optimal control problem is said to be regular, and the output
feedback controller is given by:
(
v =(A + BF +KC
1
) v − Ky,
Σ
c
:
(3.42)
u =
F
v,
where
F = −(D
2
D
2
)
−1
(D
2
C
2
+ B
T
P),
(3.43)
K = −(QC
1
+ ED
1
)(D
1
D
1
)
−1
,
(3.44)
and P ≥ 0andQ ≥ 0 are respectively the solutions of the following algebraic
Riccati equations (ARE),
A
T
P + PA+ C
2
C
2
−(PB+ C
2
D
2
)(D
2
D
2
)
−1
(D
2
C
2
+ B
T
P)=0,
(3.45)
QA
T
+ AQ + EE
T
−(QC
1
+ ED
1
)(D
1
D
1
)
−1
(D
1
E
T
+ C
1
Q)=0.
(3.46)
Moreover, the infimum γ
∗
2
of the H
2
norm of the closed-loop transfer matrix
T
zw
(Σ×Σ
c
)isgivenby
∗
2
γ
:= inf{ T
zw
(Σ×Σ
c
)
2
|Σ
c
internally stabilizes Σ},
=
{trace(E
T
PE)+trace[(A
T
P + PA+ C
2
C
2
)Q]}
1/2
.
(3.47)
When the problem is singular, the so-called perturbation approach can be
used by adding some small values to z, and redefininganewD and E,
⎡
⎤
⎡
⎤
⎡
⎤
z
x
u
C
2
I
0
D
2
0
I
⎣
⎦
⎣
⎦
⎣
⎦
z
:=
=
x +
u,
(3.48)
E =[ E I 0],
(3.49)
D
1
=[ D
1
0 I ] ,
(3.50)
