Environmental Engineering Reference
In-Depth Information
4.4 H
'
Optimal Control Background
Consider an LTI system with the following state space realization
x
¼
Ax
þ
B
w
w
þ
B
u
u
z
¼
C
z
x
þ
D
zw
w
þ
D
zu
u
y
¼
C
y
x
þ
D
yw
w
þ
D
yu
u
ð
4
:
8
Þ
n
y
n
u
. The signal u is the control input
and w is the disturbance. The signal y is the controlled variable and z is a fictitious
output signal that serves to state the control objectives. The signal z is commonly
named as the performance output.
It is assumed that system in Eq.
4.8
is stabilizable and detectable. That is, there
exists a control law u
¼
K
ð
s
Þ
y that stabilizes the closed loop system
n
n
, D
zw
2
n
z
n
w
where A
2
R
R
and D
yu
2
R
T
zw
ð
s
Þ¼
G
zw
ð
s
Þþ
G
zu
ð
s
Þ
K
ð
s
Þð
I
þ
G
yu
ð
s
Þ
K
ð
s
ÞÞ
1
G
yw
ð
s
Þ
with I the identity matrix and G(s) the transfer function of the system in Eq.
4.8
partitioned as
The c-suboptimal H
?
synthesis problem consists in finding an internally sta-
bilizing control law u
¼
K
ð
s
Þ
y that guarantees an
1
-norm of the closed-loop
transfer function from the disturbance w to the performance output z lower than c.
Being T
zw
ð
s
Þ
being the closed-loop transfer function from w to z, the control
objective can be formalized as
k
T
zw
k
1
\c
;
ð
4
:
9
Þ
where
kk
1
denotes the infinity norm. For a stable system with transfer function
G(s), the
1
-norm is defined as
k
G
ð
s
Þk
1
¼
max
x
r
max
ð
G
ð
jx
ÞÞ
where r
max
is the maximum singular value and x is the frequency [
10
]. In other
words, the
1
-norm is basically the maximum gain of the frequency response of
the transfer function G(s).