Environmental Engineering Reference
In-Depth Information
In this chapter, a static output-feedback controller u
¼
Ky is to be determined
under constrained information, i.e., a zero-nonzero structure on the control gain
matrix by utilizing
LMI
s such that the following requirements are met:
1. The following closed-loop system is asymptotically stable;
¼
:
A
cl
B
cl
A
t
þ
B
t
KC
t
B
1
ð
6
:
8
Þ
C
cl
0
C
z
þ
D
z
KC
t
0
2. Under zero initial condition, the closed-loop system satisfies
z
ð
t
k
2
\cx
ð
t
k
2
for any nonzero x
ð
t
Þ2
L
2
0
;
1
½
Þ
where c is a positive scalar.
6.3 Controller Design
H
1
control is chosen because of its ability to minimize any energy-bounded
disturbance on the controlled output. Also, since the linear model is of low order
and the nonlinear model is of high order, this so-called advanced control design
technique can catch a part of the unmodeled dynamics. The main issue in this
section is to design a controller, which is able to handle constrained information,
i.e., only a part of the available information will be used to calculate the control
signal. In this case, that is to design a decentralized controller. What is meant with
decentralized control, is that none of the calculated control signals should directly
interfere with each other. The considered system consists of three inputs and three
outputs, this indicates that the gain matrix is square and of dimension
R
3
3
.By
imposing a diagonal structure on the gain matrix, this is possible. The solution
found in [
20
], makes it possible to impose zero-nonzero constraints on the
LMI
variables. The control problem is formulated in terms of
LMI
s, and are solved
using YALMIP [
17
] interfaced with MATLAB.
The
H
1
performance constraints for a state-feedback system formulated in
terms of
LMI
s are as follows:
\0
;
Þ þ
c
2
B
1
B
1
sym A
t
X
þ
B
t
Y
ð
C
z
X
þ
D
z
Y
I
ð
6
:
9
Þ
X [ 0
:
Remark 1 In a manner similar to [
3
], it is possible to present a new
H
1
perfor-
mance criterion for the robust stability analysis of the system (
6
) with norm-
bounded time-varying parameter uncertainties in the state-space matrices.
For the state-feedback case, the gain matrix is calculated as K
¼
YX
1
. In the
output-feedback case, the state gain matrix factors as the product K
¼
KC
t
, where
