Linear Matrix Inequalities in Control

Basic LMI tools for control (8)
Bounded real lemma
The system definition
x Ax Bu
G(
s)
sup ( G(
s))
sup ( G(
j ))
Re( s)
0
R
y Cx Du
x(0)
0
G(
s)
1
C(
sI A)
B
Minimize γ 2 subject to:
T
T
T
A P PA C C PB C D
T T
T 2
B P D C D D I
D
Find (the upper bound)
0; P
P
T
0
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Basic LMI tools for control (9)
S-procedure (simplified)
If there exist symmetric matrices T 0 ,...,T p and nonnegative
numbers
1
0,...,
p
0 such that
T
0
i
p
1
iT i
0
T
T
then x T0 x 0; x 0if
x Ti
x 0; i 1, ,
p.
For p=1 the reverse is true.
KSO materiały do wykładu 2008/09 38
Basic LMI tools for control (10)
S-procedure example
There are two condtions given for P
P
T
0,
0 and any
T
T
A P PA
T
B P
PB
0
0
T C T C
T
A P
PA
T
B P
T
C C
PB
I
0; P
P
T
T .
Using S-procedure one can obtain LMIs:
0;
0
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