Linear Matrix Inequalities in Control

LMI Primer (5)
The Schur complement lemma:
Converts convex nonlinear inequality
R(
x)
where
0,
Q(
x)
1 T
S(
x)
R(
x)
S(
x)
T
T
Q( x)
Q(
x)
, R(
x)
R(
x)
0 and S(
x)
into the equivalent LMI
Q(
x)
T
S(
x)
S(
x)
R(
x)
Very important lemma!
0
0
depend affinely on x
KSO materiały do wykładu 2008/09 25
The condition
( A(
x))
1
I
A(
x)
T
LMI Primer (6)
Maximum singular value: is defined as:
T
( A(
x))
max(
A(
x)
A(
x)
)
where matrix A(x) depends affinely on x
( A(
x))
1 can be expressed as a LMI
A(
x)
A(
x)
A(
x)
0
I
T
I I
A(
x)
A(
x)
T
0
KSO materiały do wykładu 2008/09 26
whereP
LMI Primer (7)
Algebraic Riccati inequality:
T
1 T
A P PA PBR B P Q 0
P
T
T
0; Q Q ; R 0;
can be expressed as a LMI
T
A P PA
T
B P
Q
PB
R
0;
P
0.
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