Linear Matrix Inequalities in Control

Basic LMI tools for control (2)
Stability
Equivalence between LMI definition and matrix formula
p1
p2
Example for n=2 P
0
p p
A T P PA
2a
p1
a
11
12
a12
0
a
a
11
12
a
a
21
22
11
p1
p
2
2
2a21
p2
a a
22
p2
p
3
3
a11
a
2a
12
p1
p
2
22
p2
p
3
a
a
11
21
0
p3
a
21
a
a
12
22
a
2a
21
22
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Basic LMI tools for control (3)
Stability
Additional conditions on s-plane regions
Guaranteed damping
C stab
{ s C | re(
s)
} s s 2 0
Maximal damping and oscillation
r s q
C stab
{ s C | s q r}
0
s q r
Vertical strip
( s s)
2
2
0
C stab
{ s C |
1
re(
s)
2}
0
0 ( s s)
2
1
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Basic LMI tools for control (4)
Stability
Theorem:
All eigenvalues of the real matrix A are in the region
described by C
stab
{ s C|
P Qs
T
Q s 0}
with P
T
P
if and only if there exists symmetric matrix X=X T >0
p X
p
11
k1
X
q11AX
q AX
k1
T
q11XA
T
q XA
1k
T
p1
k
X q1
k
AX qk1XA
T
p X q AX q XA
kk
kk
kk
0
Equivalent LMI using Kronecker multiplication:
P X Q AX
Q
T
XA
T
0
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11