Linear Matrix Inequalities in Control

Optimisation approach (2)
scalar signal norms
L
L 2
norm - amplitude
f ( t)
sup f ( t)
f
t 0
norm - energy
1
( d
2
2
2
f t)
dt fˆ(
)
2
0
2
KSO materiały do wykładu 2008/09 7
Optimisation approach (3)
system norm H
System description (causal, with L 2 gain γ>=0)
x ( t)
Ax(
t)
Bf ( t)
y(
t)
Cx(
t)
Df ( t)
H(
s)
C(
sI
Norm definition L 2 gain
T
T
2
2
y(
t)
dt
c
0
0
H sup H(
j
R
2
f ( t)
dt
Norm interpretation
) H
sup max
R
( H(
j
A)
))
1
B
D
KSO materiały do wykładu 2008/09 8
Optimisation approach (4)
system norm H calculations
Theorem
Let A be a Hurwitz (stable) matrix. Then the L 2 gain of the
system is less than γ if and only if max(
D)
and the matrix
F
2
A B(
I
T T
C C C D(
T 1 T
D D)
D C
2 T 1 T
I D D)
D C
2 T 1 T
B(
I D D)
B
T T 2 T
A C D(
I D D)
1
B
does not have eigenvalues on the imaginary axis.
Hint : use bisection method.
KSO materiały do wykładu 2008/09 9
3