Linear Matrix Inequalities in Control

Synthesis (18)
Gain Scheduled H-inf Controller
• Gain scheduled H-inf controller can be
calculated in three steps:
1. After the application of elimination lemma one can
obtain Lapunov matrices X and Y for all vertices of
hypercube.
2. For each vertex of hypercube calculate the vertex
controller.
3. For each value of parameter calculate the
corresponding controller and use it in the control.
KSO materiały do wykładu 2008/09 79
Synthesis (19)
Elimination Lemma
• Definition. An orthogonal complement of given
matrix N is any maximal rank matrix, such that
T
N N 0 and N N 0.
n k n m
T n n
• Lemma. Let M R , N R , H H R
The following statements are equivalent:
m k
1. There exists X R such that
T T T
NXM MX N H 0
2. The following two conditions hold
N HN
T
M HM
T
0
0
KSO materiały do wykładu 2008/09 80
Synthesis (20)
Gain Scheduled H-inf Controller
Step 1. Solve 2*2 L +1 LMIs for X and Y Lapunov matrices
NX
0
0
I
T
T
XA A X
T
B1
X
C
1
XB1
I
D
11
T
C1
T
D11
I
NX
0
0
I
0
Where:
N X
null( C 2
D21
)
)
NY
0
0
I
T
T
YA AY
C Y
B
1
T
1
T
YC1
I
D
T
11
B1
D11
I
NY
0
0
I
0
N
Y
null
T T
( B2
D12
X
I
I
Y
0
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