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IF
ϑ
1
(τ )
is M
i
1
AND
...
AND
ϑ
p
(τ )
is M
ip
(6.12)
THEN u
(τ )
=
K
i
x
(τ )
that is, to calculate the state-feedback gains
K
i
such that the closed-loop poles of
(
6.2
) are robustly placed in
D
, independently of the values taken by the uncertain
matrices
B
i
.
The main motivation for seeking pole clustering in specific regions of the com-
plex plane is that, by constraining the eigenvalues to lie in a prescribed region, a
satisfactory transient response can be ensured.
The following theorem is valid:
A
i
and
Theorem 1
be an LMI region and assume that, for each uncertain LTI subsys-
tem in
(
6.1
)
described by the matrix pair
Let
D
(
A
i
,
B
i
)
as in
(
6.6
)
-
(
6.9
)
, a state-feedback
X
T
gain K
i
and a Lyapunov matrix X
=
>
0
have been obtained such that:
α
kl
X
+
β
kl
A
Ni
+
A
ij
X
+
B
Nt
+
B
tj
i
+
B
Nt
+
B
tj
i
T
+
β
lk
A
Ni
+
A
ij
X
<
0
(6.13)
1
≤
k
,
l
≤
m
for each i
K
i
X.
Moreover, assume that a single Lyapunov matrix X has been used to solve this
problem for all the subsystems. Then, the TS states-feedback controller
(
6.12
)
places
the closed-loop poles of
(
6.1
)
in
,
t
=
1
,...,
N and j
=
1
,...,
M, where
i
=
D
, independently of the values taken by the uncertain
matrices
A
i
and
B
t
.
Proof
Due to a basic property of matrices (Horn and Johnson
1990
), any linear
combination of (
6.13
) with non-negative coefficients is negative definite. Hence,
using the linear combination given by (
6.8
)-(
6.9
) leads to:
1
η
ij
M
+
β
kl
A
Ni
+
A
ij
X
+
B
Nt
+
B
tj
i
α
kl
X
j
=
+
B
Nt
+
B
tj
i
T
+
β
lk
A
Ni
+
A
ij
X
<
0
(6.14)
1
≤
k
,
l
≤
m
that can be rewritten, taking into account that
j
=
1
η
ij
=
1 and through simple
mathematical manipulation, in the following form:
+
β
kl
A
Ni
+
A
i
X
+
B
Nt
+
B
t
i
α
kl
X
+
B
Nt
+
B
t
i
T
+
β
lk
A
Ni
+
A
i
X
<
0
(6.15)
1
≤
k
,
l
≤
m
Then, considering twice the linear combination given by the coefficients
ρ
i
(ϑ(τ))
in (
6.4
), the following can be written:
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