Information Technology Reference
In-Depth Information
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:
Search WWH ::




Custom Search