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5.5.2 Observer Design
State-feedback controllers, like the one proposed in the above section, require the
complete knowledge of the whole system state. However, measuring all the states is
often not possible, so a state observer is needed in order to estimate the unmeasurable
states.
Consider the fuzzy polynomial system:
r
˙
=
1 μ i (
)
p i (
)
=
(
)
x
z
x
y
C
x
(5.62)
i
=
n is the state vector and z , in this case, are the premise variables of the
fuzzy model (states or known inputs
where x
∈ R
). Then, a state observer in order to estimate
state x from the measurement y ,isontheform
w
i = 1 μ i ( ˆ
r
ˆ
x
=
z
)(
p i ( ˆ
x
) +
L i (
y
, ˆ
x
)(
y
−ˆ
y
))
(5.63)
where
. L i are polynomial observer gains, on
the measurements and estimated states, to be computed. Also, consider the assump-
tions:
x denotes estimated state and
ˆ
z
ˆ
= ( ˆ
x
,w)
Initial estimation error is equal to initial state, as
x
ˆ
(
0
) =
0 is freely assignable.
System trajectories are ensured to remain in the modeling region ( 5.40 ) (the system
is stable or there exist a control law stabilizing it)
Premises z are measurable, i.e., z
z
There exist a maximum allowed region where the estimation error must remain
e T S e e
χ e ={
:
>
}
S e
χ e (
)
e
1
0
0 and
0
(5.64)
Theorem 5.3 A fuzzy-polynomial state observer can be computed by solving the
following SOS problem:
x T Px
ε(
x
) x
(5.65)
2 e T P p i (
)
2 e T
e T Pe
x
)
p i ( ˆ
x
(
H i (
y
, ˆ
x
)(
C
(
x
)
C
( ˆ
x
)))
2
α
m
e T S e e
s ij (
x
, ˆ
x
)
g j (
x
) λ i (
x
, ˆ
x
)(
1
) x , x
i
:
1
,...,
r
(5.66)
j
=
1
n
where e
=
x
−ˆ
x
∈ R
is the estimation error,
α>
0 is the decay rate and
P 1 H i (
(
s ij i ) x , x . The observer gains can be obtained by L i (
y
, ˆ
x
) =
y
, ˆ
x
)
.
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