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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
=ˆ
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
,λ
i
)
∈
x
,
x
. The observer gains can be obtained by L
i
(
y
,
ˆ
x
)
=
y
,
ˆ
x
)
.
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