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These parameters de
ne completely the given observer:
8
<
_
z
ðÞ
¼
P
2
h
i
x
ð
t
ðÞ
H
i
z
ðÞþ
L
i
y
ðÞþ
J
i
u
ðÞþ
M
i
y
ðÞ
u
ðÞ
ð
Þ
:
i¼1
^
ðÞ
¼
ðÞ
ðÞ
xt
zt
Ey t
To show the effectiveness of the proposed observer, simulation results are
presented in Figs.
1
,
2
and
3
for the input signal u
ð
t
Þ
¼
0
:
5 sin
ð
0
:
5
p
t
Þ
and for
the unknown inputs d(t)
is a rectangular signal of amplitude 0.5 applied
for 1
5. The evolution of the state estimation error with unmeasurable
decision variables are given in these Figs.
1
,
2
and
3
.
It can be deduced from Figs.
1
,
2
and
3
that the state estimation error converges
asymptotically tends to zero in spite of the presence of the unknown input and
unmeasurable premise variables.
Then, the fuzzy bilinear state and their estimation are given in the following
Figs.
4
,
5
and
6
.
Figures
4
,
5
and
6
show respectively the evolution of the state variables x
1
, x
2
and x
3
of the considered system and their corresponding observer estimation x
1
, x
2
and
:
5
t
2
:
T
0. Based on
Figs.
4
,
5
and
6
it can seen that the estimated state converges globally asymptot-
ically to the real state of nonlinear system.
^
x
3
with the initial conditions x
0
¼
½
0
:
50
:
50
:
5
and
^
x
0
¼
Fig. 1 Trajectories of state estimation error between x
1
and its estimate with unmeasurable
decision variables
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