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Lyapunov function (it can quickly exhaust computational resources); [
b
] Remodel
the nonlinear system in a smaller region in order to reduce the “distance” between
the vertex models.
5.4.2 Local stability analysis
Apart from the above alternatives, consider now
={
x
:
h
1
(
x
)
≥
0
,...,
h
i
(
x
)
≥
0
known polynomials. Then, local stability problem is based on mod-
ifying conditions (
5.27
) and (
5.28
)(or(
5.29
) alternatively) in order to make them
hold locally only in a region of interest. This can be addressed using Lemma 5.2, as
next example illustrates.
}
, being
h
i
(
x
)
Example 5.8
Consider the system:
x
1
=−
˙
3
x
1
+
0
.
5
x
2
x
2
=
(
−
˙
2
+
3sin
(
x
1
))
x
2
(5.32)
with the objective of finding a decreasing Lyapunov function in the square region
x
1
−
π
2
x
2
−
π
2
={
x
:
≤
0
,
≤
0
}
(5.33)
(
x
1
)
ensuring maximal decay rate. Under usual TS modeling, as sin
ranges between
−
1 and 1, we have
−
5
x
2
≤
(
−
2
+
3sin
(
x
1
))
x
2
≤
x
2
, so we would obtain
x
˙
=
i
=
1
μ
i
A
i
x
, with:
−
−
30
.
5
5
01
30
.
A
1
=
A
2
=
(5.34)
0
−
5
and, as
A
2
is not stable, the TS approach would fail in proving stability.
However, using the 1th-order Taylor expansion of sin
(
x
1
)
, we can show that there
exists a fuzzy model so that sin
(
x
1
)
=
μ
1
·
0
·
x
1
+
μ
2
·
1
·
x
1
. In this way, replacing
2
2
2
1
2
2
μ
1
=
σ
1
,
μ
2
=
σ
2
, and noting that
σ
+
σ
=
1, we obtain the fuzzy-polynomial
model:
1
2
x
1
=
(
−
˙
3
x
1
+
0
.
5
x
2
)(σ
+
σ
)
1
2
2
x
1
x
2
x
2
=−
˙
2
x
2
(σ
+
σ
)
+
σ
3
(5.35)
)
=
v
1
x
1
+
v
2
x
1
x
2
+
v
3
x
2
with
Looking for a quadratic Lyapunov function
V
(
x
v
i
∈ R
decision variables, and a decay-rate bound
α
, condition
,σ)
=−
V
2
1
2
Q
(
x
−
2
α
V
(σ
+
σ
2
)>
0
(5.36)
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