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Lemma 5.4
(discrete-time (Papachristodoulou and Prajna
2005
))
Global stability
of a system
(
5.16
)
will be proved if a polynomial Lyapunov function V
(
)
x
can be
found such that
(
5.27
)
holds and
V
(
x
)
−
V
(
˜
p
(
x
,σ))
−
ε
2
(
x
)
∈
x
,σ
(5.29)
Indeed, setting
V
(
x
)
to be an arbitrary polynomial in the state (but
not
in the
dV
dx
. Hence
d
dt
memberships),
is also a vector of polynomials in the variables
x
and
σ
in (
5.28
) and
V
k
in (
5.29
) are polynomial. If
V
is linear in some decision
variables (the natural choice are the polynomial coefficients), expressions (
5.27
),
(
5.28
) and (
5.29
) can be introduced into SOS packages in order to get values of the
decision variables fulfilling the constraints. Also, expressions (
5.28
) and (
5.29
) need
to be modified for actual computations, making it homogeneous in the memberships
(i.e., all the monomials must have the same degree in
V
=
V
k
+
1
−
σ
). It can be achieved by
multiplying anything by
i
=
1
σ
2
(which is equal to one, anyway) as many times as
i
needed (Sala
2009
).
Example 5.7
Recall system (
5.26
) and consider now a Lyapunov function
V
in
the form of a fourth degree polynomial in the state variables
x
1
,
x
2
,
x
3
and tolerances:
(
x
)
x
1
+
x
2
+
x
3
)ε
2
(
2
1
2
ε
1
(
x
)
=
0
.
01
(
x
)
=
ε
1
(
x
)(σ
+
σ
2
)
(5.30)
Conditions of Lemma 5.3 can be introduced in YALMIPSOS as discussed in
AppendixB. Note that the derivative of the Lyapunov function will then be non-
positive for any value of
μ
i
.
Using SeDuMi, the solver finds, in 0.4 s, a 4th-degree Lyapunov function:
σ
i
, i.e., for any non-negative
49
x
1
+
453
x
3
+
875
x
1
+
18
x
2
+
532
x
2
V
(
x
)
=
24
.
6
.
50
.
19
.
19
.
0182
x
3
+
555
x
1
x
2
+
138
x
1
x
3
+
683
x
2
x
3
+
0
.
33
.
6
.
1
.
(5.31)
which is, evidently, SOS and whose time derivative with changed sign is also SOS.
Luckily, the proposed system is asymptotically stable in all state space (as it is easy to
derive from system dynamic equations). However,
V
, or the equivalent discrete
V
,
n
(for instance if
V
is not an even degree polynomial).
Therefore, proving global stability for fuzzy polynomial systems is usually difficult.
often are not SOS for all
x
∈ R
Remark 5.2
Despite setting up global conditions, as Taylor-series polynomial fuzzy
models are only valid locally in most cases, stability or decay-rate performance is
not proved in the whole state space where the SOS conditions hold (unless
n
).
= R
The actually proved local domain of attraction (denoted as
D
) is the largest invariant
set
V
(
x
)
≤
v
0
,
v
0
constant, contained in
. The set can actually be a very small
fraction of
so ill-shaped Lyapunov functions might be useless in practice.
In case that global Lyapunov function search is unsatisfactory but the linearized
system around
x
0 is exponentially stable (so the nonlinear system is “locally sta-
ble”, first Lyapunov theorem), there exist some options: [
a
] Try with a higher degree
=
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