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5.3 Polynomial Sum-of-Squares Optimization: An Outline
Once fuzzy polynomial models are available, as the derivatives (increments in
discrete-time case) of functions of the state will involve polynomials, stability analy-
sis and control design (to be later discussed) require proving positiveness or nega-
tiveness of some polynomials (in several variables, actually state-space ones). These
issues have been developed in literature at the beginning of the past decade, conform-
ingwhat it is now called Sum-Of-Squares (SOS) programming and optimization. The
basic ideas are now outlined.
5.3.1 Notation
Arbitrary degree polynomials in variables “
z
” will be denoted as
R
z
. For instance,
x
1
x
2
+
x
2
−
2
x
1
+
the polynomials in
x
=
(
x
1
,
x
2
,
x
3
)
such as
p
(
x
)
=
5
−
x
3
will be
said to verify
p
(
x
)
∈
R
x
.The
n
-dimensional vectors of polynomials will be written
n
as
z
. For instance, the right-hand side of state-space equations of a polynomial
3rd-order system will be a polynomial vector in
R
3
x
.
In matrices, the corresponding element of a symmetric expression will be omitted
and denoted as
R
(
∗
)
. For instance,
(
∗
)
denotes
B
(
z
)
on both of the expressions below:
A
T
(
z
)(
∗
)
T
PB
,
∗
)
(
z
)
(5.20)
B
(
z
)
C
(
z
)
5.3.2 Sum of Squares Polynomials
The key idea in the SOS approach is trying to find an expression of a polynomial as
the sum of squares of simpler polynomials. The notation
z
will be used to denote
the set of sum-of-squares (SOS) polynomials in the variables “
z
”.
An even-degree polynomial
p
(
z
)
is SOS if and only if there exist a vector of mono-
T
Hm
mials
m
;
in this way, SOS problems can be solved via semidefinite programming (SDP, LMI)
tools searching for
H
(Löfberg
2009
; Prajna et al.
2004a
). Evidently, all SOS poly-
nomials are non-negative, but the converse is not true (Blekherman
2006
).
(
z
)
and a positive definite matrix
H
0 such that
p
(
z
)
=
m
(
z
)
(
z
)
z
1
−
4
z
1
z
2
+
2
z
1
+
Example 5.3
The above software finds that polynomial
p
(
z
)
=
4
z
1
z
2
−
12
z
1
z
2
+
z
1
+
16
z
1
z
2
−
16
z
2
can be written as:
8
z
1
z
2
+
⎛
⎞
⎡
⎤
1
−
4
−
21
z
1
z
2
z
1
z
2
z
1
)
=
z
1
z
2
z
1
z
2
z
1
⎝
⎠
⎣
⎦
−
416 8
−
4
p
(
z
(5.21)
−
28 4
−
2
1
−
4
−
21
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