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(2σ
j
ζ
j
−
1)
+
Ψ
4
,
j
(
x
F
j
≥
∀
j
∈
Ξ
J
,
F
j
Ψ
4
,
j
(
x
,
χ
)+
,
χ
)
0
,
(10.37)
−
σ
j
Γ
1
K
1j
PO
−
σ
j
K
2j
OO
−
(1+2τ
j
ζ
j
)
+
Ψ
4
,
j
(
x
G
j
≥
∀
j
∈
Ξ
J
,
G
j
Ψ
4
,
j
(
x
,
χ
)+
,
χ
)
0
,
(10.38)
τ
j
Γ
1
K
1j
PO
τ
j
K
2j
OO
(1
−
κ)
+
Ψ
5
(
x
K
≥
KΨ
5
(
x
,
χ
)+
,
χ
)
0
,
(10.39)
0
−
P
0
O
κ
I
with
⎛
⎝
⎞
⎠
x
2
−
x
1
0
...
0
0
O
,
χ)
x
3
0
−
x
1
...
0
0
x
(
x
)
Ψ
1
(
x
,
χ
)
,
x
n
00
...
0
−
x
1
Θ
−
Θ
(
x
I
n
0
x
3
−
x
2
...
0
0
Ψ
1
(
x
,
χ)
)
.
.
.
O
Ψ
2
(
x
,
χ
)
,
Λ
x
(
x
)
,
Ω
1
(
x
,
χ
)
Γ
1
Ω
2
(
x
,
χ
x
)
0
x
n
0
...
0
−
x
2
−
Γ
1
.
.
.
Ψ
3
(
x
,
χ
)
,
(10.40)
0
Ψ
1
(
x
,
χ
(
0
Υ
1j
(
x
,
χ)
Γ
1
)
Υ
2j
(
x
,
χ)
,
00 0
...
x
n
−
x
(
n
−
1)
Ψ
3
(
x
,
χ
)
O
Ψ
4
,
j
(
x
,
χ
)
−
x
0
−
E
x
n
)
.
Γ
1
Ψ
5
(
x
,
χ
)
,
and
x
=(
x
1
...
0
Ψ
1
(
x
,
χ
)
O
Recall that a criterion to be optimized under the above set of LMIs can be introduced
along the lines of Section 10.3.1.2.
Proof.
From the developments of Section 10.4.2.1, the properties underlying the
definition of the multicriteria basin of attraction
˜
E
have been turned into inequalities
of the form (10.14), which depend quadratically on the vector functions
φ
1
(
x
,
χ
),
φ
2
(
x
). Moreover, the matrix functions defined in (10.40) depend
affinely on their arguments and satisfy
,
χ
), ...,
φ
5
(
x
,
χ
˜
∀
(
x
,
χ
)
∈
X×X
χ
,
Λ
x
(
x
)=
0
;
∀
l
∈
Ξ
5
,
Ψ
l
(
x
,
χ
)
φ
l
(
x
,
χ
)=
0
.
(10.41)
So,
Λ
x
(
x
) and
Ψ
l
(
x
,
χ
),
l
∈
Ξ
5
, are linear annihilators of
x
and
φ
l
(
x
,
χ
), respectively.
The LMIs (10.34)-(10.39) on the matrix
) and
on other decision variables straightly follow from the application of Lemma 10.2
to (10.23)-(10.33). Importantly, the conservativeness of these sufficient conclusions
similar to (10.15) can be all the more reduced as the number of lines of
P
entailed in the definition of
V
P
(
x
,
χ
φ
l
(
.,.
),
l
∈
Ξ
5
, is important. This is why
Λ
x
(
x
) appears in
φ
l
(
.,.
),
l
∈
Ξ
5
.
10.4.3
LMI Conditions for Multicriteria Analysis Based on
PW-BQLFs
The counterparts of the results developed in Section 10.4.2 when using piecewise
biquadratic Lyapunov functions are hereafter sketched out. In addition to the above
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