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˜
∈
Ξ
S
×
Ξ
S
. To ensure that the time-derivative
V
(
X
s
b
, (
s
a
,
s
b
)
.,.
) along the system
˜
s
a
s
b
's, the continuity of
V
(
X
.,.
trajectories is well-defined on all
∂
) can be imposed.
10.3.2
Matrix Inequalities and Related Important Lemmas
Definition 10.3 (LMIs [2]).
A constraint
(
x
)
on the real-valued vector or matrix
decision variable
x
is an
LMI
on
x
if it writes as the negative or positive definiteness
of an affine matrix combination of the entries x
1
,...,
L
A
i
x
n
of
x
, i.e. if (with
A
i
=
given real matrices)
L
(
x
) :
A
0
+
x
1
A
1
+
···
+
x
n
A
n
≤
0
.
(10.10)
LMI constraints are convex. Consequently, the feasibility of a set of LMIs as well
as the minimum of a convex criterion subject to LMIs are convex problems, whose
solutions can be worked out numerically in polynomial time with an arbitrary pre-
cision. So, such problems are considered as solved. The versatility of semidefinite
programming in engineering was acknowledged more than a decade ago [2, 16].
Thanks to the equivalent representations of (10.4)-(10.5) to be introduced in
Section 10.4.1, the rules (10.6)-(10.9) will be turned into inclusion relationships
between sets defined by quadratic functions. The following lemmas bridge the gap
with LMIs.
1
F
l
1
,
Lemma 10.1 (S-procedure [2]).
Consider quadratic functions f
l
(
ξ
)
F
l
.Then
l
∈
Ξ
L
, with
F
l
=
{
ξ
:
f
l
(
ξ
)
≤
0
, ∀
l
∈
Ξ
L
}⊂{
ξ
:
f
0
(
ξ
)
≤
0
}
(10.11)
is true if
L
l
=1
τ
l
f
l
(ξ)
≤
0
.
∃
τ
1
≥
0
,...,
τ
L
≥
0:
∀
ξ
,
f
0
(
ξ
)
−
(10.12)
When the entries of
ξ
are independent,
(10.12)
is equivalent to the LMI on
τ
1
,...,
τ
L
:
L
l
=1
τ
l
F
l
≤
0
.
τ
1
≥
0
,...,
τ
L
≥
0
and
F
0
−
(10.13)
If the entries of
are related, a less conservative sufficient condition can be got.
Lemma 10.2 ([24, 8]).
Consider two vectors
x
ξ
˜
,
χ
in given convex polytopes
X ,X
χ
.
˜
Σ
τ
l
,
P
,...
(
.,.
)=
Σ
τ
l
,
P
,...
(
.,.
) :
n
σ
×
n
Define a matrix function
X×X
χ
−→
R
σ
, with affine
dependency on its arguments and on the decision variables
τ
l
,
P
,...
Consider the
˜
following constraint on a prescribed nonlinear vector function
σ
(
.,.
) :
X×X
χ
−→
n
R
σ
:
X×X
χ
,
σ
(
x
˜
∀
(
x
,
χ
)
∈
,
χ
)
Σ
τ
l
,
P
,...
(
x
,
χ
)
σ
(
x
,
χ
)
≤
0
.
(10.14)
By convexity, if the LMIs
Σ
τ
l
,
P
,...
(
x
,
χ
)
≤
0
on
τ
l
,
P
,...
hold at all the vertices
(
x
,
χ
)
∈
(
˜
˜
V
X×X
χ
)
, then they also hold on
X×X
χ
and
(10.14)
is satisfied. Yet, if an affine
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