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10.3
Theoretical Foundations of the Method
Lyapunov theory and LMIs constitute the cornerstone of the proposed stability anal-
ysis of the rational closed-loop servo (10.4) subject to the rational constraints (10.5).
The fundamentals issues are hereafter outlined.
10.3.1
Elements of Lyapunov Theory
This section introduces the concept of multicriteria basin of attraction, and discusses
the difficulties raised by robotics in its calculation.
10.3.1.1
Fundamentals of the Definition of a Lyapunov Function
˜
Definition 10.2 (Multicriteria Basin of Attraction).
Aset
is said a
multicriteria
basin of attraction
if it is a domain of initial conditions from which the state tra-
jectories of the closed-loop system
(10.4)
converge to the equilibrium
x
∗
=
0
while
wholly lying in the admissible subset of the state space defined by the criteria
(10.5)
.
E
˜
˜
A function
V
(
.,.
) :
X×X
χ
−→
R
is sought for, such that
E
{
x
:
V
(
x
,
χ
)
≤
1
,
∀
χ
∈X
χ
}
is a multicriteria basin of attraction. The type of
V
(
.,.
) is selected be-
forehand. Its DOF constitute a matrix
P
,tobetunedsothat
V
(
.,.
)=
V
P
(
.,.
) satisfies
the following three rules:
˜
X×X
χ
so that
x
∗
=
0
is locally asymptoti-
1.
V
P
(
.,.
) is a Lyapunov function on
cally stable,
i.e. V
P
(
.,.
) is continuously differentiable and
×X
χ
,
˜
∀
(
x
,
χ
∈
X\{
}
V
P
(
x
,
χ
>
∀
χ
∈X
χ
,
,
χ
)
0
)
0;
V
P
(
0
)=0 ;
(10.6)
∈
}
×X
χ
,
˜
V
P
(
x
V
P
(
0
∀
(
x
,
χ
)
X\{
0
,
χ
)
<
0;
∀
χ
∈X
χ
,
,
χ
)=0;
(10.7)
˜
is a basin of attraction of
x
∗
= 0 for the unconstrained problem as soon as
2.
E
˜
∀
(
x
,
χ
) :
V
P
(
x
,
χ
)
≤
1
,
x
∈
X
;
(10.8)
3. the boundedness (10.5) of the additional variables is ensured when
Z
j
(
x
∀
j
∈
Ξ
J
, ∀
(
x
,
χ
) :
V
P
(
x
,
χ
)
≤
1
,
ζ
j
≤
,
χ
)
x
≤
ζ
j
.
(10.9)
Figure 10.1 illustrates the above conditions when
V
(
.
) and
Z
j
(
.
) depend only on
x
.
Therein,
˜
˜
x
:
Z
j
(
x
)
x
.
Some level sets of the Lyapunov function are sketched, together with the basin
˜
A
terms the admissible set and is defined by
A
{
∈
[
ζ
j
,
ζ
j
]
}
˜
˜
˜
E
{
x
:
V
(
x
)
≤
1
}
. Note that
E
lies both into
X
and
A
.
10.3.1.2
Additional Guidelines
Additional arguments can lead to more purposeful multicriteria basins of attrac-
tion. For instance, when the aim is to analyze the feedback system for initial
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