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˜
E
˜
∂
A
0
˜
A
˜
X
Fig. 10.1
Illustration of the basics of a Lyapunov solution
sensor-target situations in a given set
X
0
, extra conditions must be put on the ma-
˜
˜
trix
, with
X
c
0
the domain of checked initial values of the controller state vector. An opti-
mization problem on
P
in addition to (10.6)-(10.9) so as to ensure that
X
0
X
0
×X
c
0
∈
E
P
under the above constraints an also guide the search. For
˜
instance, min
P
f
(
P
) enables the heuristic maximization of the “size” of
E
defined
from
V
(
x
)=
x
P
x
,
P
>
0, if
f
(
P
) is set to trace(
P
) [2]. In the same vein, consider-
˜
λ
h
,
χ
E
ing min
P
max
χ
max
h
V
(
) can help in heuristically maximizing the extent of
˜
{
λ
h
}⊂
X
towards a set of selected points
[12].
10.3.1.3
Difficulties Raised by the Robotics Context
Despite the above control problem looks fairly usual, the robotics context raises
sharp issues. Indeed, the admissible areas of the state space in regard to the cri-
teria are seldom symmetric with respect to
0
, nor convex, nor even connected. A
fundamental consequence is that ensuring (10.9) may be far too conservative for
some classes of Lyapunov functions. For instance, it was proved in [11] that using
quadratic Lyapunov functions of the form
V
(
x
)=
x
P
0, leads to ex-
tremely poor conclusions even for very simple problems, unless nontrivial theoreti-
cal enhancements are devised. Widening the class of Lyapunov function candidates
can be a solution, yet there is generally a balance between the obtained versatility
and the induced computational complexity. The easiest that can be done given a
class of Lyapunov functions is to define
x
, with
P
>
˜
E
∪
r
∈
Ξ
R
(
˜
˜
E
as the union
E
)
r
of
R
multi-
criteria basins of attraction (
˜
E
)
r
independently computed under the assumption of
˜
polytopes (
)
r
.
Piecewise Lyapunov functions can be envisaged as an alternative with limited
complexity. First,
X×X
χ
)
r
and/or of the sets of directions of extent (
{
λ
h
}
˜
˜
n
R
is partitioned into
S
distinct convex cones
C
1
,...,
C
S
join-
ing at
0
. Roughly speaking, a set of matrices
P
1
,...,
P
S
is then sought for so that
˜
˜
˜
(i)
V
P
s
(
.,.
) is a Lyapunov function on each
X
X ∩
C
s
,
s
∈
Ξ
S
, (ii) the conse-
s
˜
˜
˜
quent set
E
{
x
∈
C
s
:
V
(
x
,
χ
)
≤
1
,∀
χ
∈X
χ
}
lies into
X
and is admissible in
s
˜
regard to the criteria. Let
V
(
.,.
) :
X×X
χ
−→
R
be the piecewise function whose
˜
S
.Then
˜
˜
X
.,.
∈
Ξ
E
∪
E
restriction to each
s
is equal to
V
s
(
),
s
=
s
becomes a mul-
s
∈
Ξ
S
P
,...,
P
.,.
ticriteria basin of attraction if extra constraints are put on
S
so that
V
(
)
1
˜
˜
is continuous on all the interfaces
∂
X
s
a
s
b
between pairs of adjacent cells
X
s
a
and
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