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the system under consideration reads
1
T
+
A
,
B
B
,
ω
,
x
=
(
z
x
)
x
+
2
(
z
x
)
(15.16)
where the acceleration of the camera
T
is the control vector.
15.2.2
Problem Formulation
To cope with constraint (15.13), the control law we consider has the form
=
K
1
K
2
∈
ℜ
T
=
sat
u
0
(
3
×
6
K
x
) with
K
.
(15.17)
Hence, the closed-loop system reads
x
=
A
(
z
,
x
)
x
+
B
1
sat
u
0
(
K
x
)+
B
2
(
z
,
x
)
ω
.
(15.18)
Relative to the closed-loop system (15.18), one has to take into account the con-
straints (15.3) and (15.12), which means that the state
x
must belong to the polyhe-
dral set
(
x
)=
x
β
β
1
3
u
1
1
3
u
1
6
;
Ω
∈
ℜ
−
x
.
(15.19)
The problem we intend to solve with respect to the closed-loop system (15.18),
subject to constraints (15.19), can be summarized as follows.
K
S
S
Problem 15.1.
Determine a gain
1
, as large as possible,
such that, in spite of conditions C1 and C4, the constraints C2 and C3 are satisfied
and:
and two regions
0
and
•
(internal stability)
when
0
the closed-loop trajectories of
system (15.18) converge asymptotically to the origin; and
ω
= 0, for any
x
(0)
∈S
•
(input-to-state stability)
when
ω
= 0, the closed-loop trajectories remain bounded
in
S
1
for any
x
(0)
∈S
1
and for all
ω
satisfying (15.11).
The first item of Problem 15.1 corresponds to address the case of a fix target [22],
[11]. Indeed, this case consists of stabilizing a camera in front of a target, and de-
termining the associated region of stability in spite of unknown value of the target
points depth, bounds on admissible feature errors which guarantee visibility, and
limits on the camera velocity and acceleration. The second item of Problem 15.1
corresponds to the tolerance of the closed-loop system with respect to the perturba-
tion
. In this case, the objective is to ensure that the closed-loop trajectories are
bounded for any disturbance
ω
ω
satisfying (15.11).
15.3
Preliminary Results
Let us specify some properties of the closed-loop system (15.18). By defining the
decentralized dead-zone nonlinearity
φ
(
K
x
)=
sat
u
0
(
K
x
)
−
K
x
,definedby
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