OPTIMAL CONTROL APPLIED TO OPTIMIZATION
OF MOBILE SWITCHING SURFACES
PART II : APPLICATIONS
C
´
eline Qu
´
emard*
Jean-Claude Jolly*
*LISA-FRE 2656 CNRS
62, avenue Notre-Dame du Lac - F49000 Angers
Keywords:
Hybrid Dynamical System, Optimization, Mobile Switching Surface, Thermostat with Anticipative Resis-
tance, Car with Two Gears, Robot, Obstacle Avoidance, Target Approach.
Abstract:
To reinforce interest of a general optimization algorithm obtained in a previous paper (Jolly et al., 2005),
we consider three applications : an original one about control of cycles for a thermostat with anticipative
resistance, a classical one with a new resolution for a car with two gears and a last one about an obstacle-
avoidance problem in robotics. For the ﬁrst case, we optimize the adjustment of thermostat thresholds to
control at best the room temperature. For the second case, we optimize the switching times to stop the car as
near as possible of chosen points and this, in a minimum time . In the last example, we optimize parameters
of the switching surfaces in order that the robot reaches a chosen target without meeting a mobile obstacle.
1 INTRODUCTION
In (Jolly et al., 2005), we have found results on the
question of optimization of switching surfaces for a
hybrid dynamical system (h.d.s), generalizing what
was in (Wardi et al., 2004).
Here, we consider three applications that underline
interest of these theorical results. The ﬁrst, somewhat
original, is one of a thermostat with anticipative resis-
tance controlling a convector in a same room (C
´
ebron,
2000), (Qu
´
emard et al., 2005). In this example, we
optimize the adjustment of thermostat thresholds to
control at best the room temperature. This applica-
tion can be taken as a pattern for h.d.s leading to some
cycle solutions.
The second application is one of a car with two
gears (Gapaillard, 2003), (Hedlund and Rantzer,
2002). We optimize the switching times, ﬁrstly, to
stop the car as near as possible of a ﬁrst desired des-
tination and then, after a new start-up, to stop the car
as near as possible of a ﬁnal destination and this, in a
minimum time. Interest of this classical h.d.s problem
for us is to bring a new resolution improving numeri-
cal performance.
The last application solves an obstacle avoidance
problem in robotics (Boccadoro, 2004). Here, we
optimize parameters of the switching surfaces in or-
der that a robot reaches a pre-speciﬁed target without
never meating a given mobile obstacle. Compared to
(Boccadoro, 2004) where the considered obstacle is
ﬁxed, this example underlines interest of mobility for
switching surfaces in applications.
In section 2, we brieﬂy present the theorical algo-
rithm found in (Jolly et al., 2005). From section 3 to
section 5, we detail each application presented above.
Section 6 concludes the paper.
2 OPTIMIZATION ALGORITHM
REMINDER
Let t
0
, x
0
= x(t
0
) ∈ R
n
be given initial time
and state. Here, we consider a h.d.s which sustains
switchings at increasing times t
1
, ..., t
N
in [t
0
, t
N+1
]
(t
N+1
is the ﬁnal time) so that for i = 1, ..., N + 1,
state x
i
= x(t
i
) belongs to a given mobile surface
parameterized by a
i
∈ R
r
i
and of equation:
Ψ
i
(x
i
, t
i
, a
i
) = 0, (1)
where Ψ
i
is from C
1
class with values in R. In
[t
0
, t
N+1
], state x(t) is supposed to be continuous and
in [t
i−1
, t
i
], i = 1, ..., N +1, state x(t) complies with
dynamical system:
˙x = f
i
(x, t), (2)
372
Quémard C. and Jolly J. (2005).
OPTIMAL CONTROL APPLIED TO OPTIMIZATION OF MOBILE SWITCHING SURFACES PART II : APPLICATIONS.
In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and
Control, pages 372-377
DOI: 10.5220/0001184703720377
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