The main goal of this paper is to design a state feedback control that makes a point mass track a non-Zeno reference trajectory in a planar billiard. This objective is achieved by first determining a continuous-time dynamical model, whose trajectories approximate the solutions of the hybrid system. Hence, a state feedback that makes the hybrid system track a reference trajectory of the continuous-time one is proposed. Finally, these two techniques are combined in order to find a state feedback that achieves tracking of the trajectories of the unforced system. Examples are reported all throughout the paper to illustrate the theoretical results.

References

References
1.
Galeani
,
S.
,
Menini
,
L.
,
Potini
,
A.
, and
Tornambe
,
A.
,
2008
, “
Trajectory Tracking for a Particle in Elliptical Billiards
,”
Int. J. Control
,
81
(
2
), pp.
189
213
.
2.
Sinai
,
Y. G.
,
1970
, “
Dynamical Systems With Elastic Reflections
,”
Russ. Math. Surv.
,
25
(
2
), pp.
137
189
.
3.
Bunimovich
,
L. A.
,
Sinai
,
Y. G.
, and
Chernov
,
N. I.
,
1991
, “
Statistical Properties of Two-Dimensional Hyperbolic Billiards
,”
Russ. Math. Surv.
,
46
(
4
), pp.
47
106
.
4.
Burago
,
D.
,
Ferleger
,
S.
, and
Kononenko
,
A.
,
1998
, “
Uniform Estimates on the Number of Collisions in Semi-Dispersing Billiards
,”
Ann. Math
,
147
(
3
), pp.
695
708
.
5.
Lehman
,
R.
, and
White
,
C.
,
2002
, “
Hyperbolic Billiard Paths
,”
Math. Sci. Tech. Rep.
,
61
, pp. 1–32.https://scholar.rose-hulman.edu/math_mstr/61/
6.
Chernov
,
N.
, and
Markarian
,
R.
,
2006
, “
Chaotic Billiards
,”
Math. Surv. Monographs
,
127
, pp. 1–278.
7.
Gibbs
,
J. W.
,
2014
,
Elementary Principles in Statistical Mechanics
,
Courier Corp
, Mineola, NY.
8.
Steiner
,
F.
,
1994
, “
Quantum Chaos
,” preprint
arXiv:chao-dyn/9402001
.https://arxiv.org/abs/chao-dyn/9402001
9.
Chernov
,
N. I.
,
1991
, “
New Proof of Sinai's Formula for the Entropy of Hyperbolic Billiard Systems. Application to Lorentz Gases and Bunimovich Stadiums
,”
Funct. Anal. Applic
,
25
(
3
), pp.
204
219
.
10.
Bardos
,
C.
,
Caflisch
,
R. E.
, and
Nicolaenko
,
B.
,
1986
, “
The Milne and Kramers Problems for the Boltzmann Equation of a Hard Sphere Gas
,”
Commun. Pure Appl. Math.
,
39
(
3
), pp.
323
352
.
11.
Brogliato
,
B.
,
Niculescu
,
S.-I.
, and
Orhant
,
P.
,
1997
, “
On the Control of Finite-Dimensional Mechanical Systems With Unilateral Constraints
,”
IEEE Trans. Autom. Control
,
42
(
2
), pp.
200
215
.
12.
Brogliato
,
B.
, and
Zavala Rio
,
A.
,
2000
, “
On the Control of Complementary-Slackness Juggling Mechanical Systems
,”
IEEE Trans. Autom. Control
,
45
(
2
), pp.
235
246
.
13.
Brogliato
,
B.
,
2000
,
Impacts in Mechanical Systems: Analysis and Modelling
, Springer-Verlag, Berlin.
14.
Heemels
,
W. P. M. H.
, and
Brogliato
,
B.
,
2003
, “
The Complementarity Class of Hybrid Dynamical Systems
,”
Eur. J. Control
,
9
(
2–3
), pp.
322
360
.
15.
Brogliato
,
B.
,
2003
, “
Some Perspectives on the Analysis and Control of Complementarity Systems
,”
IEEE Trans. Autom. Control
,
48
(
6
), pp.
918
935
.
16.
Brogliato
,
B.
,
2016
,
Nonsmooth Mechanics
, Springer-Verlag, London.
17.
Brogliato
,
B.
,
2017
, “
Feedback Control of Multibody Systems With Joint Clearance and Dynamic Backlash: A Tutorial
,”
Multibody Syst. Dyn.
,
42
(
3
), pp.
283
315
.
18.
Sanfelice
,
R. G.
,
Teel
,
A. R.
, and
Sepulchre
,
R.
,
2007
, “
A Hybrid Systems Approach to Trajectory Tracking Control for Juggling Systems
,”
46th IEEE on Decision and Control
(
CDC
), New Orleans, LA, Dec. 12–14, pp.
5282
5287
.
19.
Goebel
,
R.
,
Sanfelice
,
R. G.
, and
Teel
,
A. R.
,
2009
, “
Hybrid Dynamical Systems
,”
IEEE Control Syst. Mag.
,
29
(
2
), pp.
28
93
.
20.
Rizzi
,
A. A.
,
Whitcomb
,
L. L.
, and
Koditschek
,
D. E.
,
1992
, “
Distributed Real-Time Control of a Spatial Robot Juggler
,”
Computer
,
25
(
5
), pp.
12
24
.
21.
Forni
,
F.
,
Teel
,
A. R.
, and
Zaccarian
,
L.
,
2013
, “
Follow the Bouncing Ball: Global Results on Tracking and State Estimation With Impacts
,”
IEEE Trans. Autom. Control
,
58
(
6
), pp.
1470
1485
.
22.
Rijnen
,
M.
,
Saccon
,
A.
, and
Nijmeijer
,
H.
,
2015
, “
On Optimal Trajectory Tracking for Mechanical Systems With Unilateral Constraints
,”
54th IEEE Conference on Decision Control
(
CDC
), Osaka, Japan, Dec. 15–18, pp.
2561
2566
.
23.
Tanwani
,
A.
,
Brogliato
,
B.
, and
Prieur
,
C.
,
2014
, “
On Output Regulation in State-Constrained Systems: An Application to Polyhedral Case
,”
IFAC Proc.
,
47
(3), pp.
1513
1518
.
24.
Tanwani
,
A.
,
Brogliato
,
B.
, and
Prieur
,
C.
,
2016
, “
Observer Design for Unilaterally Constrained Lagrangian Systems: A Passivity-Based Approach
,”
IEEE Trans. Autom. Control
,
61
(
9
), pp.
2386
2401
.
25.
Menini
,
L.
, and
Tornambe
,
A.
,
2003
, “
Control of (Otherwise) Uncontrollable Linear Mechanical Systems Through Non-Smooth Impacts
,”
Syst. Control Lett.
,
49
(
4
), pp.
311
322
.
26.
Menini
,
L.
,
Possieri
,
C.
, and
Tornambe
,
A.
,
2015
, “
On the Computation of the Continuous-Time Reference Trajectory for Mechanical Juggling Systems
,” IEEE 54th Annual Conference on Decision and Control (
CDC
), Osaka, Japan, Dec. 15–18, pp.
145
150
.
27.
Tornambe
,
A.
,
1993
, “
Global Output Tracking of Polynomial Reference Signals for a Class of Single-Input Single-Output Nonlinear Systems
,”
IEE Proc. Control Theory Appl.
,
140
(2), pp.
93
98
.
28.
Menini
,
L.
,
Possieri
,
C.
, and
Tornambè
,
A.
,
2018
, “
Dead–Beat Regulation of Mechanical Juggling Systems
,”
Asian J. Control
,
20
(
1
), pp.
1
11
.
29.
Brogliato
,
B.
,
Niculescu
,
S.
, and
Monteiro-Marques
,
M.
,
2000
, “
On Tracking Control of a Class of Complementary-Slackness Hybrid Mechanical Systems
,”
Syst. Control Lett.
,
39
(
4
), pp.
255
266
.
30.
Miranda-Villatoro
,
F.
,
Brogliato
,
B.
, and
Castaños
,
F.
,
2016
, “
Multivalued Robust Tracking Control of Lagrange Systems: Continuous and Discrete–Time Algorithms
,”
IEEE Trans. Autom. Control
,
62
(
9
), pp.
4436
4450
.
31.
Halpern
,
B.
,
1977
, “
Strange Billiard Tables
,”
Trans. Am. Math. Soc
,
232
, pp.
297
305
.
32.
Vanderbei
,
R.
,
1996
,
Linear Programming: Foundations and Extensions
,
Springer
,
New York
.
33.
Wright
,
S.
, and
Nocedal
,
J.
,
1999
,
Numerical Optimization
, Vol.
35
, Springer-Verlag, New York, pp.
67
68
.
34.
Menini
,
L.
, and
Tornambe
,
A.
,
2000
, “
The Use of the Barrier Method for the Impact Analysis in Mechanical Systems
,”
IFAC Proc. Vol.
,
33
(13), pp. 77–82.
35.
Goebel
,
R.
,
Sanfelice
,
R. G.
, and
Teel
,
A. R.
,
2012
,
Hybrid Dynamical Systems
,
Princeton University Press
, Princeton, NJ.
36.
Goebel
,
R.
, and
Sanfelice
,
R.
,
2016
, “
How Well-Posedness of Hybrid Systems Can Extend Beyond Zeno Times
,”
55th IEEE Conference on Decision and Control
(
CDC
), Las Vegas, NV, Dec. 12–14, pp.
598
603
.
37.
Curtiss
,
D. R.
,
1918
, “
Recent Extensions of Descartes' Rule of Signs
,”
Ann. Math.
,
19
(4), pp.
251
278
.
38.
Montano
,
O. E.
,
Orlov
,
Y.
, and
Aoustin
,
Y.
,
2016
, “
Nonlinear H∞-Control Under Unilateral Constraints
,”
Int. J. Control
,
89
(
12
), pp.
2549
2571
.
You do not currently have access to this content.