Master's Thesis Measures and LMIs for Optimal Control of Piecewise-Affine Dynamical Systems

56 Pages·2012·1.97 MB·English

Czech Technical University in Prague
Faculty of Electrical Engineering
Department of Control Engineering
Master’s Thesis
Measures and LMIs for Optimal Control of
Piecewise-A(cid:14)ne Dynamical Systems
Mohamed Rasheed-Hilmy Abdalmoaty
Supervisor:
Ing. Didier Henrion, Ph.D.
Faculty of Electrical Engineering,
Czech Technical University in Prague,
Technick(cid:19)a 2, CZ-16626 Prague; Czech Republic
National Center for Scienti(cid:12)c Research (CNRS),
Laboratory of Analysis and Architecture of Systems (LAAS),
7 avenue du colonel Roche, F-31077 Toulouse; France.
Prague, June 2012

Abstract
The project considers the class of deterministic continuous-time optimal control problems
(OCPs) with piecewise-a(cid:14)ne (PWA) vector (cid:12)elds and polynomial data. The OCP is relaxed
as an in(cid:12)nite-dimensional linear program (LP) over space of occupation measures. The LP
is then written as a particular instance of the generalized moment problem which is then
approached by an asymptotically converging hierarchy of linear matrix inequality (LMI)
relaxations. The relaxed dual of the original LP gives a polynomial approximation of the
value function along optimal trajectories. Based on this polynomial approximation, a novel
suboptimal policy is developed to construct a state feedback in a sample-and-hold manner.
The results show that the suboptimal policy succeeds in providing a stabilizing suboptimal
state feedback law that drives the system relatively close to the optimal trajectories and
respects the given constraints.

Acknowledgments
First of all, I would like to thank my supervisor, Didier Henrion, for giving me the chance
to work on this interesting topic and for his ceaseless support and guidance. I’d also like to
thank Luis Rodrigues, from Concordia University in Montr(cid:19)eal in Canada, for his technical
advice and insight regarding PWA systems.
MythanksalsogotoeveryonewhoisinvolvedintheSpaceMasterprogramatLule(cid:23)a Uni-
versity of Technology in Sweden, and Julius-Maximilian’s University of Wu(cid:127)rzburg in Ger-
many. In the Department of Control Engineering of the Czech Technical University I’d like
tothankProf. JanS(cid:20)techa, MartinHrom(cid:20)c(cid:19)(cid:16)k, Tom(cid:19)a(cid:20)sHani(cid:20)s, andZden(cid:20)ekHur(cid:19)akfortheirteach-
ing e(cid:11)orts and the technical discussions we had throughout my stay in the department. I’m
specially grateful to Jana Nov(cid:19)akov(cid:19)a and Martin Hrom(cid:20)c(cid:19)(cid:16)k for supporting my CTU scholarship
application, without which I probably wouldn’t have been able to complete SpaceMaster.
Last but not least I want to express my deepest gratitude to my parents for their caring,
encouragement and constant support.