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Sujet de Thèse : Systèmes commutés avec contraintes de langage
Dates : 2017/10/01 - 2020/09/30
Directeur(s) CRAN : Marc JUNGERS
Description : Language constrained switched systems.

1 Motivation
Hybrid systems are dynamical systems that are characterized by heterogenous interactions between a
dynamical automaton and a differential dynamic (in continuous-time) or a difference dynamic (in discretetime).
A significant class of hybrid systems is the class of switched ones. A switched system consists in a
finite family of differential processes orchestrated by an algorithm pointing which process is active at each
time [13]. Such a system is suitable to model a large field of real processes and applications. For instance, one
can cite automotive and aeronautic industries, mobile telecommunications, medical industries and health,
energy management or networked control systems. More precisely concrete examples could be :
— A car integrates more than 80 embedded controllers, which govern subsystems such that the motor
engine, the wheels, the suspension, the electric windows. In function of the task to perform, the active
mode is different.
— A power converter is generally modeled as a switched system, where each dynamic is defined by the
position of the electric switches (transistors and diodes).
— The schedule issue or time-management, that is the organisation in time of a list of tasks, may be
viewed also as a switched system activating each task or not [18]. A concrete example is the dynamical
power allocation in mobile telecommunication
— In medicine, several medical treatments may be possible and mixing/switching them may be more
efficient by avoiding some undesirable effects [10].
— In a multiagent system, the communication topology defining the communication between the agents
may be time-varying and regarded as a switching structure.

2 Recent researches
Switched systems are by essence is a time-varying ones : they offer rich behaviors, which can be not
intuitive. Actually switching between stable modes may potentially render unstable the whole system. In the
same way, this is possible to stabilize a system by switching between unstable modes [14]. A great part of
the literature in control system theory is focused on the study of switched systems, and mainly on the ones
with linear modes in discrete time, of the form :
x(k+1) = A_(sigma(k))x(k); (1)
with the switching rule sigma: Nset ->{1,…,N} and with A_i that are given.
Sufficient conditions are widely-known to guarantee their asymptotical stability [1] (ensuring the convergence
to the origin for any arbitrary switching rule ()) but also their asymptotical stabilization [6] (ensuring
that there exists at least a switching law () allowing the convergence to the origin). Recently necessary
and sufficient conditions for the stabilizability of the switched system (1) have been provided in [3, 4].

A strong assumption of these studies concerning the stabilizability is that at each time, it may be possible
to choose any mode. This assumption is not realistic in numerous applications with constraints [2] : for an
electrical circuit, the position of an electrical switch should be maintained during a given duration avoiding
too frequent switches (this duration is called the dwell time) ; constraints concerning the sequence of modes
(restriction of the successors of a given mode) may be imposed or concerning the availability of resources.
Lately the idea of considering a language generated a non-deterministic automaton has been provided to
generically model the constraints of a switching law sigma(.). The first results dealing with the stabilizability of
system (1) with constraints have been already published in [5, 11].

3 Topic of the PhD proposal
The PhD research consists in consolidating and extending the recent results [5, 11] described above. In
the context of the two last PhD theses [8] (awarded by the best PhD thesis grant in control system theory
in France 2013) and [15], that have been co-advised by Marc Jungers and Jamal Daafouz, full professor at
University of Lorraine.
The main objectives aim at providing a solution to the both following strong challenges :
1. go beyond the notion of stability by taking into account the performance aspects with the notion of
consistency for switched systems when the switching rule should satisfy some constraints (a switching
law is called consistent if it leads to a performance level better or equal to the best performance
obtained by each isolated mode [7, 17]).
2. extend the results to the case of switched systems with modes that are not linear and consider for
instance switched Lur’e systems [9, 12, 16], with dedicated tools.
4 Background of the candidate and contact
We are looking for a candidate with a MS or engineer degree having a good background in control theory.
The applicant should be interested in theoretical and practical aspects. A pronounced taste for handle
dynamical systems, matrices and graphs is a welcome asset. The working language can be either English or
French. The standard duration of a PhD thesis in France is 36 months.
For more information and complete topic, please contact by email marc.jungers@univ-lorraine.fr,
http://w3.cran.univ-lorraine.fr/perso/marc.jungers/
Mots clés : Théorie du contrôle des systèmes; systèmes commutés, systèmes non linéaires, performances, language
Département(s) :
Contrôle Identification Diagnostic
Financement : Demande de financement au CONACYT (réservé aux étudiants mexicains)