Ph. D. Project
Title:
Input redundancy for hybrid systems
Dates:
2022/10/01 - 2025/09/30
Supervisor(s): 
Description:
When a single actuator is not capable to deliver a sufficiently large effort, when fault tolerant comes into
play or to provide better performances, a system is frequently designed with a larger number of actuators
than strictly necessary to be controlled. Such a system is known as control objective does not determine
solely the input. In fact, the system is not left-invertible [1]. With respect to the control objective, the
system gets additionnal degrees of freedom. On the one hand, these degrees of freedom can be used for
secendary objectives in order to enhance the system performances. On the other hand, they induce
additional difficulties related to the stabilization. The aim of the control allocation techniques is to use the
degrees of freedom both for the performance improvement and for stability guarantees (see [2] for a state
of art about control allocation methods).

Lately, the input redondancy has been redefined and entirely characterized in the context of linear time
invariant (LTI) system (paper provisionally accepted at Systems and Control Letters [3]). The
characterization are based on the geometric control theory [1], which is well suited to the LTI context.
Among the definitions, a three kind taxonomy has been proposed in order to distinguish the distinct
origins of the input redundancy.

Furthermore, many existing applications (such as power converters [4], communications over networks,
mechanical systems) present at the same time continuous and discrete phenomena. They can be
modelised by the powerful formalism of the hybrid theory, or as switched systems. Obviously, the
continuous/discrete heterogeneity does not prevent the system from input redundancy.

This thesis is focused on the study of input redundancy for hybrid systems. The goal is to entirely
characterize this notion in order to be able to design tailored control allocation methods (for instance
bumpless transfer [7], or energy minimization [8]). The notion of input redundacy in this context is rather
involved, as well as stability or stabilisation considerations [9, 10] for hybrid systems. To the best of our
knowledge, their exist merely characterization of left invertibility for the switched systems [11]. The
research work will start from the switched systems while considering increasing difficulties assumptions
(arbitrary switching law or to be chosen, linear or non-linear modes for example).

[1] W. Murray Wonham. Linear multivariable control : a geometric approach. Springer-Verlag New York,
2012.
[2] Tor A. Johansen et Thor I. Fossen. "Control allocation-a survey". In : Automatica (2013), p. 1087-1103.
[3] Jérémie Kreiss et Jean-François Trégouët. "Input Redundancy : Definitions, Taxonomy and
Characterizations. Part I : Unconstrained Dynamics". In : soumis à IEEE Systems and Control Letters (2021).
[4] Jérémie Kreiss, Marc Bodson, Romain Delpoux, Jean-Yves Gauthier, Jean-François Trégouët et Xuefang
Lin-Shi. "Optimal control allocation for the parallel interconnection of buck converters". In : Control
Engineering Practice (2021), p. 104727.
[5] Daniel Liberzon. Switching in systems and control. Springer Science & Business Media, 2003.
[6] Rafal Goebel, Ricardo G Sanfelice et Andrew R Teel. Hybrid dynamical systems. Princeton University
Press, 2012.
[7] Luca Zaccarian et Andrew R Teel. "The L2 (l2) bumpless transfer problem for linear plants : Its definition
and solution". In : Automatica (2005), p. 1273-1280.
[8] J. Kreiss, J. Trégouët, D. Eberard, R. Delpoux, J. Gauthier et X. Lin-Shi. "Hamiltonian Point of View on
Parallel Interconnection of Buck Converters". In : IEEE Transactions on Control Systems Technology (2020),
p. 1-10.
[9] Mirko Fiacchini et Marc Jungers. "Necessary and sufficient condition for stabilizability of discrete-time
linear switched systems : A set-theory approach". In : Automatica (2014), p. 75-83.
[10] Mirko Fiacchini, Antoine Girard et Marc Jungers. "On the stabilizability of discrete-time switched linear
systems : Novel conditions and comparisons". In : IEEE Transactions on Automatic Control (2015), p. 1181-
1193.
[11] Mustafa Devrim Kaba et MK Camlibel. "On the left-invertibility of switched linear systems". In : IFAC
Proceedings Volumes (2010), p. 350-355.
Keywords:
input redundancy, hybrid systems, geometric control theory
Conditions:
Duration : 3 years ;
Employer : Université de Lorraine ;
Place : CRAN, Nancy ;
Remuneration : around 1 769 euros bruts for a month ;
Expected candidate : We are lookin for a candidate which is strongly motivated, with a MASTER degree in
control theory or applied mathematics. The candidate must also be interested for methodological research.
Department(s): 
Control Identification Diagnosis
Funds:
Doctoral contract from Université de Lorraine