Trainee Project
Input redundancy for switched systems
2022/02/01 - 2022/07/31
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 an over-actuated system or an input redundant system and owns the following property : its 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 secondary 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 MASTER thesis is focused on the study of input redundancy for switched systems. The goal is to provide characterizations
propoerties for 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, there exists merely characterization of left invertibility for
the switched systems [11]. The research work will start from the switched systems where the switched law is already set.

[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.
input redundancy, hybrid systems, geometric control theory
Duration : 6 months ;
Employer : Université de Lorraine ;
Place : CRAN, Nancy ;
Remuneration : 577 euros per month ;
Expected profile : We are looking for strongly motivated candidate in Master 2 with a good knowledge of control theory and
mathematics. The candidate must be interested in methodological research and have the possibility to pursue with a PhD thesis.

For more informations, you can contact Jérémie Kreiss (
Control Identification Diagnosis
Gratification Université de Lorraine via the project ANR HANDY