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Ph. D. Project

Dates:

2022/10/13 - 2025/10/12

Student:

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.

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

Department(s):

Control Identification Diagnosis |

Publications: