PhD position at Ecole des Mines de Douai, Department of Computer Sciences and Control, 59508 Douai, France. Website : http://www.mines-douai.fr
Topic : Identification of hybrid systems : identifiability, persistence of excitation and convergence analysis
Some keywords : Hybrid systems, switched systems, piecewise affine systems, system identiication, identifiability, convergence analysis, stability theory,

General description of the project: A dynamic hybrid system is a physical, natural or artificial process in which there are discrete-event phenomena interacting with continuous phenomena. Examples of such systems arise naturally in various engineering fields, typically genetic regulatory networks study, air traffic management, nonlinear systems control, manufacturing processes modeling, computer vision, etc. From a mathematical viewpoint, a hybrid system consists of a finite set of dynamical subsystems (which can be linear or nonlinear) that interact over time via a certain switching signal. Analysis, control or simulation of such systems require the knowledge of a mathematical model describing their dynamics. The model is not always available from first principles of physics and has to be identified from measurements. The problem of interest in this project concerns the estimation, from input-output observations, of the parameters describing the constituent subsystems. See the references [10, 5, 3, 1, 2, 4, 6, 7] for a sample of relevant works. The main challenge with this problem is that one does not know a priori which subsystem has generated which data. A certain number of algorithms with more or less efficiency have been developed over the last decade to solve this identification problem. Our goal in this research program is twofold :

1) study the identifiability of hybrid systems when represented by ARX models. This part is concerned with the question of whether the hybrid system identification problem is well-posed. It relates to the formal and yet fundamental question of under which conditions a hybrid model structure can be uniquely inferred from empirical input-output data. A closely related question is : in case a given model structure is identifiable, how to design the experiment that will produce rich enough information allowing the computation of that model. Some earlier results on this topic have been published in [8, 9] regarding the case of linear switched state-space realizations.

2) study the convergence properties of a class of iterative/recursive algorithms. In the light of the identifiability conditions, this part will focus on characterizing correctness of some methods from the existing literature that is, their ability, under appropriate conditions, to reach the true model. Because most of the existing algorithms have an iterative or a recursive nature, it is of vital importance to know whether they are able to recover correctly the parameters of the model they are designed for. Note that for the majority of the state-of-art methods that deal with hybrid systems, convergence analysis is still a largely open research problem. In this project we will try to derive, for a specific class of algorithms, conditions under which convergence can be guaranteed. Required skills and application : The applicant must hold a M.Sc in engineering or applied mathematics with a strong focus on control systems theory. A good knowledge of the software Matlab is desirable. To apply to this position, please send a motivational letter, a detailed curriculum vitae, names and addresses of at least two references and a copy of official transcripts for your MSc program with corresponding grades.

If you need more information on the position, please contact us.

Duration : 3 years, starting in september 2011

Supervision :
Pr Stéphane Lecoeuche,
Ecole des Mines de Douai
Unité de Recherche en Informatique et Automatique, URIA
59508 Douai, France.
E-mail : stephane.lecoeuche@mines-douai.fr
Tél. : (+33) (0) 3 27 71 24 45.

and

Dr Laurent Bako
Ecole des Mines de Douai
Unité de Recherche en Informatique et Automatique, URIA
59508 Douai, France.
E-mail : laurent.bako@mines-douai.fr
Tél. : (+33) (0) 3 27 71 21 27

Références
[1] L. Bako. Identification of switched linear systems via sparse optimization. Automatica, 47 :668–677, 2011.
[2] L. Bako, K. Boukharouba, E. Duviella, and S. Lecoeuche. A recursive identification algorithm for switched linear/affine models. Nonlinear Analysis : Hybrid Systems, 5 :242–253, 2011.
[3] A. Bemporad, A. Garulli, S. Paoletti, and A. Vicino. A bounded-error approach to piecewise affine system identification. IEEE Transactions on Automatic Control, 50 :1567–1580, 2005.
[4] K. Boukharouba, L. Bako, and S. Lecoeuche. Identification of piecewise affine systems based on dempster-shafer theory. In IFAC Symposium on System Identification, Saint Malo, France, 2009.
[5] G. Ferrari-Trecate, M. Muselli, D. Liberati, and M. Morari. A clustering technique for the identification of piecewise affine systems. Automatica, 39 :205–217, 2003.
[6] A. L. Juloski, S. Weiland, and W. Heemels. A bayesian approach to identification of hybrid systems. IEEE Transactions on Automatic Control, 50 :1520–1533, 2005.
[7] F. Lauer, G. Bloch, and R. Vidal. A continuous optimization framework for hybrid system identification. Automatica, 47 :608–613, 2011.
[8] M. Petreczky and L. Bako. On the notion of persistence of excitation for linear switched systems. In Submitted to CDC’2011, 2011.
[9] M. Petreczky, L. Bako, and J. H. van Schuppen. Identifiability of discrete-time linear switched systems. In Hybrid Systems : Computation and Control, Stockholm, Sweden, 2010.
[10] R. Vidal, S. Soatto, Y. Ma, and S. Sastry. An algebraic geometric approach to the identification of a class of linear hybrid systems. In Conference on Decision and Control, Maui, Hawaii, USA, 2003.