Master internship proposal
Observer Design for Nonlinear Systems via Time-Scaling
Subject: Observer Design for Nonlinear Systems with Unknown Parameters via Regular and Singular
Time-Scaling
Student profile: Master student or final year of an engineering school (bac+5), with experience in
mathematical modelling, control theory, and Matlab/Simulink.
Salary: 500-600 €, according to the working days of the month (minimum statutory allowance ap-
plicable in public institutions).
Work place: L2S, CentraleSupélec, 3 rue Joliot Curie, 91190 Gif-sur-Yvette, France.
Keywords: Observers, nonlinear control, systems with unknown parameters, simulations.
Employer: CentraleSupélec. Legal status: Public engineering school (Grand Établissement); Member
of Université Paris-Saclay.
Advisors: Florentina Nicolau, L2S, CentraleSupélec, and ENSEA,
(email: florentina.nicolau@centralesupelec.fr)
William Pasillas-Lépine, L2S, CentraleSupélec,
(email: william.pasillas-lepine@centralesupelec.fr)
Context
In many dynamical systems, the variables that can be directly measured with physical sensors are not
necessarily the variables of interest for monitoring, diagnosis, or control. Software-based reconstruction
of unmeasured states and parameters can therefore provide a valuable complement to physical measure-
ments. An observer can be viewed as a software sensor: it consists of a dynamical model of the system
coupled with measured outputs through suitable correction terms, with the objective of making the ob-
server synchronize asymptotically with the actual system.
A specific approach was proposed in [1], to construct observers for a particular class of nonlinear sys-
tems. It addresses systems for which the error dynamics can be linearized after a regular or singular
time-scaling. This method relies on a switching observer that combines different observer dynamics de-
pending on the operating region of the system. This approach makes it possible to construct observers for
systems for which standard observer-design techniques may not be directly applicable.
The objective of this internship is to investigate extensions of this methodology and to assess its appli-
cability to more complex nonlinear systems, in particular to systems with unknown parameters.
Objectives:
The internship will focus on the design and analysis of observers based on regular and singular time-
scaling. The work will be organized around the following objectives:
1. Study of the existing methodology. The student will first review the theory of nonlinear observers,
asymptotic stability of observation errors and time-scale transformations. For instance, the observer
proposed in [1, 2] will be implemented and tested on some representative nonlinear systems.
2. Extension to more general systems. Then, the student will study an observer approach based on
least-squares identification [3], applied to a representation of the system in a new time scale. Partic-
ular attention will be paid to the inversion of the time transformation, which is required to recover
the system state in the original time scale. This work will serve as a starting point for investigating
more advanced observer designs, in particular the switching observer proposed in [1, 2], and for
extending the time-scaling approach to nonlinear systems with unknown or uncertain parameters.
3. Numerical validation. The proposed methods will be implemented in MATLAB/Simulink and com-
pared with existing observer techniques. Numerical experiments will be conducted on one or more
applications, potentially including automotive systems, electromechanical systems, or biological pro-
cesses.
Contacts
Please send a CV and a motivation letter to:
florentina.nicolau@centralesupelec.fr and william.pasillas-lepine@centralesupelec.fr
with object [Stage Observers]
The position is available immediately and applications will be accepted until this position is filled. A
National Security clearance is needed, and it can require approximately 2 months.
References
[1] Hoang, T. B., Pasillas-Lépine, W., and Respondek, W (2014). A switching observer for systems with
linearizable error dynamics via singular time-scaling. In MTNS 2014.
[2] Aguado-Rojas, M., Hoang, T. B., Pasillas-Lépine, W., Loria, A., Respondek, W. (2021). A switching
observer for a class of nonuniformly observable systems via singular time-rescaling. IEEE Transactions on
Automatic Control, 66(12), 6071-6076.
[3] Ling, K. V., Wu, X. J., Lim, K. W. (1996). State observers for single rate and multirate controllers. IFAC
Proceedings Volumes, 29(1), 1321-1326.
