Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-20T15:33:17.041Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogeneous approximations and local observer design

Published online by Cambridge University Press:  03 June 2013

Tomas Ménard ,
Emmanuel Moulay  and
Wilfrid Perruquetti
Tomas Ménard
Affiliation:
GREYC, UMR CNRS 6072, Université de Caen, 6 Bd du Maréchal Juin, BP 5186–14032 Caen Cedex, France. [email protected]
Emmanuel Moulay
Affiliation:
Xlim-SIC, UMR CNRS 6172, Université de Poitiers, Bvd Marie et Pierre Curie, BP 30179, 86962 futuroscope Chasseneuil, France; [email protected]
Wilfrid Perruquetti
Affiliation:
NON-A, INRIA Lille Nord Europe and LAGIS UMR CNRS 8219, Ecole Centrale de Lille, BP 48, 59651 Villeneuve D’Ascq, France; [email protected]
Get access

Abstract

This paper is concerned with the construction of local observers for nonlinear systems without inputs, satisfying an observability rank condition. The aim of this study is, first, to define an homogeneous approximation that keeps the observability property unchanged at the origin. This approximation is further used in the synthesis of a local observer which is proven to be locally convergent for Lyapunov-stable systems. We compare the performance of the homogeneous approximation observer with the classical linear approximation observer on an example.

Keywords

Homogeneityapproximationslocal observer
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 906 - 929
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrieu, V. and Praly, L., On the existence of a Kazantzis–Kravaris/Luenberger observer. SIAM J. Control Optim. 45 (2006) 432–456. Google Scholar
Andrieu, V., Praly, L. and Astolfi, A., Homogeneous approximation, recursive observer design, and output feedback. SIAM J. Control Optim. 47 (2008) 1814–1850. Google Scholar
Besançon, G. and Ticlea, A., An immersion-based observer design for rank-observable nonlinear systems. IEEE Trans. Autom. Control 52 (2007) 83–88. Google Scholar
Bianchini, R.M. and Stefani, G., Graded approximations and controllability along a trajectory. SIAM J. Control Optim. 28 (1990) 903–924. Google Scholar
G. Conte, C.H. Moog and A.M. Perdon, Algrebraic Methods for Nonlinear Control Systems. Springer, London, 2nd (2007).
S. Diop and M. Fliess, Nonlinear observability, identifiability and persistent trajectory, in IEEE 30th Conference on Decision and Control (1991) 714–719.
Fliess, M., Join, C. and Sira-Ramirez, H., Nonlinear estimation is easy. Int. J. Model. Identification Control. 4 (2008) 12–27. Google Scholar
Gauthier, J.P. and Bornard, G., Observability for any u(t) of a class of nonlinear systems. IEEE Trans. Autom. Control 26 (1981) 922–926. Google Scholar
Gauthier, J.P., Hammouri, H. and Othman, S., A simple observer for nonlinear systems applications to bioreactors. IEEE Trans. Autom. Control 37 (1992) 875–880. Google Scholar
Glumineau, A., Moog, C.H. and Plestan, F., New algebraic-geometric conditions for the linearization by input-output injection. IEEE Trans. Autom. Control 41 (1996) 598–603. Google Scholar
R.W Goodman, Nilpotent Lie groups: structure and applications to analysis, Lectures Note Math. Springer-Verlag, Berlin, New-York 562 (1976).
Hermes, H., Control systems which generate decomposable lie algebras. J. Differ. Equ. 44 (1982) 166–187. Google Scholar
Hermes, H., Nilpotent and high-order approximations of vector-field systems. SIAM Rev. 33 (1991) 238–264. Google Scholar
Kawski, M., Homogeneous stabilizing feedback laws. Control Theory and Advanced Technology 6 (1990) 497–516. Google Scholar
Kazantzis, N. and Kravaris, C., Nonlinear observer design using Lyapunov’s auxiliary theorem. Syst. Control Lett. 34 (1998) 241–247. Google Scholar
H.K. Khalil, High-gain observers in nonlinear feedback control. In International Conference on Control, Automation and system (2008).
Krener, A.J. and Isidori, A., Linearization by output injection and nonlinear observers. Syst. Control Lett. 3 (1983) 47–52. Google Scholar
Krener, A.J. and Respondek, W., Nonlinear observers with linearizable error dynamics. SIAM J. Control Optim. 23 (1985) 197–216. Google Scholar
Li, J. and Qian, C., Global finite-time stabilization by dynamic output feedback for a class of continuous nonlinear systems. IEEE Trans. Autom. Control 51 (2006) 879–884. Google Scholar
Menard, T., Moulay, E. and Perruquetti, W., A global high-gain finite-time observer. IEEE Trans. Autom. Control 55 (2010) 1500–1506. Google Scholar
Nam, K., An approximate nonlinear observer with polynomial coordinate transformation maps. IEEE Trans. Autom. Control 42 (1997) 522–527. Google Scholar
Phelps, A.R., On constructing nonlinear observers. SIAM J. Control Optim. 29 (1991) 516–534. Google Scholar
Rothschild, L. and Stein, E., Hypoelliptic differential operators and nilpotent groups. Acta Mathematica 137 (1976) 247–320. Google Scholar
Souleiman, I., Glumineau, A. and Schreier, G., Direct transformation of nonlinear systems into state affine miso form for observer design. IEEE Trans. Autom. Control 48 (2003) 2191–2196. Google Scholar
G. Stefani, Polynomial approximations to control systems and local controllability, in Proc. of the 24th IEEE Conference on Decision and Control (1985) 33–38.
Sundarapandian, V., Local observer design for nonlinear systems. Math. Comput. Model. 35 (2002) 25–36. Google Scholar
Sussmann, H.J., A general theorem on local controllability. SIAM J. Control Optim. 25 (1987) 158–194. Google Scholar