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.
Mots-clés : homogeneity, approximations, local observer
@article{COCV_2013__19_3_906_0,
author = {M\'enard, Tomas and Moulay, Emmanuel and Perruquetti, Wilfrid},
title = {Homogeneous approximations and local observer design},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {906--929},
publisher = {EDP-Sciences},
volume = {19},
number = {3},
year = {2013},
doi = {10.1051/cocv/2012038},
mrnumber = {3092367},
zbl = {1269.93015},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2012038/}
}
TY - JOUR AU - Ménard, Tomas AU - Moulay, Emmanuel AU - Perruquetti, Wilfrid TI - Homogeneous approximations and local observer design JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 906 EP - 929 VL - 19 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012038/ DO - 10.1051/cocv/2012038 LA - en ID - COCV_2013__19_3_906_0 ER -
%0 Journal Article %A Ménard, Tomas %A Moulay, Emmanuel %A Perruquetti, Wilfrid %T Homogeneous approximations and local observer design %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 906-929 %V 19 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012038/ %R 10.1051/cocv/2012038 %G en %F COCV_2013__19_3_906_0
Ménard, Tomas; Moulay, Emmanuel; Perruquetti, Wilfrid. Homogeneous approximations and local observer design. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 906-929. doi : 10.1051/cocv/2012038. http://www.numdam.org/articles/10.1051/cocv/2012038/
[1] and , On the existence of a Kazantzis-Kravaris/Luenberger observer. SIAM J. Control Optim. 45 (2006) 432-456. | MR | Zbl
[2] , and , Homogeneous approximation, recursive observer design, and output feedback. SIAM J. Control Optim. 47 (2008) 1814-1850. | MR | Zbl
[3] and , An immersion-based observer design for rank-observable nonlinear systems. IEEE Trans. Autom. Control 52 (2007) 83-88. | MR | Zbl
[4] and , Graded approximations and controllability along a trajectory. SIAM J. Control Optim. 28 (1990) 903-924. | MR | Zbl
[5] , and , Algrebraic Methods for Nonlinear Control Systems. Springer, London, 2nd (2007). | MR | Zbl
[6] and , Nonlinear observability, identifiability and persistent trajectory, in IEEE 30th Conference on Decision and Control (1991) 714-719.
[7] , and , Nonlinear estimation is easy. Int. J. Model. Identification Control. 4 (2008) 12-27.
[8] and , Observability for any u(t) of a class of nonlinear systems. IEEE Trans. Autom. Control 26 (1981) 922-926. | MR | Zbl
[9] , and , A simple observer for nonlinear systems applications to bioreactors. IEEE Trans. Autom. Control 37 (1992) 875-880. | MR | Zbl
[10] , and , New algebraic-geometric conditions for the linearization by input-output injection. IEEE Trans. Autom. Control 41 (1996) 598-603. | MR | Zbl
[11] , Nilpotent Lie groups: structure and applications to analysis, Lectures Note Math. Springer-Verlag, Berlin, New-York 562 (1976). | MR | Zbl
[12] , Control systems which generate decomposable lie algebras. J. Differ. Equ. 44 (1982) 166-187. | MR | Zbl
[13] , Nilpotent and high-order approximations of vector-field systems. SIAM Rev. 33 (1991) 238-264. | MR | Zbl
[14] , Homogeneous stabilizing feedback laws. Control Theory and Advanced Technology 6 (1990) 497-516. | MR
[15] and , Nonlinear observer design using Lyapunov's auxiliary theorem. Syst. Control Lett. 34 (1998) 241-247. | MR | Zbl
[16] , High-gain observers in nonlinear feedback control. In International Conference on Control, Automation and system (2008). | Zbl
[17] and , Linearization by output injection and nonlinear observers. Syst. Control Lett. 3 (1983) 47-52. | MR | Zbl
[18] and , Nonlinear observers with linearizable error dynamics. SIAM J. Control Optim. 23 (1985) 197-216. | MR | Zbl
[19] and , Global finite-time stabilization by dynamic output feedback for a class of continuous nonlinear systems. IEEE Trans. Autom. Control 51 (2006) 879-884. | MR
[20] , and , A global high-gain finite-time observer. IEEE Trans. Autom. Control 55 (2010) 1500-1506. | MR
[21] , An approximate nonlinear observer with polynomial coordinate transformation maps. IEEE Trans. Autom. Control 42 (1997) 522-527. | MR | Zbl
[22] , On constructing nonlinear observers. SIAM J. Control Optim. 29 (1991) 516-534. | MR | Zbl
[23] and , Hypoelliptic differential operators and nilpotent groups. Acta Mathematica 137 (1976) 247-320. | MR | Zbl
[24] , and , Direct transformation of nonlinear systems into state affine miso form for observer design. IEEE Trans. Autom. Control 48 (2003) 2191-2196. | MR
[25] , Polynomial approximations to control systems and local controllability, in Proc. of the 24th IEEE Conference on Decision and Control (1985) 33-38.
[26] , Local observer design for nonlinear systems. Math. Comput. Model. 35 (2002) 25-36. | MR | Zbl
[27] , A general theorem on local controllability. SIAM J. Control Optim. 25 (1987) 158-194. | MR | Zbl
Cité par Sources :
