# Genericity of observability and the existence of asymptotic observers

Banach Center Publications (1995)

- Volume: 32, Issue: 1, page 227-244
- ISSN: 0137-6934

## Access Full Article

top## Abstract

top## How to cite

topGauthier, J., and Kupka, I. "Genericity of observability and the existence of asymptotic observers." Banach Center Publications 32.1 (1995): 227-244. <http://eudml.org/doc/262690>.

@article{Gauthier1995,

abstract = {In this paper, we deal with the genericity of the observability property and the existence of asymptotic observers for nonlinear systems. In the case where the number of outputs is larger than the number of inputs and the state space is compact, we prove that observability in a very strong sense (more or less, observability for each sufficiently differentiable input) is generic. This is obtained by using standard (but not easy) transversality arguments. For the inputs that are bounded with their derivatives up to some order, we prove the generic existence of an asymptotic observer with arbitrary exponential decay of the error.},

author = {Gauthier, J., Kupka, I},

journal = {Banach Center Publications},

keywords = {generic properties; observability; asymptotic observers},

language = {eng},

number = {1},

pages = {227-244},

title = {Genericity of observability and the existence of asymptotic observers},

url = {http://eudml.org/doc/262690},

volume = {32},

year = {1995},

}

TY - JOUR

AU - Gauthier, J.

AU - Kupka, I

TI - Genericity of observability and the existence of asymptotic observers

JO - Banach Center Publications

PY - 1995

VL - 32

IS - 1

SP - 227

EP - 244

AB - In this paper, we deal with the genericity of the observability property and the existence of asymptotic observers for nonlinear systems. In the case where the number of outputs is larger than the number of inputs and the state space is compact, we prove that observability in a very strong sense (more or less, observability for each sufficiently differentiable input) is generic. This is obtained by using standard (but not easy) transversality arguments. For the inputs that are bounded with their derivatives up to some order, we prove the generic existence of an asymptotic observer with arbitrary exponential decay of the error.

LA - eng

KW - generic properties; observability; asymptotic observers

UR - http://eudml.org/doc/262690

ER -

## References

top- [A] D. Aeyels, Generic observability of differentiable systems, SIAM J. Control Optim. 19 (1981), 1-15. Zbl0474.93016
- [AR] R. Abraham and J. Robbin, Transversal Mappings and Flows, Benjamin, 1967. Zbl0171.44404
- [GHK] J. P. Gauthier, H. Hammouri and I. Kupka, Observers for nonlinear systems, IEEE CDC Conference, Brighton, 1991, 1483-1489.
- [GHO] J. P. Gauthier, H. Hammouri and S. Othman, A simple observer for nonlinear systems, application to bioreactors, IEEE Trans. Automat. Control 37 (1992), 875-880. Zbl0775.93020
- [GK] J. P. Gauthier and I. Kupka, Observability and observers for nonlinear systems, SIAM J. Control Optim. 32 (1994), 975-995. Zbl0802.93008
- [HG1] H. Hammouri and J. P. Gauthier, Bilinearization up to output injection, Systems Control Letters 11 (1988), 139-149. Zbl0648.93024
- [HG2] H. Hammouri and J. P. Gauthier, Global time varying linearization up to output injection, SIAM J. Control 6 (1992), 1295-1310. Zbl0771.93033
- [HIR] M. W. Hirsch, Differential Topology, Springer, 1976.
- [KI] A. Krener and A. Isidori, Linearization by output injection and nonlinear observers, Systems Control Letters 3 (1983), 47-52. Zbl0524.93030
- [KR] A. Krener and W. Respondek, Nonlinear observers with linearizable error dynamics, SIAM J. Control Optim. 23 (1985), 197-216. Zbl0569.93035
- [L] S. Łojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. PISA, 1964, 449-474. Zbl0128.17101
- [LU] D. G. Luenberger, Observers for multivariable systems, IEEE Trans. Automat. Control 11 (1966), 190-197.
- [S] H. Sussmann, Single input observability of continuous time systems, Math. Systems Theory 12 (1979), 263-284. Zbl0422.93019
- [T] F. Takens, Detecting strange attractors in turbulence, in: Dynamic Systems and Turbulence, Warwick 1980, Springer, Berlin, 1981, 366-381.
- [TC] K. Tchon, On solvability of several affine systems, Systems Control Letters 4 (1984), 373-379. Zbl0544.93031
- [TOU] J. C. Tougeron, Idéaux de fonctions différentiables, Springer, 1972. Zbl0251.58001
- [WH] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89. Zbl0008.24902