# State observers for nonlinear systems with smooth/bounded input

Alfredo Germani; Costanzo Manes

Kybernetika (1999)

- Volume: 35, Issue: 4, page [393]-413
- ISSN: 0023-5954

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topGermani, Alfredo, and Manes, Costanzo. "State observers for nonlinear systems with smooth/bounded input." Kybernetika 35.4 (1999): [393]-413. <http://eudml.org/doc/33436>.

@article{Germani1999,

abstract = {It is known that for affine nonlinear systems the drift-observability property (i. e. observability for zero input) is not sufficient to guarantee the existence of an asymptotic observer for any input. Many authors studied structural conditions that ensure uniform observability of nonlinear systems (i. e. observability for any input). Conditions are available that define classes of systems that are uniformly observable. This work considers the problem of state observation with exponential error rate for smooth nonlinear systems that meet or not conditions of uniform observability. In previous works the authors showed that drift-observability together with a smallness condition on the input is sufficient to ensure existence of an exponential observer. Here it is shown that drift- observability implies a kind of local uniform observability, that is observability for sufficiently small and smooth input. For locally uniformly observable systems two observers are presented: an exponential observer that uses derivatives of the input functions; an observer that does not use input derivatives and ensures exponential decay of the observation error below a prescribed level (high-gain observer). The construction of both observers is straightforward. Moreover the state observation is provided in the original coordinate system. Simulation results close the paper.},

author = {Germani, Alfredo, Manes, Costanzo},

journal = {Kybernetika},

keywords = {uniform observability; drift-observability; affine nonlinear system; uniform observability; drift-observability; affine nonlinear system},

language = {eng},

number = {4},

pages = {[393]-413},

publisher = {Institute of Information Theory and Automation AS CR},

title = {State observers for nonlinear systems with smooth/bounded input},

url = {http://eudml.org/doc/33436},

volume = {35},

year = {1999},

}

TY - JOUR

AU - Germani, Alfredo

AU - Manes, Costanzo

TI - State observers for nonlinear systems with smooth/bounded input

JO - Kybernetika

PY - 1999

PB - Institute of Information Theory and Automation AS CR

VL - 35

IS - 4

SP - [393]

EP - 413

AB - It is known that for affine nonlinear systems the drift-observability property (i. e. observability for zero input) is not sufficient to guarantee the existence of an asymptotic observer for any input. Many authors studied structural conditions that ensure uniform observability of nonlinear systems (i. e. observability for any input). Conditions are available that define classes of systems that are uniformly observable. This work considers the problem of state observation with exponential error rate for smooth nonlinear systems that meet or not conditions of uniform observability. In previous works the authors showed that drift-observability together with a smallness condition on the input is sufficient to ensure existence of an exponential observer. Here it is shown that drift- observability implies a kind of local uniform observability, that is observability for sufficiently small and smooth input. For locally uniformly observable systems two observers are presented: an exponential observer that uses derivatives of the input functions; an observer that does not use input derivatives and ensures exponential decay of the observation error below a prescribed level (high-gain observer). The construction of both observers is straightforward. Moreover the state observation is provided in the original coordinate system. Simulation results close the paper.

LA - eng

KW - uniform observability; drift-observability; affine nonlinear system; uniform observability; drift-observability; affine nonlinear system

UR - http://eudml.org/doc/33436

ER -

## References

top- Besancon G., Hammouri H., 10.1016/0167-6911(96)00043-6, Systems Control Lett. 29 (1996), 9–19 (1996) Zbl0866.93013MR1416747DOI10.1016/0167-6911(96)00043-6
- Ciccarella G., Mora M. Dalla, Germani A., 10.1080/00207179308934406, Internat. J. Control 57 (1993), 3, 537–556 (1993) MR1205006DOI10.1080/00207179308934406
- Mora M. Dalla, Germani A., Manes C., 10.1016/S0362-546X(97)00184-3, Nonlinear Anal.: Theory, Methods Appl. 30 (1997), 7, 4485–4496 (1997) MR1603593DOI10.1016/S0362-546X(97)00184-3
- Mora M. Dalla, Germani A., Manes C., Exponential Observer for Smooth Nonlinear Systems, Internal Report R. 96-11 of Dept. of Electrical Eng., Univ. of L’Aquila, 1996. Submitted for publication (1996)
- Esfandiari F., Khalil H. K., 10.1080/00207179208934355, Internat. J. Control 56 (1992), 5, 1007–1037 (1992) Zbl0762.93069MR1187838DOI10.1080/00207179208934355
- Gauthier J. P., Bornard G., 10.1109/TAC.1981.1102743, IEEE Trans. Automat. Control 26 (1981), 4, 922–926 (1981) Zbl0553.93014MR0635851DOI10.1109/TAC.1981.1102743
- Gauthier J. P., Hammouri H., Othman S., 10.1109/9.256352, IEEE Trans. Automat. Control 37 (1992), 875–880 (1992) MR1164571DOI10.1109/9.256352
- Gauthier J. P., Kupka I., 10.1137/S0363012991221791, SIAM J. Control Optim. 32 (1994), 4, 975–994 (1994) Zbl0802.93008MR1280224DOI10.1137/S0363012991221791
- Isidori A., Nonlinear Control Systems, Springer–Verlag, Berlin 1989 Zbl0931.93005MR1015932
- Khalil H. K., Esfandiari F., 10.1109/9.237658, IEEE Trans. Automat. Control 38 (1993), 9, 1412–1415 (1993) Zbl0787.93079MR1240837DOI10.1109/9.237658
- Krener A., Respondek W., 10.1137/0323016, SIAM J. Control Optim. 23 (1985) 197–216 (1985) Zbl0569.93035MR0777456DOI10.1137/0323016
- Raghavan S., Hedrick J. K., 10.1080/00207179408923090, Internat. J. Control 59 (1994), 2, 515–528 (1994) Zbl0802.93007MR1261285DOI10.1080/00207179408923090
- Teel A., Praly L., 10.1016/0167-6911(94)90029-9, Systems Control Lett. 22 (1994), 313–325 (1994) Zbl0820.93054MR1274906DOI10.1016/0167-6911(94)90029-9
- Thau F. E., 10.1080/00207177308932395, Internat. J. Control 17 (1973), 471–479 (1973) DOI10.1080/00207177308932395
- Tornambè A., High–gain observers for non–linear systems, Internat. J. Systems Sci. 23 (1992), 9, 1475–1489 (1992) Zbl0768.93013MR1181806
- Tsinias J., 10.1016/0167-6911(89)90030-3, Systems Control Lett. 13 (1989), 135–142 (1989) Zbl0684.93006MR1014239DOI10.1016/0167-6911(89)90030-3
- Tsinias J., 10.1016/0167-6911(90)90092-9, Systems Control Lett. 14 (1990), 411–418 (1990) Zbl0698.93004MR1057160DOI10.1016/0167-6911(90)90092-9

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