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Semi-global finite-time observers for nonlinear systems. (English) Zbl 1153.93332

Summary: It is well known that high gain observers exist for single output nonlinear systems that are uniformly observable and globally Lipschitzian. Under the same conditions, we show that these systems admit semi-global and finite-time converging observers. This is achieved with a derivation of a new sufficient condition for local finite-time stability, in conjunction with applications of geometric homogeneity and Lyapunov theories.

MSC:

93B07 Observability
93C15 Control/observation systems governed by ordinary differential equations
93C10 Nonlinear systems in control theory
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[1] Andrieu, V., Praly, L., & Astolfi, A. (2007). Homogeneous observers with dynamic high gains. The 7th IFAC symposium on nonlinear contr. syst.; Andrieu, V., Praly, L., & Astolfi, A. (2007). Homogeneous observers with dynamic high gains. The 7th IFAC symposium on nonlinear contr. syst.
[2] Bestle, D.; Zeitz, M., Canonical form observer design for non-linear time-variable systems, International Journal of Control, 38, 2, 419-431 (1983) · Zbl 0521.93012
[3] Bhat, S.; Bernstein, D., Finite-time stability of continuous autonomous systems, SIAM Journal of Control and Optimization, 38, 3, 751-766 (2000) · Zbl 0945.34039
[4] Bhat, S.; Bernstein, D., Geometric homogeneity with applications to finite-time stability, Mathematics of Control, Signals, and Systems, 17, 101-127 (2005) · Zbl 1110.34033
[5] Engel, R.; Kreisselmeier, G., A continuous-time observer which converges in finite time, IEEE Transactions on Automatic Control, 47, 7, 1202-1204 (2002) · Zbl 1364.93084
[6] Gauthier, J. P.; Hammouri, H.; Othman, S., A simple observer for nonlinear systems applications to Bioreactors, IEEE Transactions on Automatic Control, 37, 6, 875-880 (1992) · Zbl 0775.93020
[7] Gauthier, J. P.; Kupka, I. A.K., Observability and observers for nonlinear systems, SIAM Journal of Control and Optimization, 32, 4, 975-994 (1994) · Zbl 0802.93008
[8] Hammouri, H.; Targui, B.; Armanet, F., High gain observer based on a triangular structure, International Journal of Robust and Nonlinear Control, 12, 6, 497-518 (2002) · Zbl 1006.93007
[9] Haskara, I.; Ozguner, U.; Utkin, V., On sliding mode observers via equivalent control approach, International Journal of Control, 71, 6, 1051-1067 (1998) · Zbl 0938.93505
[10] Hermann, R.; Krener, A. J., Nonlinear controllability and observability, IEEE Transactions on Automatic Control, 22, 5, 728-740 (1997) · Zbl 0396.93015
[11] Hong, Y.; Huang, J.; Xu, Y., On an output feedback finite-time stabilization problem, IEEE Transactions on Automatic Control, 46, 2, 305-309 (2001) · Zbl 0992.93075
[12] Hong, Y., Finite-time stabilization and stabilizability of a class of controllable systems, Systems & Control Letters, 46, 4, 231-236 (2002) · Zbl 0994.93049
[13] Isidori, A., Nonlinear control systems (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0569.93034
[14] Kazantzis, N.; Kravaris, C., Nonlinear observer design using Lyapunov’s auxiliary theorem, Systems & Control Letters, 34, 5, 241-247 (1998) · Zbl 0909.93002
[15] Kou, S. R.; Elliott, D. L.; Tarn, T. J., Exponential observers for nonlinear dynamic systems, Information and Control, 29, 204-216 (1975) · Zbl 0319.93049
[16] Krener, A. J., (Nonlinear stabilizability and detectability. Nonlinear stabilizability and detectability, Systems and networks: Mathematical theory in nonlinear control theory (1986), Reidel, Dordrecht), 89-98
[17] Krener, A. J.; Respondek, W., Nonlinear observers with linearizable error dynamics, SIAM Journal of Control and Optimization, 23, 2, 197-216 (1985) · Zbl 0569.93035
[18] Krener, A. J.; Isidori, A., Linearization by output injection and nonlinear observers, Systems & Control Letters, 3, 1, 47-52 (1983) · Zbl 0524.93030
[19] Menold, P. H., Findeisen, R., & Allgöwer, F. (2003a). Finite time convergent observers for linear time-varying systems. In Proceedings of the 11th Mediterranean conference on control and automation; Menold, P. H., Findeisen, R., & Allgöwer, F. (2003a). Finite time convergent observers for linear time-varying systems. In Proceedings of the 11th Mediterranean conference on control and automation
[20] Menold, P. H., Findeisen, R., & Allgöwer, F. (2003b). Finite time convergent observers for nonlinear systems. In Proceedings of the 42nd IEEE conference on decision and controlVol. 6; Menold, P. H., Findeisen, R., & Allgöwer, F. (2003b). Finite time convergent observers for nonlinear systems. In Proceedings of the 42nd IEEE conference on decision and controlVol. 6
[21] Michalska, H.; Mayne, D., Moving horizon observers and observer-based control, IEEE Transactions on Automatic Control, 40, 6, 995-1006 (1995) · Zbl 0832.93007
[22] Perruquetti, W.; Floquet, T.; Moulay, E., Finite time observers and secure communication, IEEE Transactions on Automatic Control, 53, 1, 356-360 (2008) · Zbl 1367.94361
[23] Qian, C.; Lin, W., Non-lipschitz continuous stabilizer for nonlinear systems with uncontrollable unstable linearization, Systems & Control Letters, 42, 3, 185-200 (2001) · Zbl 0974.93050
[24] Qian, C.; Lin, W., A continuous feedback approach to global strong stabilization of nonlinear systems, IEEE Transactions on Automatic Control, 46, 7, 1061-1079 (2001) · Zbl 1012.93053
[25] Raghavan, S.; Hedrick, J. K., Observer design for a class of nonlinear systems, International Journal Control, 59, 2, 515-528 (1994) · Zbl 0802.93007
[26] Rajamani, R.; Cho, Y., Existence and design of observers for nonlinear systems: Relation to distance to unobservability, International Journal Control, 69, 5, 717-731 (1998) · Zbl 0933.93019
[27] Sauvage, F.; Guay, M.; Dochain, D., Design of a nonlinear finite-time converging observer for a class of nonlinear systems, Journal of Control Science and Engineering, 2007, 1-9 (2007) · Zbl 1229.93091
[28] Shim, H.; Son, Y. I.; Seo, J. H., Semi-global observer for multi-output nonlinear systems, Systems & Control Letters, 42, 3, 233-244 (2001) · Zbl 0985.93006
[29] Thau, F. E., Observing the state of nonlinear dynamic systems, International Journal Control, 17, 3, 471-479 (1973) · Zbl 0249.93006
[30] Xia, X.-H.; Gao, W.-B., On exponential observers for nonlinear systems, Systems & Control Letters, 11, 4, 319-325 (1988)
[31] Xia, X.-H.; Gao, W.-B., Nonlinear observer design by observer error linearization, SIAM Journal Control and Optimization, 27, 1, 199-216 (1989) · Zbl 0667.93014
[32] Xia, X.; Zeitz, M., On nonlinear continuous observers, International Journal of Control, 66, 6, 943-954 (1997) · Zbl 0872.93016
[33] Yoshizawa, T., Stability theory by Lyapunov’s second method (1966), The Mathematical Society of Japan: The Mathematical Society of Japan Tokyo · Zbl 0144.10802
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