We propose a new type of Proportional Integral (PI) state observer for a class of nonlinear systems in continuous time which ensures an asymptotic stable convergence of the state estimates. Approximations of nonlinearity are not necessary to obtain such results, but the functions must be, at least locally, of the Lipschitz type. The obtained state variables are exact and robust against noise. Naslin's damping criterion permits synthesizing gains in an algebraically simple and efficient way. Both the speed and damping of the observer response are controlled in this way. Model simulations based on a Sprott strange attractor are discussed as an example.
@article{bwmeta1.element.bwnjournal-article-amcv23z2p383bwm,
author = {Beno\^\i t Schwaller and Denis Ensminger and Birgitta Dresp-Langley and Jos\'e Ragot},
title = {State estimation for a class of nonlinear systems},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {23},
year = {2013},
pages = {383-394},
zbl = {1282.93247},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv23z2p383bwm}
}
Benoît Schwaller; Denis Ensminger; Birgitta Dresp-Langley; José Ragot. State estimation for a class of nonlinear systems. International Journal of Applied Mathematics and Computer Science, Tome 23 (2013) pp. 383-394. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv23z2p383bwm/
[000] Abdessameud, A. and Khelfi, M.F. (2006). A variable structure observer for the control of robot manipulators, International Journal of Applied Mathematics and Computer Science 16(2): 189-196. | Zbl 1111.93056
[001] Bastin, G. and Gevers, M. (1988). Stable adaptive observers for nonlinear time-varying systems, IEEE Transactions on Automatic Control 33(7): 650-658. | Zbl 0676.93037
[002] Besançon, G., Voda, A. and Jouffroy, G. (2010). A note on state and parameter estimation in a Van der Pol oscillator, Automatica 46(10): 1735-1738. | Zbl 1203.93032
[003] Besançon, G., Zhang, Q. and Hammouri, H. (2004). High gain observer based state and parameter estimation in nonlinear systems, 6th IFAC Symposium on Nonlinear Control Systems, NOLCOS, Stuttgart, Germany, pp. 325-332.
[004] Bestle, D. and Zeitz, M. (1983). Canonical form observer design for non-linear time-variable systems, International Journal of Control 38(2): 419-431. | Zbl 0521.93012
[005] Bodizs, L., Srinivasan, B. and Bonvin, D. (2011). On the design of integral observers for unbiased output estimation in the presence of uncertainty, Journal of Process Control 21(3): 379-390.
[006] Boizot, N., Busvelle, E. and Gauthier, J. (2010). An adaptive high-gain observer for nonlinear systems, Automatica 46(9): 1483-1488. | Zbl 1201.93122
[007] Boutat, D., Benali, A., Hammouri, H. and Busawon, K. (2009). New algorithm for observer error linearization with a diffeomorphism on the outputs, Automatica 45(10): 2187-2193. | Zbl 1179.93050
[008] Boutayeb, M. and Aubry, D. (1999). A strong tracking extended Kalman observer for nonlinear discrete-time systems, IEEE Transactions on Automatic Control 44(8): 1550-1556. | Zbl 0957.93086
[009] Chen, W., Khan, A.Q., Abid, M. and Ding, S.X. (2011). Integrated design of observer based fault detection for a class of uncertain nonlinear systems, International Journal of Applied Mathematics and Computer Science 21(3): 423-430, DOI: 10.2478/v10006-011-0031-0. | Zbl 1234.93036
[010] Ciccarella, G., Mora, M. D. and Germani, A. (1993). A Luenberger-like observer for nonlinear systems, International Journal of Control 57(3): 537-556. | Zbl 0772.93018
[011] Efimov, D. and Fridman, L. (2011). Global sliding-mode observer with adjusted gains for locally Lipschitz systems, Automatica 47(3): 565-570. | Zbl 1216.93019
[012] Farza, M., M'Saad, M., Triki, M. and Maatoug, T. (2011). High gain observer for a class of non-triangular systems, Systems and Control Letters 60(1): 27-35. | Zbl 1207.93014
[013] Feki, M. (2009). Observer-based synchronization of chaotic systems, Chaos, Solitons and Fractals 39(3): 981-990.
[014] Fliess, M. (1990). Generalized controller canonical forms for linear and nonlinear dynamics, IEEE Transactions on Automatic Control 35(9): 994-1001. | Zbl 0724.93010
[015] Fuller, A. (1992). Guest editorial, International Journal of Control 55(3): 521-527.
[016] Gauthier, J. and Bornard, G. (1981). Observability for any u(t) of a class of nonlinear systems, IEEE Transactions on Automatic Control AC-26(4): 922-926. | Zbl 0553.93014
[017] Gauthier, J., Hammouri, H. and Othman, S. (1992). A simple observer for nonlinear systems: Applications to bioreactors, IEEE Transactions on Automatic Control 37(6): 875-880. | Zbl 0775.93020
[018] Ghosh, D., Saha, P. and Chowdhury, A. (2010). Linear observer based projective synchronization in delay Roessler system, Communications in Nonlinear Science and Numerical Simulation 15(6): 1640-1647. | Zbl 1221.34138
[019] Gille, J., Decaulne, P. and Pélegrin, M. (1988). Systèmes asservis non linéaires, 5 ième edn, Dunod, Paris, pp. 110-146. | Zbl 0189.45701
[020] Gißler, J. and Schmid, M. (1990). Vom Prozeß zur Regelung. Analyse, Entwurf, Realisierung in der Praxis, 1st Edn., Siemens, Berlin/Munich, pp. 350-399.
[021] Hermann, R. and Krener, A. (1977). Nonlinear controllability and observability, IEEE Transactions on Automatic Control AC-22(5): 728-740. | Zbl 0396.93015
[022] Hui, S. and Żak, S.H. (2005). Observer design for systems with unknown inputs, International Journal of Applied Mathematics and Computer Science 15(4): 431-446. | Zbl 1127.93018
[023] Humbert, C. and Ragot, J. (1970). Comportement asymptotique des rapports caractéristiques dans l'optimisation quadratique, Revue A 12(1): 32-34.
[024] Ibrir, S. (2009). Adaptive observers for time-delay nonlinear systems in triangular form, Automatica 45(10): 2392-2399. | Zbl 1179.93047
[025] Kim, Y.C., Keel, L.H. and Manabe. S.M. (2002). Controller design for time domain specifications, Proceedings of the 15th Triennial World Congress, Barcelona, Spain, pp. 1230-1235.
[026] Khémiri, K., Ben Hmida, F., Ragot, J. and Gossa, M. (2011). Novel optimal recursive filter for state and fault estimation of linear stochastic systems with unknown disturbances, International Journal of Applied Mathematics and Computer Science 21(4): 629-637, DOI: 10.2478/v10006-011-0049-3. | Zbl 1283.93282
[027] Krener, A. and Isidori, A. (1983). Linearization by output injection and nonlinear observers, Systems & Control Letters 3(1): 47-52. | Zbl 0524.93030
[028] Luenberger, D. (1966). Observers for multivariable systems, IEEE Transactions on Automatic Control AC-11(2): 190-197.
[029] Lyapunov, A. (1892). The General Problem of Motion Stability, Annals of Mathematics Studies, Vol. 17, Princeton University Press, (translated into English in 1949). | Zbl 0041.32204
[030] Morales, A. and Ramirez, J. (2002). A PI observer for a class of nonlinear oscillators, Physics Letters A 297(3-4): 205-209. | Zbl 0995.65057
[031] Naslin, P. (1960). Nouveau critère d'amortissement, Automatisme 5(6): 229-236. | Zbl 0095.31804
[032] Naslin, P. (1963). Polynomes normaux et critère algébrique d'amortissement, Automatisme 8(6): 215-223.
[033] Raghavan, S. and Hedrick, J. (1994). Observer design for a class of nonlinear systems, International Journal of Control 59(2): 515-528. | Zbl 0802.93007
[034] Röbenack, K. and Lynch, A.F. (2004). An efficient method for observer design with approximately linear error dynamics, International Journal of Control 77(7): 607-612. | Zbl 1062.93008
[035] Röbenack, K. and Lynch, A. (2006). Observer design using a partial nonlinear observer canonical form, International Journal of Applied Mathematics and Computer Science 16(3): 333-343. | Zbl 1136.93313
[036] Shuang, C., Guodong, L. and Xianyong, F. (WCE 2010). Parameters identification of nonlinear DC motor model using compound evolution algorithms, Proceedings of the World Congress on Engineering, London, UK, pp. 15-20.
[037] Sprott, J. (1994). Some simple chaotic flows, Physical Review E 50(2): 647-650.
[038] Söffker, D., Yu, T. and Müller, P. (1995). State estimation of dynamical systems with nonlinearities by using proportional-integral observers, International Journal of Systems Science 26(9): 1571-1582. | Zbl 0830.93009
[039] Tornambè, A. (1992). High-gain observers for non-linear systems, International Journal of Systems Science 23(9): 1475-1489. | Zbl 0768.93013
[040] Veluvolu, K., Soh, Y. and Cao, W. (2007). Robust observer with sliding mode estimation for nonlinear uncertain systems, IET Control Theory and Applications 1(5): 1533-1540. | Zbl 1134.93032
[041] Zeitz, M. (1987). The extended Luenberger observer for nonlinear systems, Systems and Control Letters Archive 9(2): 149-156. | Zbl 0624.93012
[042] Zheng, G., Boutat, D. and Barbot, J. (2009). Multi-output dependent observability normal form, Nonlinear Analysis: Theory, Methods and Applications 70(1): 404-418. | Zbl 1152.93015