×

Discrete-time normal form for left invertibility problem. (English) Zbl 1298.93094

Summary: This paper deals with the design of quadratic and higher order normal forms for the left invertibility problem. The linearly observable case and one-dimensional linearly unobservable case are investigated. The interest of such a study in the design of a delayed discrete-time observer is examined. The example of the Burgers map with unknown input is treated and a delayed discrete-time observer is designed. Finally, some simulated results are commented.

MSC:

93B10 Canonical structure
93B07 Observability
93C55 Discrete-time control/observation systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bazet, A.; Johari Majd, V., An alternative way to test the criteria ensuring the existence of discrete-time normal form, Syst Control Lett, 56, 315-319 (2007) · Zbl 1113.93073
[2] Barbot J-P, Belmouhoub I, Boutat-Baddas L. Observability quadratic normal forms. In: Kang W, Xiao M, Borges C. (ed), New Trends in Nonlinear Dynamics & Control, & their Applications, LINCIS 295. Springer-Verlag: Berlin, pp. 3-17.; Barbot J-P, Belmouhoub I, Boutat-Baddas L. Observability quadratic normal forms. In: Kang W, Xiao M, Borges C. (ed), New Trends in Nonlinear Dynamics & Control, & their Applications, LINCIS 295. Springer-Verlag: Berlin, pp. 3-17. · Zbl 1203.93037
[3] Barbot, J.-P.; Monaco, S.; Normand-Cyrot, D., Quadratic forms and approximate feedback linearization in discrete-time, Int J Control, 67, 567-586 (1997) · Zbl 0886.93038
[4] Barbot, J.-P.; Monaco, S.; Normand-Cyrot, D., On the observer for differential/difference representations of discrete-time dynamics, (European Control Conference. European Control Conference, Kos, Grèce (2007)), 5783-5788
[5] Belmouhoub, I.; Djemai, M.; Barbot, J.-P., An example of nonlinear discrete-time synchronization of chaotic systems for secure communication, (European Control Conference. European Control Conference, Cambridge (2003)) · Zbl 1365.93051
[6] Belmouhoub, I.; Djemai, M.; Barbot, J.-P., Cryptography by discrete-time hyperchaotic systems, (IEEE Conference on Decision and Control. IEEE Conference on Decision and Control, Hawaï (2003)) · Zbl 1365.93051
[7] Boutat, D.; Barbot, J.-P., Poincaré Normal form for a class of driftless system in one-dimensional submanifold neighborhood, Mathematics of Control, Signals, and Systems, 256-274 (2002), MCSS 15 · Zbl 1049.93009
[8] Boutat-Baddas, L.; Boutat, D.; Barbot, J.-P.; Tauleigne, R., Quadratic observability normal form, (Proceedings of IEEE—CDC (2001)) · Zbl 1074.93506
[9] Boutat-Baddas, L.; Barbot, J.-P.; Boutat, D.; Tauleigne, R., Observability bifurcation versus observing bifurcations, (Proceedings of IFAC World Congress (2002)) · Zbl 1074.93506
[10] Brunovsky, P., A classification of linear controllable systems, Kybernetika, 6, 173-188 (1970) · Zbl 0199.48202
[11] Fliess M, Fuchshumer S, Schöberl M, Schlacher K, Sira-Ramirez H. An introduction to algebraic discretetime linear parametric identification with a concrete application. J Européen des Systèmes Automatisés (JESA) 2008; 42: 210-232, http://hal.inria.fr/inria-00188435/fr/; Fliess M, Fuchshumer S, Schöberl M, Schlacher K, Sira-Ramirez H. An introduction to algebraic discretetime linear parametric identification with a concrete application. J Européen des Systèmes Automatisés (JESA) 2008; 42: 210-232, http://hal.inria.fr/inria-00188435/fr/
[12] Floquet, T.; Barbot, J.-P., State and unknown input estimation for linear discrete-time systems, Automatica, 42, 1883-1889 (2006) · Zbl 1222.93215
[13] Gordillo, F.; Salas, F.; Ortega, R.; Aracil, J., Hopf bifurcation in indirect field-oriented control of induction motors, Automatica, 38, 829-835 (2002) · Zbl 1007.93054
[14] Gu, G.; Sparks, A.; Kang, W., Bifurcation analysis and control for model via the projection method, (Proceedings 1998 of ACC (1998)), 3617-3621
[15] Hamzi, B.; Barbot, J.-P.; Monaco, S.; Normand-Cyrot, D., Nonlinear discrete-time control of systems with a Naimar-Sacker bifurcation, Syst Control Lett, 44, 245-258 (2001) · Zbl 0986.93058
[16] Hamzi, B.; Krener, A. J.; Kang, W., The controlled center dynamics of discrete-time control bifurcations, Syst Control Lett, 55, 585-596 (2006) · Zbl 1129.93501
[17] Hamzi, B.; Tall, I. A., Normal forms for nonlinear discretetime control systems, (Proceedings of 42nd IEEE Conference on Decision and Control (2003)), 1357-1361, 2
[18] Huijberts, H. J.C.; Lilge, T.; Nijmeijer, H., Nonlinear discrete-time synchronization via extended observers, Int J Bifurcation Chaos, 11, 1997-2006 (2001)
[19] Korsch, H. J.; Jodl, H.-J., Chaos a program collection for the PC (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0909.58026
[20] Kang, W., Bifurcation and normal form of nonlinear control system: Part I and II, SIAM J Control Optim, 36, 193-232 (1998)
[21] Kang, W.; Krener, A. J., Extended quadratic controller normal form and dynamic state feedback linearization of non linear systems, SIAM J Control Optim, 30, 1319-1337 (1992) · Zbl 0771.93034
[22] Krener, A. J., Approximate linearization by state feedback and coordinate change, Syst Control Lett, 5, 181-185 (1984) · Zbl 0555.93027
[23] Krener, A. J., Feedback linearization mathematical control theory, (Bailleul, J.; Willems, J-C., Mathematical Control Theory (1998), Springer), 66-98 · Zbl 0927.93021
[24] Lilge, T., Nonlinear discrete-time observers for synchronization problems, New Direction in nonlinear Observer Design, 491-510 (1999), LNCIS 244 · Zbl 0929.93009
[25] Monaco, S.; Normand-Cyrot, D., The immersion under feedback of a multidimensional discrete-time nonlinear system into a linear one, Int J Control, 38, 245-261 (1983) · Zbl 0566.93012
[26] Monaco, S.; Normand-Cyrot, D., Zero dynamics of sampled nonlinear systems, Syst Control Lett, 11, 229-234 (1988) · Zbl 0664.93037
[27] Monaco, S.; Normand-Cyrot, D., Functional expansions for nonlinear discrete-time systems, Math Systems Theory, 21, 235-254 (1989) · Zbl 0676.93042
[28] Monaco, S.; Normand-Cyrot, D., Multirate three axes attitude stabilization of spacecraft, (Proceedings of the 28th IEEE Conference Decision and Control. Proceedings of the 28th IEEE Conference Decision and Control, Tampa, CA, USA (1989)) · Zbl 0738.93017
[29] Monaco, S.; Normand-Cyrot, D., Normal forms and approximated feedback linearization in discrete-time, Syst Control Lett, 55, 71-80 (2006) · Zbl 1129.93336
[30] Monaco S, Normand-Cyrot D. Controller and observer normal forms in discrete-time. In: Astolfi A, Marconi L (eds), Analysis and Design of Nonlinear Control Systems. Springer-Verlag, pp. 377-396.; Monaco S, Normand-Cyrot D. Controller and observer normal forms in discrete-time. In: Astolfi A, Marconi L (eds), Analysis and Design of Nonlinear Control Systems. Springer-Verlag, pp. 377-396. · Zbl 1189.93083
[31] Nijmeijer, H.; Mareels, I. M.Y., An observer looks at synchronization, IEEE Trans Circuits Syst 1, Fundam Theory Appl, 44, 882-891 (1997)
[32] Perruquetti, W.; Barbot, J.-P., Sliding Mode Control in Engineering. Control Eng Series (2002), Marcel Dekker: Marcel Dekker New York
[33] Poincaré H. Les Méthodes nouvelles de la mecanique céleste. Gauthier Villard 1899 Réedition 1987, B. bibliothèque scientifique A. Blanchard.; Poincaré H. Les Méthodes nouvelles de la mecanique céleste. Gauthier Villard 1899 Réedition 1987, B. bibliothèque scientifique A. Blanchard. · JFM 30.0834.08
[34] Rapaport, A.; Maloum, A., Embedding for exponential observers of nonlinear systems, (39th CDC Conference (CDROM) (2000)) · Zbl 1048.93013
[35] Respondek, W., Right and left invertibility of nonlinear control systems, (Sussmann, H. J., Nonlinear Controllability and Optimal Control (1990), Marcel Dekker), 133-176
[36] Sain, M. K.; Massey, J. L., Invertibility of linear timeinvariant dynamical systems, IEEE Trans Autom Control, 14, 141-149 (1969)
[37] Silverman, L. M., Inversion of multivariable linear systems, IEEE Trans Autom Control, 14, 270-276 (1969)
[38] Singh, S. N., A modified algorithm for invertibility in nonlinear systems, IEEE Trans Autom Control, 26, 595-598 (1981) · Zbl 0488.93026
[39] Tall, I. A.; Respondek, W., Normal forms and invariants of nonlinear single-input systems with noncrollable linearization, IFAC NOLCOS (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.