Map invariance and the state reconstruction problem for nonlinear discrete-time systems

European Journal of Control

Volumn 15, Issue 2, 2009, Pages 105-119

  Kazantzis, Nikolas ^a  

^a Worcester Polytechnic Institute   (United States)

Author keywords

Discrete time systems; Invariant manifolds; Map invariance; Nonlinear observers; Nonlinear systems; Reduced order observer design

Indexed keywords

DIGITAL CONTROL SYSTEMS;

DISCRETE - TIME SYSTEMS; DISCRETE TIME-DOMAIN; DISCRETE-TIME DOMAIN; INVARIANT MANIFOLDS; MAP INVARIANCE; NON-LINEAR OBSERVER; NONLINEAR DISCRETE-TIME SYSTEMS; OBSERVERS DESIGNS; REDUCED-ORDER OBSERVER DESIGN; REDUCED-ORDER OBSERVERS;

DISCRETE TIME CONTROL SYSTEMS;

EID: 77955310892     PISSN: 09473580     EISSN: None     Source Type: Journal    
DOI: 10.3166/ejc.15.105-119     Document Type: Article

References (65)

1. Andrieu V, Praly L. On the existence of a Kazantzis-Kravaris/Luenberger observer. SIAM J Contr Optim 2006; 45: 432.
2. Arcak M, Nesic D. A framework for nonlinear sampleddata observer design via approximate discrete- time models and emulation. Automatica 2004; 40: 1931.
3. Arnold VI. GeometricalMethods in the Theory of Ordinary Differential Equations. New York, NY: Springer-Verlag, 1983.
4. Astolfi A, Ortega R. Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems. IEEE Trans Autom Contr 2003; 48: 590.
5. Astolfi A, Praly L. Global complete observability and output-to-state stability imply the existence of a globally convergent observer. Proceedings of the 42nd IEEE Conference on Decision and Control, 2003.
6. Banaszuk A, Hauser J. Feedback linearization of transverse dynamics for periodic orbits. Syst Contr Lett 1995; 26: 95.
7. Barbot J-P, Monaco S, Normand-Cyrot D. On the observer for differential-difference representations of discrete-time systems. Proceeedings of the European Control Conference, 2007.
8. Bastin G, Dochain D. Online Estimation and Adaptive Control of Bioreactors. Elsevier: New York, 1990.
9. Besançon G, Bornard G. Time-varying linearization up to output injection of discrete-time nonlinear systems. Proceedings of the 1995 European Control Conference, 1995.
10. Besançon G. On output transformations for state linearization up to output injection. IEEE Trans Autom Contr 1999; 50: 2059.
11. Boutayeb M, Darouach M. A reduced-order observer for nonlinear discrete-time systems. Syst Contr Lett 2000; 39: 141.
12. Byrnes CI, Delli Priscoli F, Isidori A. Output Regulation of Uncertain Nonlinear Systems. Boston, MA: Birkhauser, 1997.
13. Califano C, Monaco S, Normand-Cyrot D. On the observer design in nonlinear discrete-time systems. Syst Contr Lett 2003; 49: 255-265.
14. Carr J. Applications of Center Manifold Theory. New York, NY: Springer-Verlag, 1981.
15. Castillo B, DiGennaro S, Monaco S, Normand-Cyrot D. Nonlinear regulation for a class of discrete- time systems. Syst Contr Lett 1993; 20: 57.
16. Chen CT. Linear System Theory and Design. New York, NY: Holt, Rinehart and Winston, 1984.
17. Christofides PD. Nonlinear and Robust Control of PDE Systems. Boston, MA: Birkhauser, 2001.
18. Chung C-T, Grizzle JW. Sampled-data observer error linearization. Automatica 1990; 26: 997.
19. Ciccarela G, Dalla Mora M, Germani A. Observers for discrete-time nonlinear systems. Syst Contr Lett 1993; 20: 373.
20. Colonius F, Kliemann W. The Dynamics of Control. Boston, MA: Birkhauser, 2000.
21. Dabroom AM, Khalil HK. Output feedback sampleddata control of nonlinear systems using high-gain observers. IEEE Trans Autom Contr 2001; 46: 1712.
22. Elaydi SN. An Introduction to Difference Equations. New York, NY: Springer-Verlag, 1999.
23. Foias C, Sell R, Titi ES. Exponential tracking and approximation of inertial manifolds for dissipative equations. J Dynam Diff Equat 1989; 1: 199.
24. Gantmacher FR. The Theory of Matrices. New York, NY: Chelsea Publishing Company, 1960.
25. Germani A, Manes C, Palumbo P. Polynomial extended Kalman filter. IEEE Trans Autom Contr 2005; 50: 2059.
26. Gorban A, Kazantzis N, Kevrekidis I, Ottinger HC, Theodoropoulos C (eds). Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena. Heidelberg: Springer-Verlag, 2006.
27. Guckenheimer J, Holmes PJ. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York, NY: Springer-Verlag, 1983.
28. Hamzi B, Monaco S, Normand-Cyrot D. Quadratic stabilization of systems with periodic doubling bifurcation. Proceedings of the 41st IEEE Conference on Decision and Control, 2002.
29. Hamzi B, Kang W, Krener A. Control of center manifolds. Proceedings of the 42nd IEEE Conference on Decision and Control, 2003.
30. Hamzi B, Kang W, Krener A. Controlled center dynamics. SIAM Multisc Model Simul 2005; 3: 838.
31. Huijberts HJ, Lilge T, Nijmeijer H. Nonlinear discretetime synchronization via extended observers. Int J Bifur Chaos 2001; 11: 1997.
32. Kang W. Bifurcation and normal form of nonlinear control systems. SIAM J Contr Optim 1998; 36: 193.
33. Kang W, Krener AJ. Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems. SIAM J Contr Optim 1992; 30: 1319.
34. Kaper T, Koppel N, Jones CKRT. Tracking invariant manifolds up to exponentially small errors. SIAM J Math Anal 1996; 7: 58.
35. Karafyllis I, Kravaris C. On the observer problem for discrete-time control systems. IEEE Trans Autom Contr 2007; 52: 12.
36. Karagiannis D, Astolfi A. Nonlinear observer design using invariant manifolds and applications. Proceedings of the 44th IEEE Conference on Decision and Control, 2005.
37. Kazantzis N, Kravaris C. Nonlinear observer design using Lyapunov's auxiliary theorem. Syst Contr Lett 1998; 34: 241.
38. Kazantzis N, Kravaris C. Discrete-time nonlinear observer design using functional equations. Syst Contr Lett 2001; 42: 81.
39. Kazantzis N, Huynh N, Good T. A model-based characterization of the long-term asymptotic behavior of nonlinear discrete-time processes using invariance functional equations. Comp Chem Engn 2005; 29: 2346.
40. Kazantzis N, Demetriou M. Singular control invariance PDEs for nonlinear systems. SIAM Multisc Model Simul 2005; 3: 731.
41. Kazantzis N. On invariant manifolds of nonlinear discrete-time input-driven dynamical systems. Phys Lett A 2001; 292: 107.
42. Kazantzis N. A new approach to the zero-dynamics assignment problem for nonlinear discrete-time systems using functional equations. Syst Contr Lett 2004; 51: 311.
43. Kazantzis N. Invariance inducing control of nonlinear discrete-time dynamical systems. J Nonl Sci 2004; 13: 579.
44. Kokotovic PV, Khalil HK, O'Reilly JO. Singular Perturbation Methods in Control: Analysis and Design. Academic Press, 1986.
45. Kosnakoglu C, Serrani A. Attenuation of oscillations in Galerkin systems using center-manifold techniques. Eur J Contr 2007; 13: 529.
46. Kouvaritakis B, Wang W, Lee YI. Observers in nonlinear model-based predictive control. Int J Rob Nonl Contr 2000; 10: 749.
47. Kravaris C, Niemiec M, Kazantzis N. Singular PDEs and the assignment of zero dynamics in nonlinear systems. Syst Contr Lett 2004; 51: 67.
48. Kravaris C, Thompson S, Schwank J. Higher-order corrections to the picriterion using Center Manifold theory. Proceedings of the 41st IEEE Conference on Decision and Control, 2002.
49. Krener AJ, Isidori A. Linearization by output injection and nonlinear observers. Syst Contr Lett 1983; 3: 47.
50. Lin W, Byrnes CI. Design of discrete-time nonlinear control systems via smooth feedback. IEEE Trans Autom Contr 1994; 39: 2340.
51. Lin W, Byrnes CI. Remarks on linearization of discretetime autonomous systems and nonlinear observer design. Syst Contr Lett 1995; 25: 31.
52. Luenberger DG. Observing the state of a linear system. IEEE Trans Milit Electr 1963; 8: 74.
53. Lyapunov AM. The General Problem of the Stability of Motion. London, UK: Taylor & Francis, 1992.
54. Moraal PE, Grizzle JE. Observer design for nonlinear systems with discrete-time measurements. IEEE Trans Autom Contr 2003; 40: 622.
55. Nielsen C, Maggiore M. Transverse feedback linearization of multi-input systems. Proceedings of the 44th IEEE Conference on Decision and Control, 2005.
56. Smajdor W. Local analytic solutions of the functional equation Φ(z)=h(z, Φ[f(z)]) in multidimensional spaces. Aequationes Math 1968; 1: 20.
57. Talasila V, Astolfi A, Clemente-Gallardo J. Controlled invariant manifolds: A normally-hyperbolic viewpoint. Proceedings of the 45th IEEE Conference on Decision and Control, 2006.
58. Teman R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. New York, NY: Springer-Verlag, 1988.
59. van der Schaft AJ. On a state space approach to nonlinear H1 control. Syst Contr Lett 1991; 16: 1.
60. Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York, NY: Springer-Verlag, 1990.
61. Xiao M, Krener A. Nonlinear observer design in the Siegel domain. SIAM J Contr Optim 2002; 41: 932.
62. Xiao M, Kazantzis N, Kravaris C, Krener AJ. Nonlinear discrete-time observer design with linearizable error dynamics. IEEE Trans Autom Contr 2003; 48: 622.
63. Xiao M. A direct method for the construction of nonlinear discrete-time observer with linearizable error dynamics. IEEE Trans Autom Contr 2006; 51: 128.
64. Xiao M. The global existence of nonlinear observers with linear error dynamics: a topological point of view. Syst Contr Lett 2006; 55: 849.
65. Xiao M, Kazantzis N, Kravaris C. A new nonlinear discrete-time observer design method with assignable error dynamics. Proceedings of the 2007 European Control Conference, 2007.