Feedback classification of nonlinear single-input control systems with controllable linearization: Normal forms, canonical forms, and invariants

SIAM Journal on Control and Optimization

Volumn 41, Issue 5, 2003, Pages 1498-1531

  Tall, Issa Amadou ^a   Respondek, Witold ^a  

^a IRISA   (France)

Author keywords

Canonical forms; Feedback equivalence; Invariants; Normal forms

Indexed keywords

FEEDBACK CONTROL; INVARIANCE; LINEARIZATION; MATHEMATICAL TRANSFORMATIONS;

FEEDBACK EQUIVALENCE;

NONLINEAR CONTROL SYSTEMS;

EID: 0141534415     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012900381753     Document Type: Article

References (38)

1. V.I. ARNOLD, Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd ed., Springer-Verlag, New York, 1988.
2. V. BOGAEVSKI AND A. POVZNER, Algebraic Methods in Nonlinear Physics, Springer-Verlag, New York, 1991.
3. B. BONNARD, Feedback equivalence for nonlinear systems and the time optimal control problem, SIAM J. Control Optim., 29 (1991), pp. 1300-1321.
4. B. BONNARD, Quadratic control systems, Math. Control Signals Systems, 4 (1991), pp. 139-160.
5. A. BRESSAN AND F. RAMPAZZO, On differential systems with quadratic impulses and their applications to Lagrangian mechanics, SIAM J. Control Optim., 31 (1993), pp. 1205-1230.
6. E. CARTAN, Théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Gauthier-Villars, Paris, 1937.
7. S. ǑELIKOVSKY AND H. NIJMEIJER, Equivalence of nonlinear systems to triangular form: The singular case, Systems Control Lett., 27 (1996), pp. 135-144.
8. M. FLIESS, J. LÉVINE, P. MARTIN, AND P. ROUCHON, Flatness and defect of nonlinear systems: Introductory theory and examples, Internat. J. Control, 6 (1995), pp. 1327-1361.
9. R.B. GARDNER, The Method of Equivalence and Its Applications, CBMS Reg. Conf. Ser. in Appl. Math. 58, SIAM, Philadelphia, 1989.
10. R.B. GARDNER, W.F. SHADWICK, AND G.R. WILKENS, A geometric isomorphism with applications to closed loop controls, SIAM J. Control Optim., 27 (1989), pp. 1361-1368.
11. R. GARDNER AND W. SHADWICK, The GS algorithm for exact linearization to Brunovský normal form, IEEE Trans. Automat. Control, 37 (1992), pp. 224-230.
12. R.B. GARDNER, W.F. SHADWICK, AND G.R. WILKENS, Feedback equivalence and symmetries of Brunovský normal forms, Contemp. Math., 97 (1989), pp. 115-130.
13. L.R. HUNT AND R. SU, Linear equivalents of nonlinear time varying systems, in Proceedings of the 4th International Symposium on Mathematical Theory of Networks and Systems, Santa Monica, CA, 1981, pp. 119-123.
14. L.R. HUNT, R. SU, AND G. MEYER, Design for multi-input nonlinear systems, in Differential Geometric Control Theory, R.W. Brockett, R. Millman, and H.J. Sussmann, eds., Birkhäuser, Boston, 1983, pp. 268-298.
15. B. JAKUBCZYK, Equivalence and invariants of nonlinear control systems, in Nonlinear Control-lability and Optimal Control, H.J. Sussmann, ed., Marcel Dekker, New York, Basel, 1990, pp. 177-218.
16. B. JAKUBCZYK, Feedback invariants, critical trajectories, and Hamiltonian formalism, in Nonlinear Control in the Year 2000, Vol. 1, Lecture Notes in Control and Inform. Sci. 258, A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek, eds., Springer-Verlag, London, 2001, pp. 545-568.
17. B. JAKUBCZYK, Critical Hamiltonians and feedback invariants, in Geometry of Feedback and Optimal Control, B. Jakubczyk and W. Respondek, eds., Marcel Dekker, New York, Basel, 1998, pp. 219-256.
18. B. JAKUBCZYK AND W. RESPONDEK, On linearization of control systems, Bull. Acad. Polon. Sci. Sér. Sci. Math., 28 (1980), pp. 517-522.
19. B. JAKUBCZYK AND W. RESPONDEK, Feedback classification of analytic control systems in the plane, in Analysis of Controlled Dynamical Systems, B. Bonnard et al., eds., Birkhäuser, Boston, 1991, pp. 262-273.
20. T. KAILATH, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.
21. W. KANG, Extended controller form and invariants of nonlinear control systems with single input, J. Math. System Estimation Control, 6 (1996), pp. 27-51.
22. W. KANG, Quadratic normal forms of nonlinear control systems with uncontrollable linearization, in Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, 1995, pp. 608-612.
23. W. KANG, Bifurcation and normal form of nonlinear control systems, Part I, SIAM J. Control Optim., 36 (1998), pp. 193-212.
24. W. KANG, Bifurcation and normal form of nonlinear control systems, Part II, SIAM J. Control Optim., 36 (1998), pp. 213-232.
25. W. KANG, Normal form, invariants, and bifurcations of nonlinear control systems in the particle deflection plane, in Dynamics, Bifurcations and Control, Lecture Notes in Control and Inform. Sci. 273, F. Colonius and L. Grüne, eds., Springer-Verlag, Berlin, Heidelberg, 2002, pp. 67-87.
26. W. KANG AND A.J. KRENER, Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems, SIAM J. Control Optim., 30 (1992), pp. 1319-1337.
27. A.J. KRENER, Approximate linearization by state feedback and coordinate change, Systems Control Lett., 5 (1984), pp. 181-185.
28. I. KUPKA, On feedback equivalence, in Differential Geometry, Global Analysis, and Topology, CMS Conf. Proc. 12, AMS, Providence, RI, 1991, pp. 105-117.
29. 3, in Geometry of Feedback and Optimal Control, B. Jakubczyk and W. Respondek, eds., Marcel Dekker, New York, 1998, pp. 347-382.
30. W. RESPONDEK AND I.A. TALL, How many symmetries does admit a nonlinear single-input control system around an equilibrium?, in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, 2001, pp. 1795-1800.
31. W. RESPONDEK AND I.A. TALL, Nonlinearizable single-input control systems do not admit stationary symmetries, Systems Control Lett., 46 (2002), pp. 1-16.
32. W. RESPONDEK AND M. ZHITOMIRSKII, Feedback classification of nonlinear control systems on 3-manifolds, Math. Control Signals Systems, 8 (1995), pp. 299-333.
33. eme cycle, Université de Dakar, Senegal, 1999.
34. I.A. TALL AND W. RESPONDEK, Normal forms, canonical forms, and invariants of single single-input control systems under feedback, in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, 2000, pp. 1625-1630.
35. I.A. TALL AND W. RESPONDEK, Transforming a single-input nonlinear system to a feedforward form via feedback, in Nonlinear Control in the Year 2000, Vol. 2, Lecture Notes in Control and Inform. Sci. 259, A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek, eds., Springer-Verlag, London, 2000, pp. 527-542.
36. I.A. TALL AND W. RESPONDEK, Feedback equivalence to feedforward forms for nonlinear single-input control systems, in Dynamics, Bifurcations and Control, Lecture Notes in Control and Inform. Sci. 273, F. Colonius and L. Grüne, eds., Springer-Verlag, Berlin, Heidelberg, 2002, pp. 269-286.
37. I.A. TALL AND W. RESPONDEK, Feedback equivalence to feedforward forms for nonlinear single-input systems, in Dynamics, Bifurcations, and Control, Lecture Notes in Control and Inform. Sci. 273, F. Colonius and L. Grüne, eds., Springer-Verlag, Berlin, Heidelberg, 2002, pp. 269-286.
38. M. ZHITOMIRSKII AND W. RESPONDEK, Simple germs of corank one affine distributions, in Singularities Symposium - Łojasiewicz 70, Banach Center Pub. 44, B. Jakubczyk, W. Pawlucki, and J. Stasica, eds., Polish Acad. Sci., Warsaw, 1998, pp. 269-276.