Differential flatness and defect: an overview

Fliess, Michel ; Lévine, Jean ; Martin, Philippe ; Rouchon, Pierre

Banach Center Publications, Tome 31 (1995), p. 209-225 / Harvested from The Polish Digital Mathematics Library

Résumé

We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra which suits well to the fact that, in accordance with Willems' standpoint, flatness and defect are best defined without distinguishing between input, state, output and other variables. We treat an example of non-flat system, the variable-length pendulum. A high frequency control strategy is proposed such that the averaged system becomes flat.

Détail

Publié le : 1995-01-01

Zbl 1010.93504

EUDML-ID : urn:eudml:doc:262868

Bibliographie

[000] [1] J. Baillieul, Stable average motions of mechanical systems subject to periodic forcing, reprint, 1993. | Zbl 0793.70020

[001] [2] J. Bentsman, Vibrational control of a class of nonlinear multiplicative vibrations, IEEE Trans. Automat. Control 32 (1987), 711-716. | Zbl 0631.93056

[002] [3] A. Bressan and F. Rampazzo, On differential systems with quadratic impulses and their applications to Lagrangian mechanics, SIAM J. Control Optim. 31 (1993), 1205-1220. | Zbl 0780.34055

[003] [4] E. Cartan, Sur l'intégration de certains systèmes indéterminés d'équations différentielles, J. Reine Angew. Math. 145 (1915), 86-91; also in: Oeuvres Complètes, part II, vol. 2, CNRS, Paris, 1984, 1164-1174. | Zbl 45.0472.03

[004] [5] P. J. Cassidy, Differential algebraic groups, Amer. J. Math. 94 (1972), 891-954. | Zbl 0258.14013

[005] [6] B. Charlet, J. Lévine and R. Marino, On dynamic feedback linearization, Systems Control Letters 13 (1989), 143-151. | Zbl 0684.93043

[006] [7] B. Charlet, J. Lévine and R. Marino, Sufficient conditions for dynamic state feedback linearization, SIAM J. Control Optim. 29 (1991), 38-57. | Zbl 0739.93021

[007] [8] D. Claude, Everything you always wanted to know about linearization, in: M. Fliess and M. Hazewinkel (ed.), Algebraic and Geometric Methods in Nonlinear Control Theory, Reidel, Dordrecht, 1986, 181-226. | Zbl 0607.93027

[008] [9] P. M. Cohn, Free Rings and their Relations, 2nd ed., Academic Press, London, 1985. | Zbl 0659.16001

[009] [10] J. M. Coron, Linearized control systems and applications to smooth stabilization, SIAM J. Control Optim. 1994. | Zbl 0796.93097

[010] [11] E. Delaleau et M. Fliess, Algorithme de structure, filtrations et découplage, C.R. Acad. Sci. Paris Sér. I 315 (1992), 101-106. | Zbl 0791.68113

[011] [12] M. D. Di Benedetto, J. W. Grizzle and C. H. Moog, Rank invariants of nonlinear systems, SIAM J. Control Optim. 27 (1989), 658-672. | Zbl 0696.93033

[012] [13] S. Diop, Elimination in control theory, Math. Control Signals Systems 4 (1991), 17-32. | Zbl 0727.93025

[013] [14] S. Diop, Differential-algebraic decision methods and some applications to system theory, Theoret. Comput. Sci. 98 (1992), 137-161. | Zbl 0768.93014

[014] [15] M. Fliess, Automatique et corps différentiels, Forum Math. 1 (1989), 227-238. | Zbl 0701.93048

[015] [16] M. Fliess, Generalized controller canonical forms for linear and nonlinear dynamics, IEEE Trans. Automat. Control 35 (1990), 994-1001. | Zbl 0724.93010

[016] [17] M. Fliess, Some basic structural properties of generalized linear systems, Systems Control Letters 15 (1990), 391-396. | Zbl 0727.93024

[017] [18] M. Fliess, A remark on Willems' trajectory characterization of linear controllability, ibid. 19 (1992), 43-45. | Zbl 0765.93003

[018] [19] M. Fliess and S. T. Glad, An algebraic approach to linear and nonlinear control, in: H. J. Trentelman and J. C. Willems (eds.), Essays on Control: Perspectives in the Theory and its Applications, Birkhäuser, Boston, 1993, 223-267. | Zbl 0838.93021

[019] [20] M. Fliess and M. Hasler, Questioning the classical state space description via circuit examples, in: M. A. Kashoek, J. H. van Schuppen, and A. C. M. Ran (eds.), Realization and Modelling in System Theory, MTNS'89, volume I, Birkhäuser, Boston, 1990, 1-12.

[020] [21] M. Fliess, J. Lévine, P. Martin and P. Rouchon, On differentially flat nonlinear systems, in: Proc. IFAC-Symposium NOLCOS'92, Bordeaux, 1992, 408-412. | Zbl 0776.93038

[021] [22] M. Fliess, J. Lévine, P. Martin and P. Rouchon, Sur les systèmes non linéaires différentiellement plats, C. R. Acad. Sci. Paris Sér. I 315 (1992), 619-624. | Zbl 0776.93038

[022] [23] M. Fliess, J. Lévine, P. Martin and P. Rouchon, Défaut d'un système non linéaire et commande haute fréquence, ibid. 316 (1993), 513-518. | Zbl 0777.93044

[023] [24] M. Fliess, J. Lévine, P. Martin and P. Rouchon, Linéarisation par bouclage dynamique et transformations de Lie-Bäcklund, ibid. 317 (1993), 981-986. | Zbl 0796.93042

[024] [25] M. Fliess, J. Lévine, P. Martin and P. Rouchon, Towards a new differential geometric setting in nonlinear control, in: Proc. Internat. Geometric Coll., Moscow, May 1993. | Zbl 0885.93014

[025] [26] M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of nonlinear systems: introductory theory and examples, Internat. J. Control, 1995. | Zbl 0838.93022

[026] [27] M. Fliess, J. Lévine and P. Rouchon, A simplified approach of crane control via a generalized state-space model, in: Proc. 30th IEEE Control Decision Conf., Brighton, 1991, 736-741.

[027] [28] M. Fliess, J. Lévine, and P. Rouchon, A generalized state variable representation for a simplified crane description, Internat. J. Control 58 (1993), 277-283. | Zbl 0782.93049

[028] [29] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983. | Zbl 0515.34001

[029] [30] R. Hartshorne, Algebraic Geometry, Springer, New York, 1977.

[030] [31] D. Hilbert, Über den Begriff der Klasse von Differentialgleichungen, Math. Ann. 73 (1912), 95-108; also in: Gesammelte Abhandlungen, Vol. III, Chelsea, New York, 1965, 81-93.

[031] [32] A. Isidori, Control of nonlinear systems via dynamic state feedback, in: M. Fliess and M. Hazewinkel (eds.), Algebraic and Geometric Methods in Nonlinear Control Theory, Reidel, 1986.

[032] [33] A. Isidori, Nonlinear Control Systems, 2nd ed., Springer, New York, 1989.

[033] [34] A. Isidori, C. H. Moog, and A. De Luca, A sufficient condition for full linearization via dynamic state feedback, in: Proc. 25th IEEE Conf. Decision Control, 1986, 203-208.

[034] [35] N. Jacobson, Basic Algebra, I and II, 2nd ed., Freeman, New York, 1985. | Zbl 0557.16001

[035] [36] B. Jakubczyk, Remarks on equivalence and linearization of nonlinear systems, in: Proc. IFAC-Symposium NOLCOS'92, Bordeaux, 1992, 393-397.

[036] [37] B. Jakubczyk, Invariants of dynamic feedback and free systems, in: Proc. ECC'93, Groningen, 1993, 1510-1513.

[037] [38] J. Johnson, Kähler differentials and differential algebra, Ann. of Math. 89 (1969), 92-98. | Zbl 0179.34302

[038] [39] J. Johnson, Order for systems of differential equations and a generalization of the notion of differential ring, J. Algebra 78 (1982), 91-119. | Zbl 0496.12019

[039] [40] T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, N.J., 1980.

[040] [41] E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973. | Zbl 0264.12102

[041] [42] E. R. Kolchin, Differential Algebraic Groups, Academic Press, Orlando, 1985. | Zbl 0556.12006

[042] [43] C.-W. Li and Y.-K. Feng, Functional reproducibility of general multivariable analytic nonlinear systems, Internat. J. Control 45 (1987), 255-268. | Zbl 0611.93038

[043] [44] P. Martin, Contribution à l'étude des systèmes diffèrentiellement plats, PhD thesis, École des Mines de Paris, 1992.

[044] [45] S. M. Meerkov, Principle of vibrational control: theory and applications, IEEE Trans. Automat. Control 25 (1980), 755-762. | Zbl 0454.93021

[045] [46] H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems, Springer, New York, 1990. | Zbl 0701.93001

[046] [47] J. B. Pomet, A differential geometric setting for dynamic equivalence and dynamic linearization, this volume, 319-339. | Zbl 0838.93019

[047] [48] J. F. Ritt, Systems of differential equations i. theory of ideals, Amer. J. Math. 60 (1938), 535-548. | Zbl 0019.11601

[048] [49] J. F. Ritt, Differential Algebra, Amer. Math. Soc., New York, 1950.

[049] [50] A. Seidenberg, Some basic theorems in differential algebra (characteristic p, arbitrary), Trans. Amer. Math. Soc. 73 (1952), 174-190. | Zbl 0047.03502

[050] [51] W. F. Shadwick, Absolute equivalence and dynamic feedback linearization, Systems Control Letters 15 (1990), 35-39. | Zbl 0704.93037

[051] [52] E. D. Sontag, Finite dimensional open loop control generator for nonlinear control systems, Internat. J. Control 47 (1988), 537-556. | Zbl 0641.93035

[052] [53] E. D. Sontag, Universal nonsingular controls, Systems Control Letters 19 (1992), 221-224.

[053] [54] H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations 12 (1972), 95-116. | Zbl 0242.49040

[054] [55] H. J. Sussmann and W. Liu, Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories, in: Proc. 30th IEEE Control Decision Conf., Brighton, 1991, 437-442.

[055] [56] J. C. Willems, Paradigms and puzzles in the theory of dynamical systems, IEEE Trans. Automat. Control 36 (1991), 259-294. | Zbl 0737.93004

[056] [57] D. J. Winter, The Structure of Fields, Springer, New York, 1974. | Zbl 0292.12101

[057] [58] V. V. Zharinov, Geometrical Aspect of Partial Differential Equations, World Scientific, Singapore, 1992. | Zbl 0763.58002