Abstract
We study the problem of transforming a single-input nonlinear control system to a strict feedforward form via a static state feedback. We provide checkable necessary and sufficient conditions to bring the homogeneous terms of any fixed degree of the system into a homogeneous strict feedforward form. If those conditions are satisfied, this leads to a constructive procedure which transforms the system, step by step, into a strict feedforward form. We explain our method by showing how it works for terms of degree three. We also illustrate it by analyzing three-dimensional systems.
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References
V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Second Edition, Springer-Verlag, 1988.
A. Astolfi and F. Mazenc, A geometric characterization of feedforward forms, in Proc. MTNS'2000, Perpignan, France, 2000.
S. Battilotti, Semiglobal stabilization of uncertain block-feedforward systems via measurement feedback, in Proc. of NOLCOS'98, Enschede, the Netherlands, 1998, pp. 342–347.
M. Fliess, J. Lévine, P. Martin, and P. Rouchon, Flatness and defect of nonlinear systems: Introductory theory and examples, International Journal of Control, 6 (1995), pp. 1327–1361.
M. Jankovic, R. Sepulchre, and P. Kokotovic, Constructive Lyapunov stabilization of nonlinear cascade systems, IEEE Trans. Automat. Control, 41 (1996), pp. 1723–1735.
T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.
W. Kang, Extended controller form and invariants of nonlinear control systems with single input, J. of Mathem. Systems, Estimat. and Control, 4 (1994), pp.253–256.
W. Kang and A.J. Krener, Extended quadratic controller normal form and dynamic feedback linearization of nonlinear systems, SIAM J. Control and Optim., 30 (1992), pp. 1319–1337.
A.J. Krener, Approximate linearization by state feedback and coordinate change, Systems Control Letters, 5 (1984), pp. 181–185.
A. Marigo, Constructive necessary and sufficient conditions for strict triangularizability of driftless nonholonomic systems,in Proc. 34 th CDC, Phoenix, Arizona, USA, 1999, pp. 2138–2143.
F. Mazenc and L. Praly, Adding integrations, saturated controls, and stabilization for feedforward systems, IEEE Trans. Automat. Control, 41 (1996), pp. 1559–1578.
F. Mazenc and L. Praly, Asymptotic tracking of a reference state for systems with a feedforward structure, Automatica, 36 (2000), pp. 179–187.
Sepulchre R., Janković M., and Kokotović P. Constructive Nonlinear Control, Springer, Berlin-Heidelberg-New York, 1996.
I.A. Tall and W. Respondek, Transforming nonlinear single-input control systems to normal forms via feedback, in Proc. MTNS'2000, Perpignan, France, 2000.
I.A. Tall and W. Respondek, Feedback equivalence to a strict feedforward form for nonlinear single-input systems, in preparation.
A. Teel, Feedback stabilization: nonlinear solutions to inherently nonlinear problems, Memorandum UCB/ERL M92/65.
A. Teel, A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE Trans Autom Control, 41 (1996), pp. 1256–1270.
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© 2001 Springer-Verlag London Limited
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Tall, I.A., Respondek, W. (2001). Transforming a single-input nonlinear system to a strict feedforward form via feedback. In: Isidori, A., Lamnabhi-Lagarrigue, F., Respondek, W. (eds) Nonlinear control in the year 2000 volume 2. Lecture Notes in Control and Information Sciences, vol 259. Springer, London. https://doi.org/10.1007/BFb0110323
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DOI: https://doi.org/10.1007/BFb0110323
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