Continuous Stabilizing of First Order Single Input Nonlinear Systems

Abstract

In this paper, stabilizability of first order nonlinear systems by a smooth control law is investigated. The main results are presented by the examples and finally summarized in a lemma. The proof for the lemma is according to Sontag’s formula. In addition, it is explained that using weak control Lyapunov functions in Sontag’s formula generates (possibly nonsmooth) the control law, which globally stabilizes the system-globally asymptotic stability needs more investigation.

KeywordsControl Lyapunov FunctionInverse OptimalitySontag’s Formula

R. Sepulchre, M. Jankovic and P. V. Kokotovic, “Constructive Nonlinear Control,” Springer Verlag, London, 1997.
J. A. Primbs, V. Nevistic and J. C. Doyle, “A Receding Horizon Generalization of Pointwise Min Norm Controllers,” IEEE Transactions on Automatic Control, Vol. 45, No. 5, May 2000, pp. 898-909. doi:org/10.1109/9.855550
M. Krstic, I. Anellakopoulos and P. V. Kokotovic, “Nonlinear and Adaptive Control Design,” John Wiley & Sons, New York, 1995.
E. D. Sontag, “A ‘Universal’ Construction of Artstein’s Theorem on Nonlinear Stabilization,” System & Control letters, Vol. 13, No. 2, 1989, pp. 117-123. doi:10.1016/0167-6911(89)90028-5
F. Ceragioli, “Some Remarks on Stabilization by Means of Discontinuous Feedbacks,” System & Control Letters, Vol. 45, No. 4, 2002, pp. 271-281. doi:org/10.1016/S0167-6911(01)00185-2
F. A. C. C. Fontes, “Discontinuoud Feedbacks, Discontinuous Optimal Controls, and Continuous-Time Model Predictive Control,” International Journal of Robust Control and Nonlinear Control, Vol. 13, No. 3-4, 2003, pp. 191-209.
R. W. Brockett, R. S. Millman and H. S. Sussmann, “Differential Geometric Control Theory,” Birkhouser, Boston, 1983.
A. Bacciotti and L. Rosier, “Liapunov Functions and Stability in Control Theory,” Springer-Verlag, London, 2001.

Menu

Continuous Stabilizing of First Order Single Input Nonlinear Systems

Abstract

References (8)