Non-smooth stabilizers for nonlinear systems with uncontrollable unstable linearization

We prove that every chain of odd power integrators perturbed by a C/sup 1/ triangular vector field can be stabilized in the large via continuous state feedback, although it is not stabilizable, even locally, by any smooth state feedback. The proof is constructive and accomplished by developing a machinery-a continuous type of adding a power integrator-that enables one to explicitly design a C/sup 0/ globally stabilizing feedback law as well as a C/sup 1/ control Lyapunov function which is positive definite and proper.

[1]  L. Rosier.  Homogeneous Lyapunov function for homogeneous continuous vector field , 1992 .

[2]  Henry Hermes,et al.  Nilpotent and High-Order Approximations of Vector Field Systems , 1991, SIAM Rev..

[3]  Andrea Bacciotti,et al.  Local Stabilizability of Nonlinear Control Systems , 1991, Series on Advances in Mathematics for Applied Sciences.

[4]  H. Sussmann.  A general theorem on local controllability , 1987 .

[5]  M. Kawski.  Stabilization of nonlinear systems in the plane , 1989 .

[6]  Wolfgang Hahn,et al.  Stability of Motion , 1967 .

[7]  Eduardo D. Sontag,et al.  FEEDBACK STABILIZATION OF NONLINEAR SYSTEMS , 1990 .

[8]  M. Kawski.  Homogeneous Stabilizing Feedback Laws , 1990 .

[9]  Eduardo Aranda-Bricaire,et al.  Constructive nonsmooth stabilization of triangular systems , 1999 .

[10]  W. P. Dayawansa,et al.  Asymptotic stabilization of a class of smooth two-dimensional systems , 1990 .

[11]  J. Coron,et al.  Adding an integrator for the stabilization problem , 1991 .

[12]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[13]  Wei Lin,et al.  Robust regulation of a chain of power integrators perturbed by a lower‐triangular vector field , 2000 .

[14]  Gianna Stefani.  Polynomial approximations to control systems and local controllability , 1985, 1985 24th IEEE Conference on Decision and Control.

[15]  Wei Lin,et al.  Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems , 2000 .

[16]  W. P. Dayawansa,et al.  Recent Advances in The Stabilization Problem for Low Dimensional Systems , 1992 .

[17]  J. Tsinias,et al.  Explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization , 1999 .