Smooth global stabilization for a class of nonlinear systems using homogeneity with monotone degrees

This paper considers the problem of designing C1 (continuously differentiable) state feedback stabilizers for a class of 3-dimensional nonlinear systems whose linearizations around the origin may contain uncontrollable modes. Based on a new definition of homogeneity with monotone degrees, we not only propose conditions of constructing C1 and C∞ (smooth) controllers, but also provide explicit design schemes for such systems. Several examples are investigated to show the advantages of the generalized homogeneity compared to the traditional unified homogeneity.

[1]  Riccardo Marino,et al.  Nonlinear control design: geometric, adaptive and robust , 1995 .

[2]  A. Isidori,et al.  Asymptotic stabilization of minimum phase nonlinear systems , 1991 .

[3]  D. Mayne.  Nonlinear and Adaptive Control Design [Book Review] , 1996, IEEE Transactions on Automatic Control.

[4]  A. Isidori.  Nonlinear Control Systems , 1985 .

[5]  H. Shim,et al.  Non-smooth feedback stabilizer for strict-feedback nonlinear systems not even linearizable at the origin , 2005, Proceedings of the 2005, American Control Conference, 2005..

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

[7]  A. Isidori,et al.  H∞ control via measurement feedback for general nonlinear systems , 1995, IEEE Trans. Autom. Control..

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

[9]  Chunjiang Qian,et al.  An expanded method to robustly stabilize uncertain nonlinear systems , 2008, Commun. Inf. Syst..

[10]  Wei Lin,et al.  A continuous feedback approach to global strong stabilization of nonlinear systems , 2001, IEEE Trans. Autom. Control..

[11]  Frédéric Mazenc,et al.  Stabilization of feedforward systems approximated by a non-linear chain of integrators , 1997 .

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

[13]  A. Bacciotti,et al.  Liapunov functions and stability in control theory , 2001 .

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