Global stabilization of a class of upper‐triangular systems with unbounded or uncontrollable linearizations

In this paper, we consider the problem of global stabilization for a class of upper-triangular systems which have unbounded or uncontrollable linearizations around the origin. The explicit formula of the control law is designed in two steps: First, we use the generalized adding a power integrator technique to design a homogeneous controller which locally stabilizes the upper-triangular systems. Then, we integrate a series of nested saturation functions with the homogeneous controller and adjust the saturation level to ensure global asymptotic stability of the closed-loop systems. Owing to the versatility of the generalized adding a power integrator technique, our controller not only can be used to stabilize more general upper-triangular systems by relaxing the current conditions used in existing results, but also is able to lead to a stronger result of finite-time stabilization under appropriate conditions. Copyright © 2010 John Wiley & Sons, Ltd.

[1]  P. Kokotovic,et al.  Global asymptotic stabilization of the ball-and-beam system , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

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

[3]  Lorenzo Marconi,et al.  Input disturbance suppression for a class of feedforward uncertain nonlinear systems , 2002, Syst. Control. Lett..

[4]  Wei-Song Lin,et al.  Synthesis of upper-triangular non-linear systems with marginally unstable free dynamics using state-dependent saturation , 1999 .

[5]  Frédéric Mazenc,et al.  Tracking trajectories of the cart-pendulum system , 2003, Autom..

[6]  C. Qian,et al.  Global output feedback stabilization of upper‐triangular nonlinear systems using a homogeneous domination approach , 2006 .

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

[8]  L. Praly,et al.  Adding integrations, saturated controls, and stabilization for feedforward systems , 1996, IEEE Trans. Autom. Control..

[9]  Dennis S. Bernstein,et al.  Geometric homogeneity with applications to finite-time stability , 2005, Math. Control. Signals Syst..

[10]  Chunjiang Qian,et al.  A universal method for robust stabilization of nonlinear systems: unification and extension of smooth and non-smooth approaches , 2006, 2006 American Control Conference.

[11]  Jie Huang,et al.  Disturbance Attenuation of Feedforward Systems With Dynamic Uncertainty , 2008, IEEE Transactions on Automatic Control.

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

[13]  M. P. Tzamtzi,et al.  An explicit formula of bounded feedback stabilizers for feedforward systems , 2001, Syst. Control. Lett..

[14]  Mrdjan Jankovic,et al.  Integrator forwarding: A new recursive nonlinear robust design , 1997, Autom..

[15]  A. Teel.  A nonlinear small gain theorem for the analysis of control systems with saturation , 1996, IEEE Trans. Autom. Control..

[16]  A. Teel.  Global stabilization and restricted tracking for multiple integrators with bounded controls , 1992 .

[17]  Ye Xudong,et al.  Brief Universal stabilization of feedforward nonlinear systems , 2003 .

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

[19]  Wei Lin,et al.  Using small feedback to stabilize a wider class of feedforward systems 1 , 1999 .

[20]  A. Isidori,et al.  Robust global stabilization of a class of uncertain feedforward nonlinear systems , 2000 .

[21]  Shihua Li,et al.  Global stabilization of feedforward systems with lower-order vector field , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[22]  Nami Nakamura,et al.  Global inverse optimal controller with guaranteed convergence rate for input-affine nonlinear systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[23]  Dennis S. Bernstein,et al.  Finite-Time Stability of Continuous Autonomous Systems , 2000, SIAM J. Control. Optim..