Smooth Feedback, Global Stabilization, and Disturbance Attenuation of Nonlinear Systems with Uncontrollable Linearization

Problems of global asymptotic stabilization and disturbance attenuation are addressed for a class of highly nonlinear systems that are comprised of a lower dimensional zero-dynamics subsystem and a chain of power integrators perturbed by a nontriangular vector field. It is shown in this paper that global stabilization and disturbance attenuation are solvable by smooth state feedback if one takes full advantage of the characteristics of the system in the feedback design to dominate the nonlinearity rather than to cancel it. A systematic design procedure which is based upon, but generalizes, the recent technique of adding a power integrator is developed for the explicit construction of the smooth controllers. Several examples are presented to demonstrate the key features of the proposed nonlinear control schemes.

[1]  H. Sussmann.  Limitations on the stabilizability of globally-minimum-phase systems , 1990 .

[2]  Riccardo Marino,et al.  Nonlinear control design , 1995 .

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

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

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

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

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

[8]  A. Isidori,et al.  New results and examples in nonlinear feedback stabilization , 1989 .

[9]  C. Qian,et al.  Almost disturbance decoupling for a chain of power integrators perturbed by a lower-triangular vector field , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[10]  John Tsinias,et al.  Sufficient lyapunov-like conditions for stabilization , 1989, Math. Control. Signals Syst..

[11]  A. Isidori.  Global almost disturbance decoupling with stability for non minimum-phase single-input single-output , 1996 .

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

[13]  Wei Lin,et al.  Global L 2-gain design for a class of nonlinear systems 1 1 Supported in part by NSF under grant ECS-9412340 and by MURST. , 1998 .

[14]  Miroslav Krstic,et al.  Nonlinear and adaptive control de-sign , 1995 .

[15]  Wei Lin,et al.  Robust passivity and feedback design for minimum-phase nonlinear systems with structural uncertainty , 1999, Autom..

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

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

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

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

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

[21]  Wei Lin,et al.  Global Robust Stabilization of Minimum-Phase Nonlinear Systems with Uncertainty , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

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

[23]  Arjan van der Schaft,et al.  Non-linear dynamical control systems , 1990 .

[24]  A. Schaft.  L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences 218 , 1996 .