Orbital Stabilization of Nonlinear Systems via the Immersion and Invariance Technique

Immersion and Invariance is a technique for the design of stabilizing and adaptive controllers and state observers for nonlinear systems recently proposed in the literature. In all these applications the problem is translated into stabilization of equilibrium points. Motivated by some modern applications we show in this paper that the technique can also be used to treat the problem of orbital stabilization, where the final objective is to generate periodic solutions that are orbitally attractive. The feasibility of our result is illustrated with some classical mechanical engineering examples.

[1]  Daniel E. Koditschek,et al.  Hybrid zero dynamics of planar biped walkers , 2003, IEEE Trans. Autom. Control..

[2]  Luca Consolini,et al.  Virtual Holonomic Constraints for Euler-Lagrange Systems , 2010 .

[3]  Alexander L. Fradkov,et al.  Introduction to Control of Oscillations and Chaos , 1998 .

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

[5]  Stefano Stramigioli,et al.  Port-Based Asymptotic Curve Tracking for Mechanical Systems , 2004, Eur. J. Control.

[6]  A. Pritsker,et al.  Smooth , 1974 .

[7]  Romeo Ortega,et al.  Immersion and Invariance Stabilization of Nonlinear Systems Via Virtual and Horizontal Contraction , 2017, IEEE Transactions on Automatic Control.

[8]  P. Olver.  Nonlinear Systems , 2013 .

[9]  Antonio T. Alexandridis,et al.  Studying Complexity for a Modified Dissipative Hamiltonian System: From Lyapunov Stability at the Origin to a Limit Cycle and Chaos , 2013 .

[10]  P. Mattavelli,et al.  Dissipativity-based adaptive and robust control of UPS in unbalanced operation , 2001, 2001 IEEE 32nd Annual Power Electronics Specialists Conference (IEEE Cat. No.01CH37230).

[11]  Carlos Canudas-de-Wit,et al.  Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach , 2005, IEEE Transactions on Automatic Control.

[12]  Javier Aracil,et al.  A family of smooth controllers for swinging up a pendulum , 2008, Autom..

[13]  Alessandro Astolfi,et al.  Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one , 2004, Proceedings of the 2004 American Control Conference.

[14]  Rodolphe Sepulchre,et al.  Analysis of Interconnected Oscillators by Dissipativity Theory , 2007, IEEE Transactions on Automatic Control.

[15]  Alessandro Astolfi,et al.  Nonlinear and adaptive control with applications , 2008 .

[16]  Mark W. Spong,et al.  The swing up control problem for the Acrobot , 1995 .

[17]  Alessandro Astolfi,et al.  Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems , 2001, IEEE Trans. Autom. Control..

[18]  Leonid B. Freidovich,et al.  Transverse Linearization for Controlled Mechanical Systems With Several Passive Degrees of Freedom , 2010, IEEE Transactions on Automatic Control.

[19]  Javier Aracil,et al.  Stabilization of oscillations through backstepping in high-dimensional systems , 2005, IEEE Transactions on Automatic Control.

[20]  Luca Consolini,et al.  Dynamic virtual holonomic constraints for stabilization of closed orbits in underactuated mechanical systems , 2017, Autom..

[21]  Romeo Ortega,et al.  Orbital Stabilization of Nonlinear Systems via Mexican Sombrero Energy Shaping and Pumping-and-Damping Injection , 2019, Autom..

[22]  David J. Hill,et al.  Speed Gradient Control and Passivity of Nonlinear Oscillators , 1995 .

[23]  Romeo Ortega,et al.  Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment , 2002, IEEE Trans. Autom. Control..

[24]  C. Byrnes.  Output Regulation of Uncertain Nonlinear Systems , 1997 .