Combining a backstepping controller with a local stabilizer

We consider nonlinear control systems for which there exist some structural obstacles to the design of classical continuous stabilizing feedback laws. More precisely, it is studied systems for which the backstepping tool for the design of stabilizers can not be applied. On the contrary, it leads to feedback laws such that the origin of the closed-loop system is not globally asymptotically stable, but a suitable attractor (strictly containing the origin) is practically asymptotically stable. Then, a design method is suggested to build a hybrid feedback law combining a backstepping controller with a locally stabilizing controller. The results are illustrated for a nonlinear system which, due to the structure of the system, does not have a priori any globally stabilizing backstepping controller.

[1]  Arthur J. Krener,et al.  Backstepping design with local optimality matching , 2001, IEEE Trans. Autom. Control..

[2]  P. Kokotovic,et al.  Constructive Lyapunov stabilization of nonlinear cascade systems , 1996, IEEE Trans. Autom. Control..

[3]  P. Morin,et al.  Robust stabilization of driftless systems with hybrid open-loop/feedback control , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[4]  Julio H. Braslavsky,et al.  Global and semi-global stabilizability in certain cascade nonlinear systems , 1996, IEEE Trans. Autom. Control..

[5]  Alessandro Astolfi,et al.  Robust stabilization of chained systems via hybrid control , 2003, IEEE Trans. Autom. Control..

[6]  Christophe Prieur.  Uniting Local and Global Controllers with Robustness to Vanishing Noise , 2001, Math. Control. Signals Syst..

[7]  R. Freeman,et al.  Robust Nonlinear Control Design: State-Space and Lyapunov Techniques , 1996 .

[8]  P.V. Kokotovic,et al.  The joy of feedback: nonlinear and adaptive , 1992, IEEE Control Systems.

[9]  Christophe Prieur,et al.  Uniting two Control Lyapunov Functions for affine systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[10]  Rafal Goebel,et al.  Solutions to hybrid inclusions via set and graphical convergence with stability theory applications , 2006, Autom..

[11]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[12]  Rafal Goebel,et al.  Smooth patchy control Lyapunov functions , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[13]  R. Sanfelice,et al.  Hybrid dynamical systems , 2009, IEEE Control Systems.

[14]  J. Hespanha,et al.  Hybrid systems: Generalized solutions and robust stability , 2004 .

[15]  J. Coron,et al.  Lyapunov design of stabilizing controllers for cascaded systems , 1991 .

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

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

[18]  P. Kokotovic,et al.  Global stabilization of partially linear composite systems , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[19]  João Pedro Hespanha,et al.  Postprints from CCDC Title Hysteresis-based switching algorithms for supervisory control of uncertain systems Permalink , 2002 .

[20]  Rafal Goebel,et al.  Hybrid Feedback Control and Robust Stabilization of Nonlinear Systems , 2007, IEEE Transactions on Automatic Control.

[21]  Frédéric Grognard,et al.  Global stabilization of feedforward systems with exponentially unstable Jacobian linearization , 1999 .

[22]  Andrew R. Teel,et al.  Trading the stability of finite zeros for global stabilization of nonlinear cascade systems , 2002, IEEE Trans. Autom. Control..

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

[24]  João Pedro Hespanha,et al.  Stabilization of nonholonomic integrators via logic-based switching , 1999, Autom..

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

[26]  Jean-Pierre Aubin,et al.  Impulse differential inclusions: a viability approach to hybrid systems , 2002, IEEE Trans. Autom. Control..

[27]  A. F. Filippov.  Equations with the Right-Hand Side Continuous in x and Discontinuous in t , 1988 .

[28]  Mrdjan J. Jankovic,et al.  Constructive Nonlinear Control , 2011 .

[29]  Rodolphe Sepulchre.  Slow peaking and low-gain designs for global stabilization of nonlinear systems , 2000, IEEE Trans. Autom. Control..

[30]  Joao P. Hespanha.  Stabilization of Nonholonomic Integrators via Logic-Based Switching , 1996 .

[31]  P. Kokotovic,et al.  The peaking phenomenon and the global stabilization of nonlinear systems , 1991 .

[32]  J.P. Hespanha,et al.  Towards the supervisory control of uncertain nonholonomic systems , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).