Relaxed and hybridized backstepping

In the present work, we consider nonlinear control systems for which there exist structural obstacles to the design of classical continuous backstepping feedback laws. We conceive 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. A design method is suggested to build a hybrid feedback law combining a backstepping controller with a locally stabilizing controller. A constructive approach is also suggested employing a differential inclusion representation of the nonlinear dynamics. The results are illustrated for a nonlinear system which, due to its structure, does not have {\it a priori} any globally stabilizing backstepping controller.