The feasibility issue in trajectory tracking by means of regions-of-attraction-based gain scheduling

Linear control theory has been long established and a myriad of techniques are available for designing controllers for linear systems in view of conflicting performance requirements. On the other hand, nonlinear control techniques are often tailored to specific applications and versatile nonlinear control frameworks are still on their infancy. A common approach is to resort to local linearized descriptions at desired set-points over a given desired trajectory and employ linear tools. Furthermore, to enforce stability when switching controllers, the regions-of-attraction approach has gained recent attention. This paper questions whether such method - when applied to a well-posed smooth nonlinear controllable system - always yields a sequence of controllers that successfully tracks a given reference equilibrium trajectory, and an analytic counter-example is provided and thoroughly discussed. Finally, our case study additionally shed light on how gain scheduling fails to track particular trajectories for certain globally controllable systems.