Fixed point controllers and stabilization of the cart-pole system and the rotating pendulum

We consider stabilization of nonlinear systems in a special normal form as the cascade of a nonlinear subsystem and a linear subsystem. These systems do not possess any particular triangular structure. Despite this fact, we show how a backstepping type procedure applied to these systems naturally leads to a fixed point equation in the control input. We give conditions for well-posedness of these fixed point equations and show how these fixed points called Fixed Point Controllers (FPC) can be used for stabilization of cascade nonlinear systems. As special cases, we apply our results to semiglobal stabilization of two complex under-actuated nonlinear systems, namely the cart-pole system and the rotating pendulum.