Bounded control based on saturation functions of nonlinear under-actuated mechanical systems: The cart-pendulum system case

We are concerned in this paper by bounded control of nonlinear underactuated dynamical systems. We focus our exposition on a feedback-based stabilization bounded control action shaped by saturation functions. A simple stabilizing controller for the well-known cart-pendulum system is then designed in this paper. Our control strategy describes in lumped linear time-invariant terms the concerned underactuated nonlinear system as a cascade nonlinear dynamical system consisted of a simple chain of four integrators with a high-order smooth nonlinear perturbation, and assumes initialization of the resulting underactuated system in the upperhalf plane. Our proposed feedback-based regulation design procedure involves the simultaneous combination of two control actions: one bounded linear and one bounded quasilinear. Control boundedness is provided in both involved control actions by specific saturation functions. The first bounded control action brings the non-actuated coordinate near to the upright position and keep it inside of a well-characterized small vicinity, whereas the second bounded control action asymptotically brings the whole state of the dynamical system to the origin. The necessary closed-loop stability analysis uses standard linear stability arguments as well as the traditional well-known Lyapunov method and the LaSalle's theorem. Our proposed control law ensures global stability of the closed-loop system in the upper half plane, while avoiding the necessity of solving either partial differential equations, nonlinear differential equations or fixed-point controllers. We illustrate the effectiveness of the proposed control strategy via numerical simulations.