Experimental evaluation of regulated non-linear under-actuated mechanical systems via saturation-functions-based bounded control: the cart-pendulum system case

The authors are concerned in this study by bounded control of single input non-linear under-actuated mechanical systems. The authors focus the exposition on a feedback-based stabilisation-bounded control action shaped by saturation functions, and the proposed approach was illustrated via the design and the experimental evaluation of a simple stabilising controller for the cart-pendulum system, a well-known control benchmark. The proposed simple control strategy is built around a lumped linear continuous time-invariant description of the concerned under-actuated non-linear system. Namely, a model consisted of a cascade non-linear dynamical system constituted by a chain of four integrators affected by a high-order smooth non-linear perturbation. Assuming initialisation of the under-actuated system to the upper-half plane, the proposed feedback-based regulation design procedure involves the simultaneous combination of two control actions: one bounded linear and one bounded quasi-linear. Control boundedness is provided in both involved control actions by specifically designed saturation functions. The first bounded control action brings the non-actuated coordinate near to the upright position and keep it inside of a well characterised 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 invariance principle. The proposed control law ensures global stability of the closed-loop system in the upper-half plane.