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. [ABSTRACT FROM AUTHOR]