A Proportional–Integral–Derivative type regulation scheme with non-Lipschitz control actions for mechanical systems with constrained inputs

Motivated by the benefits that recent finite-time continuous control approaches have proven to give rise, this work aims to design a proportional–integral–derivative (PID) type control scheme for the global regulation of constrained-input mechanical systems that incorporates design features characteristic of such finite-time continuous algorithms. This is proven to be achieved through a more general PID type control structure that incorporates exponential weights on the P and D type terms, through which such control actions are permitted to loose Lipschitz-continuity at the desired equilibrium values. This entails an important challenge consisting on the introduction of an appropriate analytical framework and the development of a suitable closed-loop analysis through which the resulting design is properly supported. The study is complemented by experimental tests which show that appropriate (less-than-unity) values on the incorporated exponential weights indeed give rise to closed-loop improvements characteristic of finite-time continuous control approaches, such as reduction of overshoot on the position error responses and of the control effort, alleviating such a performance adjustment task.