A generalised proportional-derivative type scheme with multiple saturating structure for the finite-time and exponential tracking continuous control of Euler–Lagrange systems with bounded inputs

A generalised proportional-derivative type scheme for the finite-time and exponential tracking continuous control of Euler–Lagrange systems with input constraints is developed. Its generalised design permits the choice among multiple saturating structures. Suitable functions are involved for the appropriate shaping of every error correction term, and complementarily of their addition, to achieve the concerned type of convergence (in addition to the input saturation avoidance). Such a shaping is carried out through control parameters that act as exponential weights on the gained configuration and generalised velocity errors and on their joint action. Compared to previous finite-time approaches, such exponential weights are required to satisfy generalised comparative conditions, avoiding a fixed equivalence relation among any of them. This permits to give rise to a wider spectrum of finite-time convergent closed-loop trajectories and gives an optional type of (unconventional) exponential convergence (in addition to the conventional one). The resulting extended conditions on the exponential weights and the generalised saturating structure enlarge the controller potential for performance improvement or adjustment. The generalised design is supported through a more general analysis based on more general strict Lyapunov functions, stating more solid analytical bases for possible design extensions on the accomplishment of other control objectives. The study is further supported through experimental tests on a 3-degree-of-freedom robot manipulator.