Free-will Arbitrary Time Terminal Sliding Mode Control

In this technical note, free-will arbitrary time terminal sliding mode control design is proposed. With such a design, it is possible to control the time duration of sliding in the sliding phase. Under the assumption that the chosen arbitrary time is greater than the reaching phase time ( $t_{r}$ ), it is shown that the $n^{\textrm {th}}$ order chain of integrators converges to zero within the selected arbitrary time. An algorithm is also proposed for the case when $t_{r}$ is not known to the designer. Stability analysis based on Lyapunov theory has been provided.