Optimal terminal sliding-mode control for second-order motion systems

Terminal sliding mode (TSM) control algorithm and its non-singular refinement have been elaborated for two decades and belong, since then, to a broader class of the finite-time controllers, which are known to be robust against the matched perturbations. While TSM manifold allows for different forms of the sliding variable, which are satisfying the $q/p$ power ratio of the measurable output state, we demonstrate that $q/p=0.5$ is the optimal one for the second-order Newton's motion dynamics with a bounded control action. The paper analyzes the time-optimal sliding surface and, based thereupon, claims the optimal TSM control for the second-order motion systems. It is stressed that the optimal TSM control is fully inline with the Fuller's problem of optimal switching which minimizes the settling time, i.e. with time-optimal control of an unperturbed double-integrator. Is is also shown that for the given plant characteristics, i.e. the overall inertia and control bound, there is no need for additional control parameters. The single surface design parameter might (but not necessarily need to) be used for driving system on the boundary layer of the twisting mode, or for forcing it to the robust terminal sliding mode. Additional insight is given into the finite-time convergence of TSM and robustness against the bounded perturbations. Numerical examples with different upper-bounded perturbations are demonstrated.