Time-varying sliding modes for second-order systems

Three types of optimal, continuously time-varying sliding mode for robust control of second-order uncertain dynamic systems subject to input constraint are presented. Two of the modes incorporate straight sliding lines, and the third uses the so-called terminal slider, that is a curve that guarantees system error convergence to zero in finite time. At first, all three lines adapt themselves to the initial conditions of the system, and afterwards they move in such a way that, for each of them, the integral of the absolute value of the systems error is minimised over the whole period of the control action. By this means, insensitivity of the system to external disturbances and parameter uncertainties is guaranteed from the very beginning of the proposed control action, and the system error convergence rate can be increased. Performance of the three control algorithms is compared, and the Lyapunov theory is used to prove the existence of a sliding mode on the lines