Proper Implicit Discretization of the Super-Twisting Controller -- without and with Actuator Saturation

The discrete-time implementation of the super-twisting sliding mode controller for a plant with disturbances with bounded slope, zero-order hold actuation, and actuator constraints is considered. Motivated by restrictions of existing implicit or semi-implicit discretization variants, a new proper implicit discretization for the super-twisting controller is proposed. This discretization is then extended to the conditioned super-twisting controller, which mitigates windup in presence of actuator constraints by means of the conditioning technique. It is proven that the proposed controllers achieve best possible worst-case performance subject to similarily simple stability conditions as their continuous-time counterparts. Numerical simulations and comparisons demonstrate and illustrate the results.