Difference Equation With Minima-Based Discrete-Time Sliding Mode Control

This note proposes a reaching law based on a difference equation with minima, for discrete-time sliding mode control, providing a solution that has properties of modified Gao's reaching law when far away from the origin and Utkin's equivalent control law when close to the origin. On one hand, the proposed approach eliminates chattering and makes the system states stay on the sliding hyperplane perfectly, while on the other hand, it restricts the rate of change of the sliding variable by tuning a design parameter. As a result, the control law derived from this reaching law is not as aggressive as it is for Gao's reaching law, for very large initial conditions. The proposed reaching law based discrete-time sliding mode control is designed for both unperturbed and perturbed systems. For the unperturbed system, the sliding variable is made zero in finite time, and for the perturbed system, the sliding variable remains in the neighborhood of the sliding hyperplane. An example is simulated to show the effectiveness of the proposed reaching law.