The main obstacles for application of Sliding Mode Control are two interconnected phenomena: chattering and high activity of control action. It is well known that the amplitude of chattering is proportional to the magnitude of a discontinuous control. These two problems can be handled simultaneously if the magnitude is reduced to a minimal admissible level defined by the conditions for the sliding mode to exist. Here an adaptation methodology is discussed for obtaining the minimum possible value of control based on two approaches developed in recent publications:
The σ - adaptation
, providing adequate adjustments of the magnitude of a discontinuous control within the “reaching phase”, that is, when the state trajectories are out of a sliding surface;
or the adaptation within a sliding mode (on a sliding surface), based on the, so-called,
obtained by the direct measurements of the output signals of a first-order low-pass filter containing in the input the discontinuous control with the specially adapted magnitude value.
The application of these methodologies is illustrated by the corresponding adaptive versions of the
controller. It is shown that the adaptation based on the equivalent control application enables reducing of the control action magnitude to the minimum possible value keeping the property of finite-time convergence.