On plant dynamics transformation through addition of equivalent control

Equivalent control is a fundamental notion in sliding mode (SM) control theory. In designs of SM control, the control function is often formulated as the sum of the equivalent and discontinuous (or Lipschitz-discontinuous) control. In the present paper, this control arrangement is referred to as the augmentation of plant dynamics by means of the equivalent control. It is shown in the present paper that for the considered class of systems, that includes linear plants and nonlinear affine in control plants, with linear sliding surface, the above-noted approach results in the input–output dynamics of the augmented plant dynamics being reduced to those of an integrator. This offers a simplified way of convergence and stability analysis, consisting of the following two steps. First, the convergence of the sliding variable in a SM system has to be proved for the SM controller plus integrator system. And second, the stability of internal dynamics of the plant has to be proved. An explanation of the phenomenon of reducing the input–output plant dynamics to those of an integrator is provided. An approach to the analysis of internal stability of the augmented plant is developed.