On phase deficit of homogeneous sliding mode control

The notion of the phase deficit recently proposed in the literature is determined for systems with homogeneous higher order sliding mode (HOSM) controllers. The previously formulated criterion based on the phase deficit gives necessary conditions of finite-time convergence. A few homogeneous control algorithms of orders from two to five are analyzed. It is shown that, when the phase deficit is positive, the system controlled by an HOSM algorithm is capable of producing finite time convergence. However, under some controller parameters, a homogeneous HOSM controller may produce asymptotic conversion or even nonvanishing oscillations. Necessary conditions for finite-time convergence, infinite-time convergence, and finite-frequency oscillations are obtained through the developed approach.