The frequency-domain analysis of discrete variable structure control and chattering-free criteria

Conventionally, the discrete variable structure control always suffers from the chattering phenomenon. Therefore, in this paper, we shall show that, for the discrete variable structure control, there always exist a limit cycle if the traditional nonlinearity of the sign function is used. This limit cycle behaviour is just the chattering phenomenon in the variable structure control theory. In particular, the frequency of this limit cycle is identical to the Nyquist frequency in the sampling theorem. In addition, a theorem related to the chattering-free design for the multiple-input multiple-output variable structure control is proposed. If the nonlinear function is memoryless and belongs to the sector 0,2), the chattering phenomenon can be eliminated. If the width of the boundary layer is enough, the reaching condition as well as chattering-free design can be attained; otherwise, it will result in the chattering phenomenon.