On introducing integral action in sliding mode control

An ideal sliding mode control (SMC) can guarantee asymptotic tracking with zero steady-state error for a wide class of nonlinear systems. When implemented using continuous approximations, the controller can no longer guarantee zero steady-state error. Instead, it guarantees ultimate boundedness with an ultimate bound that is proportional to the width of the boundary layer. To reduce the steady-state error, we have to reduce the width of the boundary layer, but a too small boundary layer gets us back into chattering and excitation of high-frequency modes, which are the reasons why the ideal SMC was approximated in the first place. For constant, approximately constant, or eventually constant exogenous signals, we can achieve zero steady-state error by introducing integral action in the controller. The traditional way of introducing integral action usually leads one to deterioration of the transient response of the system. In this paper we show how to introduce integral action in such a way as to recover the transient performance of ideal SMC while achieving zero steady-state error. We give both regional as well as global results.