A transformed Lure problem for sliding mode control and chattering reduction

The sliding mode control problem is investigated with the frequency domain approach. The authors show that the sliding mode control can be transformed into a Lure problem. With this formulation, it is shown that the conventional sliding mode control with a nonlinear sign function can only guarantee to make the overall system stable but not asymptotically stable. The authors also consider a practical situation where the sign nonlinearity may contain a hysteresis loop. For this situation, they show that even if the hysteresis loop is very small, there still exists a limit cycle. However, chattering phenomenon caused by the limit cycle behavior can be eliminated by a certain class of nonlinearities