Control of nonlinear systems on Poincaré section via quasi-sliding mode method

The order of some dynamical systems can be reduced by finding a good Poincaré section for them. In such a case the problem of controlling the main continuous system, can be reduced into controlling the discrete Poincaré map of the system which is usually nonlinear. In this paper, this idea is implemented for a simple nonlinear system. A quasi-sliding mode controller is used to stabilize the unstable fixed point of the discrete system and the control signal obtained, is applied to the main continuous system, where a corresponding limit cycle is observed. The process happens to be useful in understanding the necessary conditions for the existence of periodic responses in the system.