approximate functions

Many of the physical phenomena, like friction, backlash, drag, and etc., which appear in mechanical systems are inherently nonlinear and have destructive effects on the control systems behavior. Generally, they are modeled by hard nonlinearities. In this paper, two different methods are proposed to cope with the effects of hard nonlinearities which exist in friction various models. Simple inverted pendulum on a cart (SIPC) is considered as a test bed system, too. In the first technique, a nonlinear optimal controller based on approximate solution of Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) is designed for the system and finally an adaptive anti disturbance technique is proposed to eliminate the friction destructive effects. In the second one, three continuous functions are used to approximate hard nonlinearities when they are augmented to system model. These techniques are compared with each other using simulations and their effectiveness are shown.