Sliding Mode Control and Feedback Linearization for Non-Regular Systems

This paper presents an approach for sliding mode control(SMC) and feedback linearization(FBL) of systems with relative order singularities. Traditionally, SMC and FBL are designed by taking derivatives of the output until the control signal appears(at the r 1 derivative). When a system does not have a well-defined relative degree (r1), the coefficient that multiplies u vanishes for some region of the state-space S1. In this instance, conventional SMC and FBL techniques fail. The presented approach differentiates further the output until the control input appears again (the secondary relative degree, r2) and a differential equation in u is acquired. It may be possible to solve for a dynamic compensator, or in the neighborhood of the singularity, N1, the equations degenerate to a polynomial form. Preliminary results show that at the singularity region, S1, the control-derivative term disappears and the differential equation is degenerated to a center manifold defined by a polynomial (quadratic in general) equation on u. The solution to the quadratic equations is discussed. When this equation has only real roots, the system is well defined at the singularity. A switching controller can be designed to switch from the r 1 controller when system is far away from the singularity to the r th 2 controller when the system is in the neighborhood of the singularity. We demonstrate the controller applied to the ball and beam system.