Asymptotic harmonic generator and its application to finite time orbital stabilization of a friction pendulum with experimental verification

A second order sliding mode approach to orbital stabilization is presented and tested on a friction pendulum, operating under uncertain conditions. The quasihomogeneous control synthesis is utilized to design a variable structure controller that drives the pendulum to a model orbit in finite time in spite of the presence of external disturbances with an a priori known magnitude bound. A well-known Van der Pol oscillator is modified to be introduced into the synthesis as a reference model. This modification is made to shape the oscillator limit cycle to a harmonic one, thereby yielding an asymptotic harmonic generator of the periodic motion. The parameters of the asymptotic harmonic generator are shown to specify amplitude and frequency of the limit cycle production and damping of non-linear oscillations of the generator. The resulting closed-loop system is capable of moving from one orbit to another by changing these parameters dynamically. Performance issues of the controller constructed are illustrated in an experimental study.