On control of servo systems affected by friction forces

We consider the position control of a servo-system containing friction. In designing a control law which compensates the effects caused by friction the first step is to choose an appropriate friction model. The friction force strongly depends on the relative velocity, and for different ranges of relative velocity, the characteristics of the friction force are quite different. The classical methods divide the friction into a stiction zone for zero velocity and Coulomb and viscous friction for nonzero velocity. However, this model fails to predict the low frequency oscillations, say, in a precision servo task, and is difficult to simulate. In order to solve these problems, Karnopp´s model is proposed where the stiction zone is broadened to an interval around zero velocity. In simulations Karnopp´s model has successfully predicted the low frequency oscillations. We show that by using Karnopp´s model, most conventional continuous feedback control laws can not be used to solve a set-point control problem. One possible explanation is that in Karnopp´s model several aspects of friction are not included. Another is that this problem in nature does not have a continuous solution. In fact, a discontinuous control law based on Karnopp´s model was designed by Baril (1993). The control law has been successfully implemented in a servo motor system and no limit cycle is observed in the closed-loop system during the experiment