| Whenever two mechanical surfaces in a dynamical system are forced together in direct contact, a friction force is induced at the point of contact which acts to resist the relative motion of the surfaces. The result is often a stick-slip friction force that is highly nonlinear, and generally has an undesirable influence on the system dynamics. Its primary function is to remove useful energy from the system. With a representative model of this phenomena, control schemes can be developed and evaluated to compensate for the undesirable effects. In this dissertation, a model is presented which, when incorporated into a dynamic system model, will provide the stick-slip dynamics observed in practice. Using this model, classical control techniques are evaluated for their effectiveness.; A Proportional+Derivative (PD) control law is the simplest classical control law to regulate the position of a one-degree-of-freedom (1-DOF) system. A stable set of multiple equilibrium points is found to exist for the 1-DOF system under PD control, and the steady-state error is guaranteed to be bounded. Integral control (PID) is added in an attempt to remove the steady-state error, but certain types of slipping force models are shown to promote the generation of limit cycles. Neither of these classical techniques are able to effectively regulate the position of the 1-DOF system.; A nonlinear compensation force for stick-slip friction is developed to supplement a PD control law applied to the 1-DOF mechanical system. The choice of a discontinuous compensation force is motivated by the requirement that the desired reference be a unique equilibrium point of the system. The stick-slip friction force, modelled with a sticking force and a slipping force, generates discontinuous state derivatives. A Lyapunov function is introduced to prove global asymptotic stability of the desired reference using a modification of the direct method for discontinuous systems. Stability is verified numerically as well as experimentally. The nonlinear compensation force is robust with respect to the character of the slipping force which is assumed to lie within piecewise linear bounds. Exact knowledge of the static friction force levels is not required, only upper bounds for these levels. |
