Control Systems with Friction
Abstract: Friction-related problems are frequently encountered in control systems. This thesis treats three aspects of such problems: modeling, analysis, and friction compensation. A new dynamic friction model is presented and investigated. The model is described by a first order nonlinear differential equation with a reasonable number of parameters, yet it captures most of the experimentally observed friction phenomena. The model is suitable both for simulation purposes and control design. Analysis of friction-generated limit cycles in control systems is the second topic of the thesis. A distinction is made between limit cycles with and without periods of sticking. Oscillations without sticking where the velocity is zero only for single time instants can be treated as oscillations in relay-feedback systems for which tools are available. These tools are in the thesis extended to oscillations with sticking where the velocity is kept at zero for a period of time by the friction. The new tools give a procedure for exact computation of shape and stability of limit cycles caused by friction. The procedure requires the solution of a nonlinear equation system and that the feasibility of the solution is checked. The method is applied to several examples and comparisons are made with describing function analysis. The thesis also treats friction compensation based on the new model. A friction force observer is developed which enables model based friction compensation. The observer can be combined with traditional linear compensators. Stability theorems are given which allows a wide range of controller designs. The compensation scheme is applied to an example where the performance is studied with respect to model errors and disturbances. The resulting control error is thoroughly investigated. It is described how a simple statistical analysis of the error can give information on the success of the friction compensation. Furthermore the error during zero velocity crossings provides information on how model parameters should be changed.
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