Analysis of piecewise linear variable gain controllers for linear motion systems

This report concerns variable gain control (VGC), a specific class of nonlinear control. It is investigated if there exist disturbance situations where VGC utilizing a tri-modal nonlinearity leads to improved performance compared to VGC using the more common bi-modal nonlinearities. A model derived from a wafer stage of a wafer scanner is used for this investigation. The steady-state errors, resulting from a certain disturbance situation, are calculated using the MTF algorithm. The performance is quantified using the performance indicator J, which allows for emphasis on either the low-frequent or the high-frequent content present in this steady-state error. Two different tri-modal nonlinearities were constructed. In the first case, performance in time-domain was indisputably better using tri-modal VGC than a bi-modal VGC. In the second case, the difference in performance was more subtle. If the emphasis is on low-frequency content of the steady-state error, the bi-modal VGC performs better, whereas the tri-modal VGC performs better if the high-frequency content is of higher importance. It can thus be concluded that there are disturbance situations where a tri-modal VGC outperforms a bimodal VGC. It should be noted that these results are simulation results. Practical research should be done to verify this conclusion. Future work should include the mentioned practical research, but also the effect of incorporating a nonlinearity with a negative slope in the VGC. This could lead to improved performance in a variety of disturbance situations