Issues in Linear Feedback Control Design with Notch Filters

In this note, we describe some design issues that should be carefully considered when one designs a linear feedback controller which includes a notch filter. Notch filters are generally used to compensate the effects of resonant modes that may result in large magnitude peaks in the frequency domain. It is common that practical engineers prefer to add such a notch filter after having previously designed a feedback controller in the absence of the filter. Apparently the resulting performance does not seem to be different from when a feedback controller is designed for a plant that has been previously compensated by a notch filter. In order to compare the performance of both approaches, a low-order controller design method that uses partial model matching ideas has been applied to a linear time invariant (LTI) model. The results suggest that frequency domain characteristics such as the sensitivity function and stability margins will tend to be similar for both approaches. On the other hand, when notch filter compensation is used before designing a feedback controller there is a tendency to achieving much better time responses.