Formulas for asymmetric lead and lag compensators

Lead and lag compensators are commonly used for classical loop shaping and as weighting functions for automated controller synthesis algorithms. The existing formulas for these compensators provide the capability of selecting phase at the frequency of maximum phase lead such that the phase peak or notch is symmetrical on the Bode plot phase chart. This paper develops easy-to-use formulas for lead compensators for choosing the phase at a frequency that is either before, or after the frequency of the maximum phase. For second order compensators, formulas are also provided to specify the degree of asymmetry of the phase peak. Combining these formulas leads to a method for choosing the phase at the frequency of maximum phase for an asymmetric phase peak. An application to a robust controller design problem is shown.