Simultaneous Compensation of the Gain, Phase, and Phase-Slope

This paper deals with the problem of simultaneous compensation of the gain, phase, and phase-slope at an arbitrary frequency by using a fractional-order lead/lag compensator. The necessary and sufficient conditions for feasibility of the problem are derived. Also, the number of existing solutions (i.e., the number of distinct fractional-order lead/lag compensators satisfying the considered compensation requirements) is analytically found. Moreover, as a sample application, it is shown that the obtained results for the considered compensation problem are helpful in tuning fractional-order lead/lag compensators for simultaneously achieving desired phase margin, desired gain cross frequency, and flatness of the Bode phase plot of the loop transfer function at this frequency.