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A novel conformal mapping based Fractional Order (FO) methodology isdeveloped in this paper for tuning existing classical (Integer Order)Proportional Integral Derivative (PID) controllers especially for sluggish andoscillatory second order systems. The conventional pole placement tuning viaLinear Quadratic Regulator (LQR) method is extended for open loop oscillatorysystems as well. The locations of the open loop zeros of a fractional order PID(FOPID or PI{\lambda}D{\mu}) controller have been approximated in this papervis-\`a-vis a LQR tuned conventional integer order PID controller, to achieveequivalent integer order PID control system. This approach eases theimplementation of analog/digital realization of a FOPID controller with itsinteger order counterpart along with the advantages of fractional ordercontroller preserved. It is shown here in the paper that decrease in theintegro-differential operators of the FOPID/PI{\lambda}D{\mu} controller pushesthe open loop zeros of the equivalent PID controller towards greater dampingregions which gives a trajectory of the controller zeros and dominant closedloop poles. This trajectory is termed as "M-curve". This phenomena is used todesign a two-stage tuning algorithm which reduces the existing PID controller'seffort in a significant manner compared to that with a single stage LQR basedpole placement method at a desired closed loop damping and frequency.

A Conformal Mapping Based Fractional Order Approach for Sub-optimal Tuning of PID Controllers with Guaranteed Dominant Pole Placement