  A novel conformal mapping based Fractional Order (FO) methodology is
developed in this paper for tuning existing classical (Integer Order)
Proportional Integral Derivative (PID) controllers especially for sluggish and
oscillatory second order systems. The conventional pole placement tuning via
Linear Quadratic Regulator (LQR) method is extended for open loop oscillatory
systems 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 paper
vis-\`a-vis a LQR tuned conventional integer order PID controller, to achieve
equivalent integer order PID control system. This approach eases the
implementation of analog/digital realization of a FOPID controller with its
integer order counterpart along with the advantages of fractional order
controller preserved. It is shown here in the paper that decrease in the
integro-differential operators of the FOPID/PI{\lambda}D{\mu} controller pushes
the open loop zeros of the equivalent PID controller towards greater damping
regions which gives a trajectory of the controller zeros and dominant closed
loop poles. This trajectory is termed as "M-curve". This phenomena is used to
design a two-stage tuning algorithm which reduces the existing PID controller's
effort in a significant manner compared to that with a single stage LQR based
pole placement method at a desired closed loop damping and frequency.
