  An optimal trade-off design for fractional order (FO)-PID controller is
proposed in this paper with a Linear Quadratic Regulator (LQR) based technique
using two conflicting time domain control objectives. The deviation of the
state trajectories and control signal are automatically enforced by the LQR. A
class of delayed FO systems with single non-integer order element has been
controlled here which exhibit both sluggish and oscillatory open loop
responses. The FO time delay processes are controlled within a multi-objective
optimization (MOO) formulation of LQR based FOPID design. The time delays in
the FO models are handled by two analytical formulations of designing optimal
quadratic regulator for delayed systems. A comparison is made between the two
approaches of LQR design for the stabilization of time-delay systems in the
context of FOPID controller tuning. The MOO control design methodology yields
the Pareto optimal trade-off solutions between the tracking performance for
unit set-point change and total variation (TV) of the control signal. Numerical
simulations are provided to compare the merits of the two delay handling
techniques in the multi-objective LQR-FOPID design, while also showing the
capability of load disturbance suppression using these optimal controllers.
Tuning rules are then formed for the optimal LQR-FOPID controller knobs, using
the median of the non-dominated Pareto solution to handle delays FO processes.
