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An optimal trade-off design for fractional order (FO)-PID controller isproposed in this paper with a Linear Quadratic Regulator (LQR) based techniqueusing two conflicting time domain control objectives. The deviation of thestate trajectories and control signal are automatically enforced by the LQR. Aclass of delayed FO systems with single non-integer order element has beencontrolled here which exhibit both sluggish and oscillatory open loopresponses. The FO time delay processes are controlled within a multi-objectiveoptimization (MOO) formulation of LQR based FOPID design. The time delays inthe FO models are handled by two analytical formulations of designing optimalquadratic regulator for delayed systems. A comparison is made between the twoapproaches of LQR design for the stabilization of time-delay systems in thecontext of FOPID controller tuning. The MOO control design methodology yieldsthe Pareto optimal trade-off solutions between the tracking performance forunit set-point change and total variation (TV) of the control signal. Numericalsimulations are provided to compare the merits of the two delay handlingtechniques in the multi-objective LQR-FOPID design, while also showing thecapability of load disturbance suppression using these optimal controllers.Tuning rules are then formed for the optimal LQR-FOPID controller knobs, usingthe median of the non-dominated Pareto solution to handle delays FO processes.

Multi-objective LQR with Optimum Weight Selection to Design FOPID Controllers for Delayed Fractional Order Processes