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

TL;DR

This paper introduces a conformal mapping based fractional order method for tuning PID controllers, improving ease of implementation and control performance for oscillatory systems by leveraging a novel zero placement trajectory called the "M-curve."

Contribution

It develops a new fractional order tuning approach using conformal mapping and zero trajectory analysis, enhancing classical pole placement methods for better control of oscillatory systems.

Findings

The proposed method effectively approximates FOPID zeros with integer PID zeros.

Decreasing fractional operators shifts zeros towards greater damping, forming the "M-curve."

Two-stage tuning reduces effort compared to single-stage LQR tuning.

Abstract

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.