Comparison of Distributed-Order and Fractional-Order PID with symmetric Bode plot for position control of inertial loads

Distributed Order PID (DOPID) is a possible method, alternative to Fractional Order PID (FOPID), to extend the tuning options of the classical PID control by utilizing fractional calculus. With the classical FOPID scheme, the PI<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sup>D<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">µ</sup>, two variable real orders, λ and µ, are introduced and can be tuned to improve the system behaviour. On the contrary, with the DOPID approach fractional-order terms are added to the integer-order ones to extend the tuning possibilities. In particular, with the PII<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup>DD<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> scheme, the half-derivative and half-integral terms are added. Recently, a Bode plot-based method has been proposed for the PII<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup>DD<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> tuning, which derives the control parameters from those of a given PID without requiring optimization techniques. In the proposed work, this Bode-plot tuning method is extended to the PI<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sup>D<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">µ</sup>. Simulation results show that the proposed method, with the hypothesis of symmetry of the magnitude Bode plot, allows to achieve better performance in combination with the DOPID rather than with the FOPID.