Comparative evaluation of rational approximation techniques for $$PI^{\lambda }D^{\mu }$$ controller implementation on programmable logic controllers

This paper presents a comparative study of rational approximation techniques for the implementation of $$PI^{\\lambda }D^{\\mu }$$ controllers with a focus on practical realisations. Implementation and evaluation of a broad family of methods like Oustaloup’s Recursive Approximation (ORA), frequency-domain Vector Fitting (VF), Matsuda least-squares ladder, Continued Fraction Expansion (CFE/Tustin) and a proposed Modified SBL (M-SBL) refinement has been conducted. Performance is assessed using in-band magnitude and phase errors, RMS and peak error metrics, numerical conditioning, pole behaviour, and achievable model order. Results show marked differences in suitability. Matsuda’s ladder (selected at moderate ladder depth) attains the best in-band accuracy and complexity trade-off. Three practical challenge clusters are identified. There is a trade-off between matching the magnitude and keeping the phase accurate. The model complexity matters as many methods become very high order and are hard to implement in practice. Some methods are numerically fragile, meaning small numerical errors can lead to unstable or unreliable behaviour. The results show that even when a method matches the frequency response well, it may still need a very high-order model. This makes such methods inefficient for real controllers. Therefore, the paper provides guidance on choosing controller approximations that are not only accurate but also stable, low-order, and reliable for practical implementation.