[Paper Review] The modelling and analysis of fractional-order control systems in frequency domain

Ivo Petráš, Ľ. Dorčák+3 more|ArXiv.org|Aug 24, 2000

Advanced Control Systems Design4 citations

TL;DR

This paper proposes a frequency-domain framework for modeling and analyzing fractional-order control systems using fractional transfer functions, enabling stability analysis via Bode and Nyquist plots. It demonstrates that fractional-order controllers (e.g., $PI^λ D^δ$) offer enhanced flexibility and robustness over classical PID controllers, with a numerical example confirming stability through frequency response analysis.

ABSTRACT

This paper deals with fractional-order controlled systems and fractional-order controllers in the frequency domain. The mathematical description by fractional transfer functions and properties of these systems are presented. The new ways for modelling of fractional-order systems are illustrated with a numerical example and obtained results are discussed in conclusion.

Motivation & Objective

• To develop a frequency-domain modeling approach for fractional-order control systems using fractional transfer functions.
• To extend classical control theory—particularly stability analysis—using Bode and Nyquist methods for fractional-order systems.
• To demonstrate the design and analysis of fractional-order $PI^\lambda D^\delta$ controllers for improved dynamical performance.
• To validate the proposed framework through a numerical example with specified stability and damping measures.
• To show that fractional-order controllers provide robustness and qualitatively different dynamical behavior compared to integer-order counterparts.

Proposed method

• Model fractional-order systems using a generalized transfer function: $G_s(j\omega) = \frac{\sum_{k=0}^{m} b_k (j\omega)^{\alpha_k}}{\sum_{k=0}^{n} a_k (j\omega)^{\beta_k}}$, where $\alpha_k, \beta_k \in \mathbb{R}$.
• Define the fractional-order $PI^\lambda D^\delta$ controller in the frequency domain as $G_c(j\omega) = K + \frac{T_i}{(j\omega)^\lambda} + T_d (j\omega)^\delta$, with $\lambda, \delta \geq 0$.
• Use a least-squares optimization criterion $Q = \sum_{m=0}^{M} W^2(\omega_m) |F(\omega_m) - G_s(j\omega_m)|^2$ to identify system parameters from experimental frequency response data.
• Apply Bode and Nyquist frequency response methods to assess stability: a system is stable if the Nyquist plot encircles the critical point $(-1, i0)$ in the correct direction.
• Utilize stability measures ($S_t = 2.0$) and damping ratio ($\xi = 0.4$) to tune controller parameters for desired dynamic behavior.
• Reformulate the controller as $G_c(j\omega) = C \frac{((j\omega)/\omega_n)^{\delta+\lambda} + 2\xi (j\omega)^\lambda / \omega_n + 1}{(j\omega)^\lambda}$ to facilitate design with complex poles and zeros.

Experimental results

Research questions

• RQ1 How can fractional-order systems be effectively modeled in the frequency domain using generalized transfer functions?
• RQ2 What are the stability conditions for fractional-order control systems, and how can they be assessed using Bode and Nyquist plots?
• RQ3 How does the $PI^\lambda D^\delta$ controller improve control performance compared to classical PID controllers in fractional-order systems?
• RQ4 What is the impact of non-integer orders $\lambda$ and $\delta$ on system dynamics and robustness?
• RQ5 Can frequency-domain methods be reliably extended to analyze and design fractional-order control systems with memory and hereditary behavior?

Key findings

• The fractional-order system with transfer function $G_s(j\omega) = \frac{1}{0.8(j\omega)^{2.2} + 0.5(j\omega)^{0.9} + 1}$ was successfully modeled and analyzed in the frequency domain.
• The $PD^\delta$ controller with $G_c(j\omega) = 50.0 + 5.326(j\omega)^{1.286}$ was designed to achieve a stability measure $S_t = 2.0$ and damping ratio $\xi = 0.4$.
• Nyquist and Bode plots (Figs. 2 and 3) confirmed the stability of the closed-loop system, with the Nyquist curve passing to the right of the critical point $(-1, i0)$.
• The proposed framework enables robust controller design through frequency-domain analysis, with the $PI^\lambda D^\delta$ controller offering greater flexibility than classical PID.
• Fractional-order controllers exhibit robustness and can induce qualitatively different dynamical phenomena not achievable with integer-order controllers.
• Parameter identification via minimization of the quadratic error criterion $Q$ effectively captures the system's frequency response from experimental data.