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

Ivo Petráš, Ľ. Dorčák+1 more|ArXiv.org|Jun 26, 2000

Advanced Control Systems DesignEngineering7 references18 citations

TL;DR

This paper proposes a framework for modeling and simulating fractional-order control systems in the discrete domain using fractional difference equations derived from discrete approximations of fractional calculus. By applying the Grünwald-Letnikov and Riemann-Liouville definitions via Z-transform and short memory principles, the authors develop a general discrete transfer function model for fractional-order controllers and systems, demonstrating improved robustness and flexibility over classical PID control through a numerical example with a stable closed-loop system.

ABSTRACT

This paper deals with fractional-order controlled systems and fractional-order controllers in the discrete domain. The mathematical description by the fractional difference equations and properties of these systems are presented. A practical example for modelling the fractional-order control loop is shown and obtained results are discussed in conclusion.

Motivation & Objective

• To develop a systematic approach for modeling fractional-order control systems in the discrete domain using fractional difference equations.
• To extend classical PID control by introducing a generalized fractional-order $PI^\lambda D^\delta$ controller structure with arbitrary real-order integration and differentiation.
• To enable practical digital implementation of fractional-order controllers through discrete approximations of fractional operators.
• To demonstrate the feasibility and advantages of fractional-order control in discrete-time systems through a numerical simulation example.

Proposed method

• Utilizes the Grünwald-Letnikov and Riemann-Liouville definitions of fractional calculus for discrete approximation of fractional-order operators.
• Applies the short memory principle and power series expansion (PSE) to compute binomial coefficients for fractional-order difference equations.
• Derives discrete transfer functions using the Z-transform of the discrete fractional operator $\omega(z^{-1})$, where $\omega(z^{-1}) = \frac{1 - z^{-1}}{T}$.
• Constructs the discrete fractional-order difference equation by inverse Z-transform of the system and controller transfer functions.
• Employs the trapezoidal (Tustin) rule as an alternative generating function for improved approximation accuracy in discrete fractional systems.
• Uses recursive computation of binomial coefficients via the recurrence $c_j^{(\alpha)} = \left(1 - \frac{1 \pm \alpha}{j}\right) c_{j-1}^{(\alpha)}$ for efficient numerical implementation.

Experimental results

Research questions

• RQ1 How can fractional-order control systems be accurately modeled in the discrete domain using fractional difference equations?
• RQ2 What discrete approximation methods are suitable for implementing fractional-order operators in digital control systems?
• RQ3 How does the generalized $PI^\lambda D^\delta$ controller improve system performance compared to classical PID in discrete-time systems?
• RQ4 What is the impact of fractional-order integration and differentiation on system stability and transient response in a closed-loop configuration?

Key findings

• The discrete fractional-order system model is derived from the Z-transform of the fractional operator, enabling simulation via a recursive difference equation.
• The fractional-order $PD^\delta$ controller with $\delta = 1.286$ and $T_d = 5.326$ achieves a stability measure $S_t = 2.0$ and damping ratio $\xi = 0.4$ in the closed-loop system.
• The resulting discrete difference equation (15) accurately simulates the closed-loop response with $w_k = 1$ for $k \geq 2$, showing stable convergence.
• The proposed method allows for robust digital implementation of fractional-order controllers by leveraging Z-transform and short memory approximation.
• The framework supports flexible controller design with arbitrary real-order integration and differentiation, enhancing system tuning capabilities beyond classical PID.