[Paper Review] The fractional - order controllers: Methods for their synthesis and application

Ivo Petráš|ArXiv.org|Apr 11, 2000

Coding theory and cryptographyComputer Science10 references210 citations

TL;DR

This paper proposes a modified root locus method for synthesizing fractional-order controllers (e.g., PI^λD^δ) to improve control performance in fractional-order systems. By directly designing controllers for non-integer order dynamics—rather than approximating them as integer-order systems—it achieves better stability, damping, and robustness, as demonstrated by simulations showing faster response and reduced overshoot compared to integer-order PID controllers.

ABSTRACT

This paper deals with fractional-order controllers. We outline mathematical description of fractional controllers and methods of their synthesis and application. Synthesis method is a modified root locus method for fractional-order systems and fractional-order controllers. In the next section we describe how to apply the fractional controller on control systems.

Motivation & Objective

• To address the limitations of approximating fractional-order systems with integer-order models during controller design.
• To develop a systematic method for synthesizing fractional-order controllers (e.g., PI^λD^δ) that match the true dynamics of fractional-order systems.
• To improve control performance, stability, and robustness in systems with non-integer order dynamics.
• To provide a practical design framework for fractional-order controllers using numerical and analytical tools from fractional calculus.

Proposed method

• Adapts the classical root locus method to fractional-order systems by extending it to handle non-integer derivatives and integrals in the controller and plant transfer functions.
• Uses the Riemann-Liouville and Grünwald definitions of fractional derivatives for numerical computation, with a 'short memory' principle to reduce computational load.
• Employs the Laplace transform of fractional derivatives and the Mittag-Leffler function for analytical solution of fractional-order differential equations.
• Derives a generalized characteristic equation (14) involving fractional powers of the Laplace variable, solved in the complex plane to determine controller parameters.
• Applies a discrete-time control algorithm using binomial coefficients derived from the Grünwald formula to implement the fractional-order controller numerically.
• Validates the method through simulation of unit-step responses comparing fractional-order and integer-order controllers on a fractional-order system.

Experimental results

Research questions

• RQ1 Can a modified root locus method be effectively applied to design fractional-order controllers for fractional-order systems?
• RQ2 Does direct design on fractional-order systems yield better control performance than controller design based on integer-order approximations?
• RQ3 How do fractional-order controllers (e.g., PI^λD^δ) improve stability and damping measures compared to standard integer-order PID controllers?
• RQ4 What is the impact of non-integer differentiation and integration orders on system robustness and transient response?
• RQ5 Can numerical implementation of fractional-order controllers be made efficient using the 'short memory' principle and binomial coefficient recurrence?

Key findings

• The fractional-order PD^δ controller with K=20.5, T_d=5.79, and δ=0.95 achieved superior transient response compared to the integer-order PD controller on the original fractional-order system.
• The integer-order PD controller, designed on an approximated integer-order model, failed to achieve desired performance when applied to the true fractional-order system, showing significant overshoot and slower settling.
• The fractional-order controller reduced overshoot and stabilized the system faster, demonstrating improved dynamical properties in the closed-loop system.
• The use of fractional-order controllers led to better robustness, as they were less sensitive to parameter variations in the system and controller.
• The simulation results confirmed that approximating fractional-order systems with integer-order models for controller design leads to inadequate performance, validating the need for direct fractional-order controller synthesis.
• The proposed method successfully determined controller parameters (K, T_d, δ, T_i, λ) by solving the generalized characteristic equation (14) in the complex plane for desired stability and damping measures.