[Paper Review] Approximation of a Fractional Order System by an Integer Order Model Using Particle Swarm Optimization Technique

Deepyaman Maiti, Amit Konar|ArXiv.org|Nov 1, 2008

Advanced Control Systems Design10 references16 citations

TL;DR

This paper proposes a novel method to approximate fractional-order systems using integer-order models via Particle Swarm Optimization (PSO). By minimizing the sum of squared errors between the fractional-order system's response and the integer-order model's output, the PSO algorithm efficiently identifies high-accuracy integer-order approximations, demonstrating strong performance in system identification for control applications.

ABSTRACT

System identification is a necessity in control theory. Classical control theory usually considers processes with integer order transfer functions. Real processes are usually of fractional order as opposed to the ideal integral order models. A simple and elegant scheme is presented for approximation of such a real world fractional order process by an ideal integral order model. A population of integral order process models is generated and updated by PSO technique, the fitness function being the sum of squared deviations from the set of observations obtained from the actual fractional order process. Results show that the proposed scheme offers a high degree of accuracy.

Motivation & Objective

• To address the challenge of modeling real-world processes that exhibit fractional-order dynamics using classical integer-order control models.
• To develop a systematic and computationally efficient method for approximating fractional-order systems with integer-order transfer functions.
• To improve the accuracy of system identification in control theory by leveraging metaheuristic optimization.
• To validate the effectiveness of PSO in minimizing the error between actual fractional-order system responses and approximated integer-order models.

Proposed method

• A population of candidate integer-order transfer functions is initialized to represent potential approximations of the fractional-order system.
• The Particle Swarm Optimization (PSO) algorithm is employed to iteratively update the parameters of these models to minimize a fitness function.
• The fitness function is defined as the sum of squared deviations between the frequency-domain or time-domain responses of the actual fractional-order system and the candidate integer-order models.
• The algorithm evaluates each candidate model's performance based on its response similarity to the original system across a set of observed data points.
• Convergence is achieved when the PSO algorithm identifies a set of integer-order model parameters that minimize the error metric.
• The final model is selected based on the lowest fitness value, representing the best approximation of the fractional-order system.

Experimental results

Research questions

• RQ1 Can a population-based metaheuristic like PSO effectively approximate a fractional-order system with an integer-order model?
• RQ2 How accurately can PSO minimize the error between the frequency or time response of a fractional-order system and its integer-order approximation?
• RQ3 What is the performance of the PSO-based approximation method compared to traditional system identification techniques?
• RQ4 How robust is the PSO-based approach across different types of fractional-order systems?

Key findings

• The PSO-based method achieves a high degree of accuracy in approximating fractional-order systems using integer-order models.
• The fitness function, based on sum of squared deviations, effectively guides the PSO algorithm toward optimal parameter configurations.
• The proposed scheme demonstrates strong convergence and stability in identifying accurate integer-order approximations.
• The results confirm that PSO is a viable and effective optimization technique for system identification tasks involving fractional-order dynamics.