Towards High-Performance DC Motor Control: Fractional Modelling and FOPID Optimisation Cover

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Towards High-Performance DC Motor Control: Fractional Modelling and FOPID Optimisation

Power Electronics and Drives

Volume 10 (2025): Issue 1 (January 2025)

By: Bilel Kanzari and Adel Taeib

Open Access

|Jan 2026

Figures & Tables

System identification using algorithms. ABC, artificial bee colony; ACO, ant colony optimisation; GA, genetic algorithm; PSO, particle swarm optimisation.

Closed loop system with FOPID controller. FOPID, fractional-order proportional-integral-derivative.

Optimisation structure with algorithm of tuning FOPID control parameters. FOPID, fractional-order proportional-integral-derivative.

XK-AUT1003A prototype model.

Block diagram of the experimental setup.

Open-loop step response-based identification of a DC motor. PSO, particle swarm optimisation.

Comparison of open-loop step responses: integer vs. fractional models. ABC, artificial bee colony; ACO, ant colony optimisation; GA, genetic algorithm; PSO, particle swarm optimisation.

Validation of model.

Convergence behaviour and statistical error comparison of optimisation algorithms. ABC, artificial bee colony; ACO, ant colony optimisation; GA, genetic algorithm; PSO, particle swarm optimisation.

Effort made control inputs with GIO and GFO. ABC, artificial bee colony; ACO, ant colony optimisation; GA, genetic algorithm; PSO, particle swarm optimisation.

DC motor closed-loop response under FOPID-PSO control and effort evaluation. FOPID, fractional-order proportional-integral-derivative; PSO, particle swarm optimisation.

Closed-loop dynamics with disturbances introduced at 2 s and 2.5 s. PSO, particle swarm optimisation.

DC motor parameters_

Parameter	Value	Unit
Rated voltage	12	V
No-load speed	3,000	RPM
No-load current	0.1	A
Rated torque	0.05	Nm
Rotor resistance	2	Ω
Rotor inductance	15	mH
Rotor inertia	2.1 × 10⁻⁵	kg/m²
Command voltage range	−5 to +5	V
Measured speed range	0–3,000	RPM

Control effort final values for integer and fractional models_

Method	PSO	GA	ACO	ABC
‖u‖_IO (V)	6.6222	9.1140	8.3212	9.0678
‖u‖_FO (V)	6.4791	9.1226	8.4645	8.8118

Characteristic parameters of a DC motor_

Parameters	Symbol
Moment of inertia of rotor	T
Motor viscous friction	b
Electromotive force constant	Ke
Motor torque constant	Kt
Electric resistance	R
Electric inductance	L
Power gain	P

Adjustment of FOPID using the first method for open-loop response-based parameter_

Parameters to use when 0.1 ≤ T ≤ 5
	P	I	λ	D	μ
1	−1.0574	0.6014	1.1857	0.8796	0.2778
L	24.5420	0.4025	−0.3464	−15.0846	−2.1522
T	0.3544	0.7921	−0.0492	−0.0771	0.0675
L²	−46.7325	−0.4508	1.7377	28.0388	2.4387
T²	−0.0021	0.0018	0.0006	−0.0000	−0.0013
LT	−0.3106	−1.2050	0.0380	1.6711	0.0021

Integer and fractional model with optimal parameter set_

(a)
	b	a₂	a₁	a₀	Error (%)
ABC	703.040	0.1030	4.0911	5.3434	1.00
ACO	568.216	0.4270	4.7000	4.1927	2.34
GA	1100.24	0.7100	6.6710	8.3864	0.88
PSO	715.489	0.4595	4.3391	5.4522	0.79

Adjustment of parameter for FOPID by the second method using the closed-loop response as a basis_

	P	I	λ	D	μ
1	0.4139	0.7067	1.3240	0.2293	0.8804
Kcr	0.0145	0.0101	−0.0081	0.0153	−0.0048
Pcr	−0.1584	−0.0049	−0.0163	0.0936	0.0061
1/ Kcr	−0.4384	−0.2951	0.1393	−0.5293	0.0749
1/ Pcr	−0.0855	−0.1001	0.0791	−0.0440	0.0810

Optimisation algorithm parameter settings_

Algorithm	Population size	Number of iterations	Main control parameters
PSO	50	100	w = 0.7, c₁ = 1.5, c₂ = 1.5
GA	40	100	Crossover = 0.8, Mutation = 0.05
ABC	30	100	Limit = 50, food sources = 15
ACO	25	100	α = 1, β = 2, ρ = 0.5
Number of decision variables		Lower bound	Upper bound
4 (b,a₂,a₁,a₀)		[0, 0, 0, 0]	[1,000, 1, 10, 10]
6 (b,a₂,α₂,a₁, α₁, a₀)		[0, 0, 0, 0, 0, 0]	[1,500, 1, 10, 10, 5, 10]
5(K_p,K_I,λ,K_d,μ)		[0, 0, 0, 0, 0]	[200, 200, 2, 10, 2]

Performance assessment of controllers applied to integer and fractional models_

	K_p	K_I	λ	K_d	μ	Overshoot (%)	Risetime (s)	Settling time (s)
ABC_IO	140.275	187.65	1.5861	0.6831	0.9582	2.0510	1.6875	3.5448
ACO_IO	90.420	45.157	0.9424	2.6584	0.8912	3.5231	2.3654	2.5447
GA_IO	42.895	63.845	1.2045	2.9821	0.9994	1.7548	1.5483	0.9543
PSO_IO	70.548	30.448	1.0054	3.584	1.2854	1.4545	0.5898	1.0911
ZN1_IO	91.578	70.6554	0.9582	5.3254	1.1415	10.554	3.1554	7.5897
ZN2_IO	101.05	60.545	1.0254	4.5544	1.2054	11.215	4.114411	8.1545
ABC_FO	70.6225	100.01	1.0025	0.6831	1.1532	0.0000	1.5487	1.9269
ACO_FO	48.600	87.617	0.9596	0.1190	0.8729	0.0000	1.5307	1.8454
GA_FO	74.064	83.584	1.1099	1.8281	0.8895	0.0000	1.6928	1.0516
PSO_FO	49.219	65.352	1.0823	2.0012	0.9899	0.0000	1.5280	1.0911
ZN1_FO	50.215	70.225	1.2015	4.5454	0.1521	4.4544	1,454.5	1.8474
ZN2_FO	60.124	100.55	1.1024	5.1544	0.9554	7.5441	1,5485	1.77555

DOI: https://doi.org/10.2478/pead-2025-0030 | Journal eISSN: 2543-4292 | Journal ISSN: 2451-0262

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Language: English

Page range: 448 - 466

Submitted on: Sep 7, 2025

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Accepted on: Nov 15, 2025

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Published on: Jan 16, 2026

Published by: Wroclaw University of Science and Technology

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

fractional PID controller,

GA,

DC motor speed control,

Related subjects:

Computer sciences,

Artificial intelligence,

Electrical engineering,

© 2026 Bilel Kanzari, Adel Taeib, published by Wroclaw University of Science and Technology
This work is licensed under the Creative Commons Attribution 4.0 License.

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