Mathematical modeling and analysis of controlled fractional RC and RLC circuits with generalized memory kernels and uncertainty quantification

Ishtiaq Ali; Ishtiaq Ali

doi:10.3934/math.20261128

AIMS Mathematics

2026, Volume 11, Issue 9: 28318-28347. doi: 10.3934/math.20261128

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Mathematical modeling and analysis of controlled fractional RC and RLC circuits with generalized memory kernels and uncertainty quantification

Ishtiaq Ali ^,

Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa, 31982, Saudi Arabia

Received: 22 July 2026 Revised: 21 August 2026 Accepted: 26 August 2026 Published: 04 September 2026
MSC : 26A33, 34A08, 34C60

This paper presents a common kernel-dependent mathematical framework for modeling and analyzing controlled fractional resistor–capacitor (RC) and resistor–inductor–capacitor (RLC) circuits using several singular and nonsingular memory kernels. The proposed models describe hereditary and nonlocal memory effects in transient, oscillatory, and damping behaviors under constant and periodic excitations, with a characteristic dimensional scaling parameter introduced to preserve physical consistency in the fractional-order formulation. Analytical solutions are derived using Laplace transform techniques and generalized Mittag–Leffler functions, providing explicit representations of the circuit responses for different fractional operators. A fractional-order proportional–integral (PI) controller is incorporated to suppress oscillations, reduce overshoot, enhance damping, and accelerate convergence toward the corresponding equilibrium or bounded periodic regime. To assess the robustness of the proposed models, uncertainty quantification based on Monte Carlo (MC) simulations is employed under stochastic variations in fractional orders, relaxation parameters, damping coefficients, oscillation frequencies, and controller gains. Numerical results demonstrate the significant influence of the memory kernels and fractional orders on the dynamical behavior of the circuits, and the proposed control strategy consistently improves transient responses. Furthermore, the uncertainty analysis shows that the stochastic responses remain bounded and closely follow the corresponding deterministic controlled trajectories, indicating robust behavior of the proposed mathematical framework under the considered parameter perturbations.
- mathematical modeling,
- generalized memory kernels,
- fractional RC and RLC circuits,
- fractional–order control,
- dynamical analysis,
- uncertainty quantification
Citation: Ishtiaq Ali. Mathematical modeling and analysis of controlled fractional RC and RLC circuits with generalized memory kernels and uncertainty quantification[J]. AIMS Mathematics, 2026, 11(9): 28318-28347. doi: 10.3934/math.20261128

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Abstract

This paper presents a common kernel-dependent mathematical framework for modeling and analyzing controlled fractional resistor–capacitor (RC) and resistor–inductor–capacitor (RLC) circuits using several singular and nonsingular memory kernels. The proposed models describe hereditary and nonlocal memory effects in transient, oscillatory, and damping behaviors under constant and periodic excitations, with a characteristic dimensional scaling parameter introduced to preserve physical consistency in the fractional-order formulation. Analytical solutions are derived using Laplace transform techniques and generalized Mittag–Leffler functions, providing explicit representations of the circuit responses for different fractional operators. A fractional-order proportional–integral (PI) controller is incorporated to suppress oscillations, reduce overshoot, enhance damping, and accelerate convergence toward the corresponding equilibrium or bounded periodic regime. To assess the robustness of the proposed models, uncertainty quantification based on Monte Carlo (MC) simulations is employed under stochastic variations in fractional orders, relaxation parameters, damping coefficients, oscillation frequencies, and controller gains. Numerical results demonstrate the significant influence of the memory kernels and fractional orders on the dynamical behavior of the circuits, and the proposed control strategy consistently improves transient responses. Furthermore, the uncertainty analysis shows that the stochastic responses remain bounded and closely follow the corresponding deterministic controlled trajectories, indicating robust behavior of the proposed mathematical framework under the considered parameter perturbations.

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