Fractional Order Calculus Research Articles

Modeling Thermal Impedance of IGBT Devices Based on Fractional Calculus Techniques

The thermal impedance characteristics of insulated gate bipolar transistor (IGBT) modules are critical for the thermal management and design of electronic devices. This paper proposes a fractional-order equivalent thermal impedance model, which is inspired by the correlation between multi-time-scale dissipation characteristics of heat conduction processes and fractional calculus. The fractional-order equivalent thermal impedance model is derived based on the connection between fractional-order calculus and the Foster thermal network model in mathematical operations, with only two parameters to be identified: heat capacity C and fractional order α. Moreover, this paper provides a parameter identification method for the proposed fractional-order equivalent thermal impedance model based on the multi-objective particle swarm optimization (MOPSO) algorithm. In order to validate the effectiveness and superiority of this work, experiments and comparative works are provided in this paper. The results indicate that the fractional-order equivalent thermal impedance model can accurately describe the frequency domain characteristic curves of the thermal impedance of the Foster thermal network model for IGBT modules, with the difference between the amplitude frequency characteristics not exceeding 1 dB and the difference between the phase frequency characteristics not exceeding 1° within the operating frequency range of (1 kHz, 1 MHz).

Nonlinear Dynamic Analysis of Riemann–Liouville Fractional-Order Damping Giant Magnetostrictive Actuator

Abstract In view of the large errors in the integer-order prediction model of the current giant magnetostrictive actuator (GMA), existing studies have shown that the fractional-order theory can improve the classical integer-order error situation. To this end, the Riemann–Liouville (R–L) fractional-order calculus theory is applied to the damping part of the GMA system; based on the averaging method and the power series method, the analytical and numerical solutions of the system are obtained, respectively, the motion of the GMA system is obtained through simulation, the parameters affecting the main resonance response of the system are analyzed as well as the motion characteristics of the system under the parameters, and the bifurcation and chaotic characteristics of the system are analyzed qualitatively and quantitatively. It is shown that the fractional-order model can improve the prediction accuracy of the system, the fractional order has a significant effect on the motion of the system, and the interval of the periodical motion parameter is less than an integer when the order of the damping term is (0,1), and the system can be induced to shift to periodic motion by changing the parameters.

A Special Note According to Possible Applications of Fractional-Order Calculus for Various Special Functions

The main aim of this special study is to recall certain information about fractional (arbitrary) order calculus, which has wide and fruitful applications in science and engineering. Then, it aims to consider various essential definitions related to fractional order integrals and derivatives for stating and proving some results, as well as to present some of their possible applications to the attention of related researchers.

Fractional order calculus enhanced dung beetle optimizer for function global optimization and multilevel threshold medical image segmentation

Fractional order calculus enhanced dung beetle optimizer for function global optimization and multilevel threshold medical image segmentation

Multi-Buoy Deployment Method Based on an Improved Tuna Swarm Optimizer Enhanced with Fractional-Order Calculus Method for Marine Observation

Ocean buoys play a critical role in marine hydrological, water quality, and meteorological monitoring, with applications in navigation, environmental observation, and communication. However, accurately modeling and deploying a multi-buoy system in the complex marine environment presents significant challenges. To address these challenges, this study proposes an enhanced deployment strategy using the tuna swarm optimizer enhanced with the fractional-order calculus method for marine observation. The proposed method first introduces a detailed observation model that precisely captures the performance of buoys in terms of coverage and communication efficiency. By integrating the observation coverage ratio and communication energy consumption, we establish an optimal multi-buoy deployment model. The proposed method leverages tent chaotic mapping to improve the diversity of initial solution generation and incorporates fractional-order calculus to strengthen its search capabilities. Simulation experiments and statistical analysis verify the effectiveness of the proposed deployment model, with the proposed method achieving the best performance in deploying the multi-buoy system, reaching a final fitness value of 0.190052 at iteration 449, outperforming TSA, PSO, GWO, and WOA. These results highlight the potential of the proposed method in optimizing multi-buoy system deployment in marine observation.

Fractional order calculus model-derived histogram metrics for assessing pathological complete response to neoadjuvant chemotherapy in locally advanced rectal cancer

AimThis study evaluates the value of diffusion fractional order calculus (FROC) model for the assessment of pathological complete response (pCR) after neoadjuvant chemoradiotherapy (nCRT) for locally advanced rectal cancer (LARC) by using histogram analysis derived from whole-tumor volumes. Materials and methodsNinety-eight patients were prospectively included. Every patient received MRI scans before and after nCRT using a 3.0-Tesla MRI machine. Parameters of the FROC model, including the anomalous diffusion coefficient (D), intravoxel diffusion heterogeneity (β), spatial parameter (μ), and the standard apparent diffusion coefficient (ADC), were calculated. Changes in median values (ΔX-median) and ratio (rΔX-median) were calculated. Receiver operating characteristic (ROC) curves were used for evaluating the diagnostic performance. ResultsPre-treatmentβ-10th percentile values were significantly lower in the pCR group compared to the non-pCR group (p < 0.001). The Δβ-median showed higher diagnostic accuracy (AUC = 0.870) and sensitivity (76.67 %) for predicting tumor response compared to MRI tumor regression grading (mrTRG) scores (AUC = 0.722; sensitivity = 90.0 %). DiscussionThe use of FROC alongside comprehensive tumor histogram analysis was found to be practical and effective in evaluating the tumor response to nCRT in LARC patients.

Analysis of Fractional Order-Adaptive Systems Represented by Error Model 1 Using a Fractional-Order Gradient Approach

In adaptive control, error models use system output error and adaptive laws to update controller parameters for control or identification tasks. Fractional-order calculus, involving non-integer-order derivatives and integrals, is increasingly important for modeling, estimation, and control due to its ability to generalize classical methods and offer improved robustness to disturbances. This paper addresses the gap in the literature where fractional-order gradient methods have not yet been extensively applied in identification and adaptive control schemes. We introduce a fractional-order error model with fractional-order gradient (FOEM1-FG), which integrates fractional gradient operators based on the Caputo fractional derivative. By using theoretical analysis and simulations, we confirm that FOEM1-FG maintains stability and ensures bounded output errors across a variety of input signals. Notably, the fractional gradient’s performance improves as the order, β, increases with β&gt;1, leading to faster convergence. Compared to existing integer-order methods, the proposed approach provides a more flexible and efficient solution in adaptive identification and control schemes. Our results show that FOEM1-FG offers superior stability and convergence characteristics, contributing new insights to the field of fractional calculus in adaptive systems.

Predicting the Ki-67 proliferation index in cervical cancer: a preliminary comparative study of four non-Gaussian diffusion-weighted imaging models combined with histogram analysis.

The prognosis for patients with cervical cancer (CC) is strongly correlated with the Ki-67 proliferation index (PI). However, the Ki-67 PI obtained through biopsy has certain limitations. The non-Gaussian distribution diffusion model of magnetic resonance imaging (MRI) may play an important role in characterizing tissue heterogeneity. At present, there are limited data available concerning the prediction of Ki-67 PI using models based on histogram features of non-Gaussian diffusion distribution. This study aimed to determine whether preoperative histogram features from multiple non-Gaussian models of diffusion-weighted imaging can predict the Ki-67 PI in patients with CC. Our cross-sectional prospective study recruited a total of 53 patients suspected of having CC who underwent 3.0-T MRI at Sun Yat-sen Memorial Hospital of Sun Yat-sen University between January 2022 and January 2023. Fifteen b values (0-4,000 s/mm2) were used for diffusion-weighted imaging. A total of nine parameters from four non-Gaussian diffusion-weighted imaging models, including continuous-time random walk (CTRW), diffusion kurtosis imaging (DKI), fractional order calculus (FROC), and intravoxel incoherent motion (IVIM), were used. Whole-tumor volumetric histogram analysis of these parameters was then obtained. In logistic regression, significant histogram characteristics were statistically examined across two groups to build the final prediction model. To assess diagnostic parameters of the proposed model in the diagnosis of the Ki-67 PI, along with the sensitivity, specificity, and diagnostic accuracy of these various parameters from the four models, receiver operating feature analysis was applied. Among the 53 patients (55.3±9.6 years, ranging from 23 to 79 years) included in the study, 15 had a Ki-67 PI ≤50% and 38 had a Ki-67 PI >50%. Univariable analysis determined that 12 histogram features were statistically different between the two groups. In multivariable logistic regression, we ultimately selected 6 histogram features to construct the final prediction model, with CTRW_α_10th percentile [odds ratio (OR) =0.955; 95% confidence interval (CI): 0.92-0.99; P=0.019], CTRW_α_robust mean absolute deviation (OR =0.893; 95% CI: 0.81-0.99; P=0.028), and CTRW_α_uniformity (OR =0.000, 95% CI: 0.00-0.90, P=0.047) being the independent predictive variables. The area under the curve of the combined prediction model was 0.845 (95% CI: 0.74-0.95), with a sensitivity of 78.9% (95% CI: 0.63-0.90), a specificity of 86.7% (95% CI: 0.60-0.98), an accuracy of 81.1% (95% CI: 0.68-0.91), a positive predictive value of 93.8% (95% CI: 0.79-0.99), and a negative predictive value of 61.9% (95% CI: 0.38-0.82). The histogram features of multiple non-Gaussian diffusion-weighted imaging can help to predict the Ki-67 PI of CC, providing a new method for the noninvasive evaluation of critical biological features of CC.

Improved Performance of the Permanent Magnet Synchronous Motor Sensorless Control System Based on Direct Torque Control Strategy and Sliding Mode Control Using Fractional Order and Fractal Dimension Calculus

This article starts from the premise that one of the global control strategies of the Permanent Magnet Synchronous Motor (PMSM), namely the Direct Torque Control (DTC) control strategy, is characterized by the fact that the internal flux and torque control loop usually uses ON–OFF controllers with hysteresis, which offer easy implementation and very short response times, but the oscillations introduced by them must be cancelled by the external speed loop controller. Typically, this is a PI speed controller, whose performance is good around global operating points and for relatively small variations in external parameters and disturbances, caused in particular by load torque variation. Exploiting the advantages of the DTC strategy, this article presents a way to improve the performance of the sensorless control system (SCS) of the PMSM using the Proportional Integrator (PI), PI Equilibrium Optimizer Algorithm (EOA), Fractional Order (FO) PI, Tilt Integral Derivative (TID) and FO Lead–Lag under constant flux conditions. Sliding Mode Control (SMC) and FOSMC are proposed under conditions where the flux is variable. The performance indicators of the control system are the usual ones: response time, settling time, overshoot, steady-state error and speed ripple, plus another one given by the fractal dimension (FD) of the PMSM rotor speed signal, and the hypothesis that the FD of the controlled signal is higher when the control system performs better is verified. The article also presents the basic equations of the PMSM, based on which the synthesis of integer and fractional controllers, the synthesis of an observer for estimating the PMSM rotor speed, electromagnetic torque and stator flux are presented. The comparison of the performance for the proposed control systems and the demonstration of the parametric robustness are performed by numerical simulations in Matlab/Simulink using Simscape Electrical and Fractional-Order Modelling and Control (FOMCON). Real-time control based on an embedded system using a TMS320F28379D controller demonstrates the good performance of the PMSM-SCS based on the DTC strategy in a complete Hardware-In-the-Loop (HIL) implementation.

Some operators of fractional calculus and their applications regarding various complex functions analytic in certain domains

Abstract. In this academic research note, some familiar operators prearranged by fractional-order calculus will first be introduced and various characteristic properties of those operators will next be propounded. Through the instrumentality of various earlier results associating with both those operators and some complex- exponential forms, and also in the light of certain special information in [1], [20], [17] and [38], an extensive result together with a variety of its implications consisting of several exponential type inequalities will then be determined. A number of its possible implications will extra be pointed out. Mathematics Subject Classification (2010): 26A33, 30A10, 34A40, 35A30, 41A58, 30C45, 30C55, 30C80, 33D15, 26E05, 33E20. Keywords: Complex plane, domains, regular functions, complex exponential, series expansions, fractional calculus, operators of fractional calculus, exponential type inequalities, differential inequalities.

Predictive value of mono-exponential and multiple mathematical models in locally advanced rectal cancer response to neoadjuvant chemoradiotherapy.

This prospective study aimed to assess the predictive value of mono-exponential and multiple mathematical diffusion-weighted imaging (DWI) models in determining the response to neoadjuvant chemoradiotherapy (nCRT) in patients with locally advanced rectal cancer (LARC). The study included 103 LARC patients scheduled for preoperative chemoradiotherapy between December 2021 and June 2023 Magnetic resonance imaging (MRI) scans were performed using a 3.0-T MR scanner, encompassing sagittal, axial, and oblique coronal T2-weighted images without fat saturation, along with DWI perpendicular to the rectum's long axis. Various DWI parameters, including apparent diffusion coefficient (ADC), stretched exponential model (SEM), continuous-time random-walk model (CTRW), and fractional-order calculus model (FROC), were measured. The pathologic complete response (pCR) rate and tumor downstaging (T-downstage) rate were determined. After nCRT, SEM-α, SEM-DDC, CTRW-α, CTRW-β, CTRW-D, FROC-β, and ADC values were significantly higher in the pCR group compared to the non-pCR group (all P < 0.05). SEM-DDC, CTRW-α, CTRW-D, FROC-β, FROC-µ, and ADC values were significantly higher in the T-downstage group (ypT0-1) than in the non-T-downstage group (ypT2-4) (P < 0.05). The combination of CTRW (α + β + D) exhibited the best diagnostic performance for assessing pCR after nCRT (AUC = 0.840, P < 0.001). Pre-nCRT CTRW (α + β) demonstrated a predictive AUC of 0.652 (95%CI: 0.552-0.743), 90.3% sensitivity, and 43.1% specificity for pCR. Regarding T-downstage assessment after nCRT, the combination of CTRW (α + D) yielded the best diagnostic performance (AUC = 0.877, P = 0.048). In LARC patients, imaging markers derived from CTRW show promise in predicting tumor response before nCRT and assessing pCR after nCRT.

Advanced diffusion-weighted MRI models for preoperative prediction of lymph node metastasis in resectable gastric cancer.

To investigate the potential of six advanced diffusion-weighted imaging (DWI) models for preoperative prediction of lymph node metastasis (LNM) in resectable gastric cancer (GC). Between Nov 2022 and Nov 2023, standard MRI scans were prospectively performed in consecutive patients with endoscopic pathology-confirmed gastric adenocarcinoma who were referred for direct radical gastrectomy. Six DWI models, including fractional order calculus (FROC), continuous-time random walk (CTRW), diffusion kurtosis imaging (DKI), intravoxel incoherent motion (IVIM), the mono-exponential model (MEM) and the stretched exponential model (SEM) were computed. Surgical pathologic diagnosis of LNM was the reference standard, and patients were classified into LNM-positive or LNM-negative groups accordingly. The morphological features and quantitative parameters of the DWI models in different LNM categories were analyzed and compared. Multivariable logistic regression was used to screen significant predictors. Receiver-operating characteristic curves and the area under the curve (AUC) were plotted to evaluate the performances, the Delong test was performed to compare the AUCs. In the LNM-positive group, tumor thickness and kurtosis (DKI_K) were significantly higher, while anomalous diffusion coefficient (CTRW_D), diffusivity (DKI_D), diffusion coefficient (FROC_D), pseudodiffusion coefficient (IVIM_D*), perfusion fraction (IVIM_f), and ADC were lower compared to the LNM-negative group. Clinical tumor staging (cT) and CTRW_D were independent predictors. Their combination demonstrated a superior AUC of 0.930, significantly higher than that of individual parameters. Tumor thickness, DKI_K, CTRW_D, DKI_D, FROC_D, IVIM_D*, IVIM_f and ADC were associated with LNM status. The combination of independent predictors of cT and CTRW_D further enhanced the performance.

Hybrid feedback control of PMSG-based WECS with multilevel flying-capacitor inverter enhanced by fractional order extremum seeking

This paper proposes an original hybrid control strategy for a three-phase multilevel flying capacitor inverter (FCI) used in wind energy conversion system (WECS). A state feedback law, computed using a Lyapunov function and an invariance principle, is designed to manage grid current control and capacitor voltage balancing, guaranteeing the asymptotic stability of the closed-loop system. Additionally, a fractional-order extremum seeking control (FOESC) algorithm for Maximum Power Point Tracking (MPPT) is introduced, which optimizes power capture by adjusting turbine speed from the DC side without requiring a detailed system model (model-free approach). The use of fractional-order calculus leads to improved convergence and more seamless performance while preserving the robust characteristics of the system. The proposed control strategies are validated through simulation, demonstrating improved transient response and stability under varying wind conditions and parameter uncertainties. These results highlight the potential of advanced control algorithms to enhance the efficiency and reliability of wind energy systems, contributing to global efforts to achieve sustainable energy goals.

Fixed-time fractional-order sliding mode controller with disturbance observer for U-tube steam generator

Water level control of a U-tube steam generator plays a critical and challenging role in ensuring safe and reliable operation for a pressurized-water reactor-type nuclear power plant, as both excessively high and low water level in the U-tube steam generator would affect the normal operation of the nuclear power plant and even lead to unplanned shutdown events. This work presents a fixed-time fractional-order sliding mode water level controller coupled with a fixed-time disturbance observer, aiming to further improve the water level control performance under full operating conditions, considering uncertainties and actuator saturation. The proposed scheme is formulatedbased on the popular Irving water level model with uncertainties and the pre-existing actual water level control framework, while at the same time incorporating the benefits of the state-of-the-art anti-windup techniques, disturbance observer-based feedforward compensation techniques, fractional-order calculus, sliding mode control techniques, and fixed-time stability theorem, such that the transient water level response and steady-state control accuracy can be further enhanced even in the presence of uncertainties and actuator saturation phenomenon. It is theoretically proved that under the proposed scheme, both the estimation error of the uncertainties and the water level control error can be driven to zero within a fixed time regardless of their initial values by Lyapunov’s theorem. Meanwhile, simulation studies, involving the comparisons with a real-world adopted water level controller and a pre-existing integer-order sliding mode water level controller coupled with an uncertainty estimator, reveal the feasibility and superiority of the proposed approach.

Cluster synchronization of fractional-order coupled genetic regulatory networks via pinning control

Cell-to-cell interaction and quorum sensing are universal in real life and have an extremely important biological significance. Besides, fractional-order calculus has tremendous advantages in describing the memory of neurons and the inheritance of genes. In this article, the cluster synchronization of fractional-order coupled genetic regulatory networks (GRNs) is explored by applying pinning control technique. First of all, a type of coupled fractional-order models about GRNs is proposed based on the quorum sensing and cluster expression characteristics of genes. Secondly, by designing pinning control strategies and fractional-order inequalities, several sufficient criteria are constructed to reach cluster synchronization. Furthermore, the selection rule of pining nodes is provided to conduct what nodes can be selected to be pinned and how many nodes are pinned to achieve cluster synchronization. In addition, the fractional-order adaptive control strategy is designed to regulate control gains. Several numerical results are provided lastly to confirm the theoretical analysis.

Assessment of prognostic indicators and KRAS mutations in rectal cancer using a fractional-order calculus MR diffusion model: whole tumor histogram analysis.

This study aims to explore the relationship between apparent diffusion coefficient (ADC) and fractional-order calculus (FROC)-specific parameters with prognostic indicators and Kirsten rat sarcoma viral oncogene homologue (KRAS) mutation status in rectal cancer. One hundred fifty-eight patients with rectal cancer were retrospectively enrolled. Histogram measurements of ADC, diffusion coefficient (D), intravoxel diffusion heterogeneity (β), and a microstructural quantity (μ) were estimated for the whole-tumor volume. The relationships between histogram measurements and prognostic indicators were evaluated. The efficacy of histogram measurements, both conducted singly and in conjunction, for evaluating different KRAS mutation statuses was also assessed. The performance of mean and median histogram measurements in evaluating various KRAS mutation statuses was assessed using Receiver Operating Characteristic (ROC) curve analysis. A p-value of less than 0.05 was considered statistically significant. The histogram measurements of ADC, D, β, and μ differed significantly between well-moderately differentiated groups and poorly differentiated groups, T1-2 and T3-4 subgroups, lymph node metastasis (LNM)-negative and LNM-positive subgroups, extranodal extension (ENE)-negative and ENE-positive subgroups, tumor deposit (TD)-negative and TD-positive subgroups, and lymphovascular invasion (LVI)-negative and LVI-positive subgroups. The combination of Dmean, βmean, and μmean achieved the highest performance [The area under the ROC curve (AUC) = 0.904] in evaluating the KRAS mutation status. When assessing parameters from the FROC model as potential biomarkers through histograms, they surpass traditional ADC values in distinguishing prognostic indicators and determining KRAS mutation status in rectal cancer.

Dynamic analysis of planetary gear transmission based on Lagrange interpolation polynomials

Abstract This paper proposes a novel dynamic model considering manufacturing errors and eccentricity errors to analyze the dynamics of planetary gear transmission (PGT). The dynamic model is established based on the fractional-order calculus (FOC) and solved by an enhanced fourth-order Lagrange interpolation polynomials (LIP) method. Three numerical examples and the vibration experiments of planetary gear transmission are employed for verification. The comparison results indicate that the proposed solution method has higher solution accuracy and efficient than the existing algorithms in solving fractional equations, and the relative errors of the proposed solution method in three examples are 0.32%, 0.78% and 0.16%, respectively. The proposed dynamic model of PGT has better agreement with the experimentally measured signal compared with the integer-order dynamic model, and the maximum error and average error of the characteristic frequency amplitude between the proposed dynamic model and the measured signal are 4.76% and 3.57%, respectively. The proposed method contributes to the theoretical foundation for the signal monitoring of PGT.

Value of fractional-order calculus (FROC) model diffusion-weighted imaging combined with simultaneous multi-slice (SMS) acceleration technology for evaluating benign and malignant breast lesions

BackgroundThis study explores the diagnostic value of combining fractional-order calculus (FROC) diffusion-weighted model with simultaneous multi-slice (SMS) acceleration technology in distinguishing benign and malignant breast lesions.Methods178 lesions (73 benign, 105 malignant) underwent magnetic resonance imaging with diffusion-weighted imaging using multiple b-values (14 b-values, highest 3000 s/mm2). Independent samples t-test or Mann-Whitney U test compared image quality scores, FROC model parameters (D,, ), and ADC values between two groups. Multivariate logistic regression analysis identified independent variables and constructed nomograms. Model discrimination ability was assessed with receiver operating characteristic (ROC) curve and calibration chart. Spearman correlation analysis and Bland-Altman plot evaluated parameter correlation and consistency.ResultsMalignant lesions exhibited lower D, and ADC values than benign lesions (P < 0.05), with higher values (P < 0.05). In SSEPI-DWI and SMS-SSEPI-DWI sequences, the AUC and diagnostic accuracy of D value are maximal, with D value demonstrating the highest diagnostic sensitivity, while value exhibits the highest specificity. The D and combined model had the highest AUC and accuracy. D and ADC values showed high correlation between sequences, and moderate. Bland-Altman plot demonstrated unbiased parameter values.ConclusionSMS-SSEPI-DWI FROC model provides good image quality and lesion characteristic values within an acceptable time. It shows consistent diagnostic performance compared to SSEPI-DWI, particularly in D and values, and significantly reduces scanning time.

A hybrid fractional order LMS algorithm for power system harmonic estimation

This article proposes a novel multi-innovation fractional order least mean square algorithm for estimating harmonic parameters in power systems with unknown amplitudes and phases. The algorithm incorporates fractional order calculus with variable initial values, which, in contrast to the fixed initial value approach, overcomes the non-locality issue, and converges to its true extreme value. In addition, by extending scalar innovation term to a vector form, the algorithm fully utilizes innovation within a fixed window to improve computational accuracy. The convex combination between integer order and novel fractional order ensures that the performance of the algorithm will not change significantly with the fractional order. Finally, the effectiveness is verified by numerical simulations and comparisons.

A Hybrid Control Framework for Chemical Processes with Long Time Delay: Theory and Experiments.

This paper proposes a hybrid control framework based on internal model concepts, sliding mode control methodology, and fractional-order calculus theory. As a result, a modified Smith predictor (SP) is proposed for nonlinear systems with significant delays. The particular predictive approach enhances the sliding mode control (SMC) controller's transient responses for dead-time processes, and the SMC gives the predictive structure robustness for model mismatches by combining the previous methods with fractional order concepts; the result is a dynamical sliding mode controller. A numerical example is considered to evaluate the performance of the proposed approach, where a step change, external disturbance, and parametric uncertainty test are performed. A real application in the TCLab Arduino kit is presented; the proposed method presented good performance with a little amount of chattering, and in the disturbance rejection case, the overshoot increased with an aggressive response; in both cases, better tuning parameters can improve the process response and the controller action.