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A review and evaluation of numerical tools for fractional calculus and fractional order control
Zhuo Li, Lu Liu, Sina Dehghan, YangQuan Chen, Dingyu Xue
TL;DR
Fractional calculus and fractional-order control have created demand for reliable numerical tools, but available resources are scattered across applications and implementations. This paper surveys Matlab-based tools, describes their algorithms and usage, and evaluates their accuracy and performance to guide tool selection. The comparison identifies different strengths across tool categories while highlighting practical trade-offs and accuracy limitations.
Problem
Growing use of fractional calculus increases the need to identify suitable numerical tools for fractional integration, differentiation, and fractional-order system simulation.
Method
The paper collects Matlab-based tools into a dashboard, describes their usage and algorithms, and evaluates accuracy and performance across selected numerical and control tools.
Results
The comparison finds that glfdiff and FIT are most accurate for function integration or differentiation, while NILT provides higher accuracy for control-system simulation.
Takeaways & Limitations
Tool selection should consider application-specific accuracy together with capabilities such as Simulink integration and nonlinear-system simulation.
Takeaways & Limitations
Numerical computation of Mittag-Leffler functions introduces error because their analytical expression is an infinite-term summation.
Abstract
from arXiv · showhide
In recent years, as fractional calculus becomes more and more broadly used in research across different academic disciplines, there are increasing demands for the numerical tools for the computation of fractional integration/differentiation, and the simulation of fractional order systems. Time to time, being asked about which tool is suitable for a specific application, the authors decide to carry out this survey to present recapitulative information of the available tools in the literature, in hope of benefiting researchers with different academic backgrounds. With this motivation, the present article collects the scattered tools into a dashboard view, briefly introduces their usage and algorithms, evaluates the accuracy, compares the performance, and provides informative comments for selection.
1. Introduction
The paper surveys Matlab-based numerical tools for fractional calculus and fractional order control, combining tool descriptions with quantitative evaluation and practical selection guidance. It organizes scattered resources into a dashboard view and compares their usage, accuracy, features, advantages, and drawbacks.
- Scope and motivation: The survey collects Matlab-based tools for fractional integration, differentiation, and fractional order control applications.It presents the available resources in a consolidated dashboard view.
- Scope and motivation: Table 1 summarizes the tools' usage, algorithms, accuracy, unique features, advantages, and drawbacks.
- Paper organization: The paper reviews 20 selected numerical tools, evaluates quantitative performance in three categories, and comments on tool selection based on empirical use.
2. Review and description
The reviewed tools cover fundamental fractional-calculus computations and fractional-order control, with different modeling, approximation, simulation, and usability characteristics. Their practical capabilities are accompanied by implementation and compatibility limitations.
- Review scope: The review emphasizes fractional integration/differentiation, fractional differential-equation tools, and fractional-order control applications.Automatic control receives additional focus because it was among the earliest engineering areas to use fractional calculus.
- @fotf: @fotf extends Matlab transfer-function functionality to fractional-order models and supports time delay, but not direct transfer-function matrices.MIMO relationships can instead be established manually through Simulink block diagrams.
- @fotf: @fotf's sampling time has a relatively large impact on accuracy.The paper notes this issue in the validation comments and mentions an upcoming update.
- Ninteger: Ninteger approximates fractional-order integrators and differentiators with integer-order transfer functions and provides CRONE, Carlson, and Matsuda frequency-domain methods.
- Ninteger: Ninteger provides Simulink blocks and a graphical interface for fractional-order PID design, but conflicts with some built-in functions in Matlab 2013a or later without editing.
2.3 ooCroneToolbox
The CRONE toolbox supports broad fractional-order engineering workflows, including MIMO transfer functions, identification, and robust control analysis and design. The reviewed ecosystem also includes FOMCON and multiple numerical implementations for Mittag-Leffler functions and related generalized functions.
- CRONE Toolbox: CRONE supports MIMO fractional transfer functions and is used for fractional-order system identification and robust control analysis and design.The toolbox extends beyond basic fractional-order system simulation.
- CRONE Toolbox: CRONE cannot incorporate time delay directly into generated fractional-order transfer functions.The paper also notes that manually multiplying the delay by the fractional transfer-function object does not work because the relevant operation is not overloaded.
- FOMCON: FOMCON combines algorithms from FOTF, Ninteger, and CRONE and adds a graphical interface for fractional-order controller design.
- FOMCON: FOMCON adds string-based transfer-function input and conversions between fractional-order transfer-function and state-space objects.
- M-L functions: The reviewed M-L functions compute one- to four-parameter Mittag-Leffler forms using precision, error-tolerance, fast-truncation, or C-MEX implementations.Generalized hypergeometric functions can also compute generalized Mittag-Leffler functions under certain conditions.
- M-L functions: Generalized hypergeometric-function implementations provide an alternative route for computing generalized Mittag-Leffler functions under certain conditions.
2.6 NILT
The section presents numerical Laplace-transform inversion and related discretization tools, emphasizing time-delay handling and careful alignment of sampled time vectors for error comparisons.
- Numerical Laplace-transform inversion becomes necessary when analytical solutions are difficult and transform tables do not cover arbitrary cases.
- NILT tools support direct time-delay inputs in the form exp(-Ls), although INVLAP() can show a glitch at the end of the delay.
- NILT may compute one fewer point than regularly spaced baseline vectors because its script uses linspace(0,tm,M).
- Analytical comparisons should use NILT’s time vector, such as t=linspace(0,tm,M), to align timestamps and sample counts.
- DFOD: DFOD provides fractional differentiator and integrator approximations through IIR and FIR implementations based on continued-fraction or power-series expansions.
- Discretization: Other discretization approaches include Prony’s technique, direct discretization, and binomial expansion of the backward difference.
2.8 IRID
IRID is a family of approximation functions for fractional-order transfer functions and operators, including fractional powers, distributed-order integration, filters, and differentiators.
- IRID comprises functions designed for different approximation purposes within the impulse response invariant discretization family.
- fod() approximates the continuous irrational transfer function s^α with a discrete-time finite-dimensional z transfer function.
- doi() approximates a distributed-order integrator involving fractional powers integrated over an order range.
- dolp() approximates the continuous-time fractional low-pass filter 1/(τs + 1)^α.
- fsof() approximates a fractional second-order filter, while irid() approximates the Bode’s Ideal Cut-Off transfer function.
- foc() approximates fractional-order differentiators 1/s^α using the Oustaloup-Recursive-Approximation method.
2.10 fderiv
The fderiv-related tools compute fractional derivatives and integrals using G-L finite differences, Fourier expansion, quadrature, and large-scale Riemann–Liouville implementations.
- fderiv() computes the fractional derivative of order α for a function r(t) using the Grünwald-Letnikov definition.
- An improved fgl-deriv() implementation uses vectorization for faster Matlab computation.
- glfdiff(y,t,α) calculates the αth derivative from signal and time vectors using a forward finite-difference approximation of the G-L definition.
- A Fourier-series-based function computes an αth-order derivative or integral over a specified range using Gauss-Legendre quadrature.
- FIT computes Riemann-Liouville fractional integration and differentiation for large datasets and supports parallel GPU execution.
2.14 DFOC
The section surveys digital fractional-order control and simulation tools, covering PID design, optimal control, variable-order differentiation, and educational state-space simulation.
- DFOC: DFOC provides a transfer function for a digital fractional-order PID controller from specified parameters.
- FOPID: The FOPID toolbox designs robust PI^αD^β controllers by converting tuning into a five-parameter optimization using non-smooth H∞ synthesis.
- Related tools: A Sysquake-based interactive fractional-order PID tool exists, but it is not reviewed in detail because it runs outside Matlab.
- FOCP: The FOCP approach converts fractional optimal-control problems to integer-order form through rational approximation from an SVD-derived Hankel matrix and optimizes them with RIOTS 95.
- FSST: FSST is a Matlab/Simulink toolkit for fractional-order discrete state-space education, including state-space and fractional Kalman-filter examples.
- Variable-order differentiation: Dedicated variable-order tools include triangular-strip matrix methods, the fvoderiv Simulink toolkit, and vod, which supports Riemann-Liouville, Caputo, and G-L definitions.
2.20 FO root locus
The paper reviews Matlab tools for plotting fractional-order root loci in both the Laplace s-plane and transformed w-plane. In the demonstrated example, branches from different Riemann-sheet layers aggregate and then bifurcate as system gain varies.
- FO root locus: Three Matlab-based scripts and the @fotf toolbox provide root-locus plotting for fractional-order transfer functions.The tools include frlocus(), forlocus(), and code associated with Machado et al.
- FO root locus: Figure 5 compares the root locus in the Laplace s-plane with its representation in the w = s^1.3 plane.
- FO root locus: Two root-locus branches appear on the first Riemann-sheet layer in the example, with another branch originating from the next sheet.
- FO root locus: As gain varies, the branches aggregate at (−1.25 + 1.1i), after which one approaches the open-loop zero and the other tends to infinity.
3. Evaluation and comparison
The evaluation compares fractional-order tools on benchmark problems using analytical Matlab solutions as baselines and examines numerical accuracy, runtime, and simulation behavior. Results show method- and setting-dependent performance, with some comparisons limited when analytical solutions are unavailable.
- Benchmark design: The benchmarks cover transfer-function responses, function integration/differentiation, and fractional-order state-space simulation.The evaluation includes step and impulse responses, derivative/integral problems, and toolbox-based state-space simulations.
- Benchmark design: The integral absolute error S = ∫_0^T |e(t)|dt quantifies accuracy against Matlab analytical solutions used as the baseline.For the impulse response of the half-order integrator, the infinite first point is excluded from error calculation.
- Comparison I: fod() accuracy depends on finite-dimensional transfer-function order and sampling time; its reported table-2 errors use a 10th-order approximation.At some high orders, rank deficiency occurs during prony() within irid.
- Comparison II: diffint() performs worse than the other integration/differentiation methods for the identity polynomial despite using many Fourier and Gaussian coefficients, but performs better on a third-order polynomial.The comparison also reports computational error and average elapsed time over 20 runs.
- Comparison III: In state-space simulation, FSST with a 1-second step size differs more from the other toolbox results, but unavailable analytical solutions prevent a quantitative accuracy ranking.The comparison uses direct and Simulink-based fractional-order integrator implementations.
4. Comments for selection
Tool selection depends on the task: glfdiff and FIT perform best for function integration/differentiation accuracy, while NILT provides higher accuracy for control-system simulation. Other toolboxes offer implementation advantages such as Simulink integrator blocks.
- Tool selection: glfdiff and FIT outperform other tools in accuracy for integrating or differentiating a function.
- Tool selection: NILT provides higher accuracy for control-system simulation in the reported comparisons.
- Tool selection: ninteger and CRONE offer Simulink integrator blocks, enabling simulation of nonlinear systems.
5. Conclusion
The paper reviews Matlab-based numerical tools for fractional calculus and fractional-order control, evaluates selected tools quantitatively, and provides guidance for choosing tools for specific applications.
- Conclusion: The paper presents a comprehensive review of Matlab-based numerical tools for fractional calculus and fractional-order control.
- Conclusion: Quantitative evaluation of selected tools is conducted to support their comparison.
- Conclusion: The summarized descriptions and numerical comparisons are intended to guide tool selection for specific applications.