1,457
Views
116
CrossRef citations to date
0
Altmetric
Original Articles

A review and evaluation of numerical tools for fractional calculus and fractional order controls

, , , &
Pages 1165-1181 | Received 12 Jul 2015, Accepted 21 Nov 2015, Published online: 06 Jan 2016

References

  • Apkarian, P., & Noll, D. (2006). Nonsmooth H∞ synthesis. IEEE Transactions on Automatic Control, 51(1), 71–86.
  • Axtell, M., & Bise, M.E. (1990). Fractional calculus applications in control systems. National aerospace and electronic conference (pp. 563–566), New York, NY.
  • Bagley, R.L., & Calico, R.A. (1989). Fractional order state equations for the control of viscoelastically damped structure. Journal of Guidance, 14(2), 304–310.
  • Barbosa, R., & Machado, J.A.T. (2006). Implementation of discrete-time fractional-order controllers based on LS approximations. Acta Polytechnica Hung, 3(4), 5–22.
  • Barrowes, B. (2005). Generalized hypergeometric function [Online] (Matlab Central). Retrieved from http://www. mathworks.com/matlabcentral/fileexchange/5616
  • Bayat, F.M. (2007). Fractional differentiator [Online] (Matlab Central). Retrieved from http://www.mathworks.com/ matlabcentral/fileexchange/13858
  • Bayat, F.M. (2008). Root-locus plot of fractional order systems [Online] (Matlab Central). Retrieved from http://www. mathworks.com/matlabcentral/fileexchange/20577
  • Bode, H. (1945). Network analysis and feedback amplifier design. New York, NY: D. Van Nostrand Company.
  • Bohannan, G.W. (2008). Analog fractional order controller in temperature and motor control applications. Journal of Vibration and Control, 14(9), 1487–1498.
  • Branc̆ík, L. (1999). An improved numerical inversion of two-dimensional Laplace transforms with application to transient analysis of transmission lines. Proceedings EDS’99 Brno, Czech Republic.
  • Branc̆ík, L. (2001). Utilization of quotient-difference algorithm in FFT-based numerical ILT method. Proceedings of the 11th international Czech-Slovak scientific conference Radioelektronika, Czech Republic.
  • Caponetto, R., Dongola, G., Fortuna, L., & Petráš, I. (2010). Fractional order systems: Modelling and control applications. Hackensack, NJ: World Scientific.
  • Chaurasia, V., & Pandey, S. (2010). On the fractional calculus of generalized Mittag–Leffler function. SCIENTIA Series A: Mathematical Sciences, 20, 113–122.
  • Chen, D., Chen, Y.Q., & Xue, D. (2011). Digital fractional order Savitzky–Golay differentiator. IEEE Transactions on Circuits and Systems – II: Express Briefs, 58(11), 758–762.
  • Chen, Y.Q. (2003). Oustaloup-recursive-approximation for fractional order differentiators [Online] (Matlab Central). Retrieved from http://www.mathworks.com/ matlabcentral/fileexchange/3802
  • Chen, Y.Q. (2008a). Generalized Mittag–Leffler function [Online] (Matlab Central). Retrieved from http://www. mathworks.com/matlabcentral/fileexchange/20849
  • Chen, Y.Q. (2008b). Impulse response invariant discretization of fractional order integrators/differentiators [Online] (Matlab Central). Retrieved from http://www.mathworks. com/matlabcentral/fileexchange/21342
  • Chen, Y.Q., Petráš, I., & Xue, D. (2009, June). Fractional order control – a tutorial. American control conference, 2009. ACC’09 (pp. 1397–1411), St. Louis, MO.
  • Chen, Y.Q., & Vinagre, B.M. (2003). A new IIR-type digital fractional order differentiator. Signal Processing, 83, 2359–2365.
  • Corless, R.M., & Jeffrey, D.J. (1998). Graphing elementary Riemann surface. SIGSAM Bulletin, 32(1), 11–17.
  • Das, S., & Pan, I. (2012). Fractional order signal processing: Introductory concepts and applications. Berlin: Springer.
  • de Hoog, F., Knight, J., & Stokes, A.N. (1982). An improved method for numerical inversion of Laplace transforms. SIAM Journal on Scientific Computing, 3(3), 357–366.
  • Ding, Z., Granger, C.W., & Engle, R.F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1, 83–106.
  • Dzieliński, A., & Sierociuk, D. (2008). Simulation and experimental tools for fractional order control education. Proceedings IFAC World Congress (pp. 11 654–11 659), Seoul, Korea.
  • Farkas, H.M., & Kra, I. (1980). Riemann surfaces (2nd ed.). New York, NY: Springer-Verlag.
  • Garrappa, R. (2014). The Mittag–Leffler function [Online] (Matlab Central). Retrieved from http://www.mathworks. com/matlabcentral/fileexchange/48154
  • Germano, G. (2008). Mittag–Leffler random number generator [Online] (Matlab Central). Retrieved from http://www. mathworks.com/matlabcentral/fileexchange/19392
  • Guzman, J.L., Astrom, K.J., Dormido, S., Hagglund, T., Berenguel, M., & Piguet, Y. (2008). Interactive modules for PID control. EEE Control Systems Magazine I, 28(5), 118–134.
  • Huntley, J. (2012). Generation of random variates [Online] (Matlab Central). Retrieved from http://www.mathworks. com/matlabcentral/fileexchange/35008
  • Jiang, C., Hartley, T.T., Carletta, J., & Veillette, R.J. (2013). A systematic approach for implementing fractional-order operators and systems. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 3(3), 301–312.
  • Jonathan. (2014). Fractional derivative [Online] (Matlab Central). Retrieved from http://www.mathworks. com/matlabcentral/fileexchange/45982
  • Juraj. (2011). Numerical inversion of Laplace transforms in Matlab [Online] (Matlab Central). Retrieved from http:// www.mathworks.com/matlabcentral/fileexchange/32824
  • Lachhab, N., Svaricek, F., Wobbe, F., & Rabba, H. (2013). Fractional order PID controller (FOPID) – toolbox. 2013 European control conference (ECC) (pp. 3694–3699), Zuich, Switzerland.
  • Lewis, T.G. (2014). Book of extremes: Why the 21st century isn’t like the 20th century. Switzerland: Springer International Publishing.
  • Li, H., Luo, Y., & Chen, Y.Q. (2010). A fractional order proportional and derivative (FOPD) motion controller: Tuning rule and experiments. IEEE Transactions on Control Systems Technology, 18(2), 516–520.
  • Li, Y., Sheng, H., & Chen, Y.Q. (2010, July). Impulse response invariant discretization of a generalized commensurate fractional order filter. Proceedings of the 8th World Congress on Intelligent Control and Automation, Jinan, China.
  • Li, Z. (2015). Fractional order root locus [online] (Matlab Central). Retrieved from http://www.mathworks. com/matlabcentral/fileexchange/50458
  • Li, Z., & Chen, Y.Q. (2014). Identification of linear fractional order systems using the relay feedback approach. 2014 American control conference (ACC), Portland, OR.
  • Liang, J. (2005). Control of linear time-invariant distributed parameter systems: From integer order to fractional order (Doctoral dissertation). Logan, UT: Utah State University.
  • Lorenzo, C.F., & Hartley, T.T. (2002). Variable order and distributed order fractional operators. Nonlinear Dynamics, 29(1), 57–98.
  • Luo, Y., & Chen, Y.Q. (2013). Fractional order motion controls. West Sussex: John Wiley & Sons.
  • Machado, J.A.T. (2011). Root locus of fractional linear systems. Communications in Nonlinear Science and Numerical Simulation, 16, 3855–3862.
  • Machado, J.T., Kiryakova, V., & Mainardi, F. (2011). Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation, 16, 1140–1153.
  • Magin, R.L. (2006). Fractional calculus in bioengineering. Redding, CA: Begell House.
  • Malkiel, B.G. (1999). A random walk down wall street (7th ed.). New York, NY: W.W. Norton & Company.
  • Marinov, T.M., Ramirez, N., & Santamaria, F. (2013a). Fractional integration toolbox. Fractional Calculus & Applied Analysis, 16(3), 670–681.
  • Marinov, T.M., Ramirez, N., & Santamaria, F. (2013b). Fractional integration toolbox (FIT) [Online] (Santamaria Lab). Retrieved from http://www.cbi.utsa.edu/FIT
  • Miller, K.S., & Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations (1st ed.). New York, NY: Wiley-Interscience.
  • Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., & Feliu, V. (2006). Fractional order systems and controls: Fundamentals and applications. London: Springer-Verlag.
  • Mukhopadhyay, S. (2008). Mittag-leffler function, M-file, cmex DLL, and S-function [Online] (Matlab Central). Retrieved from http://www.mathworks.com/matlabcentral/ fileexchange/20731
  • Narayanana, V.A., & Prabhu, K. (2003). The fractional fourier transform: Theory, implementation and error analysis. Microprocessors and Microsystems, 27, 511–521.
  • Oustaloup, A., Levron, F., Mathieu, B., & Nanot, F.M. (2000). Frequency-band complex noninteger differentiator: Characterization and synthesis. IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications, 47(1), 25–39.
  • Oustaloup, A., Melchior, P., Lanusse, P., Cois, O., & Dancla, F. (2000). The CRONE toolbox for matlab. In Computer-aided control system design (pp. 190–195), Anchorage, AK. doi:10.1109/CACSD.2000.900210
  • Ozaktas, H.M., Zalevsky, Z., & Kutay, M.A. (2001). The fractional Fourier transform. Hoboken, NJ: John Wiley & Sons.
  • Papazafeiropoulos, G. (2014). Fractional differentiation and integration [Online] (Matlab Central). http:// www.mathworks.com/matlabcentral/fileexchange/45877
  • Petráš, I. (2003a). Digital fractional order differentiator/ integrator – IIR type [Online] (Matlab Central). Retrieved from http://www.mathworks.com/matlabcentral/fileexchange/ 3672
  • Petráš, I. (2003b). Digital fractional order differentiator/integrator – FIR type [Online] (Matlab Central). Retrieved from http://www.mathworks.com/ matlabcentral/fileexchange/3673,
  • Petráš, I. (2011a). Digital fractional order differentiator/integrator – new IIR type [Online] (Matlab Central). http://www.mathworks.com/matlabcentral/fileexchange/ 31358
  • Petráš, I. (2011b). Discrete fractional-order PID controller [Online] (Matlab Central). Retrieved from http:// www.mathworks.com/matlabcentral/fileexchange/33761
  • Petráš, I. (2011c). Fractional derivatives, fractional integrals, and fractional differential equations in Matlab. In Engineering education and research using MATLAB (pp. 239–264). Winchester: InTech.
  • Petráš, I. (2011d). Fractional-order nonlinear systems: Modeling, analysis and simulation. Berlin: Springer Science & Business Media.
  • Pisoni, E., Visioli, A., & Dormido, S. (2009). An interactive tool for fractional order PID controllers. IECON ’09. 35th annual conference of IEEE. Porto, Portugal.
  • Podlubny, I. (1999a). Fractional differential equations. Waltham, MA: Academic Press.
  • Podlubny, I. (1999b). Fractional-order systems and PIλDμ controllers. IEEE Transactions on Automatic Control, 44(1), 208–214.
  • Podlubny, I. (2000). Matrix approach to discrete fractional calculus. Factional Calculus and Applied Analysis, 29(4), 281–296.
  • Podlubny, I. (2005). Mittag–Leffler function [Online] (Matlab Central). Retrieved from http://www.mathworks.com/ matlabcentral/fileexchange/8738
  • Podlubny, I. (2008). Matrix approach to discretization of ODEs and PDEs of arbitrary real order [Online] (Matlab Central). Retrieved from http://www.mathworks. com/matlabcentral/fileexchange/22071
  • Prabhakar, T. (1971). A singular integral equation with a generalised Mittag–Leffler function in the kernel. Yokohama Mathematical Journal, 19, 7–15.
  • Sabatier, J., Agrawal, O., & Machado, J.T. (Eds.). (2007). Advances in fractional calculus: Theoretical developments and applications in physics and engineering. Berlin: Springer.
  • Sheng, H., Li, Y., & Chen, Y.Q. (2011). Application of numerical inverse laplace transform algorithms in fractional calculus. Journal The Franklin Institute, 348, 315–330.
  • Sierociuk, D. (2003). Fractional states-space toolkit (FSST). [Online] http://www.ee.pw.edu.pl/ dsieroci/fsst/fsst.htm
  • Sierociuk, D. (2012). Fractional variable order derivative Simulink toolkit [Online] (Matlab Central). Retrieved from http://www.mathworks.com/matlabcentral/fileexchange/ 38801
  • Tepljakov, A. (2011). Fractional-order calculus based identification and control of linear dynamic systems ( Master's thesis). Tallinn: Tallinn University of Technology.
  • Tepljakov, A., Petlenkov, E., & Belikov, J. (2011). FOMCON: Fractional-order modeling and control toolbox for MATLAB. 18th international conference on “Mixed Design of Integrated Circuits and Systems”, Gliwice, Poland.
  • The CRONE Team (2014, February). The CRONE toolbox homepage [Internet]. Retrieved from http://www.ims-bordeaux.fr/CRONE/toolbox
  • Tricaud, C. (2009). Solution of fractional optimal control problems [Online] (Matlab Central). Retrieved from http://www.mathworks.com/matlabcentral/fileexchange/ 22196
  • Tricaud, C., & Chen, Y.Q. (2008). Solving fractional order optimal control problems in RIOTS_95 – a general-purpose optimal control problem solver. Proceedings of the 3rd IFAC workshop on fractional differentiation and its applications, Ankara, Turkey.
  • Tricaud, C., & Chen, Y.Q. (2009). Solution of fractional order optimal control problems using SVD-based rational approximations. The 2009 American control conference (ACC) (pp. 1430–1435), St. Louis, MO.
  • Valério, D. (2009). Variable order derivatives [Online] (Matlab Central). Retrieved from http://www.mathworks. com/matlabcentral/fileexchange/24444
  • Valério, D., & da Costa, J.S. (2004). Ninteger: A non-integer control toolbox for Matlab. 1st IFAC workshop on fractional differentiation and applications (pp. 208–213), Bordeaux, France.
  • Valério, D., & da Costa, J.S. (2005, September). Time-domain implementation of fractional order controllers. IEE Proceedings – Control Theory and Applications, 152(5), 539–552.
  • Valsa, J., & Branc̆ík, L. (1998). Fractional order state equations for the control of viscoelastically damped structure. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 11(3), 153–166.
  • West, B.J. (2006). Where medicine went wrong. Singapore: World Scientific.
  • West, B.J., Turalska, M., & Grigolini, P. (2014). Complex networks: From social crises to neuronal avalanches (pp. 509–524). Hoboken, NJ: Wiley-VCH Verlag GmbH.
  • Xue, D., & Chen, Y.Q. (2014a). Modeling, analysis and design of control systems in Matlab and Simulink. Singapore: World Scientific.
  • Xue, D., & Chen, Y.Q. (2014b). System simulation techniques with Matlab and Simulink. Singapore: John Wiley & Sons.
  • Xue, D., Chen, Y.Q., & Atherton, D. (2009). Linear feedback control – analysis and design with Matlab 6.5. Hoboken, NJ: Society for Industrial and Applied Mathematics.
  • Yin, C., Chen, Y.Q., & Zhong, S.M. (2014). Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems. Automatica, 50, 3173–3181.
  • Yousfi, N., Melchior, P., Rekik, C., Derbel, N., & Oustaloup, A. (2012). Design of centralized CRONE controller combined with MIMO-QFT approach applied to non-square multivariable systems. International Journal of Computer Applications, 45, 6–14.
  • Zhao, T., Chen, Y.Q., & Li, Z. (2014). Fractional order nonlinear model predictive control using RIOTS_95. The international conference on fractional differentiation and its applications (ICFDA), Catania, Italy.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.