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Volume 3 Issue 4
Oct.  2016

IEEE/CAA Journal of Automatica Sinica

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Mohammad Saleh Tavazoei, "Criteria for Response Monotonicity Preserving in Approximation of Fractional Order Systems," IEEE/CAA J. Autom. Sinica, vol. 3, no. 4, pp. 422-429, Oct. 2016.
Citation: Mohammad Saleh Tavazoei, "Criteria for Response Monotonicity Preserving in Approximation of Fractional Order Systems," IEEE/CAA J. Autom. Sinica, vol. 3, no. 4, pp. 422-429, Oct. 2016.

Criteria for Response Monotonicity Preserving in Approximation of Fractional Order Systems

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This work was supported by the Research Council of Sharif University of Technology G930720

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  • In approximation of fractional order systems, a significant objective is to preserve the important properties of the original system. The monotonicity of time/frequency responses is one of these properties whose preservation is of great importance in approximation process. Considering this importance, the issues of monotonicity preservation of the step response and monotonicity preservation of the magnitude-frequency response are independently investigated in this paper. In these investigations, some conditions on approximating filters of fractional operators are found to guarantee the preservation of step/magnitude-frequency response monotonicity in approximation process. These conditions are also simplified in some special cases. In addition, numerical simulation results are presented to show the usefulness of the obtained conditions.

     

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  • [1]
    Oldham K B, Spanier J. The Fractional Calculus:Theory and Applications of Differentiation and Integration to Arbitrary Order. UK:Dover Publications Inc., 2006.
    [2]
    Ortigueira M D. An introduction to the fractional continuous-time linear systems:the 21st century systems. IEEE Circuits and Systems Magazine, 2008, 8(3):19-26 doi: 10.1109/MCAS.2008.928419
    [3]
    Cafagna D. Fractional calculus:a mathematical tool from the past for present engineers. IEEE Industrial Electronics Magazine, 2007, 1(2):35-40 doi: 10.1109/MIE.2007.901479
    [4]
    Tavazoei M S. From traditional to fractional PI control:a key for generalization. IEEE Industrial Electronics Magazine, 2012, 6(3):41-51 doi: 10.1109/MIE.2012.2207818
    [5]
    Ma C B, Hori Y. Fractional-order control:theory and applications in motion control. IEEE Industrial Electronics Magazine, 2007, 1(4):6-16 doi: 10.1109/MIE.2007.909703
    [6]
    Tavakoli-Kakhki M, Tavazoei M S. Proportional stabilization and closedloop identification of an unstable fractional order process. Journal of Process Control, 2014, 24(5):542-549 doi: 10.1016/j.jprocont.2014.02.019
    [7]
    Tavakoli-Kakhki M, Tavazoei M S. Estimation of the order and parameters of a fractional order model from a noisy step response data. Journal of Dynamic Systems, Measurement and Control, 2014, 136(3):031020 doi: 10.1115/1.4026345
    [8]
    Elwakil A S. Fractional-order circuits and systems:an emerging interdisciplinary research area. IEEE Circuits and Systems Magazine, 2010, 10(4):40-50 doi: 10.1109/MCAS.2010.938637
    [9]
    Elwakil A, Maundy B, Fortuna L, Chen G R. Guest editorial fractionalorder circuits and systems. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 2013, 3(3):297-300 doi: 10.1109/JETCAS.2013.2273856
    [10]
    Yang C, Yu H, Shang Y, Fei W. Characterization of CMOS metamaterial transmission line by compact fractional-order equivalent circuit model. IEEE Transactions on Electron Devices, 2015, 62(9):3012-3018 doi: 10.1109/TED.2015.2458931
    [11]
    Ortigueira M D, Ionescu C M, Machado J T, Trujillo J J. Fractional signal processing and applications. Signal Processing, 2015, 107:197 doi: 10.1016/j.sigpro.2014.10.002
    [12]
    Cuesta E, Kirane M, Malik S A. Image structure preserving denoising using generalized fractional time integrals. Signal Processing, 2012, 92(2):553-563 doi: 10.1016/j.sigpro.2011.09.001
    [13]
    Efe MÖ. ADALINE based robust control in robotics:a Riemann-Liouville fractional differintegration based learning scheme. Soft Computing, 2009, 13(1):23-29 doi: 10.1007/s00500-008-0289-9
    [14]
    Rekanos I T, Yioultsis T V. Approximation of Grünwald-Letnikov fractional derivative for FDTD modeling of cole-cole media. IEEE Transactions on Magnetics, 2014, 50(2):181-184 doi: 10.1109/TMAG.2013.2281998
    [15]
    Du Y C, Chen W L, Lin C H, Kan C D, Wu M J. Residual stenosis estimation of arteriovenous grafts using a dual-channel phonoangiography with fractional-order features. IEEE Journal of Biomedical and Health Informatics, 2015, 19(2):590-600 doi: 10.1109/JBHI.2014.2328346
    [16]
    Tavazoei M S. Reduction of oscillations via fractional order pre-filtering. Signal Processing, 2015, 107:407-414 doi: 10.1016/j.sigpro.2014.03.008
    [17]
    Muresan C I, Dulf E H, Prodan O. A fractional order controller for seismic mitigation of structures equipped with viscoelastic mass dampers. Journal of Vibration and Control, 2014. doi:10.1177/1077546314557553, to be published
    [18]
    Mescia L, Bia P, Caratelli D. Fractional derivative based FDTD modeling of transient wave propagation in Havriliak-Negami media. IEEE Transactions on Microwave Theory and Techniques, 2014, 62(9):1920-1929 doi: 10.1109/TMTT.2014.2327202
    [19]
    Di Paola M, Pirrotta A, Valenza A. Visco-elastic behavior through fractional calculus:an easier method for best fitting experimental results. Mechanics of Materials, 2011, 43(12):799-806 doi: 10.1016/j.mechmat.2011.08.016
    [20]
    Maione G. High-speed digital realizations of fractional operators in the delta domain. IEEE Transactions on Automatic Control, 2011, 56(3):697-702 doi: 10.1109/TAC.2010.2101134
    [21]
    Oustaloup A, Levron F, Mathieu B, Nanot F M. Frequency-band complex noninteger differentiator:characterization and synthesis. IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications, 2000, 47(1):25-39 doi: 10.1109/81.817385
    [22]
    Maione G. Closed-form rational approximations of fractional, analog and digital differentiators/integrators. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 2013, 3(3):322-329 doi: 10.1109/JETCAS.2013.2268949
    [23]
    Chen Y Q, Moore K L. Discretization schemes for fractional-order differentiators and integrators. IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications, 2002, 49(3):363-367 doi: 10.1109/81.989172
    [24]
    Maione G. Continued fractions approximation of the impulse response of fractional-order dynamic systems. IET Control Theory and Applications, 2008, 2(7):564-572 doi: 10.1049/iet-cta:20070205
    [25]
    Charef A. Analogue realisation of fractional-order integrator, differentiator and fractional PIλ Dμ controller. IEE Proceedings-Control Theory and Applications, 2006, 153(6):714-720 doi: 10.1049/ip-cta:20050019
    [26]
    Vinagre B M, Podlubny I, Hernández A, Feliu V. Some approximations of fractional order operators used in control theory and applications. Fractional Calculus and Applied Analysis, 2000, 3(3):231-248 http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.452.5372
    [27]
    Tavazoei M S, Haeri M. Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems. IET Signal Processing, 2007, 1(4):171-181 doi: 10.1049/iet-spr:20070053
    [28]
    Tavazoei M S. Comments on "Stability analysis of a class of nonlinear fractional-order systems". IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs, 2009, 56(6):519-520 http://cn.bing.com/academic/profile?id=2139986549&encoded=0&v=paper_preview&mkt=zh-cn
    [29]
    Tavazoei M S. Comments on "Chaotic characteristics analysis and circuit implementation for a fractional-order system". IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications, 2015, 62(1):329-332 http://cn.bing.com/academic/profile?id=2085157477&encoded=0&v=paper_preview&mkt=zh-cn
    [30]
    Tavazoei M S, Haeri M. Rational approximations in the simulation and implementation of fractional-order dynamics:a descriptor system approach. Automatica, 2010, 46(1):94-100 doi: 10.1016/j.automatica.2009.09.016
    [31]
    Tavazoei M S, Haeri M, Bolouki S, Siami M. Stability preservation analysis for frequency-based methods in numerical simulation of fractional order systems. SIAM Journal on Numerical Analysis, 2008, 47(1):321-338 http://cn.bing.com/academic/profile?id=1992065362&encoded=0&v=paper_preview&mkt=zh-cn
    [32]
    Siami M, Tavazoei M S, Haeri M. Stability preservation analysis in direct discretization of fractional order transfer functions. Signal Processing, 2011, 91(3):508-512 doi: 10.1016/j.sigpro.2010.06.009
    [33]
    Darbha S. On the synthesis of controllers for continuous time LTI systems that achieve a non-negative impulse response. Automatica, 2003, 39(1):159-165 doi: 10.1016/S0005-1098(02)00202-9
    [34]
    Rachid A. Some conditions on zeros to avoid step-response extrema. IEEE Transactions on Automatic Control, 1995, 40(8):1501-1503 doi: 10.1109/9.402253
    [35]
    Tavazoei M S. Overshoot in the step response of fractional-order control systems. Journal of Process Control, 2012, 22(1):90-94 doi: 10.1016/j.jprocont.2011.10.005
    [36]
    Bement M, Jayasuriya S. Construction of a set of nonovershooting tracking controllers. Journal of Dynamic Systems, Measurement, and Control, 2004, 126(3):558-567 doi: 10.1115/1.1789971
    [37]
    Tavazoei M S. On type number concept in fractional-order systems. Automatica, 2013, 49(1):301-304 doi: 10.1016/j.automatica.2012.09.022
    [38]
    Kim Y C, Keel L H, Bhattacharyya S P. Transient response control via characteristic ratio assignment. IEEE Transactions on Automatic Control, 2003, 48(12):2238-2244 doi: 10.1109/TAC.2003.820153
    [39]
    Filanovsky I M. A generalization of filters with monotonic amplitudefrequency response. IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications, 1999, 46(11):1382-1385 doi: 10.1109/81.802839
    [40]
    Kidambi S S. Simple method for design of monotonic analogue filters. Electronics Letters, 2000, 36(4):287-288 doi: 10.1049/el:20000266
    [41]
    Tavazoei M S. On monotonic and nonmonotonic step responses in fractional order systems. IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs, 2011, 58(7):447-451 doi: 10.1109/TCSII.2011.2158258
    [42]
    Tavazoei M S. Algebraic conditions for monotonicity of magnitudefrequency responses in all-pole fractional order systems. IET Control Theory and Applications, 2014, 8(12):1091-1095 doi: 10.1049/iet-cta.2013.0999
    [43]
    Tavazoei M S. Fractional/distributed-order systems and irrational transfer functions with monotonic step responses. Journal of Vibration and Control, 2014, 20(11):1697-1706 doi: 10.1177/1077546313481049
    [44]
    Tavakoli-Kakhki M, Haeri M, Tavazoei M S. Notes on the state space realizations of rational order transfer functions. IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications, 2011, 58(5):1099-1108 doi: 10.1109/TCSI.2010.2090568
    [45]
    Hartley T T, Lorenzo C F, Qammer H K. Chaos in a fractional order Chuas system. IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications, 1995, 42(8):485-490 doi: 10.1109/81.404062
    [46]
    Abate J, Choudhury G L, Whitt W. An introduction to numerical transform inversion and its application to probability models. Computational Probability. Boston:Kluwer, 1999. 257-323. http://cn.bing.com/academic/profile?id=1577201783&encoded=0&v=paper_preview&mkt=zh-cn
    [47]
    Bryan K. Elementary inversion of the Laplace transform, Preprint[Online]. available:http://www.rose-hulman.edu/bryan/invlap.pdf. 2006.
    [48]
    Roman S. The formula of Faa di bruno. American Mathematical Monthly, 1980, 87(10):805-809 doi: 10.2307/2320788
    [49]
    Valério D. Toolbox ninteger for MatLab[Online]. available:http://web.ist.utl.pt/duarte.valerio/ninteger/ninteger.htm. 2011.
    [50]
    Singh R R. Electrical Networks. New Delhi:Tata McGraw-Hill, 2009.
    [51]
    Valério D, da Costa J S. An Introduction to Fractional Control. Stevenage:IET, 2013.

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