Yiheng WEI, Yingdong WEI, Yuqing HOU, Xuan ZHAO

Limited frequency band diffusive representation for nabla fractional order transfer functions

Though infinite-dimensional characteristic is the natural property of nabla fractional order systems and it is the foundation of stability analysis, controller synthesis and numerical realization, there are few research focusing on this topic. Under this background, this paper concerns the diffusive representation of nabla fractional order systems. Firstly, several variants are developed for the elementary equality in frequency domain, i.e. $frac{mathrm 1}{mathrm s ^a}=int_o ^{+infty} frac{mu a(omega)}{s+omega}$dω. Afterwards, the limited frequency band diffusive representation and the unit impulse response are derived for a series of nabla fractional order transfer functions. Finally, an attempt to find the diffusive representation for general nabla fractional order transfer functions is made. Some conclusions are presented.

PDF

___

[1] Čermák J, Kisela T, Nechvátal L. Stability regions for linear fractional differential systems and their discretizations. Applied Mathematics and Computation 2013; 219: 7012-7022. doi: 10.1016/j.amc.2012.12.019
[2] Chen YQ, Wei YH, Zhou X, Wang Y. Stability for nonlinear fractional order systems: an indirect approach. Nonlinear Dynamics 2017; 89 (2): 1011-1018. doi: 10.1007/s11071-017-3497-y
[3] Cheng JF. Fractional Difference Equation Theory. Xiamen, China: Xiamen University Press, 2011.
[4] Djouambi A, Charef A, BesançOn A. Optimal approximation, simulation and analog realization of the fundamental fractional order transfer function. International Journal of Applied Mathematics and Computer Science 2007; 17 (4): 455-462. doi: 10.2478/v10006-007-0037-9
[5] Du B, Wei YH, Liang S, Wang Y. Rational approximation of fractional order systems by vector fitting method. International Journal of Control Automation and Systems 2016; 15 (1): 186-195. doi: 10.1007/s12555-015-0351-1
[6] Du FF, Erbe L, Jia BG, Peterson A. Two asymptotic results of solutions for nabla fractional (q, h)-difference equations. Turkish Journal of Mathematics 2018; 42 (5): 2214-2242. doi: 10.3906/mat-1802-49
[7] Du FF, Jia BG. Finite-time stability of a class of nonlinear fractional delay difference systems. Applied Mathematics Letters, 2019, 98: 233-239. doi: 10.1016/j.aml.2019.06.017
[8] Du FF, Jia BG. Finite time stability of fractional delay difference systems: a discrete delayed Mittag-Leffler matrix function approach. Chaos, Solitons & Fractals, 2020, 141: 110430. doi: 10.1016/j.chaos.2020.110430
[9] Goodrich C, Peterson AC. Discrete Fractional Calculus. Cham, Germany: Springer, 2015.
[10] Gorenflo R, Mainardi F. Fractional calculus: integral and differential equations of fractional order. arXiv: 0805.3823, 2008.
[11] Heleschewitz D, Matignon D. Diffusive realisations of fractional integrodifferential operators: structural analysis under approximation. IFAC Proceedings Volumes 1998; 31 (18): 227-232. doi: 10.1016/s1474-6670(17)41996-3
[12] Hinze M, Schmidt A, Leine RI. Lyapunov stability of a fractionally damped oscillator with linear (anti-)damping. International Journal of Nonlinear Sciences and Numerical Simulation 2020; 21 (5): 425-442. doi: 10.1515/ijnsns2018-0381
[13] Hinze M, Schmidt A, Leine RI. Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation. Fractional Calculus and Applied Analysis 2019; 22 (5): 1321-1350. doi: 10.1515/fca-2019-0070.
[14] Hinze M, Schmidt A, Leine RI. The direct method of Lyapunov for nonlinear dynamical systems with fractional damping. Nonlinear Dynamics 2020; 102: 2017-2037. doi: 10.1007/s11071-020-05962-3
[15] Li A, Wei YH, Wang JC, Wang Y. A numerical approximation method for fractional order systems with new distributions of zeros and poles. ISA Transactions 2020; 99: 20-27. doi: 10.1016/j.isatra.2019.09.001
[16] Liang S. Control theory of fractional order systems. PhD, University of Science and Technology of China, Hefei, China, 2015.
[17] Liang S, Liang Y. Inverse Lyapunov theorem for linear time invariant fractional order systems. Journal of Systems Science and Complexity 2019; 32 (6): 1544-1559. doi: 10.1007/s11424-019-7049-z
[18] Liang S, Peng C, Liao Z, Wang Y. State space approximation for general fractional order dynamic systems. International Journal of Systems Science 2014; 45 (10-12): 2203-2212. doi: 10.1080/00207721.2013.766773
[19] Liu X, Erbe L, Jia BG, Peterson A. Stability analysis for a class of nabla (q, h)-fractional difference equations. Turkish Journal of Mathematics 2019; 43 (2): 664-687. doi: 10.3906/mat-1811-96
[20] Montseny G. Diffusive representation of pseudo-differential time-operators. In: Fractional Differential Systems: Models, Methods and Applications; Paris, France; 1998. pp. 159-175. doi: 10.1051/proc:1998005
[21] Montseny G, Audounet J, Mbodje B. Optimal models of fractional integrators and application to systems with fading memory. In: IEEE International Conference on Systems, Man and Cybernetics; Le Touquet, France; 1993. pp. 65-70. doi: 10.1109/icsmc.1993.390826
[22] Oustaloup A. Diversity and Non-Integer Differentiation for System Dynamics. London, UK: Wiley, 2014.
[23] Oustaloup A, Levron F, Mathieu B, Nanot FM. 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.
[24] Poinot T, Trigeassou JC. A method for modelling and simulation of fractional systems. Signal Processing 2003; 83 (11): 2319-2333. doi: 10.1016/s0165-1684(03)00185-3
[25] Rapaić MR, Šekara TB, Bošković MČ. Frequency-distributed representation of irrational linear systems. Fractional Calculus and Applied Analysis 2018; 21 (5): 1396-1419. doi: 10.1515/fca-2018-0073
[26] Raynaud HF, Zergaınoh A. State-space representation for fractional order controllers. Automatica 2000; 36 (7): 1017-1021. doi: 10.1016/s0005-1098(00)00011-x
[27] Sabatier J. Beyond the particular case of circuits with geometrically distributed components for approximation of fractional order models: application to a new class of model for power law type long memory behaviour modelling. Journal of Advanced Research 2020; 25: 243-255. doi: 10.1016/j.jare.2020.04.004.
[28] Sabatier J. Fractional order models are doubly infinite dimensional models and thus of infinite memory: consequences on initialization and some solutions. Symmetry 2021; 13: 1099. doi: 10.3390/sym13061099
[29] Sabatier J. Fractional state space description: a particular case of the Volterra equations. Fractal and Fractional 2020; 4 (2): 23. doi: 10.3390/fractalfract4020023
[30] Sabatier J. Non-singular kernels for modelling power law type long memory behaviours and beyond. Cybernetics and Systems 2020; 51 (4): 383-401. doi: 10.1080/01969722.2020.1758470
[31] Sabatier J. Solutions to the sub-optimality and stability issues of recursive pole and zero distribution algorithms for the approximation of fractional order models. Algorithms 2018; 11 (7): 103. doi: 10.3390/a11070103
[32] Sabatier J, Cadavid SR, Farges C. Advantages of a limited frequency band fractional integration operator in the definition of fractional models. In: the 6th International Conference on Control, Decision and Information Technologies; Paris, France; 2019. pp. 882-887. doi: 10.1109/codit.2019.8820413
[33] Sabatier J, Farges C. Analysis of fractional models physical consistency. Journal of Vibration and Control 2017; 23 (6): 895-908. doi: 10.1177/1077546315587177
[34] Sabatier J, Farges C, Trigeassou JC. Fractional systems state space description: some wrong ideas and proposed solutions. Journal of Vibration and Control 2014; 20 (7): 1076-1084. doi: 10.1177/1077546313481839
[35] Shen A, Guo YX, Zhang QP. A novel diffusive representation of fractional calculus to stability and stabilisation of noncommensurate fractional-order nonlinear systems. International Journal of Dynamics and Control 2021. doi: 10.1007/s40435-021-00811-6
[36] Sun HG, Yong Z, Baleanu D, Wen C, Chen YQ. A new collection of real world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation 2018; 64: 213-231. doi: 10.1016/j.cnsns.2018.04.019
[37] Tapdigoglu R, Torebek B. Global existence and blow-up of solutions of the time-fractional space-involution reactiondiffusion equation. Turkish Journal of Mathematics 2020; 44 (3): 960-969. doi: 10.3906/mat-1909-65
[38] Trigeassou JC, Maamri N. Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach. London, UK: ISTE Ltd, 2019.
[39] Trigeassou JC, Maamri N. Analysis, Modeling and Stability of Fractional Order Differential Systems 2: The Infinite State Approach. London, UK: ISTE Ltd, 2019.
[40] Trigeassou JC, Maamri N, Oustaloup A. Lyapunov stability of noncommensurate fractional order systems: an energy balance approach. Journal of Computational and Nonlinear Dynamics 2016; 11 (4): 041007. doi: 10.1115/1.4031841
[41] Trigeassou JC, Maamri N, Oustaloup A. Lyapunov stability of commensurate fractional order systems: a physical interpretation. Journal of Computational and Nonlinear Dynamics 2016; 11 (5): 051007. doi: 10.1115/1.4032387
[42] Trigeassou JC, Maamri N, Oustaloup A. The infinite state approach: origin and necessity. Computers & Mathematics with Applications 2013; 66 (5): 892-907. doi: 10.1016/j.camwa.2012.11.020
[43] Trigeassou JC, Maamri N, Sabatier J, Oustaloup A. A Lyapunov approach to the stability of fractional differential equations. Signal Processing 2011; 91 (3): 437-445. doi: 10.1016/j.sigpro.2010.04.024
[44] Wei YH. Adaptive control for uncertain fractional order systems. PhD, University of Science and Technology of China, Hefei, China, 2015.
[45] Wei YH, Chen YQ. Converse Lyapunov theorem for nabla asymptotic stability without conservativeness. IEEE Transactions on Systems, Man, and Cybernetics: Systems 2021. doi: 10.1109/tsmc.2021.3051639
[46] Wei YH, Chen YQ, Wei YD, Zhang XF. Consistent approximation of fractional order operators. Journal of Dynamic Systems Measurement and Control 2021; 143 (8): 084501. doi: 10.1115/1.4050393
[47] Wei YH, Chen YQ, Wang JC, Wang Y. Analysis and description of the infinite-dimensional nature for nabla discrete fractional order systems. Communications in Nonlinear Science and Numerical Simulation 2019; 72: 472-492. doi: 10.1016/j.cnsns.2018.12.023
[48] Wei YH, Chen YQ, Wang Y, Chen YQ. Some fundamental properties on the sampling free nabla Laplace transform. In: ASME 2019 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference; Anaheim, USA; 2019. pp. 1-9. doi: 10.1115/detc2019-97351
[49] Wei YH, Gao Q, Peng C, Wang Y. A rational approximate method to fractional order systems. International Journal of Control Automation and Systems 2014; 12 (6): 1180-1186. doi: 10.1007/s12555-013-0109-6
[50] Wei YH, Tse PW, Du B, Wang Y. An innovative fixed-pole numerical approximation for fractional order systems. ISA Transactions 2016; 62: 94-102. doi: 10.1016/j.isatra.2016.01.010
[51] Wei YH, Wang JC, Liu TY, Wang Y. Fixed pole based modeling and simulation schemes for fractional order systems. ISA Transactions 2019; 84: 43-54. doi: 10.1016/j.isatra.2018.10.001
[52] Wei YH, Wang JC, Tse PW, Wang Y. Modelling and simulation of nabla fractional dynamic systems with nonzero initial conditions. Asian Journal of Control 2021; 2 3(1): 525-535. doi: 10.1002/asjc.2232
[53] Wei YH, Zhang H, Hou YQ, Cheng K. Multiple fixed pole based rational approximation for fractional order systems. Journal of Dynamic Systems Measurement and Control 2021; 143 (6): 061008. doi: 10.1115/1.4049557