Abstract
An identification method for fractional order models with time delay is presented. The proposed method, based on the output error optimization, simultaneously estimates model orders, coefficients and time delay from a single noisy step response. Analytical expressions for logarithmic derivatives of the step input are derived to evaluate the Jacobian and the Hessian required for the Newton’s algorithm for optimization. A simplified initialization procedure is also outlined that assumes an integral initial order and uses estimated coefficients as the initial guess. Simulation results are presented to demonstrate the efficacy of the proposed approach. Convergence of the Newton’s method and the Gauss–Newton scheme are also studied in simulation. Identification results from noisy step response data for time delay models with different structures are presented.
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References
S. Ahmed, Parameter and delay estimation of fractional order models from step response, in 9th IFAC Symposium on Advanced Control of Chemical Processes, Whistler, BC, Canada (2015), pp. 942–947
S. Ahmed, B. Huang, S.L. Shah, Parameter and delay estimation of continuous-time models using a linear filter. J. Process Control 16(4), 323–331 (2006)
M. Aoun, R. Malti, F. Levron, A. Oustaloup, Synthesis of fractional Laguerre basis for system approximation. Automatica 43, 1640–1648 (2007)
D. Babusci, G. Dattoli, On the logarithm of the derivative operator arXiv e-prints (2011)
A. Benchellal, T. Poinot, C. Trigeassou, Approximation and identification of diffusive interfaces by fractional systems. Signal Process. 86(10), 2712–2727 (2006)
Y. Chen, I. Petras, D. Xue, Fractional order control—A tutorial, in 2009 American Control Conference, St. Louis, USA (2009), pp. 1397–1411
E.K. Chong, S.H. Zak, An Introduction to Optimization (Wiley, New York, 1996)
O. Cois, A. Oustaloup, T. Poinot, J. Battaglia, Fractional state variable filter for system identification by fractional model, in European Control Conference, Porto, Portugal (2001)
J.E. Diamessis, A new method for determining the parameters of physical systems, in Proceedings of the IEEE (1965), pp. 205–206
S.M. Fahim, S. Ahmed, S.A. Imtiaz, Fractional order model identification using the sinusoidal input. ISA Trans. 83, 35–41 (2018). https://doi.org/10.1016/j.isatra.2018.09.009
R. Fletcher, Practical Methods of Optimization. Vol. 1: Unconstrained Optimization (Wiley, New York, 1980)
J.D. Gabano, T. Poinot, H. Kanoun, Identification of a thermal system using continuous linear parameter-varying fractional modelling. IET Control Theory Appl. 5(7), 889–899 (2011)
E.V. Hayngworth, K. Goldbe, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, chap. Bernoulli and Euler Polynomials- Riemann Zeta Function, U.S. Department of Commerce, NIST, Washington DC (1972) pp. 803–820
J. Hines, Operator mathematics II. Math. Mag. 28(4), 199–207 (1955)
K. Leyden, B. Goodwine, Fractional-order system identification for health monitoring. Nonlinear Dyn. 92(3), 1317–1334 (2018)
B. Lurie, Three-parameter tunable tilt-integral-derivative (TID) controller. US Patent 5371670 (1994)
R. Malti, S. Victor, A. Oustaloup, Advances in system identification using fractional models. J. Comput. Nonlinear Dyn. 3(2), 0214011–7 (2008)
R. Malti, S. Victor, A. Oustaloup, H. Garnier, An optimal instrumental variable method for continuous-time fractional order model identification, in Proceedings 17th IFAC World Congress, Seoul, Korea (2008), pp. 14379–14384
A.K. Mani, M.D. Narayanan, M. Sen, Parametric identification of fractional-order nonlinear systems. Nonlinear Dyn. 93(2), 945–960 (2018)
R. Mansouri, M. Bettayeb, S. Djennoune, Approximation of high order integer systems by fractional order reduced parameter models. Math. Comput. Model. 51, 53–62 (2010)
W.F. Mascarenhas, Newton iterates can converge to non-stationary points. Math. Program. Ser. A 112, 327–334 (2008)
C. Monje, Y. Chen, B. Vinagre, D. Xue, V. Feliu-Batlle, Fractional-order Systems and Controls (Springer, London, 2010)
M. Muddu, A. Narang, S. Patwardhan, Development of ARX models for predictive control using fractional order and orthonormal basis filter parameterization. Ind. Eng. Chem. Res. 48(19), 8966–8979 (2009)
C.I. Muresan, S. Folea, I.R. Birs, C. Ionescu, A novel fractional-order model and controller for vibration suppression in flexible smart beam. Nonlinear Dyn. 93(2), 525–541 (2018). https://doi.org/10.1007/s11071-018-4207-0
A. Narang, S.L. Shah, T. Chen, Continuous-time model identification of fractional-order models with time delays. IET Control Theory Appl. 15(5), 900–912 (2010)
P. Nazarian, M. Haeri, M.S. Tavazoei, Identifiability of fractional-order systems using input output frequency contents. ISA Trans. 49, 207–214 (2010)
K. Oldham, J. Myland, J. Spanier, An Atlas of Functions, 2nd edn. (Springer, New York, 2000)
K. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York, 1974)
A. Oustaloup, X. Moreau, M. Nouillant, The CRONE suspension. Control Eng. Pract. 4(8), 1101–1108 (1996)
I. Podlubny, Fractional order systems and \( {PI}^{\lambda } {D}^{\mu }\) controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)
T. Poinot, J.C. Trigeassou, Identification of fractional systems using an output-error technique. Nonlinear Dyn. 38(1), 133–154 (2004)
H. Raynaud, A. ZergaInoh, State-space representation for fractional order controllers. Automatica 36, 1017–1021 (2000)
L. Sersour, T. Djamah, M. Bettayeb, Nonlinear system identification of fractional wiener models. Nonlinear Dyn. 92(4), 1493–1505 (2018). https://doi.org/10.1007/s11071-018-4142-0
M. Tavakoli-Kakhki, M. Haeri, M. Tavazoei, Simple fractional order model structures and their applications in control system design. Eur. J. Control 16(6), 680–694 (2010)
M. Tavakoli-Kakhki, M. Haeri, M.S. Tavazoei, Over- and under-convergent step responses in fractional-order transfer functions. Trans. Inst. Meas. Control 32(4), 376–394 (2010)
M. Tavakoli-Kakhki, M. Tavazoei, Estimation of the order and parameters of fractional order models from a noisy step response data. J. Dyn. Syst. Meas. Control 136(3), 0310201–6 (2014)
M.S. Tavazoei, Overshoot in the step response of fractional-order control systems. J. Process Control 22, 90–94 (2012)
D. Valerio, M.D. Ortigueira, J.S. da Costa, Identifying a transfer function from a frequency response. J. Comput. Nonlinear Dyn. 3(2), 0212071–7 (2008)
S. Victor, R. Malti, Model order identification for fractional-order models, in European Control Conference, Zurich, Switzerland (2013), pp. 3470–3475
S. Victor, R. Malti, H. Garnier, A. Oustaloup, Parameter and differential order estimation of fractional-order models. Automatica 49, 926–935 (2013)
P.C. Young, Optimal IV identification and estimation of continuous-time TF models, in IFAC World Congress, Barcelona, Spain (2002), pp. 337–358
L. Yuan, Q. Yang, C. Zeng, Chaos detection and parameter identification in fractional-order chaotic systems with delay. Nonlinear Dyn. 73(1), 439–448 (2013)
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The author acknowledges the financial supports from Research and Development Corporation (RDC) of Newfoundland and Labrador and Natural Sciences and Engineering Research Council (NSERC) of Canada.
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Ahmed, S. Step Response-Based Identification of Fractional Order Time Delay Models. Circuits Syst Signal Process 39, 3858–3874 (2020). https://doi.org/10.1007/s00034-020-01344-7
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DOI: https://doi.org/10.1007/s00034-020-01344-7