Identifying a non-commensurable fractional transfer function from a frequency response
Introduction
Given a frequency response, Levy׳s identification method allows finding a transfer function to model it [1]. Levy׳s method has been enhanced with several improvements [2], [3], [4], and adapted to the identification of commensurable fractional transfer functions [5], [6], [7].
This paper extends the method to non-commensurable fractional models. Identifying this type of models from time responses is a problem already addressed in the literature: Refs. [8], [9], [10], [11] address identification from time responses; to the best of our knowledge, only [12] concerns frequency responses, presenting what is termed below as the first formulation with summed matrices of Levy׳s method. The present paper gives explicit expressions for the calculations using five different variations of Levy׳s method (Section 2), addresses the possible use of weights (Section 3), comments on the way to determine the fractional orders involved (Section 4, that includes sensitivity analyses for the commensurable and the non-commensurable cases), and concludes with application examples (Section 5).
All methods described in this paper are implemented as part of the freely available toolbox Ninteger for Matlab [13].
Section snippets
Levy׳s method for fractional plants
Let us suppose we have the frequency response of a plant , known at f frequencies . We want to fit a model given byThe n+1 denominator coefficients are , with . The m+1 numerator coefficients are . The n+1 denominator orders are , with . The m+1 numerator orders are , with .
Let the numerator and the denominator of the frequency response of (1) be
Equal weights
The original version of Levy׳s method weights all frequencies equally: .
Correcting the high-frequency bias
Because of the way frequency ωp appears in the equations, terms for higher frequencies carry a greater weight, and as a consequence models resulting from Levy׳s method are often better fitted to high-frequency data than to low-frequency data. Intuitively it is seen that this can be corrected using frequency-dependent weights , verifyingso as to increase the influence of low-frequency data.
A
Determining the fractional orders
In the integer case, and , so the only parameters that have to be found are m and n. There may be theoretical reasons to fix these values, or an inspection of the data may suggest which values are to be used, or else several possible combinations of values may be tried and the best results retained. To ensure causality, the restriction must be verified.
In the fractional commensurable case, and , so there is now an additional parameter to determine: the commensurability
Examples
The methods above can be applied to the identification of a model for a viscoelastic material (Section 5.1) and of a model for the human arm at the elbow joint (Section 5.2).
Acknowledgements
Inés Tejado would like to thank the Portuguese Fundação para a Ciência e a Tecnologia (FCT) for the grant with reference SFRH/BPD/81106/2011. This work was also supported by national funding through FCT, under project Pest/OE/EME/LA0022/2011—LAETA/IDMEC/CSI, and under the joint Portuguese–Slovakian project SK-PT-0025-12.
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