Elsevier

Signal Processing

Volume 107, February 2015, Pages 254-264
Signal Processing

Identifying a non-commensurable fractional transfer function from a frequency response

https://doi.org/10.1016/j.sigpro.2014.03.001Get rights and content

Highlights

  • Levy׳s identification method is expanded to non-commensurable fractional models.

  • Numerical and empirical ways of finding the orders involved are justified.

  • Models for the human arm are found as an application.

Abstract

This paper extends Levy׳s identification method to non-commensurable fractional models. This identification method allows finding a transfer function that models a frequency response. Explicit expressions for the calculations, using five different variations of the method, are given. A sensitivity analysis supports an empirical way of finding the orders involved. Two application examples, concerning the identification of a model for a viscoelastic material and a model the human arm, are given.

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 G(jωp), known at f frequencies ωp,p=1,2,,f. We want to fit a model given byG˜(s)=k=0mbksβkk=0naksαk=b0+k=1mbksβk1+k=1naksαkThe n+1 denominator coefficients are a=[a0a1an], with a0=1. The m+1 numerator coefficients are b=[b0b1bm]. The n+1 denominator orders are α=[α0α1αn], with α0=0. The m+1 numerator orders are β=[β0β1βm], with β0=0.

Let the numerator and the denominator of the frequency response of (1) beN(jωp)=k=0mbk(jωp)βkD(jωp

Equal weights

The original version of Levy׳s method weights all frequencies equally: w(ωp)=1,p.

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 w(ωp), verifyingw(ω1)w(ω2)w(ωf)so as to increase the influence of low-frequency data.

A

Determining the fractional orders

In the integer case, αk=k and βk=k, 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 nm must be verified.

In the fractional commensurable case, αk=kα and βk=kα, 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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