Duarte Valério
IDMEC/IST,
Technical University of Lisbon,
Avenida Rovisco Pais,
1049-001 Lisboa, Portugal
e-mail: dvalerio@dem.ist.utl.pt
Manuel Duarte Ortigueira
UNINOVA and DEE of Faculdade de Ciências e
Tecnologia of UNL,
Campus da FCT da UNL,
Quinta da Torre, 2825-114 Monte da Caparica,
Portugal
e-mail: mdortigueira@uninova.pt
José Sá da Costa
IDMEC/IST,
Technical University of Lisbon,
Avenida Rovisco Pais,
1049-001 Lisboa, Portugal
e-mail: sadacosta@dem.ist.utl.pt
Identifying a Transfer Function
From a Frequency Response
In this paper, the classic Levy identification method is reviewed and reformulated using a
complex representation. This new formulation addresses the well known bias of the clas-
sic method at low frequencies. The formulation is generic, coping with both integer order
and fractional order transfer functions. A new algorithm based on a stacked matrix and
its pseudoinverse is proposed to accommodate the data over a wide range of frequencies.
Several simulation results are presented, together with a real system identification. This
system is theArchimedes Wave Swing, a prototype of a device to convert the energy of sea
waves into electricity. DOI: 10.1115/1.2833906
Introduction
The identification of linear systems is an interesting subject that
deserved a lot of attention in the past 1. Identification in the
frequency domain is a particular case with great interest in appli-
cations. Algorithms are traditionally based on Levy’s work 2,a
least-squares based algorithm formulated in a real framework by
separating the real and imaginary parts of the transfer function.
This led to a formulation with lengthy expressions and to results
not equally good at all frequencies 3. This frequency depen-
dence has been faced 3 by using an iterative method; another
alternative, without iterations, was also proposed 4. Both adap-
tations modify the basic algorithm introducing weights.
Here, we reformulate the original approach in a completely
complex framework, leading to a set of normal equations from
which we could remove the frequency dependency. Expressions
obtained are shorter. Thus, we obtain two sets of linear equations
for each frequency. The formulation is given for the general case
of a fractional, commensurate transfer function.
The use of measurements corresponding to several frequencies
has been addressed in the past by an average of the coefficients
computed from each frequency. This procedure is neither robust
nor sensible to order changes. To obtain a more reliable algorithm,
we propose a new algorithm with two steps: firstly, collect the
matrices corresponding to the different frequencies in a stacked
matrix; secondly, compute the coefficients of the model by using
the pseudoinverse. To illustrate the behavior of this new formula-
tion, we present some simulation results and also a real practical
example.
Adapting Levy’s Identification Method for Fractional
Orders
Original Formulation. Let us suppose we have a plant de-
scribed by a linear system with a transfer function G and a corre-
sponding frequency response G j and that we want to model it
using a transfer function
G
ˆ
s =
b
0
+ b
1
s
q
+ b
2
s
2q
+ ¯ + b
m
s
mq
a
0
+ a
1
s
q
+ a
2
s
2q
+ ¯ + a
n
s
nq
=
k=0
m
b
k
s
kq
k=0
n
a
k
s
kq
1
where m and n are the preassigned orders of the numerator and
denominator, and q is the fractional derivative order. Without
loosing generality, we set a
0
= 1. The frequency response of 1 is
given by
G
ˆ
j =
k=0
m
b
k
j
kq
1+
k=1
n
a
k
j
kq
=
N j
D j
=
+ j
+ j
2
where N and D are complex valued and , , , and the real
and imaginary parts thereof are real valued. From 2 we see that
=
k=0
m
b
k
Re j
kq
=
k=0
n
a
k
Re j
kq
=1+
k=1
n
a
k
Re j
kq
3
=
k=0
m
b
k
Im j
kq
=
k=0
n
a
k
Im j
kq
=
k=1
n
a
k
Im j
kq
The error between model and plant, for a given frequency , will
be
j = G j -
N j
D j
4
Minimizing the error power would be an obvious but difficult way
of adjusting the parameters of 1. Instead of this, Levy’s method
minimizes the square of the norm of
E j =
def
jD j = G jD j - N j 5
This leads to a set of normal equations, easy to solve. Omitting the
frequency argument to simplify the notation, we have
Contributed by the Design Engineering Division of ASME for publication in the
JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 6,
2007; final manuscript received November 30, 2007; published online February 4,
2008. Review conducted by J. A. Tenreiro Machado. Paper presented at the ASME
2007 Design Engineering Technical Conferences and Computers and Information in
Engineering Conference DETC2007, Las Vegas, NV, September 4–7, 2007.
Journal of Computational and Nonlinear Dynamics APRIL 2008, Vol. 3 / 021207-1
Copyright © 2008 by ASME
Downloaded 05 Feb 2008 to 193.136.128.14. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm