基于频率响应的传递函数辨识
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denominator, and q is the fractional derivative order. Without loosing generality, we set a0 = 1. The frequency response of ͑1͒ is given by
Gˆ ͑j͒
=
1
͚km=0bk͑ j͒kq + ͚kn=1ak͑j͒kq
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 classic 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 the Archimedes Wave Swing, a prototype of a device to convert the energy of sea waves into electricity. ͓DOI: 10.1115/1.2833906͔
def
E͑j͒ = ⑀͑j͒D͑j͒ = G͑j͒D͑j͒ − N͑j͒
͑5͒
This leads to a set of normal equations, easy to solve. Omitting the frequency argument to simplify the notation, we have
E = GD − N = ͓Re͑G͒ + j Im͑G͔͒͑ + j͒ − ͑␣ + j͒ = ͓Re͑G͒ − Im͑G͒ − ␣͔ + j͓Re͑G͒ + Im͑G͒ − ͔ ͑6͒
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.
Journal of Computational and Nonlinear Dynamics Copyright © 2008 by ASME
APRIL 2008, Vol. 3 / 021207-1
Downloaded 08 Nov 2011 to 211.68.3.254. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
m
͚ ␣͑͒ = bk Re͓͑j͒kq͔ k=0
n
n
͚ ͚ ͑͒ = ak Re͓͑j͒kq͔ = 1 + ak Re͓͑j͒kq͔
k=0
k=1
m
͑3͒
͚ ͑͒ = bk Im͓͑j͒kq͔
k=0
n
n
͚ ͚ ͑͒ = ak Im͓͑j͒kq͔ = ak Im͓͑j͒kq͔
=
N͑j͒ D͑j͒
=
␣͑͒ ͑͒
ห้องสมุดไป่ตู้+ +
j͑͒ j͑͒
͑2͒
where N and D are complex valued and ␣, , , and ͑the real and imaginary parts thereof͒ are real valued. From ͑2͒ we see that
Adapting Levy’s Identification Method for Fractional Orders
Original Formulation. Let us suppose we have a plant described by a linear system with a transfer function G and a corresponding frequency response G͑j͒ and that we want to model it using a transfer function
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 ͑DETC2007͒, Las Vegas, NV, September 4–7, 2007.
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
Introduction
The identification of linear systems is an interesting subject that deserved a lot of attention in the past ͓1͔. Identification in the frequency domain is a particular case with great interest in applications. Algorithms are traditionally based on Levy’s work ͓2͔, 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 ͓3͔. This frequency dependence has been faced ͓3͔ by using an iterative method; another alternative, without iterations, was also proposed ͓4͔. Both adaptations modify the basic algorithm introducing weights.
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
Gˆ ͑s͒
=
b0 + b1sq + b2s2q + a0 + a1sq + a2s2q +
¯ ¯
+ bmsmq + ansnq
=
͚km=0bkskq ͚kn=0akskq
͑1͒
where m and n are the preassigned orders of the numerator and
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
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 formulation, we present some simulation results and also a real practical example.
k=0
k=1
The error between model and plant, for a given frequency , will be
⑀͑
j͒
=
G͑
j͒
−
N͑j͒ D͑j͒
͑4͒
Minimizing the error power would be an obvious but difficult way of adjusting the parameters of ͑1͒. Instead of this, Levy’s method minimizes the square of the norm of
Gˆ ͑j͒
=
1
͚km=0bk͑ j͒kq + ͚kn=1ak͑j͒kq
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 classic 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 the Archimedes Wave Swing, a prototype of a device to convert the energy of sea waves into electricity. ͓DOI: 10.1115/1.2833906͔
def
E͑j͒ = ⑀͑j͒D͑j͒ = G͑j͒D͑j͒ − N͑j͒
͑5͒
This leads to a set of normal equations, easy to solve. Omitting the frequency argument to simplify the notation, we have
E = GD − N = ͓Re͑G͒ + j Im͑G͔͒͑ + j͒ − ͑␣ + j͒ = ͓Re͑G͒ − Im͑G͒ − ␣͔ + j͓Re͑G͒ + Im͑G͒ − ͔ ͑6͒
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.
Journal of Computational and Nonlinear Dynamics Copyright © 2008 by ASME
APRIL 2008, Vol. 3 / 021207-1
Downloaded 08 Nov 2011 to 211.68.3.254. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
m
͚ ␣͑͒ = bk Re͓͑j͒kq͔ k=0
n
n
͚ ͚ ͑͒ = ak Re͓͑j͒kq͔ = 1 + ak Re͓͑j͒kq͔
k=0
k=1
m
͑3͒
͚ ͑͒ = bk Im͓͑j͒kq͔
k=0
n
n
͚ ͚ ͑͒ = ak Im͓͑j͒kq͔ = ak Im͓͑j͒kq͔
=
N͑j͒ D͑j͒
=
␣͑͒ ͑͒
ห้องสมุดไป่ตู้+ +
j͑͒ j͑͒
͑2͒
where N and D are complex valued and ␣, , , and ͑the real and imaginary parts thereof͒ are real valued. From ͑2͒ we see that
Adapting Levy’s Identification Method for Fractional Orders
Original Formulation. Let us suppose we have a plant described by a linear system with a transfer function G and a corresponding frequency response G͑j͒ and that we want to model it using a transfer function
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 ͑DETC2007͒, Las Vegas, NV, September 4–7, 2007.
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
Introduction
The identification of linear systems is an interesting subject that deserved a lot of attention in the past ͓1͔. Identification in the frequency domain is a particular case with great interest in applications. Algorithms are traditionally based on Levy’s work ͓2͔, 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 ͓3͔. This frequency dependence has been faced ͓3͔ by using an iterative method; another alternative, without iterations, was also proposed ͓4͔. Both adaptations modify the basic algorithm introducing weights.
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
Gˆ ͑s͒
=
b0 + b1sq + b2s2q + a0 + a1sq + a2s2q +
¯ ¯
+ bmsmq + ansnq
=
͚km=0bkskq ͚kn=0akskq
͑1͒
where m and n are the preassigned orders of the numerator and
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
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 formulation, we present some simulation results and also a real practical example.
k=0
k=1
The error between model and plant, for a given frequency , will be
⑀͑
j͒
=
G͑
j͒
−
N͑j͒ D͑j͒
͑4͒
Minimizing the error power would be an obvious but difficult way of adjusting the parameters of ͑1͒. Instead of this, Levy’s method minimizes the square of the norm of