Elsevier

Automatica

Volume 42, Issue 1, January 2006, Pages 63-75
Automatica

Box–Jenkins identification revisited—Part I: Theory

https://doi.org/10.1016/j.automatica.2005.09.004Get rights and content

Abstract

In classical time domain Box–Jenkins identification discrete-time plant and noise models are estimated using sampled input/output signals. The frequency content of the input/output samples covers uniformly the whole unit circle in a natural way, even in case of prefiltering. Recently, the classical time domain Box–Jenkins framework has been extended to frequency domain data captured in open loop. The proposed frequency domain maximum likelihood (ML) solution can handle (i) discrete-time models using data that only covers a part of the unit circle, and (ii) continuous-time models. Part I of this series of two papers (i) generalizes the frequency domain ML solution to the closed loop case, and (ii) proves the properties of the ML estimator under non-standard conditions. Contrary to the classical time domain case it is shown that the controller should be either known or estimated. The proposed ML estimators are applicable to frequency domain data as well as time domain data.

Introduction

System identification is a powerful technique for building accurate models of complex systems from noisy data. According to the particular application one either has full control over the excitation, or has to live with the operational perturbations. In the first case, it is advised to use periodic excitation signals because it strongly simplifies the identification problem. For example, handling noisy input/noisy output data is as simple as handling known input/noisy output data (Pintelon & Schoukens, 2001). A non-parametric noise model is then obtained in a pre-processing step and is used as weighting for the identification of the parametric plant model. In the second case, the perturbations are often random and (parts of it) even may be unmeasurable, for example, the wind excitation on bridges, buildings and airplanes. It is typically assumed that the measurable part of the input is known exactly and that noisy observations of the output are available (Ljung, 1999). A parametric plant and noise model are then identified simultaneously from sampled input/output signals. This case is studied in this series of two papers.

Since the frequency content of a sampled signal covers the whole unit circle, the classical time domain Box–Jenkins approach (Box & Jenkins, 1970) identifies the discrete-time plant and noise models from DC (f=0) to Nyquist (=half the sampling frequency fs). Often one is only interested in the plant characteristics on a fraction of the unit circle, or one would like to remove the effect of slow trends and/or high-frequency disturbances. The classical approach consists in applying a prefilter to the input/output data (Ljung, 1999). The prefiltering does not affect the input/output relation, and is equivalent to dividing the noise model by the prefilter characteristics. However, to preserve the efficiency (open/closed loop) and the consistency (closed loop only) of the plant estimates, the parametric noise model should be flexible enough to follow the prefiltered error spectrum accurately (Ljung, 1999). As such, it will try to cancel the effect of the prefilter. Hence, through the prefilter/noise model selection a compromise must be made between the suppression of the undesired frequency band(s) and the loss in efficiency and/or consistency of the plant estimates. These conflicting demands can be avoided by performing the filtering in the frequency domain: the plant and noise models are identified in the frequency band(s) of interest only. Another advantage of the frequency domain approach is that it is equally simple to identify continuous-time models as discrete-time ones. Moreover in a lot of applications like, for example, modal analysis in mechanical engineering, electro-chemical impedance spectroscopy, and modeling of high-frequency devices, the data are collected by network analysers which provide the frequency domain spectra to the users rather than the original time signals. It can be concluded that there is a need for a frequency domain Box–Jenkins framework.

In Ljung, 1993, Ljung, 1999 a frequency domain Box–Jenkins framework has been developed for data collected in open loop. The proposed frequency domain maximum likelihood (ML) estimator can handle discrete-time and continuous-time models on arbitrary frequency grids. The main contributions of this series of two papers are the following: (i) The frequency domain ML solution is extended to the closed-loop case. A surprising result is that the controller should be either known (see also McKelvey, 2000) or estimated. (ii) It is shown that the ML cost function can be reduced to a quadratic form. As a consequence the classical Newton–Gauss-based iterative schemes can still be used for calculating the ML estimates. It also allows to prove the asymptotic properties of the ML estimator under non-standard conditions. Some properties have already been shown in McKelvey and Ljung (1997) and McKelvey (2002) for discrete-time noise models in an open-loop setting (see Section 4.4). (iii) Throughout Part I and II, the connection with the classical prediction error method is established. (iv) Illustration on a real life problem.

Section snippets

Plant model

It is well known that the zero-order-hold (ZOH) assumption (the input u(t) is piecewise constant and the measurement setup contains no anti-alias filters), and the band limited (BL) assumption (all acquisition channels of the measurement setup contain anti-alias filters), lead in a natural way to a discrete-time (DT) and continuous-time (CT) representation of the plant. The input U(k) and output Y(k) discrete Fourier transform (DFT) spectra of the input u(n) and output y(n) samplesX(k)=N-1/2n=0

Closed-loop framework

The closed-loop setup of Fig. 1 is defined by the following assumptions:

Assumption 5 Closed loop

The input/output data U(k), Y(k) are related to the reference signal R(k) and the driving white noise source E(k) as U(k)=11+G(Ωk)M(Ωk)R(k)-M(Ωk)H(Ωk)1+G(Ωk)M(Ωk)E(k),Y(k)=G(Ωk)1+G(Ωk)M(Ωk)R(k)+H(Ωk)1+G(Ωk)M(Ωk)E(k),where G(Ω), H(Ω) and M(Ω) are rational transfer functions in Ω.

Assumption 6 Independence reference signal and process noise

The reference signal R(k) is independent of the process noise V(k).

It will be shown that the ML estimator needs the controller knowledge if only a

Maximum likelihood cost function

Consider the parametric models G(Ω,θ)(3) and H(Ω,θ)(6), withθ=[aT,bT,cT,dT]T,where b,a and c,d are vectors containing the numerator and denominator coefficients of G(Ω,θ) and H(Ω,θ), respectively, and assume that the frequency domain data U(k), Y(k) is available at DFT frequencies fk=kfs/N, kK, whereK{0,1,2,,N/2}.Assume furthermore that the controller is known.

Assumption 7 Known controller

The controller transfer function M0(Ω) is known.

Note that under Assumption 4 (the input and output are observed without errors) the

Maximum likelihood cost function

From Section 4 it follows that if the controller is unknown, it must be estimated to avoid a bias error in the plant model. When identifying simultaneously the plant and the controller, Y(k) is a noisy observation of the true plant output, and U(k) is a noisy observation of the true controller output. Hence, similarly to the identification of the plant, the following assumptions are needed to identify the controller.

Assumption 15 Signal model with existing nth order moments

The reference signal R(k) in Fig. 1 can be written asR(k)=L(Ωk)W(k),where Ω=z-1

Conclusions

Summarized, Part I of this series of two papers has the following contributions:

  • identification in closed loop on an arbitrary frequency grid with known and unknown controller

  • ML properties are proven under non-standard conditions

  • discussion of the differences and similarities with the classical time domain DT modeling.

The following practical advice results. Beside the input u(t) and output y(t) of the plant, it is strongly recommended to store also the reference signal r(t) in a feedback

Acknowledgements

This work is sponsored by the Fund for Scientific Research (FWO-Vlaanderen), the Flemish Government (GOA-ILiNoS) and the Belgian Government (IUAP V/22).

Rik Pintelon was born in Gent, Belgium, on December 4, 1959. He received the degree of electrical engineer (burgerlijk ingenieur) in July 1982, the degree of doctor in applied sciences in January 1988, and the qualification to teach at university level (geaggregeerde voor het hoger onderwijs) in April 1994, all from the Vrije Universiteit Brussel (VUB), Brussels, Belgium. From October 1982 till September 2000 he was a researcher of the Fund for Scientific Research—Flanders at the VUB. Since

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Rik Pintelon was born in Gent, Belgium, on December 4, 1959. He received the degree of electrical engineer (burgerlijk ingenieur) in July 1982, the degree of doctor in applied sciences in January 1988, and the qualification to teach at university level (geaggregeerde voor het hoger onderwijs) in April 1994, all from the Vrije Universiteit Brussel (VUB), Brussels, Belgium. From October 1982 till September 2000 he was a researcher of the Fund for Scientific Research—Flanders at the VUB. Since October 2000 he is professor at the VUB in the Electrical Measurement Department (ELEC). His main research interests are in the field of parameter estimation/system identification, and signal processing.

Johan Schoukens is born in Belgium in 1957. He received the degree of engineer in 1980 and the degree of doctor in applied sciences in 1985, both from the Vrije Universiteit Brussel, Brussels, Belgium. He is presently professor at the Vrije Universiteit Brussel. The prime factors of his interest are in the field of system identification for linear and nonlinear systems, and growing tomatoes in his green house.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Brett Ninness under the direction of Editor T. Söderström.

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