Brief paperBox-Jenkins identification revisited—Part II: Applications☆
Introduction
A frequency domain Box-Jenkins (BJ) framework has been developed in Part I (Pintelon & Schoukens, 2005) for data collected in open or closed loop. The resulting maximum likelihood estimators can handle discrete-time (DT) and continuous-time (CT) models on arbitrary frequency grids (part(s) of the unit circle). In Part II of this series of two papers we
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discuss the numerical issues of the minimization of the maximum likelihood (ML) cost functions described in Part I: choice and implementation of the parameter constraints, Newton–Gauss optimization of the ML cost functions, and generation of starting values (see Sections 2.1 and 2.2),
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calculate the uncertainty of the estimated noise model (see Section 2.3),
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illustrate some of the theoretical results (consistency, identification in feedback, and bias) proven in Part I (see Section 3),
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show the usefulness and feasibility of the frequency domain (CT) BJ modeling approach on a real life problem: the modeling of a nonlinear electrical device (see Section 4).
Section snippets
Identification in feedback with known controller
In Part I it has been shown that the estimated plant and noise models are independent of the particular parameter constraint(s) chosen. Possible parameter constraints are listed in Table 1 of Part I, for example, for the CT-BJ model structure the constraints areFixing a coefficient to one has the disadvantage that it deteriorates the numerical conditioning in case the true value of the coefficient is zero. Norm one constraints do not have this disadvantage and are therefore
Simulation example
The simulation set up (see Fig. 1 of Part I) consists of a second-order DT plant model , and a second-order monic DT noise model , in a unity feedback setting (). The reference signal and the driving white noise source are white Gaussian DT noise processes. The input/output DFT spectra and are calculated from time domain samples, and DFT lines () are used to identify the plant and noise model
Real measurement example
The nonlinear electrical circuit of Fig. 2 simulates a second-order nonlinear mechanical system with a hardening spring. Using normally distributed excitations, the main error source within a linear system identification framework is the stochastic nonlinear contribution that satisfies the mixing condition of Assumption 12 of Part I (see Pintelon & Schoukens, 2001). Two experiments have been performed, the first with a white band-limited normally distributed signal and the second with a white
Conclusion
The usefulness/feasibility of (CT) BJ modeling in a limited frequency band (part of the unit circle) has been demonstrated via simulations and a real measurement example, and the practical implementation differences/similarities with the classical prediction error method have been discussed in detail (global minima of the cost function, Newton–Gauss minimization algorithm, choice/implementation of the parameter constraints, uncertainty calculation of the estimated noise model).
Acknowledgements
This work is sponsored by the Fund for Scientific Research (FWO-Vlaanderen), the Flemish Government (GOA-ILiNoS) and the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science Policy programming (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.
Yves Rolain is presently active at the VUB in the Electrical Measurement Department (ELEC). His main research interests are nonlinear micro-wave measurement techniques, applied digital signal processing, parameter estimation/system identification, and biological agriculture.
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
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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.