318 IEEE TRANSACTIONS ON INSTRUMENTATION ANDMEASUREMENT, VOL. 54, NO. 1, FEBRUARY 2005 On the Frequency Scaling in Continuous-Time Modeling Rik Pintelon, Fellow, IEEE, and Istvan Kollár, Fellow, IEEE Abstract—When identifying continuous-time systems in the Laplace domain, it is indispensable to scale the frequency axis to guarantee the numerical stability of the normal equations. Without scaling, identification in the Laplace domain is often impossible even for modest model orders of the transfer function. Although the optimal scaling depends on the system, the model, and the excitation signal, the arithmetic mean of the maximum and minimum angular frequencies in the frequency band of interest is commonly used as a good compromise as shown in the following references: J. Schoukens and R. Pintelon, Identification of Linear Systems: A Practical Guideline to Accurate Modeling (London, U.K.: Pergamon), R. Pintelon and J. Schoukens, System Identi- fication: A Frequency Domain Approach, (Piscataway, NJ: IEEE Press, 2001), and I. Kollár, R. Pintelon, Y. Rolain, J. Schoukens, G. Simon, “Frequency domain system identification toolbox for Matlab: Automatic processes—From data to model,” in Proc. 13th IFAC Symp. System Identification, Rotterdam, The Netherlands, Aug. 27–29, 2003, pp. 1502– 1506. In this paper, we show: 1) that the optimal frequency scaling also strongly depends on the estima- tion algorithm and 2) that the median of the angular frequencies is a better compromise for improving the numerical stability than the arithmetic mean. Index Terms—Continuous-time modeling, frequency domain, system identification. I. PROBLEM STATEMENT C ONSIDER the identification of rational transfer function models in the Laplace variable with (1) starting from measured input/output spectra or frequency response functions . The goal is to estimate the numerator and denomi- nator coefficients for a given value of the order of the numerator and denominator polynomials. Since parametrization (1) is not identifiable ( for any nonzero real number ), the parameter vector should be constrained, for example, or . From a numerical Manuscript received July 11, 2003; revised March 16, 2004. This work was supported by the Fund for Scientific Research (FWO-Vlaanderen), by the Flemish Government (GOA-IMMI), and by the Belgian Program on Interuni- versity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy programming (IUAP V/22). R. Pintelon is with the Electrical Measurement Department, Vrije Universiteit Brussel, 1050 Brussels, Belgium (e-mail: Rik.Pintelon@vub.ac.be). I. Kollár is with the Management Information Systems Department, Budapest University of Technology and Economics, 1521 Budapest, Hungary. Digital Object Identifier 10.1109/TIM.2004.838916 point of view, it is often better to use the full overparametrized form (1) in combination with the 2-norm constraint (see [2, ch. 18]). This approach will be used throughout the paper. Most algorithms estimate by minimizing (in each step) a “quadratic-like” cost function (2) where superscript is the Hermitian transpose (transpose complex conjugate), and . The by 1 vector of the residuals is some kind of measure of the differ- ence between the measurements and the model. is a (non)linear function of the model parame- ters and the measurements at frequency (input/output spectra or frequency response functions ). Often, a Newton-Gauss type algorithm [4] is used to find the minimizer of (2). Rewriting (2) as , where stacks the real and imaginary parts on top of each other, (3) the th iteration step of this algorithm is given by (4) with and the Jacobian of the vector . The numerical stability (sensi- tivity) of the solution of the normal equation (4) is basically limited by the numerical precision of the calculation of . The latter is quantified by the condition number of , where [5]. The solution of (4) can be cal- culated without forming the product explicitly by solving the overdetermined set of equations (5) using a singular value decomposition or a QR-factorization of the matrix (see [2], [5]). The numerical precision of (5) is basically limited by . Summarized, the solution of (4) and (5) is numerically reliable if, respectively, and (6) 0018-9456/$20.00 © 2005 IEEE