On the frequency scaling in continuous-time modeling

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 Identification: A Frequency Domain Approach, (Piscataway, NJ: IEEE Press, 2001), and I. Kolla´r, 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. We show: 1) that the optimal frequency scaling also strongly depends on the estimation algorithm and 2) that the median of the angular frequencies is a better compromise for improving the numerical stability than the arithmetic mean.