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Fluids Engineering Quarteroni , G. Finally, Section 5 provides concluding remarks as well as future studies. Interpolation and the least squares method are two commonly used techniques for curve fitting and modeling. Interpolation has normal polynomials and spline representations, while the least squares curve fitting also uses a simple polynomial and is a general approach that permits additional choices of least squares fitting functions such as Chebyshev or spline series.
Other fitting methods in the literature include the spline function, which is a curve built from polynomial segments that are subjected to the conditions at their joints. The common model names and their associated equations can be classified as shown in Table 1. Table 1 provides a list of the common fitting equations that MATLAB is based on; some of the listed equations may fit one set of data, but not others, and the fitting accuracy depends on the selection of the equation order.
There is no dominant fitting approach for the types of observations, even for physical applications. Therefore, it is essential to consider the performance of several statistical methods to fit a particular power curve to select the best one for a given measured data set. A brief classification of most power curve models, with the position of the proposed algorithm included among them, is shown in Fig 1.
Vector fitting VF is a robust numerical method for extracting a rational approximation in the frequency domain. VF is a state equation approximation; thus, the time and the frequency domain approximations are parameterized exactly the same as in the solution of the state matrix. The only difference in the calculation is the basis used during the least squares of the method. The definition of a vector fitting VF algorithm is a mathematical method for sampled response-matching system identification, in which the algorithm includes an iterative procedure for solving linear least squares by using partial fraction expansion roots.
This technique covers a wide range of variation in frequency domains from 0-GHz , avoiding ill-conditioned computations and therefore working in a more robust and efficient manner. VF is used in modeling of different electrical systems [ 3 ] and other areas such as in filter design [ 4 , 5 ], in power network analysis [ 4 , 6 ] and in the field of electromagnetic simulation [ 7 ]. VF was an idea first introduced for the modeling of power transmission lines as transient response-measured data in [ 8 ]. The fundamental idea of VF is to update the proposed or initialized poles with estimated values as an improved set of poles through the technique of pole relocation, improving the approximation iteratively.
VF has the capability to estimate an original system as a new model by means of partial fraction roots of real, complex conjugate, or mixed poles. For better integration and performance of VF, several simplifications and expansions have been considered with special classification [ 9 — 17 ]. The Bode equation vector fitting BEVF method is a modified algorithm in which the vector fitting VF algorithm has been adopted due to its robust numerical fitting property.
However, VF provides a state space modeling representation for measured input data in the frequency domain; therefore, the proposed effort investigates the ability of this method to develop a new mathematical model by relying on mathematical assumptions, set as adaptors, to match any measured or calculated data points using the VF process. This stage represents the assumptions for synthesizing the input data to frequency-domain based algorithm VF algorithm in this case and to initialize the parameters required for the second stage.
Appendix A shows the software steps with example of empirical data. The data are now ready to be processed by the conventional VF algorithm. VF provides a model system expressed by its state space. Therefore, the VF algorithm can be defined as a method to determine the state space representation of measured or calculated data points [ 18 ]. The polynomial expression of the transfer function can be expressed as:.
The rational approximate function of 2 can be expressed as:. Based on 1 , which represents the assumption of data, the algorithm transforms the state space model into a Rational Approximation Function RAF , which can be seen in 3 , and then transforms each fraction of 3 into a sum of 1 st order fractions. The aim of transforming the fractions into a sum of 1 st order fractions is to be able to easily transform them back into either Bode magnitude form or Bode phase form, as explained in Fig 2. Furthermore, the matching process is based on the assumption of the input data being either in Polar or Cartesian form and then being processed as a complex quantity with zero imaginary parts, as described above in 1.
As explained in [ 18 ], the VF method first recognizes the poles by solving the problem in the least squares sense of the linear problem:. All poles and residues in 5 and 6 are real or complex conjugate pairs, while k is a real number.
The assumption that VF adopts is that the poles of h s , given in 3 , should be equal to the zeros that can be calculated as the eigenvalues of a matrix as in 7 [ 19 ], page This pole relocation process typically converges in 2—3 iterations. After identifying the poles, the residues of 3 are calculated by solving the equivalent least squares LS problem with the new known poles.
Furthermore, the objective of the vector fitting method is to calculate the unknown poles, residues and constant of the rational approximation function equivalent to 3. Referring to 8 , the general numerical expression of r n , and p n , can be expressed in complex form as:. The same applies for im p n and re p n.
Applying the inverse Laplace transform on 8 yields h t in 10 :. Weak attenuations guarantee that the least squares LS problem is solved, and the allocation of the pair over the frequency range reduces the probability that the poles must be relocated a long distance, avoiding the need for many iterations.
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In some instances, a logarithmic distribution results in a faster convergence. It is worth mentiond that the VF has been addressed in time-based environment as a Time-Domain VF TDVF method for macromodeling the multiport linear system and supported with accurate validations, as described in [ 20 ]. This research proposes not only different approach towards the macromodeling of experimentally-based data system in time-domain, but also generalize the VF algorithm to be applied for all power curves and measured data within any domain. The main aim of the proposed modifications to the VF is to generalize the VF algorithm from the s-domain frequency environment to the time-domain, or even for any domain application.
Thus, the RAF must be converted back into a time-based formula. The final step is to retrieve the originality of the input or measured data points of the modeled system; this step depends on the preliminary assumption that was adopted in 1 to describe the measured data.