# 8

# Process Identification Methods for Frequency Response Models

A Fourier series is one of the most important representations for describing a periodic function. The Fourier series and Fourier transform have been widely used to identify process models. This chapter introduces several process identification methods to estimate the frequency response data of the process.

## 8.1 Fourier Series

The Fourier series is an important basic theory needed in deriving and analyzing process identification methods. In this section, the formulas to calculate the Fourier coefficients of the Fourier series are derived.

Assume that the periodic function has a period *p*. It is proven that all the data of the periodic function can be represented by the following Fourier series (Kreyszig, 2006):

where the coefficients are called the Fourier coefficients. Then, how to calculate the Fourier coefficients for the given periodic function? Let us derive the formula.

Formula to obtain *a*_{0} Let us integrate both sides of (8.1) from *t* = −*p*/2 to *t* = *p*/2. Then, (8.2) is obtained:

It is straightforward to derive (8.3) from (8.2):

Formula to obtain *a*_{m}, m = 1, 2, … Let us integrate both sides of (8.1) from *t* = −*p*/2 to *t*