Model Identification and Data Analysis

5Identification of Input–Output Models

5.1 Introduction

In this chapter, we present two of the most widely used methods for the estimation of dynamical models in input–output form, the least squares and the maximum likelihood (ML) methods. The former is conceived for the estimation of AR or ARX models and the latter for the estimation of ARMA or an ARMAX models.

5.2 Estimating AR and ARX Models: The Least Squares Method

Since the AR model is a special case of the ARX model, we focus on the latter class of models,

where, as usual, is the system input, is the system output, and . By introducing the vector of parameters

and the vector of the observations

(5.1)

the model can be written as

The associated predictor is

so that the prediction error is given by

(5.2)

Assuming as estimation ...