# System identification using ARMA models

## System identification

System identification falls in to two categories - parametric and non-parametric. When using parametric methods, it is implicitly assumed that the system being investigated adheres to a certain model (e.g. first-order plus dead-time, second-order, etc.) - and the task of identification boils down to finding the 'parameters' for that model. Non-parametric methods, in contrast, make no such assumption - they involve analysis of the frequency response of the system instead.

Parametric methods tend to yield better results than non-parametric methods, as long as the correct model is assumed. If the wrong model is assumed however (e.g., one tries to identify the parameters of a third-order system using a first-order model), the parametric method might yield nonsensical values.

## ARMA models

In the case of parametric identification, ARMA models a typical choice for linear systems. ARMA stands for Auto-Regressive Moving-Average. I.e., it is assumed that the system consists of an 'auto-regressive' component (the denominator coefficients in the system transfer function) and a 'moving-averager' component (the numerator coefficients). Once the order of the system (i.e. how many numerator / denominator coefficients there are) is known, the next step in system identification is to determine what the value of these coefficients - these are the 'parameters' of our 'model'. One could begin this process by 'guessing' some parameter values, and comparing the output of our 'guess' model with that of the actual system. The closer the correspondence between the output of the model and the output of the actual process, the better the model.

## Least-squares estimation

Through trial-and-error, one might eventually find models that closely match the output. This would be far too time-consuming, however. Instead, engineers use matrix mathematics and a technique called 'least-squares estimation'.

We studied this issue as part of our control engineering module in our final year, and used an ARMA model in conjunction with least-squares-estimation to determine the parameters of a first-order system. These issues are discussed more in the PDF below (an assignment report).

Here's the PDF: System identification.