Lecture 17: ARMAX and other Linear Model Structures

1. Lecture 17:ARMAX and other Linear Model Structures Dr Martin Brown Room: E1k, Control Systems Centre Email: martin.brown@manchester.ac.uk Telephone: 0161 306 4672 http://www.eee.manchester.ac.uk/intranet/pg/coursematerial/

2. L17: Resources & Learning Objectives • Core texts • Ljung, Chapters 2, 3 & 4 • In this lecture we’re looking at the basic ARMAX model structure and considering • How it differs from ARX representation • What disturbance signals can be modelled • How the parameters are represented and estimated • Other discrete time polynomials models

4. Not Gaussian, Additive Disturbances • The disturbances are characterised by the fact that the value is not known beforehand, however it is important for making predictions about future values. • Use a probabilistic framework to describe disturbances, and generally describe e(t) by its mean and variance (iid). • The modelling of the transfer function h, can give dynamic disturbance terms: • where m is small and r~N(0,s2) v(t)

9. Example: ARMAX Model • First order model • We assume that e(t) is normal, iid noise. This is not true for v(t) = e(t)+0.2e(t-1), hence we can’t use an ARX model and must use a first order ARMAX system. • The poles of the disturbance->output and the control->output are both given by A=1-0.5q-1 • The zeros of the disturbance->output are given by C=1+0.2q-1 • The zeros of the control->output are given by B=q-1 • In forming a prediction, we use e(t)=y(t)-y(t), hence the model is non-linear in its parameters. ^

13. L17 Summary • Whilst much of this course has concentrated on a simple ARX model, this is very limiting in the type of disturbances that can be modelled. • ARMAX, Output Error, Box-Jenkins … models all generalise the basic ARX transfer function and can disturbance/noise terms with dynamics • However, the parameter estimation problem is no longer a quadratic optimization process and iterative algorithms must be used.

14. L17 Lab