EE-M /7, EF L17 1/12, v1.0 Lecture 17: ARMAX and other Linear Model Structures Dr Martin Brown Room: E1k, Control Systems Centre Telephone:

EE-M /7, EF L17 2/12, v1.0 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 1.How it differs from ARX representation 2.What disturbance signals can be modelled 3.How the parameters are represented and estimated 4.Other discrete time polynomials models

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EE-M /7, EF L17 4/12, v1.0 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  is small and r~N(0,  2 ) v(t)v(t)

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EE-M /7, EF L17 9/12, v1.0 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. ^

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EE-M /7, EF L17 13/12, v1.0 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.

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