Stochastic models

Abstract

All science is concerned in some way with prediction, since the ultimate test of a scientific theory is its ability to predict the results of experiments which have not yet been carried out. In the context of engineering systems, a model is some description of a system which enables us to predict its behaviour when it is subjected to certain classes of inputs. Models may be divided into two categories, internal and external Internal models describe the complete structure of a system, possibly including parts of it which do not contribute directly to observable outputs, whereas external models are concerned solely with describing the input/output behaviour of the system. There are two ways in which models may be arrived at: by an analysis of the components of the system using physical laws, or by a ‘black box’ approach whereby the contents of the ‘box’ are inferred from experimental data. In the former case the laws involved are those of Newtonian mechanics, electromagnetism, thermodynamics, etc. In elementary situations such as, say, describing the motion of a pendulum, Newtonian mechanics gives such good predictions that the distinction between ‘model’ and ‘system’ is almost forgotten. However, in more complicated cases — describing the motion of an aircraft, for example — it will be clear that the equations one writes down are only approximations, valid over a certain range of operating conditions. Models arrived at in this way are generally in the first instance internal ones, in that they involve the ‘states’ of various components comprising the system regardless of whether these states are ‘observable’. An external model — which is, after all, less detailed — can often be derived from a given internal model; we study this question in Section 2.4 below. On the other hand, a model obtained by the black-box approach is necessarily external since no other information is available about the system than its input/output behaviour.