A method of analysing the behaviour of linear systems in terms of time series

A simple method of analysing the behaviour of linear systems is outlined, in which time functions are represented by the sequences of numbers giving the heights of successive equally spaced ordinates, or an equivalent. Such sets of numbers, represented by a single symbol, or regarded as single, “multi-place” numbers, may be “multiplied,” “divided,” “added” or “subtracted” by stated rules so that the numerical implications of any operational equation may be found by direct calculation. The rules for combining such multi-place numbers are found to be substantially identical with those of ordinary multiplication, addition, etc., of decimal numbers, except that “carrying” is barred. The method does not require the equations or the parameters of the system to be known in detail, and is therefore specially advantageous in the analysis of performance on the basis of test or statistical data, the differential equations of the system being unknown or excessively complicated.As examples of applications in which the method is useful or suggestive, the paper includes some discussion of its use in the following problems:(a) In analysing the performance of systems in which definite time delays occur, such as the response-time of a human operator.(b) In the design of means of stabilizing closed-cycle control systems.The approach has also been adapted to deal with generally linear systems in which a single non-linear relationship occurs, but this application is not developed in the present paper.