# Where did Arnold Tustin first introduce the bilinear transform?

It's well known that the bilinear transform is also known as Tustin's Method. As far as I know, Arnold Tustin really did introduce the idea into the control systems literature, so the name isn't just a case of Stigler's Law. For example, I managed to find the following reference:

Tustin in the UK developed the bilinear transformation for time series models, while Oldenbourg and Sartorius also used difference equations to model such systems. $$

What isn't clear is where he first introduced the idea--even when browsing the titles of his publications. I'm guessing that it only became known as the bilinear transform later on, so he probably didn't use that terminology. I'd like to read his exposition of the technique. Does anyone know where he first published it?

1. Bissel, C.C. A History of Automatic Control. link.
• Arnold Tustin was my great uncle. I'd be grateful if someone can give me a quick layman's explanation of the bilinear transform (also known as Tustin's method). Unfortunately, I'm not a mathematician so can't really understand the explanation given on Wikipedia. I hope someone can tell me why this calculation is significant, whether it is still used and for what purpose. Jan 2 '20 at 19:29
• Richard, not being a mathematician, do you understand the concept of integration or the definite integral? and, if "yes", then do you understand the Riemann summation as a numerical approximation to the definite integral? then, if "yes", do you know about the so-called Trapazoid rule to numerical integration? That's the beginning. But to understand what DSPers are doing with the bilinear transform (a.k.a. Tustin's method) there is a little more to it. It has to do with the difference between continuous-time systems and discrete-time systems. Jan 2 '20 at 19:52
• and the Bilinear Transform is definitely used, currently. especially by us audio guys. send me an email to rbj@audioimagination.com and we can discuss this a little. Jan 2 '20 at 19:53

Even though I couldn't find a decent table of contents, I took a shot on ordering a source book of papers edited by Tustin (i.e. Automatic and Manual Control), thinking he may have included something relevant. The direct source wasn't in there, but in a paper by B. M. Brown, Application of Finite Difference Operators to Linear Systems, the following citation is given:

The simplest of these [approximations in the form of rational fractions] has been derived ad hoc by $Tustin^6$. It is the time series equivalent of the operator $${2\over{\delta}} \space {1 - E^{-1} \over {1 + E^{-1}}}$$

This can be written as

$${2\over{\delta}} {E - 1 \over{ E + 1 }} \space = \space {1\over{\delta}} {\Delta \over { 1 + 1/2 \Delta} } \space = \space {1\over{\delta}} (\Delta - 1/2 \Delta^2 + 1/4 \Delta^3 - \space \ldots \space)$$

That looks like the bilinear transform. The actual reference is a little vague,

$$^{6} \space Tustin, A. \space \space J. Inst. Elect. Engrs. \space 94 \space (1947) \space 130.$$

but I'm pretty sure it corresponds to A method of analysing the behaviour of linear systems in terms of time series. That paper has a pretty detailed exposition, and Tustin doesn't seem to cite any of his other works as references, so it's reasonable to assume that's where he first introduced it. What's really interesting is the notation he used, $${2\over{\delta}} { [1,-1] \over{ [1, 1] } }.$$

• I'm impressed at the amount of effort you put into finding this. You must be a real engineering history buff. Oct 11 '11 at 23:44
• @JasonR Thanks :) I'm mostly interested in how people made the mathematical connections to introduce novel ideas into a field (w.r.t. R&D applications). Oct 11 '11 at 23:48
• @datageist: I second JasonR's comment! Well done.
– Peter K.
Oct 12 '11 at 8:57