# Rigorous proof of the good regulator theorem

As an applied mathematician with an interest in control theory, I have read several research papers that explicitly use the good regulator theorem of Conant and Ashby 1 which states that:

Every good regulator of a system must be a model of that system.

As pointed out by the authors of [3], the importance and generality of this theorem in control theory makes it comparable in importance to Einstein's $$E=mc^2$$ for physics. However, as John C. Baez carefully argues in a blog post titled The Internal Model Principle it's not clear that Conant and Ashby's paper demonstrates what it sets out to prove. I'd like to add that many other researchers, besides myself, share John C. Baez' perspective.

At present, I think an information-theoretic approach may be used to demonstrate a general version of the good regulator theorem and in the near future I will probably attempt such a demonstration. Meanwhile, might there be control theorists on the MathOverflow that know of a proof of the good regulator theorem that is both clear and rigorous?

Such a publication, if it exists, would be of interest to applied mathematicians working on control-theoretic problems, researchers in the area of behavioural neuroscience as well as artificial intelligence researchers.

References:

1. Roger C. Conant and W. Ross Ashby, Every good regulator of a system must be a model of that system), International Journal of Systems Science 1 (1970), 89–97.
2. B. A. Francis and W. M. Wonham, The internal model principle of control theory, Automatica 12 (1976) 457–465
3. Daniel L. Scholten, Every good key must be a model of the lock it opens (the Conant & Ashby Theorem revisited), 2010.