This post was written during the Dovetail research fellowship. Thanks to Alex, Dalcy, and Jose for reading and commenting on the draft.

The Internal Model Principle (IMP) is often stated as "a feedback regulator must incorporate a dynamic model of its environment in its internal structure" which is one of those sentences where every word needs a footnote. Recently, I have been trying to understand the IMP, to see if it can tell us anything useful for understanding the Agent-like Structure Problem. In particular, I was interested whether the IMP can be considered as selection theorem. In this post, I will focus on explaining the theorem itself and save its application to the Agent-like Structure Problem for future posts (Edit: I have now written a post about this here)^[1].

I have written this post to summarise what I understand of the Internal Model Principle and I have tried to emphasise intuitive explanations. For a more detailed and mathematically formal distillation of the IMP, I recommend Jose's post on the subject.

This post focuses on the 'Abstract Internal Model Principle' and is based on the paper 'Towards an Abstract Internal Model Principle' by Wonham and the first chapter of the book 'Supervisory Control of Discrete-Event Systems' by Wonham and Cai. There also exists a version of the IMP that is framed in the more traditional language of control theory (using differential equations, transfer functions etc.) which is described in another paper, but I will not focus on it here. The authors imply that this version of the IMP is just a special case of the Abstract IMP but I haven't verified this. From now on, I will use the term 'IMP' to refer to the Abstract IMP.

The mathematical prerequisites for reading this post are roughly 'knows what a set is' and 'knows what a function is'^[2]. The paper and book chapter use a lot of algebraic formalism and lattice theory notation in order to ~~look intimidating~~ be mathematically rigourous. The book chapter is also used to introduce a lot of other concepts for use later in the book but which aren't strictly necessary for just understanding the IMP. I think that by sacrificing these elements we can buy a lot of clarity at the cost of very little rigour. By explaining the IMP in this way, I hope that we can see what it is actually saying, rather than being blinded by mathematical notation.

Let's begin!

T

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