Good Regulator Theorem (Conant & Ashby, 1970)

"Every Good Regulator of a System Must Be a Model of That System." To regulate a system well, the regulator is forced to be a model of it. Control is modeling.

Setup

Disturbances D affect an essential variable Z (room temp, drone attitude); a regulator R observes and acts to keep Z constant despite D — i.e. minimize the outcome entropy H(Z). Theorem: the optimal R is necessarily a model of the system — a homomorphism from system states to R's states.

Why (intuition)

To hold Z steady as disturbances vary, R must produce, for each system state, exactly the counteracting response. The map "which state → which cancelling response" is a model of the system's input→output structure. You can't regulate what you can't map.

Proof skeleton (information-theoretic)

optimal regulation = minimize H(Z);
key lemma: randomness in the regulator only adds entropy to the outcome — you can't make Z more deterministic by injecting noise → the optimal regulator is a deterministic function ρ: S→R;
ρ deterministic + outcome a deterministic function of (S,R) + outcome forced constant → ρ must make R's states correspond (homomorphically) to S's → R is a model of S.

Lineage

Ancestor: Ashby's Law of Requisite Variety ("only variety can destroy variety") — the regulator needs enough variety to match disturbances (ashby);
Good Regulator sharpens it: not just enough variety, but the right structure — a model;
Rigorous control cousin: the Internal Model Principle (Francis–Wonham, 1976) — to asymptotically track/reject a signal class, the controller must embed a model of the signal generator;
Modern echo: Friston's Free Energy Principle / active inference.

Honest caveat

The "model" proved is a weak formal notion (a mapping/homomorphism), not a rich predictive model; the original proof has been criticized. But the core intuition — regulating well = encoding the system's structure — is robust and widely influential (often merged with the Internal Model Principle).