The good regulator is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

conceived by Roger C. Conant and W. Ross Ashby that is central to

cybernetics Cybernetics is a wide-ranging field concerned with circular causality, such as feedback, in regulatory and purposive systems. Cybernetics is named after an example of circular causal feedback, that of steering a ship, where the helmsperson ma ...

. Originally stated that "every good regulator of a system must be a model of that system", but more accurately, every good regulator must contain a model of the system. That is, any regulator that is maximally simple among optimal regulators must behave as an image of that system under a

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

; while the authors sometimes say 'isomorphism', the mapping they construct is only a homomorphism.

Theorem

This theorem is obtained by considering the

entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...

of the variation of the output of the controlled system, and shows that, under very general conditions, that the entropy is minimized when there is a (deterministic) mapping

h:S\to R

from the states of the

system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and express ...

to the states of the regulator. The authors view this map

h

as making the regulator a 'model' of the system. With regard to the brain, insofar as it is successful and efficient as a regulator for survival, it must proceed, in learning, by the formation of a model (or models) of its environment. The theorem is general enough to apply to all regulating and self-regulating or

homeostatic In biology, homeostasis (British also homoeostasis) (/hɒmɪə(ʊ)ˈsteɪsɪs/) is the state of steady internal, physical, and chemical conditions maintained by living systems. This is the condition of optimal functioning for the organism an ...

s. The theorem does not explain what it takes for the system to become a good regulator. In cybernetics, the problem of creating good regulators is addressed by the ethical regulator theorem, and by the theory of practopoiesis. The construction of good regulators is a general problem for any system (e.g., an automated information system) that regulates some domain of application. When restricted to the

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

(ODE) subset of

control theory Control theory is a field of mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive ...

, it is referred to as the internal model principle, which was first articulated in 1976 by B. A. Francis and W. M. Wonham. In this form, it stands in contrast to classical control, in that the classical

feedback loop Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled c ...

fails to explicitly model the controlled system (although the classical controller may contain an implicit model).Jan Swevers,
Internal model control (IMC)
{{Webarchive, url=https://web.archive.org/web/20170830033243/https://people.mech.kuleuven.be/~jswevers/h04x3a/lecture_c9_c10.pdf , date=2017-08-30 ", 2006

References

Cybernetics

Theorem

See also

References