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 th ...
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 m ...
. 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 structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
; 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 thermodynam ...
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 from the
states of the
system
A system is a group of Interaction, 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 (systems), environment, is described by its boundaries, ...
to the states of the regulator. The authors view this map
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 and ...
system
A system is a group of Interaction, 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 (systems), environment, is described by its boundaries, ...
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
Ethical Regulator Theorem
Mick Ashby's ethical regulator theorem builds upon the Conant-Ashby good regulator theorem, which is ambiguous because being good at regulating does not imply being good ethically. "The ethical regulator theorem claims ...
theorem, and by the theory of
practopoiesis
An adaptive system is a set of interacting or interdependent entities, real or abstract, forming an integrated whole that together are able to respond to environmental changes or changes in the interacting parts, in a way analogous to either conti ...
. 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 w ...
(ODE) subset of
control theory
Control theory is a field of mathematics that deals with the 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 the system to a ...
, 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
See also
*
Analogy#Mathematics
*
Map–territory relation
The map–territory relation is the relationship between an object and a representation of that object, as in the relation between a geographical territory and a map of it. Polish-American scientist and philosopher Alfred Korzybski remarked that ...
*
Variety (cybernetics)
In cybernetics, the term variety denotes the total number of distinguishable elements of a set, most often the set of states, inputs, or outputs of a finite-state machine or transformation, or the binary logarithm of the same quantity. Variety ...
*
Internal model (motor control)
Internal may refer to:
*Internality as a concept in behavioural economics
*Neijia, internal styles of Chinese martial arts
*Neigong or "internal skills", a type of exercise in meditation associated with Daoism
*''Internal (album)'' by Safia, 2016
...
*
Ethical regulator
Ethical Regulator Theorem
Mick Ashby's ethical regulator theorem builds upon the Conant-Ashby good regulator theorem, which is ambiguous because being good at regulating does not imply being good ethically. "The ethical regulator theorem claims ...
*
W. Ross Ashby
References
Cybernetics