[From Bruce Abbott (960131.1630 EST)]

Now that we have a new term -- retrofaction -- to replace the term

"control," we will need to rewrite our tutorials accordingly. I thought I'd

take a stab at it.

An advantage of the new term is that it allows the term "control" to be used

as in EAB. There, to control is to influence or determine the state of a

variable. I will use "control" in that sense here.

THE RETROFACTION SYSTEM

A prototypical retrofaction system is diagrammed below:

r

>

p = k1*i p v e a = k2*e

+-->[ if ]--------->[comp]--------->[ af ]--+

> e = r-p |

i | i = a'-d' a' a' = k3*a | a

+---------------[rfv]<-------[ ef ]---------+

^

d' = k4*d | d'

d ----->[ df ]-----+

In the diagram, rfv is the retrofacted variable and d is the disturbance

variable. The disturbance variable produces an effect on the rfv through

the disturbance function df, whose output is d'. In the absence of

retrofaction, the disturbance would exert strong control over the the value

of the rfv. The effect of d on rfv is given by d'. The disturbance effect

combines with the environmental effect of action, a', to control the state

of the retrofacted variable. This then becomes the input (i) to the

retrofaction input function (if), which converts the input into the

retrofacted perception signal (p). The parameter k1 is the retrofaction

input fraction (or input gain).

The retrofacted perception signal is subtracted from the retrofaction

reference signal (r) in the comparator to produce the error signal (e). The

error signal in turn controls the retrofaction action (a) via the

retrofaction action function (af); the parameter k2 is called the

retrofaction action fraction (or output gain); it determines the magnitude

of action (output) produced by a given magnitude of error signal. The

action affects the state of an environmental variable a' via the

retrofaction feedback function (ef), also called the environment function;

the parameter k3 determines how much effect (a') the action will have on the

retrofacted variable.

For the purpose of the present analysis, all variables will be assumed to

exert their effects simultaneously around the loop. For all variables

within the loop, it is easy to see that each variable both controls and is

controlled by every other variable in the loop, including itself.

We can analyze this diagram into two parts; one part the "system" and the

other the "environment." The system consists of everything on the top part

of the diagram, beginning at the input function and ending at the action

function. The remainder of the diagram lies in the environment of the

system. From these two parts we can derive two functions. The retrofaction

forward function (forward system function) shows how the input variable

controls the action variable (assuming a constant reference). In this

example it is:

(1) a = (r - k1*i)*k2.

The retrofaction feedback function (environment function) shows how the

action variable controls the input variable (assuming a constant disturbance):

(2) i = k3*a - k4*d.

However, both a and i are dependent variables. What is needed are equations

for predicting both a and i from the values of the independent variables (r

and d) and the paramater values. These equations can be derived by

substitution. Substituting the right side of Equation 2 for i in Equation 1

gives:

k2*r + k1*k2*k4*d

(3) a = -----------------.

1 + k1*k2*k3

Substituting the right side of Equation 1 for a in Equation 2 gives:

k2*k3*r - k4*d

(4) i = --------------

1 + k1*k2*k3

The term k1*k2*k3 is the product of all the gains around the loop and is

termed the retrofaction fraction, otherwise known as loop gain or system

sensitivity. The higher the loop gain, the better the retrofaction, in the

sense that the retrofacted variable is retrofacted closer to its

retrofaction reference value. In fact, the ability of the disturbance to

control the retrofacted variable will be only 1/(1 + k1*k2*k3) of its effect

in the absence of retrofaction. Retrofaction progressively reduces the

control exerted on the retrofacted variable by the disturbance as the loop

gain increases; At high gain the rfv can seem almost immune to the

disturbance, so long as the system can produce sufficient action to

compensate for the disturbance effect on the rfv.

Equations (3) and (4) give the equilibrium (steady-state) values for a and

i; more sophisticated analyses are required to describe the dynamic behavior

of the system. Also, much more complex systems are possible, including

those with nonlinear components (all the component functions given in this

example were linear).

Sometimes analysts will unwittingly solve Equation 2 for a, yielding:

i + k4*d

(5) a = --------.

k3

The equation seems to show how input controls output through the mediation

of the system, but in reality it only describes the environment. It is in

fact the inverse environment function (or inverse retrofaction feedback

function) and does not describe the system at all.

I hope I've been able to clarify an otherwise murky subject. If I've made

any glaring errors, I'm _confident_ that they will be pointed out to me in

short order!

[Note: Terms like "retrofaction feedback fraction" are very new (I just

invented them) and currently in use only by the most "in" of the "in" crowd,

so if you want yourself to be understood by lesser mortals I would strongly

recommend using the older terms, such as loop gain, input function, and the

like until you are sure you have joined the ranks of the retrofaction

faction. (In fact, I suspect that cooler heads will prevail and my proposed

new terms will be subjected to some much-deserved ridicule and soundly

rejected, so I suggest taking a firm wait-and-see attitude for now.) For

more information about the application for retrofaction to human behavior,

please see the recently retitled _Action: the Retrofaction of Perception_ by

William T. Powers, or A:RP.]

Regards,

Bruce