That feed back was present in the ordinary pendulum (see Ex. 3/6/
Between its position and its momentum— a “feedback” that,
From the practical point of view, is somewhat mystical. To this the
Mathematician retorts that if feedback is to be considered present
Only when there is an actual wire or nerve to represent it, then the
Theory becomes chaotic and riddled with irrelevancies.
In fact, there need be no dispute, for the exact definition of
“feedback” is nowhere important. The fact is that the concept of
“feedback”, so simple and natural in certain elementary cases,
Becomes artificial and of little use when the interconnexions
Between the parts become more complex. When there are only
Two parts joined so that each affects the other, the properties of the
Feedback give important and useful information about the proper-
Ties of the whole. But when the parts rise to even as few as four,
If every one affects the other three, then twenty circuits can be
Traced through them; and knowing the properties of all the twenty
Circuits does not give complete information about the system.
Such complex systems cannot be treated as an interlaced set of
More or less independent feedback circuits, but only as a whole.
For understanding the general principles of dynamic systems,
Therefore, the concept of feedback is inadequate in itself. What is
Important is that complex systems, richly cross-connected inter-
Nally, have complex behaviours, and that these behaviours can be
Goal-seeking in complex patterns.
Ex. 1: Trace twenty circuits in the diagram of Fig. 4/11/1:
Ex. 2: A machine with input a, has the transformation
x' = y – α z
T: y' = 2z
z' = x + α
What machine (as transformation) results if its input α is coupled to its out-
put z, by α =– z?
Ex. 3: (Continued.) will this second machine behave differently from the first
one when the first has α held permanently at– 1 ?
Ex. 4: A machine has, among its inputs, a photoelectric cell; among its outputs a
Lamp of variable brightness. In Condition I there is no connexion from lamp
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To cell, either electrical or optical. In Condition 2 a mirror is placed so that
Variations in the lamp’s brightness cause variations in the cell’s potential (i.e.
So that the machine can “see itself”). Would you expect the behaviours in
Conditions 1 and 2 to differ? (Hint: compare with Ex. 3.)
INDEP E NDENC E W IT HI N A W HOLE
In the last few sections the concept of one machine or part
Or variable “having an effect on” another machine or part or vari-
Able has been used repeatedly. It must now be made precise, for it
Is of profound importance. What does it mean in terms of actual
Operations on a given machine? The process is as follows.
Suppose we are testing whether part or variable i has an imme-
Diate effect on part or variable j. Roughly, we let the system show
Its behaviour, and we notice whether the behaviour of part j is
Changed when part i’s value is changed. If part j’s behaviour is
Just the same, whatever i’s value, then we say, in general, that i
Has no effect on j.
To be more precise, we pick on some one state S (of the whole
System) first. With i at some value we notice the transition that
Occurs in part j (ignoring those of other variables). We compare
This transition with those that occur when states S1, S2, etc.— other
Than S— are used, in which S1, S2, etc. differ from S only in the
Value of the i-th component. If S1, S2, etc., give the same transition
In part j as S, then we say that i has no immediate effect on j, and
Vice versa. (“Immediate” effect because we are considering j’s
Values over only one step of time.)
Next consider what the concept means in a transformation. Sup-
Pose its elements are vectors with four components (u,x,y,z), and
That the third line of the canonical equations reads
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y' = 2uy – z.
This tells us that if y is at some value now, the particular value it
Will be at the next step will depend on what values u and z have,
55
Fig. 4/11/1
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A N I N T R O D UC T I O N T O C Y B E R NE T I C S
TH E MA C HI N E WI TH I N PUT
But will not depend on what value x has. The variables u and z are
Said to have an immediate effect on y.
It should be noticed, if the rigour is to be maintained, that the
Presence or absence of an immediate effect, of u on y say, can be
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