TH E MA C HI N E WI TH I N PUT
Under the control of whoever arranges the coupling.) Let us fur-
Ther suppose— this is essential to the orderliness of the coupling—
That the two machines P and R work on a common time-scale, so
That their changes keep in step.
It will now be found that the two machines form a new machine
Of completely determined behaviour. Thus, suppose the whole is
Started with R at a and P at i. Because P at i., the R- transformation
Will be R2 (by Z). This will turn a to b; P’s i will turn to k; so the
States a and i have changed determinately to b and k. The argu-
Ment can now be repeated. With P at k, the R-transformation will
Again (by Z) be R2 ; so b will turn (under R2 ) to a, and k will turn
(under P) to i. This happens to bring the whole system back to the
Initial state of (a,i), so the whole will evidently go on indefinitely
Round this cycle.
The behaviour of the whole machine becomes more obvious if
We use the method of S.3/5 and recognise that the state of the
Whole machine is simply a vector with two components (x,y),
Where x is one of a, b, c, d and y is one of i, j, k. The whole
Machine thus has twelve states, and it was shown above that the
State (a,i) undergoes the transitions
(a,i) → (b,k) → (a,i) → etc.
Ex. 1: If Q is the transformation of the whole machine, of the twelve states (x,y),
Complete Q.
Ex. 2: Draw Q’s kinematic graph. How many basins has it?
Ex. 3: Join P and R by using the transformation Y
Y: state of P: ↓ i j k
value of α :1 2 3
What happens when this machine is started from (a,i) ?
Ex. 4: If two machines are joined to form a whole, does the behaviour of the
Whole depend on the manner of coupling? (Hint: use the previous Ex.)
Ex. 5. If two machines of n1 and n2 states respectively are joined together, what
Is the maximal length of transient that the whole can produce ?
Ex. 6: If machine M has a maximal length of transient of n states, what will be
The maximal length of transient if a machine is formed by joining three M’s
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Together ?
Ex. 7: Take many parts (A, B, C, . . .) each with transformation
↓ 0 1 2
α 0 2 0
β 1 1 1
γ 2 2 2
And join them into a single long chain
input → Α →→ C → etc.,
So that A affects B, B affects C, and so on, by Z:
0 1 2Z: ↓ α β γ
If the input to A is kept at a, what happens to the states down the chain?
Ex. 8: (Continued. ) What happens if the input is now changed for one step to β
and then returned to α, where it is held?
Coupling with feedback. In the previous section, P was cou-
Pled to R so that P’s changes affected, or determined in some way,
What R’s changes would be, but P’s changes did not depend on
What state R was at. Two machines can, however, be coupled so
That each affects the other.
For this to be possible, each must have an input, i.e. parameters.
P had no parameters, so this double coupling cannot be made
Directly on the machines of the previous section. Suppose, then,
That we are going to couple R (as before) to S, given below:
↓
a
c
b
d
b
d
a
c
c
d
d
d
d
b
c
b
↓
e
f
F f
E f
F f
F e
S could be coupled to affect R by Y(if R’s parameter is α):
e fY: state of S: : ↓ 3 1 value of α
and R to affect S by X (if S’s parameter is β):
X: state of R: ↓ a b c d3 1 1 2 value of β :
To trace the changes that this new whole machine (call it T) will
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Undergo, suppose it starts at the vector state (a,e). By Y and X, the
Transformations to be used at the first step are R3 and S3. They, act-
Ing on a and e respectively, will give d and f; so the new state of
The whole machine is (d,f). The next two transformations will be
R1 and S2, and the next state therefore (b,f); and so on.
R1
R2
R3
Ex. 1: Construct T’s kinematic graph.
Ex. 2: Couple S and R in some other way.
Ex. 3: Couple S and R so that S affects R but R does not affect S. (Hint: Consider
the effect in X of putting all the values of β the same.
S1
S2
S3
S4
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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
Algebraic coupling. The process of the previous sections, by
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