4/2 AN INTRODUCTION TO CYBERNETICS 



A real machine whose behaviour can be represented by such a set 

 of closed single-valued transformations will be called a transducer 

 or a machine with input (according to the convenience of the context). 

 The set of transformations is its canonical representation. The 

 parameter, as something that can vary, is its input. 



Ex. I: US is I I * 



b a, 



how many other closed and single-valued transformations can be formed 

 on the same two operands ? 



Ex. 2: Draw the three kinematic graphs of the transformations Ry, Ri, Ri 

 above. Does change of parameter- value change the graph ? 



Ex. 3: With R (above) at /?i, the representative point is started at c and allowed 

 to move two steps (to RiHc)); then, with the representative point at this 

 new state, the transformation is changed to Rz and the point allowed to 

 move two more steps. Where is it now ? 



Ex. 4: Find a sequence of R's that will take the representative point (i) from d 

 to a, (ii) from c to a. 



Ex. 5: What change in the transformation corresponds to a machine having 

 one of its variables fixed? What transformation would be obtained if the 

 system 



x'= — X + 2y 



y'= X - y 

 were to have its variable x fixed at the value 4? 



Ex. 6 : Form a table of transformations affected by a parameter, to show that a 

 parameter, though present, may in fact have no actual eff'ect. 



4/2. We can now consider the algebraic way of representing a 

 transducer. 

 The three transformations 



Ri:n' = n+\ R2: n = n + 2 Ry. n' = n + 3 



can obviously be written more compactly as 



R^: n = n + a, 



and this shows us how to proceed. In this expression it must be 

 noticed that the relations of n and a to the transducer are quite 

 different, and the distinction must on no account be lost sight of. 

 n is operand and is changed by the transformation; the fact that it is 

 an operand is shown by the occurrence of n' . a is parameter and 

 determines which transformation shall be applied to n. a must 

 therefore be specified in value before w's change can be found. 



When the expressions in the canonical representation become 

 more complex, the distinction between variable and parameter can 

 be made by remembering that the symbols representing the operands 

 will appear, in some form, on the left, as x' or dxjdt; for the trans- 



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