THE ERROR-CONTROLLED REGULATOR 12/15 



12/14. Regulation by vetoer. The basic formulation of S.11/4 is 

 of extremely wide applicability. Perhaps its most important par- 

 ticular case occurs when both T and R are machines (determinate or 

 Markovian) and when the values of E depend on the various states 

 of equilibrium that T may come to, with 17 as some state (or states) 

 that have some appropriate or desired property. Most physical 

 regulators are of this type. If R and T are Markovian machines, 

 the bringing of T to a desired state of equilibrium tj by the action of 

 R can readily be achieved if advantage is taken of the fundamental 

 fact that if two machines (such as T and R are now assumed to be) 

 are coupled, the whole can be at a state of equilibrium only when 

 each part is itself at a state of equilibrium, in the conditions provided 

 by the other. The thesis was stated in S.5/13 for the determinate 

 machine, but it is just as true for the Markovian. 



Let the regulator R be built as follows. Let it have an input 

 that can take two values, ^ and y. When its input is j8 (for "bad") 

 let }io state be one of equilibrium, and when its input is y (for "good") 

 let them all be equilibrial. Now couple it to T so that all the states 

 in 7] are transformed, at i?'s input, to the value y, and all others to the 

 value jS. Let the whole follow some trajectory. The only states of 

 equilibrium the whole can go to are those that have i? at a state of 

 equilibrium (by S.5/13); but this implies that i?'s input must be at 

 y, and this implies that T's state must be at one of rj. Thus the 

 construction of R makes it a vetoer of all states of equilibrium in T 

 save those in -q. The whole is thus regulatory; and as T and R are 

 here Markovian, the whole will seem to be hunting for a "desirable" 

 state, and will stick to it when found. R might be regarded as 

 "directing" T's hunting. 



(The possibility that T and R may become trapped in a stable 

 region that contains states not in 77 can be made as small as we 

 please by making R large, i.e. by giving it plenty of states, and by 

 seeing that its /3-matrix is richly connected, so that from any state it 

 has some non-zero probability of moving to any other state.) 



Ex. 1 : What, briefly, must characterise the matrix y, and what jS ? 

 *Ex. 2: Show that the thesis of S.5/13 is equally true for the Markovian machine. 



12/15. The homeostat. In this form we can get another point of 

 view on the homeostat. In S.5/14 (which the reader should read 

 again) we considered it as a whole which moved to an equilibrium, 

 but there we considered the values on the stepping-switches to be 

 soldered on, given, and known. Thus fi's behaviour was deter- 

 minate. We can, however, re-define the homeostat to include the 



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