Thus, if the whole is at a state of equilibrium, each part must be
In a state of equilibrium in the conditions provided by the other.
The argument can also be reversed. Suppose A and B are at
States of equilibrium, and that each state provides, for the other
82
System, an input-value that makes the other’s state to be one of
Equilibrium. Then neither can change, and the whole cannot
Change; and thus the whole must be at a state of equilibrium.
Thus each implies the other. Formally: the whole is at a state of
Equilibrium if and only if each part is at a state of equilibrium in
The conditions provided by the other part. (If there are several
Parts the last word is merely changed to “parts”.)
Power of veto. The same thesis can be stated more vividly
Making it more useful conceptually. Suppose A and B are coupled
And suppose we are interested only in the occurrence of a state of
Equilibrium (not of cycles). When the whole is started from some
Initial state, and goes along some trajectory, A and B will pass
Through various states. Suppose it happens that at some moment
B’s state provides conditions that make A’s present state one of
Equilibrium. A will not change during the next step. If B is not
Itself at a state of equilibrium in the conditions provided by A, it
Will move to a new state. A’s conditions will thereby be changed,
Its states of equilibrium will probably be changed, and the state it
Is at will probably no longer be one of equilibrium. So A will start
Moving again.
Picturesquely, we can say that A proposed a state of equilibrium
(for A was willing to stop), but B refused to accept the proposal,
Or vetoed the state. We can thus regard each part as having, as it
Were, a power of veto over the states of equilibrium of the whole.
No state (of the whole) can be a state of equilibrium unless it is
Acceptable to every one of the component parts, each acting in the
Conditions given by the others.
Ex.: Three one-variable systems, with Greek-letter parameters, are:
x' = – x + α ,y' = 2 β y + 3,z' =– γ z + δ .
Can they be coupled so as to have a state of equilibrium at (0,0,0)? (Hint:
|
|
|
What value would β have to have ?)
The homeostat. This principle provides a simple way of
Looking at the homeostat and of understanding its working. It can
Be regarded as a part A coupled to a part B (Fig. 5/14/1).
Part A consists essentially of the four needles (with ancillary
Coils, potentiometers, etc.) acting on one another to form a
Four-variable system to which B’s values are input. A’s state is
Specified by the positions of the four needles. Depending on the
Conditions and input, A may have states of equilibrium with the
Needles either central or at the extreme deviation.
Part B consists essentially of a relay, which can be energised or
83
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
STA BI LIT Y
Not, and four stepping-switches, each of which can be in any one
Of 25 positions (not shown accurately in the Figure). Each posi-
tion carries a resistor of some value. So B has 2 × 25 × 25 × 25 ×
I.e. 781250, states. To this system A is input. B has been built
So that, with the relay energised, none of B’s states is equilibrial
(i.e. the switches keep moving), while, with the relay not ener-
Gised, all are equilibrial (i.e. all switches stay where they are).
Finally, B has been coupled to A so that the relay is non-ener-
Gised when and only when A is stable at the central positions.
Fig. 5/14/1
When a problem is set (by a change of value at some input to A
Not shown formally in the Figure), A has a variety of possible
States of equilibrium, some with the needles at the central posi-
Tions, some with the needles fully diverged. The whole will go to
Some state of equilibrium. An equilibrium of the whole implies
Дата добавления: 2019-11-16; просмотров: 216; Мы поможем в написании вашей работы! |
Мы поможем в написании ваших работ!