System follow its own laws for some time and see whether the sys-
Tem does or does not come back to the same state”. In algebraic
Form, we start with a state of equilibrium a, displace the system to
State D(a), and then find TD(a), T2D(a), T3D(a), and so on; and we
Notice whether this succession of states does or does not finish as
A, a, a, .... More compactly: the state of equilibrium a in the sys-
Tem with transformation T is stable under displacement D if and
Only if
nlim T D ( a ) = a
Try this formulation with the three standard examples. With the
cube, a is the state with angle of tilt = 0 °. D displaces this to, say,
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n →∞
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
5 °, and T eventually will bring this back to 0 °. With the cone (hav-
Ing transformation U, say) D can be the same displacement, but
the limit, whatever it is, of UnD(a) is certainly not a tilt of 0 °, the
Equilibrium is unstable. With the billiard ball, at position a, the
Dynamic laws will not bring it back to a after displacement, so it
Is not stable by the definition given here. It has the peculiarity,
However, that the limit is D(a); i.e. it retains the displacement,
Annulling it nor exaggerating it. This is the case of neutral equi-
Librium.
It will be noticed that this study of what happens after the sys-
Tem has been displaced from a is worth making only if a is a state
Of equilibrium.)
Ex. 1: Is the state of equilibrium c stable to T under the displacement D if T
Given by:
a b c d e ↓
Tc d c a e
Db a d e d
Ex. 2: (Continued.) What if the state of equilibrium is e?
Ex. 3: The region composed of the set of states b, c and d is stable under U:
a b c d e f ↓
Ud c b b c a
Eb e ff f d
What is the effect of displacement E, followed by repeated action of U?
(Hint: Consider all three possibilities.)
In general, the results of repeated application of a transforma-
Tion to a state depend on what that state is. The outcome of the test
Of finding what is
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Nlim T ( x )
Will thus depend in general on which state is x. Thus if there are
Two disturbances available, D and E, and D takes a to b, while E
Takes a to c (no order being implied between a, b and c) the limits
Of TnD(a) and TnE(a) may be different.
Thus the result of a test for stability, carried out in the manner
Of S.5/6, may give different results according to whether the dis-
Placement is D or E. The distinction is by no means physically
Unreasonable. Thus a pencil, balanced on its square-cut base, may
be stable to D, if D is a displacement of 1 ° from the vertical, but
may be unstable to E, if E is a displacement of 5 °.
The representation given in S.5/6 thus accords with common
Practice. A system can be said to be in stable equilibrium only if
Some sufficiently definite set of displacements D is specified. If
The specification is explicit, then D is fully defined. Often D is not
Given explicitly but is understood; thus if a radio circuit is said to
Be “stable”, one understands that D means any of the commonly
Occurring voltage fluctuations, but it would usually be understood
To exclude the stroke of lightning. Often the system is understood
To be stable provided the disturbance lies within a certain range
What is important here is that in unusual cases, in biological sys-
Tems for instance, precise specification of the disturbances D, and
Of the state of equilibrium under discussion a, may be necessary
If the discussion is to have exactness.
The continuous system. In the previous sections, the states
Considered were usually arbitrary. Real systems, however, often
Show some continuity, so that the states have the natural relation-
Ship amongst themselves (quite apart from any transformation im-
Posed by their belonging to a transducer) that two states can be
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“near” or “far from” one another.
With such systems, and a state of equilibrium a, D is usually
Defined to be a displacement, from a, to one of the states “near” a.
If the states are defined by vectors with numerical components,
I.e. based on measurements, then D often has the effect of adding
small numerical quantities δ 1, δ 2, .., δ n to the components, so that
the vector (x1, . . ., xn) becomes the vector (x1 + δ 1, . . ., xn + δ n).
In this form, more specialised tests for stability become possi-
Ble. An introduction to the subject has been given in Design...
79
n →∞
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