1.04 - Feedback and Oscillation

classes ::: Cybernetics, Cybernetics, or Control and Communication in the Animal and the Machine, Norbert Wiener, chapterA patient comes into a neurological clinic. He is not paralyzed,,
children :::
branches :::
see also :::

bookmarks: Instances - Definitions - Quotes - Chapters - Wordnet - Webgen

object:1.04 - Feedback and Oscillation
subject class:Cybernetics
book class:Cybernetics, or Control and Communication in the Animal and the Machine
author class:Norbert Wiener
class:chapterA patient comes into a neurological clinic. He is not paralyzed,
and he can move his legs when he receives the order. Neverthe-
less, he suffers under a severe disability. He walks with a peculiar
uncertain gait, with eyes downcast on the ground and on his
legs. He starts each step with a kick, throwing each leg in suc-
cession in front of him. If blindfolded, he cannot stand up, and
totters to the ground. What is the matter with him?
Another patient comes in. While he sits at rest in his chair,
there seems to be nothing wrong with him. However, offer him
a cigarette, and he will swing his hand past it in trying to pick
it up. This will be followed by an equally futile swing in the
other direction, and this by still a third swing back, until his
motion becomes nothing but a futile and violent oscillation.
Give him a glass of water, and he will empty it in these swings
before he is able to bring it to his mouth. What is the matter
with him?
Both of these patients are suffering from one form or another
of what is known as ataxia. Their muscles are strong and healthy
enough, but they are unable to organize their actions. The first
patient suffers from tabes dorsalis. The part of the spinal cord
which ordinarily receives sensations has been damaged or130
Chapter IV
destroyed by the late sequelae of syphilis. The incoming mes-
sages are blunted, if they have not totally disappeared. The
receptors in the joints and tendons and muscles and the soles
of his feet, which ordinarily convey to him the position and
state of motion of his legs, send no messages which his central
nervous system can pick up and transmit, and for information
concerning his posture he is obliged to trust to his eyes and the
balancing organs of his inner ear. In the jargon of the physiolo-
gist, he has lost an important part of his proprioceptive or kin-
esthetic sense.
The second patient has lost none of his proprioceptive sense.
His injury is elsewhere, in the cerebellum, and he is suffering
from what is known as a cerebellar tremor or purpose tremor. It
seems likely that the cerebellum has some function of propor-
tioning the muscular response to the proprioceptive input, and
if this proportioning is disturbed, a tremor may be one of the
results.
We thus see that for effective action on the outer world it is
not only essential that we possess good effectors, but that the
performance of these effectors be properly monitored back to
the central nervous system, and that the readings of these moni-
tors be properly combined with the other information coming
in from the sense organs to produce a properly proportioned
output to the effectors. Something quite similar is the case in
mechanical systems. Let us consider a signal tower on a railroad.
The signalman controls a number of levers which turn the sema-
phore signals on or off and which regulate the setting of the
switches. However, it does not do for him to assume blindly that
the signals and the switches have followed his orders. It may
be that the switches have frozen fast, or that the weight of a
load of snow has bent the signal arms, and that what he hasFeedback and Oscillation
131
supposed to be the actual state of the switches and the signals—
his effectors—does not correspond to the orders he has given.
To avoid the dangers inherent in this contingency, every effec-
tor, switch or signal, is attached to a telltale back in the signal
tower, which conveys to the signalman its actual states and per-
formance. This is the mechanical equivalent of the repeating of
orders in the navy, according to a code by which every subordi-
nate, upon the reception of an order, must repeat it back to his
superior, to show that he has heard and understood it. It is on
such repeated orders that the signalman must act.
Notice that in this system there is a human link in the chain
of the transmission and return of information: in what we shall
from now on call the chain of feedback. It is true that the signal-
man is not altogether a free agent; that his switches and signals
are interlocked, either mechanically or electrically, and that he
is not free to choose some of the more disastrous combinations.
There are, however, feedback chains in which no human element
intervenes. The ordinary thermostat by which we regulate the
heating of a house is one of these. There is a setting for the desired
room temperature; and if the actual temperature of the house is
below this, an apparatus is actuated which opens the damper, or
increases the flow of fuel oil, and brings the temperature of the
house up to the desired level. If, on the other hand, the tempera-
ture of the house exceeds the desired level, the dampers are turned
off or the flow of fuel oil is slackened or interrupted. In this way
the temperature of the house is kept approximately at a steady
level. Note that the constancy of this level depends on the good
design of the thermostat, and that a badly designed thermostat
may send the temperature of the house into violent oscillations
not unlike the motions of the man suffering from cerebellar
tremor.132
Chapter IV
Another example of a purely mechanical feedback system—
the one originally treated by Clerk Maxwell—is that of the gov-
ernor of a steam engine, which serves to regulate its velocity
under varying conditions of load. In the original form designed
by Watt, it consists of two balls attached to pendulum rods
and swinging on opposite sides of a rotating shaft. They are
kept down by their own weight or by a spring, and they are
swung upward by a centrifugal action dependent on the angu-
lar velocity of the shaft. They thus assume a compromise posi-
tion likewise dependent on the angular velocity. This position
is transmitted by other rods to a collar about the shaft, which
actuates a member which serves to open the intake valves of the
cylinder when the engine slows down and the balls fall, and to
close them when the engine speeds up and the balls rise. Notice
that the feedback tends to oppose what the system is already
doing, and is thus negative.
We have thus examples of negative feedbacks to stabilize tem-
perature and negative feedbacks to stabilize velocity. There are
also negative feedbacks to stabilize position, as in the case of
the steering engines of a ship, which are actuated by the angular
difference between the position of the wheel and the position
of the rudder, and always act so as to bring the position of the
rudder into accord with that of the wheel. The feedback of vol-
untary activity is of this nature. We do not will the motions of
certain muscles, and indeed we generally do not know which
muscles are to be moved to accomplish a given task; we will, say,
to pick up a cigarette. Our motion is regulated by some measure
of the amount by which it has not yet been accomplished.
The information fed back to the control center tends to
oppose the departure of the controlled from the controlling
quantity, but it may depend in widely different ways on thisFeedback and Oscillation
133
departure. The simplest control systems are linear: the output
of the effector is a linear expression in the input, and when we
add inputs, we also add outputs. The output is read by some
apparatus equally linear. This reading is simply subtracted from
the input. We wish to give a precise theory of the performance
of such a piece of apparatus, and, in particular, of its defective
behavior and its breaking into oscillation when it is mishandled
or overloaded.
In this book, we have avoided mathematical symbolism
and mathematical technique as far as possible, although we
have been forced to compromise with them in various places,
and in particular in the previous chapter. Here, too, in the rest
of the present chapter, we are dealing precisely with those mat-
ters for which the symbolism of mathematics is the appropriate
language, and we can avoid it only by long periphrases which
are scarcely intelligible to the layman, and which are intelligible
only to the reader acquainted with mathematical symbolism by
virtue of his ability to translate them into this symbolism. The
best compromise we can make is to supplement the symbolism
by an ample verbal explanation.
Let f(t) be a function of the time t where t runs from −∞ to ∞;
that is, let f(t) be a quantity assuming a numerical value for each
time t. At any time t, the quantities f(s) are accessible to us when
s is less than or equal to t but not when s is greater than t. There
are pieces of apparatus, electrical and mechanical, which delay
their input by a fixed time, and these yield us, for an input f(t),
an output f(t − τ), where τ is the fixed delay.
We may combine several pieces of apparatus of this kind,
yielding us outputs f(t − τ 1 ), f(t − τ 2 ), ..., f(t − τ n ). We can multiply
each of these outputs by fixed quantities, positive or negative.
For example, we may use a potentiometer to multiply a voltage134
Chapter IV
by a fixed positive number less than 1, and it is not too difficult
to devise automatic balancing devices and amplifiers to multiply
a voltage by quantities which are negative or are greater than 1.
It is also not difficult to construct simple wiring diagrams of cir-
cuits by which we can add voltages continuously, and with the
aid of these we may obtain an output
n
∑ a f ( t − τ
k
k
)
(4.01)
1
By increasing the number of delays τ k and suitably adjusting the
coefficients a k , we may approximate as closely as we wish to an
output of the form
∫
∞
0
a ( τ ) f ( t − τ ) d τ
(4.02)
In this expression, it is important to realize that the fact that
we are integrating from 0 to ∞, and not from −∞ to ∞, is essen-
tial. Otherwise we could use various practical devices to operate
on this result and to obtain f(t + σ), where σ is positive. This,
however, involves the knowledge of the future of f(t); and f(t)
may be a quantity, like the coordinates of a streetcar which may
turn off one way or the other at a switch, which is not deter-
mined by its past. When a physical process seems to yield us an
operator which converts f(t) to
∫
∞
−∞
a ( τ ) f ( t − τ ) d τ
(4.03)
where a(τ) does not effectively vanish for negative values of τ,
it means that we have no longer a true operator on f(t), deter-
mined uniquely by its past. There are physical cases where this
may occur. For example, a dynamical system with no input may
go into permanent oscillation, or even oscillation building up
to infinity, with an undetermined amplitude. In such a case,Feedback and Oscillation
135
the future of the system is not determined by the past, and we
may in appearance find a formalism which suggests an operator
dependent on the future.
The operation by which we obtain Expression 4.02 from f(t)
has two important further properties: (I) it is independent of a
shift of the origin of time, and (2) it is linear. The first property
is expressed by the statement that if
∞
g ( t ) = ∫ α ( τ ) f ( t − τ ) d τ
(4.04)
0
then
∞
g ( t + σ ) = ∫ α ( τ ) f ( t + σ − τ ) d τ
0
(4.05)
The second property is expressed by the statement that if
g ( t ) = Af 1 ( t ) + Bf 2 ( t )
(4.06)
then
∫
∞
0
a ( τ ) g ( t − τ ) d τ
∞ ∞
0 0
= A ∫ a ( τ ) f 1 ( t − τ ) d τ + B ∫ a ( τ ) f 2 ( t − τ ) d τ
(4.07)
It may be shown that in an appropriate sense every operator
on the past of f(t) which is linear and is invariant under a shift of the
origin of time is either of the form of Expression 4.02 or is a limit of
a sequence of operators of that form. For example, f ′(t) is the result
of an operator with these properties when applied to f(t), and
f ′ ( t ) = lim ∫
 → 0
∞
0
1  τ 
a   f ( t − τ ) d τ
 2   
(4.08)
where
0  x < 1
 1

a ( x ) =  − 1 1  x  2
 0 2  x

(4.09)136
Chapter IV
As we have seen before, the functions e zt are a set offunctions
f(t) which are particularly important from the point of view of
Operator 4.02, since
e z ( t − τ ) = e zt �
e − z τ
(4.10)
and the delay operator becomes merely a multiplier dependent
on z. Thus Operator 4.02 becomes
∞
e zt ∫ a ( τ ) e − z τ d τ
(4.11)
0
and is also a multiplication operator dependent on z only. The
expression
∫
∞
0
a ( τ ) e − z τ d τ = A ( z )
(4.12)
is said to be the representation of Operator 4.02 as a function of
frequency. If z is taken as the complex quantity x + iy, where x
and y are real, this becomes
∫
∞
0
a ( τ ) e − x τ e − iu τ d τ
(4.13)
so that by the well-known Schwarz inequality concerning inte-
grals, if y > 0 and
∫
∞
0
a ( τ ) d τ < ∞
2
(4.14)
we have
∞
∞
2
A ( x + iy )    ∫ a ( τ ) d τ ∫ e − 2 x τ d τ  
0
 0

1 ∞
2
=   ∫ a ( τ ) d τ  
 2 x 0

1
2
1
2
(4.15)
This means that A(x + iy) is a bounded holomorphic func-
tion of a complex variable in every half-plane x   > 0, and thatFeedback and Oscillation
137
the function A(iy) represents in a certain very definite sense the
boundary values of such a function.
Let us put
u + iv = A ( x + iy )
(4.16)
where u and v are real. The x + iy will be determined as a function
(not necessarily single-valued) of u + iv. This function will be
analytic, though meromorphic, except at the points u + iv cor-
responding to points z = x + iy, where ∂A(z)/∂z = 0. The boundary
x = 0 will go into the curve with the parametric equation
u + iv = A ( iy )
( y real )
(4.17)
This new curve may intersect itself any number of times.
In general, however, it will divide the plane into two regions.
Let us consider the curve (Eq. 4.17) traced in the direction in
which y goes from −∞ to ∞. Then if we depart from Eq. 4.17
to the right and follow a continuous course not again cutting
Eq. 4.17, we may arrive at certain points. The points which are
neither in this set nor on Eq. 4.17 we shall call exterior points.
The part of the curve (Eq. 4.17) which contains limit points of
the exterior points we shall call the effective boundary. All other
points will be termed interior points. Thus in the diagram of
Fig. 1, with the boundary drawn in the sense of the arrow, the
interior points are shaded and the effective boundary is drawn
heavily.
The condition that A be bounded in any right half-plane will
then tell us that the point at infinity cannot be an interior point. It
may be a boundary point, although there are certain very defi-
nite restrictions on the character of the type of boundary point
it may be. These concern the “thickness" of the set of interior
points reaching out to infinity.138
Chapter IV
Fig. 1
Now we come to the problem of the mathematical expression
of the problem of linear feedback. Let the control flow chart—
not the wiring diagram—of such a system be as shown in Fig. 2.
Here the input of the motor is Y, which is the difference between
the original input X and the output of the multiplier, which
multiplies the power output AY of the motor by the factor λ.
Thus
Y = X − λ AY
(4.18)
and
Y =
X
1 + λ A
(4.19)
so that the motor output is
AY = X
A
1 + λ A
(4.20)
The operator produced by the whole feedback mechanism is
then A/(1 + λA). This will be infinite when and only when A = −1/λ.
The diagram (Eq. 4.17) for this new operator will be
u + iv =
A ( iy )
1 + λ A ( iy )
(4.21)Feedback and Oscillation
139
Fig. 2
and ∞ will be an interior point of this when and only when −1/λ is an
interior point of Eq. 4.17.
In this case, a feedback with a multiplier λ will certainly pro-
duce something catastrophic, and, as a matter of fact the catas-
trophe will be that the system will go into unrestrained and
increasing oscillation. If, on the other hand, the point −1/λ is an
exterior point, it may be shown that there will be no difficulty,
and the feedback is stable. If −1/λ is on the effective boundary,
a more elaborate discussion is necessary. Under most circum-
stances, the system may go into an oscillation of an amplitude
which does not increase.
It is perhaps worth considering several operators A and the
ranges of feedback which are admissible under them. We shall
consider not only the operations of Expression 4.02 but also their
limits, assuming that the same argument will apply to these.
If the operator A corresponds to the differential operator, A(z)
= z, as y goes from −∞ to ∞, A(y) does the same, and the interior140
Chapter IV
points are the points interior to the right, half-plane. The point
−1/λ is always an exterior point, and any amount of feedback is
possible. If
A ( z ) =
1
1 + kz
(4.22)
the curve (Eq. 4.17) is
u + iv =
1
1 + kiy
(4.23)
or
u =
1
1 + k 2 y 2
v =
− ky
1 + k 2 y 2
(4.24)
which we may write
u 2 + v 2 = u
(4.25)
This is a circle with radius 1/2, and center at (1/2, 0). It is
described in the clockwise sense, and the interior points are
those which we should ordinarily consider interior. In this case
too, the admissible feedback is unlimited, as −1/λ is always out-
side the circle. The a(t) corresponding to this operator is
a ( t ) = e − t k k
(4.26)
Again, let
1  2
A ( z ) =  
 1 + kz  
(4.27)
Then Eq. 4.17 is
2
( 1 − kiy )
 1 
=
u + iv = 
 1 + kiy  
( 1 + k 2 y 2 ) 2
2
(4.28)Feedback and Oscillation
141
and
u =
1 − k 2 y 2
( 1 + k
2
y
v =
,
)
2 2
− 2 ky
( 1 + k 2 y 2 ) 2
(4.29)
This yields
u 2 + v 2 =
1
(4.30)
( 1 + k 2 y 2 )
2
or
y =
− v
(4.31)
( u 2 + v 2 ) 2 k
Then


k 2 v 2
v 2
u = ( u 2 + v 2 )  1 −
 = ( u 2 + v 2 ) −
2
2
2
2 2
4 ( u + v 2 )
4 k ( u + v )  
 
(4.32)
In polar coordinates, if u = ρ cos φ, v = ρ sin φ, this becomes
ρ cos φ = ρ 2 −
1 cos 2 φ
sin 2 φ
= ρ 2 − +
4
4
4
(4.33)
or
ρ −
1
cos φ
=±
2
2
(4.34)
That is,
ρ
1
2
= − sin
φ
,
2
ρ
1
2
= cos
φ
2
(4.35)
It can be shown that these two equations represent only one
curve, a cardioid with vertex at the origin and cusp pointing to
the right. The interior of this curve will contain no point of the
negative real axis, and, as in the previous case, the admissible
amplification is unlimited. Here the operator a(t) is142
Chapter IV
a ( t ) =
t − tk
e
k 2
(4.36)
Let
1  3
A ( z ) =  
 1 + kz  
(4.37)
Let p and φ be defined as in the last case. Then
1
ρ 3 cos
1
φ
φ
1
+ i ρ 3 sin =
3
3 1 + kiy
(4.38)
As in the first case, this will give us
2
ρ 3 cos 2
φ
φ
φ
2
2
+ ρ 3 sin 2 = ρ 3 cos
3
3
3
(4.39)
That is,
ρ
1
3
= cos
φ
3
(4.40)
which is a curve of the shape of Fig. 3. The shaded region repre-
sents the interior points. All feedback with coefficient exceeding
1/8 is impossible. The corresponding a(t) is
t 2 − t k
e
(4.41)
2 k 3
Finally, let our operator corresponding to A be a simple delay
a ( t ) =
of T units of time. Then
A ( z ) = e − Tz
(4.42)
Then
u + iv = e − Tiy = cos Ty − i sin Ty
(4.43)
The curve (Eq. 4.17) will be the unit circle about the origin,
described in a clockwise sense about the origin with unit velocity.Feedback and Oscillation
143
Fig. 3
The inside of this curve will be the inside in the ordinary sense,
and the limit of feedback intensity will be 1.
There is one very interesting conclusion to be drawn from
this. It is possible to compensate for the operator 1/(1 + kz) by
an arbitrarily heavy feedback, which will give us an A/(1 + λA) as
near to 1 as we wish for as large a frequency range as we wish. It
is thus possible to compensate for three successive operators of
this sort by three—or even two—successive feedbacks. It is not,
however, possible to compensate as closely as we wish for an
operator 1/(1 + kz) 3 , which is the resultant of the composition of
three operators 1/(1 + kz) in cascade, by a single feedback, The
operator 1/(1 + kz) 3 may also be written144
1 d 2
1
2 k 2 dz 2 1 + kz
Chapter IV
(4.44)
and may be regarded as the limit of the additive composition of
three operators with first-degree denominators, It thus appears
that a sum of different operators, each of which may be compen-
sated as well as we wish by a single feedback, cannot itself be so
compensated.
In the important book of MacColl, we have an example of a
complicated system which can be stabilized by two feedbacks
but not by one. It concerns the steering of a ship by a gyro-
compass. The angle between the course set by the quartermaster
and that shown by the compass expresses itself in the turning
of the rudder, which, in view of the headway of the ship, pro-
duces a turning moment which serves to change the course of
the ship in such a way as to decrease the difference between the
set course and the actual course. If this is done by a direct open-
ing of the valves of one steering engine and closing of the valves
of the other in such a way that the turning velocity of the rud-
der is proportional to the deviation of the ship from this course,
let us note that the angular position of the rudder is roughly
proportional to the turning moment of the ship and thus to its
angular acceleration. Hence the amount of turning of the ship is
proportional with a negative factor to the third derivative of the
deviation from the course, and the operation which we have to
stabilize by the feedback from the gyrocompass is kz 3 , where k is
positive. We thus get for our curve (Eq. 4.17)
u + iv = − kiy 3
(4.45)
and, as the left half-plane is the interior region, no servomecha-
nism whatever will stabilize the system.Feedback and Oscillation
145
In this account, we have slightly oversimplified the steer-
ing problem. Actually there is a certain amount of friction, and
the force turning the ship does not determine the acceleration.
Instead, if θ is the angular position of the ship and φ that of the
rudder with respect to the ship, we have
d 2 θ
d θ
= c 1 φ − c 2
dt
dt 2
(4.46)
and
u + iv = − k 1 iy 3 − k 2 y 2
(4.47)
This curve may be written
v 2 = − k 3 u 3
(4.48)
which still cannot be stabilized by any feedback. As y goes from
−∞ to ∞, u goes from ∞ to −∞, and the inside of the curve is to
the left.
If, on the other hand, the position of the rudder is propor-
tional to the deviation of the course, the operator to be stabilized
by feedback is k 1 z 2 + k 2 z, and Eq. 4.17 becomes
u + iv = − k 1 y 2 + k 2 iy
(4.49)
This curve may be written
v 2 = − k 3 u
(4.50)
but in this case, as y goes from −∞ to ∞, so does v, and the curve
is described from y = −∞ to y = ∞. In this case, the outside of the
curve is to the left, and unlimited amount of amplification is
possible.
To achieve this we may employ another stage of feedback. If
we regulate the position of the valves of the steering engine, not
by the discrepancy between the actual and the desired course but146
Chapter IV
by the difference between this quantity and the angular position
of the rudder, we shall keep the angular position of the rudder
as nearly proportional to the ship’s deviation from true course as
we wish, if we allow a large enough feedback—that is, if we open
the valves wide enough. This double feedback system of control
is in fact the one usually adopted for the automatic steering of
ships by means of the gyrocompass.
In the human body, the motion of a hand or a finger involves
a system with a large number of joints. The output is an additive
vectorial combination of the outputs of all these joints. We have
seen that, in general, a complex additive system like this cannot
be stabilized by a single feedback. Correspondingly, the volun-
tary feedback by which we regulate the performance of a task
through the observation of the amount by which it is not yet
accomplished needs the backing up of other feedbacks. These we
call postural feedbacks, and they are associated with the general
maintenance of tone of the muscular system. It is the volun-
tary feedback which shows a tendency to break down or become
deranged in cases of cerebellar injury, for the ensuing tremor
does not appear unless the patient tries to perform a voluntary
task. This purpose tremor, in which the patient cannot pick up
a glass of water without upsetting it, is very different in nature
from the tremor of Parkinsonism, or paralysis agitans, which
appears in its most typical form when the patient is at rest, and
indeed often seems to be greatly mitigated when he attempts to
perform a specific task. There are surgeons with Parkinsonism
who manage to operate quite efficiently. Parkinsonism is known
not to have its origin in a diseased condition of the cerebellum,
but to be associated with a pathological focus somewhere in the
brain stem. It is only one of the diseases of the postural feed-
backs, and many of these must have their origin in defects ofFeedback and Oscillation
147
parts of the nervous system situated very differently. One of the
great tasks of physiological cybernetics is to disentangle and iso-
late loci of the different parts of this complex of voluntary and
postural feedbacks. Examples of component reflexes of this sort
are the scratch and the walking reflex.
When feedback is possible and stable, its advantage, as we
have already said, is to make performance less dependent on the
load. Let us consider that the load changes the characteristic A
by dA. The fractional change will be dA/A. If the operator after
feedback is
B =
A
C + A
(4.51)
we shall have
dB
=
B
C
C
− d  1 + 
dA

dA C
A  = A 2
=
C
C
A A + C
1 +
1 +
A
A
(4.52)
Thus feedback serves to diminish the dependence of the system
on the characteristic of the motor, and serves to stabilize it, for
all frequencies for which
A + C
> 1
C
(4.53)
This is to say that the entire boundary between interior and
exterior points must lie inside the circle of radius C about the
point −C. This will not even be true in the first of the cases we
have discussed. The effect of a heavy negative feedback, if it is
at all stable, will be to increase the stability of the system for
low frequencies, but generally at the expense of its stability for
some high frequencies. There are many cases in which even this
degree of stabilization is advantageous.148
Chapter IV
A very important question which arises in connection with
oscillations due to an excessive amount of feedback is that of
the frequency of incipient oscillation. This is determined by the
value of y in the iy corresponding to the point of the boundary
of the inside and outside regions of Eq. 4.17 lying furthest from
the left on the negative u-axis. The quantity y is of course of the
nature of a frequency.
We have now come to the end of an elementary discussion of
linear oscillations, studied from the point of view of feedback.
A linear oscillating system has certain very special properties
which characterize its oscillations. One is that when it oscillates,
it always can and very generally—in the absence of independent
simultaneous oscillations— does oscillate in the form
A sin ( Bt + C ) e Dt
(4.54)
The existence of a periodic non-sinusoidal oscillation is always
a suggestion at least that the variable observed is one in which
the system is not linear. In some cases, but in very few, the sys-
tem may be rendered linear again by a new choice of the inde-
pendent variable. Another very significant difference between
linear and non-linear oscillations is that in the first the ampli-
tude of oscillation is completely independent of the frequency;
while in the latter, there is generally only one amplitude, or at
most a discrete set of amplitudes, for which the system will oscil-
late at a given frequency, as well as a discrete set of frequen-
cies for which the system will oscillate. This is well illustrated
by the study of what happens in an organ pipe. There are two
theories of the organ pipe—a cruder linear theory, and a more
precise non-linear theory. In the first, the organ pipe is treated
as a conservative system. No question is asked about how the
pipe came to oscillate, and the level of oscillation is completelyFeedback and Oscillation
149
indeterminate. In the second theory, the oscillation of the organ
pipe is considered as dissipating energy, and this energy is con-
sidered to have its origin in the stream of air across the lip of
the pipe. There is indeed a theoretical steady-state flow of air
across the lip of the pipe which does not interchange any energy
with any of the modes of oscillation of the pipe, but for certain
velocities of air flow this steady-state condition is unstable. The
slightest chance deviation from it will introduce an energy input
into one or more of the natural modes of linear oscillation of
the pipe; and up to a certain point, this motion will actually
increase the coupling of the proper modes of oscillation of the
pipe with the energy input. The rate of energy input and the
rate of energy output by thermal dissipation and otherwise have
different laws of growth, but, to arrive at a steady state of oscil-
lation, these two quantities must be identical. Thus the level of
the non-linear oscillation is determined just as definitely as its
frequency.
The case we have examined is an example of what is known
as a relaxation oscillation: a case, that is, where a system of equa-
tions invariant under a translation in time leads to a solution
periodic—or corresponding to some generalized notion of peri-
odicity—in time, and determinate in amplitude and frequency
but not in phase. In the case we have discussed, the frequency
of oscillation of the system is close to that of some loosely cou-
pled, nearly linear part of the system. B. van der Pol, one of the
chief authorities on relaxation oscillations, has pointed out that
this is not always the case, and that there are in fact relaxation
oscillations where the predominating frequency is not near
the frequency of linear oscillation of any part of the system.
An example is given by a stream of gas flowing into a chamber
open to the air and in which a pilot light is burning: when the150
Chapter IV
concentration of gas in the air reaches a certain critical value, the
system is ready to explode under ignition by the pilot light, and
the time it takes for this to happen depends only on the rate of
flow of the coal gas, the rate at which air seeps in and the prod-
ucts of combustion seep out, and the percentage composition of
an explosive mixture of coal gas and air.
In general, non-linear systems of equations are hard to solve.
There is, however, a specially tractable case, in which the system
differs only slightly from a linear system, and the terms which
distinguish it change so slowly that they may be considered sub-
stantially constant over a period of oscillation. In this case, we
may study the non-linear system as if it were a linear system
with slowly varying parameters. Systems which may be studied
this way are said to be perturbed secularly, and the theory of
secularly perturbed systems plays a most important role in gravi-
tational astronomy.
It is quite possible that some of the physiological tremors
may be treated somewhat roughly as secularly perturbed lin-
ear systems. We can see quite clearly in such a system why the
steady-state amplitude level may be just as determinate as the
frequency. Let one element in such a system be an amplifier
whose gain decreases as some long-time average of the input of
such a system increases. Then as the oscillation of the system
builds up, the gain may be reduced until a state of equilibrium
is reached.
Non-linear systems of relaxation oscillations have been stud-
ied in some cases by methods developed by Hill and Poincaré. 1
The classical cases for the study of such oscillations are those
in which the equations of the systems are of a different nature;
especially where these differential equations are of low order.
There is not, as far as I know, any comparable adequate study ofFeedback and Oscillation
151
the corresponding integral equations when the system depends
for its future behavior on its entire past behavior. However, it is
not hard to sketch out the form such a theory should take, espe-
cially when we are looking only for periodic solutions. In this
case, the slight modification of the constants of the equation
should lead to a slight, and therefore nearly linear, modification
of the equations of motion. For example, let Op[f(t)] be a func-
tion of t which results from a non-linear operation on f(t), and
which is affected by a translation. Then the variation of Op[f(t)],
δOp[f(t)] corresponding to a variational change δf(t) in f(t) and a
known change in the dynamics of the system, is linear but not
homogeneous in δf(t), though not linear in f(t). If we now know
a solution f(t) of
Op [ f ( t ) ] = 0
(4.55)
and we change the dynamics of the system, we obtain a linear
non-homogeneous equation for δf(t). If
∞
f ( t ) = ∑ a n e in λ t
(4.56)
−∞
and f(t) + δf(t) is also periodic, being of the form
∞
f ( t ) + δ f ( t ) = ∑ ( a n + δ a n ) e in ( λ + δλ ) t
(4.57)
−∞
then
∞ ∞
−∞ −∞
δ f ( t ) = ∑ δ a n e i λ nt + ∑ a n e i λ nt in δλ t
(4.58)
The linear equations for δf(t) will have all coefficients develop-
able into series in e tλnt , since f(t) can itself be developed in this
form. We shall thus obtain an infinite system of linear non-
homogeneous equations in δa n + an, δλ, and λ, and this system152
Chapter IV
of equations may be solvable by the methods of Hill. In this
case, it is at least conceivable that by starting with a linear equa-
tion (non-homogeneous) and gradually shifting the constraints
we may arrive at a solution of a very general type of non-linear
problem in relaxation oscillations. This work, however, lies in
the future.
To a certain extent, the feedback systems of control discussed
in this chapter and the compensation systems discussed in the
previous one are competitors. They both serve to bring the
complicated input-output relations of an effector into a form
approaching a simple proportionality. The feedback system, as
we have seen, does more than this, and has a performance rela-
tively independent of the characteristic and changes of charac-
teristic of the effector used. The relative usefulness of the two
methods of control thus depends on the constancy of the char-
acteristic of the effector. It is natural to suppose that cases arise
in which it is advantageous to combine the two methods. There
are various ways of doing this. One of the most simple is that
illustrated in the diagram of Fig. 4.
Fig. 4Feedback and Oscillation
153
Fig. 5
In this, the entire feedback system may be regarded as a larger
effector, and no new point arises, except that the compensa-
tor must be arranged to compensate what is in some sense the
average characteristic of the feedback system. Another type of
arrangement is shown in Fig. 5.
Here the compensator and effector are combined into one
larger effector. This change will in general alter the maximum
feedback admissible, and it is not easy to see how it can ordinar-
ily be made to increase that level to an important extent. On the
other hand, for the same feedback level, it will most definitely
improve the performance of the system. If, for example, the
effector has an essentially lagging characteristic, the compensa-
tor will be an anticipator or predictor, designed for its statistical
ensemble of inputs. Our feedback, which we may call an antici-
patory feedback, will tend to hurry up the action of the effector
mechanism.
Feedbacks of this general type are certainly found in human
and animal reflexes. When we go duck shooting, the error which
we try to minimize is not that between the position of the gun
and the actual position of the target but that between the posi-
tion of the gun and the anticipated position of the target. Any
system of anti-aircraft fire control must meet the same problem.
The conditions of stability and effectiveness of anticipatory154
Chapter IV
feedbacks need a more thorough discussion than they have yet
received.
Another interesting variant of feedback systems is found in
the way in which we steer a car on an icy road. Our entire con-
duct of driving depends on a knowledge of the slipperiness of
the road surface, that is, on a knowledge of the performance
characteristics of the system car–road. If we wait to find this out
by the ordinary performance of the system, we shall discover
ourselves in a skid before we know it. We thus give to the steer-
ing wheel a succession of small, fast impulses, not enough to
throw the car into a major skid but quite enough to report to our
kinesthetic sense whether the car is in danger of skidding, and
we regulate our method of steering accordingly.
This method of control, which we may call control by infor-
mative feedback, is not difficult to schematize into a mechanical
form and may well be worthwhile employing in practice. We
have a compensator for our effector, and this compensator has
a characteristic which may be varied from outside. We super-
impose on the incoming message a weak high-frequency input
and take off the output of the effector a partial output of the
same high frequency, separated from the rest of the output by
an appropriate filter. We explore the amplitude-phase relations
of the high-frequency output to the input in order to obtain
the performance characteristics of the effector. On the basis of
this, we modify in the appropriate sense the characteristics of
the compensator. The flow chart of the system is much as in the
diagram of Fig. 6.
The advantages of this type of feedback are that the compen-
sator may be adjusted to give stability for every type of constant
load; and that, if the characteristic of the load changes slowly
enough, in what we have called a secular manner, in comparisonFeedback and Oscillation
155
Fig. 6
with the changes of the original input, and if the reading of the
load condition is accurate, the system has no tendency to go
into oscillation. There are very many cases where the change of
load is secular in this manner. For example, the frictional load of
a gun turret depends on the stiffness of the grease, and this again
on the temperature; but this stiffness will not change apprecia-
bly in a few swings of the turret.
Of course, this informative feedback will work well only if the
characteristics of the load at high frequencies are the same as, or
give a good indication of, its characteristics at low frequencies.
This will often be the case if the character of the load, and
hence of the effector, contains a relatively small number of vari-
able parameters.
This informative feedback and the examples we have given of
feedback with compensators are only particular cases of what is
a very complicated theory, and a theory as yet imperfectly stud-
ied. The whole field is undergoing a very rapid development. It
deserves much more attention in the near future.156
Chapter IV
Before we end this chapter, we must not forget another
important physiological application of the principle of feed-
back. A great group of cases in which some sort of feedback is
not only exemplified in physiological phenomena but is abso-
lutely essential for the continuation of life is found in what is
known as homeostasis. The conditions under which life, espe-
cially healthy life, can continue in the higher animals are quite
narrow. A variation of one-half degree centigrade in the body
temperature is generally a sign of illness, and a permanent varia-
tion of five degrees is scarcely consistent with life. The osmotic
pressure of the blood and its hydrogen-ion concentration must
be held within strict limits. The waste products of the body must
be excreted before they rise to toxic concentrations. Beside all
these, our leucocytes and our chemical defenses against infec-
tion must be kept at adequate levels; our heart rate and blood
pressure must neither be too high nor too low; our sex cycle
must conform to the racial needs of reproduction; our calcium
metabolism must be such as neither to soften our bones nor to
calcify our tissues; and so on. In short, our inner economy must
contain an assembly of thermostats, automatic hydrogen-ion-
concentration controls, governors, and the like, which would
be adequate for a great chemical plant. These are what we know
collectively as our homeostatic mechanism.
Our homeostatic feedbacks have one general difference from
our voluntary and our postural feedbacks: they tend to be slower.
There are very few changes in physiological homeostasis—not
even cerebral anemia—that produce serious or permanent dam-
age in a small fraction of a second. Accordingly, the nerve fibers
reserved for the processes of homeostasis—the sympathetic and
parasympathetic systems—
are often non-
myelinated and are
known to have a considerably slower rate of transmission thanFeedback and Oscillation
157
the myelinated fibers. The typical effectors of homeostasis—
smooth muscles and glands—are likewise slow in their action
compared with striped muscles, the typical effectors of volun-
tary and postural activity. Many of the messages of the homeo-
static system are carried by non-nervous channels—the direct
anastomosis of the muscular fibers of the heart, or chemical mes-
sengers such as the hormones, the carbon dioxide content of the
blood, etc.; and, except in the case of the heart muscle, these
too are generally slower modes of transmission than myelinated
nerve fibers.
Any complete textbook on cybernetics should contain a thor-
ough detailed discussion of homeostatic processes, many indi-
vidual cases of which have been discussed in the literature with
some detail. 2 However, this book is rather an introduction to the
subject than a compendious treatise, and the theory of homeo-
static processes involves rather too detailed a knowledge of gen-
eral physiology to be in place here.

questions, comments, suggestions/feedback, take-down requests, contribute, etc
contact me @ integralyogin@gmail.com or
join the integral discord server (chatrooms)
if the page you visited was empty, it may be noted and I will try to fill it out. cheers

OBJECT INSTANCES [0] - TOPICS - AUTHORS - BOOKS - CHAPTERS - CLASSES - SEE ALSO - SIMILAR TITLES

TOPICS

SEE ALSO

AUTH

BOOKS

IN CHAPTERS TITLE

1.04_-_Feedback_and_Oscillation

IN CHAPTERS CLASSNAME

IN CHAPTERS TEXT

1.04_-_Feedback_and_Oscillation

PRIMARY CLASS

chapterA_patient_comes_into_a_neurological_clinic._He_is_not_paralyzed,

SIMILAR TITLES

DEFINITIONS

QUOTES [0 / 0 - 0 / 0]

KEYS (10k)

NEW FULL DB (2.4M)

*** NEWFULLDB 2.4M ***

IN CHAPTERS [0/0]

WORDNET

IN WEBGEN [10000/0]

change font "color":
change "background-color":
change "font-family":
change "padding":
change "table font size":
last updated: 2022-02-04 06:20:35
254912 site hits