PCT and CT

[From Bill Powers (931228.0750 EST)]

Osmo Eerola (931228.0845 GMT) --

Cannot see why "multiple input" systems could not control
"single output".

... and ...

... it is very well possible that multiple inputs can control a
few outputs - the output O(t) is a function of several inputs
I1(t), I2(t),...,In(t). There can even be several sensed
(perceived) variables of a process and all they control the
ouput in combination in a closed loop manner.

The most important parts of your statement are "is a function of"
and "in combination." A function or combination of a set of
inputs has just one value at a time. Thus what seem to be
multiple degrees of freedom when seen from outside the system
become a single degree of freedom inside the system. This is the
basis for the hierarchical perception aspect of HPCT. When
multiple inputs are reduced to a single representative signal, it
is no longer possible for each input to have a unique effect on
the output. Instead, only a function of all the inputs has a
relationship to the output.

Because of this reduction in degrees of freedom, it is possible
for the same set of inputs to be controlled in several different
and independent respects. The simplest example is a system with
two input variables. One perceptual function can compute the sum
of the two variables, and a second one the difference. Two
independent control systems can be set up, using two output
actuators, such that one controls the sensed sum while the other
controls the sensed difference. The reference signals for each
system can be adjusted independently to cause the sum and the
difference to be maintained at the specified levels,
independently of each other. One control system's error signal
affects both external variables in the same direction, and the
other affects them in opposite directions. So both actuators are
used by both control systems, but the effect is that of
independently controlling two degrees of freedom of the
environment. This is easy to simulate, by the way.

Control of this kind is handled in standard control theory, but
in a way that obscures the principle involved (at least for me).
When I see matrix control equations, I don't see any of these
basic relationships. Maybe other people do. I've never seen them
discussed in a CT text.

From me:

I think of controlling as something done by the whole closed
loop, not by any one part of it.

From you:
Sure. But one can observe the closed loop functioning by
measuring it from different parts:

That's true of the engineer looking at an artificial control
system. But now consider your own control systems (the ones
the perceptual signals, the feedback signals. You can't
experience your outputs, or the mechanisms in the external part
of the loop. Everything you know (even about the forces you
generate with your muscles) has to come to you through sensors,
so all you ever experience are the feedback signals in various
control systems.

When you look at another person behaving, you see the immediate
and indirect environmental effects of the muscle forces, but now
you have no access to the feedback signals or the other parts of
the control systems inside the other person. You can't see
directly what aspect of the environment the other person is
perceiving, so you can't see immediately what the other person is
controlling.

Building up a control-system model of human behavior requires
putting these two views together into a model that explains them
both (never forgetting that while you are watching another person
behaving, you are still observing only your own perceptual
signals). I haven't seen this discussed in CT textbooks, either.

You have applied Ct to human behaviour and call it PCT.
The main "difference" between CT and PCT is that you
control perceptions, not outputs (but actually the ouputs
are affected, too).

Yes, the outputs are affected, too, as well as the external
variables that the outputs act upon. Let's try to reach agreement
on what "output" means in PCT. The basic output of a higher
organism consists of muscle contractions. Muscle contractions are
a physical effect of neural signals that depends ONLY on the
neural signals (disturbances capable of altering the muscle
contractions independently of the neural signals' effect would
hardly ever occur, and then only under very unusual
circumstances). In general, we use the word output to mean the
immediate effect of an actuator device on the part of the
physical world that it directly acts upon.

The parallel in the world of electromechanical system would be
found in a motor used as an output device. The electronic signal
entering the motor is like the neural signal in the living
organism. This electronic signal (operating through a power
output stage) causes current to flow through the windings of the
motor, with the result that a torque is exerted on the armature.
It would be very unusual to find any external agency that could
alter this torque, so the torque (the static torque at least) is
a direct measure of the immediate effect of the electronic system
on its environment. We define that as the output of the control
system.

This lets us distinguish between the physical outputs generated
by a control system and the effects those outputs have on the
rest of the physical world. In the human system, the muscle
contraction has the immediate effect of stretching the elastic
part of the muscle between its attachments. This is a purely
passive, mechanical, physical effect involving the series spring
constant of the muscle. It is the stretching of this "spring"
that produces a force tending to pull the attachments of the
muscle together. The shortest feedback loop in the human body
involves sensors embedded in the tendons that fasten the muscles
to the bones. These tendon signals do not measure the output of
the control system (the shortening of the contractile part of the
muscle), but a physical effect of that output (the stretching of
the series spring element, and hence the force applied to the
tendon). This is a negative feedback loop, with the comparator
residing in the spinal cord as a spinal motor neuron. The error
signal is the output of this neuron, and it enters the muscle to
cause the contractile elements to shorten. The feedback from the
tendon receptor inhibits (proportionally) the output of the
spinal motor neuron, making the feedback negative

We can apply this same principle to the artificial system with a
motor as an output device. The output of the electronic control
system is the torque applied to the armature of the motor. The
immediate result is to create a twisting moment on the shaft that
connects the motor to the load. The shaft "winds up" slightly,
and this slight torsion produces a torque at the other end of the
shaft, determined by the torsional spring constant of the shaft.
That torque tends to accelerate the load.

In the human system, the actual force applied to a tendon when
the contractile fibers shorten depends not only on the
contraction, but on the angle at the joint. If the joint angle
changes, the series spring element in the muscle will lengthen or
shorten even with the contractile part in a constant state of
contraction. The joint angle, in turn, depends on external loads
and inertial forces (if the angle is changing). So the net force
sensed in the tendon is affected not only by the muscle
contraction, but by external disturbances of various kinds.

This applies in the artificial control system as well. The
angular position of the load can be disturbed by external forces
and inertial effects, independently of the motor's torque. The
acceleration of the load is thus not completely determined by the
torque applied to the armature of the motor and the springiness
of the shaft; it is also affected, just as much, by externally
applied torques and by the moment of inertia of the load. The
acceleration, velocity, and position of the load are affected
BOTH by the output of the motor (the torque applied to the
armature) AND external torques and loads.

Therefore measurements of the state of the load are not the same
as measurements of the output of the motor, with output defined
as we do in PCT. I trust that you agree with this analysis so
far. Any actual engineering design has to take these factors into
account. These details tend to get lost in conceptual diagrams,
but in building any real control system they must be considered
just as laid out here.

The tendon "reflex" is a force or torque control system. When you
press a finger against a tabletop, this force control system is
used to make the applied force vary in a precise way, with the
tabletop preventing any angular accelerations or velocities from
developing. The reason that this is a force control system is
that force is _sensed_ by the system. The muscle contraction can
have different effects under different physical circumstances; it
is the fact that tendon receptors sense force, and not velocity
or position, that makes this a force control system.

depends on what is sensed. If a strain gauge is used to detect
the torque applied by the load to something else, something that
is prevented from spinning freely, then we will have a torque
control system. If we mount an angular accelerometer on the load,
the system will be an angular acceleration control system. If a
tachometer is mounted on the load, sensing angular velocity, we
will have an angular velocity control system. If we mount a
potentiometer on the load, sensing angular position, we will have
an angular position control system. If the load is a pinion gear
engaging a rack, and we use a linear position sensor on the rack,
the system will be a linear position control system. If we let
the load be an iris diaphragm and we mount a photocell behind the
iris, we will have a light-intensity control system.

What is controlled depends on which physical effect of a control
system's output is sensed. The sensor, not the immediate output
of the system, is the determining factor. In the above paragraph,
we used exactly the same output, a torque applied to the armature
of a motor, to control six different physical variables. What
made the difference was not the kind of output used, but the kind
of sensor used, and where among all the indirect effects of the
output torque we chose to place the sensor.

By making a distinction between the output of a control system
and physical effects of that output, we separate the controlled
variable from the output. When disturbances are applied to the
controlled variable, the output can now change in a way that has
an equal and opposite effect on the controlled variable. If we
shine a light on the iris diagraphm, disturbing the illumination
of the photocell behind it, the motor will rotate and close the
diaphragm until the light intensity is at its former value. If we
increase and decrease the illumination slowly and smoothly, the
motor will rotate in one direction and the other, and the light
intensity at the photocell will not change appreciably (of course
it will change a little bit). So the output of the control system
is changing, but the _controlled effect_ of the output is not
changing.

Because the sensor determines which physical effect of the
system's output will be controlled, the controlled variable is
defined primarily by the sensor and the signal emitted by the
sensor. This is what "control of perception" means in the context
of standard control theory.

I have not seen any discussion of this kind in a textbook of
control engineering. When it comes down to a specific design,
then of course all the details I have mentioned must be
considered and the appropriate calculations must be done. But
these details get left out of conceptual discussions. In your
diagram showing G(s) and H(s) and S and P, none of the details
was considered. The output of G(s) was simply labelled S, the
controlled variable, with no provision for a disturbance that
might also affect S. When such a disturbance is drawn into the
diagram, it becomes necessary to separate the output of G(s) from
the state of S, because now the disturbance can alter S
independently of effects from the output of G(s). And when you do
that, calling S the "output" becomes less justifiable -- S is not
a measure of the actual output of the controller any more.

There is much more to say on this subject, particularly about
controlled variables that exist only as functions of multiple
external variables, but this is enough for now.

The control of perception concept is important in the analysis of
human behavior, because the physical outputs of a human being
have innumerable effects on the local environment. Most of these
effects are not under control. The only way to find out which
effects are controlled is to find out which are being represented
in the perceptions of the person, and which of those perceptions
is being compared with a reference signal to produce an error
signal that drives the outputs. This is not like the engineering
problem, where you start by knowing what is to be controlled and
design the rest of the system around that. When we apply control
theory to living systems, we do not see what is being controlled.
All we see are the physical outputs and their multiple physical
effects. That is not enough to tell us what the person is doing
-- that is, which effects of the observable actions are
intentional and under control, and which are accidental side-
effects.

Sorry to go on for so long, but it is important to me to convince
you that PCT is not JUST standard control theory. It is a
detailed and orderly way of applying CT that is not taught in any
control-system courses I have ever seen. Whether it is necessary
for engineering students I can't say, but this careful approach
is very necessary if we are to understand human control systems.
Human behavior is so complex and multidimensional that one can
easily get lost in trying to apply control theory. Without a
detailed and systematic approach, important details can easily be
overlooked, with the final result that the beautiful overall
picture doesn't come into view.

ยทยทยท

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Best,

Bill P.