# What PCT says about control (For Osmo)

[From Bill Powers (931210.0715 MST)]

Osmo Eerola (931209) --

My earliest direct experiences with the properties of negative
feedback systems (not counting my U.S. Navy experience as a
technician during World War II) began with a Philbrick analog
computer, in about 1954. Here I became acquainted with
operational amplifiers, which are high-gain (x100000 and up) DC
amplifiers with a bandwidth of perhaps 20 kilohertz and a
frequency characteristic that falls off in amplitude by 3 db per
octave. By connecting various passive components in series with
the negative input and in the path from the output to the
negative input, one can create functions ranging from simple
proportionality to first and higher integrals. By using diodes,
one can make an absolute-value function; by using an amplifier
open-loop, one can create a step function; by using a capacitor
in series with the input, one can make a derivative-taker. Adding
a feedback path to the positive input can create a flip-flop with
hysteresis. Multiplication was even possible using special
circuit elements

-FEEDBACK ELEMENT--<-
> >
> >\ |
> > \ |
> - | OP \ |
IN-->INPUT SERIES ELEMENT|---x-I--| -------------> OUT
+ | AMP/
GROUND -->| / x and I: see text
>/

FIG 1: An operational amplifier in
an analog computing setup.

My insight into "control of input" arose when I read, and
verified, that an operational amplifier in one of these computing
configurations acts to keep its negative input terminal at a
voltage matching that of its positive input terminal. When you
understand that, it's easy to see what functions are created by
various combinations of passive series input elements and
feedback elements. The current generated by a voltage applied to
the input element can be computed by realizing that the other end
of the input element is effectively tied to a fixed voltage; that
current must run through the feedback element to the output,
which immediately shows the way the output depends on the input
current, and hence on the input voltage.

It took me a little while to make the connection between
operational amplifiers and the control systems I was simulating
by using them. The engineering control-system diagrams I was
using were not organized the way the analog computing diagram was
organized, and there was an entirely different emphasis on
various aspects of the whole system. Finally, however, I realized
that the controlled variable of the control system did NOT
correspond to the output voltage of the operational amplifier,
but to the voltage at the negative input terminal. I had been
deceived by the fact that engineers talked about controlling
"outputs," stabilizing them against disturbances, while I could
see that the "output" of the operational amplifier was not
stabilized against disturbances. The disturbances were clearly
the input voltages applied to the input series element, and what
was stabilized against them was the voltage at the negative input
terminal, not the output voltage.

The reference signal was the voltage at the positive input
terminal; if that voltage varied, the voltage at the negative
input terminal would follow it accurately. The output of the
operational amplifier corresponded to something like the torque
in an output effector motor, while the feedback element
corresponded to the physical laws determining the effects of that
torque on the angular position of a shaft. Under that analogy,
the shaft position would be the controlled variable at point x,
and the sensor that feeds the shaft position back to the control
system's input would be in the position labelled I above.
Disturbances would be things like loads, and the series input
element would express the physical effects of loads on the same
shaft position.

Now consider one of the elementary physiological control systems
in a human being, the so-called "tendon reflex." It is organized
like this:

> alpha command signal
>
- + V
------>----- Spinal Motor Neuron----->-
> >
>sensory signal |
> >
> force |
TENDON <------(passive spring)---<-MUSCLE CONTRACTION
RECEPTOR<--(physical effects)-
>
disturbances

Fig. 2: Spinal control system

Clearly, this diagram can be converted into a diagram of the form
of Fig. 1 just by rotating and moving things around a little
without changing any connections. The spinal motor neuron and the
contractile part of the muscle combine to make up an operational
amplifier with a high-powered output; the negative input is the
sensory signal input to the spinal motor neuron, and the positive
input is the alpha command signal. The feedback pathway consists
of the physical effects of the muscle contraction on the tendon
receptor, and the sensory signal completes the path to the
negative input.

The only missing component is the series input element, and that
is clearly supplied by external disturbances that tend to alter
the force applied by the muscle to the tendon. While the
disturbances are not applied directly to the negative input, they
are connected though a physical path to the negative input and
that path can be represented as in Fig. 1. The relationship
between disturbances and muscle contractions is clearly that
between the operational amplifier's output and the input to the
series computing element. The form of the relationship is
determined by the forms of the functions in the feedback path and
in series with the effects of the disturbance. If the gain in the
neuromotor "op amp" is high enough, the characteristics of the
spinal neuron and muscle are of minor importance: the passive
external components determine the nature of the observed
relationship.

The physical output of this system is a contraction in a muscle,
a shortening of contractile fibers. But that is not the
controlled variable. What is controlled is the force created by
the muscle, and that force is what the tendon receptor senses.
The input to the sensor is controlled. When disturbances appear,
the output of the system, the degree of muscle contraction,
changes equally and oppositely, maintaining the sensory signal
almost unchanged and matching the alpha command signal. If the
alpha command signal changes in magnitude, the muscle contraction
will vary as required to keep the input signal tracking the alpha
command signal, and as much more as required to offset the
effects of any disturbance.

Clearly, the alpha command signal is not a command signal, but a
reference signal. The spinal motor neuron is a comparator, and
the muscle is an output function.

Compare the above two diagrams with a standard engineering
diagram of a control system, found in any elementary textbook.

+ error ("controller")
INPUT ------- Comp ------>FORWARD ELEMENT --------->OUTPUT
>- |
---<-----FEEDBACK ELEMENT<----

Fig 3. Standard engineering diagram
of a control system

How would you match Fig. 3 to Fig. 2? The feedback element
clearly includes the tendon receptor, which is affected by
contractions of the muscle that produce force through the spring
characteristics of the passive part of the muscle. Those passive
spring components would have to be part of the forward element as
the diagram is shown; there is no provision for showing the
difference between the physical action of the effector and the
resulting creation of an output effect, the force. The feedback
element would be located at the junction labeled "I" in Fig. 1.
The tendon receptor's signal is carried to the negative terminal
of the spinal motor neuron. The error signal is the output of the
spinal neuron, so the muscle corresponds to the "controller" or
forward element. The "output" of the muscle is contraction
producing a force acting on the tendon receptor, closing that
part of the loop. What, then, corresponds to the line labelled
INPUT? It is the alpha command signal, which is the reference
signal for this control system.

Where, in the spinal control system, does the alpha command
signal or reference signal come from? It does not come from any
sensory input, but from higher in the nervous system, in some
cases from the brain stem, in others from the cerebellum, and in
still others directly from the primary motor area of the cerebral
cortex. It is not an input from the environment, but the pathway
through which higher centers in the brain operate the muscles. In
other words, higher centers in the brain produce actions by
varying reference signals that enter the comparators of spinal
control systems that control sensed force. The reference signal
specifies a particular amount of force, and the feedback action
of the spinal control system alters the contraction in the muscle
until the sensed force matches the requested force. If the
requested force varies, the control action makes the sensed force
track it.

Clearly, the label "output" in the standard diagram is
misleading. The physical output of the muscle that is immediately
caused by the error signal is a contraction, a shortening of
contractile elements. That output stretches a passive spring
element, which results in production of a force. The force is the
variable that is sensed and controlled, and that accelerates the
limb, but that force is most closely connected to the input, not
the output, of the control system. Independent physical effects
can alter the force just as much as the muscle contraction can;
even changing a joint angle alters the force by stretching or
relaxing the spring element. The only aspect of muscle function
that corresponds reliably to the driving error signal is the
length of the contractile elements in the muscle: that is the
true output of the system. And that output is obviously not
controlled; any disturbance of the force can cause it to change.
The force is a consequence of applying that contraction to the
physical world in parallel with other physical processes that
also affect the force. The force is a controlled variable, but it
is not a measure of the output of the control system.

In our PCT diagram, we carefully separate the elements of the
control process so that things which are critically different are
not lumped together and represented as a single function, as
shown in the "canonical" diagram below.

>reference signal
>
perceptual signal---->---COMP------>----- error signal
> >
input funct output funct
> >
input quantity i<-----feedback funct-<--o output quantity
>
----<--disturbance function <---disturbance

Fig. 4: The standard PCT diagram of a control system

Note that the output quantity, a measure of the immediate
physical effect of the output transducer, is separated from the
input quantity by a feedback function. This feedback function
contains the physical relationship that exists between the output
quantity and its effects on other variables. In a shaft angle
control system, the output quantity would be the motor torque
which varies directly with driving current, and the feedback
function would express the things like shaft twist and moment of
inertia that are involved in translating motor torque into an
effect on shaft position at the end of the shaft. The disturbance
would be some other physical variable, such as a varying load or
an externally applied torque, while the disturbance function
expresses the effects of the disturbance on the controlled
variable, the shaft angle.

The controlled variable is treated as an _input_ quantity,
because it is directly sensed by the input function (which
includes a sensor and any signal-processing functions). It is the
input quantity that is controlled. When disturbances vary, the
output quantity varies in an equal and opposite way, keeping the
input quantity from changing (nearly, if control is good). The
overall effect is to keep the perceptual signal, which is what we
call the output of the sensor, in a continuing match with the
reference signal.

and such matters as achieving stability under various conditions.
It is about how we think about control systems, how we set up our
ideas about their organization before we begin to analyze them.
You will notice that each element of the PCT diagram in Fig. 4
has an exact counterpart in the diagram of a spinal control
system, Fig. 2. In contrast, the standard engineering diagram in
Fig. 3 is very hard to fit to Fig. 2, and in addition it leaves
false impressions about the nature of the "input" and the
"output." From the engineering diagram, one could easily get the
impression that reference signals are inputs from the
environment, and that muscle contractions are controlled outputs:
neither of those interpretations is correct.

My contention is that the engineering diagram of a control system
has been extremely misleading to scientists trying to apply
control theory to living systems. When an engineer designs an
artificial system for a specific purpose, any diagram will
suffice if its elements contain all the necessary
transformations. But in trying to analyze an already-existing
living control system, understanding is impeded if the diagram is
not organized in detail like the real system. I am sure you can
see that the PCT diagram is exactly like the engineering diagram,
except that elements and relationships that are laid out
specifically in the PCT diagram are lumped together in the
engineering diagram. All engineers understand that the "input" is
a reference signal -- but even such engineers are misled when
they think about living control systems into believing that this
"input" is a sensory input from the environment. In the living
control systems about which we know the specifics, like the
spinal reflexes, iris reflex, and others, reference inputs do NOT
come from the environment, but from systems higher in the brain's
organization.

I should mention that Fig. 2 is a simplification; there are
actually two layers of control loops in the motor reflexes, the
higher level being the stretch reflex which senses and controls
muscle length. The error signal from that system enters the
spinal motor neuron, adjusting the reference signal for the force
(acceleration) control system. The gamma efferents are the
reference signals for this second layer of control, which
controls muscle length and thus, approximately, joint angle.
Gamma and alpha efferents are often co-activated.

While the engineering diagram is sufficient for engineering
been impressed by the orderliness of the initial approach to
design. As far as I can see, there are no systematic design
principles taught to students; at least none appear in the
textbooks I have seen. The basic approach seems to be to define
what it is about the plant that one wants to control, and to find
inputs and forward transforms that will produce approximately
that effect, with feedback being used more or less as a way of
trimming the performance and almost as an afterthought
counteracting disturbances. The design phase is brief and
sketchy; the student is plunged immediately into complex
mathematics without further consideration.

I think that the principles made clear in the PCT diagram could
improve engineering design. The basic principle is clear in this
diagram: the primary consideration is to provide a sensory signal
that represents the variable in the plant that one wishes to
control, so that by comparing the signal against a reference
signal, one can generate an error signal indicating the degree to
which control has not been achieved. Then the design of the
forward part of the system can be filled in, to provide the gain
and output power necessary for good control and the filtering
necessary for stability.

With this kind of design, the reference signal represents
directly the desired state of the plant variable, and the
feedback signal directly represents the actual state of that
variable. Oddly enough, in many designs I have seen (including
modern examples), this simple concept seems not to be used. The
potential meanings of these signals are lost in the shorthand
mathematical representations of the system, and the simple logic
of control as seen under PCT is buried in the mathematics. I
suspect that in many cases this has led to awkward and
unnecessarily complex designs.

But that's really not my problem, and engineers may know of
difficulties that preclude this simple and orderly approach. All
I know is that the PCT approach makes good sense out of
behavioral organization, and the models we have built to
represent real behaviors on that basis work very well.

···

--------------------------------------------------------------
Best,

Bill P.

From Osmo Eerola [931216-09.10]

To Bill Powers (931210.0715 MST)

Neither control engineers nor life scientist nor PCTers do think
(I suppose) that the whole complexity of human behaviour can be modelled by
one simple control block or even by thousands of such serial/parallel blocks.

I believe, however, that control theory can be applied in physiological
systems and also at the behavioral level. Expanded even to international
politics (there is in Wienna an international research center 'for global system
analysis').

So, all this discussion started from the statement that PCT is a new science.

My earliest direct experiences with the properties of negative
feedback systems (not counting my U.S. Navy experience as a
technician during World War II) began with a Philbrick analog

Another army colleague, I see. Sir, I started 20 years later with op.amps.
and analogue computers. I have built system models with analogue computers
including differentiators, integrators etc. A nice tool, seldomly used

-FEEDBACK ELEMENT--<-
> >
> >\ |
> > \ |
> - | OP \ |
IN-->INPUT SERIES ELEMENT|---x-I--| -------------> OUT
+ | AMP/
GROUND -->| / x and I: see text
>/

FIG 1: An operational amplifier in
an analog computing setup.

My insight into "control of input" arose when I read, and
verified, that an operational amplifier in one of these computing
configurations acts to keep its negative input terminal at a
voltage matching that of its positive input terminal.

That is true. For a more detailed circuit design you actually have
to go inside the op. amp. block and understand what kind of
FET/bipolar transistors and in which configuration are used
e.g. in the first differential state etc.

The disturbances were clearly
the input voltages applied to the input series element, and what
was stabilized against them was the voltage at the negative input
terminal, not the output voltage.

Closed-loop negative feedback system reacts (within its limits)
to all disturbances applied anywhere within the closed loop.
But, as I mentioned in my earlier mail, sometimes the "final physical
output" (energy flow) cannot me directly measured. Then there is an
extra block (transfer function) in the schema and disturbances to this
block are out of control (see my figure).

The output of the
operational amplifier corresponded to something like the torque
in an output effector motor, while the feedback element
corresponded to the physical laws determining the effects of that
torque on the angular position of a shaft. Under that analogy,
the shaft position would be the controlled variable at point x,

Yes, we can say so. As I told we can consider the closed loop from
different points of view, but it still is the same loop! No new
science needed.

Below there is the basic block schema, but in the example the final
energy flow (output) is out of direct control. At point S the
physical output of the block G(s) is sensed and the
'perception' P is the sensed output as manipulated by the transfer
function H(s). The physical output of the G(s) further drives
the energy flow via G2(s). Of course the perception is affected or
controlled, too, since it is part of the loop!

The control of 'perception' is maybe easier to understand by means
of discrete time control: The P signal feeded to Comp. is a function
of H(ti) and G(ti), when the time index i goes from some timepoint to
the present timepoint. E.G. S(tj)=G*(Ref-H*S(ti)) , where i=j-dt

_______ Energy
I I flow
------------------I G2(s) I---->
_______ I _______ I_______I
Comp I I I I I
Ref. ------->X--------I G(s) I------I H(s) I---I
I diff I______I S I_______I P I
I I
I----------------------------------I

The physical output of this system is a contraction in a muscle,
a shortening of contractile fibers. But that is not the
controlled variable. What is controlled is the force created by
the muscle, and that force is what the tendon receptor senses.

See my figure above. It is not so seldom in artificial (man made)
control systems that you have to measure something else than the
final 'energy flow'.

Where, in the spinal control system, does the alpha command
signal or reference signal come from? It does not come from any
sensory input, but from higher in the nervous system,

Well, I think the reference signal comes always from a higher level
than from the level taken into consideration. In self-organizing
systems (like humans) the system itself can adjust the loop parameters
and the reference values, too. But even then the reference is
adjusted (typically) by a higher level part of the whole system. And
at the psycho-social-economical level of individual behaviour the 'highest'
references originate from the society (learned), though applied individually by
different individuals.

Clearly, the label "output" in the standard diagram is

It maybe is, depends on what you mean by output. In my diagram
S is the output of block G(s) and P of H(s), 'energy flow' is the
final output of the control system. But still, new science is not needed.

>reference signal
>
perceptual signal---->---COMP------>----- error signal
> >
input funct output funct
> >
input quantity i<-----feedback funct-<--o output quantity
>
----<--disturbance function <---disturbance

Fig. 4: The standard PCT diagram of a control system

If this is the standard PCT diagram I cannot see any
fundamental difference to my circuit: If we add after the
H(s) block an extra feedback transfer function (serially) we
get your schemata. And as you know there are
rules, how to simplyfy complex block schemata (transfer
functions) to simpler ones. So my old hat diagram perfectly fits
your figure 4. Disturbances in real world are present at any damn
block and wirings between them.

The overall effect is to keep the perceptual signal, which is what we
call the output of the sensor, in a continuing match with the
reference signal.

As in my diagram the signal P. You of course can call it how you want,
but believe me, as you describe it above is just what a conventional
control block does!!!! It tries to get the difference of reference and
feedback signal as zero, which equals to matching them perfectly.
You have made the hat trick by changing the point of view, but
the structure is EXACTLY the same.

From the engineering diagram, one could easily get the
impression that reference signals are inputs from the
environment, and that muscle contractions are controlled outputs:
neither of those interpretations is correct.

As I told, the origin of reference signals depends on the level we
consider the system.

I am sure you can
see that the PCT diagram is exactly like the engineering diagram,
except that elements and relationships that are laid out
specifically in the PCT diagram are lumped together in the
engineering diagram.

As I did!

All engineers understand that the "input" is
a reference signal -- but even such engineers are misled when
they think about living control systems into believing that this
"input" is a sensory input from the environment. In the living
control systems about which we know the specifics, like the
spinal reflexes, iris reflex, and others, reference inputs do NOT
come from the environment, but from systems higher in the brain's
organization.

And from the higher level of the system under consideration! Of course
my soldiers' spinal cord control is not affected directly by my commands,
but at some high level in the extremly complex hierarhical control system
my command is the reference for their behaviour.

While the engineering diagram is sufficient for engineering
been impressed by the orderliness of the initial approach to
design. As far as I can see, there are no systematic design
principles taught to students; at least none appear in the
textbooks I have seen. The basic approach seems to be to define
what it is about the plant that one wants to control, and to find
inputs and forward transforms that will produce approximately
that effect, with feedback being used more or less as a way of
trimming the performance and almost as an afterthought
counteracting disturbances. The design phase is brief and
sketchy; the student is plunged immediately into complex
mathematics without further consideration.

You ought to have a closer look at the advances in the control theory.
I learned 20 years ago simple PID-algorithms and their tuning, state
variable control, and discrete control, and adaptive control. I only
took the first course in control engineering. Afterwards fuzzy control
and self tuning systems plus non-linear control has advanced a lot.
But before you can control you have to understand the process you
are controlling. And in complex systems maths is the only way.

I think that the principles made clear in the PCT diagram could
improve engineering design. The basic principle is clear in this
diagram: the primary consideration is to provide a sensory signal
that represents the variable in the plant that one wishes to
control, so that by comparing the signal against a reference
signal, one can generate an error signal indicating the degree to
which control has not been achieved. Then the design of the
forward part of the system can be filled in, to provide the gain
and output power necessary for good control and the filtering
necessary for stability.

I am still more astonished, this is exactly what the control engineering
has done at least for the 20 years I have known something about it.

regards
Osmo Eerola