[From Bill Powers (960710.1430 MDT)]
Bruce Abbott (960710.1035 EST) --
To achieve synchrony of two (or more) oscillators, all that is
required is negative feedback.
Actually, no feedback at all is needed. Synchrony of free-running
oscilllators can be achieved simply by perturbing them at intervals
close to the free-running frequency, like pushing on a pendulum at
nearly the same point in each cycle.
The reason I don't think that menstrual cycles are under control is
precisely that they can be synchronized by such minor influences as
pheremones. If they were actually under control (with respect to
frequency or phase) it would be difficult to change them. The fact that
a few molecules of a chemical can alter the cycle doesn't show that the
chemicals are superpowerful; it just shows that what they're affecting
isn't hard to change.
···
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Hans Blom, 960710 --
Uh? Another misunderstanding? What you call "disturbance", I call
"noise". For me, this translation is automatic; the literature that
I have read does not talk about "disturbance" but about "noise".
As Rick Marken pointed out, we're talking about different things.
Disturbances do not have to be randomly distributed or have a mean value
of zero as you usually seem to assume; in fact most of them aren't of
that nature, because they're caused by perfectly regular and systematic
physical processes. Some of them, like gravity, are constant.
At any rate, what I think of as "noise" in perception would be simply
uncertainty about the state of the perception. In my experience,
perceptions are not in themselves uncertain -- for example, when I
perceive the brightness of a light, I perceive a steady or smoothly
changing value of brightess. I do not perceive a brightness b, plus or
minus some band of uncertainty. When I perceive a table, I do not
perceive a range of shapes (whether I'm filling in parts of the scene
from imagination or not), but simply the shape that I see. I can't find
any unpredictable fluctuations in that perception.
When disturbances do occur, I know of them through obvious changes in my
perceptions. The television set may unaccountably become dimmer. I am
not uncertain about that dimming; to me it seems that the brightess of
the screen has decreased from one value to another, and there is no
uncertainty about either value. And I can correct the change in
brightess by using the brightess control, bringing it back to the level
I prefer -- again, without any uncertainty.
Try to look again. Now with the mindset: what do my sensors truly
capture at this moment. That is extremely hard; we just don't
operate that way. And so we confuse what we perceive with what we,
sort of, reconstruct out of something internal that has been fed by
earlier observations.
You're talking about the mechanisms that (hypothetically) give rise to
perception, not about the experience of perceptions. If I hallucinate
the smell of pizza, the hallucinated smell is perfectly clear and not at
all noisy. What I'm talking about is the direct experience of the world
of perception, without any obbligato (a persistent or background motif)
of commentary or explanation or conjecture about what is causing it. I
don't experience any reconstruction; it may be going on, but it's not
part of the world of perception. I experience only the outcome of
whatever processes lead to perception. And that outcome is not normally
obscured, fuzzy, irregular, uncertain, or noisy. It is what it is.
When a perception IS noisy, it's clearly not normal. The room is nearly
blacked out so all objects are barely discernible, or the music is
overlaid with hisses, pops, crackles, and squeaks, or the TV picture is
full of snow and drifting bands of light and dark. Noise appears in
perception when it should: when the signals are close to the lower
threshold of detection, or when other signals are interfering
significantly. And then it is perfectly clear to the perceiver that
something is preventing normal perception.
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Hans Blom, 960709b --
Glad to know I got somewhere near an understanding of the strategy of
the Kalman Filter.
Overall, you understand the approach but your reluctance to use (or
inexperience with) probabilities may make that it _seems_ beyond
your comprehension.
From what I can see, the basic strategy doesn't depend on probabilities,
but on partial derivatives. The probabilistic aspects of it come in when
the data contain a lot of noise, so there is some doubt about whether
the extrapolation is valid. In that case, the method lowers the weight
given to the predicted value and increases the weight of the measured
value. That seems simple enough to me. The concept isn't what throws me;
it's the supercompact notation, and the fact that the method is
described in terms of multiple variables. I never could keep subscripts
straight, and I always had trouble with matrix manipulations because I
couldn't visualize what was going on.
Basically, the method is a "real time" curve fit. You may be
acquainted with the least squares method of fitting a best straight
line through a number of points, resulting in two parameters: a
slope and an offset.
Yes, that's basically how Welch and Bishop explained it: it's a running
least-squares computation that is always as up-to-date as it can be.
They also say "The Kalman filter estimates a process by using a form of
feedback control."
I remember telling you, somewhen in the remote past, that my home
heating system does not have a thermostat. It is a feedforward
system that adjust the temperature of the circulating water to the
outside temperature, which is measured. (That such a system works
as well as it does may depend on the fact that in the part of
Europe where I live weather changes are not extreme and houses have
better insulation.)
I think it also arises from the fact that Europeans are much more rugged
than Americans, and don't demand very much of their thermostats. I'm
curious, though: just where does your system measure the temperature?
And how does it handle days when sunshine is pouring in the windows, as
opposed to days when it's cloudy? What happens when it's cold outside
and the wind is blowing? What happens when you build a fire in the
fireplace? What happens when someone opens a door?
Now assume that this adjustment takes place in real time, so that
the operating curve is always well tuned. At the lowest level, we
still have a feedforward control system, but one level up is the
(control?) system that continually adjusts the operating curve.
This is, in short, how Kalman Filter based control systems work. As
you see, this is different from the PCT paradigm.
Yes, it is. It seems to me that this system must be quite slow to
respond to novel circumstances.
For this model to operate, the form of the real environment function
must be known
Here, "known" must be interpreted in the same sense as we "know" a
straight line from the computed offset and slope fitted through a
collection of points. So we'd better say that it must have been
"modelled" than to say that it must be "known".
No, it means more than that. You assume a certain form of the
environment equation, which may or may not be appropriate. The offsets
and slopes of which you speak concern only the parameters in the
equation; you have to have nearly the right form of the equation, with
general parameters, before you can start adjusting parameters. If you
assume that the plant response is proportional, but it really involves
an integral or a derivative, no amount of adjusting the model parameters
will give you good control. The only way around this is to start with a
model that is completely general, containing all possible integrals and
derivatives and polynomials in every variable, each with an adjustable
parameter, and then hope that your method will discard those elements of
the general equation that don't apply.
Let's speculate a bit about what an internal model in a human could
be used for, apart from moment-to-moment control. Basically, what
is available is a description of _what can be done_: imagine a
pattern of actions and observe, again in imagination, what the
outcome will be in terms of perceptions. In short, an internal
model allows planning: imagine action (t) for t=0 to t=T, and you
will, within the accuracy of the model, observe perception (t) for
t=0 to t=T.
Yes -- that is roughly what I said about planning in B:CP. I agree that
this sort of thing must happen, at the higher levels of perception.
There are clearly some advantages to the Kalman Filter approach, in
terms of achieving continued operation after loss of feedback
information. However, the most critical question in relation to
models of living systems is to what extent living systems have this
capacity.
Do you know of another mechanism that "explains" planning? You
might, of course, that planning does not exist. But that does not
solve the problem.
I agree that this mechanism is probably involved in conscious planning.
I suspect it is not done by using the computational methods of the
Kalman Filter. I do not agree that it is involved in lower-level control
processes.
Obviously, if control processes by living systems degenerate when
feedback is lost, a model that continues to operate properly would be
the wrong model.
Some time ago, I proposed a test: have the feedback signal go away
occasionally in a tracking experiment and observe the subject's
behavior. What happens? A) The subject stops, complaining that he
cannot track when no feedback is available. B) The cursor remains
at a constant position as long as there is no feedback. C) The
cursor follows a non-constant trajectory that is some sort of
extrapolation of its previous path. D) ... Only A would indicate
that no internal model exists. B would occur with the simplest
model thinkable: a zero order hold. C might be a first or higher
order hold.
What happens in a tracking experiment is normally A. The subjects say
"Are you crazy? How can I track when I can't see the cursor?" Once in a
while we see B): the subject freezes. And occasionally we see C. In all
cases, tracking immediately (or in a second or two) deteriorates, errors
growing tenfold or more.
If the break in visibility is brief enough, subjects will continue to
move the handle and, if no large change in the disturbance occurs during
that time, the actual cursor position will look like an extrapolation of
the curve and may remain close to the target. However, the time over
which this happens is so brief that it can be accounted for without
invoking an internal model. I would look more seriously for an internal
model if there were evidence that the model took over and produced some
extended pattern of output changes. But that doesn't happen. What seems
to happen is that the last velocity of the handle simply continues.
... control _theory_ might shed some light on what is possible and
what not, and therefore suggest some of the tests that might be
performed. Bruce Gregory describes this aspect of the value of a
theory extremely well.
I agree to both sentences.
I am convinced that in some kind of behaviors, probably at the higher
levels, a model similar to the one Hans proposes might be
appropriate. The greatest disadvantage of this model, however, is that
it requires the nervous system to know the form of an environmental
feedback function, and to be able to compute its exact inverse, or at
least one valid inverse.
This is a misunderstanding. In the first place, "smooth" functions
can be well approximated by only a few terms of their series
expansion, and many of the "laws of nature" appear to be smooth.
That means that a simple model can usually provide a pretty
realistic approximation of reality. Second, exactness is not
required, precisely because control goes on: errors will be
corrected upon their perception. Control -- the continuous
availability of feedback information -- allows simple
approximations to be effective.
I like this much better than what you've been saying. I simply can't
believe that the brain is doing all those computations that appear in
your programs. But if there is a simple approximation to these methods
that can be done without such elaborate arithmetic, I would be much more
interested. I agree that the closed-loop aspect of the process will
forgive many inaccuracies and approximations.
This kind of knowledge is hard to account for in lower-
level systems -- it's hard to imagine the midbrain or the brainstem
computing the inverse of a complicated function
Hardly more difficult than imagining the brain to compute a multi-
plication or an integral...
Far more difficult, because approximate multiplication and integration
can be done by analog methods, even by a single neuron, while computing
the general inverse of a function requires symbol manipulation and the
formal operations of mathematics.
Few people can thread a needle or drive a car blindfolded.
But _all_ people who drive a car occasionally take their eyes off
the road without causing an immediate accident.
They do this not by controlling a model but simply by holding the
steering wheel in a fixed position. You are dodging my example, though.
Can't you explain why a person can't thread a needle or drive a car
blindfolded?
---------------------------------------------
Regarding the artificial cerebellum model:
I have found a more interesting version of this model that I'll send to
you first. If I send you an executable version, do you have a machine
you can run it on (a PC)? If not, getting it running might be a pain,
because I use two machine-language routines for the algorithm itself --
when I worked this out, back in the mid-80s, Pascal and C were just too
slow on the machine I then had. Perhaps I could try rewriting that part
in Pascal, if you can't run the object code immediately.
In the version I want to send you, the dynamic model of the arm from
Little Man v2 is used, and a two-level control system operates each
joint. The lower level controls (angular) velocity by varying torque,
and the higher level controls position by varying the reference level
for velocity. A total of four "artificial cerebellum" modules are used
for the four output functions of the control systems, so the entire
model adapts from scratch.
Let me know if you'd rather have an all-Pascal version; if so, I don't
know if I can get the revised program running before we leave for the
CSG meeting (the 16th). If you can use object code, I'll send a MIME64
version of a self-extracting .zip file (along with all the Pascal source
code and the assembler listings).
----------------------------
I want to bring up a point about hierarchical control. If you'll look at
the first-level test modes of Little Man V2, brought up with F1, F2, and
F3, you'll see that the behavior of the joint angles becomes a very
simple function of the reference signals. The perceptions representing
joint angle respond to changes in the perceptual signals as if there
were a single-pole Tchebyschev (sp?) filter connecting them.
Without the first-level control systems, the arm would behave like a
very wobbly mass on a spring, with not much damping. A step-change in
the driving signal to the motor neuron would result in a large damped
oscillation of position, and thus of the signals representing position
and velocity. These oscillations would be complex because of coupling to
other limb segments.
It's possible, of course, that an adaptive system at a higher level
could still control the position signals it receives by emitting the
right driving signals to the motor neurons. But the model that this
system would need would have to represent the second-order differential
equation that describes the arm's response to driving forces, as well as
all the nonlinear and cross-coupling terms that arise because of
linkages to other arm segments. The reference signal would have to pass
through the inverse function, too, so the system would have to know the
form of the inverse function as well as the forward function.
However, with the lower feedback control systems in place, the higher
system experiences a much simpler relationship between the driving (now
reference) signals and the resulting position signals. A step-change in
the reference signal results in a quick asymptotic rise of the position
signal to the specified level (of nearly the form 1 - exp(-kt)). The
interactions among the control systems operating different joints all
but disappear; each joint control system operates independently of the
others, or appears to do so from the standpoint of the higher system.
All signs of underdamping are gone.
In short, the model that the next level of control uses can be a simple
first-order linear differential equation: almost trivially simple in
comparison with what would be required without the lower-level control
systems.
Your remarks on using simplified models reminded me of this. As I say, I
simply can't believe that the brain actually performs all the
computations that the Kalman Filter approach seems to call for. But I
could believe that some simple approximation to the Kalman approach
might actually exist in the nervous system, particularly if there were a
way to simplify the environment that a given control system has to deal
with -- meaning, of course, to simplify the MODEL that the control
system needs in order to operate adaptively in the Kalman manner.
What would make me sit up and take notice would be a hierarchical model
in which some VERY simple approximation to the Kalman method is used to
do some model-based controlling of an environment simplified by lower-
level control systems.
-----------------------------------------
Without nonlinearities, the "plant" in the A.C. demonstration is
described bym(d2x/dt^2) - k1(dx/dt) - k2(x - x0) = [driving force].
What is the inverse of this equation that you would use in your
model?
I don't understand the question. Do you want an explicit expression
for x as a function of the driving force? Solve the differential
equation. It's a simple second order system. What's the problem?
Actually I said this the wrong way. Given the behavior of x, the above
equation allows us to calculate the driving force u(t) that would
produce that behavior. That is already the inverse of the forward model
The forward model would be the solution of the above differential
equation: it would express x as a function of the driving force, u. The
general solution is a rather complex thing which I don't need to go look
up, I hope. But it is the model of the environment. You need both that
model and the inverse expression to use your "feed-forward" method. The
forward model is used to find the parameter values; the inverse model
(given above) is used to convert the reference signal into the output u.
---------------------------------
If I had the equations of motion of the jointed stick, I'd love to
try a PCT model of this [balancing three jointed sticks]. I'm not very
handy with dynamical equations -- I'd have to have them given to me.
Each stick is described by a formula like the one you give above.
Not quite -- the sticks interact with each other. You need all the
cross-coupling terms and all that, which I'm not up to producing.
Perhaps you could find the actual equations in some report on this
stick-balancing feat.
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Best to all,
Bill P.