# Problems with simulation of velocity feedback model

Hello, all --

This is not so much a progress report as a tentative admission of failure. The "hierarchical" feedback model of tracking, at least the simulation I am working with, will not stabilize when there is a transport lag in the velocity feedback path. Max Mulder, I believe, pointed out the fact that the velocity feedback in the simulation had no lag in it and predicted trouble when the lag was included. He was right.

I think I have painted myself into a corner without meaning to. This all began when I said that there is a way of organizing a control system with a load (plant) consisting of a mass on a spring with damping, which would continue to control over a wide range of mass, spring constant, and damping coefficient. I demonstrated this model with a simulation.

This quickly brought up the McRuer crossover model, which claims that human controllers adapt to changing plant characteristics so as to produce a constant overall closed-loop performance like that of a system with a first-order lag and a time delay. My demonstration showed that the appearance of adaptation could be produced by a model having fixed values of both position and velocity feedback (meaning no adaptation) over a wide range of load characteristics. As I have pointed out several times, this did not imply that no adaptation took place in the human controller. It merely said that any actual adaptation was not necessarily reflected accurately in the apparent closed-loop properties of the human controller when analyzed for the input-output transfer function. However, the idea of an "adaptive illusion" was seductive enough to tempt both me and Rick Marken to start talking as if this illusion could account for everything that was observed, and this is what is turning out not to be correct -- at least so far.

For those not familiar with frequency-domain analysis, here is the basic problem.

There are two kinds of lag that can show up in a physical system: an integral lag, and a transport lag.

An integral lag shows up when there is something some part of the system produces an output that is the accumulated sum of its inputs over time, like a bucket filling with water. The output is the integral of the input. When an input signal is applied to a functional unit with this characteristic, the output of the unit begins to change at a rate proportion to the magnitude of the input signal. It takes time for the output to change from one value to another, and this appears as a lag in the response.

If the input signal is a sine wave (which takes us into the frequency domain), the output signal will also be a sine wave. The relationship between the input and output sine waves depends on the frequency of the sinusoidal variations. In a pure (not "leaky") integrator, the output sine wave always lags 90 degrees behind the input sine wave (one full cycle is 360 degres, so this means a quarter of a cycle), and as the frequency of the input sine wave changes, the _amplitude_ of the output sine wave steadily decreases. In fact, it decreases, as engineers like to say, at 3 db per octave, or 20 db per decade, which are merely ways of saying that the amplitude is inversely proportional to the frequency. Double the frequency, halve the amplitude. Multiply the frequency by 10, output amplitude drops by a factor of 10.

In a control system, an integral lag of 90 degrees is ideal. A control loop containing a single 90-degree lag will be stable for any loop gain at all. It won't control well at very high frequencies, but that's all right: real control systems can't do that either.

The other kind of lag, the transport lag, is not so benign. A transport lag simply delays a signal without changing it in any other way. The output amplitude is the same as the input amplitude at all frequencies. If the input frequency is 1 Hz (cycle per second) and the transport lag is 1/4 second, the output frequency will be 1 Hz, the output amplitude will be the same as the input amplitude, and the output sine wave will lag 1/4 cycle or 90 degrees behind the input sine wave. So far so good.

But suppose the input frequency doubles. The output amplitude and frequency will still be the same, but now the same time delay amounts to 1/2 cycle of the sine wave, which is varying twice as fast. Instead of lagging 90 degrees, the output will now lag the input by 180 degrees. Worse, the amplitude of the output of this part of the system will not drop off at all. If this is part of a closed-loop control system, the result will be just as if the feedback were switched from negative to positive. Instead of controlling, the system will start oscillating and drive itself to destruction. Well, it won't control, and that's for sure.

When there is a combination of a transport lag and an integral lag, it is possible to prevent these oscillations. In the integral-lag part of the system, the amplitude becomes smaller as the frequency increases. If the amplitude is small enough when the frequency becomes just high enough to produce a total of 180 degrees of phase shift, the loop gain will be too small to sustain oscillations, and the control system will be stable. For any amount of transport lag, it is always possible to adjust the amount of integral lag to achieve overall stability.

In the models of human tracking performance that we have used in the past, we have found that the best fit of model performance to real human performance involves a transport lag of 0.1 to 0.15 seconds, and a leaky integral lag with a time constant of about 6 seconds. This is exactly McRuer's crossover model, in this case with a plant that is simply a constant multiplier: cursor position is proportional to the controlling hand position. We haven't explored other plant characteristics to speak of.

So we know that a control system with a transport lag can be stable if it has an integral lag with the right input-output sensitivity. However, when a model contains velocity feedback, everything changes, or so it seems right now. If the delay is perceptual, it will apply to perception of velocity as well as perception of position. It's possible that the delay for perception of velocity is shorter than the delay for perceiving position, and it's also possible that in the simulation, the velocity calculation requires much finer increments of the time than I am using, but I don't want to grasp at straws here. The fact is that my cut-and-try approach to modeling is leading to confusion, and what we need is a more rigorous approach, the approach that John Flach and his colleagues have learned to apply -- analytical mathematics.

I suspect that there is an arrangement similar to the model we have been looking at for a year that will work properly. But finding it requires more horsepower than I can supply.

Maybe the most important thing we can salvage from this interesting go-around is a much-modified version of what we so bravely called "the adaptive illusion." Clearly, any good control system with fixed properties will operate with reasonable accuracy and speed over _some_ range of "properties of the external world," or "the plant" as control engineers say. Within this range, it is not necessary for the control system to adapt itself in any way to its environment. The question is. how wide is the range? And the second question is, is there a design that will increase this range? I think both questions, put in that way, are answerable.

When the effect of transport lags is properly taken into account, we will be able to trust the model and know what designs will work and not work. Until then, I think it's time to retract all bets and wait for further developments.

Best,

Bill P.

Bill and others,

I think there is still an interesting story here. It has to do with measurement and inference with closed-loop systems.

Even for people who have worked on closed-loop systems for many years -- the properties of this
system can still hold surprises. Our intuitions about measurement have been formed within a scientific
culture based on simple models of linear causality. These intuitions often get us into trouble when the
loop is closed and the causality is "circular." Even when we explicitly reject the linear causal model -- we can
still fall victim to the intuitions that surround us and that have become synonymous with "science" in the
Western culture.

Max and I will work on producing a draft of what we think is the real story here. This will take time, as we are
both juggling many other projects. Don't expect anything before Jan (when I will be resident at Delft -- and Max
and I can interact in real-time to finish the manuscript). We will send the draft to Bill and Rick for their input.

Despite the difficulties -- I think that it is good to see the PCT movement address some of the empirical research
that has been generated in the area of "manual control." I look forward to continued discussions.

John

Hello, all --

This is not so much a progress report as a tentative admission of failure. The "hierarchical" feedback model of tracking, at least the simulation I am working with, will not stabilize when there is a transport lag in the velocity feedback path. Max Mulder, I believe, pointed out the fact that the velocity feedback in the simulation had no lag in it and predicted trouble when the lag was included. He was right.

I think I have painted myself into a corner without meaning to. This all began when I said that there is a way of organizing a control system with a load (plant) consisting of a mass on a spring with damping, which would continue to control over a wide range of mass, spring constant, and damping coefficient. I demonstrated this model with a simulation.

This quickly brought up the McRuer crossover model, which claims that human controllers adapt to changing plant characteristics so as to produce a constant overall closed-loop performance like that of a system with a first-order lag and a time delay. My demonstration showed that the appearance of adaptation could be produced by a model having fixed values of both position and velocity feedback (meaning no adaptation) over a wide range of load characteristics. As I have pointed out several times, this did not imply that no adaptation took place in the human controller. It merely said that any actual adaptation was not necessarily reflected accurately in the apparent closed-loop properties of the human controller when analyzed for the input-output transfer function. However, the idea of an "adaptive illusion" was seductive enough to tempt both me and Rick Marken to start talking as if this illusion could account for everything that was observed, and this is what is turning out not to be correct -- at least so far.

For those not familiar with frequency-domain analysis, here is the basic problem.

There are two kinds of lag that can show up in a physical system: an integral lag, and a transport lag.

An integral lag shows up when there is something some part of the system produces an output that is the accumulated sum of its inputs over time, like a bucket filling with water. The output is the integral of the input. When an input signal is applied to a functional unit with this characteristic, the output of the unit begins to change at a rate proportion to the magnitude of the input signal. It takes time for the output to change from one value to another, and this appears as a lag in the response.

If the input signal is a sine wave (which takes us into the frequency domain), the output signal will also be a sine wave. The relationship between the input and output sine waves depends on the frequency of the sinusoidal variations. In a pure (not "leaky") integrator, the output sine wave always lags 90 degrees behind the input sine wave (one full cycle is 360 degres, so this means a quarter of a cycle), and as the frequency of the input sine wave changes, the _amplitude_ of the output sine wave steadily decreases. In fact, it decreases, as engineers like to say, at 3 db per octave, or 20 db per decade, which are merely ways of saying that the amplitude is inversely proportional to the frequency. Double the frequency, halve the amplitude. Multiply the frequency by 10, output amplitude drops by a factor of 10.

In a control system, an integral lag of 90 degrees is ideal. A control loop containing a single 90-degree lag will be stable for any loop gain at all. It won't control well at very high frequencies, but that's all right: real control systems can't do that either.

The other kind of lag, the transport lag, is not so benign. A transport lag simply delays a signal without changing it in any other way. The output amplitude is the same as the input amplitude at all frequencies. If the input frequency is 1 Hz (cycle per second) and the transport lag is 1/4 second, the output frequency will be 1 Hz, the output amplitude will be the same as the input amplitude, and the output sine wave will lag 1/4 cycle or 90 degrees behind the input sine wave. So far so good.

But suppose the input frequency doubles. The output amplitude and frequency will still be the same, but now the same time delay amounts to 1/2 cycle of the sine wave, which is varying twice as fast. Instead of lagging 90 degrees, the output will now lag the input by 180 degrees. Worse, the amplitude of the output of this part of the system will not drop off at all. If this is part of a closed-loop control system, the result will be just as if the feedback were switched from negative to positive. Instead of controlling, the system will start oscillating and drive itself to destruction. Well, it won't control, and that's for sure.

When there is a combination of a transport lag and an integral lag, it is possible to prevent these oscillations. In the integral-lag part of the system, the amplitude becomes smaller as the frequency increases. If the amplitude is small enough when the frequency becomes just high enough to produce a total of 180 degrees of phase shift, the loop gain will be too small to sustain oscillations, and the control system will be stable. For any amount of transport lag, it is always possible to adjust the amount of integral lag to achieve overall stability.

In the models of human tracking performance that we have used in the past, we have found that the best fit of model performance to real human performance involves a transport lag of 0.1 to 0.15 seconds, and a leaky integral lag with a time constant of about 6 seconds. This is exactly McRuer's crossover model, in this case with a plant that is simply a constant multiplier: cursor position is proportional to the controlling hand position. We haven't explored other plant characteristics to speak of.

So we know that a control system with a transport lag can be stable if it has an integral lag with the right input-output sensitivity. However, when a model contains velocity feedback, everything changes, or so it seems right now. If the delay is perceptual, it will apply to perception of velocity as well as perception of position. It's possible that the delay for perception of velocity is shorter than the delay for perceiving position, and it's also possible that in the simulation, the velocity calculation requires much finer increments of the time than I am using, but I don't want to grasp at straws here. The fact is that my cut-and-try approach to modeling is leading to confusion, and what we need is a more rigorous approach, the approach that John Flach and his colleagues have learned to apply -- analytical mathematics.

I suspect that there is an arrangement similar to the model we have been looking at for a year that will work properly. But finding it requires more horsepower than I can supply.

Maybe the most important thing we can salvage from this interesting go-around is a much-modified version of what we so bravely called "the adaptive illusion." Clearly, any good control system with fixed properties will operate with reasonable accuracy and speed over _some_ range of "properties of the external world," or "the plant" as control engineers say. Within this range, it is not necessary for the control system to adapt itself in any way to its environment. The question is. how wide is the range? And the second question is, is there a design that will increase this range? I think both questions, put in that way, are answerable.

When the effect of transport lags is properly taken into account, we will be able to trust the model and know what designs will work and not work. Until then, I think it's time to retract all bets and wait for further developments.

Best,

Bill P.

John M. Flach
Professor, Department of Psychology
Wright State University
3640 Colonel Glenn Hwy.
Dayton, OH 45435-0001
Phone: (937) 775-2391
Fax: (937) 775-3347
www.psych.wright.edu

"Science is built of facts the way a house is built of bricks, but an accumulation of facts is no more science than a pile of bricks is a house" Henri Poincare

···

On Monday, September 8, 2003, at 10:34 AM, Bill Powers wrote:

Hi Bill et al --

This is not so much a progress report as a tentative admission of failure.
...

This quickly brought up the McRuer crossover model, which claims that human
controllers adapt to changing plant characteristics so as to produce a
constant overall closed-loop performance like that of a system with a
first-order lag and a time delay. My demonstration showed that the
appearance of adaptation could be produced by a model having fixed values
of both position and velocity feedback (meaning no adaptation) over a wide
range of load characteristics. As I have pointed out several times, this
did not imply that no adaptation took place in the human controller. It
merely said that any actual adaptation was not necessarily reflected
accurately in the apparent closed-loop properties of the human controller
when analyzed for the input-output transfer function. However, the idea of
an "adaptive illusion" was seductive enough to tempt both me and Rick
Marken to start talking as if this illusion could account for everything
that was observed, and this is what is turning out not to be correct -- at
least so far.

I don't think we implied that the illusion could account for "everything that was
observed". But if such an implication was taken, why is that also not a problem
for McRuer et al, who not only implied but claimed that, in fact, adaptation
could account for the apparent adaptation that was observed.

So we know that a control system with a transport lag can be stable if it
has an integral lag with the right input-output sensitivity. However, when
a model contains velocity feedback, everything changes, or so it seems
right now.

I put a perceptual lag into my baseball catching model, which controls a velocity,
and the model remains stable. Maybe it works because it's not a hierarchical
model but a transport lag seems to have no effect on its stability.

The fact is that my cut-and-try
approach to modeling is leading to confusion, and what we need is a more
rigorous approach, the approach that John Flach and his colleagues have
learned to apply -- analytical mathematics.

In this situation analytical methods might, indeed, help. But it seems to me that
analytical methods can lead to as much confusion as do computer models, at least
in certain areas. For example, the people looking a control using analytical
mathematics have not discovered (or, at least, have not written about) the
behavioral (now "response") illusion, the test for the controlled variable or,
most important, the fact that it is a perceptual representation of external
circumstances that is controlled by a control system.

Maybe the most important thing we can salvage from this interesting
go-around is a much-modified version of what we so bravely called "the

Maybe we should have said "possible adaptive illusion". I never saw it as
anything other than a demonstration of principle: in principle, a hierarchical
control model could show the same kind of Bode curves as those seen for a single

Clearly, any good control system with fixed properties
will operate with reasonable accuracy and speed over _some_ range of
"properties of the external world," or "the plant" as control engineers
say. Within this range, it is not necessary for the control system to adapt
itself in any way to its environment. The question is. how wide is the
range? And the second question is, is there a design that will increase
this range? I think both questions, put in that way, are answerable.

Right.

When the effect of transport lags is properly taken into account, we will
be able to trust the model and know what designs will work and not work.
Until then, I think it's time to retract all bets and wait for further
developments.

Sounds good to me. My main interest is still in determining what perceptual
variables living control systems control.

Best regards

Rick

···

--
Richard S. Marken, Ph.D.
Senior Behavioral Scientist
The RAND Corporation
PO Box 2138
1700 Main Street
Santa Monica, CA 90407-2138
Tel: 310-393-0411 x7971
Fax: 310-451-7018
E-mail: rmarken@rand.org