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.